Handbook of Thermal Science and Engineering [1 ed.] 9783319266947, 9783319266954

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Francis A. Kulacki Editor-in-Chief Sumanta Acharya · Yaroslav Chudnovsky Renato Machado Cotta · Ram Devireddy Vijay K. Dhir · M. Pinar Mengüç Javad Mostaghimi · Kambiz Vafai  Section Editors

Handbook of Thermal Science and Engineering

Handbook of Thermal Science and Engineering

Francis A. Kulacki Editor-in-Chief

Sumanta Acharya • Yaroslav Chudnovsky Renato Machado Cotta • Ram Devireddy Vijay K. Dhir • M. Pinar Mengüç Javad Mostaghimi • Kambiz Vafai Section Editors

Handbook of Thermal Science and Engineering With 1375 Figures and 184 Tables

Editor-in-Chief Francis A. Kulacki Department of Mechanical Engineering University of Minnesota Minneapolis, MN, USA Section Editors Sumanta Acharya Armour College of Engineering Department of Mechanical, Materials and Aerospace Engineering Illinois Institute of Technology Chicago, IL, USA

Yaroslav Chudnovsky Gas Technology Institute Des Plaines, IL, USA

Renato Machado Cotta Universidade Federal do Rio de Janeiro – UFRJ Rio de Janeiro, RJ, Brazil

Ram Devireddy Department of Mechanical Engineering Louisiana State University Baton Rouge, LA, USA

Vijay K. Dhir Mechanical and Aerospace Engineering University of California Los Angeles Los Angeles, CA, USA

M. Pinar Mengüç Cekmeköy Campus Özyegin University Çekmeköy - Istanbul, Turkey

Javad Mostaghimi Centre for Advanced Coating Technologies Department of Mechanical and Industrial Engineering Faculty of Applied Science + Engineering University of Toronto Toronto, ON, Canada

Kambiz Vafai Department of Mechanical Engineering University of California Riverside, CA, USA

ISBN 978-3-319-26694-7 ISBN 978-3-319-26695-4 (eBook) ISBN 978-3-319-28573-3 (print and electronic bundle) https://doi.org/10.1007/978-3-319-26695-4 Library of Congress Control Number: 2018935388 # Springer International Publishing AG, part of Springer Nature 2018 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Printed on acid-free paper This Springer imprint is published by the registered company Springer International Publishing AG part of Springer Nature. The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

In memoriam: Professor Emil Pfender We dedicate the section on plasma heat transfer to the memory of Professor Emil Pfender (1925–2016) of the Department of Mechanical Engineering, University of Minnesota, for his outstanding and lasting contributions to the field of thermal plasma heat transfer and materials processing by thermal plasmas. During his lifetime, Professor Pfender spearheaded pioneering studies on particle heat and mass transfer in thermal plasmas, anode boundary layer, and free burning arcs including the electrode regions, as well as nonequilibrium effects in arc plasma torches. He studied extensively plasma synthesis of ultrafine powders, later called “nanoparticles,” and developed processes for deposition of thin films, e.g., diamond films, by thermal plasma technology. His extensive research on plasma spray coating process has had applications ranging from jet engine turbine blades and combustors to medical hip implants. Professor Pfender received a Diploma in Physics in 1953, followed by Dr. Ing. in Electrical Engineering in 1959 at the Technical University of Stuttgart. He then became Chief Assistant and Lecturer at the Institute for Gaseous Electronics at the same university. He spent a year (1961) as a Visiting Scientist at the Plasma Physics Branch of the Air Force Research Laboratories, at Wright Patterson Air Force Base in Ohio. In 1964, Professor Pfender was recruited by Professor Ernst R.G. Eckert and joined the Department of Mechanical Engineering at the University of Minnesota, Minneapolis, Minnesota, USA. There, he established the High Temperature Laboratory (HTL), which quickly became one of the highest regarded laboratories in the field. Professor Pfender was the recipient of many honors and awards. In 1986, he was elected as a member of the US National Academy of

Engineering. He was a Fellow of the ASME, the recipient of the Alexander von Humboldt Award of the German Government, the Gold Honorary F. Krizik Medal for Merits in the Field of Technical Sciences of the Czech Academy of Sciences, the Honorary Doctor’s degree from the Technical University of Ilmenau, Germany, and the Plasma Chemistry Award from the International Union for Pure and Applied Chemistry (IUPAC). In 1980, he co-founded the Journal of Plasma Processing and Plasma Chemistry and remained its co-Editor-in-Chief until 2005. On a personal level, it was an honor to have been his student (1976–1982) and to have learned from his vast knowledge. He was generous, courteous, amiable, and a true gentleman.

Preface

Thermal engineering and science touches almost all branches of modern industrial activity, from the production and refining of mineral resources, to processing and production of basic food stuffs, to manufacturing processes, to energy conversion devices and systems, to environmental engineering, and to biological engineering. In all of these fields, technologies involve the transport of thermal energy, or heat, and in many cases mass transfer. I cannot think of an area of human activity that does not involve either the removal of thermal energy or the addition of thermal energy to an engineering process or manufactured product. The applied thermal sciences and engineering now apply to processes and systems from the near-atomic scale to the familiar macro-scales of industry and the environment. The topics in this handbook have been selected with this view in mind, and the goal has been to include topics that hitherto have not appeared in similar handbooks on heat and mass transfer in the past. The theory of heat on the macroscale is now well developed. This development began haltingly in the sixteenth century and blossomed in the nineteenth century with the expansion of process industries and the perfection of energy conversion devices and systems. The design of familiar thermal systems and equipment – heat exchangers, heat-treating equipment, and gas turbines for power and propulsion, refrigeration systems, conventional electronic cooling equipment, and energy conversion devices – all rest on this foundation. Nowadays, we have highly developed theoretical and empirical foundations for describing and reducing to practice knowledge of the basic modes of thermal energy transport: diffusion or heat conduction; convection; thermal radiation; and phase-change processes, principally boiling and condensation. Various levels of analysis and empiricism pertain to each, and some subfields remain resistant to complete mathematical description. We continue to rely on ad hoc closure models for predicting turbulent convective heat transfer coefficients, and heat at the nano-scale is a subject of fundamental investigation on the dominant transport mechanisms in various applications. But what is different today is the co-mingling of our understanding of the fundamental modes of heat and mass transfer and thermal physics with knowledge from widely different disciplines. In a real sense, the necessity of determining thermal effects across a range of processes and applications has brought transdisciplinarity to the forefront of thermal engineering. The applied thermal sciences at the micro- and nanoscales have also advanced vii

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Preface

rapidly from their theoretical and empirical foundations established in the twentieth century to where engineering applications – devices and manufactured products – are an emergent reality. While thermal energy transport at the nanoscale is certainly a focus of much applied and fundamental research today, the development of submicron sized devices means that thermal engineering of a wholly different character may well be needed, and a chapter focusing on thermal transport in micro- and nanoscale systems is included. The handbook is intended for researchers, practitioners, and graduate students. A good number of chapters are focused on fundamental descriptions of all modes of thermal energy transport, and this makes the handbook a general reference and introduction to the field. Applications to new and developing technologies and applied topics are also included. The section on heat transfer in biology and biological systems elaborates the techniques and several active topical areas at the intersection of biology and medicine with heat and mass transfer. A section on heat transfer in plasmas provides a comprehensive picture of contemporary industrial applications of ionized gases and their use in materials engineering. I extend my appreciation and thanks to all of those who have contributed to this handbook. The section editors have superbly managed an extraordinary wide range of topics, and authors of the chapters have skillfully summarized both classical and contemporary developments of their subjects. We hope the range of topics will serve not only current thermal engineers and scientists but also those to come in the years ahead. We have dedicated the section on heat transfer in plasmas to the memory of Dr. Emil Pfender. He was a colleague and friend to his colleagues and the many students who studied under him at the University of Minnesota. His research and professional contributions continue to have a major influence on the field of plasma heat transfer. University of Minnesota

Francis A. Kulacki Editor-in-Chief

Contents

Volume 1 Part I

Heat Transfer Fundamentals . . . . . . . . . . . . . . . . . . . . . . . . . .

1

1

Macroscopic Heat Conduction Formulation . . . . . . . . . . . . . . . . . . Leandro A. Sphaier, Jian Su, and Renato Machado Cotta

3

2

Analytical Methods in Heat Transfer . . . . . . . . . . . . . . . . . . . . . . . Renato Machado Cotta, Diego C. Knupp, and João N. N. Quaresma

61

3

Numerical Methods for Conduction-Type Phenomena . . . . . . . . . Bantwal R. Baliga, Iurii Lokhmanets, and Massimo Cimmino

127

4

Thermophysical Properties Measurement and Identification . . . . . Helcio R. B. Orlande and Olivier Fudym

179

5

Design of Thermal Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Yogesh Jaluria

219

6

Thermal Transport in Micro- and Nanoscale Systems . . . . . . . . . . Tanmoy Maitra, Shigang Zhang, and Manish K. Tiwari

277

7

Constructal Theory in Heat Transfer . . . . . . . . . . . . . . . . . . . . . . . Luiz A. O. Rocha, S. Lorente, and A. Bejan

329

Part II 8

9

Convective Heat Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . .

Single-Phase Convective Heat Transfer: Basic Equations and Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sumanta Acharya Turbulence Effects on Convective Heat Transfer . . . . . . . . . . . . . . Forrest E. Ames

361

363 391

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10

Contents

Full-Coverage Effusion Cooling in External Forced Convection: Sparse and Dense Hole Arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . Phil Ligrani

425

11

Enhancement of Convective Heat Transfer Raj M. Manglik

..................

447

12

Electrohydrodynamically Augmented Internal Forced Convection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Michal Talmor and Jamal Seyed-Yagoobi

479

13

Free Convection: External Surface . . . . . . . . . . . . . . . . . . . . . . . . . Patrick H. Oosthuizen

527

14

Free Convection: Cavities and Layers . . . . . . . . . . . . . . . . . . . . . . Andrey V. Kuznetsov and Ivan A. Kuznetsov

603

15

Heat Transfer in Rotating Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . Stefan aus der Wiesche

647

16

Natural Convection in Rotating Flows . . . . . . . . . . . . . . . . . . . . . . Peter Vadasz

691

17

Visualization of Convective Heat Transfer . . . . . . . . . . . . . . . . . . . Pradipta K. Panigrahi and K. Muralidhar

759

Volume 2 Part III Single-Phase Heat Transfer in Porous and Particulate Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

Applications of Flow-Induced Vibration in Porous Media . . . . . . . Khalil Khanafer, Mohamed Gaith, and Abdalla AlAmiri

19

Imaging the Mechanical Properties of Porous Biological Tissue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . John J. Pitre Jr. and Joseph L. Bull

20

21

Nanoparticles and Metal Foam in Thermal Control and Storage by Phase Change Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bernardo Buonomo, Davide Ercole, Oronzio Manca, and Sergio Nardini Modeling of Heat and Moisture Transfer in Porous Textile Medium Subject to External Wind: Improving Clothing Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nesreen Ghaddar and Kamel Ghali

805 807

831

859

885

Contents

Part IV 22

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Thermal Radiation Heat Transfer

....................

A Prelude to the Fundamentals and Applications of Radiation Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . M. Pinar Mengüç

917

919

23

Radiative Transfer Equation and Solutions . . . . . . . . . . . . . . . . . . Junming M. Zhao and Linhua H. Liu

933

24

Near-Field Thermal Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mathieu Francoeur

979

25

Design of Optical and Radiative Properties of Surfaces . . . . . . . . . 1023 Bo Zhao and Zhuomin M. Zhang

26

Radiative Properties of Gases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1069 Vladimir P. Solovjov, Brent W. Webb, and Frederic Andre

27

Radiative Properties of Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . 1143 Rodolphe Vaillon

28

Radiative Transfer in Combustion Systems . . . . . . . . . . . . . . . . . . 1173 Pedro J. Coelho

29

Monte Carlo Methods for Radiative Transfer . . . . . . . . . . . . . . . . 1201 Hakan Ertürk and John R. Howell

30

Inverse Problems in Radiative Transfer . . . . . . . . . . . . . . . . . . . . . 1243 Kyle J. Daun

Part V

Heat Transfer Equipment . . . . . . . . . . . . . . . . . . . . . . . . . . .

1293

. . . . 1295

31

Introduction and Classification of Heat Transfer Equipment Yaroslav Chudnovsky and Dusan P. Sekulic

32

Heat Exchanger Fundamentals: Analysis and Theory of Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1315 Ahmad Fakheri

33

Heat Transfer Media and Their Properties . . . . . . . . . . . . . . . . . . 1353 Igor L. Pioro, Mohammed Mahdi, and Roman Popov

34

Single-Phase Heat Exchangers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1447 Sunil S. Mehendale

35

Two-Phase Heat Exchangers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1473 Vladimir V. Kuznetsov

36

Compact Heat Exchangers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1501 Dusan P. Sekulic

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37

Evaporative Heat Exchangers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1521 Takahiko Miyazaki

38

Process Intensification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1535 Anna Lee Tonkovich and Eric Daymo

39

Energy Efficiency and Advanced Heat Recovery Technologies . . . 1593 Helen Skop and Yaroslav Chudnovsky

40

Heat Exchangers Fouling, Cleaning, and Maintenance . . . . . . . . . 1609 Thomas Lestina

Volume 3 Part VI

Heat Transfer with Phase Change . . . . . . . . . . . . . . . . . . . .

1643

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1645

41

Nucleate Pool Boiling Vijay K. Dhir

42

Transition and Film Boiling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1695 S. Mostafa Ghiaasiaan

43

Boiling on Enhanced Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1747 Dion S. Antao, Yangying Zhu, and Evelyn N. Wang

44

Mixture Boiling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1795 Mark A. Kedzierski

45

Boiling in Reagent and Polymeric Solutions . . . . . . . . . . . . . . . . . . 1823 Raj M. Manglik

46

Fundamental Equations for Two-Phase Flow in Tubes . . . . . . . . . 1849 Masahiro Kawaji

47

Flow Boiling in Tubes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1907 Yang Liu and Nam Dinh

48

Boiling and Two-Phase Flow in Narrow Channels . . . . . . . . . . . . . 1951 Satish G. Kandlikar

49

Single- and Multiphase Flow for Electronic Cooling . . . . . . . . . . . 1973 Yogendra Joshi and Zhimin Wan

50

Film and Dropwise Condensation . . . . . . . . . . . . . . . . . . . . . . . . . . 2031 John W. Rose

51

Internal Annular Flow Condensation and Flow Boiling: Context, Results, and Recommendations . . . . . . . . . . . . . . . . . . . . 2075 Amitabh Narain, Hrishikesh Prasad Ranga Prasad, and Aliihsan Koca

Contents

xiii

52

Heat Pipes and Thermosyphons . . . . . . . . . . . . . . . . . . . . . . . . . . . 2163 Amir Faghri

53

Phase Change Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2213 Navin Kumar and Debjyoti Banerjee

Volume 4 Part VII

Heat Transfer in Biology and Biological Systems . . . . . . .

2277

54

Thermal Properties of Porcine and Human Biological Systems . . . 2279 Shaunak Phatak, Harishankar Natesan, Jeunghwan Choi, Robert Sweet, and John Bischof

55

Microsensors for Determination of Thermal Conductivity of Biomaterials and Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2305 Xin M. Liang, Praveen K. Sekar, and Dayong Gao

56

Heat Transfer In Vivo: Phenomena and Models Alexander I. Zhmakin

57

Heat and Mass Transfer Processes in the Eye Arunn Narasimhan

58

Heat and Mass Transfer Models and Measurements for Low-Temperature Storage of Biological Systems . . . . . . . . . . . . . . 2417 Shahensha M. Shaik and Ram Devireddy

59

Gold Nanoparticle-Based Laser Photothermal Therapy Navid Manuchehrabadi and Liang Zhu

60

Thermal Considerations with Tissue Electroporation . . . . . . . . . . 2489 Timothy J. O’Brien, Christopher B. Arena, and Rafael V. Davalos

Part VIII

. . . . . . . . . . . . . . 2333 . . . . . . . . . . . . . . . . 2381

. . . . . . . . 2455

Heat Transfer in Plasmas . . . . . . . . . . . . . . . . . . . . . . . . . .

2521

61

Heat Transfer in DC and RF Plasma Torches . . . . . . . . . . . . . . . . 2523 Javad Mostaghimi, Larry Pershin, and Subramaniam Yugeswaran

62

Radiative Plasma Heat Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . 2599 Alain Gleizes

63

Heat Transfer in Arc Welding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2657 Anthony B. Murphy and John J. Lowke

64

Heat Transfer in Plasma Arc Cutting . . . . . . . . . . . . . . . . . . . . . . . 2729 Valerian Nemchinsky

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Contents

65

Synthesis of Nanosize Particles in Thermal Plasmas . . . . . . . . . . . 2791 Yasunori Tanaka

66

Plasma Waste Destruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2829 Milan Hrabovsky and Izak Jacobus van der Walt

67

Plasma-Particle Heat Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2885 Pierre Proulx

68

Heat Transfer in Suspension Plasma Spraying . . . . . . . . . . . . . . . . 2923 Mehdi Jadidi, Armelle Vardelle, Ali Dolatabadi, and Christian Moreau

69

Droplet Impact and Solidification in Plasma Spraying Javad Mostaghimi and Sanjeev Chandra

. . . . . . . . . 2967

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3009

About the Editor

Dr. Francis A. Kulacki is Professor of Mechanical Engineering at the University of Minnesota. He received his education in mechanical engineering at the Illinois Institute of Technology and the University of Minnesota. His research and scholarly interests include coupled heat and mass transfer in porous media, two-phase flow in micro-channels and micro-gaps, boiling of dilute emulsions, natural convection heat transfer, heat transfer in metal foams, hybrid renewable energy systems, thermal energy storage technology, energy policy, and management of technology. He is widely recognized for his development of fundamental knowledge of the natural convection in heat-generating fluids, and a wide range of fundamental experiments on convection in saturated porous media. His advisees include 20, 47 master’s degree students, and 14 undergraduate research scholars. He is Editor of the SpringerBriefs in Thermal Engineering and Applied Science, and the Springer Mechanical Engineering Series. His administrative work includes appointments as department Chair at the University of Delaware, Dean of engineering at the Colorado State University, and Dean of the Institute of Technology (now the College of Science and Engineering) at the University of Minnesota. In each of these positions, he was instrumental in initiating and expanding computer-aided engineering and technology-based instructional activities, increasing research funding, and establishing new multidisciplinary degree programs, research initiatives, centers, and specialized research facilities. He had served as Chair of the Heat Transfer Division of the American Society of Mechanical Engineers and was member of the ASME Board on Professional Development, Board xv

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About the Editor

on Engineering Education, and Board of the Center for Education. He chaired an ASME Task Force on Graduate Education and was a member of the ASME Vision 2030 project, which addressed the body of knowledge for mechanical engineers in the twenty-first century. He also chaired the Education Advisory Group of the National Society for Professional Engineers and was a member of the NSPE Task Force on Education and Registration. Dr. Kulacki has served on the advisory boards of engineering programs at Swarthmore College, the University of Kentucky, the University of Maryland Baltimore County, and Florida International University. In 1996, he was a member of the DOE Peer Review Panel on Thermal and Hydrological Impacts of the Yucca Mountain Repository. From 1998 to 2001, he was an ASME Distinguished Lecturer. From 1996 to 1998, he served as the Executive Director of the TechnologyBased Engineering Education Consortium, an initiative of the William C. Norris Institute. In 2002, he served as the Director of graduate studies for the MS in Management of Technology program at Minnesota and has lectured on energy policy and related issues in the MOT program and at the Hubert H. Humphrey Institute for Public Affairs. Dr. Kulacki is a Fellow of ASME and the American Association for the Advancement of Science. He has received the ASME Distinguished Service Award and the George Taylor Distinguished Service Award of the Institute of Technology at the University of Minnesota. In 2015, he received the ASME’s Heat Transfer Memorial Award. In 2017, he received ASME’s E. F. Church Award, which recognized his scholarly and administrative achievements in engineering education.

Section Editors

Sumanta Acharya received his Ph.D. from the University of Minnesota and his Bachelor’s degree from the Indian Institute of Technology in Mechanical Engineering. He is currently Professor and Department Chair of Mechanical, Materials and Aerospace Engineering at the Illinois Institute of Technology, Chicago. From 2010 to 2014, he served as the Program Director of the Thermal Transport Program in the Directorate of Engineering at the National Science Foundation (NSF). From 2014 to 2016, he was the Ring Companies Chair and Department Chair of the Mechanical Engineering Department at the Herff College of Engineering. His academic career prior to 2014 was at Louisiana State University (LSU) where he was the L. R. Daniel Professor and the Fritz & Francis Blumer Professor in the Department of Mechanical Engineering. He was the Founding Director in 2003 of the Center for Turbine Innovation and Energy Research (TIER), which focused on energy generation and propulsion research. His scholarly contributions include mentoring nearly 85 postdoctoral researchers and graduate students, and publishing nearly 200 refereed journal articles and book chapters and over 230 refereed conference/proceedings papers. Professor Acharya was awarded the 2015 AIAA Thermophysics Award, the 2014 AIChE Donald Q. Kern Award, the 75th ASME Heat Transfer Division Medal in 2013, and the 2011 ASME Heat Transfer Memorial Award in the Science category. He served as the Chair of the Heat Transfer Division (HTD) at ASME in 2016–2017 and currently serves in the HTD’s Executive Committee. He has served as the Associate Technical Editor (ATE) of the ASME Journal of Heat

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Transfer, ASME Journal of Energy Resources Technology, and currently is the ATE of the ASME Journal of Validation, Verification and Uncertainty Quantification. Yaroslav Chudnovsky is a senior member of the R&D Staff at the Gas Technology Institute (GTI), an independent not-for-profit R&D organization serving research, development, and training needs of industrial and energy markets since 1941. For over three decades, he successfully developed and led comprehensive R&D programs combined with teaching in enhanced heat transfer, process heating and cooling, power generation and energy harvesting, waste heat recovery and energy efficiency, wastewater reuse, advanced clean combustion, and product quality improvement, related to cost-effective industrial and commercial innovations and advanced technical solutions. He has been working for GTI Energy Utilization Group since 1995 and has a diversified practical experience in thermal-fluid and energy systems, energy efficiency, and clean environmental and industrial technologies. Prior to joining GTI Dr. Chudnovsky was a Director of Heat and Mass Transfer Research laboratory at Moscow Bauman Technical University. During his professional career, he has earned an extensive record of federal, state, and private industry funded high-risk innovative research, early-stage development, pre-commercial demonstration, cost-effective deployment, and successful commercialization of a wide spectrum of technologies. Dr. Chudnovsky earned a Ph.D. in Thermal Sciences (1990), an M.S. in Cryogenic Engineering (1982), and a B.S. in Mechanical Engineering (1980) from the Bauman Technical University. He is an Editorial Board member of a number of professional journals, Fellow of ASME, Member of ASTFE, ABS, AIChE, AIAA, and AFRC, as well as author/coauthor of over 200 professional publications including books, archival articles, conference proceedings, technical reports, and patents.

Section Editors

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Renato Machado Cotta received his B.Sc. in Mechanical/Nuclear Engineering from the Federal University of Rio de Janeiro, Brazil, in 1981, and the Ph.D. in Mechanical and Aerospace Engineering, from the North Carolina State University, Raleigh, in 1985. He joined the Mechanical Engineering Department at the Federal University of Rio de Janeiro in 1987. He has authored 500 technical papers, 9 books, and supervised 80 Ph.D. and M.Sc. theses. Dr. Cotta is a member of the Honorary Advisory Board of International Journal of Heat and Mass Transfer, International Communications in Heat and Mass Transfer, International Journal of Thermal Sciences, International Journal of Numerical Methods in Heat and Fluid Flow, and Computational Thermal Sciences. He is a Regional Editor of High Temperatures - High Pressures and Associate Editor of the Annals of the Brazilian Academy of Sciences. He served as President of Brazilian Association of Mechanical Sciences, ABCM, 2000–2001, Member of Scientific Council of the International Centre for Heat and Mass Transfer, since 1993, Executive Committee of ICHMT since 2006, presently Chairman of the Executive Committee of ICHMT, Congress Committee member of the International Union of Theoretical and Applied Mechanics (IUTAM) since 2010, and Executive Committee member of the Brazilian Academy of Sciences from 2012 to 2015. Dr. Cotta received the ICHMT Hartnett-Irvine Award in 2009 and 2015 and was elected member of the National Order of Scientific Merit, Brazil, 2009. He is an elected member of the Brazilian Academy of Sciences, 2009, National Engineering Academy of Brazil, 2011, and the World Academy of Sciences, Trieste, Italy, 2012. Dr. Cotta served as the President of the National Commission of Nuclear Energy, CNEN/Brazil, 2015–2017, and presently is Technical Counselor to the General Directorate for Nuclear and Technological Development of the Brazilian Navy.

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Section Editors

Ram Devireddy is the DeSoto Parish Chapter University Alumni Professor and the Louisiana Land and Exploration Company Endowed Chair Professor of Mechanical Engineering at Louisiana State University, Baton Rouge. Dr. Devireddy received his Ph.D. from the University of Minnesota, M.S. from the University of Colorado, Boulder, and his bachelor’s degree from the University of Madras, India, in Mechanical Engineering. He is interested in a wide variety of biological phenomena at low temperatures with emphasis on phase-change phenomena with particular emphasis on conservation of endangered species, rational design of ovarian tissue cryopreservation protocols, adult stem cell bio-preservation, tissue engineering, macro- and micro-scale simulation of bio-membrane-cryoprotective agent interactions, and nano- and macroscale heat transfer phenomena. Dr. Devireddy has coauthored several book chapters, over 80 archival journal publications, and 80 conference proceedings and abstracts. The quality of his publications has been recognized by best paper awards from the ASME Journal of Heat Transfer, Mid-West Thermal Analysis Forum, the Society of Cryobiology, and the Material Research Society. He has served as Co-chair (2008–2010) and Chair (2010–2012) of the American Society of Mechanical Engineering (ASME) Biotransport Committee, as well as the Technical Program Chair for the 2013 ASME Summer Bioengineering Conference. In 2011, he was inducted as a Fellow of the ASME. Dr. Devireddy has received numerous honors and awards including a Brains (back) to Brussels Fellowship to visit Université Catholique de Louvain, Brussels (2009), and a Japan Society for Promotion of Science (JSPS) Fellowship to visit the Yokohama National University, Tokyo (2016). Dr. Devireddy is also the recipient of the Louisiana Alumni Association Faculty Excellence Award (2013) for outstanding teaching, research, and service.

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Vijay K. Dhir is Distinguished Professor of Mechanical and Aerospace Engineering and served as Dean of UCLA’s Henry Samueli School of Engineering and Applied Science from 2003 to 2016. He received his Bachelor of Science degree from Punjab Engineering College in Chandigarh, India, and his Master of Technology degree from the Indian Institute of Technology in Kanpur, India. He received his Ph.D. from the University of Kentucky and joined the faculty at UCLA in 1974. In 2006, he was elected to the National Academy of Engineering for his work in boiling heat transfer and nuclear reactor thermal hydraulics and safety. He received the 2004 Max Jakob Memorial Award of ASME and AIChE and was delivered the 2008 ASME Thurston Lecture. He is a Fellow of ASME and the American Nuclear Society. In 2004, he was selected as an inductee into the University of Kentucky’s Engineering Hall of Distinction. He has also received the American Society of Mechanical Engineers (ASME) Heat Transfer Memorial Award in the Science category and the Donald Q. Kern Award from the American Institute of Chemical Engineers (AIChE). He is recipient of the Technical Achievement Award of the Thermal Hydraulics Division of the American Nuclear Society and twice has received the Best Paper Award for papers published in ASME Journal of Heat Transfer. He received an honorary Ph.D. in Engineering from University of Kentucky and a Lifetime Achievement Award at the ICCES conference. He is also an honorary member of ASME and received the 75th Anniversary Medal from the Heat Transfer Division of ASME. He was recognized in 2013 as Educator of the Year by the Engineering Council. Dr. Dhir leads the Boiling Heat Transfer Laboratory at UCLA, which conducts research on boiling including flow boiling, micro-gravity boiling, and nuclear reactor thermal hydraulics. More than 45 Ph.D. students and 40 M.S. students have graduated under Dhir’s supervision. He is author or coauthor of over 350 papers published in archival journals and proceedings of conferences.

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M. Pinar Mengüç completed his B.S. and M.S. degrees at the Middle East Technical University in Ankara, Turkey. He received his Ph.D. in Mechanical Engineering from Purdue University in 1985 and joined the faculty at the University of Kentucky the same year. He was promoted to the rank of Professor in 1993, and in 2008 was named as the Engineering Alumni Association Chair Professor. He was a Visiting Professor at the Universita degli Studi “Federico II,” in Napoli, Italy, in 1991, and at the Massachusetts General Hospital/Harvard University in Boston during 1998–1999. He served as an Associate Editor of the ASME Journal of Heat Transfer and is currently the Editor-in-Chief of the Journal of Quantitative Spectroscopy and Radiative Transfer. He was the Chair of five International Symposia on Radiation Transfer, organized by the International Center for Heat and Mass Transfer. Dr. Mengüç has authored/coauthored more than 125 refereed journal articles and more than 180 conference papers, book chapters, and two books, including the Sixth Edition of Thermal Radiation Heat Transfer, with Jack Howell and Robert Siegel, which appeared in 2016. He has five patents and has guided more than 65 M.S. and Ph.D. students and postdoctoral fellows. Dr. Mengüç served as the Founding Director of the Nano-Scale Engineering Certificate Program at the University of Kentucky. Since early 2009, he is at Ozyegin University in Istanbul as the Founding Director of Center for Energy, Environment, and Economy (CEEE-EÇEM) and the Founding Head of Mechanical Engineering Program. He is a Fellow of ASME and ICHMT, a Senior Member of OSA, and an elected member of the Science Academy of Turkey. Javad Mostaghimi is the Distinguished Professor in Plasma Engineering in the Department of Mechanical and Industrial Engineering at the University of Toronto and the Director of Centre for Advanced Coating Technologies (CACT). He received a B.Sc. degree from Sharif University, Iran, in 1974, and M.Sc. and Ph.D. degrees in Mechanical Engineering from the University of Minnesota in 1978 and 1982, respectively. Before joining University of Toronto in 1990, he held positions at Pratt & Whitney Canada, Longueil, Quebec, and the Department of Chemical Engineering, University of Sherbrooke, Sherbrooke, Quebec. His main research

Section Editors

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interests are the study of thermal spray coatings, and transport phenomena and electromagnetics in thermal plasma sources, in particular, the flow, temperature, and electromagnetic fields within arcs and RF inductively coupled plasmas. He has also done extensive simulation of the dynamics of droplet impact and solidification in thermal spray processes. He is a Fellow of the Royal Society of Canada, ASME, ASM, CSME, EIC, CAE, AAAS, and IUPAC. He is a recipient of the 75th Anniversary Medal of the ASME Heat Transfer Division, 2013 Robert W. Angus Medal of the CSME, 2012 Heat Transfer Memorial Award of the ASME, 2011 Jules Stachiewicz Medal of the CSME, 2010 NSERC Brockhouse Canada Prize, and 2009 Engineering Medal in R&D from the Professional Engineers of Ontario. He is a member of the Editorial Board of Plasma Chemistry and Plasma Processing and a member of the International Review Board of the Journal of Thermal Spray. Kambiz Vafai received his B.S. in Mechanical Engineering from the University of Minnesota, Minneapolis, and M.S. and Ph.D. degrees from the University of California, Berkeley. He is a Fellow of ASME, AAAS, and WIF and Associate Fellow of AIAA. He has one of the highest number of citations and h indices in several of the research areas that he has worked on in both ISI and Google Scholar metrics. He has authored over 350 journal publications, book chapters, and symposium volumes. He is currently a Distinguished Professor at the University of California, Riverside, where he started as the Presidential Chair in the Department of Mechanical Engineering. While he was at the Ohio State University, he won the Outstanding Research Awards in the assistant, associate, and full professor categories. He is the recipient of the ASME Classic Paper Award and received the ASME Memorial Award for Outstanding Contributions to and Leadership in Research on convection in porous media, convection in enclosed fluids, and flat-shaped heat pipes. He was given the International Society of Porous Media (InterPore) Highest Award in recognition of outstanding and extraordinary contributions to porous media science. He is also the recipient of the 75th Anniversary Medal of ASME Heat Transfer Division. He holds 13 US patents associated with

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electronic cooling and medical applications. Dr. Vafai has worked in many technical and scientific areas including multiphase transport, porous media, innovative heat pipes, electronics cooling, innovative microchannels, innovative biosensors, aircraft braking systems, innovative nano-fluid applications, biomedical advances, polymerase chain reaction, land mine detection, innovative high heat flux, thermal/fluid flow regulation and control, and discovery of a new set of fluid flow instabilities.

Contributors

Sumanta Acharya Armour College of Engineering, Department of Mechanical, Materials and Aerospace Engineering, Illinois Institute of Technology, Chicago, IL, USA Abdalla AlAmiri Mechanical Engineering Department, United Arab Emirates University, Al-Ain, UAE Forrest E. Ames Mechanical Engineering Deptartment, University of North Dakota, Grand Forks, ND, USA Frederic Andre Centre de Thermique et d’Energétique de Lyon, INSA de Lyon, Villeurbanne, France Dion S. Antao Device Research Laboratory, Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA, USA Christopher B. Arena Laboratory for Therapeutic Directed Energy, Department of Physics, Elon University, Elon, NC, USA Bantwal R. Baliga Department of Mechanical Engineering, Heat Transfer Laboratory, McGill University, Montreal, QC, Canada Debjyoti Banerjee Texas A&M University, College Station, TX, USA A. Bejan Department of Mechanical Engineering and Materials Science, Duke University, Durham, NC, USA John Bischof Department of Mechanical Engineering, University of Minnesota, Minneapolis, MN, USA Joseph L. Bull Biomedical Engineering Department, Tulane University, New Orleans, LA, USA Bernardo Buonomo Dipartimento di Ingegneria Industriale e dell’Informazione, Università degli Studi della Campania “Luigi Vanvitelli”, Aversa (CE), Italy Sanjeev Chandra Department of Mechanical and Industrial Engineering, University of Toronto, Toronto, ON, Canada xxv

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Contributors

Jeunghwan Choi Department of Engineering, East Carolina University, Greenville, NC, USA Yaroslav Chudnovsky Gas Technology Institute, Des Plaines, IL, USA Massimo Cimmino Department of Mechanical Engineering, Heat Transfer Laboratory, McGill University, Montreal, QC, Canada Pedro J. Coelho LAETA, IDMEC, Department of Mechanical Engineering, Instituto Superior Técnico, Universidade de Lisboa, Lisbon, Portugal Renato Machado Cotta Universidade Federal do Rio de Janeiro – UFRJ, Rio de Janeiro, RJ, Brazil Kyle J. Daun Department of Mechanical and Mechatronics Engineering, University of Waterloo, Waterloo, ON, Canada Rafael V. Davalos Bioelectromechanical Systems Laboratory, ICTAS Center for Engineered Health, Department of Biomedical Engineering and Mechanics, Virginia Tech - Wake Forest School of Biomedical Engineering and Sciences, Blacksburg, VA, USA Eric Daymo Tonkomo LLC, Gilbert, AZ, USA Ram Devireddy Department of Mechanical Engineering, Louisiana State University, Baton Rouge, LA, USA Vijay K. Dhir Mechanical and Aerospace Engineering, University of California Los Angeles, Los Angeles, CA, USA Nam Dinh North Carolina State University, Raleigh, NC, USA Ali Dolatabadi Department of Mechanical and Industrial Engineering, Concordia University, Montreal, QC, Canada Davide Ercole Dipartimento di Ingegneria Industriale e dell’Informazione, Università degli Studi della Campania “Luigi Vanvitelli”, Aversa (CE), Italy Hakan Ertürk Department of Mechanical Engineering, Boğaziçi University, Istanbul, Turkey Amir Faghri Department of Mechanical Engineering, University of Connecticut, Storrs, CT, USA Ahmad Fakheri Department of Mechanical Engineering, Bradley University, Peoria, IL, USA Mathieu Francoeur Radiative Energy Transfer Laboratory, Department of Mechanical Engineering, University of Utah, Salt Lake City, UT, USA Olivier Fudym CNRS Office for Brazil and the South Cone, Avenida Presidente Antônio Carlos, Rio de Janeiro, RJ, Brazil

Contributors

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Mohamed Gaith Department of Mechanical Engineering, Australian College of Kuwait, Kuwait City, Kuwait Dayong Gao Department of Mechanical Engineering, University of Washington, Seattle, WA, USA Nesreen Ghaddar Department of Mechanical Engineering, Faculty of Engineering and Architecture, American University of Beirut, Beirut, Lebanon Kamel Ghali Department of Mechanical Engineering, Faculty of Engineering and Architecture, American University of Beirut, Beirut, Lebanon S. Mostafa Ghiaasiaan George W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA, USA Alain Gleizes Institute LAPLACE Laboratory, CNRS and Paul Sabatier University, Toulouse, France John R. Howell Department of Mechanical Engineering, The University of Texas at Austin, Austin, TX, USA Milan Hrabovsky Institute of Plasma Physics ASCR, Prague, Czech Republic Mehdi Jadidi Department of Mechanical and Industrial Engineering, Concordia University, Montreal, QC, Canada Yogesh Jaluria Department of Mechanical and Aerospace Engineering, Rutgers, the State University of New Jersey, Piscataway, NJ, USA Yogendra Joshi George W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA, USA Satish G. Kandlikar Rochester Institute of Technology, Rochester, NY, USA Masahiro Kawaji City College of New York, New York, NY, USA University of Toronto, Toronto, ON, Canada Mark A. Kedzierski National Institute of Standards and Technology, Gaithersburg, MD, USA Khalil Khanafer Department of Mechanical Engineering, Australian College of Kuwait, Kuwait City, Kuwait Diego C. Knupp Laboratory of Experimentation and Numerical Simulation in Heat and Mass Transfer, Department of Mechanical Engineering and Energy, Polytechnic Institute, Rio de Janeiro State University, IPRJ/UERJ, Nova Friburgo, RJ, Brazil Aliihsan Koca Michigan Technological University, Houghton, MI, USA Navin Kumar Texas A&M University, College Station, TX, USA Andrey V. Kuznetsov Department of Mechanical and Aerospace Engineering, North Carolina State University, Raleigh, NC, USA

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Contributors

Ivan A. Kuznetsov Perelman School of Medicine, University of Pennsylvania, Philadelphia, PA, USA Department of Bioengineering, University of Pennsylvania, Philadelphia, PA, USA Vladimir V. Kuznetsov Department of Thermophysics of Multiphase Systems, Kutateladze Institute of Thermophysics of Siberian Branch of Russian Academy of Sciences, Novosibirsk, Russia Thomas Lestina Heat Transfer Research, Inc., Navasota, TX, USA Xin M. Liang Department of Cancer Biology, Dana-Farber Cancer Institute, Boston, MA, USA Department of Medicine, VA Boston Healthcare System, Boston, MA, USA Department of Biological Chemistry and Molecular Pharmacology, Harvard Medical School, Boston, MA, USA Phil Ligrani Propulsion Research Center, Department of Mechanical and Aerospace Engineering, University of Alabama in Huntsville, Huntsville, AL, USA Linhua H. Liu School of Energy Science and Engineering, Harbin Institute of Technology, Harbin, China Yang Liu North Carolina State University, Raleigh, NC, USA Iurii Lokhmanets Department of Mechanical Engineering, Heat Transfer Laboratory, McGill University, Montreal, QC, Canada S. Lorente Departement de Genie Civil, Institut National des Sciences Appliquées, Toulouse, France John J. Lowke CSIRO Manufacturing, Lindfield, NSW, Australia Mohammed Mahdi Faculty of Energy Systems and Nuclear Science, University of Ontario Institute of Technology, Oshawa, ON, Canada Tanmoy Maitra Nanoengineered Systems Laboratory, UCL Mechanical Engineering, University College London, London, UK Oronzio Manca Dipartimento di Ingegneria Industriale e dell’Informazione, Università degli Studi della Campania “Luigi Vanvitelli”, Aversa (CE), Italy Raj M. Manglik Thermal-Fluids and Thermal Processing Laboratory, Department of Mechanical and Materials Engineering, University of Cincinnati, Cincinnati, OH, USA Navid Manuchehrabadi Department of Mechanical Engineering, University of Minnesota at Minneapolis, Minneapolis, MN, USA Sunil S. Mehendale Michigan Technological University, Houghton, MI, USA

Contributors

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M. Pinar Mengüç Cekmeköy Campus, Özyegin University, Çekmeköy - Istanbul, Turkey Takahiko Miyazaki Faculty of Engineering Sciences, Kyushu University, Kasugashi, Fukuoka, Japan Christian Moreau Department of Mechanical and Industrial Engineering, Concordia University, Montreal, QC, Canada Javad Mostaghimi Centre for Advanced Coating Technologies, Department of Mechanical and Industrial Engineering, Faculty of Applied Science + Engineering, University of Toronto, Toronto, ON, Canada K. Muralidhar Department of Mechanical Engineering, Indian Institute of Technology Kanpur, Kanpur, India Anthony B. Murphy CSIRO Manufacturing, Lindfield, NSW, Australia Amitabh Narain Michigan Technological University, Houghton, MI, USA Arunn Narasimhan Department of Mechanical Engineering, Heat Transfer and Thermal Power Laboratory, Indian Institute of Technology Madras, Chennai, India Sergio Nardini Dipartimento di Ingegneria Industriale e dell’Informazione, Università degli Studi della Campania “Luigi Vanvitelli”, Aversa (CE), Italy Harishankar Natesan Department of Mechanical Engineering, University of Minnesota, Minneapolis, MN, USA Valerian Nemchinsky Physics Department, Keiser University, Fort Lauderdale, FL, USA Timothy J. O’Brien Bioelectromechanical Systems Laboratory, ICTAS Center for Engineered Health, Department of Biomedical Engineering and Mechanics, Virginia Tech - Wake Forest School of Biomedical Engineering and Sciences, Blacksburg, VA, USA Patrick H. Oosthuizen Department of Mechanical and Materials Engineering, Faculty of Engineering and Applied Science, Queen’s University, Kingston, ON, Canada Helcio R. B. Orlande Federal University of Rio de Janeiro, Cidade Universitária, Rio de Janeiro, RJ, Brazil Pradipta K. Panigrahi Department of Mechanical Engineering, Indian Institute of Technology Kanpur, Kanpur, India Larry Pershin University of Toronto, Toronto, ON, Canada Shaunak Phatak Department of Mechanical Engineering, University of Minnesota, Minneapolis, MN, USA

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Contributors

Igor L. Pioro Faculty of Energy Systems and Nuclear Science, University of Ontario Institute of Technology, Oshawa, ON, Canada John J. Pitre Jr. Biomedical Engineering Department, Tulane University, New Orleans, LA, USA Roman Popov Faculty of Energy Systems and Nuclear Science, University of Ontario Institute of Technology, Oshawa, ON, Canada Pierre Proulx Department of Chemical and Biotechnological Engineering, Université de Sherbrooke, Sherbrooke, QC, Canada João N. N. Quaresma School of Chemical Engineering, Universidade Federal do Pará, FEQ/UFPA, Campus Universitário do Guamá, Belém, PA, Brazil Hrishikesh Prasad Ranga Prasad Michigan Technological University, Houghton, MI, USA Luiz A. O. Rocha Departamento de Engenharia Mec^anica, Universidade Federal do Rio Grande do Sul, Porto Alegre, RS, Brazil John W. Rose School of Engineering and Materials Science, Queen Mary University of London, London, UK Praveen K. Sekar Department of Mechanical Engineering, University of Washington, Seattle, WA, USA Dusan P. Sekulic Department of Mechanical Engineering, University of Kentucky, Lexington, KY, USA School of Materials Science and Engineering, Harbin Institute of Technology, Harbin, PR, China Jamal Seyed-Yagoobi Multi-Scale Heat Transfer Laboratory, Department of Mechanical Engineering, Worcester Polytechnic Institute, Worcester, MA, USA Shahensha M. Shaik Department of Mechanical Engineering, Louisiana State University, Baton Rouge, LA, USA Helen Skop Smart Heat Corporation, Skokie, IL, USA Vladimir P. Solovjov Mechanical Engineering Department, Brigham Young University, Provo, UT, USA Leandro A. Sphaier Laboratory of Thermal Sciences – LATERMO, Department of Mechanical Engineering – TEM/PGMEC, Universidade Federal Fluminense, Niteroi, RJ, Brazil Jian Su Nuclear Engineering Department – PEN and Nanoengineering Department – PENT, COPPE, Federal University of Rio de Janeiro, Rio de Janeiro, RJ, Brazil Robert Sweet Department of Urology, University of Washington, Seattle, WA, USA

Contributors

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Michal Talmor Multi-Scale Heat Transfer Laboratory, Department of Mechanical Engineering, Worcester Polytechnic Institute, Worcester, MA, USA Yasunori Tanaka Faculty of Electrical and Computer Engineering, Kanazawa University, Kakuma, Kanazawa, Japan Manish K. Tiwari Nanoengineered Systems Laboratory, UCL Mechanical Engineering, University College London, London, UK Anna Lee Tonkovich Tonkomo LLC, Gilbert, AZ, USA Peter Vadasz Department of Mechanical Engineering, Northern Arizona University, Flagstaff, AZ, USA Rodolphe Vaillon CETHIL, UMR 5008, Univ Lyon, CNRS, INSA-Lyon, Université Claude Bernard Lyon 1, Villeurbanne, France Radiative Energy Transfer Laboratory, Department of Mechanical Engineering, University of Utah, Salt Lake City, UT, USA Izak Jacobus van der Walt R&D Plasma Development, The South African Nuclear Energy Corporation, Plelindaba, North West Province, South Africa Armelle Vardelle European Ceramic Center, Laboratoire Sciences des Procédés Céramiques et de Traitements de Surface, University of Limoges, Limoges Cedex, France Zhimin Wan George W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA, USA Evelyn N. Wang Device Research Laboratory, Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA, USA Brent W. Webb Mechanical Engineering Department, Brigham Young University, Provo, UT, USA Stefan aus der Wiesche Department of Mechanical Engineering, Muenster University of Applied Sciences, Steinfurt, Germany Subramaniam Yugeswaran University of Toronto, Toronto, ON, Canada Shigang Zhang Nanoengineered Systems Laboratory, UCL Mechanical Engineering, University College London, London, UK Zhuomin M. Zhang George W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA, USA Bo Zhao George W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA, USA

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Contributors

Junming M. Zhao School of Energy Science and Engineering, Harbin Institute of Technology, Harbin, China Alexander I. Zhmakin SoftImpact Ltd., St. Petersburg, Russia Ioffe Institute, St. Petersburg, Russia Liang Zhu Department of Mechanical Engineering, University of Maryland Baltimore County, Baltimore, MD, USA Yangying Zhu Device Research Laboratory, Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA, USA

Part I Heat Transfer Fundamentals

1

Macroscopic Heat Conduction Formulation Leandro A. Sphaier, Jian Su, and Renato Machado Cotta

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Basic Heat Conduction Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 General Mathematical Formulation for Heat Conduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Isotropic Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Anisotropic Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Boundary and Initial Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Heat Conduction Equation in Different Coordinate Systems . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Regular and Irregular Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Dimensionless Form of the Heat Conduction Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8 Nonlinear Heat Conduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Extended Heat Conduction Formulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Heat Conduction in Heterogeneous Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Heat Conduction in Multilayered Composite Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Heat Conduction with Phase Change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Hyperbolic Heat Conduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Conjugate Conduction-Convection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Drying . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 Heat Conduction with Mass Transfer and Physical Adsorption . . . . . . . . . . . . . . . . . . . . . . .

4 6 6 8 9 9 12 14 15 17 19 19 21 23 26 27 29 33

L. A. Sphaier (*) Laboratory of Thermal Sciences – LATERMO, Department of Mechanical Engineering – TEM/PGMEC, Universidade Federal Fluminense, Niteroi, RJ, Brazil e-mail: [email protected] J. Su Nuclear Engineering Department – PEN and Nanoengineering Department – PENT, COPPE, Federal University of Rio de Janeiro, Rio de Janeiro, RJ, Brazil e-mail: [email protected] R. M. Cotta Universidade Federal do Rio de Janeiro – UFRJ, Rio de Janeiro, RJ, Brazil e-mail: [email protected] # Springer International Publishing AG, part of Springer Nature 2018 F. A. Kulacki (ed.), Handbook of Thermal Science and Engineering, https://doi.org/10.1007/978-3-319-26695-4_3

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4 Lumped and Improved-Lumped Formulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Lumped-Capacitance Formulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Lumped-Differential Formulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Improved Lumped-Differential Formulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

In this chapter, mathematical formulations of macroscopic heat conduction are derived from the First Law of Thermodynamics. Specific forms of the heat conduction equation in isotropic media are given in Cartesian, Cylindrical, and Spherical coordinates systems, as well as in a general orthogonal coordinate system. Heat conduction equations in anisotropic media and in heterogeneous media are then derived. Mathematical formulations of one-dimensional transient heat conduction with phase change and in multilayered composite media are presented. Finally, classical and improved lumped parameter formulations for transient heat conduction problems are analyzed more closely. The so-called Coupled Integral Equations Approach (CIEA) is reviewed as a problem reformulation and simplification tool in heat and mass diffusion. The averaged temperature and heat flux, in one or more space coordinates, are approximated by Hermite formulae for integrals, yielding analytic relations between boundary and average temperatures, to be used in place of the usual plain equality assumed in the classical lumped system analysis. The accuracy gains achieved through the improved lumped-differential formulations are then illustrated through a few typical examples.

1

Introduction

The classical heat conduction theory was fully established much before heat transfer physics could explain energy transport, storage, and conversion through the principal microscopic energy carriers. By the beginning of the nineteenth century, when heat conduction theory was in fact constructed, the physics behind heat transfer within a body was not unanimously explained. One line of academic thinking would defend that heat was some kind of invisible fluid or microscopic particles that would diffuse within the body undergoing heat transfer, also known as “caloric.” Another current would trust that heat was a result of vibrations and other motions of matter at the atomic and molecular levels (Narasimhan 1999). Jean-Baptiste Fourier (Auxerre, 1768–Paris, 1830), in his classical work of 1807 “Théorie de la Propagation de la Chaleur dans les Solides” (Grattan-Guinness and Ravetz 1972), laid the foundations of macroscopic heat conduction theory, by first proposing the empirically based constitutive equation relating temperature gradient and heat flux, afterwards named as Fourier’s law of heat conduction. This proposition, in combination with the first law of thermodynamics and some additional assumptions, allowed him to derive a fairly general partial differential

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Macroscopic Heat Conduction Formulation

5

equation that represents the transient heat conduction phenomena, and governs the temperature distribution in space and time within a body undergoing heat transfer. However, his pioneering work was not immediately accepted by the scientific community, and it was not until the consolidation of his findings, in his treatise “Théorie Analytique de la Chaleur” of 1822 (Fourier 1822), that the theory was more widely accepted. Already in Chap. I, and first section of his groundbreaking treatise, at the statement of the object of the work, it can be read: The effects of heat are subject to constant laws which cannot be discovered without the aid of mathematical analysis. The object of the theory which we are about to explain is to demonstrate these laws; it reduces all physical researches on the propagation of heat, to problems of the integral calculus whose elements are given by experiment. No subject has more extensive relations with the progress of industry and the natural sciences; for the action of heat is always present, it penetrates all bodies and spaces, it influences the processes of the arts, and occurs in all the phenomena of the universe.

In this very first paragraph, Fourier makes it clear that the material that would follow in his consolidating work, would be strongly based on mathematics, but built on top of experimental observations that led to the establishment of the fundamental law. He was also visionary in terms of the importance of his findings, in all areas of fundamental sciences and practical applications, in light of the presence of heat transfer in most natural or man-promoted phenomena. In the sequence, Fourier also summarizes the motivation for his mathematical modeling effort, relating the physics behind heat transfer within a body to the need of finding steady or transient temperature distributions for any such process, so as to understand the phenomena behind it: When heat is unequally distributed among the different parts of a solid mass, it tends to attain equilibrium, and passes slowly from the parts which are more heated to those which are less; and at the same time it is dissipated at the surface, and lost in the medium or in the void. The tendency to uniform distribution and spontaneous emission, which acts at the surface of bodies, changes continually the temperature at their different points. The problem of the propagation of heat consists in determining what is the temperature at each point of a body at a given instant, supposing that the initial temperatures are known. The following examples will more clearly make known the nature of these problems.

Somehow, the first two paragraphs of Fourier’s treatise of 1822 (Fourier 1822) already introduce the notion that problem formulation is a major step in the handling of heat conduction. Thus, a compilation on modern heat conduction theory should start from the classical formulation known as Fourier’s equation, and then advance toward generalizations that extend the applicability of Fourier’s theory to other phenomena that are not accounted for in the original formulation, such as in dealing with anisotropic media, phase change, heterogeneous media, conjugated heat transfer problems, hyperbolic heat conduction, and simultaneous heat and mass transfer. A number of textbooks and reference works in heat conduction, among them refs.(Carslaw and Jaeger 1959; Arpaci 1966; Özişik 1968; Luikov 1968; Özişik 1993; Myers 1998; Yener et al. 2017; Mikhailov and Özişik 1984; Poulikakos 1993; Cotta and Mikhailov 1997), have

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extended the applicability of Fourier’s work along these almost 200 years after his treatise, by both providing extended heat conduction formulations and deriving analytical solutions to the resulting partial differential equations. It is not surprising that Fourier’s heat conduction theory then triggered the interpretation of different diffusion processes in the form of constitutive equations relating fluxes with potential gradients, such as in Ohm’s law (1827), Fick’s law (1855), and Darcy’s law (1856), among others. The present chapter first provides a brief introduction on the derivation of the heat conduction equation and then offers a systematic revision of the most important extensions to this basic formulation. In addition, the last section deals with lumpeddifferential formulations, when full or partial lumping of the partial differential heat conduction equation can lead to significant simplification of the problem to be mathematically solved. Special emphasis is given to an improved lumpeddifferential formulation methodology, known as the Coupled Integral Equations Approach (CIEA) (Mennig and Özişik 1985; Aparecido et al. 1989; Cotta et al. 1990; Scofano Neto and Cotta 1993; Traiano et al. 1997; Cheroto et al. 1997; Corrêa and Cotta 1998; Cotta 1998; Alves et al. 2000; Regis et al. 2000; Reis et al. 2000; Su 2001; Su and Cotta 2001; Cotta et al. 2003; Su 2004; Ruperti et al. 2004; Dantas et al. 2007; Pontedeiro et al. 2008; Su et al. 2009; Tan et al. 2009; Naveira et al. 2009; Naveira-Cotta et al. 2010; da Silva and Sphaier 2010; An and Su 2011; Knupp et al. 2012; Sphaier and Jurumenha 2012; An and Su 2013; De Souza et al. 2015; An and Su 2015; Moreira et al. 2015; de Souza et al. 2016). This approach is based on integral equations for the temperature and heat flux averaged over one or more space variables, combined with approximate Hermite formulas for integration, to offer improved accuracy, but at the same level of complexity, with respect to the classical lumped analysis procedure. The aim is to reduce, whenever possible, the number of space coordinates to be considered in the heat conduction formulation, however without neglecting its contribution to the overall heat transfer process, by taking into consideration the influence of the boundary conditions in that particular coordinate direction. The main difference is that the classical lumping procedure essentially introduces a zeroth order approximation to the temperature gradient within the medium along the specific coordinate(s), while the improved approach provides higher order approximations. The discussion of analytical solution methodologies to any such formulations is then postponed to the following chapter.

2

Basic Heat Conduction Theory

2.1

General Mathematical Formulation for Heat Conduction

The first law of thermodynamics (Cengel and Boles 1998; Sontag and Van Wylen 1991) for a material volume, or closed system, can be written in a general rate form as: dE _ ¼ Q_  W, dt

(1)

1

Macroscopic Heat Conduction Formulation

7

where E is the total energy of the system, Q_ is the rate of heat transfer to the system, and W_ is the rate of work done by the system. Separating E in terms of the internal energy U, the mechanical energy Em and other forms of energy Eo gives: dU dEm dEo þ ¼ Q_  W_  , dt dt dt

(2)

where the last term in the previous equation represents the conversion of other forms of energy, such as chemical, nuclear, electrical, or electromagnetic, into thermal energy. It is commonly denoted as the rate of thermal energy generation E_ g ¼ dEo =dt, such that Eq. (2) can be rewritten as: dU dEm þ ¼ Q_  W_ þ E_ g: dt dt

(3)

Assuming rigid body motion (negligible deformation), the rate of work term cancels out the rate of change in mechanical energy, such that Eq. (3) is reduced to dU ¼ Q_ þ E_ g , dt

(4)

where, as there is no deformation, any heat generation by friction would be only possible at the volume boundaries. Let ρ be the specific mass (or density) of the solid and u be the specific internal energy, the internal energy U in a volume V is given by ð U¼

V

ρu dV :

(5)

The rate of heat addition to the system through its surface, S is written in terms of the heat flux vector: Q_ ¼ 

ð S

q00  n dA

(6)

whereas the rate of energy generation can be written in terms of the volumetric rate 000 of energy generation g_ E_ g ¼

ð V

g_ 000 dV :

(7)

Substituting these quantities in Eq. (4) leads to d dt

ð V

ð ρ u dV

¼

S

q00  n dA þ

ð V

g_ 000 dV :

Using Gauss’ divergence theorem, Eq. (8) can be rewritten as

(8)

8

L. A. Sphaier et al.

ð  V

@ ðρuÞ þ ∇  q00  g_ 000 Þ dV @t

¼ 0:

(9)

Considering that this equation must be valid for an arbitrary volume, the integrand must be zero. Hence, one arrives at a general energy balance equation in differential form: @ ðρuÞ ¼ ∇  q00 þ g_ 000 : @t

(10)

Since mass conservation for the considered system requires that @ρ/@t = 0, and the internal energy for an incompressible substance can be written as a function of temperature T only, the previous balance can be written in terms of the rate of change of temperature: ρc

@T ¼ ∇  q00 þ g_ 000 , @t

(11)

where c = cp = cv (as an incompressible substance is considered) is the specific heat of the medium.

2.2

Isotropic Materials

For an isotropic material, Fourier’s law (Fourier 1822)1 in terms of a scalar thermal conductivity k is used to relate the heat flux vector to the temperature gradient: q00 ¼ k∇T,

(12)

such that Eq. (11) can be rewritten as: ρc

@T ¼ ∇  ðk∇T Þ þ g_ 000 : @t

(13)

For temperature independent and uniform thermal conductivity, Eq. (13) is generally written as 1 @T g_ 000 ¼ ∇2 T þ , α @t k in which α = k/(ρcp) is the thermal diffusivity. If a steady-state problem is involved, Eq. (14) is reduced to

1

English translation reprint in (Fourier 1878).

(14)

1

Macroscopic Heat Conduction Formulation

9

g_ 000 ∇2 T ¼  , k

(15)

which is known as a Poisson equation. Additionally, if there is no heat generation, the Laplace equation is obtained: ∇2 T ¼ 0:

2.3

(16)

Anisotropic Materials

In an anisotropic medium, the heat flux vector q00 is not aligned with the temperature gradient vector through a scalar coefficient, but through a second-order tensor 000

q ¼ K  ∇T,

(17)

such that Eq. (11) can be rewritten as: ρc

@T ¼ ∇  ðK  ∇T Þ þ g_ 000 , @t

(18)

where the thermal conductivity tensor K is a symmetric tensor generally written as: 2

k11 K ¼ 4 k21 k31

k12 k22 k32

3 k13 k23 5: k33

(19)

In the special case of orthotropic materials, the conductivity tensor K assumes a diagonal form: 2

k11 K ¼ 40 0

2.4

0 k22 0

3 0 0 5: k33

(20)

Boundary and Initial Conditions

Boundary conditions for the heat conduction equation are commonly of the first-type (or Dirichlet), second-type (or Neumann), or third-type (or Robin). The Dirichlet condition involves prescribing the temperature at the surface S : T ðx, tÞ ¼ T s ðx, tÞ, for x  S , where Ts corresponds to the known surface temperature.

(21)

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L. A. Sphaier et al.

The Neumann condition involves prescribing the heat flux at the surface: q00  n ¼ q00s ðx, tÞ,

for

xS ,

(22)

where qs00 corresponds to the supplied power per unit area at the surface. Finally, the Robin condition involves writing the heat flux at the surface in terms of a temperature difference, using Newton’s law of cooling:   q00  n ¼ h T  T f ,

for

xS ,

(23)

where h is a heat transfer coefficient and Tf is the external ambient fluid temperature. This condition is associated with convection heat transfer at the surface. For small values of h, this condition approaches the adiabatic condition (thermal insulation), whereas large h values approach the Dirichlet boundary condition. Besides the three classical boundary conditions, a variety of other boundary conditions can occur. One typical example is found when there is radiation heat transfer at the surface, which can lead to   q00  n ¼ eσ T 4  T 4s ,

for

xS ,

(24)

where σ is the Stephan Boltzmann constant, e is the surface emissivity for a gray body, and Ts is the radiation sink temperature to which the body emits radiation. In many applications, both convective and radiative heat transfer occur at the body surface. In these cases, the combined convective-radiation boundary condition should be specified as     q00  n ¼ h T  T f þ eσ T  T 4s ,

for

xS :

(25)

It should be noted that in many cases the environmental fluid temperature Tf differs from the radiation sink temperature Ts. Hence, it is often convenient to introduce the adiabatic surface temperature Ta, defined by the nonlinear algebraic equation     h T a  T f þ eσ T 4a  T 4s ¼ 0:

(26)

The boundary condition (25) can be rewritten with the use of the adiabatic surface temperature   q00  n ¼ hðT  T a Þ þ eσ T 4  T 4a ,

for

xS :

(27)

The use of the adiabatic surface temperature Ta has the advantage of specifying only one boundary temperature in numerical solutions, instead of two boundary temperatures. The nonlinear boundary condition Eq. (27) can be linearized by using first-order Taylor’s expansion as

1

Macroscopic Heat Conduction Formulation

11

q00  n ¼ hðT  T a Þ þ 4eσT 3a ðT  T a Þ,

for

xS :

(28)

The quantity 4e σ T 3a can be considered to be an equivalent to a radiative heat transfer coefficient hrad, with the boundary condition Eq. (29) rewritten as q00  n ¼ hðT  T a Þ þ hrad ðT  T a Þ ¼ hcomb ðT  T a Þ,

for

xS ,

(29)

where the combined convective-radiative heat transfer coefficient hcomb = h + hrad. Besides the boundary conditions, interfacial conditions should be specified if there is an interface between two regions in which there is discontinuity in the thermophysical properties: ni  ðka ∇T a Þ ¼ ni  ðkb ∇T b Þ,

on

 ni  ðka ∇T a Þ ¼ hi ðT a  T b Þ,

on

S i, S i,

(30) (31)

where S i is the interface surface, ni is the normal vector of the interface pointing from Region a to Region b, and hi is the interfacial heat transfer coefficient between Region a and Region b. Equation (30) represents the heat flux continuity condition, while Eq. (31) expresses the effect of a thermal contact resistance. If the interfacial heat transfer coefficient is sufficiently high, the temperature difference across the interface becomes negligible, and the interfacial boundary condition Eq. (31) can be replaced by a temperature continuity condition, as Ta ¼ Tb,

on

S i:

(32)

Interfacial condition Eq. (32) is also called the perfect thermal contact condition. Besides boundary conditions, the classical Fourier heat conduction formulation, as given by Eqs. (13) or (18), also requires an initial condition to be prescribed. Due to the first-order nature in time of the problem, one simply needs to provide the initial temperature distribution throughout the domain: T ðx, t ¼ 0Þ ¼ T 0 ðxÞ,

for

xV :

(33)

When hyperbolic heat conduction models are considered, the conduction problem becomes of second order in time, and thus an additional initial condition is required for the rate of temperature change. This can be written as  @T  ¼ f 0 ðxÞ, @t t ¼ 0

for

xV :

(34)

in which f0 is the initial rate of temperature change. Several other boundary condition expressions could be provided, such as finite thermal capacitance at the boundary (fourth and fifth kinds), wall conjugation (transversal resistance or axial wall conduction), coupled heat and mass transfer,

12

L. A. Sphaier et al.

and interface phase change. These shall be partially examined along with the extensions to the classical heat conduction equation in Sect. 3.

2.5

Heat Conduction Equation in Different Coordinate Systems

The heat conduction equation can assume different forms, according to the chosen coordinate system. Using the isotropic heat conduction formulation, three common forms are presented. In the Cartesian system, Eq. (13) is rewritten as: ρc

      @T @ @T @ @T @ @T ¼ k k k þ þ þ g_ 000: @t @x @x @y @y @z @z

(35)

If the Polar-Cylindrical system is adopted, Eq. (13) is written in the form: ρc

      @T 1 @ @T 1 @ @T @ @T ¼ kr k k þ 2 þ þ g_ 000: @t r @r @r r @ϕ @ϕ @z @z

(36)

Using the Polar-Spherical system, Eq. (13) is written as:       @T 1 1@ 1 @ @T 1 @ @T 2 @T ¼ kr k sin θ k ρc þ 2 þ 2 2 @t r 2 @r @r r sin θ @θ @θ r sin θ @ϕ @ϕ þ g_ 000 :

(37)

While the previous forms are the most common ones, other coordinate systems can be adopted. Consider a general orthogonal coordinate system (u1, u2, u3), which is related to the Cartesian coordinate system through x ¼ xðu1 , u2 , u3 Þ,

y ¼ yðu1 , u2 , u3 Þ,

z ¼ zðu1 , u2 , u3 Þ:

(38)

In this system, the differential length ds is obtained as ðdsÞ2 ¼ ðdxÞ2 þ ðdyÞ2 þ ðdzÞ2 ¼ a21 ðdu1 Þ2 þ a22 ðdu2 Þ2 þ a23 ðdu3 Þ2 ,

(39)

where  a2i

¼

@x @ui

2



@y þ @ui

2



@z þ @ui

2 :

(40)

Then, the heat conduction equation in a general orthogonal coordinate system is given by ρc

       @T 1 @ a @T @ a @T @ a @T ¼ k 2 k 2 k 2 þ þ þ g_ 000 , @t a @u1 @u2 @u3 a1 @u1 a2 @u2 a3 @u3

(41)

1

Macroscopic Heat Conduction Formulation

13

where a ¼ a1 a2 a3 :

(42)

In the case of a general anisotropic medium, for the Cartesian coordinate system, the relations between the components of the heat flux vector and the temperature gradient are given by   @T @T @T þ k12 þ k13 q00x ¼  k11 , @x @y @z   @T @T @T 00 þ k22 þ k23 , qy ¼  k21 @x @y @z   @T @T @T þ k32 þ k33 : q00z ¼  k31 @x @y @z

(43a)

(43b)

(43c)

For the Polar-Cylindrical coordinate system, the heat flux components are written as: 

@T k12 þ ¼  k11 @r r  @T k22 þ q00ϕ ¼  k21 @r r  @T k32 þ q00z ¼  k31 @r r q00r

 @T @T þ k13 , @ϕ @z  @T @T þ k23 , @ϕ @z  @T @T þ k33 , @ϕ @z

(44a)

(44b)

(44c)

whereas for the Polar-Spherical coordinate system, the heat flux components are given by  @T k12 @T k13 þ þ q00r ¼  k11 @r r sin θ @ϕ r  @T k22 @T k23 þ þ q00ϕ ¼  k21 @r r sin θ @ϕ r  @T k32 @T k33 þ þ q00θ ¼  k31 @r r sin θ @ϕ r

 @T , @θ  @T , @θ  @T : @θ

(45a)

(45b)

(45c)

The general anisotropic case can be simplified to other forms if the coordinate system axes are aligned with the principal directions of the conductivity tensor, and one recovers the orthotropic heat conduction equation. For this situation, the thermal

14

L. A. Sphaier et al.

conductivity tensor is reduced to a diagonal form, and the heat flux and heat conduction equations are simplified accordingly. For the Cartesian coordinate system, the heat conduction equation for orthotropic medium is written as: ρc

      @T @ @T @ @T @ @T ¼ k11 k22 k33 þ þ þ g_ 000 , @t @x @x @y @y @z @z

(46)

whereas for the Polar-Cylindrical system the resulting form is given by:       @T 1 @ @T 1 @ @T @ @T ¼ k11 r k22 k33 ρc þ 2 þ þ g_ 000 , @t r @r @r r @ϕ @ϕ @z @z

(47)

and the following form is obtained for the Polar-Spherical system: ρc

    @T 1 @ @T 1 @ @T 1 ¼ 2 k11 r 2 k22 sin θ þ 2 þ 2 2 @t r @r @r r sin θ @θ @θ r sin θ   @ @T k33  þ g_ 000 : @ϕ @ϕ

(48)

When one-dimensional formulations are considered, and the relevant spatial coordinate in polar systems is the radial direction, a general heat conduction formulation valid for all three systems is commonly adopted. These cases are written in the general form: ρc

  @T 1 @ @T ¼ p k11 r p þ g_ 000 , @t r @r @r

(49)

where p = 0, 1, 2 indicates Cartesian, Polar-Cylindrical, and Polar-Spherical coordinate systems, respectively.

2.6

Regular and Irregular Domains

The choice of a coordinate system for the formulation of a heat conduction problem will have a great impact on the required solution techniques. Regular domains are the preferred choice, for this allows many problems to be solved with analytical techniques, whereas a heat conduction formulation within an irregular domain will require fully numerical or hybrid analytical-numerical methods. A regular domain is here understood as a configuration for which the boundary surfaces of the domain can be described by a simple equation in which a coordinate variable is constant and there is no dependence on other spatial variables. Consequently, the normal vector to this surface is also constant and has a single component in the corresponding coordinate direction. As an example, consider a twodimensional linear heat conduction problem defined within a semicircular region 0  r  a and 0  ϕ  π:

1

Macroscopic Heat Conduction Formulation

15

    @T 1 @ @T 1 @ @T ¼ kr k ρc þ 2 þ g_ 000 , @t r @r @r r @ϕ @ϕ

(50a)

T ða, ϕ, tÞ ¼ T a ,

for

0  ϕ  π,

and

t > 0,

(50b)

T ðr, 0, tÞ ¼ T a ,

for

0  r  a,

and t > 0,

(50c)

T ðr, π, tÞ ¼ T a ,

for

0  r  a,

and

t > 0,

(50d)

0  ϕ  π:

(50e)

T ðr, ϕ, 0Þ ¼ T 0 ,

for

0  r  a,

and

This comprises a regular domain formulation, as the first boundary condition is defined at a surface described by r = a. Similarly, the second and third boundary conditions are defined at surfaces described by constant values of the angular variable (ϕ = 0 and ϕ = π, respectively). If this same problem were written in the Cartesian system, the resulting formulation would be given by:     @T @ @T @ @T ¼ k k ρc þ þ g_ 000 , @t @x @x @y @y T ðx, y1 ðxÞ, tÞ ¼ T a, T ðx, 0, tÞ ¼ T a, T ðx, y, 0Þ ¼ T 0,

for

for for

a  x  a, a  x  a,

a  x  a,

and

and

(51a) t > 0,

(51b)

t > 0,

(51c)

0  y  y1 ðxÞ,

(51d)

and

which is in an irregular domain form, as the upper boundary y = y1(x), which for this pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi case would be given by y1 ðxÞ ¼ a2  x2 , the value of y at the boundary varies with x. This poses a difficulty for classical analytical methods, as this boundary condition cannot be separated by traditional approaches. While this example illustrates a domain that can be regular if a proper choice of coordinate system is adopted, there are cases in which the description of the boundaries cannot be done in a regular domain form. Although these problems require more complex solution strategies, a combination of analytical and numerical techniques have been shown to provide effective solutions (Aparecido et al. 1989; Barbuto and Cotta 1997; Pérez Guerrero et al. 2000; Sphaier and Cotta 2000, 2002), as shall be further discussed in ▶ Chap. 3, “Numerical Methods for Conduction-Type Phenomena.”

2.7

Dimensionless Form of the Heat Conduction Equation

The heat conduction equation can be normalized by introducing appropriate dimensionless variables and parameters. As an example, consider Eq. (13) in Cartesian coordinates for heat conduction in an isotropic material with constant properties and no heat generation:

16

L. A. Sphaier et al.

1 @T @ 2 T @ 2 T @ 2 T ¼ 2þ 2þ 2: α @t @x @y @z

(52)

The dimensionless temperature θ is defined as θ¼

T  T ref , ΔT ref

(53)

where Tref is a reference temperature, such as a constant external fluid temperature Tf, and ΔTref is a reference temperature difference, such as the difference between a uniform initial temperature T0 and the external temperature Tf. If the length scale of the solid body in consideration is L, the dimensionless spatial coordinates are defined as x ξ¼ , L

η¼

y , L

z ζ¼ , L

(54)

whereas the dimensionless time is defined as τ¼

αt , L2

(55)

which is also known as the Fourier number. Using the previously defined nondimensional quantities, the heat conduction equation is finally written in a normalized form: @θ @ 2 θ @ 2 θ @ 2 θ ¼ þ þ : @τ @ξ2 @η2 @ζ 2

(56)

Boundary conditions are also normalized with the same parameters and additional dimensionless quantities. For a Dirichlet condition at the surface S , the normalized equation is directly given in terms of a prescribed dimensionless surface temperature θs: θðξ, η, ζ, τÞ ¼ θs ðξ, η, ζ, τÞ,

at

S ,

(57)

A dimensionless Neumann condition prescribing the heat flux at the surface is written as: 

@θ ¼ ϕs , @n

at

S ,

(58)

where n is the normal vector in the above normal derivative, and the dimensionless heat flux ϕs is defined as ϕs ¼

q00s L : kΔT ref

(59)

1

Macroscopic Heat Conduction Formulation

17

If the problem possesses no characteristic temperature difference, ΔTref is commonly chosen based on the other dimensional quantities such that ϕs = 1. Finally, a dimensionless Robin condition involving convective heat transfer to an external fluid at a constant temperature is written as: 

@θ ¼ Biθ, @n

S ,

at

(60)

where Bi is the Biot number, defined as Bi ¼

hL , k

(61)

which indicates the ratio between the thermal resistance due to conduction heat transfer into the solid and convective heat transfer at the solid surface. Small values of the Biot number indicate relatively higher resistance in convecting heat to the fluid and lower to conduct into the solid, whereas large Biot values indicate relatively lower resistance in convecting heat to the fluid and higher in conducting to the solid.

2.8

Nonlinear Heat Conduction

The nonlinear heat conduction Eq. (13) can be linearized by using Kirchoff transformation, by introducing a modified temperature (Carslaw and Jaeger 1959; Todreas and Kazimi 2012) θ¼

1 k0

ðT

kðT Þ dT,

(62)

T0

where k0 is the thermal conductivity at a reference temperature T0. The gradient of the modified temperature θ is given by ∇θ ¼

1 ∇ k0

ðT

kðT Þ dT ¼

T0

1 kðT Þ∇T: k0

(63)

Thus k0 ∇θ ¼ kðT Þ∇T,

(64)

g_ 000 1 @θ ¼ ∇2 θ þ , αðT Þ @t k0

(65)

and Eq. (13) becomes

where the thermal diffusivity is defined by

18

L. A. Sphaier et al.

αðT Þ ¼

ρcðT Þ : k ðT Þ

(66)

It is interesting to note that, for steady state, Eq. (65) becomes a linear conduction equation: k0 ∇2 θ þ g_ 000 ¼ 0:

(67)

Moreover, if the temperature dependence of the thermal diffusivity is negligible, Eq. (65) also becomes a linear equation for transient heat conduction: 1 @θ g_ 000 ¼ ∇2 θ þ , α @t k0

(68)

where α is an average thermal diffusivity of the solid in the temperature range under study. When the thermal diffusivity has a fairly uniform spatial behavior, but its variation in time cannot be disregarded, another transformation in time can be introduced to linearize the left hand side term of Eq. (65): τ¼

1 α0

ð1

αðtÞdt,

(69)

0

such that Eq. (65) becomes a linear equation for transient heat conduction: 1 @θðτÞ g_ 000 ¼ ∇2 θðτÞ þ : α0 @τ k0

(70)

The Kirchoff transformation does not affect the linearity of boundary conditions of first or second kinds. However, in the case of a third kind boundary condition, except for a very specific functional behavior of the heat transfer coefficient, the transformed boundary condition results to be nonlinear (Özişik 1993). Several other transformations and linearization procedures can be found in the literature for nonlinear heat conduction, as reviewed in (Özişik 1993). For instance, for one-dimensional transient heat conduction in a semi-infinite domain, an independent variables transformation, known as the Boltzmann transformation, can be used to transform the partial differential equation into an ordinary differential equation. The one-dimensional heat conduction equation in the Cartesian coordinate system is written as   @T @ @T ρc ¼ k : @t @x @x Then, the following similarity variable is introduced:

(71)

1

Macroscopic Heat Conduction Formulation

19

x ηðx, tÞ ¼ pffi , t

(72)

such that Eq. (71) becomes a nonlinear ordinary differential equation: ρc

  η dT d dT ¼ k : 2 dη dη dη

(73)

Modern hybrid and purely numerical methods, are readily applicable to handle such nonlinear formulations.

3

Extended Heat Conduction Formulations

3.1

Heat Conduction in Heterogeneous Media

Consider a macroscopic region composed of n subregions of different materials with contact thermal resistance and heat generation. The microscopic heat conduction equations and interface conditions are given by @T i ¼ ∇:ðki ∇T i Þ þ g_ 000 in V i , @t   nij  ðki ∇T i Þ ¼ nij  kj ∇T j , on S ij   nij  ðki ∇T i Þ ¼ hij T i  T j , on S ρi cp, i

i ¼ 1, . . . , n

(74a)

i, j ¼ 1, . . . , n ,

(74b)

i,

i, j ¼ 1, . . . , n :

ij

(74c)

Equations (74b) and (74c) are defined on the interface S ij between subregions i and j, nij is the unit vector normal to the interface from subregion i to subregion j, hij 000 is the interfacial heat transfer coefficient between subregions i and j, and g_ i is the volumetric heat generation rate in subregion V i . As a particular case of heterogeneous media, one considers a macroscopic region composed by only two subregions: a matrix V f , and randomly distributed spherical insertions V s. It is assumed that there is internal heat generation only in the spherical particles, as for example in fuel elements for high temperature gas-cooled nuclear reactors. The microscopic heat conduction equations for the two regions are given by: ρf cpf ρs cps 000

  @T f ¼ ∇  kf ∇T f , @t

@T s ¼ ∇  ðks ∇T s Þ þ g_ 000 s , @t

in

V in

f

V

,

(75a) s,

where g_ s is the volumetric heat generation rate in the fuel pellet.

(75b)

20

L. A. Sphaier et al.

Fig. 1 Representative element volume (REV) in heterogeneous medium (Adapted from Hsu 1999)

y z x

(REV)

 dp

Considering microscopically perfect thermal contact between the two subregions, the interface conditions written as: T f ¼ T s , on S fs ,   nfs  kf ∇T f ¼ nfs  ðks ∇T s Þ, on

(76a) S

fs :

(76b)

Using the idea of a representative element volume (REV), or a cube with dimensions l much larger than that of the particle diameter dp, as displayed in Fig. 1, and applying the volumetric averaging over the microscopic energy equations (75a) and (75b), one arrives at the following macroscopic equations (Hsu 1999): ρf C pf ρsC ps

      @ ϕ Tf þ ∇: kf Λfs þ Qfs, ¼ ∇: kf ∇ ϕ T f @t

(77)



  @ ½ð1  ϕÞhT s i ¼ ∇ : ks ∇ ðð1  ϕÞhT s iÞ  ∇: ks Λfs  Qfs þ ð1  ϕÞg_ 000 s , (78) @t

where ϕ is the porosity and 〈Tj〉 is the volumetrically averaged intrinsic temperature of subregion j in the REV, defined as

 Tj ¼

1 V

REV

ð V

T j dV , REV

Λfs represents the effect of thermal tortuosity (Hsu 1999):

(79)

1

Macroscopic Heat Conduction Formulation

Λfs 

ð

1 V

21

REV

S

nfs T f dS ¼ fs

1 V

REV

ð S

nfs T s dS ,

(80)

fs

and Qfs represents the interfacial heat transfer (Hsu 1999) Qfs 

1 V

REV

ð S

  nfs  kf ∇T f dS ¼ fs

1 V

REV

ð S

nfs  ðks ∇T s Þ dS ,

(81)

fs

where V REV is the volume of the representative element volume, and S fs is the interfacial surface within the representative element volume. Hsu (1999) showed that:    Λfs ¼ G ∇ T f  σ∇ hT s i , 

 Qfs ¼ hfs afs hT s i  T f ,

(82a) (82b)

where G and hfs are, respectively, the coefficients of thermal tortuosity and macroscopic interfacial heat transfer, afs is the specific interfacial area and σ  ks =kf :

(83)

Substituting Eqs. (82a) and (82b) in Eqs. (77) and (78) leads to:          @ ϕ Tf þ ∇  k f G ∇ T f  σ ∇ hT s i ρf C pf ¼ ∇ : kf ∇ ϕ T f @t 

 þ hfs afs hT s i  T f ,   @ ½ð1  ϕhT s iÞ ¼ ∇ : ks ∇ ðð1  ϕÞhT s iÞ ρsC ps @t 

    ∇  ks G ∇ T f  σ ∇ hT s i  hfs afs hT s i  T f þ ð1  ϕÞg_ 000 s :

(84a)

(84b)

The volume averaged two-temperature formulation can be used to model transient heat conduction in heterogeneous solid media or porous media when the two phases are not in local thermal equilibrium.

3.2

Heat Conduction in Multilayered Composite Media

Heat conduction in composite media can be modeled by Eq. (74a) and interfacial conditions Eqs. (74b) and (74c). As a special case, one-dimensional transient heat conduction in multilayer composite medium consisting of M parallel or concentric layers, subjected to convective heat transfer at the left side and combined convective and radiative heat transfer at the right side, is illustrated, as shown in Fig. 2. The layers are labeled as 1 to M from the left to right. Let x be the coordinate normal to the layers, xi, i = 1,2,...,M, represent its value at the left surface of each layer, and xM+1 be the coordinate value at the right surface of M-th layer. It is assumed that the

22

L. A. Sphaier et al.

Fig. 2 Multilayered composite medium

1

1

2

2

i

3

i+1

i+1

i

M-1

M

M-1

i+2

M

M+1

thermophysical properties of the layers are homogeneous, isotropic, and independent of temperature. The volumetric heat generation rate in the j-th layer is g_ 000: The thermal contact resistance between adjacent i-th and (i + 1)-th layer at interface xi+1, i = 1, . . ., M  1, is represented by Rci. Initially, all the layers are at a specified uniform temperature T0. At t = 0, the composite is exposed to an environment at a constant fluid temperature Tm with a constant convective heat transfer coefficient h1 at the left side, and an environment at a constant fluid temperature Tf with a constant heat transfer coefficient h2 and a constant radiation sink temperature Ts at the right side. The mathematical formulation of the one-dimensional transient heat conduction problem is given by   @T i αi @ @T i xp ¼ p , @t x @x @x

in

xi < x < xiþ1 , i ¼ 1, 2, . . . , M,

for

t > 0, (85)

where p = 0,1,2 indicates slab, cylinder, and sphere, respectively. The boundary and interface conditions are as follows: k1 kM

@T 1 ¼ h1 ðT m  T 1 Þ, @x

at

    @T M ¼ h2 T M  T f þ eσ T 4M  T 4s , @x

ki

@T i T i  T iþ1 ¼ , @x Rci

x ¼ x1 , at

for

x ¼ xMþ1 ,

t > 0,

(86a)

for

t > 0, (86b)

at x ¼ xiþ1 , i ¼ 1,2, . .., M  1, for t > 0,

(86c)

1

Macroscopic Heat Conduction Formulation

ki

@T i @T iþ1 ¼ kiþ1 , @x @x

23

at x ¼ xiþ1 , i ¼ 1, 2, .. .,M  1, for t > 0,

(86d)

and the initial conditions for each layer T i ðx, 0Þ ¼ T 0 , in xi < x < xiþ1 , i ¼ 1, 2, . . . , M,

(87)

where Ti(x,t) is the temperature in the i-th layer, t the time, αi(=ki/( ρici)) the thermal diffusivity, ki the thermal conductivity, ρi the density, ci the specific heat, e the surface emissivity, and σ the Stefan-Boltzmann constant. The boundary condition Eq. (86b) can be rewritten with use of the adiabatic surface temperature kM

3.3

  @T M ¼ h2 ðT M  T a Þ þ eσ T 4M  T 4a , @x

at

x ¼ xMþ1 , for

t > 0:

(88)

Heat Conduction with Phase Change

Three-dimensional heat conduction with phase change of a pure substance in a domain V is governed by the following equation: ρi c p , i

@T i ¼ ∇  ðki ∇T i Þ þ g_ 000 i , in @t

V

i,

i ¼ s, l,

(89)

and the Stefan condition for a sharp phase change interface (Özişik 1993) n  ðks ∇T s Þ  n  ðkl ∇T l Þ ¼ ρl Lvm  n, at S

m,

(90)

where L is the latent heat of fusion (J/kg), vm is the interfacial normal velocity (m/s) of the moving phase change interface S m , and n is the normal vector at a point on the interface pointing from the solid region toward the liquid region. Equation (90) states that the difference between the conduction heat flux of solid and liquid phases at the interface equals the rate of latent heat released during solidification per unit interface area. As a pure substance is considered, the temperature at the sharp phase change interface equals to the melting temperature Tm: Ts ¼ Tl ¼ Tm,

at

S

m:

(91)

As a special case, consider the one-dimensional time-dependent solid-liquid phase change in a slab, cylindrical, or spherical layer, with r being the spatial coordinate normal to the thickness of the layer. The internal coordinate or radius of the layer is ri, and the external coordinate or radius of the layer is ro. The position of the phase change interface is s(t). The structure is initially at a uniform temperature Ti and at the phase change interface the temperature is the melting temperature,

24

L. A. Sphaier et al.

Fig. 3 Cylindrical geometry of phase change where the liquid and solid phases have an internal heat generation

Tm. For a melting problem, Ti < Tm, therefore, initially the material is completely solid. For a solidification problem, Ti > Tm, the material is completely liquid. A schematic of this geometry is illustrated in Fig. 3. The heat conduction equation for both the solid and liquid phases is given by:   g_l000 @T l ðr, tÞ 1 @ @T l ðr, tÞ ¼ αl p rp þ : , @t r @r @r ρl c l   g_ 000 @T s ðr, tÞ 1 @ @T s ðr, tÞ ¼ αs p rp þ s @t r @r @r ρs c s

in

r i < r < sðtÞ,

(92a)

in

sðtÞ < r < r 0 ,

(92b)

where Ts(r, t) and Tl(r, t) are the temperatures of the solid and liquid phases, respectively, αs = ks/(ρscs) and αl = kl/(ρlcl) are the thermal diffusivities of solid 000 and liquid phases, respectively, g_ 000 s and g_ l are the internal heat generation in both phases, ρ is the density, and c is the specific heat. The geometrical characteristics of the considered structures are defined by the different values of p, for which p = 0, 1, 2 indicate slab, cylinder, and sphere, respectively. By performing an energy balance between the solid and liquid phase along the phase change front, the interface equation is obtained:

1

Macroscopic Heat Conduction Formulation

25

  @T s  @T l  ds ks  kl  ¼ ρs L , dt @r r¼sðtÞ @r r¼sðtÞ

(93)

where L is the latent heat of fusion. Equation (93) represents the differential equation governing the motion of the phase change front. To use this equation, it is necessary to determine simultaneously the temperature profiles in both the solid and liquid regions. Consider a melting problem caused by the internal heat generation, with the external boundary subject to convective cooling. The liquid phase exists from the left boundary to the phase change interface. Without loss of generality, one considers @T/@r|r=ri = 0, which represents an adiabatic condition, a symmetric centerline, or the center of a solid cylinder or sphere. At the interface, the temperature is at the constant melting temperature, T(s(t)) = Tm. At t = 0, the external wall is exposed to an environment at a constant fluid temperature Tf with a constant convective heat transfer coefficient h. In summary, the initial and boundary conditions are given by T s ðr, 0Þ ¼ T l ðr, 0Þ ¼ T i ,

(94a)

sð0Þ ¼ 0,

(94b)

T s ðsðtÞ, tÞ ¼ T l ðsðtÞ, tÞ ¼ T m ,

(94c)

@T l ¼ 0, @r ks

at

  @T s ¼ h Ts  Tf , @r

r ¼ ri , at

r ¼ ro ,

(94d) (94e)

together with the Stefan interface condition, Eq. (93). Contrary to pure substances for which the solid-liquid phase change occurs at a defined melting temperature with a sharp phase interface, phase change in substances such as alloys occurs over an extended temperature range in a mushy region between the solid and liquid regions. For phase change problems in such substances, the enthalpy formulation is in general preferred instead of the temperature formulation given by Eq. (89). This formulation can be written as follows (Özişik 1993): ρ

@H ¼ ∇  ðk∇T Þ þ g_ 000 , @t

(95)

where k is the mixture thermal conductivity usually given as k ¼ ð1  eÞks þ ekl ,

(96)

and e is the volume fraction of the liquid. Assuming linear release of latent heat over the mushy region, the specific enthalpy is related to the temperature through

26

L. A. Sphaier et al.

8 c T for > < p T  TS H ¼ cp T þ L for > TL  TS : cp T þ L for

T < TS, TS  T  TL,

(97)

T > TL,

where TS and TL are the solidus and liquidus temperatures of the material, respectively, and the specific heats of the solid and liquid are considered as the same. Reversely, the temperature can be determined as a function of the specific enthalpy: 8 H > > > > cp > > < H ðT  T Þ þ LT L S S T¼ > c ð T  T Þ þ L p L S > > > HL > > : cp

for

H < cp T S ,

for

cp T S  H  cp T þ L,

for

H > cp T þ L:

(98)

It should be noted that the enthalpy formulation may be used for phase change problems in pure substances as well, avoiding the difficulty with the moving boundary problems associated with temperature formulation. For pure substances with a single phase change temperature Tm, the enthalpy is related to the temperature through  H¼

cp T cp T þ L

for for

T < Tm, T  Tm:

(99)

Reversely, the temperature can be determined as a function of the specific enthalpy: 8 H > > > > < cp T ¼ Tm > > HL > > : Tm þ cp

3.4

for

H < cp T m ,

for

cp T m  H  cp T m þ L,

for

H > cp T m þ L:

(100)

Hyperbolic Heat Conduction

Non-Fourier hyperbolic models have been employed to investigate the conduction heat transfer phenomenon in a few special applications, such as in some situations related to pulsating laser heating, rapidly contacting surfaces in electronic devices, and heat transfer in nanosystems. For such situations, the classical model, in which the speed of propagation of thermal waves is infinite, is not adequate, leading to a nonrealistic notion of energy diffusion. In order to overcome this limitation, when required, a hyperbolic heat conduction model (Vernotte 1958; Cattaneo 1958; Chester 1963) can be used to account for the effect of finite-speed thermal transfers,

1

Macroscopic Heat Conduction Formulation

27

modifying the Fourier model to include a relaxation time that reflects the temporary delay of the thermal wave. For an isotropic medium with uniform thermal conductivity, this results in 1 @ 2 T 1 @θ ¼ ∇2 T, þ C2 @t2 α @t

(101)

in which C is the speed of second sound (Chester 1963). Naturally, for infinite C this model reduces to the classical Fourier heat conduction model. When this heat conduction model is considered, the relation of the heat flux vector with the temperature field is written as: τ0

@q00 þ q00 ¼ k∇T, @t

(102)

for an isotropic material. The parameter τ0 is a relaxation time, which can be written as τ0 = α/C2. Equation (102) is also reduced to the Fourier law (Eq. (12)) for an infinite value of C. While the boundary conditions for this type of problem are similar to the ones used for the classical Fourier heat conduction problem, an additional initial condition must be prescribed for the time-derivative of the temperature field.

3.5

Conjugate Conduction-Convection

In the heat conduction formulations considered in the previous sections, whenever a convection boundary condition was present, a heat transfer coefficient was assumed to be known. In general, this parameter value is offered in the form of theoretical or empirical correlations, in terms of a dimensionless quantity that includes the desired heat transfer coefficient (Nusselt number) and collapses information on the fluid, geometry, flow, temperature itself, and additional effects. On the other hand, such convection correlations are in general constructed for limiting situations and restricted conditions, such as for a prescribed surface temperature or heat flux. Therefore, the more general situation of both the fluid and the solid wall mutually influencing the heat exchange is in general not possible to treat in such a simplified form. In such cases when the temperature distributions in both media need to be solved for simultaneously, known as conjugated problems (Perelman 1961), the energy equations are then coupled through the solid-fluid interfaces, and also requiring the solution of the associated fluid flow problem. Three-dimensional conjugate conduction-convection is governed by the following energy equations for the solid and fluid, in laminar flow, respectively: ρs c p , s

@T s ¼ ∇  ðks ∇T s Þ þ g_ s 000 , @t

in

V

s,

(103a)

28

L. A. Sphaier et al.

    @T f þ ν  ∇T f ¼ ∇  ks ∇T f þ g_ 000 ρf cp, f f , @t and the conditions on the solid-fluid interface, S Ts ¼ Tf ,

on

V

in

f,

(103b)

i

S i,

(104a)

and ni  ðks ∇T s Þ ¼ ni  ðkl ∇T l Þ,

on S i :

(104b)

Conjugate heat transfer of laminar internal or external flow with longitudinal wall conduction has been studied by approximate analytical methods (Luikov et al. 1971; Mori et al. 1994), numerical methods (Barozzi and Pagliarini 1985; Campo and Schuler 1988), and hybrid methods (Guedes et al. 1991, 1994; Naveira et al. 2009; Naveira-Cotta et al. 2010; Nunes et al. 2010; Knupp et al. 2012, 2015). The coupled conduction-convection problem is in general governed by four parameters, namely, Péclet number, aspect ratio, Biot number, and a wall-to-fluid conductance parameter. Two approaches have been used: fully differential two- or three-dimensional heat conduction analysis and lumped-differential two- or one-dimensional thin wall analysis that lumps transversally (or radially) the temperature distribution at the duct wall (Shah and London 1978). As an example of a conjugated problem formulation, consider steady-state laminar forced convection within the thermal entrance region of a circular duct for the flow of a Newtonian fluid with constant properties. The effects of viscous dissipation, natural convection, and fluid axial conduction are assumed negligible. Both radial and axial heat conduction in the duct wall are considered, characterizing the so-called conjugate heat transfer problem. The mathematical formulation for the solid region is given by   1@ @T s ðr, zÞ @ 2 T s ðr, zÞ r ¼ 0, þ r @r @r @z2

0 < z < L ,

r1 < r < r2 ,

(105a)

with the boundary conditions

ks

@T s ðr, 0Þ ¼ 0, @z

r1 < r < r2 ,

(105b)

@T s ðr, L Þ ¼ 0, @z

r1 < r < r2 ,

(105c)

@T s ðr 2 , zÞ ¼ hex ðT s ðr 2 , zÞ  T 1 Þ, @r

0 0,

(118b)

where the boundary conditions (for τ > 0) are given by. @θ ¼ Q, @η

at

@ϕ @θ  Pn ¼ 0, @η @η

Fig. 4 Geometry for the drying of a moist porous medium

η ¼ 0, at

η ¼ 0,

(118c) (118d)

Flow of dry air

Insulated

Moist porous sheet

m' = 0

Supplied heat flux

Insulated

1

Macroscopic Heat Conduction Formulation

33

@θ þ Biq θ ¼ Biq  ð1  eÞKoLuBim ð1  ϕÞ, @η @ϕ þ Bi m ϕ ¼ Bi m  PnBiq ðθ  1Þ, @η

η ¼ 1,

at

η ¼ 1,

at

(118e) (118f)

and the initial conditions by: θðη, 0Þ ¼ ϕðη, 0Þ ¼ 0,

for

0 < η < 1,

(118g)

where the employed dimensionless quantities are defined as: θðη, τÞ ¼

T ðx, tÞ  T 0 , Ts  T0 Q¼

Lu ¼

αm , α

Ko ¼

q_ 00 l , k ðT s  T 0 Þ

Pn ¼ δ

r ð u0  u Þ , c ðT s  T 0 Þ

ϕðη, τÞ ¼

Ts  T0 , u0  u

u0  uðx, tÞ , u0  u

τ¼

αt , l2

η¼

Biq ¼

hl , k

Bim ¼

X , l

(119a  b) (119c  e)

hm l , km

(119f  i)

Bi m ¼ Bim ð1  ð1  eÞPnKoLuÞ:

(119j  k)

The physical quantities appearing in the dimensionless groups of Eqs. (119) are the heat transfer coefficient (h), the mass transfer coefficient (hm), the mass conductivity (km), the thickness of the porous medium (l), the prescribed heat flux ðq_ 00 Þ, the temperature of the surrounding air (Ts), the uniform initial temperature in the medium (T0), the moisture content in equilibrium with the surrounding air (u ), and the uniform initial moisture content in the medium (u0). The mass conductivity is analogous to the thermal conductivity; it is given in terms of the mass diffusivity as km = αmcmρ0, where cm is a specific mass capacity, analogous to the specific heat. The unit of the specific mass capacity is kg/kg  M, where  M denotes a mass transfer degree. The unit of the mass conductivity is kg/ms  M (Luikov 1966). Lu, Pn, and Ko denote the Luikov, Posnov, and Kossovitch numbers, respectively. The Luikov number is a ratio between the mass and heat diffusivities. Thus, it compares how the mass transfer and the heat transfer processes develop. The Posnov number indicates the change in moisture content resulting from the temperature gradient relative to the total moisture change. The Kossovitch number indicates how the heat expended in the evaporation of the liquid compares to that expended for the heating of the wet body.

3.7

Heat Conduction with Mass Transfer and Physical Adsorption

Problem formulations for heat and mass transfer with physical adsorption arise in different applications, such as desiccant dehumidification (Sphaier and Worek 2004,

34

L. A. Sphaier et al.

2009), adsorbed gas storage (Barbosa Mota et al. 1997; Vasiliev et al. 2000; BastosNeto et al. 2005; Hirata et al. 2009; Sacsa Diaz and Sphaier 2011), and gas dehydration (Benther and Sphaier 2015). The involved problem is one of heat and mass diffusion in a porous medium and can include advection effects as well. Similar to the drying model presented in the previous section, a general formulation for heat and mass diffusion in porous sorbents is composed mainly of a mass transfer equation and an energy transfer equation. The former is written as a mass conservation balance for the adsorbate contained in the porous medium, whose porosity is expressed as e: e

@ρg @ρ þ ð1  eÞ a ¼ ∇  j00g  ∇  j00a  ∇  j00T , @t @t

(120)

which corresponds to a statement that the adsorbate storage in gaseous ( ρg) and adsorbed ( ρa) forms results from the net mass inflow due to gas-phase and surface diffusion, as well as thermo-diffusion, which are assumed to occur in parallel. Accordingly, the quantities j00g , j00a and j00T represent the mass flux vectors corresponding to diffusion of vapor in the gas-phase, diffusion in the adsorbed phase (on the surface of pores), and thermo-diffusion (also known as thermophoresis or simply as Soret effect). If advective effects are considered, j00g will also involve mass flux to flow within the pores. In addition to the previous equation, the energy transport equation is written as: ρe c e

@T @p þ cf j00g  ∇T ¼ ∇  q00  ∇  q00j þ e þ g_ 000 sor isor , @t @t

(121)

where T and p are the temperature and pressure distributions in porous medium, q00 is the heat flux due to Fourier conduction, and q00j represents the heat flux due to the diffusion thermo-effect, or simply to the Dufour effect. The volumetric heat capacity ρece is an effective one, which includes contribution from the adsorbate in both adsorbed and gas phases, the porous sorbent itself, and any additional gases that may be mixed with the adsorbate but are inert to the adsorption process. This is a common example for desiccant dehumidification, when a mixture of dry air and water is used, but only water vapor is adsorbed. The specific heat cf is also an effective property of the fluid mixture that is being transported by advection. The next-to-last term on the right-hand side of Eq. (121) represents the heating due to compression in the gaseous phase, in which an ideal gas model has been considered. Moreover, the last term on the right-hand side of Eq. (121) represents heating effects associated with the sorption phenomenon, where isor is the differential heat of sorption and g_ 000 sor is the volumetric rate of sorption, given by g_ 000 sor ¼ ð1  eÞ

@ρa þ ∇  j00a : @t

(122)

1

Macroscopic Heat Conduction Formulation

35

Although boundary conditions are required for solving the presented heat and mass balance equations, additional equations are needed for relating the dependent variables that are in excess to the number of equations. These are an equation of state for relating the gas-phase density with pressure and temperature, and a constitutive equation for relating the adsorbed phase concentration with the partial pressure of the adsorbate and the mixture temperature Tf. The heat and mass fluxes in the porous sorbent material are written based on Fourier’s and Fick’s laws expressions: q00f ¼ ke ∇T,

(123)

j00g ¼ D g ∇ρg ,

(124)

j00a ¼ D a ∇ρa ,

(125)

where ke is the effective thermal conductivity of the porous medium, whereas D g and D a are effective mass diffusivities in the gas phase and in the adsorbed phase. If Soret and Dufour effects are taken into account, these fluxes also need to be defined. In a general way, the diffusion flux due to the Soret effect may be written as (Bird et al. 2002; Kays et al. 2004): j00T ¼ D Tg ∇ðlogT Þ ¼

D Tg ke T

q00

(126)

where D Tg is the effective thermal diffusion coefficient of the porous medium. In a similar fashion, the heat flux due to a concentration difference, comprising the Dufour effect, may be expressed as (Kays et al. 2004): q00j ¼ λj j00g ,

(127)

in which λj is a coefficient that relates the heat flux due to concentration difference to the mass flux j00g : Finally, the last constitutive equation is termed the sorption equilibrium relation or isotherm and is written as:   ρa ¼ ρa ρg , T

(128)

or, alternatively, as the pressure can be written as a function of ρg and T ρa ¼ ρa ðp, T Þ

(129)

A typical sorption isotherm is the Langmuir (Gregg and Sing 1982; Ruthven 1984) equation: ρa ¼ ρa, max

bp , 1 þ bp

(130)

36

L. A. Sphaier et al.

in which b is a coefficient, which generally depends on temperature. The quantity ρa, max represents the maximum adsorbate uptake, which can also depend on temperature. The presented equations can be used to model a variety of processes involving heat and mass transfer and physical adsorption. The type of fluid and the adsorbent material will naturally have a great influence on the solution, through physical properties and the mentioned constitutive equations. Typical examples of adsorbents are silica-gel, activated carbon, and zeolites.

4

Lumped and Improved-Lumped Formulations

After examining the classical heat conduction theory and a number of extensions to the original formulation proposed by Fourier, this section deals with the reformulation and simplification of heat conduction problems. The aim is to reduce the number of spatial variables in the associated partial differential equations, whenever feasible, beyond the mere elimination of one or more independent variables due to their negligible influence in the overall heat transfer process. The concept is to mix lumped and local information within the same problem statement, averaging the temperature distribution in those space coordinates along which the temperature gradients are small and eventually moderate. First, the lumpedcapacitance formulation is examined, when the averaging process is undertaken over the whole body volume, and an ordinary differential formulation for the transient volume averaged temperature is achieved. The classical lumped parameter approximation is adopted, equating the boundary and averaged temperatures. Then, this same idea is adopted in deriving lumped-differential formulations, when the averaging process is undertaken only for those coordinates along which gradients are sufficiently small. This approach is illustrated for the classical fin or extended surface equation. Then, the Coupled Integral Equations Approach (CIEA) is reviewed, when improved lumped-differential formulations are obtained by avoiding the classical lumped parameter approximation, but instead deriving a more accurate relation between boundary and average temperatures, obtained from approximating, with the aid of Hermite formulae, the integral equations that define the average temperature and heat flux. The CIEA allows for the derivation of reduced models as simple as those achievable by the classical lumped system analysis, but allowing for more accuracy, and applicability within a wider range of the governing parameters. Finally, this section ends with demonstrations of the improved lumped-differential formulations in a variety of examples and extended heat conduction formulations.

4.1

Lumped-Capacitance Formulations

The simplest possible lumped approximation, known as the Lumped-capacitance formulation can be obtained from the general heat conduction equation if a volumetric average is considered:

1

Macroscopic Heat Conduction Formulation

ð

1 V

T¼

37

V

TdV :

(131)

Then, integrating equation (13) over the entire volume gives: ð V

@T dV ρc @t

ð ¼

ð

00

V

∇:q dV

þ

V

g_ 000 dV :

(132)

Considering uniform properties and invoking Gauss’ divergence theorem yields: ρcV

dT ¼ dt

ð S

q00  n dA þ g000 V

(133)

where g000 is the average volumetric heat generation rate. Finally, the surface integral must be modified by substituting the boundary condition. Dirichlet boundary conditions would lead to a trivial solution. Neumann conditions lead to a simplified case as the surface integral would simply lead to total heat transfer rate at the surface. However, Robin boundary condition leads to:   q00  n ¼ h T  T f ,

for

xS ,

(134)

such that Eq. (133) is modified to (assuming a constant heat transfer coefficient h): dT ρcV ¼ h dt

ð

 S

 T  T f dA þ g000 V ,

(135)

which can be written as ρcV

  dT ¼ hA T~  T~f þ g000 V dt

(136)

where T~ is the average surface temperature: 1 T~ ¼ A

ð S

T dA :

(137)

In order to complete the lumped-capacitance formulation, a relation between the average surface temperature and volume-averaged temperature must be established. The simplest relation is the so-called Classical Lumped System Approximation (CLSA), which involves assuming that T~ T , such that Eq. (136) is simplified to: ρcV

  dT ¼ hA T  T~f þ g000 V dt

which can then be readily solved for calculating T ðtÞ, yielding

(138)

38

L. A. Sphaier et al.

   A ht g000 V g000 V ~ ~ : T ðtÞ ¼ T f þ exp  T0  T f  þ ρcV A h A h

4.2

(139)

Lumped-Differential Formulations

The lumping procedure described in the previous section can be applied individually to spatial directions that possess small temperature gradients. Typical examples of such formulations are those employed in fin analyses. Considering a straight rectangular fin of thickness H, length L, and width W, as illustrated in Fig. 5, a lumpeddifferential formulation may be obtained by considering the average temperature within the cross-sectional area: 4 T ðx, tÞ ¼ HW

ð H=2 ð W=2 0

T ðx, y, z, tÞdz dy:

(140)

0

where the symmetry of the problem about y = 0 and z = 0 has been considered. If the same average is employed to Eq. (35), assuming uniform thermal capacity, one obtains ρc

  @T @ @T ¼ k @t @x @x "ð  y¼H=2 z¼W=2 # ð H=2  W=2    4 @T @T  k   þ dz þ k dy þ g000 :    HW 0 @y y0 @z 0 z¼0

(141)

where g000 represents the cross-section averaged volumetric rate of heat generation. Considering that the fin is immersed in a fluid of uniform temperature Tf and that the heat transfer coefficient h is uniform over the fin surface, the boundary conditions in the z and y direction give:

Fig. 5 Geometry for heat conduction analysis in a straight rectangular fin

y H/2 L x −H/2

1

Macroscopic Heat Conduction Formulation

k k

  @T ¼ h T  Tf , @y

  @T ¼ h T  Tf , @y

k k

39

at at

  @T ¼ h T  Tf , @z

  @T ¼ h T  Tf , @z

at at

y ¼ H=2,

(142a)

y ¼ H=2,

(142b)

z ¼ W=2,

(142c)

z ¼ W=2,

(142d)

where symmetry again allows one to substitute the boundary conditions at y =  H/2 and z =  W/2 by @T ¼ 0, @y

at

y ¼ 0,

(142e)

@T ¼ 0, @z

at

z ¼ 0,

(142f)

such that Eq. (141) is finally written as: ρc

   @T @ @T 2ð H þ W Þ  e ¼ k T  T f þ g000 , h @t @x @x HW

(143)

e is the average perimeter temperature for a given x position: where T e¼ 2 T HþW

ð W=2 0

Tjy¼H=2 dz þ

ð H=2

!  T z¼W=2 dy :

(144)

0

e T leads to Finally, the Classical Lumped System Approximation T ρc

   @T @ @T 2ð H þ W Þ  ¼ k T  T f þ g000 , h @t @x @x HW

(145)

which can be solved for the temperature distribution T. Finally, considering no heat generation, constant thermal conductivity, and a steady regime, Eq. (145) is written in the common form of the classical fin equation:   d2 T  m2 T  T f ¼ 0, 2 dx where

(146)

40

L. A. Sphaier et al.

m2 ¼

h 2ðH þ W Þ : k HW

(147)

Equation (145) provides a one-dimensional reformulation of the original transient three-dimensional problem, markedly simplifying the solution task. Such lumpeddifferential formulations are frequently employed in the realm of applications, at the price of loosing information on the temperature variations along the coordinates where the lumping is promoted. In addition, the accuracy of such approximations is strongly limited by the classical lumped system analysis hypothesis, which requires fairly smooth temperature profiles along the coordinates to be eliminated.

4.3

Improved Lumped-Differential Formulations

Improved lumped-differential formulations can be proposed to offer more accurate formulations, through the use of more informative relations between the averaged temperature and the boundary temperature, than the simplest possible expression offered by the classical lumped system analysis. These approximations arise from using Hermite (1878) approximations for an integral, which are given by the following relation: ð xi xi1

f ðxÞdx

α X

cv ðα, βÞhvþ1 f ðvÞ ðxi1 Þ þ i

v¼0

β X

cv ðβ, αÞð1Þv hvþ1 f ðvÞ ðxi Þ, (148a) i

v¼0

where hi ¼ xi  xi1 , cv ðα, βÞ ¼

(148b)

ðα þ 1Þ!ðα þ β  v þ 1Þ! , ðv þ 1Þ!ðα  vÞ!ðα þ β þ 2Þ!

(148c)

and the function f(x) and its derivatives f (v)(x) are defined for all x  [xi1, xi]. It ð vÞ will be used in what follows the shorter notation f ðvÞ ðxi1 Þ ¼ f i1 for v = 0, 1, 2, ðvÞ . . ., α and f ðvÞ ðxi Þ ¼ f i for v = 0, 1, 2, . . ., β. Different levels of approximation denoted by Hα,β can be obtained from these formulas, depending on the values of α and β, of each specific approximation. The error associated with a given Hermite approximation can be obtained from the following relation: eH α:β ¼

ð1Þαþβþ2 ðαþβþ2Þ f ðξÞ ðα þ β þ 2Þ!

ð xi

xβþ1 ðx  hi Þαþ1 dx,

(149)

xi1

where xi1  ξ  xi. The Classical Lumped System Approximation consists in using a rectangular integration rule, which corresponds to either of the relations:

1

Macroscopic Heat Conduction Formulation

ðb

41

f ðxÞ dx ðb  aÞf ðaÞ,

(150)

f ðxÞ dx ðb  aÞf ðbÞ,

(151)

a

ðb a

for which the associated error is given by eCLSA ¼

ð b  aÞ 2 0 f ðξÞ, 2

with

a  ξ  b:

(152)

Using the Hermite formula, improved approximation rules can be obtained. Two simple Hermite rules are the H0,0 and H1,1, which respectively correspond to the trapezoidal and corrected trapezoidal integration rules: H 0, 0 )

ðb

f ðxÞ dx

a

H 1, 1 )

ðb

f ðxÞ dx

a

ba ðf ðaÞ þ f ðbÞÞ, 2

ba ð b  aÞ 2 0 ðf ðaÞ þ f ðbÞÞ þ ðf ðaÞ  f 0 ðbÞÞ, 2 12

(153a)

(153b)

where the error associated with these approximations are respectively given by: eH 0, 0 ¼

ðb  aÞ3 00 f ðξÞ 12

(154)

eH 1, 1 ¼

ðb  aÞ5 ð4Þ f ðξÞ 720

(155)

with a  ξ  b. A pair of the Hα,β formulas are then used to approximate the integrals involved in the averaged temperature and heat flux. As these relations involve two coupled integral equations, the technique is commonly called the Coupled Integral Equations Approach (Aparecido and Cotta 1990; Cotta et al. 1990; Corrêa and Cotta 1998; Alves et al. 2000) and was first employed by Mennig and Özişik (1985). Two common schemes are the H0,0/H0,0 and the H1,1/H0,0 approximation schemes. The former involves using trapezoidal rules for approximating the average temperature and heat flux. Within a one-dimensional Cartesian framework, these involve: ðb

ba ðT ða, tÞ þ T ðb, tÞÞ, 2 a     ðb @T b  a @T  @T  dx

þ : 2 @x x¼a @x x¼b a @x T ðx, tÞ dx

(156a)

(156b)

42

L. A. Sphaier et al.

The H1,1/H0,0 uses the corrected trapezoidal rule for approximating the integral in the average temperature: ðb a

T ðx, tÞ dx

    ba ðb  aÞ2 @T  @T  ðT ða, tÞ þ T ðb, tÞÞ þ  : 2 @x x¼a @x x¼b 12

(157)

and the trapezoidal rule for approximating the integral of the heat flux (Eq. (156b)). The integral equations are then used together with the boundary conditions to produce relations between the surface temperatures T(a,t) and T(b,t) and the average temperature T, given by: T ðtÞ ¼

1 ba

ðb

T ðx, tÞ dx:

(158)

a

Once the average temperature distribution is introduced and the integral of the heat flux is resolved, the coupled equations lead to: 1 T ðtÞ ’ ðT ða, tÞ þ T ðb, tÞÞ, 2     b  a @T  @T  T ðb, tÞ  T ða, tÞ ’ þ , 2 @x x¼a @x x¼b

(159a) (159b)

for the H0,0/H0,0 scheme and     1 ðb  aÞ @T  @T  T ðtÞ ’ ðT ða, tÞ þ T ðb, tÞÞ þ  , 2 12 @x x¼a @x x¼b     ðb  aÞ @T  @T  þ T ða, tÞ  T ðb, tÞ ’ , 12 @x x¼a @x x¼b

(160a)

(160b)

for the H1,1/H0,0. Systems (159) and (160) comprise two equations for five unknowns, these being the temperature and its space derivatives at the boundaries (four unknowns) and the average temperature. Each of these systems is solved together with the boundary conditions of the problem (two equations) and the integrated version of the governing equation, such that a total of five equations for five unknowns is obtained. For the sake of demonstrating the application of the technique, the H0,0 /H0,0 approach is employed for the same fin problem described in Sect. 3.2. As the lumping procedure is carried out in two directions, two pair of coupled integral equations are used: ð H=2 0

T dy ’

H ðT ðx, 0, z, tÞ þ T ðx, H=2, z, tÞÞ, 4

(161a)

1

Macroscopic Heat Conduction Formulation

ð W=2

T dy ’

0

43

W ðT ðx, y, 0, tÞ þ T ðx, y, W=2, tÞÞ, 4

!   H @T  @T  T ðx, H=2, z, tÞ  T ðx, 0, z, tÞ

þ , 4 @y y¼0 @y y¼H=2 !   W @T  @T  þ , T ðx, y, W=2, tÞ  T ðx, y, 0, tÞ

4 @z z¼0 @z z¼w=2

(161b)

(161c)

(161d)

Substituting the boundary conditions and eliminating the temperature distributions at y = 0 and z = 0 gives: 2 H 2 W

ð H=2

T dy ¼

Biy 4

T dz ¼

Biz 4

0

ð W=2 0

   4 1þ T ðx, H=2, z, tÞ  T f , Biy  1þ

  4 T ðx, y, W=2, tÞ  T f , Biz

(162a)

(162b)

where Biy = hH/(2k) and Biz = hW/(2k) are the Biot numbers in the y and z directions. Then, these equations can be used together with (144) to yield the e and the average following relation between the average surface temperature T temperature T: e  Tf ¼ T

    1 4W 4H þ T  Tf : ðH þ W Þ Biy þ 4 Biz þ 4

(163)

Once the previous relation has been established, the fin equation is finally written as:       @T @ @T 8 W H ρc ¼ k þ T  T f þ g000 , h @t @x @x HW Biy þ 4 Biz þ 4

(164)

such that for steady-state operation one achieves:   d2 T  m2 þ T  T f ¼ 0, dx2

(165)

in which m+ is a modified fin parameter defined as m2þ ¼

  h 8 W H þ , k HW Biy þ 4 Biz þ 4

(166)

where it is interesting to note that m+ ! m, as the values of Biy and Biz become smaller.

44

L. A. Sphaier et al.

As one observes, the presented relations are as simple as the Classical Lumped System Approximation, with just some changes on the final formulation coefficients, but can produce a notable improvement in the results. Once lumped and improvedlumped formulations have been discussed, the following subsections present a few examples of application of the improved-lumped formulations involving heat conduction.

4.3.1 Combined Convective and Radiative Cooling of a Wall Consider the thermal behavior of a wall subjected to convective heat transfer at one side and combined convective and radiative heat transfer at the other side (Tan et al. 2009). The wall is modeled as a one-dimensional slab of finite thickness, L, initially at a uniform temperature Ti. It is assumed that the thermophysical properties of the wall are homogeneous, isotropic, and independent of temperature. The wall at x = 0 is exposed to an environment of constant fluid temperature Tm with constant convective heat transfer coefficient h1, while the wall at x = L to an environment of constant fluid temperature Tf with constant heat transfer coefficient hs and constant radiation sink temperature Ts. The mathematical formulation of the problem is given by ρcp

@T @2T ¼k 2, @t @x

in

T ðx, 0Þ ¼ T i k k

@T ¼ h1 ðT m  T Þ, @x

0 < x < L, in at

    @T ¼ h2 T  T f þ eσ T 4  T 4s , @x

for t > 0,

0 < x < L, x ¼ 0, at

(167b)

for t > 0,

x ¼ L,

(167a)

for t > 0,

(167c) (167d)

where T is the temperature, t the time, x the spatial coordinate, α = k/ρcp the thermal diffusivity of the wall, k the thermal conductivity, e the surface emissivity, and σ the Stefan-Boltzmann constant. The boundary condition Eq. (167d) can be rewritten using the adiabatic surface temperature: k

  @T ¼ h2 ðT  T a Þ þ eσ T 4  T 4a , @x

at

x ¼ L,

for t > 0:

(168)

The mathematical formulation (167–168) can now be rewritten in dimensionless form as follows @θ @ 2 θ ¼ , @τ @η2

in

0 < η < 1,

for τ > 0,

(169a)

1

Macroscopic Heat Conduction Formulation

 

45

θðη, 0Þ ¼ 1,

in

0 < η < 1,

(169b)

@θ ¼ Bi1 ðθm  θÞ, @η

at

η ¼ 0,

(169c)

  @θ ¼ Bi2 ðθ  θa Þ þ N rc θ4  θ4a @η

for τ > 0,

η ¼ 1,

at

for τ > 0,

(169d)

where the dimensionless parameters are defined by θ¼ Bi1 ¼

h1 L , k

T , Ti

Bi2 ¼

x η¼ , L h2 L , k

αt , L2

(170a  c)

eσLT 3i : k

(170d  f)

τ¼

N rc ¼

It can be seen that the problem is governed by five dimensionless parameters, θm, θa, Bi1, Bi2 and Nrc. The radiation-conduction parameter, Nrc that governs the radiative cooling, is conceptually analog to the Biot number, Bi, which is the governing parameter for an equivalent transient convective cooling. The lumping procedure is initiated by introducing the spatially averaged dimensionless temperature θav ðτÞ ¼

ð1

θðη, τÞdη:

(171)

0

Ð1 Once Eq. (169a) is operated on by 0 dη and the definition of the average temperature, Eq. (171), is employed, one obtains:   dθav ðτÞ @θ @θ ¼  þ  : dτ @η η¼1 @η η¼0

(172)

Finally, the boundary conditions Eqs. (169c) and (169d) are substituted, leading to h i dθav ðτÞ ¼ Bi1 ½θð0, τÞ  θm   Bi2 ½θð1, τÞ  θa   N rc θð1, τÞ4  θ4a : dτ

(173)

The plain trapezoidal rule is employed in the integrals for both average temperature and average heat flux (H0,0/H0,0 approximation), in the form 1 θav ðτÞ ffi ½θð0, τÞ þ θð1, τÞ, 2 "   # ð1 @θðη, τÞ 1 @θ @θ dη ¼ θð1, τÞ  θð0, τÞ ffi þ : @η 2 @ηη¼0 @ηη¼1 0

(174)

(175)

46

L. A. Sphaier et al.

The boundary conditions (169c) and (169d) are substituted into Eq. (175) to yield θð1,τÞ  θð0, τÞ ¼

 i 1h Bi1 ðθm  θð0,τÞÞ  Bi2 ðθð1,τÞ  θa Þ þ N rc θð1,τÞ4  θ4a : 2 (176)

The boundary temperature θ(0, τ) is solved from Eq. (174) and substituted into Eq. (176) leading to an equation that relates θ(1, τ) and θav(τ) N rc θð1,τÞ4 þ ð4 þ Bi1 þ Bi2 Þθð1,τÞ  ð4 þ 2Bi1 Þθav ðτÞ  N rc θ4a þ Bi1 θm  Bi2 θa ¼ 0: (177) The analytical solution of Eq. (177) is readily obtained by using a symbolic computation software such as Mathematica and then used to close the ordinary differential equation (173) for the average temperature, providing the H0,0/H0,0 model. The lumped model is further improved by employing the two-side corrected trapezoidal rule in the integral for the average temperature, in the form "   # 1 1 @θ  @θ  θav ðτÞ ffi ½θð0, τÞ þ θð1, τÞ þ , 2 12 @η η¼0 @ηη¼1

(178)

The boundary conditions (169c) and (169d) are substituted into Eq. (178) to yield 1 θav ðτÞ ffi ½θð0, τÞ þ θð1, τÞ 2  i 1 h Bi1 ðθm  θð0, τÞÞ þ Bi2 ðθð1, τÞ  θa Þ þ N rc θð1, τÞ4  θ4a , þ 12

(179)

while keeping the plain trapezoidal rule in the integral for heat flux (H1,1/H0,0 approximation). The boundary temperature θ(0, τ) is solved from Eq. (179) and substituted into Eq. (176), an equation relating θ(1, τ) to θav(τ) is obtained ð4 þ Bi1 ÞN rc θð1, τÞ4 þ ð12 þ 4ðBi1 þ Bi2 Þ þ Bi1 Bi2 Þθð1, τÞ ð12 þ 2Bi1 Þθav ðτÞ  ð4 þ Bi1 ÞN rc θ4a  ð4 þ Bi1 ÞBi2 θa þ 2Bi1 θm ¼ 0:

(180)

Similarly, the analytical solution of Eq. (180) is obtained and used to close the ordinary differential equation (173) for the average temperature, providing the H1,1/ H0,0 model. The solutions of classical and improved lumped models are shown in Fig. 6 in comparison with a reference finite difference solution of the original distributed model, Eqs. (167a, 167b, 167c, 167d, and 168) (Tan et al. 2009). The higher order lumped model (H1,1/H0,0 approximation) presented a good agreement with the reference finite difference solution for values of Biot numbers as high as 20.0 and

1

Macroscopic Heat Conduction Formulation

47

1 FD solution Classical H0,0 /H0,0 H1,1/H0,0

q

0.9

0.8

0.7

0.6 0

0.2

0.4

0.6

0.8

1

t

Fig. 6 Dimensionless average temperature in the wall as a function of time for Bi1 = 20 , Bi2 = 20, and Nrc = 10

Nrc as high as 10.0, and the classical lumped model already deviated from the reference solution at Bi1 = Bi2 = 1.0 and Nrc = 1.0. It is important to observe that although the lower order improved model (H0,0/H0,0) does not predict the average temperature accurately for higher values of Biot number and the radiation-conduction parameter, it predicts the correct value of the steady-state temperature.

4.3.2 Nuclear Fuel Rod Consider transient heat conduction in a cylindrical nuclear fuel rod such as those that can be found in pressurized water reactors (PWRs), boiling water reactors (BWRs), or liquid metal cooled fast breeder reactors (LMFBRs) (Regis et al. 2000). In the most general case, the problem is described with the following equations in the cylindrical coordinate system,       @T f 1 @ @T f @T f @T f @ 1 @ kf r kf kf ¼ ρf cf þ þ 2 þ g_ 000 ðr, z, θ, tÞ, r @r @z r @θ @t @r @z @θ (181) for the fuel, and ρc cc

      @T c 1 @ @T c @ @T c 1 @ @T c kc r kc kc ¼ þ þ 2 , r @r @z r @θ @t @r @z @θ

(182)

48

L. A. Sphaier et al.

for the cladding, where Tf and Tc are temperatures in fuel and cladding; ρf, ρc their densities; cf, cc the specific heats; kf, kc the respective thermal conductivities; and g_ 000 the volumetric heat generation in the fuel. The equations are to be solved together with appropriate boundary and initial conditions. The heat generation in the cladding is assumed negligible. To illustrate the main idea of the improved lumped analysis, the problem is simplified by assuming an axisymmetric temperature distribution, negligible axial heat conduction, and no spatial variation in the heat generation across the fuel rod. The thermal conductivities are assumed to be independent of temperature, although this assumption is not essential in the lumped parameter approach (Regis et al. 2000). With the above assumptions, one arrives at the following mathematical model for one-dimensional transient heat conduction in a fuel rod, ρf c f

  @T f 1 @ @T f kf r ¼ þ g_ 000 ðtÞ, 0 < r < r fo , r @r @t @r T f ðr, 0Þ ¼ T f 0 ðr Þ, kf

ρc c c

(183b)

     @T f  ¼ hg T f r fo , t  T c ðr ci , tÞ  @r r¼rfo

  @T c 1 @ @T c kc r ¼ , r @r @t @r

(183a)

r ci < r < r co ,

T c ðr, 0Þ ¼ T c0 ðr Þ,   @T f  @T c  kc ¼ kf , @r r¼rci @r r¼rfo

(183c)

(183d) (183e) (183f)

 @T c  ¼ hðT c ðr co , tÞ  T m Þ, kc @r r¼rco

(183g)

Then, by introducing the following dimensionless variables, θf ¼

Tf  Tm , T ref  T m

τ¼ Bigc ¼

kf t , ρf cf r 2co

hg r co , kc

θc ¼ K¼

Bigf ¼

Tc  Tm , T ref  T m

k f ρc c c , k c ρf c f

hg r co , kf

R¼

Bi ¼

G¼

r , r co

hr co , kc

r 2co g_ 000 , kf T ref  T m

the following normalized formulation is obtained:



(184a  c) (184d  f) (184g  i)

1

Macroscopic Heat Conduction Formulation

49

  @θf @θf 1 @ R ¼ þ GðτÞ, R @R @τ @R

0 < R < Rfo ,

θf ðR, 0Þ ¼ θf 0 ðRÞ,      @θf    ¼ Bigf θf Rfo , τ  θc ðRci , τÞ @R R¼Rfo   @θc K @ @θc R ¼ , R @R @τ @R

Rci < R < 1,

θc ðr, 0Þ ¼ θc0 ðr Þ,      @θc   ¼ Bigc θf Rfo , τ  θc ðRci , τÞ ,  @R R¼Rci  @θf   ¼ Biθc ð1, τÞ: @R R¼1

(185a) (185b) (185c)

(185d) (185e) (185f)

(185g)

The corresponding spatially averaged dimensionless temperatures are defined by: Ð Rfo θf , av ðτÞ ¼

0

2πRθf ðR, τÞdR 2 ¼ 2 2 Rfo πRfo

ð Rfo

Rθf ðR, τÞdR,

(186a)

0

Ð1

ð1 2πRθc ðR, τÞdR 2    Rθc ðR, τÞdR: (186b) ¼ θc, av ðτÞ ¼ 1  R2ci Rci π 1  R2ci  Ð R Then, by operating Eq. (185a) with 2=R2fo 0 fo R dR and using the definition of    Ð 1 average temperature (Eq. (186a)), and by operating Eq. (185d) by 2= 1  R2ci Rci R dR and using the average temperature definition (Eq. (186b)), one respectively obtains: Rci

 dθf , av ðτÞ 2 @θf  ¼ þ GðτÞ, Rfo @R R¼Rfo dτ   ! dθc, av ðτÞ 2K @θc  @θc   Rci  ¼   : dτ @R R¼Rci @R R¼1 1  R2ci

(187a)

(187b)

Now, after using the boundary conditions, one arrives at:   2Bigf   dθf , av ðτÞ ¼ θf Rfo, τ  θc ðRci , τÞ þ GðτÞ, dτ Rfo

(188a)

50

L. A. Sphaier et al.

      dθc, av ðτÞ 2K  Bigc Rci θf Rfo , τ  θc ðRci , τÞ  Bi θc ð1, τÞ : (188b) ¼  dτ 1  R2ci For the improved lumped formulation, the one-sided corrected trapezoidal rule (H1,0 approximation) is employed in the averaged temperature integral for the fuel, and the plane trapezoidal rule (H0,0 approximation) is used in the averaged temperature integral for the cladding and the heat fluxes, in the following form:    1 2 1 2  2 R θf Rfo , τ þ θf ð0, τÞRfo , θf , av ðτÞ ¼ 2 6 Rfo 3 fo ð Rfo 0

!     @θf ðR, τÞ @θf  1 @θf  dR ¼ θf Rfo , τ  θf ð0, τÞ ¼ þ , 2 @R R¼0 @R R¼Rfo @R θc, av ðτÞ ¼ 

ð1 Rci

2 1  ðRci θc ðRci , τÞ þ θc ð1, τÞÞð1  Rci Þ, 1  R2ci 2

  ! @θc ðR, τÞ 1 @θc  @θc  dR ¼ θc ð1, τÞ  θc ðRci , τÞ ¼ þ , @R 2 @R R¼Rci @R R¼1

(189a)

(189b)

(189c)

(189d)

Using boundary conditions Eqs. (185c, 185f, 185g), Eqs. (189b, 189d) become     Bigf Rfo   θf Rfo , τ  θc ðRci , τÞ , θf Rfo, τ  θf ð0, τÞ ¼  2 θc ð1, τÞ  θc ðRci , τÞ ¼ 

(190a)

      1 Bi θc ð1, τÞ þ Bigc θf Rfo , τ  θc Rci, τ 1  Rci : 2 (190b)

Equations (189a, 189c, 190a, 190b) form a system of four linear algebraic equations for four unknowns, θf (0, τ) , θf (Rfo,τ) , θc(Rciτ), and θc(1, τ) that is solved to provide the sought relations between boundary potentials and averaged potentials, besides an approximate relation for the central temperature, θf (0, τ). These relations are then used in Eqs. (188a) and (188b) to close the two ordinary differential equations for the averaged temperatures, to be solved with the initial conditions. The classical lumped model and improved lumped model (H1,0/H0,0) were used to close the ordinary differential equations (188) which were solved numerically with a fourth-order Runge-Kutta method (Regis et al. 2000). The solutions were then compared with a finite difference solution of the original partial differential equations (183) using fully implicit scheme and central difference in spatial coordinate. Figure 7 shows the averaged fuel and cladding temperatures for Bi = 20.0 obtained by the improved and classical lumped formulations, compared with the finite difference solution (K = 0.22, Bigf = 0.09834, Bigc = 0.03, Rfo = 0.833, and Rci = 0.85). As one can observe, the averaged fuel temperature obtained by the improved lumped parameter formulation is in excellent agreement with the finite difference solution, while

1

Macroscopic Heat Conduction Formulation

51

Fig. 7 Average fuel and cladding temperatures as a function of time for Bi = 20.0, with K = 0.22, Bigf = 0.09834, Bigc = 0.03, Rfo = 0.833, and Rci = 0.85

the agreement for the averaged cladding temperature is also quite reasonable. It is clear that there is a significant improvement over the classical lumped parameter formulation, especially in predicting the fuel temperatures, which are essential in the design of the nuclear fuel rod and its safety analysis.

4.3.3 Heat and Mass Transfer in Adsorbed Gas Discharge The problem of adsorbed methane discharge at constant mass flow rate for fuel consumption applications was studied in Sphaier and Jurumenha (2012). The authors started from a typical one-dimensional formulation for the problem as discussed in Sacsa Diaz and Sphaier (2011)), whose geometry is illustrated in Fig. 8. The governing equations are given by: M g Ce

@ρ g @t

þ M l

@ρ l 1 @   ^r jg, r , ¼  ^r @r @t

@T @T FoCs @   @ρ @p ^r qr þ M l i sor l þ M g ω : þ j g, r ¼  ^r @r @t @r @t @t

(191a) (191b)

for 0  r  1, where ^r ¼ r þ r i , in which r is the dimensionless radial position and r i a dimensionless geometry ratio, both defined ahead. The quantities ρ g and ρ l respectively represent the amount of gas stored in gaseous form and adsorbed form, and are functions of temperature and pressure through equations of state, whereas i sor represents the amount of latent energy associated with the adsorption process – known as the differential heat of sorption, as described in Sphaier and Jurumenha (2012).

52

L. A. Sphaier et al.

Fig. 8 Geometry for adsorbed gas discharge problem

The dimensionless volumetric heat capacity Ce, and dimensionless mass and heat fluxes are defined as Ce ¼ Cs þ M g ρ g þ c lg M l ρ l , j g, r ¼ M g ρ g σ

@p , @r

q r ¼ 

(192)

@T , @r

(193)

while the dimensionless initial and boundary conditions are written as T ðr , 0Þ ¼ 1,

(194a)

p ðr , 0Þ ¼ 0,

(194b)

j g, r ¼ 0, 

r ¼ 1,

at

  Cw 1 @T @T ¼ Bi T  T ex þ , @r Cs Fo @t j g, r ¼ j in ,

(194c) at

r ¼ 1,

r ¼ 0,

at

  @T ¼ Biin T  T in , @r

at

r ¼ 0:

(194d) (194e) (194f)

The dimensionless variables and parameters involved in the employed formulation are given by t ¼

t , tf

T ¼

T , T0

r ¼

r  ri , rw  ri

p ¼ r i ¼

p  pmin , pmax  pmin

ri , rw  ri

(195a) (195b)

1

Macroscopic Heat Conduction Formulation

j in ¼ Cs ¼ Biin ¼

53

ρg, max e ρl, max m_ 00in tf , M ¼ , Ml ¼ , pmax ðr w  r i Þ g ρmax ρmax

cs ρb , cpg ρmax

Cw ¼

hþ in ðr w  r i Þ , kb

Fo ¼

ρw c w δw , cpg ρmax r w  r i

Bi ¼

k b tf ρb cs ðr w  r i Þ

2

hð r w  r i Þ , kb ω¼

,

cl , cpg

(195d)

isor , T 0 cpg

(195e)

c lg ¼ i sor ¼

(195c)

κ  1 pmax  pmin , κ pmax

(195f)

where κ = cpg/cvg and ρmax = ρg,max. The parameters t , T , p , and r are the normalized dependent and independent variables, r i is the dimensionless inner radius, j in is the dimensionless outlet mass flow, M g and M l are storage capacity ratios for the gas phase and adsorbed phase, Cs and Cw are thermal capacity ratios, and c lg is a specific heat ratio. The remaining parameters are Biot numbers (Biin and Bi), the dimensionless heat of sorption i sor , the Fourier number Fo, and a dimensionless parameter related to the compressed gas (ω). The lumping of the one-dimensional formulation is then carried out by employing the averaging operator 2 ϕð t Þ ¼ 2r i þ 1

ð1 0

  ϕðr , t Þ r þ r i dr ,

(196)

where ϕ stands for the dependent variables (temperature and pressure) as well as other spatial-dependent quantities. As shown in Sphaier and Jurumenha (2012), the lumping procedure transforms exactly the mass conservation equation to an ODE in terms of averaged variables: M g

dρ g dt

þ M l

dρ l ¼ Ki j in : dt

(197)

The energy equation, however, requires approximations. The classical lumpedsystem analysis leads to the following ODE: 

Ce þ Kw Cw

 dT dt

    ¼ Hex T ex  T þ Hin T in  T   dρl dp þ Ml isor þ Mg ω , dt dt

Ce ¼ M g ρ g þ c lg þ M l ρ l þ Cs ,

(198) (199a)

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Hin ¼ Ki Biin FoCs , Ki ¼

2r i , 1 þ 2r i

Hex ¼ Kw BiFoCs ,   2 r i þ 1 Kw ¼ ¼ 2  Ki , 1 þ 2r i

(199b  c) (199d  e)

and Bi ex , Bi in , and C w are modified Biot numbers, and a modified capacity ratio, respectively defined as Bi ex ¼

2BiðBiin þ 2Þ , 4 þ Biin Kw þ BiKi

Bi in ¼

C w ¼ Cw

2ðBi þ 2ÞBiin , 4 þ Biin Kw þ BiKi

(199a  b)

Bi ex : Bi

(199c)

On the other hand, if the improved-lumped schemes H0,0/H0,0 or H1,1/H0,0 are employed, a two-equation ODE system for calculating the space-averaged reservoir temperature and the reservoir wall temperature is obtained. For the H0,0/H0,0 scheme, this system is given by Ce

  dT dT   þ Kw C w w ¼ H ex T ex  T þ H in T in  T dt dt   dρ dρ þ M l i sor l þ M g ω : dt dt

(200)

    Biin Kw þ 4   Cw dT w ¼ Bi T w  T ex þ Biin T  T in þ T  T w , Ki Cs Fo dt

(201)

where H in ¼ Ki Bi in FoCs , H ex ¼ Kw Bi ex FoCs :

(202a  b)

For the H1,1/H0,0 scheme, the equation for the average temperature is identical to Eq. (200); however, the modified Biot number and wall capacity ratio are now given by alternate expressions: Bi ex ¼



Biin ð4Ki  2Þ þ ð2Ki ð6 Ki þ 1Þ  5Þ Biin ð6ð3  2Kw ÞKw þ 1Þ þ 24ð2Ki  1Þ þ Biin ð32Ki Kw  22Þ þ 24ð2Ki  1ÞKw BiðBiin ð32Ki Kw  22Þ þ 24ð2Ki  1ÞKw Þ

1 ,

(203a) Bi in ¼



BiBiin ð4Ki  2Þ þ Bið2Ki ð6 Ki þ 1Þ  5Þ Biin ðBið16Ki ð2Ki  1Þ  2Þ þ 24ð2Ki  1ÞKi Þ

Biin ð6ð3  2Kw ÞKw þ 1Þ þ 24ð2Ki  1Þ þ Biin ðBið16Ki ð2Ki  1Þ  2Þ þ 24ð2Ki  1ÞKi Þ

1 ,

(203b)

1

Macroscopic Heat Conduction Formulation

C w ¼ Cw

55

Bi ex , Bi

(203c)

and the equation that must be solved together with Eq. (200) for the wall temperature is:     Cw dT w Biin ð6ð1  2Ki ÞKi þ 1Þ ¼ Bi T w  T ex þ T  T in Biin ð2  4Ki Þ  2Ki ð6Ki þ 1Þ þ 5 Cs Fo dt Biin ð6ð3  2Kw ÞKw þ 1Þ  48Kw þ 72 :  Biin ð2  4Ki Þ  2Ki ð6Ki þ 1Þ þ 5 (204) The classical-lumped system analysis (CLSA), together with the improvedlumped schemes, is solved and compared to the numerical solution of the original one-dimensional system. The results were presented in Sphaier and Jurumenha (2012), demonstrating the behavior of the different schemes for a number of cases. Two selected cases from Sphaier and Jurumenha (2012) are herein presented for illustrational purposes, as shown in Fig. 9, which was constructed using physical Fig. 9 Average temperature evolution for adsorbed gas discharge calculated with different lumped-differential formulations

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properties data for activated carbon and methane, with the values for the dimension less parameters given by C ¼ 2, Cw ¼ 2, Bi ¼ 10, Ki ¼ 0:1 and different values for Fo = 0.1 and Biin = 0. As one can observe from these results, the H1,1/H0,0 is notably superior to the other approximation schemes, predicting the average temperature evolution very close to the one calculated with the one-dimensional formulation. For the case with Fo = 0.1 and Biin = 0, the H0,0/H0,0 also provide reasonable results, showing that lower order schemes may provide suitable results under certain circumstances. Nevertheless, a proper approximation scheme should be chosen for each scenario. The average pressure evolution is not presented, because all formulations give the same result as the one-dimensional model. The Coupled Integral Equations Approach (CIEA), as described and extensively illustrated along this section, provides a systematic path for the reformulation of heat and mass transfer problems, toward the reduction of the number of independent variables in the associated differential models. Substantial savings in computational effort can be achieved if accuracy is not at a premium and once the local spatial variation of the dependent variables is not a requirement for the simulation task. Such approximate and inexpensive solutions can be especially useful in preliminary design and/or in computer intensive simulations such as those associated with properties and sources identification and optimization.

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Regis CR, Cotta RM, Su J (2000) Improved lumped analysis of transient heat conduction in a nuclear fuel rod. Int Commun Heat Mass Transf 27(3):357–366 Reis MCL, Macêdo EN, Quaresma JNN (2000) Improved lumped-differential formulations in hyperbolic heat conduction. Int Commun Heat Mass Transf 27(7):965–974 Ribeiro JW, Cotta RM (1995) On the solution of nonlinear drying problems in capillary-porous media through integral transformation of Luikov equations. Int J Numer Methods Eng 38(6):1001–1020 Ribeiro JW, Cotta RM, Mikhailov MD (1993) Integral transform solution of Luikov equations for heat and mass-transfer in capillary-porous media. Int J Heat Mass Transf 36(18):4467–4475 Ruperti NJ, Cotta RM, Falkenberg CV, Su J (2004) Engineering analysis of ablative thermal protection for atmospheric reentry: improved lumped formulations and symbolic-numerical computation. Heat Transf Eng 25(6):101–111 Ruthven DM (1984) Principles of adsorption and adsorption processes. Wiley, New York Sacsa Diaz RP, Sphaier LA (2011) Development of dimensionless groups for heat and mass transfer in adsorbed gas storage. Int J Therm Sci 50(4):599–607 Scofano Neto F, Cotta RM (1993) Improved hybrid lumped-differential formulation for double-pipe heat-exchanger analysis. J Heat Transf 115(4):921–927 Shah RK, London AL (1978) Laminar flow forced convection in ducts: a source book for compact heat exchanger analytical data. In: Irvine TF Jr, Hartnet JP (eds) Advances in heat transfer. Academic, New York Sontag RE, Van Wylen GJ (1991) Introduction to thermodynamics: classical and statistical, 3rd edn. Wiley, New York Sphaier LA, Cotta RM (2000) Integral transform analysis of multidimensional eigenvalue problems within irregular domains. Numer Heat Transf B Fund 38(2):157–175 Sphaier LA, Cotta RM (2002) Analytical and hybrid solutions of diffusion problems within arbitrarily shaped regions via integral transforms. Comput Mech 29(3):265–276 Sphaier LA, Jurumenha DS (2012) Improved lumped-capacitance model for heat and mass transfer in adsorbed gas discharge operations. Energy 44(1):978–985 Sphaier LA, Worek WM (2004) Analysis of heat and mass transfer in porous sorbents used in rotary regenerators. Int J Heat Mass Transf 47(14–16):3415–3430 Sphaier LA, Worek WM (2009) Parametric analysis of heat and mass transfer regenerators using a generalized effectiveness-NTU method. Int J Heat Mass Transf 52(9–10):2265–2272 Su J (2001) Improved lumped models for asymmetric cooling of a long slab by heat convection. Int Commun Heat Mass Transf 28(7):973–983 Su J (2004) Improved lumped models for transient radiative cooling of a spherical body. Int Commun Heat Mass Transf 31(1) Su J, Cotta RM (2001) Improved lumped parameter formulation for simplified LWR thermohydraulic analysis. Ann Nucl Energy 28(10):1019–1031 Su G, Tan Z, Su J (2009) Improved lumped models for transient heat conduction in a slab with temperature-dependent thermal conductivity. Appl Math Model 33(1):274–283 Tan Z, Su G, Su J (2009) Improved lumped models for combined convective and radiative cooling of a wall. Appl Therm Eng 29(11–12):2439–2443 Todreas NE, Kazimi MS (2012) Nuclear systems. Thermal hydraulic fundamentals, vol 1, 2nd edn. Taylor & Francis, Boca Raton Traiano FML, Cotta RM, Orlande HRB (1997) Improved approximate formulations for anisotropic heat conduction. Int Commun Heat Mass Transf 24(6):869–878 Vasiliev LL, Kanonchik L, Mishkinis D, Rabetsky M (2000) Adsorbed natural gas storage and transportation vessels. Int J Therm Sci 39(9–11):1047–1055 Vernotte P (1958) Les paradoxes de la théorie continue de l’équation de la chaleur. Comptes Rendus 246(22):3154–3155 Yener Y, Kakac S, Naveira-Cotta CP (2017) Heat conduction, 5th edn. Taylor & Francis, New York

2

Analytical Methods in Heat Transfer Renato Machado Cotta, Diego C. Knupp, and João N. N. Quaresma

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Separation of Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 One-Dimensional Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Multidimensional Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Classical Integral Transform Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Filtering Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Generalized Integral Transform Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Formal Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Reordering Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Single-Domain Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 GITT for Eigenvalue Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Advanced Topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Integral Balance Procedure for Convergence Improvement . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Integral Transforms with Convective Eigenvalue Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Integral Transforms with Nonlinear Eigenfunction Expansions . . . . . . . . . . . . . . . . . . . . . 7 Closing Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

62 64 64 69 74 78 82 82 86 88 91 106 106 111 115 123 123

R. M. Cotta (*) Universidade Federal do Rio de Janeiro – UFRJ, Rio de Janeiro, RJ, Brazil e-mail: [email protected] D. C. Knupp Laboratory of Experimentation and Numerical Simulation in Heat and Mass Transfer, Department of Mechanical Engineering and Energy, Polytechnic Institute, Rio de Janeiro State University, IPRJ/ UERJ, Nova Friburgo, RJ, Brazil e-mail: [email protected] J. N. N. Quaresma School of Chemical Engineering, Universidade Federal do Pará, FEQ/UFPA, Campus Universitário do Guamá, Belém, PA, Brazil e-mail: [email protected] # Springer International Publishing AG, part of Springer Nature 2018 F. A. Kulacki (ed.), Handbook of Thermal Science and Engineering, https://doi.org/10.1007/978-3-319-26695-4_2

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Abstract

In this chapter, classical and modern analytical methodologies for partial differential equations are reviewed, with emphasis on the linear and nonlinear heat diffusion equation. The aim is to provide both the introductory and advanced perspectives, by gradually evolving from separation of variables, to the classical integral transform method, and finally to the hybrid numerical-analytical approach known as the generalized integral transform technique (GITT), illustrating each approach through examples in the corresponding section. The last part of the chapter is dedicated to a few recent developments that have opened up new perspectives to the GITT extension and computational enhancement, including an integral balance scheme for convergence acceleration in the solution of eigenvalue problems, the proposition of convective eigenfunction basis for convection-diffusion problems, and the adoption of nonlinear eigenvalue problems in the solution of nonlinear problems.

1

Introduction

As discussed in the previous chapter, Fourier’s instigating paper of 1807 “Théorie de la Propagation de la Chaleur dans les Solides” and his treatise “Théorie Analytique de la Chaleur” of 1822 (Fourier 1822) consolidated the mathematical formulation of heat conduction phenomena in terms of a partial differential equation for the temperature distribution within a body, variable in space and time, after incorporating into the energy conservation equation, the proposed constitutive relation nowadays known as Fourier’s law. This would already be a breakthrough in science, but he went much further into proposing the analytical handling of the resulting partial differential formulation, either in the form of Fourier series or Fourier integrals, as these approaches would be coined later on. Though his treatment was not entirely formal, but a few years later formalized by Dirichlet, he was able of providing an exact solution to every each partial differential formulation presented in his compendium, including analysis of the heat conduction equation in the rectangular, cylindrical, and spherical coordinate systems. He was aware of the importance of his findings, both on physical and mathematical grounds, and on the power of the proposed formulation and solution methodologies, as can be observed from a paragraph on Chap. 1 of his masterpiece (Fourier 1822): The general equations of the propagation of heat are partial differential equations, and though their form is very simple, the known methods do not furnish any general mode of integrating them; we could not therefore deduce from them the values of the temperatures after a definite time. The numerical interpretation of the results of analysis is however necessary, and it is a degree of perfection which it would be very important to give to every application of analysis to the natural sciences. So long as it is not obtained, the solutions may be said to remain incomplete and useless, and the truth which it is proposed to discover is no less hidden in the formulae of analysis than it was in the physical problem itself. We have applied ourselves with much care to this purpose, and we have been able to overcome the difficulty in all the problems of which we have treated, and which contain the

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chief elements of the theory of heat. There is not one of the problems whose solution does not provide convenient and exact means for discovering the numerical values of the temperatures acquired, or those of the quantities of heat which have flowed through, when the values of the time and of the variable coordinates are known. Thus will be given not only the differential equations which the functions that express the values of the temperatures must satisfy; but the functions themselves will be given under a form which facilitates the numerical applications.

In this context, the concept of the “separation of variables” approach is in fact a hypothesis, which once verified allows the solution of a linear partial differential equation (PDE) to be reduced to the direct integration of ordinary differential equations (ODE), both in time and space for a transient one-dimensional formulation, then combined to satisfy the original PDE together with the corresponding boundary and initial conditions. In satisfying the initial condition, the solution of the ordinary differential equations related to the space variables needs to satisfy an orthogonality property, so that a single initial condition can be operated to obtain the infinite set of coefficients that combine the infinite number of solutions of the separated ODEs and only then leading to the PDE formal solution. This operation is equivalent to representing the initial condition as an infinite series expansion based on the combined solution of the ODEs for each space variable. Sturm-Liouville theory then followed, which was developed by the two French mathematicians between 1829 and 1837, dealing with linear second-order differential equations called eigenvalue problems and examining the behavior and properties of the solutions, called eigenfunctions and eigenvalues, and the series expansion of arbitrary functions in terms of these eigenfunctions. It then becomes evident that the solutions of the heat conduction equation through separation of variables can be interpreted as eigenfunction expansions of the unknown time- and space-dependent temperature fields, offering a new avenue for a more embracing analytical solution methodology known as the integral transform method (Koshlyakov 1936; Grinberg 1948; Olçer 1964; Mikhailov 1967, 1972; Luikov 1968, 1993; Ozisik 1968, 1993; Mikhailov and Ozisik 1984). According to Luikov (1980), the integral transform method is the most suitable approach in analytically solving nonhomogeneous linear heat conduction problems, not in principle solvable through separation of variables, including the case of time-dependent nonhomogeneous boundary conditions. The method was introduced by Koshlyakov (1936) and formalized a few years later in Grinberg (1948), where an extension to multilayer problems was also provided. The most extensive compendium on the integral transform method, together with a systematic presentation of the available solutions for seven general classes of linear diffusion problems, has been provided in Mikhailov and Ozisik (1984). There it has also been shown that the same methodology derived for finite regions could be directly employed in the analysis of semi-infinite and infinite media, through appropriate limit operations. This classical integral transform approach, however, is limited to problems that are transformable, in the sense that decoupled transformed ordinary differential systems can be obtained to complete the solution procedure, after the choice of the

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eigenvalue problem, the proposition of the eigenfunction expansion, and the integral transformation process itself have been performed. Nevertheless, an extension of this methodology was first introduced in obtaining approximate analytical solutions for non-transformable problems, such as in situations involving time-dependent coefficients in either the boundary conditions (Ozisik and Murray 1974) or in the governing partial differential equation (Mikhailov 1975). This same concept was then employed in obtaining approximate analytical solutions for moving boundary problems (Ozisik and Guçeri 1977; Bogado Leite et al. 1980). These earlier contributions provided the clue for the construction of a hybrid numerical-analytical methodology, now known as the generalized integral transform technique (GITT), extensively reviewed in Cotta (1990, 1993, 1994, 1998), Cotta and Mikhailov (1997, 2006), and Cotta et al. (2013a, 2013b, 2016a). The idea of providing a complete hybrid numerical-analytical solution through GITT for a non-transformable problem was then introduced in Cotta (1986), when a non-transformable moving boundary diffusion problem was solved by implementing an error-controlled numerical solution of the transformed ordinary differential system. A fully analytical solution was also obtained through GITT for a non-transformable problem in Cotta and Ozisik (1986), related to a periodic forced convection problem with complex coefficients. The approach was soon generalized to time-dependent coefficients (Cotta and Ozisik 1987) and to nonlinear problems (Cotta 1990). In the following years, a considerable amount of work was produced on the extension of the GITT toward different classes of diffusion and convection-diffusion problems, including the various extensions of the heat conduction equation discussed in the previous chapter. This hybrid approach provides an interesting combination of accuracy and cost-effectiveness, usually present solely in purely analytical approaches, however, with a desirable flexibility, more like the classical purely numerical methods for partial differential equations. In this chapter, the difficult task of reviewing the ideas behind the classical and modern analytical methodologies mentioned above, within a limited space, is undertaken. It has been attempted to provide both the introductory and advanced perspectives, by evolving gradually from the separation of variables to the classical integral transform method and finally to the generalized integral transform technique, illustrating each approach through examples in between the sections. Also, the opportunity was exploited to present some very recent progresses that further extend the hybrid approach in scope and computational efficiency, by considering eigenvalue problems with marked spatial variation, convective eigenfunction basis, and nonlinear eigenvalue problems.

2

Separation of Variables

2.1

One-Dimensional Formulation

As a starting point for understanding the more advanced analytical and hybrid approaches in heat conduction, consider the following one-dimensional linear diffusion problem, for the potential T(x,t), a temperature distribution in the case of a heat conduction problem, with homogeneous governing equation and boundary conditions:

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 @T ðx, tÞ @ @T ðx, tÞ ¼ kðxÞ wðxÞ  dðxÞT ðx, tÞ, x0 < x < x1 , @t @x @x  @T ðx, tÞ þ α0 T ðx0 , tÞ ¼ 0 β0 k0 @x x¼x0  @T ðx, tÞ β1 k1 þ α1 T ðx1 , tÞ ¼ 0 @x x¼x1 T ðx, 0Þ ¼ f ðxÞ

65

t>0

(1a)

(1b)

(1c) (1d)

which includes the transient, diffusion, and linear dissipation terms with the respective coefficients w(x), k(x), and d(x) and where k0 = k(x0) and k1 = k(x1). It should be observed that three different boundary condition types can be recovered at each boundary (first, second, or third kind), upon different choices of the parameters α0, α1, β0, and β1, yielding nine different combinations. For instance, in the case of a third-kind boundary condition, α0 or α1 takes the value of the corresponding heat transfer coefficient (h0 or h1), while β0 or β1 is set equal to one. It is also worth noting that problem (1) represents the diffusion equation in different coordinate systems, upon the adequate choice of the equation’s coefficients w(x), k(x), and d(x). In this sense, the coefficients w(x) and k(x) represent not only the thermal capacitance and conductivity, either constant or space variable, but also the space coordinate dependence inherent to the specific geometry. This problem can be handled through separation of variables (Ozisik 1993) by assuming the representation of the desired potential T(x,t) as a separable product of space- and time-dependent functions as follows: T ðx, tÞ ¼ ψ ðxÞΓðtÞ

(2)

Substituting Eq. 2 into Eq. 1a yields wðxÞψ ðxÞ


 dΓ ðtÞ d dψ ðxÞ ¼ ΓðtÞ kðxÞ  dðxÞψ ðxÞΓðtÞ dt dx dx

(3a)

Dividing Eq. 3a throughout by w(x)Γ(t)ψ(x) yields
 1 dΓ ðtÞ 1 d dψ ðxÞ d ðxÞ ¼ kðxÞ  ΓðtÞ dt wðxÞψ ðxÞ dx dx wðxÞ

(3b)

It can be readily observed that the L.H.S. of Eq. 3b is a function of t only, while the R.H.S. is a function of x only. Hence, since x and t are independent variables, the equality provided by Eq. 3b can only be satisfied by a constant. Furthermore, it can be argued that this constant must be negative in order to warrant that the final solution will be bounded, with Γ(t) ! 0 for t ! 1 and avoid trivial solutions for X(x) (Ozisik 1993). Assuming the constant μ2 >0, the following problem is obtained for Γ(t) employing the L.H.S. of Eq. 3b:

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1 dΓ ðtÞ dΓ ðtÞ ¼ μ2 , or þ μ2 ΓðtÞ ¼ 0, ΓðtÞ dt dt

t>0

(4a; b)

which is readily solved to obtain ΓðtÞ ¼ C eμ

2

t

(4c)

where C is the integration constant. The problem for Ψ (x) is obtained employing the R.H.S. of Eq. 3b, yielding
 1 d dψ ðxÞ d ðxÞ k ðxÞ ¼ μ2  wðxÞψ ðxÞ dx dx wðxÞ

(5a)

or in its most common form
   d dψ ðxÞ kðxÞ þ μ2 wðxÞ  dðxÞ ψ ðxÞ ¼ 0 dx dx

(5b)

The boundary conditions for Eq. 5b can be readily obtained substituting Eq. 2 into Eq. 1b, 1c, yielding    dψ ðxÞ β0 k0 þ α0 ψ x0 ¼ 0  dx x¼x0  dψ ðxÞ þ α 1 ψ ðx 1 Þ ¼ 0 β1 k 1 dx x¼x1

(5c)

(5d)

The problem given by Eq. 5b, 5c, and 5d is the one-dimensional form of the Sturm-Liouville problem (Ozisik 1993; Mikhailov and Ozisik 1984), and it is known as a differential eigenvalue problem, or simply eigenvalue problem, since an infinite number of nontrivial solutions ψ i(x) is obtained for an infinite number of particular values of μi. These discrete values μi are known as the eigenvalues, while the corresponding solutions ψ i(x) are known as the eigenfunctions. The solutions of this eigenvalue problem are tabulated in classical references for different orthogonal coordinate systems (Ozisik 1993) and can also be readily obtained employing modern symbolic computation tools, such as the built-in routine DSolve of the Mathematica system (Cotta and Mikhailov 1997; Wolfram 2016). Also, for more involved problems, when an explicit analytic solution is not readily obtainable, a hybrid numerical-analytical solution can be constructed (Cotta 1993; Naveira-Cotta et al. 2009; Cotta et al. 2016a), as shall be presented further ahead in this chapter. Some very important properties of this Sturm-Liouville problem, under appropriate restrictions on the coefficients, include: • For finite regions, the eigenvalues form a discrete spectrum of nonnegative real values, in increasing order:

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μi  ℜ,

67

0  μ1 < μ2 < μ3 . . .

with

(6)

• The eigenfunctions are orthogonal with respect to the weighting function w(x) in the problem domain, x  [x0, x1], known as the orthogonality property:

xð1

wðxÞψ i ðxÞψ j ðxÞdx ¼

0, i 6¼ j Ni , i ¼ j

(7a)

x0

where Ni is the normalization integral, or simply the norm, given by: xð1

Ni ¼

wðxÞψ 2i ðxÞdx

(7b)

x0

Recalling Eqs. 2 and 4a, b, one concludes that there are also infinite solutions Γi(t) associated with the eigenvalues μi. Hence, the general solution of this problem can be written as the linear combination of all possible solutions, as follows: T ðx, tÞ ¼

1 X

Ci eμi t ψ i ðxÞ 2

(8)

i¼1

The appropriate determination of the coefficients Ci can be achieved employing the orthogonality property as applied on the initial condition, Eq. 1d. Hence, for t = 0 T ðx, 0Þ ¼ f ðxÞ ¼ Operating on both sides of Eq. 9a with

1 X

Ðx1

C i ψ i ðxÞ

(9a)

i¼1

wðxÞψ j ðxÞðÞdx yields

x0 xð1

wðxÞψ j ðxÞf ðxÞdx ¼ x0

1 X

xð1

wðxÞψ i ðxÞψ j ðxÞdx

Ci

i¼1

(9b)

x0

Thus, from the orthogonality property of the eigenfunctions, all the terms in the summation on the R.H.S. of Eq. 9b are zero except for one term, when i = j, in which case the integral evaluates to the norm, Nj. Therefore xð1

x0

1 wðxÞψ j ðxÞf ðxÞdx ¼ Cj N j , or Cj ¼ Nj

xð1

wðxÞψ j ðxÞf ðxÞdx ¼ x0

1 f Nj j

(9c; d)

Returning to the index i, the exact analytical solution for T(x,t) is then finally obtained as

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T ðx, tÞ ¼

1 X 1 2 f i eμi t ψ i ðxÞ, Ni i¼1

(9e)

with xð1

fi ¼

xð1

wðxÞψ i ðxÞf ðxÞdx, N i ¼ x0

wðxÞψ 2i ðxÞdx

(9f; g)

x0

Example 1 One-dimensional Heat Conduction in a Slab

Consider the following dimensionless heat conduction problem to be solved as an example of the methodology above described: @T ðx, tÞ @ 2 T ðx, tÞ ¼ , @t @x2

0 < x < 1,

t>0

(10a)

T ð0, tÞ ¼ 0

(10b)

T ð1, tÞ ¼ 0

(10c)

T ðx, 0Þ ¼ f ðxÞ

(10d)

The following correspondence can be made with problem (1): w(x) = 1, k (x) = 1, d(x) = 0, x0 = 0, x1 = 1, β0 = β1 = 0, and α0 = α1 = 1. Hence, the eigenvalue problem is written as d2 ψ ð x Þ þ μ 2 ψ ðxÞ ¼ 0 dx2

(11a)

ψ ð0Þ ¼ 0, ψ ð1Þ ¼ 0

(11b; c)

The solution of problem (11) is given by ψ ðxÞ ¼ c1 cos ðμxÞ þ c2 sin ðμxÞ

(12a)

In order to satisfy the first boundary condition, Eq. 11b, c1 = 0. The second boundary condition, Eq. 11c, leads to c2 sin ðμÞ ¼ 0

(12b)

Even though c2 = 0 satisfies Eq. 12b, it leads to trivial solution and should be discarded in our analysis. The alternative is to satisfy sin ðμÞ ¼ 0

(12c)

which is satisfied for infinite discrete nonnegative values of μ, the eigenvalues

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69

μi ¼ iπ, i ¼ 1, 2, 3, . . .

(12d)

Recalling the solution for ψ(x), given by Eq. 12a setting c2 = 1, one obtains ψ i ðxÞ ¼ sin ðμi xÞ ¼ sin ðiπxÞ, i ¼ 1, 2, 3, . . .

(12e)

Hence, the solution for problem (10) is given by T ðx, tÞ ¼

1 X 2 1 f i eðiπ Þ t sin ðiπxÞ Ni i¼1

(13a)

with ð1

1 N i ¼ ½ sin ðiπxÞ dx ¼ , 2 2

ð1 fi ¼

and

0

sin ðiπxÞf ðxÞdx

(13b; c)

0

where the integrals for f i must be evaluated for the given initial condition f(x).

2.2

Multidimensional Formulation

The separation of variables methodology, above described for the one-dimensional diffusion formulation, can be readily extended to the multidimensional case. Consider the following multidimensional linear diffusion problem, defined in region V and for the boundary surface S, with homogeneous governing equation and boundary conditions: w ð xÞ

@T ðx, tÞ þ LfT ðx, tÞg ¼ 0, @t BfT ðx, tÞg ¼ 0,

x  S,

T ðx, 0Þ ¼ f ðxÞ,

x  V, t>0

xV

t>0

(14a) (14b) (14c)

where the operators L and B are given by L  ∇  ½kðxÞ∇ðÞ þ dðxÞðÞ B  αðxÞðÞ þ βðxÞkðxÞ

@ ðÞ @n

(14d) (14e)

Again, Eq. 14a includes the three physical effects, namely, the transient, diffusive, and linear dissipation terms. As for the one-dimensional case, this problem can be handled through the separation of variables method by assuming the separation of the desired potential T(x,t) into a product of space- and time-dependent functions, as follows:

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T ðx, tÞ ¼ ψ ðxÞΓðtÞ

(15)

Once Eq. 15 is substituted into Eq. 14a, the following problem is obtained for Γ(t): dΓ ðtÞ þ μ2 ΓðtÞ ¼ 0, dt

t>0

(16a)

which leads to ΓðtÞ ¼ C eμ

2

t

(16b)

The following eigenvalue problem is obtained in terms of the eigenfunctions ψ(x): Lfψ ðxÞg ¼ μ2 wðxÞψ ðxÞ, Bfψ ðxÞg ¼ 0,

xV

xS

(17a) (17b)

which is the generalized Sturm-Liouville problem, which has the same properties of its one-dimensional version, i.e., it has an infinite number of solutions ψ i(x) associated with particular values of μi, the eigenfunctions and eigenvalues, respectively. The analytical solution for the multidimensional formulation is constructed in much the same way as for the one-dimensional case, by proposing the linear combination of the infinite number of separable solutions and employing the orthogonality property of the eigenfunctions on the initial condition to obtain the expansion coefficients, in the form T ðx, tÞ ¼

1 X 1 2 f i eμi t ψ i ðxÞ Ni i¼1

(18a)

where ð N i ¼ wðxÞ½ψ i ðxÞ2 dV

(18b)

V

ð f i ¼ wðxÞf ðxÞψ i ðxÞdV

(18c)

V

The separation of variables is an effective analytical method for handling problem (14), once the coordinate system under consideration allows for separation of variables of the eigenvalue problem (17) itself (Mikhailov and Ozisik 1984). For instance, in the rectangular coordinate system, with x = {x, y, z}, with constant normalized thermophysical properties and no linear dissipation, w(x) = 1, k(x) = 1, d(x) = 0, the three-dimensional Sturm-Liouville problem, Eq. 17a, results in the Helmholtz equation given as

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Analytical Methods in Heat Transfer

∇2 ψ i ðxÞ þ μ2i ψ i ðxÞ ¼ 0

71

(19a)

The separation of the eigenfunction is then proposed as ψ i ðxÞ ¼ Xl ðxÞY m ðyÞZ n ðzÞ

(19b)

where X(x), Y( y), and Z(z) are then obtained from the corresponding one-dimensional eigenvalue problems, each one associated with their corresponding eigenvalues β, λ, and γ, respectively: d2 X l ð x Þ þ β2l Xl ðxÞ ¼ 0 dx2

(19c)

d2 Y m ð y Þ þ λ2m Y m ðyÞ ¼ 0 dy2

(19d)

d2 Z n ð z Þ þ γ 2n Zn ðzÞ ¼ 0 dz2

(19e)

while the global eigenvalue is given by μ2i ¼ β2l þ λ2m þ γ 2n

(19f)

In most heat conduction textbooks, the solution for the multidimensional temperature field T(x,t) is then presented as a triple summation (or double summation in a two-dimensional problem), accounting for the eigenvalues and eigenfunctions in each coordinate direction, in the form T ðx, y, z, tÞ ¼

1 X 1 X 1 X 2 2 2 1 f lmn eðβl þλm þγn Þt Xl ðxÞY m ðyÞZn ðzÞ N lmn l¼1 m¼1 n¼1

(20a)

with ð N lmn ¼ ½Xl ðxÞY m ðyÞZn ðzÞ2 dV

(20b)

V

ð f lmn ¼ f ðxÞXl ðxÞY m ðyÞZn ðzÞdV

(20c)

V

Although formal and exact, this representation can be misleading for computational purposes, since there is a natural tendency of truncating each of the expansions in the triple summation above to an individual truncation order, when seeking a final working expression. This is certainly not a good strategy from the computational point of view, since the total number of terms in the overall summation can be very large, being the product of the individual truncation orders, N = NxNyNz. Besides,

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some relevant terms to the final result might be omitted by this truncation process, while many others that have very little influence on the results will be included in the total N parcels. The best way to handle this computation is to maintain the single series representation of the general solution, Eq. 18a, employing an adequate reordering scheme for the global eigenvalues μi and progressively accounting for the most important terms to the final converged result (Mikhailov and Cotta 1996; Cotta and Mikhailov 1997). Then, from application of the reordering criterion to each value of the index i will correspond a combination of the three indices from the one-dimensional eigenvalue problems, (l,m,n). 2 2 2 The term eðβl þλm þγn Þt in the expansion given by Eq. 20a essentially governs the convergence rate of the infinite summation, and the global eigenvalues, computed  from the sum of the squared one-dimensional eigenvalues β2l þ λ2m þ γ 2n , should then be organized in ascending order, and all other corresponding quantities, so as to account for the most important terms progressively. Each index of the eigenquantities, i, then corresponds to a combination of individual orders (l, m, n) in such a way that μ21  μ22  μ23 . . . according to Eq. 19f. Example 2 Two-dimensional Heat Conduction in a Plate

Consider the following two-dimensional linear diffusion equation in the rectangular coordinate system, with first-kind boundary conditions: @T ðx, y,tÞ @ 2 T ðx, y, tÞ @ 2 T ðx, y, tÞ ¼ þ , 0 < x < 1, 0 < y < 1, t > 0 @t @x2 @y2

(21a)

T ¼ 0,

x ¼ 0,

0 < y < 1,

t>0

(21b)

T ¼ 0,

x ¼ 1,

0 < y < 1,

t>0

(21c)

T ¼ 0,

y ¼ 0,

0 < x < 1,

t>0

(21d)

T ¼ 0,

y ¼ 1,

0 < x < 1,

t>0

(21e)

T ¼ f ðx, yÞ,

t ¼ 0,

0 < x < 1,

00, the following one-dimensional eigenvalue problems are obtained: d2 XðxÞ þ β2 XðxÞ ¼ 0, dx2 Xð0Þ ¼ 0,

00

xV

(40a) (40b) (40c)

with P ðx, tÞ ¼ Pðx, tÞ  LfFðx; tÞg  wðxÞ

@Fðx; tÞ @t

(40d)

ϕ ðx, tÞ ¼ ϕðx, tÞ  BfFðx; tÞg

(40e)

f  ðxÞ ¼ f ðxÞ  Fðx; 0Þ

(40f)

If the filtered problem becomes completely homogeneous, even separation of variables can be readily employed. Alternatively, the formal procedure for the integral transforms solution is directly applicable, but since the filtered problem has the source terms with reduced importance in comparison with the original problem, a better convergence of the eigenfunction expansion is expected. Once the solution for the filtered problem is made available, Eq. 39 can be employed to obtain the original problem solution.

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Example 3 Nonhomogeneous Heat Conduction Problem in a Slab

Consider heat conduction in a slab in dimensional form, with internal heat generation and a time variable applied heat flux at x = 0, while at the opposite boundary x = L, heat is exchanged with the environment at temperature T1, with heat transfer coefficient h. w

@T ðx, tÞ @ 2 T ðx, tÞ ¼k þ gðx, tÞ, 0 < x < L, @t @x2  @T ðx, tÞ ¼ q0 ð t Þ k @x x¼0  @T ðx, tÞ k þ hT ðL, tÞ ¼ hT 1 @x x¼L

t>0

T ðx, 0Þ ¼ f ðxÞ

(41a) (41b)

(41c) (41d)

This problem can be readily filtered in order to yield homogeneous boundary conditions, and this option is always recommended in order to yield enhanced convergence of the eigenfunction expansion. The simplest filtering scheme to achieve just homogeneous boundary conditions in this problem would be considering a linear filter in x, according to Eq. 39: Fðx; tÞ ¼ aðtÞx þ bðtÞ

(42a)

and the two boundary conditions are then satisfied for


 k þ L að t Þ aðtÞ ¼ q0 ðtÞ=k; bðtÞ ¼ T 1  h

(42b; c)

Note that this choice of filter is equivalent to adopting the following problem formulation for the filtering function, without accounting for the equation source term: @ 2 Fðx; tÞ ¼0 @x2

(42d)

If reducing the equation source term importance is also relevant for faster convergence, a more informative filter expression could be obtained, for instance, from the analytical solution of the following quasi-steady problem:

k

@ 2 Fðx; tÞ þ gðx, tÞ ¼ 0, 0 < x < L, @x2  @Fðx; tÞ ¼ q0 ðtÞ k @x x¼0

t>0

(42e) (42f)

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81

 @Fðx; tÞ k þ hFðL; tÞ ¼ hT 1 @x x¼L

(42g)

Problem (42e–42g) allows for explicit analytical solution through direct double integration in the x variable, employing the boundary conditions. In either choice of filtering, the filtered problem becomes

w

@T  ðx, tÞ @ 2 T  ðx, tÞ ¼k þ P ðx, tÞ, 0 < x < L, @t @x2  @T  ðx, tÞ ¼0 @x x¼0  @T  ðx, tÞ k þ hT  ðL, tÞ ¼ 0 @x x¼L

t>0

T  ðx, 0Þ ¼ f  ðxÞ

(43a) (43b)

(43c) (43d)

where P ðx, tÞ ¼ w

@Fðx; tÞ ; @t

f  ðxÞ ¼ f ðxÞ  Fðx; 0Þ

(43e; f)

Performing the equivalence between the general solution and the present filtered problem, the coefficients are taken as follows: Pðx, tÞ P ðx, tÞ; wðxÞ w; kðxÞ k; dðxÞ 0;  f ð xÞ f ðxÞ; x x; V ½0, L; αð0Þ 0; αðLÞ h; βð0Þ 1; βðLÞ 1; ϕð0, tÞ 0; ϕðL, tÞ 0

(44a  m)

Following the basic procedure described for the general solution via CITT, the following eigenvalue problem is obtained: d2 ψ i ð x Þ þ μi 2 wψ i ðxÞ ¼ 0 dx2  dψ i ðxÞ ¼0 dx x¼0  dψ i ðxÞ k þ hψ i ðLÞ ¼ 0 dx x¼L

k

(45a) (45b)

(45c)

which can be readily solved analytically. ÐL Problem (43) is integral transformed with the operator ψ i ðxÞðÞdx to provide the 0

transformed decoupled ODE system:

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dT i ðtÞ þ μ2i T i ðtÞ ¼ gi ðtÞ, dt

i ¼ 1, 2, 3, . . .

T i ð 0Þ ¼ f i

(46a) (46b)

where the transformed source terms are evaluated employing Eq. 37b ðL

gi ðtÞ ¼ ψ i ðxÞP ðx, tÞdx

(46c)

0

and the transformed initial conditions are calculated with Eq. 37c: ðL

f i ¼ wψ i ðxÞf  ðxÞdx

(46d)

0

The solution of the transformed problem is readily obtained as 2 T i ðtÞ ¼ e

μ2i t 4

ðt fi þ e

3 μ2i t0

0

gi ðt Þdt

05

(46e)

0

and the final solution is obtained by invoking the inversion formula: T ðx, tÞ ¼

1 X 1 T i ðtÞψ i ðxÞ Ni i¼1

(47a)

with ðL N i ¼ wψ 2i ðxÞdx

(47b)

0

5

Generalized Integral Transform Technique

5.1

Formal Solution

Even though the classical integral transform technique is able to handle nonhomogeneous problems, it encounters limitations in handling problems that are non-transformable, in the sense that an eigenfunction expansion basis cannot be encountered that leads to a decoupled transformed system upon integral transformation. A partial differential problem can become non-transformable due to various circumstances, such as in the case of convection-diffusion problems and/or the

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presence of nonlinear terms either on the equation coefficients, source term, or in the boundary conditions. Such a limitation was recognized in the earlier works of Ozisik and Murray (1974) and Mikhailov (1975), when the presence of time-dependent coefficients on the boundary conditions and governing diffusion equation, respectively, did not allow for an exact integral transformation, thus resulting in a coupled infinite ordinary differential system for the transformed potentials. At that time, the numerical solution of such ODE systems, likely to be stiff from the numerical analysis perspective, was a major numerical task, and this possibility was not even considered. In order to overcome the limitation, a hybrid numerical-analytical solution technique, coined as the generalized integral transform technique, was then advanced by Cotta (1986) and Cotta and Ozisik (1986) and systematically developed along the following years (Cotta 1990, 1993, 1994, 1998; Serfaty and Cotta 1990; Cotta and Mikhailov 1997; Cotta and Mikhailov 2006; Cotta et al. 2013, 2016). In order to present the formal solution procedure of the generalized approach, consider the following transient nonlinear problem: w ð xÞ

@T ðx, tÞ þ LfT ðx, tÞg ¼ Pðx, t, T Þ, @t BfT ðx, tÞg ¼ ϕðx, t, T Þ,

x  S,

T ðx, 0Þ ¼ f ðxÞ,

xV

x  V, t>0

t>0

(48a) (48b) (48c)

which represents a quite general formulation, including nonlinear coefficients and convective terms, which can be gathered into the source terms P(x,t,T) and ϕ(x,t,T). Hence, the linear operators L and B are here chosen as characteristic ones, incorporating linearized information from the original nonlinear problem, and the remaining nonlinear or linear non-transformable terms are grouped into the equation and boundary source terms in order to recover the formulation given by Eq. 48a, 48b, and 48c. The GITT formal solution starts with the proposition of an eigenvalue problem, which is inherently chosen from the characteristic coefficients, and is given by Lfψ ðxÞg ¼ μ2 wðxÞψ ðxÞ, Bfψ ðxÞg ¼ 0,

xS

xV

(49a) (49b)

It is worth noting that the eigenvalue problem is an independent proposal and the recommended choice is certainly the one that captures as much information as possible about the original nonlinear coefficients and still leads to an analytically solvable eigenvalue problem. As already discussed in the previous section for the CITT, this eigenvalue problem allows for the definition of the transform and inversion formulae. In the case of the hybrid approach (GITT), there is some computational advantage in adopting a symmetric kernel in the definition of the integral transform pair (Cotta 1993). Therefore, a normalized eigenfunction is defined, which splits the norm contribution in the transform and inversion formulae, in the form

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ð Transform : T i ðtÞ ¼ wðxÞe ψ i ðxÞT ðx, tÞdV

(50a)

V

Inversion : T ðx, tÞ ¼

1 X

T i ðtÞe ψ i ðxÞ

(50b)

i¼1

where the normalized eigenfunction is given by e i ð xÞ ¼ ψ ð

ψ i ð xÞ

(51)

1=2

Ni

e i ðxÞðÞdV leads to the following transformed ODE Operating on Eq. 48a with ψ V

system, as already discussed in details for the classical integral transform technique:   dT i ðtÞ þ μ2i T i ðtÞ ¼ gi t, TðtÞ , dt

i ¼ 1, 2, 3, . . .

(52a)

with

 e i ð xÞ @ψ ð ð ϕðx, t, T Þ kðxÞ e i ð xÞ ψ   @n e i ðxÞPðx, t, T ÞdV  dS (52b) gi t, TðtÞ ¼ ψ αðxÞ þ βðxÞ V

S



TðtÞ ¼ T 1 ðtÞ, T 2 ðtÞ, T 3 ðtÞ, . . .

T

(52c)

and initial conditions ð T i ð0Þ ¼ wðxÞe ψ i ðxÞf ðxÞdV ¼ f i ,

i ¼ 1, 2, 3, . . .

(52d)

V

The inversion formula, Eq. 50b, is then recalled to  represent  the potential T(x,t), in order to evaluate the transformed source term, gi t, TðtÞ , leading to the coupled nonlinear system of ordinary differential equations, Eq. 52. For computational purposes it suffices to truncate this infinite system at a sufficiently high order N, leading to a system of N ordinary differential equations for the transformed potentials, T i , i ¼ 1, 2, 3, . . . N. The resulting system can be numerically solved by reliable built-in routines such as DIVPAG (IMSL 2014) in the IMSL library, within the FORTRAN environment, or NDSolve in the Mathematica system (Wolfram 2016), or any other wellestablished platform for stiff ODE systems, such as a few available in public domain subroutine libraries. Once the numerical solution for the transformed potentials is available, the sought potential T(x,t) can be constructed by invoking the inversion formula given by Eq. 50b, in which the remaining terms retain their analytical nature.

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In recent years, emphasis has been placed in unifying and simplifying the use of the GITT, to reach a larger number of users and offer an alternative hybrid numericalanalytical solution to their problems. The effort to integrate the knowledge on the GITT application into a symbolic-numerical algorithm resulted in the so-called unified integral transform (UNIT) code (Cotta et al. 2010, 2013, 2014; Sphaier et al. 2011). Symbolic computation has allowed, along the last three decades or so, for the advancement of various analytical and hybrid approaches that were before considered too tedious in terms of mathematical derivation involvement or even unpractical in terms of human effort. The open-source UNIT code is thus a mixed symbolic-numerical implementation on the Mathematica system (Wolfram 2016) that offers a starting development platform for researchers and engineers interested on integral transform solutions of diffusion and convection-diffusion problems. The formal solution regarding the standard procedure of the GITT approach is known as the total transformation scheme, described above, in which all spatial variables are eliminated through integral transformation. There is also the partial integral transformation scheme option of the GITT approach (Cotta and Gerk 1994; Castellões et al. 2007; Cotta et al. 2014), as an alternative solution path to problems with a strong convective direction, which is not eliminated through integral transformation, but kept within the transformed system. This alternative transformation scheme will not be formally presented, but shall be illustrated in Example 6. Example 4 One-dimensional Burgers’ Equation

Consider the one-dimensional version of the nonlinear Burgers’ equation as follows (Xu et al. 2011): @T ðx, tÞ @ 2 T ðx, tÞ @T ðx, tÞ ¼υ ,  T ðx, tÞ @t @x2 @x T ð0, tÞ ¼ 0,

0 < x < L,

t>0

T ðL, tÞ ¼ 0

(53a) (53b; c)

T ðx, 0Þ ¼ f ðxÞ (53d) Performing the equivalence between the general solution via the generalized integral transform technique and the present example, the coefficients are taken as @T ðx, tÞ ; wðxÞ 1; kðxÞ υ; @x f ðxÞ; d ðxÞ 0; V ½0, L; αð0Þ 1; αðLÞ 0; βðLÞ 0; ϕð0, tÞ 0; ϕðL, tÞ 0

Pðx, t, T Þ f ð xÞ β ð 0Þ

T ðx, tÞ

1;

(54a  i)

The corresponding eigenvalue problem is then written as υ

d2 ψ i ð x Þ þ μi 2 ψ i ðxÞ ¼ 0 dx2

(55a)

ψ i ð0Þ ¼ 0, ψ i ðLÞ ¼ 0

(55b)

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which has a straightforward analytical solution. ÐL e i ðxÞðÞdx: Then, problem (53) is integral transformed with the operator ψ 0

  dT i ðtÞ þ μ2i T i ðtÞ ¼ gi t,T , dt

i ¼ 1, 2, 3, . . . , N

(56a)

T i ð 0Þ ¼ f i

(56b)

with the transformed source terms ! ! ðL N N X X  de ψ l ðxÞ e i ðxÞ gi t,T ¼  ψ T j ðtÞe ψ j ðxÞ T l ðtÞ dx dx j¼1 l¼1 

(56c)

0

which can be written more conveniently as N X N X  gi t,T ¼  Aijl T l ðtÞT j ðtÞ,



ðL with

e i ðxÞe Aijl ¼ ψ ψ j ðxÞ

j¼1 l¼1

de ψ l ðxÞ dx dx

0

(56d; e) The transformed initial conditions are computed from ðL e i ðxÞf ðxÞdx fi ¼ ψ

(56f)

0

The system of coupled nonlinear ordinary differential equations given by Eq. 56a, 56b does not allow for analytical solution, but reliable numerical routines are available, with automatic absolute and relative errors control, as previously discussed. Once the solution is made available for the transformed potentials, the inversion formula can be readily employed in order to yield the solution for the desired potential, T(x,t). Considering a test case with L = 1, υ = 1, and f(x) = sin(πx), Table 2 illustrates the convergence behavior of the solution, employing the NDSolve routine (Wolfram 2016) with automatic absolute and relative error controls for the solution of the transformed problem, at the time value t = 0.1, for different truncation orders in the eigenfunction expansion. A very good convergence behavior is demonstrated, achieving full convergence to six significant digits at all selected positions for N = 5.

5.2

Reordering Schemes

In multidimensional applications, as previously discussed in the context of the solution through separation of variables, for the appropriate computation of these

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Table 2 Convergence behavior of the eigenfunction expansion (t = 0.1) x 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

N=2 0.109326 0.209535 0.291797 0.348059 0.371835 0.359213 0.309845 0.227583 0.120481

N=3 0.109545 0.209797 0.291892 0.347917 0.371577 0.359052 0.309909 0.227813 0.120680

N=4 0.109538 0.209792 0.291897 0.347924 0.371577 0.359046 0.309905 0.227817 0.120686

N=5 0.109538 0.209792 0.291896 0.347924 0.371578 0.359046 0.309905 0.227817 0.120687

N=6 0.109538 0.209792 0.291896 0.347924 0.371578 0.359046 0.309905 0.227817 0.120687

Xu et al. (2011) 0.10953 0.20979 0.29189 0.34792 0.37157 0.35904 0.30990 0.22781 0.12068

eigenfunction expansions, one should adopt a reordering of terms according to their individual contribution to the final numerical result. For the homogeneous linear problems that are handled by the classical analytical approach, the natural choice of reordering strategy is based on arranging in increasing order the global eigenvalue, which is given by the sum of the squared eigenvalues in each spatial coordinate, as illustrated for the rectangular coordinates system. In the more general hybrid approach (GITT) for nonlinear problems, since the solution structure is not known in analytic form, the parameter which shall govern this reordering scheme must be chosen with care. The most common choice of reordering strategy is still based on arranging in increasing order the sum of the global eigenvalues, which offers a good compromise between the overall convergence enhancement and simplicity in use. However, specific applications may require more elaborate reordering that accounts for the influence of transformed initial conditions and transformed nonlinear source terms in the ODE system, as discussed below. To more clearly understand alternative reordering schemes, the formal solution of the transformed potentials is examined, which is given by

T i ðtÞ ¼ f i exp



μ2i t



ðt

    þ gi t0 ,T exp μ2i ðt  t0 Þ dt0

(57a)

0

Integration by parts of Eq. 57a provides an alternative expression that allows the understanding of the influence of the transformed initial conditions and source terms in the choice of reordering schemes:   1       T i ðtÞ ¼ f i exp μ2i t þ 2 gi t,T  gi 0, T exp μ2i t μi ðt   1 dgi  2 exp μ2i ðt  t0 Þ dt0 0 μi dt 0

(57b)

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It remains clear that the squared global eigenvalues play their role in the decay of the absolute  values of the transformed potentials, both through the exponential term exp μ2i t and through the inverse term, 1=μ2i , which dominate at lower convergence rates. In principle, the reordering scheme based on the ascending sorting of the global eigenvalues should be able to account for the most important terms in the adequate reorganization of the expansion. If one considers the case of a timeindependent linear source term, the last integral term in Eq. 57b vanishes, leading to     g  T i ðtÞ ¼ f i exp μ2i t þ 2i 1  exp μ2i t μi

(57c)

One then concludes that the decay of the transformed initial condition and the evolution of the transformed source term play a complementary role in the selection of terms in the eigenfunction expansion for a fixed truncation order. Clearly, the influence of the initial condition on convergence rates should be more significant at the early transient, decaying with the exponential term, while the source term contribution appears to be more relevant as the steady state is approached, governed by the squared eigenvalue inverse term. In principle, for a certain fixed truncation order, there could be some extra terms that are not being accounted for when just the squared global eigenvalue criterium is being considered, while their importance could become evident once the expansion reordering strategy is expanded by merging with the ordered sequences from an initial condition and a source term criterium. In the first case, for the lowest time value of interest, t = tmin, the criterion that reorders the terms based on the decay of the is based on sorting  initial conditions  in decreasing order from the expression f i exp μ2i tmin . In the second case, for the general situation of a nonlinear transformed source term, the estimation of the terms’ importance is more difficult, since it would require the knowledge of the steady-state solution, which can be a major task in the more general nonlinear situation and generally is not known a priori. One possible approach is to consider the limiting case of an uniform unitary source term, representing, for instance, its normalized maximum value and analyzing ð the reordering of terms in descending absolute value e i ðxÞdV. Therefore, combining the three criteria, and based on the expression μ12 ψ i

V

eliminating the duplicates with respect to the reordering scheme based on the squared global eigenvalues, extra terms may be added to the initially reordered terms that may still contribute to the final result.

5.3

Single-Domain Formulation

The solutions presented above, by any of the considered methodologies, are valid for any arbitrary region V with boundary surface S. In addition, the coefficients associated with each of the equation operators were allowed to be arbitrarily variable with the space coordinates. These two general characteristics of the obtained solutions naturally

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a

S l,m

Sl

Vm V1

Tk,l (x,t ) x

S

b

c S

Tk (x,t ) X

V

Tk (x,t ) X

Fig. 1 (a) Diffusion or convection-diffusion in a complex multidimensional configuration with nV subregions; (b) single-domain representation keeping the original external surface; (c) singledomain representation considering a regular overall domain that envelopes the original one (Cotta et al. 2016a)

lead to the possibility of treating complex configurations, involving different subregions and materials in perfect contact, by rewriting the problem formulation in the form of a single domain with spatially variable coefficients. Let us consider a general transient diffusion problem defined in a complex multidimensional configuration that is represented by nV different subregions with volumes Vl, l=1,2, . . . , nV, with potential and flux continuity at the interfaces, as illustrated in Fig. 1a (Cotta et al. 2016a). Different potentials can be simultaneously determined in each subregion, Tk,l(x, t), k=1,2, . . ., nT, for instance, temperature, concentrations, velocity components, and pressure, governed in the corresponding subregion through a general formulation including equation and boundary source terms, respectively, Pk , l (x, t, T) and ϕk, l(x, t, T). w k , l ð xÞ

  @T k, l ðx, tÞ ¼ ∇  kk, l ðxÞ∇T k, l ðx, tÞ  dk, l ðxÞT k, l ðx, tÞ þ Pk, l ðx, t, TÞ, @t x  V l , t > 0, k ¼ 1, 2, . . . , nT , l ¼ 1, 2, . . . , nV (58a)

with initial, interface, and boundary conditions given, respectively, by

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T k, l ðx, 0Þ ¼ f k, l ðxÞ, T k, l ðx, tÞ ¼ T k, m ðx, tÞ,

x  Vl

x  Sl, m ,

(58b) t>0

(58c)

@T k, l ðx, tÞ @T k, m ðx, tÞ ¼ k k , m ð xÞ , x  Sl , m , t > 0 @n @n

 @ αk, l ðxÞ þ βk, l ðxÞkk, l ðxÞ T k, l ðx, tÞ ¼ ϕk, l ðx, t, TÞ, x  Sl , t > 0 @n k k , l ð xÞ

(58d) (58e)

where n denotes the outward-drawn normal to the interfaces, Sl,m, and external surfaces, Sl. The source terms may incorporate nonlinear coefficients, convective terms, or any remaining terms that were chosen not to be represented in the characteristic linear coefficients that will provide the eigenfunction expansion basis. With the so called single-domain formulation, the generalized integral transform technique (GITT) can be directly applied to solve system (58) above, without the need of constructing an individual eigenfunction expansion for each potential at each subregion and then coupling all such transformed systems and potentials. In this case, one single transformed system and one single set of transformed potentials are obtained by employing the appropriate orthogonality property. Figure 1 provides two possibilities for representation of the single domain, either by keeping the original overall domain after definition of the space variable coefficients, as shown in Fig. 1b, or, depending on the boundary conditions, by considering a regular overall domain that envelopes the original one, as shown in Fig. 1c. Irregular domains can be directly integral transformed, as demonstrated in several previous works, for example, in Aparecido et al. (1989), Aparecido and Cotta (1990), Castellões et al. (2010), Monteiro et al. (2010), Pérez Guerrero et al. (2000), and Sphaier and Cotta (2002), and, in principle, there is no need to consider the second representation possibility pointed out above. However, some computational advantages may be achieved by enveloping the original irregular domain by a simple regular region. Therefore, it is possible to rewrite problem (58) as a single-domain formulation with space variable coefficients and source terms given by w k ð xÞ

@T k ðx, tÞ ¼ ∇  ðkk ðxÞ∇T k ðx, tÞÞ  dk ðxÞT k ðx, tÞ þ Pk ðx, t, TÞ, x  V, t > 0 @t (59a)

with initial and boundary conditions given, respectively, by xV T k ðx, 0Þ ¼ f k ðxÞ,

 @ αk ðxÞ þ βk ðxÞkk ðxÞ T k ðx, tÞ ¼ ϕk ðx, t, TÞ, x  S, @n where

(59b) t>0

(59c)

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V¼

nV X

91

Vl,

S¼

nV X

l¼1

Sl

(59d)

l¼1

and where the potentials vector is given by T ¼ fT 1 , T 2 , . . . , T k , . . . , T n T g

(59e)

The space variable coefficients in Eq. 59, besides the new equation and boundary source terms and initial conditions, now without the subscript l for the subregions Vl, incorporate the abrupt transitions among the different subregions and permit the representation of system (59) as a single-domain formulation. It is worth noting that the preferred eigenvalue problem choice for the solution of problem (59) is unlikely to allow for explicit analytical solutions, due to the presence of arbitrarily spatially varying coefficients. Hence, in the next section the application of the GITT to the solution of eigenvalue problems is reviewed.

5.4

GITT for Eigenvalue Problems

The GITT can be directly employed in the solution of eigenvalue problems with arbitrarily variable coefficients, as proposed in Cotta (1993), later on extended and applied (Sphaier and Cotta 2002; Naveira-Cotta et al. 2009; Knupp et al. 2015a; Cotta et al. 2016a). The GITT reduces the differential eigenvalue problem into a standard algebraic eigenvalue problem, which can be solved by existing routines for matrix eigensystem analysis. For this purpose, the eigenfunctions of the original differential eigenvalue problem are expressed by eigenfunction expansions based on a simpler auxiliary eigenvalue problem, for which analytical solutions are readily available. Consider the following eigenvalue problem defined in region V and boundary surface S (Knupp et al. 2015a; Cotta et al. 2016a): Lψ ðxÞ ¼ μ2 wðxÞψ ðxÞ, Bψ ðxÞ ¼ 0,

xV

xS

(60a) (60b)

where the operators L and B are given by L ¼ ∇  ðkðxÞ∇Þ þ dðxÞ

(60c)

@ @n

(60d)

B ¼ αðxÞ þ βðxÞkðxÞ

and w(x), k(x), and d(x) are known functions in region V and α(x) and β(x) are known functions on the boundary surface S. The idea is to propose a solution for problem (60) in the form of an eigenfunction expansion, which is based on a simpler auxiliary eigenvalue problem, given by

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^ ð xÞ ¼ λ 2 w ^ ðxÞΩðxÞ, LΩ ^ ðxÞ ¼ 0, BΩ

xV

(61a)

xS

(61b)

^ and B ^ are the simpler operators where L   ^ ¼ ∇  ^k ðxÞ∇ þ d^ ðxÞ L

(61c)

^ ¼ αðxÞ þ βðxÞ^k ðxÞ @ B @n

(61d)

^ ðxÞ, ^k ðxÞ, and d^ ðxÞ are known coefficients in V and S, chosen so as the auxiliary and w eigenvalue problem given by Eq. 61 offers a known solution for the eigenvalues λ and corresponding eigenfunctions Ω(x). In principle, the boundary coefficients, α(x) and β(x), could also be different for the auxiliary problem, as derived in Knupp et al. (2015a) and Cotta et al. (2016a), but this is not a usual choice, since it would lead to an undesirable slower convergence behavior. Therefore, making use of the eigenfunction orthogonality property, the following integral transform pair is proposed: ð transform : ψ i ¼

e i ðxÞψ ðxÞdV ^ ð xÞ Ω w

(62a)

V

inverse : ψ ðxÞ ¼

1 X

e i ðxÞψ i Ω

(62b)

i¼1

where the normalized eigenfunctions and normalization integrals are Ωi ð xÞ e i ð xÞ ¼ p ffiffiffiffiffiffiffiffi , Ω N Ωi

with

ð ^ ðxÞΩ2i ðxÞdV N Ωi ¼ w

(62c; d)

V

Equation 60a, 60b can then be rewritten as   ^ ð xÞ ¼ L ^  L ψ ðxÞ þ μ2 wðxÞψ ðxÞ, Lψ

xV

(63a)

  ^ ð xÞ ¼ B ^  B ψ ðxÞ, x  S Bψ (63b) Ð e i ðxÞðÞdV , to yield the transformed Equation 60a is now operated on with V Ω algebraic system ð ð    2 e e ^ ^ ¼ γ i B  B ψ ðxÞdS þ Ω i ðxÞ L  L ψ ðxÞdV þ μ Ω i ðxÞwðxÞψ ðxÞdV, ð

λ2i ψ i



S

V

V

i ¼ 1, 2, . . . (64a)

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e e i ðxÞ  ^k ðxÞ @ Ω i ðxÞ Ω @n γi ¼ αðxÞ þ βðxÞ

(64b)

After substituting the inversion formula into Eq. 64a, the resulting algebraic system is written in matrix form, truncated to the Mth term, as ðA þ CÞfψg ¼ μ2 Bfψg

(65a)

with the elements of the M  M matrices given by ð ð     e j ðxÞdS  Ω e i ð xÞ L e j ðxÞdV ^ B Ω ^ L Ω Aij ¼  γ i B

(65b)

V

S

Cij ¼ λ2i δij

(65c)

e i ð xÞ Ω e j ðxÞdV Bij ¼ wðxÞΩ

(65d)

ð V

where δij is the Kronecker delta. From the relation ð V

ð ð   e j ðxÞ @Ω ^ ^ e e j ðxÞdV e e i ðxÞ  ∇Ω e dS  ^k ðxÞ∇Ω Ω i ðxÞ∇  k ðxÞ∇Ω j ðxÞ dV ¼ k ðxÞΩ i ðxÞ @n V

S

(66a) the elements of A can be calculated through e # ðΩ e i ðxÞ  ^k ðxÞ @ Ω i ðxÞ " e j ðxÞ   @Ω @n βðxÞ kðxÞ  ^k ðxÞ dS @n αðxÞ þ βðxÞ S

ð  S

ð þ



ð e    e i ðxÞ @ Ω j ðxÞ dS þ kðxÞ  ^k ðxÞ ∇Ω e i ð xÞ  ∇ Ω e j ðxÞdVþ kðxÞ  ^k ðxÞ Ω @n



 e i ðxÞΩ e j ðxÞdV dðxÞ  d^ ðxÞ Ω

V

V

(66b) Thus, the eigenvalue problem given by Eq. 60 is reduced to an algebraic eigenvalue problem, Eq. 65, which can be solved for the eigenvalues μ and the corresponding eigenvectors ψ i, to find the original eigenfunction, Eq. 63b. The

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Fig. 2 Schematic representation of the conjugated heat transfer problem in parallel-plate channel (Knupp et al. 2012)

user-prescribed accuracy is then achieved by increasing the truncation order M. Example 5 Conjugated Heat Transfer in Parallel-Plate Channel

Consider laminar incompressible internal flow of a Newtonian fluid between parallel plates, for steady-state and undergoing convective heat transfer due to a prescribed temperature, Tw, at the external face of the channel wall. The channel wall is considered to participate on the heat transfer problem through transversal heat conduction only. The fluid flows with a known fully developed velocity profile uf ( y), and with an inlet temperature Tin. Figure 2 shows a schematic representation of the conjugated conduction-convection problem. For the sake of simplicity, it is assumed that the flow is dynamically developed and thermally developing and axial conduction is neglected both in the fluid stream and along the walls. The formulation of the conjugated problem as a single-region model that accounts for the heat transfer phenomena at both the fluid flow and the channel solid wall is achieved by making use of coefficients represented as space variable functions, where abrupt transitions occur at the fluid-solid wall interface, in dimensionless form, as U ðY Þ


 @θ ðY, ZÞ @ @θ ¼ K ðY Þ , @Z @Y @Y

0 < Y < 1,

Z>0

θ ðY, Z ¼ 0Þ ¼ θZ¼0 ¼ 1  @θ  ¼ 0, θ ðY ¼ 1, ZÞ ¼ 0 @Y Y¼0 where the following dimensionless parameters have been employed

(67a) (67b) (67c; d)

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z=yw z y u T  T in k ¼ ;Y ¼ ;U ¼ ;θ ¼ ;K ¼ yw 4uav kf Re Pr yw Pe T w  T in kf uav 4yw ν uav 4yw ; Pr ¼ ; Pe ¼ Re Pr ¼ ;α ¼ Re ¼ α ν α wf Z¼

(68a  i)

and the following filter was used in order to homogenize the boundary conditions: θðY, ZÞ ¼ 1 þ θ ðY, ZÞ (68j) The eigenvalue problem has been formulated by directly applying separation of variables to problem (67) so that all the information concerning the transition of the two domains is represented within the eigenvalue problem, by means of the space variable coefficients K(Y) and U(Y ). Thus
 d dψ ðY Þ K ðY Þ i þ μi 2 U ðY Þψ i ðY Þ ¼ 0 dY dY  dψ i  ¼ 0, ψ i ð1Þ ¼ 0 dY Y¼0

(69a)

(69b; c)

Problem (69) does not allow for an explicit analytic solution, but the GITT itself can be used in order to provide a hybrid numerical-analytical solution. The GITT is here employed in the solution of this eigenvalue problem via the proposition of a simpler auxiliary eigenvalue problem and expanding the unknown eigenfunctions in terms of the chosen basis. The chosen auxiliary problem is given by d2 Ωn ð Y Þ þ λ2n Ωn ðY Þ ¼ 0 dY 2  dΩn ðY Þ ¼ 0, Ωn ð1Þ ¼ 0 dY Y¼0

(70a) (70b; c)

The proposed expansion of the original eigenfunction is then given by ψ i ðY Þ ¼

1 X

e n ðY Þψ i, n , Ω

inverse

(71a)

transform

(71b)

n¼1

ψ i, n ¼

ð1

e n ðY ÞdY, ψ i ðY ÞΩ

0

e n ðY Þ are the normalized eigenfunctions given by where Ω ð1 Ωn ð Y Þ e Ω n ðY Þ ¼ pffiffiffiffiffiffiffiffi , N Ωn ¼ Ω2n ðY ÞdY N Ωn 0

(72a; b)

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The integral transformation of the eigenvalue problem with space variable Ð1 e n ðY ÞðÞdY , to coefficients is then performed by operating on Eq. 69a with 0 Ω yield the following algebraic problem in matrix form ðA  νBÞψ ¼ 0,



ψ ¼ ψ n, m ; B ¼ Bn, m , Bn, m ¼

A ¼ An, m ; An, m ¼

with ð1

νi ¼ μi 2

(73a)

e n ðY ÞΩ e m ðY ÞdY U ðY ÞΩ

0

ð1

d e m ðY Þ Ω dY 0

(73b  d)

! e n ðY Þ dΩ K ðY Þ dY dY

ð1 e m ðY Þ d Ω e n ðY Þ dΩ dY ¼  K ðY Þ dY dY

(73e; f)

0

The algebraic problem (73a) can be numerically solved to provide results for the eigenvalues and eigenvectors, upon truncation to a sufficiently large finite order M, and then combined by the inverse formula to provide the desired original eigenfunctions. Once the solution of the eigenvalue problem (69) is made available, the original problem (67) becomes completely transformable, and the final solution is then obtainable by separation of variables and given by θðY, ZÞ ¼ 1 þ θ ðY, ZÞ ¼ 1 þ 

N X



e i ðY Þ θZ¼0, i eμi Z ψ 2

i¼1

ð1

(74)

θZ¼0, i ¼ U ðY Þe ψ i ðY ÞθZ¼0 dY 0

e i ðY Þ are given by where ψ ð1 ψ i ðY Þ e i ðY Þ ¼ pffiffiffiffiffi , N i ¼ UðY Þ½ψ i ðY Þ2 dY ψ Ni

(75a; b)

0

The present test case for conjugated heat transfer also allows for exact solution via the classical integral transform technique, as detailed in Knupp et al. (2012), and is here used as benchmark to perform a critical analysis of the single-domain formulation strategy. Figure 3a, b illustrates the behavior of the space variable coefficients that are feeding the single-region model, U(Y ) and K(Y ), as space variable functions

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Fig. 3 Representation of the space variable coefficients (Knupp et al. 2012)

97

a 0.4

U(Y)

0.3

0.2 Fluid

Solid

0.1

0 0

b

0.2

0.4

0.6

0.8

1

Y 1

K(Y)

0.8

0.6 Fluid

Solid

0.4

0.2 0

0.2

0.4

0.6

0.8

1

Y

where the region from Y = 0 to Y = Yi = 0.5 corresponds to the fluid flow domain and the region from Y = Yi = 0.5 to Y = 1 corresponds to the channel wall. In the test case here presented, the dimensionless thermal conductivity has been calculated motivated by an application with a microchannel made of polyester resin (ks = 0.16 W/m C), with water as the working fluid (kf = 0.64 W/m C), so that ks/ kf = 0.25. Table 3 illustrates the excellent convergence behavior of the first ten eigenvalues associated with the original eigenvalue problem, and Fig. 4 depicts the convergence behavior of the tenth eigenfunction, for different truncation orders in

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Table 3 Convergence behavior of the first ten eigenvalues (Knupp et al. 2012) Eigenvalue μi 1 2 3 4 5 6 7 8 9 10

N = 30 1.89403 14.3682 27.3581 40.3901 53.4355 66.4872 79.5423 92.5995 105.658 118.718

Fig. 4 Convergence behavior of the tenth eigenfunction of problem (69), for different truncation orders, M = 10, 12, 14, and 16 (Knupp et al. 2012)

N = 60 1.89112 14.3671 27.3574 40.3896 53.4352 66.4869 79.5421 92.5994 105.658 118.718

N = 90 1.89014 14.3667 27.3571 40.3894 53.4350 66.4868 79.5420 92.5993 105.658 118.718

N = 120 1.88965 14.3665 27.3570 40.3893 53.4349 66.4867 79.5419 92.5992 105.658 118.718

4 3

ξ10(Y)

2 1 0 M = 10 M = 12 M = 14 M = 16

–1 –2 0

0.2

0.4

0.6

0.8

1

Y

Eq. 71a, M = 10, 12, 14, and 16, where it can be noticed that with only 16 terms the tenth eigenfunction is fully converged to the graph scale. The comparison of the single-domain solution with the exact solution at Z = 0.01 is shown in Table 4, in which the single-domain formulation solution has been obtained with M = 50 terms in the eigenvalue problem solution and N = 5 terms in the temperature expansion, achieving full convergence to the five digits shown. Accuracy improvement can be achieved by increasing the truncation orders, especially for the eigenvalue problem solution, M. This same conjugated problem is again considered in Sect. 6, Example 7, when an integral balance procedure for convergence enhancement is implemented, further reducing the deviations in Table 4, even at lower truncation orders. The

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Table 4 Comparison between the single-domain formulation (with M = 50) and the exact solution for the temperature at Z = 0.01 (Knupp et al. 2012) Y 0.00 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50

Single domain 0.01042 0.01525 0.02146 0.03043 0.04225 0.05685 0.07400 0.09325 0.1140 0.1360

Exact sol. 0.01041 0.01523 0.02143 0.03040 0.04219 0.05678 0.07390 0.09312 0.1138 0.1353

Relative deviation (%) 0.086 0.11 0.12 0.13 0.13 0.14 0.14 0.14 0.13 0.53

same approach can be readily employed in the solution of more general models of conjugated problems, including irregular geometries, as demonstrated in Knupp et al. (2013a, b, 2015a, b) and illustrated below. Example 6 Conjugated Heat Transfer in Arbitrarily Shaped Channels: GITT with Partial Transformation Scheme and Single-Domain Formulation

Consider conjugated heat transfer in a multi-stream heat exchanger with five microchannels, as shown in Fig. 5 (Knupp et al. 2015a; Cotta et al. 2016a). Steady-state laminar incompressible fully developed flow of a Newtonian fluid is here considered in each of the five microchannels shown in Fig. 5, assuming their cross-sectional shape remains uniform along the flow direction. The momentum equation in the longitudinal direction (z) is given, in the single-region formulation, by

 @ @uðx, yÞ @ @uðx, yÞ Cðx, yÞ νðx, yÞ νðx, yÞ ¼0 þ  @x @x @y @y ρðx, yÞ   uð0, yÞ ¼ uðLx , yÞ ¼ uðx, 0Þ ¼ u x, Ly ¼ 0

(76a) (76b)

with ( Cðx, yÞ ¼

@p ¼ Δp=Lz , in the fluid stream @z 0, in the solid region

(76c)

where the space variable kinematic viscosity and density, v(x,y) and ρ(x,y), are given, respectively, by νðx, yÞ ¼

νf , in the fluid stream νs ! 1, in the solid region

(76d)

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Fig. 5 Schematic representation of the multi-stream heat exchanger cross section (Knupp et al. 2015a)

ρðx, yÞ ¼

ρf , in the fluid stream ρs , in the solid region

(76e)

where νf and ρf represent the fluid kinematic viscosity and density and νs and ρs represent the equivalent solid properties. For the kinematic viscosity corresponding to the solid region, νs, the value should be high enough so that the calculated velocities in the solid region will become zero. The following integral transform pair is defined to proceed with the GITT solution of Eq. 76a, 76b, 76c, 76d, and 76e: Lðy Lðx

e χ i ðx, yÞuðx, yÞdxdy

Transform : ui ¼

(77a)

0 0

Inverse : uðx, yÞ ¼

1 X

e χ i ðx, yÞui

(77b)

i¼1

with χ ðx, yÞ e χ i ðx, yÞ ¼ ipffiffiffiffiffi , N i ¼ Ni

Lðx Lðy

χ 2i ðx, yÞdxdy

(77c; d)

0 0

where the normalized eigenfunctions e χ i ðx, yÞ and corresponding eigenvalues ηi are calculated from the solution of the following eigenvalue problem:

 @ @χ i ðx, yÞ @ @χ i ðx, yÞ νðx, yÞ νðx, yÞ þ þ η2i χ i ¼ 0 @x @x @y @y   χ ð0, yÞ ¼ χ ðLx , yÞ ¼ χ ðx, 0Þ ¼ χ x, Ly ¼ 0

(78a) (78b)

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Even though it is not possible to obtain an explicit analytic solution for this eigenvalue problem, the GITT itself can be employed in order to provide a hybrid numerical-analytical solution, as previously described. Then, problem Ð1 Ð1 e χ i ðx, yÞðÞdxdy, yield(76) becomes fully transformable when operated with 0 0

ing a linear decoupled algebraic problem for the transformed velocities, with the following analytical solution: 1 ui ¼ 2 ηi

ð1 ð1 e χ i ðx, yÞ 0 0

Cðx, yÞ dxdy, ρðx, yÞ

i ¼ 1, 2, . . .

(79)

The transformed potentials ui can then be readily substituted into the inverse formula given by Eq. 77b, yielding an analytic representation for the velocity field u(x,y). It should be noted that the bounding domain for the analysis of the flow problem is not required to be the same as the actual physical domain, including the whole solid region. Once the fully developed velocity field in the microchannels is computed, the conjugated heat transfer problem involving thermally developing flow is handled. The transient single-domain formulation for the energy balance, with the convection and axial diffusion terms merged into the equation source term, is given as wðx, yÞ



 @T ðx, y, z, tÞ @ @T @ @T ¼ kðx, yÞ kðx, yÞ þ þ Pðx, t, T Þ @t @x @x @y @y

(80a)

with Pðx, t, T Þ ¼ uðx, yÞwðx, yÞ

@T @2T þ kðx, yÞ 2 @z @z

(80b)

where the space variable coefficients are given by

uf ðx, yÞ, fluid stream u ! 0, solid region wf , fluid stream wðx, yÞ ¼ ws , solid region kf , fluid stream kðx, yÞ ¼ ks , solid region

uðx, yÞ ¼

(80c)

(80d)

(80e)

The boundary conditions, made homogeneous through filtering in both the x and y directions, are given by

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 @ α þ βkðx, yÞ T ðx, y, z, tÞ ¼ 0, @n

x, y  Sxy ,

0 < z < Lz ,

t>0

(80f)

where Sxy is the outside wall surface, with boundary conditions of first or third kind, depending on the chosen values of the coefficients α and β. The initial condition is written as T ðx, y, z, 0Þ ¼ f ðx, y, zÞ,

0 < y < Ly ,

0 < x < Lx ,

0 < z < Lz

(80g)

while the boundary conditions in the longitudinal direction are taken as  @T  T ðx, y, 0, tÞ ¼ T in ðx, yÞ,  ¼ 0, 0 < x < Lx , 0 < y < Ly , t > 0 @z z¼Lz (80h; i) Those terms with partial derivatives with respect to the longitudinal coordinate z were collected in the source term P(x,t,T ). The partial transformation scheme is then employed (Cotta and Gerk 1994; Castellões et al. 2007; Cotta et al. 2014), eliminating by integral transformation only the coordinates x and y, in which the diffusive effects are predominant, but not applying the integral transformation over the coordinate z. Hence, the following integral transform pair is adopted: Lðy Lðx

Transform : T i ðz, tÞ ¼

wðx, yÞe ψ i ðx, yÞT ðx, y, z, tÞdxdy

(81a)

0 0

Inverse : T ðx, y, z, tÞ ¼

1 X

e i ðx, yÞT i ðz, tÞ ψ

(81b)

i¼1

e i ðx, yÞ and corresponding eigenvalues μi are obtained where the eigenfunctions ψ from the solution of the following eigenvalue problem:

 @ @ψ ðx, yÞ @ @ψ ðx, yÞ kðx, yÞ i kðx, yÞ i þ þ μ2i wðx, yÞψ i ¼ 0 @x @x @y @y

 @ α þ βkðx, yÞ ψ ðx, yÞ ¼ 0, x, y  Sxy @n i

(82a)

(82b)

with normalized eigenfunctions and norms written as ψ i ðx, yÞ ffiffiffiffiffi , N i ¼ e i ðx, yÞ ¼ p ψ Ni

Lðy Lðx

wðx, yÞψ 2i ðx, yÞdxdy

(83a; b)

0 0

The GITT is employed in the solution of the eigenvalue problem (82), transforming the original differential problem into an algebraic eigensystem.

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e i ðx, yÞ are obtained, problem (80) is integral transAfter the eigenfunctions ψ L Ðy LÐx e i ðx, yÞðÞdxdy , yielding the transformed ψ formed through the operator 0

0

one-dimensional partial differential system:   @T i ðz, tÞ þ μ2i T i ¼ gi z, t,T , i ¼ 1, 2, . . . @t 1 X   @T j gi z, t,T ¼  @z j¼1



1 X @2Tj j¼1

@z2

Lðy Lðx

(84a)

ψ j ðx, yÞdxdy wðx, yÞuðx, yÞe ψ i ðx, yÞe

0 0

L ðy Lðx

kðx, yÞe ψ i ðx, yÞe ψ j ðx, yÞdxdy

(84b)



T ¼ T1, T2, T3, . . .

(84c)

0 0

where

with the transformed boundary and initial conditions Lðy Lðx

T i ð0, tÞ ¼ T in 

wðx, yÞe ψ i ðx, yÞTin ðx, yÞdxdy, 0 0

 @T i  ¼0 @z z¼Lz

(84d; e)

Lðy Lðx

T i ðz, 0Þ ¼ f i ðzÞ 

wðx, yÞe ψ i ðx, yÞf ðx, y, zÞdxdy

(84f)

0 0

The transformed system given by Eq. 84a, 84b, 84c, 84d, 84e, and 84f is numerically solved for the transformed potentials T i ðz, tÞ, with i = 1, 2, . . ., N. The built-in routine of the Mathematica system (Wolfram 2016), NDSolve, is employed to numerically solve the coupled system of one-dimensional PDEs, providing reliable solutions under automatic absolute and relative errors control. The Mathematica routine automatically provides an interpolating function object that approximates the z and t variables behavior of the solution in a continuous form. Then, the inversion formula, Eq. 81b, is recalled to provide the temperature field T(x,y, z, t). A test case for this three-dimensional conjugated problem is here considered, with Tin = 0; first-kind boundary conditions, T = 1 (α = 1 and β = 0); and dimensionless solid thermal conductivity, ks = 0.25, and fluid thermal conductivity, kf = 1 (Knupp et al. 2015a). The wall and fluid temperature profiles are plotted in Fig. 6a, b, comparing the GITT solution with the commercial CFD solver COMSOL™ Multiphysics solution, with the non-isothermal flow package, showing the calculated

104

a

1 z = 0.5

0.8 T(x,y = 0.2,z)

Fig. 6 Comparisons between GITT and COMSOL solutions for the fluid and wall temperature profiles for the multi-stream heat exchanger (Knupp et al. 2015a): (a) along x, at y = 0.2, and (b) along y at x = 0, for different longitudinal positions z

R. M. Cotta et al.

z = 0.3

z = 0.2

0.6

0.4

z = 0.1 GITT COMSOL

0.2 0

0.2

0.4

0.6

0.8

1

x

b

1 z = 0.5

0.8 T(x = 0,y,z)

z = 0.3

0.6

z = 0.2

0.4 z = 0.1

GITT COMSOL

0.2 0

0.1

0.2 y

0.3

0.4

temperatures for different longitudinal positions (z) across the transversal directions: (a) along x, at y = 0.2, and (b) along y at x = 0. It may be observed the excellent adherence to the graphical scale between the two sets of results. Clearly, the transitions between the solid and fluid regions are accurately accounted for by the single-domain formulation and the corresponding eigenfunctions. Such results demonstrate that the multi-stream heat exchanger conjugated problem can be adequately modeled as a single domain, by just properly defining the space variable coefficients, so as to capture the microchannels geometry and their transitions within the substrate. Numerical results for the temperature distribution are also presented in tabular form, Tables 5 and 6, illustrating the convergence behavior of the GITT solution as a function of the number of terms in the temperature field expansion, where it can be observed a full convergence of at least two significant digits, in the worst case, within the truncation orders considered. Again, the adherence to the COMSOL solution can be confirmed in tabular form.

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Table 5 Convergence behavior of the steady-state temperature profile for increasing truncation order of the expansion (N ), with fixed M = 120 terms in the eigenvalue problem solution (multistream heat exchanger), at (a) z = 0.1; (b) z = 0.2 (Knupp et al. 2015a) (a) N N = 75 N = 85 N = 95 N = 105 N = 115 COMSOL (b) N N = 75 N = 85 N = 95 N = 105 N = 115 COMSOL

T(x,y = 0.2, z = 0.1) x = 0.0 x = 0.3 0.3537 0.3642 0.3533 0.3640 0.3532 0.3646 0.3527 0.3661 0.3519 0.3631 0.35 0.36

x = 0.6 0.3979 0.3975 0.3988 0.3986 0.3998 0.40

x = 0.9 0.5828 0.5824 0.5825 0.5825 0.5824 0.58

T(x,y = 0.2, z = 0.2) x = 0.0 x = 0.3 0.6238 0.6314 0.6236 0.6312 0.6235 0.6315 0.6232 0.6325 0.6224 0.6298 0.62 0.63

x = 0.6 0.6686 0.6681 0.6687 0.6686 0.6697 0.67

x = 0.9 0.8202 0.8192 0.8192 0.8193 0.8193 0.82

Table 6 Convergence behavior of the steady-state temperature profile for increasing truncation order of the expansion (N ), with fixed M = 120 terms in the eigenvalue problem solution (multistream heat exchanger), at (a) z = 0.1; (b) z = 0.2 (Knupp et al. 2015a) (a) N N = 75 N = 85 N = 95 N = 105 N = 115 COMSOL (b) N N = 75 N = 85 N = 95 N = 105 N = 115 COMSOL

T(x = 0, y, z = 0.1) y = 0.1 0.5070 0.5066 0.5069 0.5064 0.5063 0.51

y = 0.2 0.3537 0.3533 0.3532 0.3527 0.3519 0.35

y = 0.3 0.4913 0.4909 0.4916 0.4930 0.4927 0.50

T(x = 0, y, z = 0.2) y = 0.1 0.7403 0.7403 0.7403 0.7400 0.7399 0.74

y = 0.2 0.6238 0.6236 0.6235 0.6232 0.6224 0.62

y = 0.3 0.7265 0.7267 0.7270 0.7278 0.7276 0.73

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Advanced Topics

After reviewing the fundamental aspects behind the classical analytical methodologies for diffusion problems (separation of variables and integral transforms) and the basic ideas behind the more applicable hybrid numerical-analytical approach (generalized integral transform technique), it is now of interest to the more advanced reader to examine a few recent developments that have opened up new perspectives to the GITT extension, including an integral balance scheme for the convergence enhancement in the solution of eigenvalue problems with variable coefficients, the proposition of convective eigenfunction basis for convection dominated problems, and the adoption of nonlinear eigenvalue problems in the solution of nonlinear problems. As has been understood from the previous formal analysis, the eigenfunction expansion convergence rate is essentially controlled by the decay of the transformed potentials with the increasing eigenvalue order, which are ultimately governed by the relative magnitude of the source terms. While filtering strategies can somehow reduce the importance of the source terms, it is also desirable to enrich the spatial representation of the original problem operators within the eigenvalue problem formulation, which may then incorporate terms that would otherwise be moved into the source term representation. However, when handling the eigenvalue problem solution by the GITT itself, the convergence of the eigenvalues and eigenfunctions may eventually be more demanding than the original potential convergence. Therefore, the following three advanced topics aim at both improving convergence and incorporating further information to the eigenvalue problem, leading to representative gains in both overall accuracy and convergence acceleration.

6.1

Integral Balance Procedure for Convergence Improvement

For enhanced convergence of eigenfunction expansions, it is desirable to include as much information as possible from the original PDE operators coefficients within the eigenvalue problem. This is particularly important when multiple spatial scales and/or very abrupt variations of the coefficients need to be handled. The GITT has been demonstrated above as a reliable tool for the solution of such eigenvalue problems with arbitrarily variable coefficients. However, in many cases, the auxiliary eigenvalue problem that provides the basis for the expansion is constructed with coefficients that are not very informative, since it may result not solvable in analytic explicit form, as desirable for the integral transformation algorithm. On the other hand, very simple noninformative auxiliary coefficients may lead to slowly converging expansions for the original eigenfunctions. In such cases, an integral balance procedure can be particularly beneficial in accelerating the convergence of the eigenfunction expansions. Analytical expressions can be derived (Cotta et al. 2016b), explicitly accounting for the space variable

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107

coefficients of the original problem, within the resulting functional form of the redefined inverse formulae for the eigenfunctions. In order to illustrate this procedure, consider the x-direction one-dimensional version of the SturmLiouville problem (Cotta et al. 2016b):

   d dΧ ðxÞ kx ðxÞ þ γ 2 wx ðxÞ  dx ðxÞ Χ ðxÞ ¼ 0, dx dx

x0  x  x1

(85a)

αx, 0 Χ ðxÞ  βx, 0 kx ðxÞ

dΧ ðxÞ ¼ 0, dx

x ¼ x0

(85b)

αx, 1 Χ ðxÞ þ βx, 1 kx ðxÞ

dΧ ðxÞ ¼ 0, dx

x ¼ x1

(85c)

The integral balance procedure is a convergence acceleration technique (Cotta and Mikhailov 1997) that is here aimed at obtaining eigenfunction expansions of improved convergence behavior for both the eigenfunction and its derivatives. The basic idea behind this integral balance is that through integration over the spatial domain, new working expressions may benefit from the better convergence characteristics of the integrals of eigenfunction expansions. It consists of the double integration of the original equation that governs the potential for which the convergence improvement is being sought. Through integration of the original equation, Eq. 85a, an improved expression for the eigenfunction derivative is obtained, and a second integration then offers an improved relation for computation of the eigenfunction itself. However, the problem boundary conditions need to be accounted for, so that the eigenfunctions and respective derivatives at the boundaries can be eliminated from the new derived expressions, thus avoiding the use of the inverse Ð x formula at the boundaries. The first step is thus the integration of Eq. 85a with x0 ð:Þdx0 to find  dΧ ðxÞ 1 dΧ ðxÞ 1 2 1 ¼ k x ðx0 Þ γ Iwx0 ðxÞ þ Id x ðxÞ   dx k x ðxÞ dx x0 kx ðxÞ k x ðxÞ 0

(86a)

where Iwx0 ðxÞ ¼

ðx

wx ðx0 ÞΧ ðx0 Þdx0 ; Id x0 ðxÞ ¼

x0

Now, integrating Eq. 86a with

Ð x1 x

ðx

dx ðx0 ÞΧ ðx0 Þdx0

ð:Þdx0 yields

 dΧ ðxÞ Χ ðxÞ ¼ Χ ðx1 Þ  k x ðx 0 Þ Ikx ðxÞ þ γ 2 Iwkx1 ðxÞ  Idkx1 ðxÞ dx x0 1 where

(86b; c)

x0

(87a)

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ð x1

1 dx0 ; Iwkx1 ðxÞ ¼ Ikx1 ðxÞ ¼ 0 k ð x xÞ x ð x1 1 Id ðx0 Þdx0 Idkx1 ðxÞ ¼ 0 Þ x0 k ð x x x

ð x1 x

1 Iwx ðx0 Þdx0 ; kx ðx0 Þ 0

(87b  d)

A similar procedure could be implemented by starting with the integration from x to x1 and then integrating the resulting expression from x0 to x, as detailed in Cotta et al. (2016b), with identical final result. Writing Eq. 49a for x = x1 and Eq. 50a for x = x0, the following two equations relating the boundary derivatives and potentials are obtained:   dΧ ðxÞ dΧ ðxÞ ¼ k ð x Þ  γ 2 Iwx0 ðx1 Þ þ Id x0 ðx1 Þ x 0 dx x1 dx x0  dΧ ðxÞ Χ ðx0 Þ ¼ Χ ðx1 Þ  kx ðx0 Þ Ikx ðx0 Þ þ γ 2 Iwkx1 ðx0 Þ  Idkx1 ðx0 Þ dx x0 1 k x ðx 1 Þ

(88a)

(88b)

These two relations, together with the boundary conditions, Eq. 85b, 85c, provide   four equations for determination of the boundary quantities Χ ðx0 Þ, Χ ðx1 Þ, dΧdxðxÞ , x0  dΧ ðxÞ , which can be readily solved through symbolic computation in the more  dx x1

general situation, that include first- to third-kind boundary conditions, to yield X ðx0 Þ ¼  X ðx1 Þ ¼

β0 ½β1 ðId x0 ðx1 Þ  γ 2 Iwx0 ðx1 ÞÞ þ α1 ðIdkx1 ðx0 Þ  γ 2 Iwkx1 ðx0 ÞÞ α0 β1 þ α1 ðβ0 þ α0 Ikx1 ðx0 ÞÞ

(89a)

β1 ½α0 Idkx1 ðx0 Þ  ðβ0 þ α0 Ikx1 ðx0 ÞÞðId x0 ðx1 Þ  γ 2 Iwx0 ðx1 ÞÞ  γ 2 α0 Iwkx1 ðx0 Þ α0 β1 þ α1 ðβ0 þ α0 Ikx1 ðx0 ÞÞ (89b)

 dX α0 ½β1 ðId x0 ðx1 Þ  γ 2 Iwx0 ðx1 ÞÞ þ α1 ðIdkx1 ðx0 Þ  γ 2 Iwkx1 ðx0 ÞÞ ¼  dx x0 ½α0 β1 þ α1 ðβ0 þ α0 Ikx1 ðx0 ÞÞkx ðx0 Þ

(89c)

 dX α1 ½α0 Idkx1 ðx0 Þ þ ðβ0 þ α0 Ikx1 ðx0 ÞÞðId x0 ðx1 Þ  γ 2 Iwx0 ðx1 ÞÞ þ γ 2 α0 Iwkx1 ðx0 Þ ¼ dx x1 ½α0 β1 þ α1 ðβ0 þ α0 Ikx1 ðx0 ÞÞkx ðx1 Þ (89d) Substituting Eq. 89a, 89b, 89c, and 89d into Eqs. 86a and 87a results in the following general expressions with enhanced convergence for the eigenfunctions and the corresponding derivatives, respectively:

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XðxÞ ¼ Idkx1 ðxÞ þ Iwkx1 ðxÞγ 2 þ fβ1 ½Idkx1 ðx0 Þα0  Id x0 ðx1 ÞðIkx1 ðx0 Þα0 þ β0 Þ þ γ 2 ðIwkx1 ðx0 Þ þ Ikx1 ðx0 ÞIwx0 ðx1 Þα0 þ Iwx0 ðx1 Þβ0 Þ

þ Ikx1 ðxÞα0 ½Idkx1 ðx0 Þα1  Iwkx1 ðx0 Þγ 2 α1 þβ1 ðId x0 ðx1 Þ  Iwx0 ðx1 Þγ 2 Þ =½Ikx1 ðx0 Þα0 α1 þ α1 β0 þ α0 β1  (90a) dXðxÞ 1 ¼ Id x0 ðxÞ  Iwx0 ðxÞγ 2 dx k ðxÞ

 Idkx1 ðx0 Þα1  Id x0 ðx1 Þβ1 þ γ 2 ðIwkx1 ðx0 Þα1 þ Iwx0 ðx1 Þβ1 Þ þ Ikx1 ðx0 Þα0 α1 þ α1 β0 þ α0 β1 (90b) which account for the local space variations of the equation and boundary condition coefficients. The expressions provided by Eq. 90a, 90b can then substitute the original inverse formulae, in the solution of the eigenvalue problem (85), thus improving the convergence rates for determination of both the eigenvalues and corresponding eigenvectors and, consequently, the original problem eigenfunctions. Example 7 Conjugated Heat Transfer in Parallel-Plate Channel – Convergence Improvement

The problem described in the single-domain formulation illustration (Example 5) is now revisited, in order to demonstrate the convergence acceleration that is achieved with the integral balance scheme applied to the eigenvalue problem solution (Cotta et al. 2016b). The equivalence between the general solution presented in this section and the conjugated problem in parallel-plate channel, problem (67), with the eigenvalue problem given by Eq. 69, is given as x Y; XðxÞ ψ ðY Þ; kx ðxÞ K ðY Þ; wx ðxÞ U ðY Þ; γ μ; d x ðxÞ 0; x0 ¼ 0; x1 ¼ 1; αx, 0 0; βx, 0 1; αx, 1 1; βx, 1

0

(91a  m)

The expressions for the eigenfunction and its derivative, as computed from the integral balance scheme, are then simplified to XðxÞ ¼ Iwkx1 ðxÞγ 2

(92a)

dXðxÞ γ2 ¼ Iwx0 ðxÞ dx k x ðxÞ

(92b)

Substituting the corresponding expressions for Iwkx1 and Iwx0(x), and employing the inversion formula for the original eigenfunctions appearing on the R.H.S. of Eq. 92a, 92b, one obtains

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X i ðxÞ ¼ γ i 2

X

Xin IBn ðxÞ

(93a)

n

dXi ðxÞ γ2 X ¼ i Xin IAn ðxÞ dx k x ðxÞ n

(93b)

with IAn(x) and IBn(x) given by ðx e n ðx0 Þdx0 , IAn ðxÞ ¼ wx ðx0 ÞΩ

ð1 IBn ðxÞ ¼ x

0

1 IAn ðx0 Þdx0 kx ðxÞ

(94a; b)

with the eigenfunctions Ω(x) calculated from a simpler auxiliary eigenvalue e n ðY Þ already defined. problem, Eq. 70, with the normalized eigenfunctions Ω Performing the integral transformation of the original eigenvalue problem; substituting the expressions for the eigenfunctions and their derivatives with improved convergence, Eq. 93a, 93b; and truncating the expansions to a finite order M yield the following algebraic eigenvalue problem: 

 A  γ2B X ¼ 0

(95a)

where ð1

e m ðxÞ dΩ dx, An, m ¼ IAn ðxÞ dx 0

ð1

e m ðxÞdx Bn, m ¼  wx ðxÞIBn ðxÞΩ

(95b; c)

0

The algebraic problem (95a) can be numerically solved to provide results for the eigenvalues γ 2 and eigenvectors Xin , upon truncation to a sufficiently large finite order M, and then employed into Eq. 93a, 93b to provide the desired eigenfunctions and their derivatives with improved convergence rates. Once these eigenfunctions and corresponding eigenvalues are made available, and recalling the correspondence provided in Eq. 91a–m, the solution for the desired potential is given by Eq. 74, as already derived in Example 5. Table 7 shows the dimensionless temperatures at different transversal positions within the fluid flow region at Z = 0.01, calculated by solving the eigenvalue problem with space variable coefficients with the traditional inverse formula, as presented in Example 5, in comparison with the solution employing the integral balance improved expressions derived in the present section. In both cases it was employed N = 5 terms in the temperature expansion, which is enough to achieve convergence of the five significant digits shown. For the traditional solution scheme, Example 5, it was employed M = 50 terms in the eigenvalue problem solution, whereas for the integral balance scheme, it was employed only M = 5 terms. The results are quite impressive, demonstrating that with a much lower

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Table 7 Comparison of exact solution, GITT via inverse formula with M = 50 (Example 5), and GITT via integral balance with M = 5 (Example 7) for dimensionless temperatures at different transversal positions within the fluid flow region, Z = 0.01 (Cotta et al. 2016b) Y 0.00 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 a

Exact solution 0.010413 0.015230 0.021439 0.030396 0.042192 0.056776 0.073900 0.093122 0.11384 0.13534

a

GITT 0.010422 0.015246 0.021465 0.030435 0.042249 0.056854 0.074001 0.093249 0.11399 0.13605

Rel. deviation GITT (%)a 0.086 0.11 0.12 0.13 0.13 0.14 0.14 0.14 0.13 0.53

GITTb Int. balance 0.010413 0.015230 0.021439 0.030396 0.042192 0.056777 0.073901 0.093123 0.11384 0.13534

Rel. deviation Int. balance (%)b 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

Example 5 Example 7

b

truncation order in the eigenfunction expansion, the integral balance scheme allowed for achieving a more accurate solution, with four to five digits of agreement with the exact solution provided.

6.2

Integral Transforms with Convective Eigenvalue Problem

When handling convection-diffusion problems, the typical solution path through the GITT is to combine the convection term with the source term, as shown in the formal solution procedure presented in Sect. 5. Although this solution path is well established and effective, a solution with enhanced convergence can be pursued once the information on the convection term can be fully or partially incorporated into the eigenvalue problem, as originally proposed in Cotta et al. (2016c) and here reviewed. Thus, consider a general multidimensional convection-diffusion equation, for the potential T(x,t), given in the region V with the position vector x: wðxÞ

@T ðx, tÞ þ uðxÞ  ∇T ðx, tÞ ¼ ∇  ½kðxÞ∇T ðx, tÞ  dðxÞT ðx, tÞ þ Pðx, t, T Þ, @t x  V, t > 0 (96)

where the linear coefficients in each operator, dependent only on the spatial variables, represent the choice of characteristic functional behaviors to be accounted for in the eigenvalue problem, while the nonlinearities and remaining terms are collected into the redefined nonlinear source term, P(x, t, T). Considering that the convective term coefficient vector u can be represented in the three-dimensional situation by the three components {ux, uy, uz}, here illustrating

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the transformation in the Cartesian coordinate system, x = {x, y, z}, then Eq. 96 can be rewritten in the generalized diffusive form as (Cotta et al. 2017) ^ ð xÞ w



 @T ðx, tÞ ^ @ ^ @T ðx, tÞ @ ^ @T ðx, tÞ ¼ k y ðxÞ^k z ðxÞ k x ð xÞ k y ð xÞ þ ^k x ðxÞ^k z ðxÞ @t @x @x @y @y 

@T ðx, tÞ @ ^ þ ^k x ðxÞ^k y ðxÞ  d^ ðxÞT ðx, tÞ k z ð xÞ @z @z ^ ðx, t, T Þ, x  V, t > 0 þP (97a)

where ^k ðxÞ ¼ ^k x ðxÞ^k y ðxÞ^k z ðxÞ; w ^ ðxÞ ¼ wðxÞ^k ðxÞ=kðxÞ; d^ ðxÞ ¼ dðxÞ^k ðxÞ=kðxÞ; ^ ðx, t, T Þ ¼ Pðx, t, T Þ^k ðxÞ=kðxÞ; u ðxÞ ¼ 1 ½uðxÞ  ∇kðxÞ; P Ð  Ð  k ð xÞ Ð   u ð x Þdx  x and ^k x ðxÞ ¼ e ; ^k y ðxÞ ¼ e uy ðxÞdy ; ^k z ðxÞ ¼ e uz ðxÞdz ; (97b  i) When the transformed diffusion coefficients are functions of only the corresponding space coordinate, or ^k x ðxÞ ¼ ^k x ðxÞ, ^k y ðxÞ ¼ ^k y ðyÞ; ^k z ðxÞ ¼ ^k z ðzÞ, with the necessary restrictions on the choices of the characteristic linear coefficients k(x) and u(x), a generalized diffusion formulation is constructed which leads to a self-adjoint eigenvalue problem and can be written in such special case as ^ ð xÞ w

  @T ðx, tÞ ^ ðx, t, T Þ, ¼ ∇  ^k ðxÞ∇T ðx, tÞ  d^ ðxÞT ðx, tÞ þ P @t

x  V,

t>0 (98a)

where ^k ðxÞ ¼ ^k x ðxÞ^k y ðyÞ^k z ðzÞ

(98b)

for which the appropriate self-adjoint eigenvalue problem would be   ^ ðxÞ  d^ ðxÞ ψ ðxÞ ¼ 0, ∇  ^k ðxÞ∇ψ ðxÞ þ μ2 w

xV

(99)

which can be directly solved by the GITT for the eigenvalues and eigenfunction, as described in Sect. 5. Problem (99) incorporates relevant information on the convective effects, as specified in the chosen linear convective term coefficients that undergo the exponential transformation, which can provide a desirable convergence enhancement effect in the corresponding integral transform solution.

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Example 8 One-dimensional Burgers’ Equation – Convergence Improvement

The example chosen to illustrate this section is the one-dimensional Burgers’ equation, as in Example 4, both in linear and nonlinear formulations (Serfaty and Cotta 1992), which allows for the analysis of the convective eigenvalue problem choice on the eigenfunction expansion convergence behavior. The convectiondiffusion problem here analyzed is given by @T ðx, tÞ @T ðx, tÞ @ 2 T ðx, tÞ þ U ðT Þ ¼ , @t @x @x2

0 < x < 1,

T ðx, 0Þ ¼ 1; T ð0, tÞ ¼ 0; T ð1, tÞ ¼ 0

t>0

(100a) (100b  d)

where the nonlinear velocity coefficient is taken as U ðT Þ ¼ u0 þ b:T

(100e)

which readily allows for the separate consideration of the linear and nonlinear effects. The convective effect will be incorporated into the eigenvalue problem in linearized form, by choosing the coefficient u0 to represent its effect. Thus, for this illustration, information on the nonlinear portion of U(T) shall not be accounted for in the convective eigenvalue problem formulation. Therefore, in the linear situation (b = 0), the convective term will be fully incorporated into the transformed eigenvalue problem. A brief comparison of convergence rates is now undertaken, by considering the integral transforms solution of Eq. 100 by both the traditional diffusive eigenvalue problem, without accounting for the convection term, which is retained in the source term, and the convective eigenvalue problem, fully or partially accounting for the convection within the eigenfunction expansion basis. Table 8 illustrates the convergence behavior of the eigenfunction expansions for T(x,t), with u0 = 10, b = 0 (linear problem) and b = 5 (nonlinear problem). Clearly, from those columns associated with the use of the convective eigenvalue problem (Conv.), one may clearly observe the marked gain in convergence rates in comparison to the diffusive alternative (Diff.), even for the nonlinear situation, when the convective eigenvalue problem does not account for the full influence of the nonlinear convection term. In the linear case, from the first two columns, for x = 0.5 and t = 0.05, it can be observed that the results through the convective eigenvalue problem are already fully converged to four significant digits at truncation orders as low as N = 4, while the solution with the diffusive eigenvalue problem needs around N = 40 terms to achieve the same level of precision. For x = 0.9 and t = 0.01, slightly larger truncation orders are required, as expected for smaller values of t, a typical behavior in eigenfunction expansions. For the convective basis, full convergence to four significant digits has already been achieved for N = 10, while truncation orders of around N = 80 are required through the purely diffusive basis. The two solutions perfectly match each other, but in this linear case the computational cost of the

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Table 8 Convergence analysis of eigenfunction expansions with convective and diffusive eigenvalue problems for the one-dimensional Burgers’ equation (Cotta et al. 2017) (a) N 2 4 6 8 10 12 14 16 18 20 30 40 50 60 70 80 90 (b) N 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

u0 = 10, b = 0 (linear problem) T(0.5,0.05) conv. T(0.5,0.05) diff. 0.3865 0.4624 0.3797 0.3562 0.3797 0.3888 0.3754 0.3820 0.3783 0.3806 0.3791 0.3801 0.3794 0.3798 0.3797 0.3797

u0 = 10, b = 5 (nonlinear problem) T(0.5,0.05) conv. T(0.5,0.05) diff. 0.2862 0.3606 0.2762 0.2496 0.2770 0.2877 0.2768 0.2718 0.2769 0.2795 0.2769 0.2753 0.2778 0.2762 0.2773 0.2765 0.2771 0.2766 0.2770 0.2767 0.2769

T(0.9,0.01) conv. 1.814 0.1994 0.7033 0.7369 0.7374 0.7374

T(0.9,0.01) diff. 0.4302 0.6538 0.7263 0.7484 0.7508 0.7464 0.7411 0.7372 0.7353 0.7350 0.7382 0.7371 0.7376 0.7373 0.7375 0.7374 0.7374

T(0.9,0.01) conv. 1.437 0.4146 0.7687 0.7980 0.8009 0.8006 0.8001 0.7998 0.7996 0.7996

T(0.9,0.01) diff. 0.4504 0.6881 0.7808 0.8125 0.8162 0.8104 0.8035 0.7984 0.7960 0.7956 0.7965 0.7977 0.7988 0.7995 0.7996

convective basis solution, which is fully explicit and analytical, is indeed negligible in comparison to the coupled transformed system solution with the usual diffusive eigenvalue problem choice. For the nonlinear one-dimensional problem, a maximum truncation order of N = 30 terms has been considered in the solution of the

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generalized diffusive eigenvalue problem and N < 30 in computing the potential expansion. From the table for the nonlinear situation, one concludes that similar observations can be drawn with respect to the comparative behavior of the convective and diffusive basis, with a somehow less marked difference in this case, since for the nonlinear case the convective eigenvalue problem does not fully account for the convective term influence, but only for a characteristic linear behavior of the velocity coefficient. For instance, at x = 0.5 and t = 0.05, convergence to four significant digits is achieved for N as low as 10 for the convective basis, while the diffusive basis requires N = 30, and for x = 0.9 and t = 0.01, the convective basis yields four significant digits at N = 18, and the diffusive basis requires at least N = 30.

6.3

Integral Transforms with Nonlinear Eigenfunction Expansions

As described in the GITT formal solution procedure, in Sect. 5, the traditional path for handling nonlinear problems is to rewrite the problem formulation grouping all of the nonlinear information on the equation and boundary conditions operators into the general nonlinear source terms. Then, the problem is reinterpreted as one of linear differential operators but with nonlinear sources, which naturally leads to a choice of basis for the eigenfunction expansions through the characteristic linear coefficients that were adopted to reformulate the problem. In Cotta et al. (2016d), a novel integral transform solution for nonlinear convection-diffusion problems was proposed, also recently employed by Pontes et al. (2017). Instead of collapsing the nonlinearities into the corresponding source terms, which nevertheless may still exist, the nonlinear coefficients are directly accounted for in the eigenvalue problem formulation, thus yielding a nonlinear eigenfunction expansion basis. In order to illustrate this procedure, consider the following general problem with nonlinear boundary conditions:

wðxÞ

@T ðx, tÞ ¼ ∇:kðxÞ∇T ðx, tÞ  dðxÞT ðx, tÞ þ Pðx, t, T Þ, @t

x  V,

t>0 (101a)

with initial and boundary conditions T ðx, 0Þ ¼ f ðxÞ, αðx, t, T ÞT ðx, tÞ þ βðx, t, T ÞkðxÞ

xV

@T ðx, tÞ ¼ ϕðx, t, T Þ, @n

(101b) x  S,

t>0

(101c)

where α and β are nonlinear boundary condition coefficients and n is the outwarddrawn normal vector to surface S. All the boundary condition coefficients and source

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terms are allowed to be nonlinear, besides being explicitly dependent also on the space and time variables for the sake of generality. Taking a different path from the usual formalism in the GITT, as presented in Sect. 5, a nonlinear eigenvalue problem that preserves the original boundary condition coefficients is now preferred, instead of the one with linear characteristic coefficients, in the form   ∇:kðxÞ∇ψ i ðx; tÞ þ μ2i ðtÞwðxÞ  dðxÞ ψ i ðx; tÞ ¼ 0,

xV

(102a)

with boundary conditions αðx, t, T Þψ i ðx; tÞ þ βðx, t, T ÞkðxÞ

@ψ i ðx; tÞ ¼ 0, @n

xS

(102b)

and the solution for the associated time-dependent eigenfunctions, ψ i(x; t), and eigenvalues, μi(t), is at this point assumed to be known. Problem (102) allows for the definition of the integral transform pair: ð T i ðtÞ ¼ wðxÞψ i ðx; tÞT ðx, tÞðx, tÞdV, transform (103a) V

T ðx, tÞ ¼

1 X i¼1

1 ψ ðx; tÞT i ðtÞ, N i ðt Þ i

inverse

and the time-dependent normalization integrals ð N i ðtÞ ¼ wðxÞψ 2i ðx; tÞdV

(103b)

(104)

V

After integral transformation, the resulting ODE system for the transformed potentials, T i ðtÞ, is written as: 1     dTi ðtÞ X þ Ai, j t,T T j ðtÞ ¼ gi t,T , dt j¼1

t > 0,

i, j ¼ 1, 2 . . .

(105a)

with initial conditions T i ð 0Þ ¼ f i

(105b)

    Ai, j t,T ¼ δij μ2i ðtÞ þ Ai, j t,T

(105c)

where

where δij is the Kronecker delta, and

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ð  1 @ t,T ¼  wðxÞ ½ψ i ðx; tÞψ j ðx; tÞdV N j ðt Þ V @t ð   gi t,T ¼ ψ i ðx; tÞPðx, t, T Þ dV Ai, j



V

0

1 @ψ i B @n C þ ϕðx, t, T Þ@ AdS αðx, t, T Þ þ βðx, t, T Þ S ð

(105d)

ψ i ðx; tÞ  kðxÞ

(105e)

ð fi ¼

wðxÞe ψ i ðx; 0Þf ðxÞdV

(105f)

V

System (105) is then numerically solved through well-established initial value problem solvers, such as the function NDSolve of the Mathematica platform (Wolfram 2016). It should be recalled that the eigenvalue problem in Eq. 102 has now to be solved simultaneously with the transformed system given by Eq. 105, yielding the time-variable eigenfunctions, eigenvalues, and norms. The GITT itself can be employed in the solution of the nonlinear eigenvalue problem, Eq. 102. The basic idea is to reduce the eigenvalue problem described by the partial differential equation into a nonlinear algebraic eigenvalue problem, which can be solved by known approaches for matrix nonlinear eigensystem analysis. The procedure is similar to the one already detailed for the solution of eigenvalue problems via GITT, but now with nonlinear coefficients in the eigenvalue problem formulation. Therefore, the eigenfunctions of the original eigenvalue problem need to be expressed by eigenfunction expansions based on a simpler auxiliary eigenvalue problem, with linear coefficients, for which analytical solutions are readily obtainable. Example 9 Heat Conduction with Nonlinear Boundary Conditions

The problem here considered, in dimensionless form, is given by @T ðx, tÞ @ 2 T ðx, tÞ ¼ , @t @x2

0 < x < 1,

t>0

(106a)

with initial and boundary conditions given, respectively, by T ðx, 0Þ ¼ 1,

0x1

@T ð0, tÞ @T ð1, tÞ ¼ 0; þ BiðT ð1, tÞÞT ð1, tÞ ¼ 0, @x @x

(106b) t>0

and for the present application, the nonlinear function Bi(T )is taken as

(106c; d)

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BiðT ð1, tÞÞ ¼ Bic T 1=3 ð1, tÞ


  γ2 2 γ þ Bir 1 þ γT ð1, tÞ þ T ð1, tÞ 1 þ T ð1, tÞ 2 2

(106e)

which represents a combined natural convection and radiation boundary condition, where ^ T ^1 ^x T αs^t hc L hr L , x ¼ , t ¼ 2 , Bic ¼ , Bir ¼ , ^0  T ^1 ks ks L L T ^ ^ ^ 1, γ ¼ T 0  T 1 hr ¼ 4eσ T ^ T1 T¼

(107a  g)

where ^ denotes the dimensional variables (temperature, time, and position) in the original formulation. The correspondence between the above formulation and the general one is given by the following relations: wðxÞ ¼ 1, kðxÞ ¼ 1, d ðxÞ ¼ 0, Pðx, t, T Þ ¼ 0, f ðxÞ ¼ 1

(108a  e)

and for the boundary conditions for x ¼ 0 : for x ¼ 1 :

αð0, t, T Þ ¼ 0, βð0, t, T Þ ¼ 1, ϕð0, t, T Þ ¼ 0 αð1, t, T Þ ¼ BiðT ð1, tÞÞ, βð1, t, T Þ ¼ 1, ϕð1, t, T Þ ¼ 0

(109a  f)

The nonlinear problem (106) was first directly solved through the traditional GITT without considering the nonlinear coefficient in the eigenvalue problem, according to Sect. 5, without filtering and adopting a linearized boundary coefficient, namely: αð1Þ ¼ Bief ¼ Bic þ Bir


  γ2  γ 1þγþ 1þ 2 2

(110a)

The above characteristic linear boundary condition coefficient choice then yields the following nonlinear source term in the nonlinear boundary condition:   ϕð1, t, T Þ ¼ Bief  BiðT ð1, tÞÞ T ð1, tÞ

(110b)

Then, the novel approach with nonlinear eigenvalue problem has been considered. Except for α(1,t,T ), all of the other coefficients are linear and remain the same in both eigenvalue problems. The nonlinear eigenvalue problem is then written as @ 2 ψ ðx; tÞ þ μ2 ðtÞψ ðx; tÞ ¼ 0, @x2

0 0 2 γ0 γ0

(32)

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a

195

22.37 22.36 22.35 22.35 22.34 22.33 22.32 22.32 22.31 22.30 22.29 22.28 22.27 22.27 22.26 22.26 22.25 22.24 22.23 22.23 22.22 22.21 22.20 22.19 22.18 22.18 0 ˚C

b

px 1800 1700 1600 1500 1400 1300 1200 1100 1000 900 800 700 600 500 400 300 200 100 0 22.18 22.20 22.22 22.24 22.26 22.28 22.30 22.32 22.34 22.36 22.38 ˚C

Fig. 1 (a) Thermal image with an infrared camera of an isothermal plate; (b) Histogram of the temperature measurements (Fonseca et al. 2014)

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and depends only on the scale parameter (centerpoint) γ 0. The mean and the variance pffiffi 2 of Rayleigh’s distribution are given by γ 0 π2 and 4π γ 0 , respectively. 2 The gamma distribution with parameters α and β, denoted as Pj ~ G(α, β), has the following density  

 Pj 1 α1 π Pj ¼ α P exp  (33) for Pj > 0 β ΓðαÞ j β with mean αβ and variance αβ2, where Γ(α) is the gamma function. For β = 1, the so-called one-parameter gamma distribution is obtained. The density that results by making α = 1 is called exponential distribution. The beta distribution Pj ~ Be(α, β) has support in 0 < Pj < 1. The density of this distribution is given by

 Γðα þ βÞ α1

β1 Pj π Pj ¼ 1  Pj in 0 < Pj < 1 ΓðαÞΓðβÞ

(34)

α with mean αþβ and variance ðαþβÞ2αβ . ðαþβþ1Þ

The probability distributions given by Eqs. 29, 30, 31, 32, 33, 34 were written for one single random variable, but they can be easily extended for multivariate cases (Beck and Arnold 1977; Lee 2004; Winkler 2003; Kaipio and Somersalo 2004; Tan et al. 2006; Calvetti and Somersalo 2007; Gamerman and Lopes 2006). The multivariate Gaussian distribution is given by Eq. 12. A multivariate prior is usually required for the solution of inverse problems in situations where the parameters represent point values of a function; a typical case involves spatially distributed functions, like a thermophysical property that varies within the medium, where the parameter Pj is then associated to an average value of the function in a finite volume resulting from the discretization of the spatial domain. Markov random fields can be used to generate priors for these situations. A collection {P1, P2, . . ., PN} is a Markov random field if the full conditional distribution of Pj depends only on its set of neighbors (Gamerman and Lopes 2006). A common use of a Markov random field is for priors that resemble Tikhonov’s regularization, written in the following general form (Kaipio and Somersalo 2004): 

 1 

~ 2 π ðPÞ / exp  γ D P  P 2

 (35)

where ||.|| denotes the L2 norm. The constant γ is a parameter associated with ~ is a reference value for P, which can be taken as a uncertainties in the prior and P null vector without loss of generality. The matrix D is such that each line of D

 ~ PP involves the parameter Pj corresponding to that line and its neighbors, in order to characterize a Markov random field. For cases that P represent point values of a one-dimensional function (such as a function varying in time or in one single spatial coordinate), matrices like those used in Tikhonov’s regularization serve well for this purpose. For example, one may use

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Thermophysical Properties Measurement and Identification

2 6 D¼6 4

1

197

3

1 1

1 ⋱

⋱ 1

7 7 with size ðN  1Þ x N 5

(36a)

1

or 2 6 D¼6 4

1

2 1

1 2 ⋱

3 1 ⋱ 1

⋱ 2

7 7 with size ðN  2Þ x N 5

(36b)

1

Equation 35 can be rewritten as   1 ~ T ~  π ðPÞ / exp  γ PP Z PP 2

(37a)

Z ¼ DT D

(37b)

where

Equation 37a is in a form similar to that of a Gaussian distribution. For this reason, it is also called a Gaussian Markov random field (Gamerman and Lopes 2006) or a Gaussian smoothness prior (Kaipio and Somersalo 2004). By comparing Eq. 37a with the canonical Gaussian multivariate distribution, one can notice that the ~ and γ 1Z1, respectively. mean and the covariance matrix of this prior are given by P Therefore, the Gaussian smoothness prior is expressed as    1=2 1 ~ T ~  exp  γ PP Z PP π ðPÞ ¼ ð2π ÞN=2 γ N=2 Z1  2

(38)

An important remark about this prior is that, with D given by Eqs. (36a, b), its variance is unbounded, since the matrix Z is singular and Z1 does not exist. Densities with unbounded variances are denoted as improper (Kaipio and Somersalo 2004). Another Markov random field prior that gives high probabilities for piecewise regular solutions with sparse gradients is based on the concept of total variation. The total variation (TV) prior satisfies these characteristics, being quite appropriate for spatially varying functions that contain large variations at few boundaries within the domain and with small variations within the regions limited by such boundaries (Kaipio and Somersalo 2004). The TV prior is given by (Kaipio and Somersalo 2004): π ðPÞ / exp½γ TV ðPÞ where

(39)

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TV ðPÞ ¼

N X j¼1

V j ð PÞ

V j ð PÞ ¼

 1 X  lij Pi  Pj  2 i  Nj

(40a; b)

being Nj the set of neighbors to Pj and lij the length of the edge between neighbors. The TV prior is improper, such as the Gaussian smoothness prior. The representation of Eq. 39 in terms of a canonical probability density would require the Ð derivation of an expression for the normalizing constant ℝN π ðPÞ dP, or, at least, practical means for its computation. Although improper priors need to be used with caution, they do not pose difficulties for the application of the Metropolis-Hastings algorithm, since the normalizing constants of such densities are cancelled when α(P| P(t)) is computed with Eq. 24. On the other hand, both the Gaussian smoothness prior and the TV prior involve an additional parameter γ that needs to be specified for the application of MCMC methods. The specification of a value for such parameter can be made by numerical experiments, with simulated experimental data that serve as a reference for the inverse problem under analysis. On the other hand, within the Bayesian framework, if a parameter is not known it shall be regarded as part of the inference problem, leading to the use of hierarchical (hyperprior) models. The parameter γ appearing in the Gaussian smoothness prior given by Eq. 38 can be treated as a hyperparameter, that is, be estimated as part of the inference problem (Kaipio and Somersalo 2004). Consider, for example, the hyperprior density for γ in the form of a Rayleigh distribution (see Eq. 32). Therefore, the posterior distribution, with the Gaussian likelihood given by Eq. 5b, can be written as: (

π ðγ,PjYÞ / γ

ðNþ2Þ=2

  ) T

 1 γ 2 1 1

T 1 ~ Z PP ~  exp  ½Y  TðPÞ W ½Y  TðPÞ  γ P  P 2 2 2 γ0

(41) On the other hand, the parameter γ appearing in the TV prior given by Eq. 39 cannot be treated as a hyperparameter. Such is the case because the normalizing constant of such prior is of difficult calculation and also depends on γ. Therefore, without the computation of the normalizing constant for this case, the effects of γ as a hyperparameter would not be correctly accounted for in the posterior distribution. Output Analysis: The analysis on a single component Pj of the vector of parameters P is considered here, by basically following references (Tan et al. 2006; n o ð1Þ ð2Þ ðnÞ Gamerman and Lopes 2006). Let Pj , Pj , . . . , Pj be a Markov chain for Pj.
 n o ðnÞ ð1Þ ð2Þ ðnÞ A function f Pj from the sample Pj , Pj , . . . , Pj is called a statistic if it does not depend on any other unknown parameters. Some useful statistics are:
 n o ðnÞ ðnÞ ð1Þ ð2Þ ðnÞ ¼ Pj, min ¼ min Pj , Pj , . . . , Pj Minimum value : f Pj

(42a)

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 n o ðnÞ ð nÞ ð1Þ ð2Þ ðnÞ Maximum value : f Pj ¼ Pj, max ¼ max Pj , Pj , . . . , Pj
 n o ðnÞ ðnÞ ð1Þ ð2Þ ðnÞ ¼ P~j ¼ med Pj , Pj , . . . , Pj Median : f Pj n
 1X ðnÞ ðnÞ ðtÞ P Mean : f Pj ¼ Pj ¼ n t¼1 j



 ðnÞ ðnÞ Variance : f Pj ¼ var Pj ¼ Since

(42b) (42c) (42d)

n
 1 X ðnÞ 2 ðtÞ Pj  P j n  1 t¼1

(42e)

n o ð1Þ ð2Þ ðnÞ Pj , Pj , . . . , Pj are realizations of a random variable, a statistic is

itself a random variable as well. A statistic of the sample will be a good representation of a statistic of the population if the sample is a good representation of the population. This certainly depends on the size n and on the n independence ofothe ð1Þ ð2Þ ðnÞ is individuals of the sample. Furthermore, since the sample Pj , Pj , . . . , Pj obtained from a Markov chain, the chain should already have reached equilibrium before statistics can be computed for the solution of the inverse problem. For this reason, states of the Markov chain are discarded before the chain reaches equilibrium, which is called the burn-in period. If m states are needed for the chain to reach equilibrium, the sample n o used for the computation of the statistics is ðmþ1Þ ðmþ2Þ ðnÞ , Pj , . . . , Pj . The index of this sample is changed from t = m + 1, Pj . . ., n to r = 1, . . ., s for simplicity in the notation, where s = n  m is the number of samples used for the computation of the statistics. n o ð1Þ ð2Þ ðsÞ is The mean of the sequence Pj , Pj , . . . , Pj ðsÞ

Pj ¼

s 1X ðr Þ P s r¼1 j

(43) ðr Þ

If the chain is ergodic this mean, based on the chain values Pj , provides a strongly consistent estimate of the mean of the limiting distribution, that is,  ðsÞ Pj ! E Pj as

s!1

(44)

This of the law of large numbers for a Markov chain. n result is the equivalent o ð1Þ ð 2Þ ðsÞ If Pj , Pj , . . . , Pj are independent samples, then the variance of the mean is h i h i var Pðj sÞ ðsÞ var Pj ¼ s

(45)

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h i n o ðsÞ ð1Þ ð2Þ ðsÞ where var Pj is the variance of Pj , Pj , . . . , Pj . On the other hand, since the samples are in general correlated, Eq. 45 is rewritten as h i h i τ var Pðj sÞ ðsÞ var Pj ¼ s

(46)

where τ is the integrated autocorrelation time (IACT), which represents the number of correlated samples between independent samples in the chain n o ð1Þ ð2Þ ðsÞ Pj , Pj , . . . , Pj . Therefore, the effective chain size, which gives the number of independent samples in the chain, is seff = s/τ. n o ð1Þ ð2Þ ðsÞ The autocovariance function of lag k of the chain Pj , Pj , . . . , Pj is defined by: h i ðr Þ ðrþkÞ Cff ðkÞ ¼ cov Pj , Pj

(47)

h i ðr Þ ðr Þ Clearly, the variance of Pj is var Pj ¼ Cff ð0Þ. The normalized autocovariance function of lag k is given by

ρff ðkÞ ¼

Cff ðkÞ Cff ð0Þ

(48)

ðr Þ

so that ρff (0) = 1, which means that Pj is perfectly correlated with itself. The calculation of the normalized autocovariance function is straightforward, since several computational packages have functions available for such a purpose. The integrated autocorrelation time is related to the normalized autocovariance function by

τ ¼1þ2

1 X

ρff ðkÞ

(49)

k¼1

For the calculation of τ, the summation in Eq. 49 needs to be truncated at a finite number of terms s  s. In fact, ρff (k) is expected to tend to zero as k increases, but it will be dominated by noise for large k. For s sufficiently large and for a uniformly ergodic chain, the distribution of h i ðsÞ Pj E½Pj  ðsÞ qffiffiffiffiffiffiffiffiffiffiffiffiffi ffi Pj , where var is given by Eq. 46, tends to a standard Gaussian  ðsÞ var Pj

distribution, with zero mean and unitary standard deviation. One can write

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201

 ðsÞ Pj  E Pj d rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h i ! N ð0, 1Þ as s ! 1 ðsÞ var Pj

(50)

where ðsÞ

Pj

h i h i τvar Pðj sÞ ðsÞ ; var Pj ¼ s

s 1X ðrÞ ¼ P ; s r¼1 j

h i ðsÞ var Pj ¼

s
 1 X ðrÞ 2 ðrÞ P j  Pj s  1 r¼1

(51a; b; c)

n o ð1Þ ð2Þ ðsÞ are the mean of Pj , Pj , . . . , Pj , the variance of this mean and the variance of n o ð1Þ ð2Þ ðsÞ Pj , Pj , . . . , Pj , respectively. The main result of Eq. 50 is that it provides a manner of presenting the solution of the inverse problem in terms of the mean of the parameter Pj, from inference over the rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n o h i ðsÞ ðsÞ ð1Þ ð2Þ ðsÞ Markov chain Pj , Pj , . . . , Pj , as Pj  C var Pj , where C is a constant ðsÞ

that defines the approximate confidence interval of Pj . For a 99% confidence interval, C = 2.576. Another usual manner is to present the solution oof the inverse n ðsÞ ð1Þ ð2Þ ðsÞ problem in terms of Pj and desired quantiles of Pj , Pj , . . . , Pj . The convergence of the Markov chain to an nequilibrium distribution can be o ð1Þ ð2Þ ðnÞ verified by plotting the chains of each parameter Pj , Pj , . . . , Pj , j = 1,. . ., N, and the posterior distribution π posterior(P(t)), t = 1, . . ., n. Geweke (1992) proposed a method for convergence diagnosis based on means computed within different ranges of the Markov chain. Let a

Pj ¼

sa s 1 X 1 X b ðr Þ ðrÞ Pj and Pj ¼ P sa r¼1 sb r¼s j

(52a; b)

where s ¼ s  sb þ 1;

sa ¼ 0:1s; sb ¼ 0:5s
 a b P j  Pj ! 0 Geweke (1992) has demonstrated that n o ð1Þ ð2Þ ðsÞ approaches equilibrium. Pj , P j , . . . , P j

(52a  c) as the chain

It is also a good practice to repeat such procedures for convergence analysis by generating Markov chains from different initial states. A method for inference from multiple chains was developed by Gelman and Rubin (1992).

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3

Applications

3.1

Thermal Conductivity of Highly Orthotropic Materials

In many industrial processes and applications, the functionalization of materials for mechanical and thermal purposes has turned out to be an important requirement. For innovation in the space industry, aeronautics, transport, efficient building design, energy production, etc., the expected performance of materials have very specific targets for the value of thermal and/or mechanical properties, such as orthotropic thermal conductivity components with a very high or low in-plane/transverse thermal conductivity. The thermal characterization of such materials often yields new metrological problems to be resolved by designing some specific characterization method. Moreover, the thermal characterization methods design is also influenced by both the structure and geometry of the available samples. The case of the thermal characterization of highly orthotropic carbon epoxy composites, aimed to be used for a cryogenic tank for satellite launchers (Battaglia et al. 2014), is presented here as an application. The proposed measurement method is derived from the hot disk (Gustavsson et al. 1994). In the usual hot disk method, the probe is inserted between two identical samples which are considered to be isotropic and semi-infinite both in the radial and longitudinal directions, such as shown in Fig. 2. The sensor is made of nickel foil in the form of a bifilar spiral covered on both sides with an insulating layer of Kapton. The thicknesses of the foil and the Kapton layer are 10 and 25 μm, respectively, and the radius of the heating area is 20 mm. The probe is used both as a heat source and temperature sensor. The sample is heated by the probe when a step of constant electrical current is applied to the nickel foil. The electrical resistance of the probe is dependent of temperature through the temperature coefficient of resistance (TCR), which is obtained by calibration as a function of temperature prior to the experiment. Thus, the temperature increase of the probe can be deduced from the electrical resistance measurement during the experiment. It is important to point out that in this case the average temperature of the heating element is obtained instead of some local temperature. Fig. 2 Hot disk probe inserted between two identical semi-infinite samples (@ Hot Disk)

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203

Fig. 3 Experiment for orthotropic with radial symmetry thin sample

The semi-infinite assumption does not hold here, due both to the small thickness of the composite sample and the high thermal conductivity values. The method is modified by coupling the sample, now considered with finite thickness, to a semi-infinite conductive material (copper or brass). It was shown by Ladevie et al. (2000) that coupling the back wall of the sample to a semi-infinite conductive material yields a perfect and simple control of the corresponding boundary condition. This method was applied for the thermal characterization of silica aerogels (Rigacci et al. 2002). Heat transfer in the experiment is thus represented by the sketch shown in Fig. 3. The governing equation is given as !    @T 1 @ @T @2T ρCp ¼ kr r þ kz ; @t r @r @r @ 2Z @T ¼0 @r @T ¼ @z

(

0 0

(55c)

T ðx, y, 0Þ ¼ T 0 ðx, yÞ for t ¼ 0, in 0 < x < Lx , 0 < y < Ly

(55d)

where C(x,y) is the local volumetric heat capacity in Jm3K1, k(x,y) is the local thermal conductivity in Wm1K1, h(x,y) is the combined convective and linearized radiative heat transfer coefficient in Wm2K1 on the top surface, and φw(x, y) is the local heat flux applied to one of the boundaries of the plate in Wm2. The axis z is oriented upward and T(x,y,t) is the averaged temperature in the z direction, such that 1 T ðx, y, tÞ ¼ e

ðe

T  ðx, y, z, tÞdz

(55e)

0

For the results presented here, temperature gradients were neglected across the plate, that is, T  ðx, y, e, tÞ  T ðx, y, tÞ

(55f)

and the resulting partial lumping formulation for the heat conduction problem is two dimensional, with spatially varying thermal conductivity and volumetric heat capacity. For situations where the temperature gradients across the plate cannot be fully neglected, an improved lumped formulation or the separation of in-plane and out-of-plane temperature distributions (Bamford et al. 2009) could be used. In this section, the solution of an inverse problem involving the simultaneous identification of both the thermophysical properties and the applied heat flux is proposed, by using experimental data provided by an infrared camera. Such

4

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211

measurement technique is quite powerful because it can provide accurate nonintrusive measurements, with fine spatial resolutions and at large frequencies. The inverse problem is solved by using the nodal strategy advanced in references (Fudym et al. 2007; Fonseca et al. 2010). For the application of the nodal strategy, Eq. 55a is written in the following nonconservative form, where the temperature gradient was neglected across the plate (see Eq. 55f):   @T 1 @kðx, yÞ @T @kðx, yÞ @T 2 ¼ aðx, yÞ∇ T þ þ @t Cðx, yÞ @x @x @y @y  H ðx, yÞðT  T 1 Þ þ Gðx, yÞ

(56a)

where aðx, yÞ ¼

kðx, yÞ , Cðx, yÞ

H ðx, yÞ ¼

hðx, yÞ , eCðx, yÞ

Gðx, yÞ ¼

φw ðx, yÞ eCðx, yÞ

(56b  d)

An explicit discretization of Eq. 56a using finite differences results in:
 n n x n y n Y nþ1 i, j ¼ Li, j ai, j þ Dxi, j δi, j þ Dyi, j δi, j  Δt T i, j  T 1 H i, j þ ΔtG i, j

(57)

where the subscripts (i,j) denote the finite-difference node at xi = iΔx, i = 1. . .ni, and yj=jΔy, j = 1. . .nj, and the superscript n denotes the time tn= nt, n = 0. . .(nt –1). The other quantities appearing in Eq. 57 are given by: nþ1 n Y nþ1 i, j ¼ T i, j  T i, j

Lni, j

¼ Δt

T ni1, j  2T ni, j þ T niþ1, j ðΔxÞ2

þ

(58a)

T ni, j1  2T ni, j þ T ni, jþ1 ðΔyÞ2

 Δt
n T iþ1, j  T ni1, j 2Δx  Δt
n T i, jþ1  T ni, j1 Dyni, j ¼ 2Δy

Dxni, j ¼

! (58b)

(58c) (58d)

δxi, j ¼

1 @k Cðx, yÞ @x

(58e)

δyi, j ¼

1 @k Cðx, yÞ @y

(58f)

Equation 58a defines the forward temperature difference in time, Eq. 58b approximates the Laplacian of temperature, while Eqs. (58c, d) are the temperature gradients, at time tn and node (i,j). Writing Eq. 57 for a given node (i,j) and all time-steps yields

212

H. R. B. Orlande and O. Fudym

Yij 5 Jij Pij

(59)

where 2

L1i, j

6 6 2 6 Jij ¼ 6 Li, j 6⋮ 4 Lni,t j

Dx1i, j

Dy1i, j

Dx2i, j ⋮ Dxni,t j

Dy2i, j ⋮ Dyni,t j


 Δt T 1i, j  T 1
 Δt T 2i, j  T 1
⋮  Δt T ni,t j  T 1

3 2 3 ai , j Y 1i, j x 6 δi, j 7 6 y 7 6 Y2 7 i , j 7 6 7 and Yij ¼ 4 Pij ¼ 6 6 δi, j 7 ⋮5 4 H i, j 5 Y ni,t j Gi, j

Δt

3

7 7 Δt 7 7 ⋮7 5 Δt

(60a)

2

(60b; c)

The vector Pi,j at node (i,j) such as defined by Eq. 60c contains the following parameters:ai, j, which is the local thermal diffusivity in m2s1; δxi, j and δyi, j, which are the local thermal conductivity gradients along the x and y directions, respectively, divided by the heat capacity; Hi, j, which is the local heat transfer coefficient divided by the local heat capacity and thickness of the plate; and Gi,j , which is the local heat flux divided by the local heat capacity and thickness of the plate, in Ks1. The vector Pi,j at node (i,j) contains five parameters involving only four independent quantities: k, C, h, and φw. All the five parameters appearing in Eq. 60c are divided by C. Hence, even if the five parameters in vector Pi,j could be estimated simultaneously, it would not be possible to retrieve from their values the thermal conductivity and volumetric heat capacity of each node (i,j). For heterogeneous media formed by adjoining different homogeneous materials with sharp interfaces, the parameters relative to the thermal conductivity gradients are nonzero only at these interfaces and zero anywhere else. Hence, for such type of material, the estimation of the thermal conductivity gradients can be avoided. Moreover, the sensitivity coefficients with respect to the heat transfer coefficient are very small for the cases of interest for this work, so that this parameter can also be neglected in the formulation of the inverse problem (Fudym et al. 2007). It then turns out that, at each pixel (i,j), the two parameters a and G are linearly independent. The sensitivity matrix defined in Eq. 60a is a function of the temperature field. As a result, the sensitivity matrix depends on the unknown parameters and the estimation problem is nonlinear. In this work a predictive error model was used, such as illustrated by Fig. 10. With the spatial resolution and frequency of measurements made available by infrared cameras, the sensitivity matrix can be approximately computed with the measurements (Orlande et al. 2011). Therefore, Eq. 59 is used as a predictor and the resulting local estimation problem is linear. Local linear estimation problems can be solved for each pixel made available by the spatial resolution of the infrared camera, or by a spatial filtering combining several pixels if desired.

4

Thermophysical Properties Measurement and Identification

System or measurements

213

ym(t)

input

+ − Predictive model

e(t)

Y(t)

Fig. 10 Predictive error model (Massard et al. 2014)

With the proposed approach, the computation of the response vector Yij with Eq. 59 becomes very fast, which is a quite desirable feature for the inverse problem solution. On the other hand, since the measurements are directly used for the computation of the sensitivity matrix, maximum likelihood estimators, like the ordinary least squares, becomes biased (Orlande et al. 2011). On the other hand, in the nodal strategy, the sensitivity matrix is approximately computed with the measurements. Therefore, for the implementation of the Metropolis-Hastings algorithm the uncertainties in the computation of the sensitivity matrix need to be taken into account. By approximating P and J as independent random variables, the sought posterior probability density is then given by π ðP, JjYÞ / π ðYjP, JÞπ ðPÞπ ðJÞ

(61)

where π(J) is the a priori distribution for the sensitivity matrix J. Such prior information can be straightforwardly computed based on the uncertainties of the measurements, which were assumed above as Gaussian, by using classical uncertainty propagation techniques. In order to challenge the present inverse problem solution approach, uniform prior distributions were used. Experimental data was obtained with a SC7600 Flir infrared camera for a homogeneous plate made of epoxy, with the thickness of 0.0011 m and cross section dimensions of 0.08 0.04 m, such as shown in the paper by Fonseca et al. 2014. A circular electrical resistance was used for heating the sample, with 25 Ω, 23.5 mm in diameter, electrically insulated with Kapton tape. The resistance was connected to a voltage source and positioned at the bottom of the plate. The power dissipated by the electrical resistance was 1.6  0.03 W for the voltage of 10 V, resulting in G = 1.93 Ks1. The prior distribution for the thermal diffusivity was assumed to be uniform in the range (8 108, 9 107) m2s1, while for Gi,j it was assumed to be uniform in the range (0, 5) Ks1. The Markov chain relative to each pixel was generated with 6000 states, while the first 1000 were discarded in the computation of the statistics of the marginal distribution of each parameter. The acceptance rate of the Metropolis-Hastings algorithm was found to be around 60%, with Gaussian proposal distributions. Based on numerical experiments, the relative standard-deviation with respect to the current state of the Markov Chain in the Gaussian proposal distribution was chosen to be 2% for aij and Gi,j, and 0.2% for the sensitivity matrix.

214 Fig. 11 Posterior distribution obtained for the thermal diffusivity at one point in a homogenoeus plate (Fonseca et al. 2014)

H. R. B. Orlande and O. Fudym Histogram of thermal diffusivity at i = 30 and j = 90 1200

1000

800

600

400

200

0

1

2

3

4

5 x 10−7

Figure 11 presents the posterior marginal distribution retrieved for the thermal diffusivity at a point in a homogeneous plate. This histogram follows a Gaussian distribution as expected, since the likelihood and the prior distributions for the sensitivity matrix are Gaussian and the prior distributions for the parameters are uniform. The Gaussian distribution for this parameter is centered around the reference value for the thermal diffusivity, which was measured independently with the Flash method. The heat flux distribution imposed by the electrical resistance and retrieved with the MCMC method is shown by Fig. 12. The qualitative analysis of this figure reveals that the MCMC method is capable of accurately recovering the spatial distribution of the heat flux parameter imposed by the electrical resistance tracks at the measurement scale used, where the sizes of the pixels were of Δx = Δy = 76.9 μm. Figure 12 shows the powerful capabilities of the MCMC method used together with the nodal approach to estimate local heat flux values at micro-scales, although no special analysis of the image provided by the infrared camera was used in the inverse problem and the local values of the parameters were estimated without taking into account any information about the values estimated at the neighboring pixels. The estimated values of the local heat flux parameter G are also in excellent agreement with the reference values obtained from the electrical power dissipated by the resistance.

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Fig. 12 Local heat flux parameter G estimation from MCMC (Fonseca et al. 2014)

4

Conclusions

Most of nowadays methods for the thermal characterization of materials in terms of their thermophysical properties rely on indirect measurements, where a mathematical model for the physical problem has to be statistically matched to the data obtained in some experimental process, through the solution of an inverse heat transfer problem. While in some cases the parameter estimation process is rather simple, other cases need to deal with the estimation of a rather large number of parameters that might represent, for example, local values of spatially varying thermophysical properties. Hence, the ill-posed character of the inverse problem becomes evident and regularization techniques need to be used to obtain solutions of an approximate well-posed problem. While classical methods for the solution of inverse problems have been successfully applied for different cases of practical interest, methods within the Bayesian framework of statistics have recently become very popular. In this framework, prior information about the unknown parameters, which is available before the experiments are conducted, can be taken into account in the inverse analysis via Bayes’ theorem. The prior information needs to be coded in the form of statistical distributions and provides regularization for the solution of the inverse problem, which is recast in the form of statistical inference from the posterior probability density. This chapter was focused on the use of Markov Chain Monte Carlo (MCMC) methods and three illustrative examples were presented, namely: the measurement of thermal conductivity components of highly orthotropic materials; a

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thermoreflectometry experiment used for the thermal characterization of metallic thin layers deposited on a silicon substrate; and the simultaneous estimation of local thermal diffusivity and heat flux from infrared camera measurements. In special, for the last application the use of a predictive error model and a nodal approach allowed the reduction of the large computational times associated with the MCMC methods, by solving linear local inverse problems for each pixel of the infrared camera.

5

Cross-References

▶ Analytical Methods in Heat Transfer ▶ Inverse Problems in Radiative Transfer ▶ Macroscopic Heat Conduction Formulation ▶ Microsensors for Determination of Thermal Conductivity of Biomaterials and Solutions ▶ Thermal Properties of Porcine and Human Biological Systems

Acknowledgments The support provided by CNRS (France), CNPq (Brazil), CAPES (Brazil), and FAPERJ (Brazil – State of Rio de Janeiro) is greatly appreciated.

References Alifanov OM (1994) Inverse heat transfer problems. Springer-Verlag, New York Alifanov OM, Artyukhin E, Rumyantsev A (1995) Extreme methods for solving ill-posed problems with applications to inverse heat transfer problems. Begell House, New York ASTM Standard C177 (2004) Standard test method for steady-state heat flux measurements and thermal transmission properties by means of the guarded-hot-plate apparatus. ASTM, West Conshohocken ASTM Standard E1461-01 (2001) Standard test method for thermal diffusivity by the flash method. ASTM, West Conshohocken Bamford M, Batsale J-C, Fudym O (2009) Nodal and modal strategies for longitudinal thermal diffusivity profile estimation. Application to the non destructive evaluation of SiC/SiC composites under uniaxial tensile tests. Infrared Phys Technol 52:1–13. https://doi.org/10.1016/j. infrared.2008.01.002 Battaglia J-L, Schick V, Rossignol C, Fudym O, Orlande H, Affonso P (2012) Global estimation of thermal parameters from a picoseconds thermoreflectometry experiment. Int J Thermal Sci 57:17–24 Battaglia J-L, Saboul M, Pailhes J, Saci A, Kusiak A, Fudym O (2014) Carbon epoxy composites thermal conductivity at 77 K and 300 K. J Appl Phys 115:223516. https://doi.org/10.1063/ 1.4882300 Bayes T (1763) An essay towards solving a problem in the doctrine of chances, by the late Rv. Mr. Bayes, F.R.S. Communicated by Mr. Price in a letter to John Cannon, M. A. and F.R.S. Phil Trans 53:370–418., 1763. https://doi.org/10.1098/rstl.1763.0053 Beck JV, Arnold KJ (1977) Parameter stimation in engineering and science. Wiley, New York Beck JV, Blackwell B, St. Clair CR (1985) Inverse heat conduction: ill-posed problems. Wiley, New York

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Nissinen A, Kolehmainen V, Kaipio J (2011a) Compensation of modeling errors due to unknown boundary domain in electrical impedance tomography. IEEE Trans Med Im 30:231–242 Nissinen A, Kolehmainen V, Kaipio J (2011b) Reconstruction of domain boundary and conductivity in electrical impedance tomography using the approximation error approach. Int J Uncertainty Quant 1:203–222 Orlande, H. R. B. (2012), Inverse problems in heat transfer: new trends on solution methodologies and applications, J Heat Transfer, v.134, p.031011-1. Orlande H, Fudym F, Maillet D, Cotta R (2011) Thermal measurements and inverse techniques. CRC Press, Boca Raton Orlande HRB, Dulikravich GS, Neumayer M, Watzenig D, Colaço M (2014) Accelerated Bayesian inference for the estimation of spatially varying heat flux in a heat conduction problem. Num Heat Transfer, Part A Appl 65:1–25 Ozisik MN, Orlande HRB (2000) Inverse heat transfer: fundamentals and applications. Taylor and Francis, New York Parker W, Jenkins R, Butler C, Abbott G (1961) Flash method of determining thermal diffusivity, heat capacity and thermal conductivity. J Appl Phys 32(9):1679–1684 Rigacci B, Ladevie H, Sallée B, Chevalier P, Achard OF (2002) Measurements of comparative apparent thermal conductivity of large monolithic silica aerogels for transparent superinsulation application. High Temp High Press 34:549–559 Sabatier PC (1978) Applied inverse problems. Springer Verlag, Hamburg Silver N (2012) The signal and the noise. Penguin Press, New York Tan S, Fox C, Nicholls G (2006) Inverse problems, Course notes for physics, vol 707. University of Auckland, New Zealand Tikhonov AN, Arsenin VY (1977) Solution of Ill-posed problems. Winston & Sons, Washington, DC Vogel C (2002) Computational methods for inverse problems. SIAM, New York Winkler R (2003) An introduction to Bayesian inference and decision. Probabilistic Publishing, Gainsville Woodbury K (2002) Inverse engineering handbook. CRC Press, Boca Raton

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Contents 1 2 3 4

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thermal Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Modeling and Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Typical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Combined Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Complex Transport Phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Multiple Scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Basic Design Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Inverse Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Feasibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Acceptable Design Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Additional Design Aspects and Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Knowledge-Based Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Concurrent Experimentation and Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Uncertainty- and Reliability-Based Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Basic Aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Lagrange Multiplier Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Search Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Response Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5 Multi-Objective Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Y. Jaluria (*) Department of Mechanical and Aerospace Engineering, Rutgers, the State University of New Jersey, Piscataway, NJ, USA e-mail: [email protected] # Springer International Publishing AG, part of Springer Nature 2018 F. A. Kulacki (ed.), Handbook of Thermal Science and Engineering, https://doi.org/10.1007/978-3-319-26695-4_67

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8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272 9 Cross-References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274

Abstract

This chapter considers the design of thermal systems, focusing on simulation, feasible design, and optimization. Though most thermal systems have been modeled and simulated extensively, the results obtained have, in many cases, not been used to design and optimize the process. This chapter reviews the basic concepts of design of thermal systems, based on simulation as well as experimentation, and discusses strategies that may be employed to design and optimize the system. Many complexities, such as strong property variations, complicated domains, conjugate mechanisms, chemical reactions, and intricate boundary conditions, are often encountered in practical thermal processes and systems. The basic approaches that may be used to accurately simulate these systems are outlined. The link between the process and the resulting output is critical, particularly in areas like manufacturing. Thus it is important to couple the modeling and experimental data with the performance and design of the system. Optimization in terms of the operating conditions, as well as of the system hardware, is needed to minimize costs and enhance product quality and system performance. Different optimization strategies that are currently available and that may be used for thermal systems are outlined. Several practical processes from a wide range of applications, such as manufacturing, thermal management of electronic systems, energy, environment, and security, are considered in greater detail to illustrate these approaches, as well as to present typical simulation and design results. Validation of the mathematical and numerical model is particularly important and is discussed in terms of existing results, as well as new experimental data. Similarly, feasibility of the process, choice of operating conditions from inverse solutions, knowledge-based design, combined experimental and numerical inputs for design, sensitivity, uncertainty, and other important aspects are presented. Most thermal systems have more than one objective of interest, leading to multi-objective optimization, which is briefly presented. The current state of the art and future needs in design of thermal systems are discussed.

1

Introduction

Processes and systems that are largely governed by the principles of heat transfer, thermodynamics, and fluid mechanics are frequently encountered. These are often referred to as thermal systems. These systems arise in a wide range of applications such as manufacturing, power generation, transportation, environmental control, and thermal management of electronic equipment. Figure 1 shows schematically many of the applications of thermal systems, indicating the extensive range of important areas where thermal systems are of particular interest. Many systems involve

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Fig. 1 A sampling of applications where thermal systems are of interest

considerable amounts of energy transfer and also have a significant effect on the environment. In addition, many applications, such as those involved in the fabrication of electronic and optical components, demand a high level of product quality and purity. Therefore, it is important to design the relevant systems and processes in order to achieve the desired output characteristics. The designs need to be optimized in order to reduce the energy consumption, enhance product quality, and reduce the environmental impact. It is also desirable to use the existing or changing environmental conditions to adjust the operating parameters in order to reduce energy consumption, for instance, in data center cooling. However, even though many thermal processes and systems have been modeled and simulated extensively, with detailed results on the effects of operating conditions and governing parameters reported in the literature, the logical extension of these results to system design and optimization has not been undertaken in many cases (Stoecker 1989; Bejan et al. 1996; Jaluria 2008). Thus, this chapter focuses on the design of thermal systems, considering simulation, acceptable design, and finally optimal design. Examples of thermal systems are taken from the various areas indicated in Fig. 1 and mentioned in the preceding. However, the basic considerations in system design and optimization are similar for most systems and form the focus of this chapter. The main steps in system design may be specified as: 1. 2. 3. 4. 5. 6.

Formulation of the design problem Conceptual design Modeling and simulation Choice of governing parameters and operating conditions Feasible or acceptable design Design optimization

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The design process starts with the formulation of the problem in terms of requirements and constraints. Fixed or given quantities must also be specified. The variables include the system hardware, such as dimensions, geometry, components, and materials, as well as the operating conditions, which include relatively easy to adjust quantities like flow rate, heat input, inlet temperature, speed, and pressure. For instance, in electronic equipment cooling, the requirement is that the temperature level be kept below a specified value for satisfactory operation (Joshi and Kumar 2012). In manufacturing, the desired characteristics of the product are specified as requirements. The constraints on space available, pressure, temperature, weight, cost, etc., must also be given. Constraints due to environmental impact of the system must also be included in many cases. Constraints also arise due to conservation principles and other basic laws, such as the second law of thermodynamics. The quantities that are given or fixed, such as materials, electronic device, available energy source, location, and dimensions, must be given, along with quantities that may be varied to obtain an acceptable design. Once all these aspects are specified, including their possible ranges, the design problem is formulated, and one can pass on to the next step of the physical design of the system. The next step is conceptual design, which is often the most difficult part of the design process. The basic concept may be based on existing or similar systems, or it may be a new or an innovative design to meet the given problem. Usually, several concepts are considered, and finally one or more designs that have the strongest chance of success are chosen. Common approaches and examples are given by Jaluria (2008). The conceptual design is then subjected to modeling and simulation to study its behavior in detail and evaluate if it would meet the design problem statement. The initial design is varied, and iterative redesigns are employed till an acceptable design that satisfies the problem is obtained. Of course, there are cases where an acceptable design is not obtained and other conceptual designs may have to be considered. Generally, a domain of acceptable designs is obtained from which readily available components, materials, and dimensions may be selected for an appropriate acceptable design. However, an acceptable design would not usually be the optimal design. Thus, optimization of the design may be undertaken to maximize a chosen quantity, such as output, product quality, and efficiency, or minimize an item, such as cost, environmental effect, and energy consumption. The quantity chosen for maximization or minimization is known as the objective function. In several cases, the optimization focuses on a single criterion, such as product quality or efficiency. However, in most thermal systems, several objective functions are of interest. In some cases, the different objective functions, such as production rate, heat transfer rate, pressure increase, and defects in the product, may be combined into a single objective function, which is then optimized. In other cases, optimization is carried out with different objective functions, yielding a Pareto front, which gives the nondominant solutions and which may then be used, with trade-offs, to achieve the desired optimal design (Deb 2001). Uncertainties that arise in material properties and in boundary conditions are also important in design and are considered. Different strategies and results are presented here. Optimization is considered in terms of both the operating conditions and the design, or hardware, parameters.

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Therefore, this chapter discusses the basic aspects in the design of thermal systems. The various steps that are involved are outlined. Since simulation forms the basis for most design problems, modeling and simulation are discussed in detail, along with typical results for systems taken from different application areas. The important ingredients and steps in developing an acceptable design are presented. This is followed by a detailed presentation on design optimization. The current and future trends and needs in this area are discussed.

2

Thermal Systems

As outlined in the preceding section and shown in Fig. 1, thermal systems and processes are of interest in many important applications, ranging from energy, manufacturing, and transportation to aerospace and heating or cooling. A few typical practical thermal systems are shown in Fig. 2. This figure shows sketches of a data center, which consists of racks of servers made up of electronic computational processing units (CPU), storage units and computer room air conditioning units (CRAC), a power plant heat rejection system to cool the condensers, a furnace for optical fiber drawing, and a chemical vapor deposition (CVD) reactor for thin film deposition. Similarly, many other thermal systems drawn for different application areas may be considered. These systems typically involve many complexities. For instance, data center cooling often encompasses complicated flow patterns, including turbulence, flow through porous media, and large variations in length scales that require multiscale modeling. Most data centers also require substantial energy consumption for cooling. Heat rejection systems are complicated because of typically large flow rates and coupling with the environment. Material properties of glass in optical fiber drawing are strong functions of temperature, and combined modes of radiation, conduction, and convection arise at various stages in the process (Paek 1999). CVD processes involve chemical kinetics, which vary strongly with temperature and concentration, and the modeling of the chemical reactions is often very complicated (Mahajan 1996). In most cases, the boundary conditions are generally complicated, and combined transport mechanisms are of interest. The governing equations are strongly coupled, and large nonlinear effects arise that make modeling and simulation quite involved. Approximations and idealizations are often used to simplify these equations. Thus, many challenges have to be overcome to accurately model practical thermal systems. Most practical processes are too complicated to be investigated by analytical methods alone, and numerical simulation and experimentation are needed to understand the underlying basic phenomena and obtain inputs that can be used for design and optimization. Complex, coupled, transport mechanisms and interacting subsystems that constitute the overall system are generally encountered. Mathematical models of the processes and systems are developed, followed by numerical modeling and simulation. The models are validated by means of available analytical, benchmark, and experimental results; see, for instance, de Vahl Davis (1983) and Roache (1998). Then, the numerical simulation is used to provide the extensive numerical

fiber

z,va

ra,ua

RF

necking region

2Rf

V0 R0

Vf

z,v

radiation + convection transport

preform

r,u

Under Floor Plenum

Server Racks

Va

L

L0

Lf

o x y

z

Ti+ΔT

d

Power plant heat rejection

b

(2) Adsorption

(1) Convection

CRAC Units

O2

Ti

(3) Reaction

SIH4

Intake

Outfall

heater

substrate

X

(6) Crystal film growth

(7) Exhaust

(5) Surface diffusion

(4) Desorption

Stratified Water body

T

Fig. 2 Examples of typical thermal processes and systems: (a) A data center, (b) heat rejection system to cool the condensers of a power plant, (c) optical fiber drawing, (d) chemical vapor deposition (CVD) system for thin film fabrication

Va

furnace

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data needed for design, control, and optimization. Limited experimental data are also often available from existing processes and specially designed experimental systems. These can be used for validation, physical insight, and development of realistic models. Frequently, the experimental data are curve fitted to obtain empirical equations that may be used with the results from analytical and numerical models for system design. The chemical vapor deposition reactor, mentioned earlier and shown in Fig. 2d, is an important thermal system, which is used extensively to obtain thin films for electronic and optical applications. Materials such as silicon, silicon carbide, gallium nitride, and gallium arsenide are employed in many applications like light-emitting diodes, solar cells, optoelectronic, high-power and high-frequency devices, and other electronic components. The systems typically involve toxic gases and substantial energy input. It is important to treat these gases before they are discharged into the environment and to determine the amount and composition of the discharges so that these may be reduced in the optimization of the system. Similarly, optimization is directed at enhancing deposition rate, while maintaining the desired quality of the product. Another interesting problem that has been mentioned is the cooling of data centers. The local environmental conditions may be employed to minimize the energy consumption. Data centers located in cooler regions can be used more efficiently than those in warmer regions, and the load may be distributed between different data centers to reduce the overall consumption and the resulting environmental impact. Some typical cases are considered later to demonstrate how this may be achieved if the changing local environmental conditions are available for different data center locations. Substantial reduction in energy consumption for the cooling of these systems is achieved. As mentioned in the preceding, the heat transfer and fluid flow considerations determine the properties of the final product, such as in crystal drawn from silicon melt, surface coating, and gel formation from the chemical conversion of a biopolymer. Thus, the desired quality and characteristics of the fabricated material or device must be part of the formulation. However, while the effort is often directed at the output characteristics, it must be noted that energy consumption, as well as the environmental effect, is quite substantial in most of these applications. Thus, the design and optimization must consider product quality on the one hand and productivity on the other, since the total energy consumption is strongly dependent on the rate at which the material and devices are fabricated. Similarly, the effect on the environment due to waste material and thermal energy must be considered.

3

Modeling and Simulation

Modeling of thermal systems involves obtaining a simplified analytical, numerical, or experimental representation of a system so that a study of the model may be used to predict the response and behavior of the given system. Simulation refers to the solution obtained when the model is subjected to changes in the governing

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parameters and operating conditions to determine the behavior and characteristics of the actual system. Though analysis may be used for a few components or for idealized circumstances, numerical modeling is essential for most practical systems. However, mathematical models form the basis for the numerical simulation and also help with the interpretation and consolidation of experimental data. Various simplifications, idealizations, and assumptions, such as axisymmetry; steady-state, lumped components; fully mixed flow; smooth surface; and uniform temperature or heat flux, are used to make the model amenable to analytical or numerical methods. A systematic approach may be used, employing characteristic dimensions, scale analysis, and simplifying assumptions, to develop a satisfactory model, as discussed in detail by Jaluria (2008). Once such a model is developed, analysis may be used in a few limited cases, otherwise numerical model is developed and employed to obtain the results needed for design. Experimental data obtained from physical models and from existing or similar systems are also often cast in terms of equations through curve fitting and are used along with simulation results for the design process. The numerical modeling of the transport phenomena involved in most practical thermal systems is complicated due to material characteristics, intricate computational regions, complex boundary conditions, and additional mechanisms such as surface tension, free surface flow, and phase change. In this section, the basic approach to the numerical simulation of such processes is discussed. Characteristic numerical results are presented and discussed in the following section for a few important practical processes, such as polymer extrusion, optical fiber drawing, enclosure fires, and cooling of electronic systems. Validation of the numerical model is a crucial aspect and is discussed in terms of comparisons with existing analytical, numerical, and experimental results. It is obviously important to obtain an accurate and valid numerical simulation in order to use the results to improve existing thermal systems and design new ones. The governing equations for thermal processes and systems are based on the conservation laws for mass, momentum, and energy. A radiative source term arises for nonopaque materials like glass, which emits and absorbs energy as function of the wavelength λ. Also, viscous dissipation is important in the flow of highly viscous materials like polymers and glass. The general equations may be written as Dρ þ ρ∇  V ¼ 0 Dt

(1)

DV ¼Fþ∇τ Dt

(2)

ρ ρ Cp

DT Dp ¼ ∇  ðk∇TÞ þ Q_ þ β T þ μΦ Dt Dt

(3)

where ρ is the density, Cp the specific heat at constant pressure, k the thermal conductivity, β the coefficient of volumetric thermal expansion, Q_ a thermal energy source per unit volume, T the temperature, t the time, p the pressure, and F the body force per unit volume. Here, D/Dt is the substantial or particle derivative, given in

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terms of the local derivatives in the flow field as D=Dt ¼ @=@t þ V  ∇. The stress tensor τ in Eq. 2 can be written in terms of the velocity V if the fluid characteristics are known. For instance, if the dynamic viscosity μ is taken as constant for a Newtonian fluid, the relationship between the shear stresses and the shear rates, given by Stokes, are employed to yield ρ

 DV μ
¼ F  ∇p þ μ∇2 V þ ∇ ∇  V Dt 3

(4)

The viscous dissipation term μΦ in Eq. 3 represents the irreversible part of the energy transfer due to the shear stress. It acts as a thermal source and is important in many practical processes such as polymer extrusion (Kokini et al. 1992; Jaluria 1996). For a Cartesian coordinate system, Φ is given by the expression "

 2  2 #     @v @w @v @u 2 @w @v 2 Φ¼2 þ þ þ þ þ þ @y @x @x @y @y @z  2
2 @u @w þ þ  ∇V @z @x @u @x

2

(5)

where u, v, and w are the velocity components in x, y, and z coordinate directions, respectively. Thus Φ is always positive and has the same effect as a volumetric heat source in the flow. Similarly, expressions for other coordinate systems may be obtained. Energy can also be released or absorbed in chemical reactions, as is the case in combustion, chemical vapor deposition, and thermal processing of reactive polymers. The equations for the chemical reactions and species have to be solved, and accurate information on the chemical kinetics is critical to the modeling of the process. Material properties are crucial to the accuracy of any numerical simulation. However, reliable and appropriate property data are often not available, and one has to depend on the available information. Very often, the data are available at conditions that are different from those for the actual process, particularly in terms of temperature. This limits the dependability and usefulness of the simulation. For example, optical fibers are usually manufactured by heating a specially fabricated silica glass preform in a cylindrical furnace to a value above its softening point Tmelt, which is around 1900 K for pure silica, and pulling it into a fiber (Li 1985; Paek 1999). An axial tension is applied at the end of the preform so that the preform is drawn into a fiber through a neck-down region, as shown in Fig. 2c. The neck-down profile depends upon the drawing conditions and the physical properties. For glass, the material properties, particularly viscosity, are strong functions of the temperature T. An equation based on the curve fit of available data for kinematic viscosity ν is written for silica, in S.I. units, as 

  Tmelt v ¼ 4545:45 exp 32 1 T

(6)

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Thus, a strong, exponential, variation of ν with temperature is obtained. Variations in all the other relevant properties of glass need to be considered as well, though the variation is not as strong as that of viscosity. Available numerical models can simulate such strong variations in the properties. However, most properties are available at only a few temperatures, particularly at room temperature. Similarly, various dopants are used to change the optical properties of the fiber. But there is lack of accurate data on the effect of these dopants on the material properties, particularly radiation properties. The radiative source term for the glass preform/fiber is nonzero because it emits and absorbs energy. The variation of the absorption coefficient with wavelength λ is important, and, again, there is lack of data for different glass compositions and over the different ranges of interest. However, the available information can be used to approximate the properties in terms of bands with constant absorption over each band. A two- or three-band absorption coefficient distribution can be effectively used (Chen and Jaluria 2009). Similarly, the transport processes in polymer processing or polymer coating involve large material property changes. Since the properties may vary with temperature and species concentration, the flow is coupled with the heat and mass transfer. Also, most of these materials are non-Newtonian, and the viscosity varies with the shear rate and thus with the flow, making the problem even more complicated. The fluid viscosity is often taken as  μ ¼ μo ð_γ=γ_ o Þn1 expðb=TÞ

(7)

where γ_ is the total strain rate, b is the temperature coefficient of viscosity, subscript o indicates reference conditions, and n is the power-law index of the fluid. The material is thus treated as a generalized Newtonian fluid (Tadmor and Gogos 1979). Other rheological models may also be used, depending on the fluid. For chemically reactive materials like food, the dependence on chemical species, such as moisture in food, and on changes in the microstructure due to chemical reactions must also be included. Then the energy transport due to chemical reactions and the species equations must also be solved, making the simulation very complicated and time consuming. Again, the property data are limited and may change from one batch to the next or with time. Therefore, in many cases, the properties are measured and provided as accurate inputs to the numerical model. Similar consideration arises for CVD systems. In most practical problems, the computational domain is very complicated. Various approaches, such as transformations and finite element methods, have been used for numerical modeling. For example, in single-screw extrusion of polymers, a rotating screw is located in a cylindrical barrel, with inflow of material at one end and a die for the outflow at the other, as shown in Fig. 3. The geometry is complicated and the rotation of the screw makes the simulation quite involved. However, by locating the coordinate system on the rotating screw and neglecting curvature effects, a steady flow in a channel with the barrel moving at the pitch angle is obtained, as shown in the figure. For a twin-screw extruder, there are two

5

Design of Thermal Systems

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Fig. 3 (a) Sketch of a single-screw extruder, (b) coordinate system with axes located on the rotating screw and curvature effects neglected, (c) screw channel and computational domain

co-rotating or counterrotating screws. Again, similar simplifications are employed, yielding two channel flows and a mixing region (Sastrohartono and Kwon 1990). Another important consideration is the accurate simulation of the boundary conditions since the results are strongly affected by the transport at the boundaries. For instance, the neck-down profile in optical fiber drawing is unknown and results from various forces acting on the fiber. The analysis must ensure that the radial and axial force balances are satisfied. The force imbalance is used to generate an iterative process to determine the profile. Once the profile is obtained, the transport in the fiber can be computed. The feasibility of the process is determined by the existence of a physically acceptable and stable neck-down profile. Similarly, in the screwextrusion of polymers, conjugate conditions due to conduction in the barrel and in the screw must be included at the channel boundaries. Many other such challenges have to be overcome to obtain an accurate simulation of the thermal system. Multiscale analysis is needed to determine defects in

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manufactured products and in simulating devices in electronic systems. Additional mechanisms like surface tension become important at small length scales and must be included. Turbulent flow has to be modeled in environmental flows, fires, data centers, and other large length scale processes and corresponding systems (Jaluria 2013). A wide variety of numerical methods is available in the literature for solving the relevant governing equations, along with the appropriate boundary conditions. For two-dimensional and axisymmetric problems, the pressure may be eliminated to obtain the vorticity equation. The stream function, vorticity, and energy equations can then be solved using a nonuniform grid, with finer grids located in regions where large gradients are expected. For three-dimensional problems, the primitive variables are used more efficiently (Jaluria and Torrance 2003; Minkowycz et al. 2006). The strong variation of viscosity with temperature in optical fiber drawing makes it necessary to employ very fine grids, linearization, decoupling of the equations, and iterative procedures, using different numerical techniques. Even a difference of a few degrees in temperature near the softening point of glass can cause substantial change in viscosity and thus in the flow field and the neck-down profile. Similar considerations apply for polymer extrusion, combustion, fires, CVD, and other thermal processes where strong property changes may occur. Before moving on to the presentation of simulation results for a few thermal systems, it is worth reiterating that these results, along with those from analysis and experimentation, are the basis for design and optimization. Thus, it is critically important to obtain accurate and validated results for wide ranges of the design parameters and operating conditions so that an acceptable design may be obtained. Various aspects of simulation are also discussed in the next section, keeping the main objective of system design in mind.

4

Simulation Results

Numerical simulation results, along with available analytical and experimental results, form the basis for the design of thermal systems. Though most of the inputs are obtained through simulation, both analytical results and experimental data, whenever available, are used for validating the model, as well as providing important physical insight. Also, in several circumstances, the numerical modeling is complicated, the basic processes may not be well understood, or the boundary conditions are not explicitly and accurately known. These include, for instance, turbulent and transitional flows, free surfaces, openings, and dynamic contact angles. In such cases, experiments may provide the needed inputs more easily and accurately. Frequently, data curve fitting is used to describe the behavior, for instance, the characteristics of a component like a pump or compressor. Approximations and simplifications may allow analysis to be used in some cases to supplement the numerical results. The transport processes in a wide range of basic and applied

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problems have been investigated and presented in the literature; see, for example, Poulikakos (1996) and Jaluria (2001, 2003) for simulations of materials processing systems. A few typical results are presented here to illustrate the basic considerations and approaches in numerical simulation.

4.1

Typical Results

An important practical thermal system that has been mentioned earlier is the screw extruder for polymers like plastics and food materials. Figure 3 gave a sketch of the extruder, a simple mathematical model to obtain the domain as a channel and the boundary conditions for channel flow. For reactive materials, the process involves convective combined heat and mass transfer, and the resulting product depends on the concentration C as well as the temperature T. The governing equations thus involve the flow equations, along with the energy and mass transfer equations. Chemical reactions occur and give rise to source terms in the energy and mass conservation equations. The properties also vary with concentration, besides the temperature and the shear for non-Newtonian materials. Figure 4 presents the computed results on the flow and the temperature for a typical case of extrusion of a power-law polymer, with n = 0.5 in Eq. 7 and a dimensionless flow rate qv, or throughput, of 0.3. In the absence of a die and thus any opposing pressure, the throughput is 0.5 for a Newtonian fluid in a corresponding shear-driven channel flow. Thus, the throughput is an indicator of the constriction imposed by the die and thus of the pressure rise in the extruder. It is seen that the flow is well layered, with little bulk mixing, and the temperature rises as the flow moves downstream due to heat input from the heaters on the barrel and viscous dissipation. The temperature can and often does exceed the barrel temperature due to viscous dissipation and heat removal, as shown in Fig. 5. Then heat removal, instead of energy input, may be needed to avoid exceeding allowable temperatures in the polymer. It is this lack of mixing, arising from high viscosity of the material, that twin-screw extruders are employed and various strategies, such as inverse screw elements and gaps in the screw, are used to promote mixing. Only one case is shown here to demonstrate the simulation. Many more results are available in the literature for a variety of single and twin-screw extruders, different dies, and various operating conditions. Another important thermal system is the one encountered in the optical fiber drawing process, which was outlined earlier. If the flow of the glass and of the aiding purge gas in a cylindrical furnace is assumed to be axisymmetric, the governing equations for the glass and the gas are then given as @v 1 @ ðruÞ þ ¼0 @z r @r      @v @v @v 1 @p 1 @ @v @u @ @v þu þv ¼ þ rυ þ υ þ2 @t @r @z ρ @z r @r @r @z @z @z

(8) (9)

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Fig. 4 Calculated velocity and temperature fields in the channel of a single-screw extruder at n = 0.5 and dimensionless throughput qv = 0.3, for typical operating conditions

     @u @u @u 1 @p 2 @ @u @ @v @u 2υu þu þv ¼ þ rυ υ þ þ  2 @t @r @z ρ @r r @r @r @z @r @z r       @T @T @T 1 @ @T @ @T þu þv rk k ρCp ¼ þ þ Φ þ Sr @t @r @z r @r @r @z @z

(10) (11)

where u and v are the velocity components in the axial and radial directions, z and r, respectively, p is the local pressure, and υ the kinematic viscosity. Figure 6 shows typical computed results in terms of the velocity and temperature distributions in the neck-down region for three furnace lengths. The glass flow is smooth and well layered because of its high viscosity. A temperature difference on the order of 100  C arises across the fiber for preform diameters of 5–10 cm. However, even this small difference is an important factor in fiber quality because of the large, exponential, dependence of viscosity on the temperature, as given in Eq. 6. The temperature affects the neck-down shape and the generation of

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Fig. 5 Isotherms and temperature distributions in a single-screw polymer extruder with greater viscous dissipation than in Fig. 4 due to higher screw rotational speed

temperature-induced defects. Larger temperature differences arise for larger preform diameters. Viscous dissipation, though relatively small, is mainly concentrated near the end of the neck-down due to the small diameter of 125 μm of the fiber, as shown in Fig. 7. Therefore, viscous dissipation must be included in the model since it is important in maintaining the temperatures above the softening point. Similarly, numerically calculated neck-down profiles, thermally induced defects, and tension in the fiber have been obtained for single- and multiple-layer optical fibers; see, for instance, Yin and Jaluria (1998) and Roy Choudhury et al. (1999) and the references given earlier. Data centers have already been mentioned as an important thermal system. Because of the stringent requirements on the maximum temperature level, the cooling systems take on substantial significance. The positioning of the server racks, the configuration and components of the cooling arrangement, the operating conditions like inlet flow rate and temperature, and the thermal load are all important aspects in the operation, design, and control of the data center (Patankar 2010; Le et al. 2011; Joshi and Kumar 2012). Figure 8 shows the calculated results on the temperature distributions in a typical data center, such as the one shown in Fig. 2a, for various scenarios, involving different server racks in operation. The steady thermal load is at 50% of the maximum allowable value for which the data center is designed. The temperatures obviously increase with higher utilization, which could be adjusted to meet the temperature requirements. Similarly, the operating conditions may be varied to operate within the allowable temperature range. Different designs, with respect to geometry, cooling system parameters, and server design, may be considered for steady as well as time-dependent load for system optimization.

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a 1.0

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Fig. 6 Calculated (a) velocity and (b) temperature fields in the glass and inert gases during drawing of a single-layer optical fiber for three furnace lengths. The vertical draw furnace is shown horizontally here for convenience.

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Fig. 7 Calculated (a) stream function, (b) vorticity, (c) viscous dissipation, and (d) temperature contours in the optical fiber drawing process with a specified neck-down profile

Fig. 8 Temperature distributions in a data center with 50% utilization. (a) Ai & Ci racks are operating, flow rate 2.195 m3/s; (b) Ai & Ci racks are operating, flow rate 2.667 m3/s; (c) Ai & Di racks are operating, flow rate 2.195 m3/s; (d) Ai & Bi racks are operating, flow rate 2.195 m3/s

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Boundary Conditions

The accuracy of any numerical simulation is strongly dependent on how accurate, realistic, and valid the imposed boundary conditions are. Even though isothermal and uniform heat flux conditions are commonly used in basic studies to understand the underlying phenomena, these conditions are generally not applicable in practical circumstances. For example, in the modeling of the solidification process in an enclosed region for casting, the coupled conduction, or conjugate transport, in the walls of the mold must be considered. The effect of the imposed conditions at the outer surface of the mold can be realistically obtained only by solving this conjugate problem. Then the temperature distribution in the mold as well as that in the solid and the liquid is obtained from which the solidification rate can be determined. Figure 9a shows the effect of conduction in the mold on the resulting temperature and velocity distributions, as well as on the time-dependent solidification process. For casting of metals, alloys, polymers, and other materials, it has been shown in the literature that it is important to model the conjugate transport in the mold walls and other boundaries in order to obtain realistic and accurate simulation results. Similarly, in the cooling of electronic systems, isolated heat sources that approximate components like electronic chips and devices are located on substrates that are conducting. Imposing adiabatic conditions on these surfaces is thus not a valid representation of the practical situation. The conduction in the walls distributes the heat input over the boundary, otherwise an erroneous concentrated heat source is approximated. This results in a substantial effect on the flow and the heat transfer. Figure 9b shows the calculated thermal and flow fields in an enclosed region with multiple, isolated, heat sources that approximate electronic devices. A recirculating flow arises due to buoyancy and the shear effect of the main forced flow, driven by a fan or a blower. This recirculating flow transports thermal energy from the sources to the main flow stream, which thus removes the energy from the system (Papanicolaou and Jaluria 1994). The system can be optimized in terms of operating conditions, like flow rate and heat input, and the design parameters, like source locations and materials used. Clearly, the walls play a significant role in the heat transfer process, and thus appropriate conjugate conditions must be considered. Similar considerations apply for fires in rooms where the walls, ceiling, and floor play an important role in the growth and spread of the fire. Many such examples arise in practical thermal systems where realistic and accurate imposition of boundary conditions is critical to an accurate numerical simulation. Of course, in some cases, the boundary conditions are not known accurately and simulation cannot be carried out. Sometimes, it may be possible to obtain experimental data that allows one to approximate the boundary condition. Sensitivity analysis may also be carried out to evaluate the importance of the unknown boundary condition and study different approximations. One circumstance in which an important boundary condition is not explicitly known is the optical fiber drawing furnace, shown in Fig. 2c. In this case, the wall temperature distribution is a critical input to the process. But this distribution is not easily obtained by modeling the heating arrangement due to the presence of many control and traverse

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b Streamlines

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(b)

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Isotherms

Design of Thermal Systems

Fig. 9 Effect of conjugate boundary conditions on the flow and heat transfer in (a) solidification process in casting, and (b) cooling of electronic devices located in an enclosed region

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subsystems or by measurements because of limited access to the furnace. An inverse calculation, which uses experimental data to determine the boundary conditions that lead to these results, is needed. Using the limited temperature data obtained from a graphite rod immersed in the furnace, this approach was used by Issa et al. (1996) to determine the wall temperature, which can then be used to accurately simulate the draw process. Typical results are seen in Fig. 10, which shows a schematic of the experimental system and the resulting wall temperature distribution from an inverse calculation based on temperature data from the immersed rod. Several such calculations and measurements were taken to obtain an essentially unique wall temperature distribution. For the simulations of flow in a room with an opening, the computational domain is generally extended and boundary conditions imposed far from the region of interest to ensure accurate simulation results. Other such approaches have been used in practical systems to obtain the relevant boundary conditions and thus accurately simulate the process.

4.3

Combined Mechanisms

In most practical thermal processes and systems, several coupled transport mechanisms generally arise and complicate the modeling and simulation. Conjugate boundary conditions were considered in the preceding section, where the effects of combined conduction and convection were discussed. Similarly, in the furnace for optical fiber drawing, thermal radiation and convection arise as coupled mechanisms, as shown in Fig. 2c. Convection arises both in the inert gas environment and in the glass, which is a subcooled liquid. Radiation is the dominant mode of transport, and the glass is largely heated up by radiation. Using radiation models such as the zonal method, the radiation transport in the glass as well as in the furnace is determined to obtain the energy absorbed by the fiber. The temperature variation in the preform/fiber depends on the combined radiation and convection, including viscous dissipation in the glass. The quality, uniformity, and rate of deposition in a CVD reactor are dependent on the heat and mass transfer and on the chemical reactions that are themselves strongly influenced by temperature and concentration levels. The flow, heat transfer, and chemical reactions in CVD reactors have been investigated by several researchers (Eversteyn et al. 1970; Chiu and Jaluria 2000; Chiu et al. 2002; Kadinski et al. 2004). Figure 11 shows typical calculated temperature and velocity distributions and the deposited film. The nonuniformity that arises, particularly near the ends of the wafer on which deposition occurs, is clearly seen. Figure 12 shows the results for the deposition of GaN on a rotating susceptor, with a flow channel to guide the downward flow of precursor and carrier gases (Meng and Jaluria 2013; Meng et al. 2015). Higher concentrations in the center lead to greater deposition, which decreases away from the center. Buoyancy effects may also become important at higher susceptor temperatures and lower inlet velocities to reduce deposition in the center. The flow arising in the reactor was found to be more

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a

TRANSLATING PLATFORM TO DATA ACQUISITION

CHUCK ROD C TYPE THERMOCOUPLE PROBE TO DATA ACQUISITION CENTORR 11B FURNACE

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Fig. 10 (a) Sketch of an optical fiber drawing furnace with a centrally positioned graphite rod instrumented for measuring its centerline temperature, (b) wall temperature obtained from a solution of the inverse problem for two diameters of the graphite rod

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a c

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stable at large rotational speeds, but a chaotic recirculation was seen to arise at low speeds. These flow patterns obviously affect the uniformity of the deposited film, resulting in a better product at higher rotational speeds. Several chemical reactions and corresponding species arise and have to be included in the analysis. Figure 12 shows the effect of susceptor rotation on deposition. A higher rotational speed tends to make the deposition more uniform and also increases the rate of deposition. The effects of other parameters such as inlet velocity, susceptor temperature, and the inlet composition of gases have been investigated in detail. The deposition of GaN is of interest due to its application in light-emitting diodes and power amplifiers. Similarly other materials like gallium arsenide, silicon carbide, and grapheme have received substantial attention in the recent years.

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Similarly, many other practical thermal systems involve combined mechanisms, and the models must include these to determine the resulting transport, temperature variation, and the flow. Combined heat and mass transfer, with or without chemical reactions, is commonly encountered. Similarly, different modes arise and the combined effects are of interest. Certainly, the involvement of chemical reactions, which often lead to stiff equations and many species, makes the simulation quite challenging. However, many systems such as CVD reactors, food extruders, and polymer processing involve chemical conversion and need to be simulated accurately to design and optimize the relevant thermal system.

4.4

Complex Transport Phenomena

In many thermal systems, several significant additional effects, such as buoyancy, complex geometry, surface tension, viscous dissipation, and free surfaces, are encountered that considerably complicate the transport phenomena being modeled. The free surface that arises during fiber drawing has already been considered. As discussed earlier, the resulting shape of these free surfaces is governed by a balance of forces due to shear, tension, gravity, and surface tension. Similarly, a force balance is used at interfaces in multilayered fibers to determine the resulting profiles. In hollow optical fiber drawing, a major concern is the collapse of the central core, which is needed for applications such as power delivery, sensors, and infrared radiation transmission. A collapse ratio C is defined to describe the collapse process of the central core as CðzÞ ¼ 1  ðR1 ðzÞ=R2 ðzÞÞ=ðR10 =R20 Þ

(12)

where R1 and R2 are given in Fig. 13a. Thus, C = 0 when the radius ratio of the final fiber equals the initial radius ratio, and C = 1 when the central cavity is closed. The effects of the feeding speed of the preform, fiber drawing speeds, furnace temperature, and preform diameter on collapse ratio have been studied in detail (Fitt et al. 2001; Jaluria and Yang 2011). Because of the size of the core, which is on the order of 10 μm, surface tension effects are important and play a significant role in the collapse. Figure 13b shows the variation of the collapse ratio with the furnace temperature at different drawing speeds. The effect of the drawing speed is relatively weak, though the preform feeding speed was found to be quite an important factor in the collapse. In order to avoid the collapse of the central cavity, the feeding speed may be increased, the furnace temperature decreased, or the preform radius ratio increased. It was also found that a higher pressure in the central cavity tends to prevent collapse. Thus, a wide variety of additional effects can arise in practical processes and complicate the transport phenomena being investigated. It is generally best to consider all the additional effects that may arise and to carry out a detailed scale analysis to determine which ones need to be retained. Then the complex process, with the important additional effects included, can be modeled and simulated.

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R10

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Fig. 13 Transport mechanisms in a (a) hollow fiber drawing process, (b) variation of the collapse ratio with furnace temperature at different drawing speeds

4.5

Multiple Scales

In the modeling of practical thermal systems, a common challenge faced is transport at very different length or time scales. Different scales may involve different mechanisms, and thus the governing equations and numerical approaches may be quite different. As an example, consider the numerical simulation of pressure-driven nitrogen slip flow in long microchannels, with conjugate heat transfer in the walls under a uniform heat flux input condition, as sketched in Fig. 14a. This problem is important in many practical circumstances such as those related to the cooling of electronic devices, localized heat input in materials processing systems, and microfluidics for biological applications. For the gas phase, the two-dimensional momentum and energy equations are solved, considering variable properties, rarefaction, which involves velocity slip, thermal creep and temperature jump, compressibility, and viscous dissipation. For conduction in the solid region, on the other hand, the energy equation is solved with variable properties. Thus, the two regions are treated with different approaches. Due to rarefaction, discontinuous boundary conditions are applied at the wall, depending on the Knudsen number, Kn = λ/H, where λ is the mean free path of the gas molecules and H is the channel height for two-dimensional rectangular channels. Continuum flow is assumed in the case of Kn  103; slip flow occurs when 103  Kn  101; transition flow arises 101  Kn  10; and free-molecular flow refers to the case of Kn > 10. For isothermal walls, the wall boundary conditions are (Sun and Jaluria 2011)     2  σv @u 3 μ @T λ þ , v ¼ 0, At y ¼ 0 and y ¼ H, u ¼ @ n w 4 ρT @ x w σv

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b

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Tg  Tw ¼

    2  σ T 2γ λ @T γ þ 1 Pr @n w σT

(13)

The coefficients σ v and σ T depend on gas properties and surface qualities. They represent the fraction of the molecule’s tangential momentum and energy loss through the interactions with the solid wall, respectively. The substrates are governed by the energy equation, 

      @ρT @ρuT @ρvT @ @T @ @T þ þ k k cp ¼ þ @t @x @y @x @x @y @y

(14)

The boundary conditions in the y-direction, i.e., at y = 0 and y = 2Hs + H, are given by q = qw, q being the heat flux and subscript w referring to the wall. The temperature at the fluid-solid interface is obtained by solving the two equations

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    2  σ T 2γ λ @T Tg  Tw ¼ γ þ 1 Pr @n w σT

(15)

qg ¼ qs where the first equation is due to the temperature jump at the gas-solid interface and the second one arises from the fact that no energy can be stored in the interface. Figure 14b and c shows the results for different substrate materials, including commercial bronze, silicon nitride, pyroceram, and fused silica. The effects of substrate axial conduction, material thermal conductivity, and substrate thickness are clearly seen. It is found that substrate axial conduction leads to a flatter bulk temperature profile along the channel, lower maximum temperature, and lower Nusselt number. The effect of substrate thickness on the conjugate heat transfer is very similar to that of the substrate thermal conductivity. That is, in terms of axial thermal resistance, the increase in substrate thickness has the same effect as that caused by an increase in its thermal conductivity. This microchannel flow may also be coupled with a thermal sink in electronic systems for thermal management, as sketched in Fig. 14d. Thus, the length scales go from a few microns in the microchannel to several centimeters in the heat sink. Comparisons are made between the numerical results and experimental data when the entire system is considered for simulation. This simple example illustrates the simulation of different components with different length scales, followed by coupling of the results to obtain the model for the entire system. This is particularly valuable in electronic thermal management systems. In materials processing systems, the characteristics and quality of the material being processed are generally determined by transport processes occurring at the micro- or nanometer scale in the material, for instance, at the solid-liquid interface in casting, over molecules involved in a chemical reaction in chemical vapor deposition, or at sites where defects are formed in an optical fiber. However, engineering systems are generally concerned with the macroscale, involving practical dimensions and boundary conditions. Therefore, different scales arise, which need to be solved by different methodologies and then coupled to obtain the overall behavior, as illustrated in the example above. Changes at the molecular level are considered in the generation of thermally induced defects in optical fiber drawing. The differential equation for the time dependence of the defect concentration is formulated in terms of the temperature. Then the transport processes in fiber drawing are linked with the defect formation equation to obtain the quality of the fiber as a function of design parameters and operating conditions (Yin and Jaluria 2000). Similarly, in food and reactive polymer extrusion, the microscopic changes in the material are linked with the operating conditions that are imposed on the system. The chemical conversion process is then represented by the chemical kinetics, which depend on the temperature and the concentration (Wang et al. 1989). These microscale conversion mechanisms are coupled with the flow and heat transfer in a screw extruder to obtain the conversion

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and product characteristics as functions of the extruder design and operating conditions. Another area where multiscale transport is of interest is related to environmental flows and systems, as shown in Fig. 15. The heat transfer near the source in a fire or in a polluting source involves much smaller length scales than those involved in the transport far downstream. In thermal discharges from power plants and industries, the length scale is of the order of a few meters at the source and of the order of several kilometers far downstream (Gebhart et al. 1984). In room fires, the source may only be a few centimeters, with the room itself being several meters in dimensions. The scales are even further apart in large-scale fires as in forest fires. Again, the modeling involved, as well as the numerical grid, near the source could be quite different from those at locations far away. For instance, radiation is particularly important near the fire source. But as one moves far away, the flow is dominated by turbulence and buoyancy. Similar considerations apply to systems used for heat and material discharges into the environment.

4.6

Validation

One of the most important considerations in modeling and simulation is that of validation. This is particularly critical in thermal systems because of simplifications and idealizations that are usually needed to make the problem amenable to a solution, lack of accurate material property data, combined mechanisms, and many other complexities in the process outlined in the preceding sections. It is important to ensure that the numerical code yields accurate results for the chosen process, as characterized by the governing equations, and that the mathematical model is an accurate representation of the physical problem. Unless the models are validated and the accuracy of the predictions established, the models cannot form a satisfactory basis for design and for the choice of appropriate operating conditions. Validation of the models is based on a consideration of the physical behavior of the results obtained; comparisons with available analytical results, often for simpler configurations; comparisons with numerical results in the literature; and comparisons with experimental results. For the numerical model itself, it is necessary to verify that the results are independent of arbitrary parameters like grid and time step, convergence criterion, and initial guess in iterative solutions (Roache 1998). As an example of validation, consider the flow in a CVD channel, as shown for the horizontal reactor in Fig. 2d. Some typical results obtained for silicon deposition are shown in Fig. 16a, indicating a comparison between numerical and experimental results. A fairly good agreement is observed, given the uncertainty in material property data and in the chemical kinetics. Similarly, two other comparisons are shown in Fig. 16b and c for the optical fiber drawing system. The first compares the calculated neck-down profile with experimental measurements and the second the meniscus at the outlet of the coating die. In both cases, the agreement is good,

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Fig. 16 Validation of the model by comparison of calculated results with experimental data for (a) a horizontal CVD reactor for silicon deposition for different models, (b) neck-down profile in optical fiber drawing, (c) lower meniscus profile at the exit of the applicator in a fiber coating process, for glycerin at fiber speed of 20 m/min

lending support to the model. In all these cases, verification of the numerical scheme through grid refinement and variation of other arbitrary parameters was undertaken. Therefore, through analytical, numerical, and experimental approaches, data on the system behavior and its response to variations in the design parameters and in the operating conditions may be obtained. The results provide physical insight into the operation of the system and also provide guidelines how the various parameters may be chosen to obtain a workable or acceptable design. The results may be stored as a data base to extract information as needed, rather than run the simulation again. Curve fitting of the data may also be used to simplify the use of these results in design. System design may now be considered, assuming that detailed results from modeling, simulation, and experimentation have been obtained to guide the selection, design, and optimization of the system.

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5

Design

5.1

Basic Design Strategy

The basic design process was briefly discussed earlier. Starting with the problem formulation, one proceeds to the conceptual design. Once a particular conceptual design is selected, it is subjected to a detailed modeling and simulation, as presented for a variety of problems in the preceding section. Experimental data are also obtained, whenever possible, and included in the results from simulation, very often in the form of correlating equations for convenience. All these results form the basis for developing the design of a given thermal system, evaluating if the design is feasible, or acceptable, and finally optimizing the design. The general strategy for obtaining an acceptable design involves starting with an initial design, evaluating if it is acceptable and, if it is not, to redesign the system by varying the parameters and evaluate it again, and so on till a design that satisfies the problem statement is obtained. The parameters include the hardware of the system, such as components, dimensions, materials, and geometry, as well as operating conditions, such as speed, heat flux, flow rate, and temperature. The hardware would often limit the range of the operating conditions, and the latter determine the appropriate hardware. Figure 17a shows the overall process in terms of a flow chart, starting with a physical system and ending with the communication of the final design for prototype development and fabrication. Figure 17b shows the basic design strategy, from the initial design, through evaluation and redesign, to the final acceptable design. The initial design is based on simulation and experimental results and involves selecting the various parameters that characterize the design. Evaluation focuses on whether the design problem statement, in terms of requirements and constraints, is satisfied. Redesign is based on considering what makes a given design unacceptable and varying appropriate parameters, as guided by available results, to obtain a better design in terms of performance. If the design-redesign process converges, an acceptable design is obtained. In most cases, a range or domain of acceptable design is obtained.

5.2

Inverse Problem

One of the most important aspects in design of a thermal system is the choice of operating conditions and design parameters in order to achieve the desired thermal process. Thus, the result is known, whereas the conditions and parameters are not. This is an inverse problem, which involves finding the conditions that would lead to a given result. As may be expected, several different conditions could lead to the same outcome. Thus, the solution of inverse problems is not unique, though the domain of uncertainty may be reduced through optimization and other techniques (Ozisik 2000; Orlande 2012). In most cases, the direct solution, with specified conditions, is solved, and the results are used to develop a scheme to converge to a solution to the inverse problem in a fairly narrow domain. An example of an

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Fig. 17 Design of a thermal system. (a) Overall design and optimization process, (b) basic design strategy

inverse problem in manufacturing is shown in Fig. 18a, which gives a sketch of the desired time-dependent temperature variation for heat treatment of materials. The desired variation, along with an acceptable envelope, is shown. For instance, annealing is used in metals and alloys like steel to relieve stresses by gradual cooling from an elevated temperature. For steel, the material has to be heated beyond its recrystallization temperature of around 723  C and then cooled very slowly to achieve the desired annealing process, which removes residual stresses and restores ductility. A batch annealing furnace for steel sheets rolled up into cylindrical coils is shown in Fig. 18b; see Jaluria (1984). The boundary conditions, particularly the time-dependent flow of flue gases from the blast furnace, need to be determined to achieve the desired temperature variation with time. This problem was numerically modeled, and the dependence of the temperature distributions on the boundary

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a

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STEEL COIL CONVECTOR PLATE

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FURNACE GASES PROTECTION HOOD BURNERS

Fig. 18 (a) Typical temperature variation needed for a heat treatment process, (b) schematic of a batch annealing furnace for cylindrical coils of steel sheets

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TEMPERATURE ( q )

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Fig. 19 Calculated variation of control thermocouple needed for achieving the desired temperature variation in the annealing process of Fig. 18, along with experimental results

conditions was established. This information was used to determine the temporal variation of a temperature sensor located at the base of the furnace that controls the flow rate of the hot gases in order to achieve the desired annealing process. Figure 19 shows the variation of this control thermocouple with time that would result in achieving the desired result. The calculated results were also confirmed by actual experimental variation needed to obtain the desired result. The inverse problem involved in determining the wall temperature distribution in the furnace for optical fiber drawing was outlined earlier (Issa et al. 1996). Centerline temperature data on rods of different materials and diameters mounted at the center within the furnace cavity were employed with an inverse calculation to obtain the furnace wall temperature. Optimization was used to narrow the domain of uncertainty and obtain an essentially unique solution, which was then used for further calculation on fiber drawing. The literature is replete with such inverse problems, and the methodologies developed may be used to select various parameters to achieve the desired result and thus obtain an acceptable design.

5.3

Feasibility

An important consideration in the design of a thermal system is the feasibility of the process, since there is often a fairly narrow range of operating conditions and corresponding design parameters over which the process is possible. Numerical simulation can play a valuable role here since it can guide the selection of the conditions that can lead to a successful design or process. In polymer extrusion, for instance, the feasibility of the process is determined largely by the flow and the pressure and temperature rise in the extruder. Using the modeling outlined earlier, the feasible domain for a co-rotating twin-screw extruder was determined for the extrusion of pure starch, as shown in Fig. 20a. An upper limit is obtained for the

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Fig. 20 Feasible domains. (a) Feasible and infeasible cases for twin-screw extrusion of starch, (b) optical fiber drawing process for solid optical fibers, showing the feasible domain in terms of constant tension contours

Mass Flow Rate(kg/h)

Feasible Domain of Numerical Simulation for Amioca in Twin-Screw Extruder (Ti=120°C Tb=150°C)

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mass flow rate, below which practical operating conditions occur and the simulation gives physically reasonable results in terms of the pressure increase downstream. Though the numerical scheme converges beyond this upper limit, the results are not physically acceptable. In actual practice, for a given screw rotational speed, the mass flow capacity of the extruder is limited by the dimensions of the channel. The mass flow rate cannot exceed this limit given by the shear-driven flow, with higher screw speeds yielding higher flow rates. If one wants to achieve mass flow rates higher than these limitations, it is necessary to impose a favorable pressure gradient to push the material down the channel. Therefore, a negative pressure gradient along the axial direction occurs in the channel. This is physically unacceptable for an extruder. For a specific screw speed, the simulation code also diverges for mass flow rates lower than the critical points shown in the figure. This is due to the stability of the extrusion process, since the pressure acts in a direction opposite to that of the drag flow due to screw rotation. When a narrow die is used, the pressure gradient in the down-channel direction becomes so large that the conventional numerical schemes fail to simulate the flow. Even with modifications in the numerical scheme, a lower limit on the mass flow rate arises in the feasible domain because of excessive pressures, temperatures, and residence times. For chemically reactive materials like food, excessive residence times can degrade the product and make it unacceptable. Similarly, in optical fiber drawing, it has been shown that, for given fiber and preform diameters and for a given draw speed, it is not possible to draw the fiber at any arbitrary furnace wall temperature distribution. If the furnace temperature is not high enough, it is found that the fiber breaks due to lack of material flow, a phenomenon that is known as viscous rupture. This is first indicated by the divergence of the numerical scheme for the profile and is then confirmed by excessive tension in the fiber. Similarly, it is determined that, for a given furnace temperature, there is a limit on the speed beyond which drawing is not possible, as this leads to rupture. Thus, as shown in Fig. 20b, a region in which drawing is feasible can be identified. For the domain in which the drawing process is feasible, the draw tension is calculated. As expected, the draw tension is small at higher temperatures and lower speeds. Similarly, different combinations of other physical and process variables may be considered to determine the feasible domain. Thus, the feasibility of the process is determined by the physical constraints, such as temperature, pressure, and tension limitations, attainment of the desired product characteristics, and availability and cost of resources and materials needed. Different criteria obviously arise for different systems and processes. But it is important to ensure that a given design is feasible and, if not, a redesign is needed to meet the given requirements and satisfy the given constraints.

5.4

Sensitivity Analysis

Because of the complexity of thermal systems, as seen from the section on modeling and simulation, it is important to focus on the dominant variables. This implies focusing on a small set of requirements and parameters. For instance, in an electronic

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system, the only requirement may be taken as the resulting maximum temperature in the system being lower than a specified value. The design variables may be the heat input and air flow rate. In order to select the dominant variables, the simulation of the system can be used to determine the sensitivity of the results to various parameters and choose the ones that are most important. During the simulation, it is common to vary one quantity, while the others are held constant. This allows the computation of sensitivity parameters @F/@x1, @G/@x1, @F/@x2, @G/@x2, . . ., where F and G are two outputs or requirements and x1, x2, x3, . . . are the design variables. By comparing the sensitivity parameters, the main variables may be chosen, keeping the total as relatively small, such as 1–3. The variables include the hardware as well as the operating conditions. The requirements must also be limited to a small number to make the problem tractable. For instance, the outlet temperature To and the heat transfer rate Q may be the two requirements in a heat exchanger, with the required values given as Tr and Qr. It is possible to follow these through the design process to judge how far a given design is from the acceptable values. The two may also be combined for simplification as " Z¼

To  Tr Tr

2

  #1=2 Q  Qr 2 þ Qr

(16)

Then, the variation of the desired characteristic parameter Z with the variables in the problem, such as flow rate, inlet temperatures, geometry, and dimensions, may be studied in detail as one proceeds with the simulation. The results obtained would allow one to focus on the dominant 1, 2, or 3 variables and proceed to obtain an acceptable design by varying these. Thus, sensitivity analysis is an important element in the design of thermal systems and is easily linked with the simulation.

5.5

Acceptable Design Domain

Following the basic design strategy, an initial design is developed on the basis of simulation results and experimental data. This initial design is evaluated to determine if it satisfies the given requirements and constraints, again by using the simulation scheme developed for the given thermal system. If it satisfies the design problem statement, an acceptable design is obtained. If it is not acceptable, the system is redesigned by varying the governing parameters. This process is repeated till an acceptable design is obtained. Several design-redesign strategies are possible and Fig. 21 shows a few of these. In Fig. 21a, the system is redesigned by varying the design parameters, carrying out the simulation and selecting appropriate parameters for acceptable design. Figure 21b shows a scheme in which several designs are developed and subjected

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255 Outputs

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Iterative Redesign Procedure Fig. 21 Different strategies for system design and redesign to obtain an acceptable design

to the operating conditions over the ranges possible for a chosen hardware. Figure 21c shows a simulation and experiment-based redesign process in which both the inputs are used to obtain different designs. The convergence of the design-redesign process may be monitored in terms of a requirement parameter Z, as defined in Eq. 16. Figure 22 shows typical convergence behavior of the process as the redesign approaches an acceptable design. The convergence with selective rules is discussed later. The convergence generally depends on the initial design. If the components, materials, dimensions, etc. are carefully chosen, the initial design may itself be acceptable. This is the approach taken in most cases since detailed simulation and experimental results provide adequate information to make such a selection. Several simple thermal systems, such as air conditioning and heat exchanger systems, are considered as examples by Jaluria (2008), and actual physical values of the hardware parameters are chosen to obtain an acceptable initial design. As an example, let us consider the design of a cooling system for electronic equipment, which consists of multiple heat sources in a channel, as shown in Fig. 23a. A vortex generator is also placed in the flow to enhance the heat transfer rate. The effects of different coolants, materials, and geometry are obtained by

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Fig. 22 Iterative convergence of the design-redesign process, monitored in terms of a design requirement parameter Z, for the design of a 1 cm. diameter circular die for polymer extrusion

numerical simulation, along with the effects of heat input and flow rate variation. The resulting temperature and flow fields are calculated and are shown for a particular circumstance. The simulation results may be used to select the hardware and the operating conditions for given heat input and constraints on flow rate, space, etc. Figure 23b and c shows the measured heat transfer rates from the two sources when the vortex generator is absent. Again, simulation and experimentation may be employed to investigate the effects of dimensions, materials, and geometry, allowing the selection of source dimensions, separation between sources, inlet flow rates, and inlet temperature to achieve the desired cooling to keep the source temperatures below a given specified value. Similar approaches may be used for more complicated systems, such as CVD reactors and data centers. In general, wide ranges of the variables would lead to acceptable design, yielding an acceptable design domain, rather than a single point. Any design in this domain would satisfy the given requirements and constraints, though it would not, in general, be the optimal design.

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Fig. 23 (a) Two isolated heat sources in a channel with a vortex generator, showing isotherms and flow streamlines, (b) measured heat transfer rate from the first heat source without a vortex generator, with flow from left to right, (c) heat transfer rate from the second source, for different source separation distances d for a given source length w. The channel and source widths are taken as large to yield a two-dimensional transport

6

Additional Design Aspects and Strategies

Several new aspects have been built into the design process in the recent years. These help in developing an initial design, in obtaining a faster convergence to an acceptable design, and also in making the design realistic and practical. These strategies can also be extended to the optimization process.

6.1

Knowledge-Based Design

An important consideration in the design of thermal systems is the use of the available knowledge base to guide the design and optimization process. This knowledge base typically includes relevant information on existing systems and processes, current practice, knowledge of an expert in the particular area, material property data, and empirical data on equipment and transport, such as pump characteristics and heat transfer correlations. It is advantageous to use any such available information in developing initial, acceptable, and optimal designs, as well as in selecting operating conditions. Though available knowledge on the system is generally used in design anyway, some focused effort has been directed in recent years at

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streamlining the design process, improving the design methodology, automating the use of existing information, and developing strategies for rapid convergence to the final design (Jaluria 2008). Consider, for instance, the initial design of a polymer extrusion die. Die design is an important consideration in plastic extrusion since the operation of the extruder system and the quality of the final product are strongly influenced by the die. Typical extrusion dies consist of different parts, such as entrance, flow channel, and exit regions, which may be fabricated as one unit or may be attached to each other through couplings and screws, and the heating/cooling arrangement to maintain a desired temperature. Two strategies may be used for generating an initial design. The first is based on a library of designs from earlier design efforts and from existing systems. The design closest to the given problem may be selected by comparing the designs in the library with the desired specifications. The second approach is based on expert rules for die design. Employing knowledge and experience used by an expert, rules may be set down to generate a design for the given requirements and constraints (Jamalabad et al. 1994). Preliminary evaluations and estimates are used to develop a possible design that is used as initial design. Of course, the user can always enter a personal selection for initial design if the output from the library or the expert rules is not satisfactory. The expert knowledge may also be used in redesign in case the initial design is not acceptable. Figure 24 shows the basic ideas schematically. Figure 22, shown earlier, also included the convergence of the redesign process if selective rules, based on the knowledge base, are used. Clearly, the convergence is accelerated. Another example is the casting of a material by pouring the molten material into an enclosed region and allowing it to solidify. A large number of design parameters arise in this problem, such as materials, initial melt pour temperature, cooling fluid and its flow rate, and dimensions. Viswanath and Jaluria (1991) considered this problem, using knowledge-based design methodology. Several models, with different simplifications, are available in the literature for the study of solidification (Ghosh and Mallik 1986). These include steady conduction heat transfer from an isothermal mold; Chvorinov model, which takes the entire thermal resistance in the mold; lumped mold; one-, two-, and three-dimensional conduction models; models with buoyancy effects; and more complicated models for alloys, defects, and complicated geometries. Each model has its own level of accuracy and range of validity. Different models may be chosen, depending on the application and materials involved. An initial design may be developed with, say, a desired solidification time for a given casting as the requirement and constraints from allowable thermal stresses and defects in the casting. The process can start with relatively simple models and move to more complicated models if the desired information on solidification time, stresses, etc. is not obtained. It may also start with the simplest model and keep on moving to models with greater complexity till the results remain essentially unchanged from one model to the next. Thus, models may be automatically selected using decision-making based on accuracy considerations. If the desired solidification time is not achieved, the pour temperature of the melt may be varied. If even

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Fig. 24 Schematic of the development of an initial design, followed by evaluation and redesign

this does not satisfy the requirement, the thickness of the mold wall may be changed. The material of the wall may also be varied, if needed. Thus, by first varying the operating conditions and then the dimensions and materials, the solidification time may be brought to a desired value, thus yielding an acceptable design. Figure 25 shows a typical run for the design of the given system, indicating model change as the design proceeds. Each successful design may be stored for help in future designs. This implies improving the design process through learning from past experiences. It is seen that the use of the available knowledge base on the system is important in the design, as well as in choosing the operating conditions for a desired product. In actual practice, this knowledge, which may simply be called experience, is commonly used by engineers involved with thermal systems and processes. Decisions are often made on the basis of what is known about existing processes, past trials, and other similar systems. Therefore, it is desirable to build this knowledge base into the design process.

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Fig. 25 Design of a casting system, using change of models during the design and optimization process

6.2

Concurrent Experimentation and Simulation

As presented in the preceding, most approaches for design of thermal systems are based on the use of numerical simulation, which is validated by experiment. Thus, a sequential numerical and experimental approach is commonly adopted. This strategy fails to use certain advantages of using both tools concurrently where the outputs from each approach drives the other to achieve better engineering design in a more efficient way. Consider, again, the design of the cooling system shown in Fig. 23a. The hardware is difficult to vary in practice, and numerical simulation can be used advantageously. However, for a given hardware, operating conditions can be easily changed in an experimental set-up, and the corresponding inputs may be obtained experimentally. Also, at low flow rates, numerical simulation is accurate and dependable, whereas in the transitional and turbulent regimes, experimental data may be more reliable. Switching from one approach to the other can then be based on flow transition to turbulence and on the design parameter being varied (Knight et al. 2002; Icoz and Jaluria 2004). Figure 26 shows a schematic of how the results from simulation and experiment are combined to build a surrogate model and how the concurrent inputs are used to achieve an acceptable or optimal design. By making the two approaches concurrent, the models are automatically validated, and the experiments can guide the selection of conditions for the next simulation in an iterative process. Similarly, simulation helps in the selection of conditions at which experimental data should be obtained, leading to faster convergence and a more realistic design. A typical set of results is shown in Fig. 27, indicating laminar simulations and experimental data in

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Fig. 26 Schematic representation of a design methodology based on concurrent simulation and experimentation

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transitional and turbulent flow, with a switch from simulation to experiment at around Re = 1500. The accuracy of simulation is high in laminar and oscillatory flow, whereas turbulent flow is better suited to experiment. Additional results have been obtained on this problem, and the concept has been shown to be a useful and efficient technique for design and optimization of practical problems.

6.3

Uncertainty- and Reliability-Based Design

An important consideration in the design of thermal systems is the presence of uncertainties that arise in various parameters. Even if an acceptable design is obtained from deterministic models, the uncertainties can cause variations that can make the design unsatisfactory. Due to the existence of the uncertainties, the traditional deterministic formulation is no longer adequate to generate safe and acceptable designs because it may lead to a design with high risk of system failure. In order to achieve high reliability in the final design, it is necessary to develop

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reliability-based design, which results in failure rate lower than an accepted level, generally taken as 0.13%. The development of the reliability-based design optimization (RBDO) algorithm evaluates the probabilities of the system failures and provides a more conservative design which ensures that the failure probabilities are subject to some acceptable level. If any uncertainties are found in the experiments, the simulations, or the manufacturing process, the information on the uncertainties is fed back to the formulation of the RBDO problems, and new appropriate conditions can be generated. Several researchers have estimated the randomness of the operating parameters in different thermal processes. These parameters are then assumed to have a distribution of values, such as a normal distribution, rather than a deterministic fixed value. The design process then employs these distributions to obtain an acceptable design, with a range of variation in the parameters needed to meet the uncertainty level. This approach has been applied to a CVD reactor and to microchannel flows (Lin et al. 2010; Zhang et al. 2014). However, much more detailed effort is needed on this aspect to obtain realistic and useful designs of thermal systems.

7

Optimization

The focus of the preceding section was on obtaining workable or acceptable designs, which satisfy the requirements for the system and do not violate the constraints. However, a design taken from the acceptable domain would generally not be the best or optimal design, as judged on the basis of a single criterion or multiple criteria. Typical criteria for optimization of thermal systems include performance, efficiency, cost, quality or performance per unit cost, output, and output per unit cost. As in other engineering systems, the need to optimize is important in thermal systems as well due to growing global competition. Also, most commonly used thermal systems have not been optimized due to the intrinsic complexity of these systems.

7.1

Basic Aspects

The optimization process requires the specification of a quantity, criterion, or function U, known as the objective function and which is to be minimized or maximized. If there are constraints in the problem, a constrained optimization has to be carried out. If the number of equality constraints m is equal to the number of independent variables n, the constraint equations may simply be solved to obtain the variables, and there is no optimization problem. If m > n, the problem is over constrained, and a unique solution is not possible, since some constraints will have to be discarded to make m  n. If m < n, an optimization problem is obtained. In thermal systems, the constraints are often given in terms of the temperature, pressure, flow rate, and heat input limitations due to material and process considerations. Conservation principles may also give rise to constraints on the processing rate, speed, tension, and other variables.

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If U is a taken as a function of the n independent variables in the problem x1, x2, x3, . . ., xn, then the objective function and the optimization process may be expressed as U ¼ Uðx1 , x2 , x3 , . . . , xn Þ ! Uopt

(17)

Here, Uopt denotes the optimal value of U. The xs represent the design variables as well as the operating conditions, which may be changed to obtain an optimal design. A minimum or a maximum in U may be sought, depending on the nature of the objective function. The process of optimization involves finding the values of the different design variables for which the objective function is minimized or maximized, without violating the constraints. Interest lies in obtaining the global optimum. However, the local optima can sometimes confuse the true optimum, making its determination difficult. It is necessary to distinguish between local and global optima so that the best design is obtained over the entire domain. The constraints in a given design problem arise due to limitations on the ranges of the physical variables and due to the basic conservation principles that must be satisfied. The constraints limit the domain in which the workable or optimal design lies. For instance, in heat treatment of steel, as shown in Fig. 18, the minimum temperature needed for the process Tmin is given, along with the maximum allowable temperature Tmax at which the material is damaged. Similarly, the maximum pressure pmax in a polymer extrusion process is fixed by strength considerations of the extruder and the minimum by the pressure needed for the flow and the process. The limitations on the dimensions define the domain in an electronic system. Many constraints arise because of the conservation laws in thermal systems. Thus, under steady-state conditions, the mass inflow into the system must equal the mass outflow. Similarly, energy balance considerations are very important in thermal systems and may limit the range of temperatures, heat fluxes, dimensions, etc. that may be used. Several such constraints are often satisfied during modeling and simulation since the governing equations are based on the conservation principles. Then the objective function being optimized has already taken these constraints into account. In such cases, only the limitations that define the boundaries of the design domain are left to be considered, and the problem is largely solved as an unconstrained optimization case. There are two types of constraints, equality constraints, Gi = 0, and inequality constraints, Hi < or > Ci, where Gi and Hi are functions of the given variables and Ci are constants or given functions. Equality constraints are most commonly obtained from conservation laws and inequalities from limitations on the domain. It is generally easier to deal with equalities than with inequalities, since many methods are available to solve different types of equations and systems of equations, whereas no such schemes are available for inequalities. Therefore, inequalities are often converted into equations by using a value larger than the constraint if a minimum is specified and a value smaller than the constraint if a maximum is given. There are several methods that may be employed for solving the mathematical problem posed for optimizing a thermal system. Each approach has its limitations

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and advantages over the others. The choice of method largely depends on the nature of the equations representing the objective function and the constraints (Arora 1989; Rao 2009). Because of the complicated nature of typical thermal systems, numerical solutions of the governing equations are often needed. However, in several cases detailed numerical and experimental results are obtained, and these are curve fitted to obtain algebraic equations to represent the characteristics of the system. Optimization of the system may then be undertaken on the basis of these relatively simple algebraic expressions and equations.

7.2

Lagrange Multiplier Method

An important method is the Lagrange multiplier method, which uses calculus for determining the optimum on the basis of derivatives of the objective function and of the constraints. The derivatives are used to indicate the location of a minimum or a maximum. At a local optimum, the slope is zero for U varying with a single design variable x1 or x2. The equations and expressions that formulate the optimization problem must be continuous and well behaved, so that these are differentiable over the design domain. This method basically converts the problem of finding the minimum or maximum into the solution of a system of algebraic equations, thus providing a convenient scheme to determine the optimum. The objective function and the constraints are combined into a new function Y, known as the Lagrange expression and defined as Y ð x 1 , x 2 , . . . , x n Þ ¼ U ð x 1 , x 2 , . . . , x n Þ þ λ1 G 1 ð x 1 , x 2 , . . . , x n Þ þλ2 G2 ðx1 , x2 , . . . , xn Þ þ . . . þ λm Gm ðx1 , x2 , . . . , xn Þ

(18)

where the λs are unknown parameters, known as Lagrange multipliers. Then, according to this method, the optimum occurs at the solution of the system of equations formed by the following equations: @Y ¼0 @x1

@Y ¼0 @x2

@Y ¼0 @xn

(19a)

@Y @Y ¼0 ¼0 @λ1 @λ2

@Y ¼0 @λm

(19b)

If the objective function U and the constraints Gi are continuous and differentiable, a system of algebraic equations is obtained. The unknowns are the m multipliers, corresponding to the m constraints, and the n independent variables. Therefore, this system may be solved to obtain the values of the independent variables that define the optimum, as well as the multipliers. Analytical methods for solving a system of algebraic equations may be employed if linear equations are obtained and/or when the number of equations is small, typically up to around 5. For nonlinear equations

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and for large sets, numerical methods are generally more appropriate. It can be shown that the Lagrange multiplier λ is the negative of the sensitivity coefficient Sc, which indicates the rate of change in the optimum with change in the constraint. This analysis can easily be extended to multiple constraints and larger number of independent variables. Though this method is applicable only in a few cases due to the general complexity of thermal systems, it allows the visualization of the general optimization procedure and identifies the basic aspects of the process. The most common strategy for thermal systems is based on search from a range of solutions, as presented in the following.

7.3

Search Methods

Search methods provide the most useful optimization strategy for thermal systems. The basic idea is to generate a number of acceptable designs, which may also be called trials or iterations, and to select the best among these. Many different approaches to carry out the search for the optimal solution have been developed. Effort is made to keep the number of trials small, often going to the next iteration only if necessary. This is an important consideration in thermal system design since each trial may take a considerable amount of computational or experimental effort. The steepest ascent/descent method and other gradient-based methods are widely used for thermal systems (Stoecker 1989; Jaluria 2008, 2009). These are hillclimbing techniques in that they attempt to move toward the peak, for maximizing the objective function, or toward the valley, for minimizing the objective function, over the shortest possible path, as sketched in Fig. 28a and b. At each step, starting with an initial trial point, the direction in which the objective function changes at the greatest rate is chosen for moving the location of the point, which represents the design on the multivariable space. The direction of the gradient vector ∇U is used since it is the direction in which U changes at the greatest rate. Thus, the number of trial runs needed to reach the optimum is kept small. However, an evaluation of the gradients is needed to determine the appropriate direction of motion. This often limits the application of the methods to problems where the gradients can be obtained accurately and easily. Numerical differentiation is generally needed for thermal systems, which are usually governed by nonlinear equations. Many methods, such as golden section, Fibonacci, and univariate search, that are based on function evaluations instead of gradients have also been developed and used, though these methods are often less efficient, in terms of trials needed, than gradient-based methods. Univariate search, which involves alternating between two variables, is sketched in Fig. 28c. In all these methods, function evaluations are used to narrow the region containing the optimum and finally selecting the optimal design from the reduced domain. Genetic algorithms are based on evolution of designs, heuristic arguments, natural selection, bio-inspired operators, and information base. They have become quite popular for thermal systems because of the typical high efficiency and rapid convergence of the algorithm, which is generally based on function evaluations rather than gradients (Rasheed et al. 1997).

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y

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The objective function U is among the most critical aspects to be decided in the optimization of thermal systems, since the optimal design is often a strong function of the chosen criterion for optimization. Frequently, optimization is carried out for different criteria, and the final design is chosen by comparison of results for different criteria. Each criterion leads to an optimization curve, and the envelope of the nondominant solutions is known as a Pareto set obtained by considering different criteria in a multi-criteria optimization problem. Then, the optimization is based on trading-off between different criteria. A practical knowledge of the system is helpful in obtaining the final solution. As an example, let us again consider the optical fiber drawing process. Following the establishment of a feasible domain, a detailed study was carried out by Cheng and Jaluria (2005) on the optimization of the process, taking the objective function as the fiber quality. This was quantified in terms of tension, thermally induced defect concentration, and velocity difference across the fiber, all these being scaled to obtain similar ranges of variation. The objective function U was taken as the square

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4.0 3.0 2.0 1.0 0.0 2200.0

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Fig. 29 Evaluation of optimal draw temperature at a draw speed of 15 m/s and the optimal draw speed at a draw temperature of 2489.78 K, obtained in the first part, by golden-section search separately for each variable

root of the sum of the squares of these three quantities and was minimized by using search methods. Figure 29 shows typical results from golden-section search for the optimal draw temperature and draw speed. The results from the first search are used in the second search, following the univariate search strategy, to obtain optimal design in terms of these two variables. Several other results were obtained on this complicated problem, particularly on furnace dimensions and operating conditions. Similarly, in a CVD system, the main aspects of interest include product quality, which is often characterized by the uniformity of the film, and the production rate, characterized by the deposition rate. These may then be represented by the percentage working area (PWA), which gives the percentage acceptable area from film uniformity, and the mean deposition rate (MDR). These quantities may be combined to yield a single objective function, which is optimized with given constraints, or each objective function may be considered separately to obtain the Pareto set. In the first case, the composite objective function may assume many possible forms, and the optimal design will generally depend on the function chosen, such as U = (PWA) x (MDR) or (product quality) x (operating cost)/(production rate). It is also possible to choose one of these objectives at a desired value and seek an optimum for the other. Figure 30 shows the results from such a strategy where the deposition rate or PWA is fixed, while the optimum is obtained for the other objective function. Again, a univariate scheme may be used to obtain the global optimum.

7.4

Response Surfaces

On the basis of simulation and experimental results, response surfaces may be generated to represent the overall behavior of the process (George and Ogot 2006). Figure 31a shows a typical response surface obtained for silicon deposition in a CVD reactor. Thus, this figure represents the dependence on the operating conditions. A corresponding optimal circumstance is shown on a two-dimensional

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Fig. 30 (a) Optimization of PWA at a chosen minimum value of the mean deposition rate, specified as a constraint, and (b) optimization of the mean deposition rate with a specified constraint on PWA

T (K)

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V*=0.76612 T*=1007.9273 V*=0.76612 T*=1007.9273 1500 1400 1300 120

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Fig. 31 Response surface model and an optimal operating point in a typical CVD process for depositing a silicon film

plot in Fig. 31b. Many different cases were investigated and the optimal conditions determined (Lin et al. 2009). A compromise response surface method (CRSM), which is a modification of the response surface method, has been developed to reduce the simulation runs needed for generating the response surface and thus increase the efficiency of the optimization strategy (George and Ogot 2006). This approach has been used for an impingement CVD reactor for gallium nitride thin film fabrication (George et al. 2015). Response surfaces were obtained for the average deposition rate and the uniformity parameter, which was defined in terms of the variation in film thickness across the susceptor. Typical results are shown in Fig. 32. Better uniformity is indicated by a lower uniformity parameter. Second-order response surfaces were used to model the deposition rate and the uniformity of the film thickness, using the inputs from the simulation. The resulting relationships were used to optimize the operating conditions, such as the susceptor temperature and flow rate.

7.5

Multi-Objective Optimization

In most practical thermal systems, several objective functions are of interest. For instance, in thermal systems for the cooling of electronic circuitry, the main design optimization objectives are maximizing the heat removal rate from the components and minimizing the pressure drop. Both of these are crucial for a successful implementation of the cooling system. Though many systems have been designed to maximize the heat removal rate, the pressure cannot be ignored, since increase in

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0.2

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Fig. 32 Response surfaces for (a) average deposition rate and (b) uniformity parameter, for the deposition of GaN in a vertical impingement CVD reactor

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Fig. 33 Multi-objective optimization with two objective functions f1 and f2, which are to be minimized, showing the nondominated designs, and the envelope of Pareto fronts for different geometric configurations

heat transfer usually comes at the cost of increasing pressure, which demands appropriate changes in the hardware such as pumps and blowers for the flow. The simple approach of combining the objectives into a single function, as outlined earlier, simplifies the problem. Weighted sums may be employed in the development of the combined objective function. This approach has been employed in many cases. If the two or more objective functions that are of interest in a given problem are considered and a strategy is developed to trade off one objective function in comparison to the others, a set of nondominated designs, termed the Pareto set, as shown in Fig. 33 for different geometrical configurations of a thermal system, is obtained. The Pareto set represents the best collection of designs generated. Then, for any design in the Pareto set, one objective function can be improved, at the expense of the other. The set of designs that constitute the Pareto set represent the formal solution to the multi-objective optimization problem. The selection of a specific design from the Pareto set is left to the designer or the engineer. Literature exists on utility theory, which seeks to provide additional insight into the decisionmaking process to assist in the selection of a specific design. Many multi-objective optimization methods are available that can be used to generate Pareto solutions. The use of this approach was demonstrated by Zhao et al. (2007) for the simple electronic system cooling problem shown in Fig. 23a. Figure 34 shows the corresponding results in terms of response curves for the two objectives, as well as the Pareto frontier for selecting the parameters through a trade-off between the two objectives. Similar results have been obtained for other thermal problems as well, such as microchannel flow systems mentioned earlier (Zhang et al. 2014). The references

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a

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Fig. 34 Response surfaces for (a) heat transfer rate, given in terms of the Stanton number, Nu/RePr, (b) pressure needed, and (c) Pareto frontier in terms of heat transfer rate and pressure drop for two heat sources in a channel flow, with L1 and L2 as the two geometry parameters

given here may be used for further information on different optimization techniques and available strategies.

8

Conclusions

This chapter discusses the main aspects that arise in the design of thermal systems. It presents the important considerations in the modeling, simulation, design, and optimization of thermal systems. Though extensive work has been done on simulation and experimentation in a wide range of thermal systems and processes, the effort

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on design has been relatively sparse. Since thermal systems generally involve many challenges in experimentation, as well as in numerical modeling, such as complicated domains, large property changes, combined mechanisms, and intricate boundary conditions, most studies use the results to understand the basic mechanisms and the system behavior. The next step of using these for design has often not been undertaken. Similarly, optimization of thermal systems has often been ignored since, once an acceptable design is obtained, the complexity of the system often dissuades further effort to achieve an optimal design. This chapter focuses on the design and optimization of thermal systems. The various steps in the design process and the basic strategies that may be adopted are discussed. Many examples are taken from different application areas, such as manufacturing, thermal management of electronic equipment, room fires, and microchannel flows, to illustrate the basic procedures and the typical results obtained. Since simulation forms a major source of inputs for design, an extensive discussion is devoted to the modeling and simulation, with validation of the model obtained by comparing with existing results and experimental data. The determination of feasibility of the process on the basis of simulation is discussed in order to achieve an acceptable design. Finally optimization, with respect to operating conditions and the system hardware, is presented in terms of conventional approaches, as well as recent ones using concurrent simulation and experimentation, response surfaces, multi-objective optimization, and knowledge base. Again, results for a few important practical problems are presented to illustrate the use of these approaches. The present state of the art is presented, along with future needs for design as well as optimization of thermal systems.

9

Cross-References

▶ Analytical Methods in Heat Transfer ▶ Constructal Theory in Heat Transfer ▶ Electrohydrodynamically Augmented Internal Forced Convection ▶ Free Convection: Cavities and Layers ▶ Free Convection: External Surface ▶ Full-Coverage Effusion Cooling in External Forced Convection: Sparse and Dense Hole Arrays ▶ Inverse Problems in Radiative Transfer ▶ Nanoparticles and Metal Foam in Thermal Control and Storage by Phase Change Materials ▶ Numerical Methods for Conduction-type Phenomena ▶ Phase Change Materials

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Acknowledgements The author acknowledges the support provided by NSF, through several grants, and by industry for the work reported here, the work done by many students, and interaction with several collaborators.

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Thermal Transport in Micro- and Nanoscale Systems Tanmoy Maitra, Shigang Zhang, and Manish K. Tiwari

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Heat Conduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Introduction to Boltzmann Transport Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Derivation of Fourier’s Law from BTE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Heat Conduction at Microscale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Thermoelectricity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Thermal Interface Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Heat Convection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Governing Equations and Dimensionless Numbers in Heat Convection . . . . . . . . . . . . 4.2 Single-Phase Convection at Microscale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Two-Phase Convection at Microscale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Summary and Future Trend . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Conduction at Microscale: Thermoelectricity and Thermal Interface Materials . . . . . 5.2 Single-Phase Convection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Phase Change Processes and Heat Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Cross-References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

Small-scale (micro-/nanoscale) heat transfer has broad and exciting range of applications. Heat transfer at small scale quite naturally is influenced – sometimes dramatically – with high surface area-to-volume ratios. This in effect means that heat transfer in small-scale devices and systems is influenced by surface treatment and surface morphology. Importantly, interfacial dynamic effects are at least non-negligible, and there is a strong potential to engineer the performance of T. Maitra · S. Zhang · M. K. Tiwari (*) Nanoengineered Systems Laboratory, UCL Mechanical Engineering, University College London, London, UK e-mail: [email protected]; [email protected]; [email protected] # Springer International Publishing AG, part of Springer Nature 2018 F. A. Kulacki (ed.), Handbook of Thermal Science and Engineering, https://doi.org/10.1007/978-3-319-26695-4_1

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such devices using the progress in micro- and nanomanufacturing technologies. With this motivation, the emphasis here is on heat conduction and convection. The chapter starts with a broad introduction to Boltzmann transport equation which captures the physics of small-scale heat transport, while also outlining the differences between small-scale transport and classical macroscale heat transport. Among applications, examples are thermoelectric and thermal interface materials where micro- and nanofabrication have led to impressive figure of merits and thermal management performance. Basic of phonon transport and its manipulation through nanostructuring materials are discussed in detail. Small-scale single-phase convection and the crucial role it has played in developing the thermal management solutions for the next generation of electronics and energy-harvesting devices are discussed as the next topic. Features of microcooling platforms and physics of optimized thermal transport using microchannel manifold heat sinks are discussed in detail along with a discussion of how such systems also facilitate use of low-grade, waste heat from data centers and photovoltaic modules. Phase change process and their control using surface micro-/nanostructure are discussed next. Among the feature considered, the first are microscale heat pipes where capillary effects play an important role. Next the role of nanostructures in controlling nucleation and mobility of the discrete phase in two-phase processes, such as boiling, condensation, and icing is explained in great detail. Special emphasis is placed on the limitations of current surface and device manufacture technologies while also outlining the potential ways to overcome them. Lastly, the chapter is concluded with a summary and perspective on future trends and, more importantly, the opportunities for new research and applications in this exciting field.

1

Introduction

Heat transfer occurs naturally between two bodies with a temperature difference. Heat transport from the hot to the cold body follows three different modes: conduction, convection, and radiation. In the conduction heat transfer, heat is transported when these two objects (hot and cold) are in intimate contact with each other; heat is transported by the vibration of the object molecules. Conduction is the most significant means of heat transfer in solids. On the other hand, heat can also be transported by the movement of the fluid from one place to another, called advection. In the convection mode, heat transfer occurs by advection in combination with conduction. The movement of the fluid can be initiated naturally, for example, by the buoyancy force originating from the density difference between hot and cold part of the fluid. The fluid movement can also be forced by external means, such as a pump, a fan, or a suction device. In the radiative mode of heat transport, the energy transfer occurs via an electromagnetic wave, without the need for a physical medium. In this chapter, the discussion is limited to the heat conduction and heat convection, only. First, the basic principle of the heat transfer at a large scale is introduced

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with exemplifying a few practical problems where heat conduction and convection are common. Second, heat conduction and convection at microscale are extensively discussed. In this section, it has been showed that how the basic heat principle is modified to describe heat transport at small scale (micro- and nanoscale regime). In the heat convection part, particularly, the discussion is split in two parts: small-scale convection without and with phase change. After the subsequent discussion of each mode of heat transfer, a few applications are given where the micro-/nanoscale heat transport is common. At the end, the chapter is concluded with a summary and perspective on the scope of future research.

2

Heat Conduction

Heat conduction is commonplace in nature and man-made applications. There are plethora of industrial processes where the heat conduction is significant, such as during casting metals and their alloys (Oksman et al. 2014), polymer melt processing, heat transfer through extended surfaces (“fins”), etc. Conduction is also employed frequently in gas turbines, air conditioning units, and cryogenic coolers where air is being used as a coolant. In these cases, heat is transported largely with the aid of convection. The low heat transfer coefficient of the air, however, limits the efficiency of these equipment, and, often, extended surface area or fins (Kraus et al. 2001a and also Free Convection-External Surface) are used on the heated side to augment the heat transfer. Fourier’s law is used to describe conduction heat transfer process at large scales (see also Part 1, “Heat Transfer Fundamentals”). In the following, the fundamental underpinnings of Fourier’s law, which is a phenomenological law, are discussed using the Boltzmann transport equation (BTE), which is the statistical description of the molecular motions that facilitate heat conduction.

2.1

Introduction to Boltzmann Transport Equation

The heat transfer processes, such as heat conduction and convection, are described by phenomenological laws, such as Fourier’s law and Newton’s law of cooling in combination with the first and second law of thermodynamics (Chen 2005). These laws are typically seen as the outcome of experimental observations (Chen 2005). However, they can be derived from the statistical description of particles (molecules) facilitating the heat transfer. The distribution of particles in a material is described by a non-equilibrium function that depends on energy, temperature, and position of particles. The corresponding equation is called Boltzmann transport equation (BTE). In the following, the BTE and the derivation of Fourier’s law are discussed briefly. A particle with wave number k, at time t, is located at a position r. Suppose, at time t + dt, the particle moves to a position r0 with a wave number of k0 due to the existence of internal and external forces (F). Then the distribution function f(r, k, t) that gives the probability of finding of a particle at a particular “state” (position and the wave number) can be solved from the following:

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  where @@ ft

@f 1 þ ∇r f v þ F ∇k f ¼ @t ℏ scat


 @f @t scat

(1)

is the variation of f due to scattering from the internal particles, such

as electrons, photons, and phonons. Particularly, phonons, which correspond to vibration modes in a crystal lattice, play a significant role in conduction heat transfer in solids. Interestingly, it was found that phonons heat conduction at nanoscale is significantly different from their bulk counterpart, will be discussed in details later. Particle scattering   is a time-dependent process, and the rigorous way to solve the scattering term @@ ft is to solve time-dependent Schrodinger equations of partiscat

cles. This approach makes Eq. 1 to an integrodifferential equation that is extremely difficult to solve. However, a relaxation time approximation often simplifies the scattering term. This approximation assumes that if a system is thrown out of equilibrium such that f  f0 is nonzero, collisions restore equilibrium with the dynamics following an exponential decay f  f0  et/τ, where f0 is the equilibrium distribution function of the particles, such as the Boltzmann, the Fermi-Dirac, and the Bose-Einstein distributions, and τ denotes relaxation time (the time constant for a particle to relax back to equilibrium state from a non-equilibrium state). Generally, τ is a function of r and momentum ( p). The relaxation time approximation linearizes the scattering term in Eq. 1.

2.2

Derivation of Fourier’s Law from BTE

The rate of energy flow per unit area can be obtained by integrating the product of f, velocity vector v(r, t), and the particle energy e( p) over the momentum space. With the introduction of a density of states, D(e), the energy flux q(r, t) can be written as ð q ¼ vðr, tÞf ðr, e, tÞeDðeÞde:

(2)

Additionally, if τ is assumed to be independent of velocity of particles, and particle frequency, in the absence of any external field (such as temperature or electric field), f can be expressed as from Eq. 1 f ¼ f0  τ

@f  τ∇r fv @t

(3)

Now, assuming f is independent of t, and from Eqs. 2 and 3, q can be expressed as ð

ð

q ¼ veDf 0 de  τ∇r f v2 eDde

(4)

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Since f0 represents the equilibrium distribution and it contributes an equal amount to the energy going in and out in all spatial directions, the first term of Eq. 4 drops out after the integration. Thus, Eq. 4 can be written as ð q ¼ τ∇r f v2 eDde Considering f being independent of T, ∇r f ¼ becomes qþτ

(5) @f @T

@q ¼ k∇T @t

dT  dr



@ f0 @T

T . Now, Eq. 5

(6)

where the total thermal conductivity(k) is defined as ð

k ¼ τv2

@f 0 De de: @T

(7)

Equation 6 is called Cattaneo’s equation, and this derivation is based on the critical assumption of ∇f  ∇f0 , i.e., the gradient of f and f0 is the same, although f changes with time. Cattaneo’s (Eq. 6) is valid for a thermal perturbation that exists for the time scale comparable to τ . However, at steady state, the second term of the left side of the Eq. 6 can be neglected, and Eq. 6 simplifies to Fourier’s law of heat conduction q ¼ k∇T

3

(8)

Heat Conduction at Microscale

The progress in micro- and nanotechnologies has led to a number of small-scale heat transfer devices where heat conduction plays an important role. Two broad application categories can be identified. The first application involves thermal management where the transport of heat generated within micro-/nanoscale devices such as sensors, transducers, integrated circuits (ICs), etc., is crucial to maintain device efficiency, functionality, and reliability. Electronics cooling and semiconductors laser (laser diodes) are a few specific examples of thermal management problems. In the second type of applications, micro- and nanostructures are employed to manipulate the heat flow to maximize energy conversion efficiency; thermoelectric energy conversion and photovoltaic power generation are typical examples. In a number of these cases, Fourier’s law is inadequate to describe the heat transport process, largely, because the heat transport at this scale is associated with several non-equilibrium processes occurring at a time scale much less than the τ and a length scale less than the mean free path. At macroscale, all particle-particle interactions occurring at a time scale less than τ are averaged out over the time scale of interest, which is typically much higher than τ

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Fig. 1 Heat transfer processes occurring at different length scales: the leftmost shows fluid flowing through a heated pipe. Here, temperature field is continuous as all particle-particle interactions are averaged out over time scale (t) much higher than the relaxation time (τ); toward the right, temperature field is not continuous anymore as the mean free path (λ) and relaxation time (τ) approach to system dimension (d ) and time scale of interest (t). Here, macroscale heat transfer analysis is not valid, and micro-/nanoscale heat transfer needs to be considered

(Fig. 1). Therefore, the temperature field can be assumed continuous. At microscale, however, the time scale of interest is in the same order of τ , and therefore, the individual particle-particle interactions need to be considered to determine the temperature field (Fig. 1). When liquid flows through a heated pipe, Fourier’s law is commonly used to determine the temperature field of the liquid, and there is no need to consider the individual interactions between the fluid molecules or between the fluid molecules and the wall, since the time scale of interest is much higher than τ. On the other hand, Fourier’s law is inadequate to study the heat transfer mechanism, for example, in an animal cell, a crystal lattice, or electron-phonon interactions in a semiconductor (Fig. 1), as the system dimension is of the order of λ .However, the general form of BTE (Eq. 1) can be utilized to study heat transport at microscale. In these cases, to describe the equilibrium distribution function ( f ) , either Fermi-Dirac or Bose-Einstein or Boltzmann distributions are used depending on type of particles under consideration. The approach enables a first principle derivation of the physical properties of materials such as thermal and electrical conductivity, heat capacity, etc. This is briefly outlined below. The Fermi-Dirac equilibrium distribution function is used to derive the electrical conductivity from BTE (Eq. 1). Considering the electric field, E and no temperature field, the expression of electrical conductivity (σ) can be derived from the equation of energy flux Eq. 2 are as follows (Chen 2005) σ¼

2e2 τðμF ÞμF DðμF Þ 3me

(9)

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where μF is the Fermi energy, e is the electronic charge, and me the mass of an electron. Here, electrons, which have energies close to Fermi energy, are contributing to the electrical conductivity of the material. Therefore, τ is calculated on the basis of only Fermi electrons. The thermal conductivity (k) , however, has two parts – electron thermal conductivity (ke) and phonon thermal conductivity (kL). In metals, electrons play an important role in heat conduction, and kL is negligible. Considering again the Fermi-Dirac distribution function, in the presence of uniform temperature gradient in one direction and with no electric field, electron thermal conductivity can be expressed as (Chen 2005) ke ¼

2 π 2 ðkB T Þ2 τðμF ÞμF DðμF Þ 3me T 3

(10)

where kB is the Boltzmann constant and T is the temperature field along one direction. Like the derivation of σ, electrons that have energies close to Fermi energy only contribute to the electronic thermal conductivity. Unlike metals, electrons play insignificant role in heat conduction in semiconductors and insulators, and therefore the total thermal conductivity, k, is approximated to the phonon thermal conductivity, kL, which can be calculated by the following equation. Considering kinetic theory, and the Bose-Einstein distributions, fBE (as phonons are Bosons), kL can be expressed as (Chen 2005) kL ¼

1 3

ð vm 0

τv2a hv

@f BE DðvÞdv @T

(11)

where vm is the maximum Debye frequency and defines the maximum frequency of the phonon in the crystal and va is the average velocity of a phonon. The detail derivation of Eqs. 10 and 11 can be found elsewhere (Chen 2005). Assuming τ and va are independent of temperature, the remaining part in the integral of Eq. 11 is the specific heat per unit volume, which can be expressed as cv ¼

ð1 0

ℏv

@f BE DðvÞdv @T

(12)

Studies show that cv has a complex dependency on T . At low temperature, cv depends on T3 . At high temperature, however, cv depends on Tn , where n = 1–1.5. The physical properties captured by Eqs. 8, 9, 10, and 11, thermal conductivity in particular, are amenable to change by suitably engineering the materials composition and/or dimension, which directly affect the density of states. This offers some unique and exciting potential applications in the field of energy and nanotechnology. In the next two sections, it will be shown how such micro- and nanostructures are employed to manipulate heat conduction at small scales in two exemplar applications, namely, thermoelectric and thermal interface materials.

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Thermoelectricity

The above microscale heat transport analysis is of direct relevance to thermoelectricity where interconversion of electrical and thermal energy is achieved without using any moving parts. The direct interconversion between heat and electricity can be uniquely valuable in a number of applications such as exploitation of waste/lowgrade heat to generate electricity, refrigeration, and cooling without moving parts, etc. Although the principle has been known since long (DiSalvo 1999), limitation of the thermoelectric materials to rare earth materials was a major shortcoming, not to mention the low energy efficiency of these devices. However, with development of micro- and nanofabrication techniques, significant advances were achieved very recently – in the last few decades (Poudel et al. 2008; Vineis et al. 2010). The idea of micro-/nanostructuring started with pioneering calculations and modeling in the early 1990s showed that due to quantum confinement of electrons in low-dimensional – micro-/nanostructured – materials, S2σ can be increased independently (Hicks and Dresselhaus 1993; Hicks et al. 1993). Interestingly, when the dimension of materials is comparable to phonon’s mean free path (50–300 nm at room temperature) (Balandin and Nika 2012), the phonon transport is dominated by the boundary scattering. Importantly, in this case, phonon thermal conductivity directly scales as the system dimension. Moreover, in low-dimensional materials, electron mobility is also adversely affected by the spatial confinement of phonons. Thus, nanostructuring and, consequently, by altering phonon’s boundary scattering can be used to engineer phonon thermal conductivity and electrical conductivity of any material (Ziman 2001). Given the fact that silicon is the second most abundant material on earth and, thanks to progress in microelectronics industry, has established micro-/nanofabrication protocols, it has been widely explored as a thermoelectric material. The performance of a thermoelectric materials is quantified by a unitless figure of 2 merit (ZT) , which is defined as σSk T , where S is the Seebeck coefficient. In most of material systems, σ, S, and k are interrelated, and therefore, each parameter cannot be changed without altering others. This interdependence and counterbalancing trends make it difficult to enhance ZT, and therefore, this is an active area of research. Advances made to this end with micro-/nanostructuring are discussed in the following subsections, which capture the broad strategies employed for improving the ZT coefficient.

3.1.1 Low-Dimensional Materials: Superlattices and Nanowires Superlattices are 2D (layered) nanostructured materials (see Fig. 2), which were the first to be used as a low-dimensional thermoelectric material. Numerous superlattice material systems such as Bi2Te3/Sb2Te3 (Venkatasubramanian et al. 2001), PbTe/ PbSexTe1-x (Vineis et al. 2010), Si/Si1-x-yGexCy (Fan et al. 2001; Vashaee and Shakouri 2007), etc., have been established to increase ZT. Some researchers have also shown that ZT can be further augmented by decreasing superlattice periods down to 3 nm (Harman et al. 1999). In this context, Luckyanova et al. were first to

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a

c

14

30K 40K 60K 80K 150K 250K 296K

12 10 8 6

Thermal conductivity (W/m-k)

Thermal conductivity (W/m-k)

b

4 2 0

1

3 5 7 Number of periods

9

14 12 10 8

1 pd 3 pd 5 pd 7 pd 9 pd

6 4 2 0

100 200 Temperature (K)

300

Fig. 2 Thermal conductivities of superlattices: (a) cross-sectional transmission electron microscopy (TEM) image of AlAs/GaA 3-period superlattice. Inset shows the high-resolution TEM image at the interface of GaAs and AlAs. (b) Variation of experimentally measured thermal conductivity at different temperatures of superlattices with varying periods. (c) Variation of thermal conductivity with temperature at different number of lattice periods (Reproduced by the permission of the American Association of the Advanced of Science (Luckyanova et al. 2012))

establish that quantum confinement in superlattices significantly affect the phonon transport (Luckyanova et al. 2012) and thus the conduction process. In fact, the phonon transport in superlattices can be either coherent or incoherent. For example, phonons traverse internal region of the film of superlattices ballistically leading to a coherent transport, whereas incoherent phonon transport occurs at the boundary of superlattices as phonons are scattered diffusely at the boundary. In most superlattices, incoherent phonon transport dominates over the coherent transport. The phase information of phonons is lost in the incoherent transport. However, the periodicity of superlattices can control the phonon transport, which in turn influences phonon’s thermal conductivity, thus raising the possibility of engineering the thermoelectric performances (Casimir 1938; Cahill et al. 2003). In fact, Luckyanova et al. (2012) showed that at a temperature below 150 K, the thermal conductivity of superlattices increases as the number of lattice periods increase, as shown in Fig. 2a, b. Figure 2c shows that thermal conductivities increases as the number of lattice periods increases up to the temperature of 150 K indicating the dominance of ballistic transport of phonons. Boundary scattering also extends to one dimension (1D, nanowires of a material). Lower thermal conductivities of nanowires compared to the bulk are achieved

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because the transport of electrons and phonons, particularly those, which have mean free path in the same order of magnitude as that of nanowires diameters, is severely impeded. However, phonons that have smaller mean free path compared to diameters of nanowires are practically unaffected. Therefore, there is a possibility to reduce further thermal conductivities of nanowires by suppressing the scattering of those short-length phonons. One possible approach to deplete those phonons is to reduce nanowires diameters. There has been active research into thinner nanowires for thermoelectrics, as, for example, group III and IV nanowires, to further enhance the quantum confinement of electrons (Li et al. 2003; Lin and Dresselhaus 2003). In this regard, experimental and theoretical study indicated that indeed higher ZT values can be obtained with thinner nanowires, although fabricating very thin nanowires, down to 5 nm, is a challenge. To this end, the prevalence of silicon (Si) in microelectronics, and the related micro-/nanomanufacturing know-how, has raised exciting possibilities of using them as efficient thermoelectric materials. Thus, the thermal and electrical properties of Si nanowires have been studied widely. Figure 3 shows an example of work by demonstrating the thermal conductivity reduction in Si nanowire. Although subsequently improved through roughening the nanowires (Fig. 3a, b, see also the next subsection), Li et al. (2003) were the first to synthesize “smooth” Si nanowires using vapor-liquid-solid technique and show that with reduction in wire diameter the thermal conductivity could be reduced by two orders of magnitude compared to bulk (black symbols in Fig. 3c).

3.1.2 Nanostructuring: At Surface and Bulk Nanostructuring is an additional strategy which can introduce numerous grain boundaries and interfaces in the materials, which enhance phonon boundary scattering and, thus, lead to a significantly reduction in thermal conductivity and enhancement of ZT (Rowe et al. 1981; Dresselhaus et al. 2007; Lan et al. 2010). Therefore, nanostructuring has been applied not only to low-dimensional materials (e.g., nanowires) but also to bulk nanograined materials (Lan et al. 2010). Hochbaum et al. (2008) showed that surface nanostructuring of silicon nanowires led to a significant reduction in thermal conductivities due to enhanced phonon boundary scattering. These rough Si nanowires were produced by electroless etching technique and roughened through surface treatment, which is evident from the high-resolution TEM image in Fig. 3b. Figure 3c shows the effect of diameters of silicon nanowires on experimentally measured thermal conductivities at different temperatures. Below 150 K and at a particular nanowire diameter, the thermal conductivities of roughened silicon nanowires are smaller compared to those nanowires that are not surface treated and produced by vapor-liquid-solid (VLS) technique (Li et al. 2003). The roughness/surface nanostructure was investigated further by Lim et al. (2012). They fabricated silicon nanowires by VLS method and subsequently employed two different etching processes – one comprising galvanic deposition of silver at the surface boundaries of nanowires followed by etching with hydrofluoric acid (HF) and the other focusing on wet etching in the mixture of HF/H2O2 and AgNO3 – induced two distinctly unique surface roughnesses. The study presented

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b

c

Vapour-liquid-solid nanowires Electroless etching nanowires

50

k (W m–1 K–1)

40

115 nm

30 56 nm

20 37 nm

10

115 nm 98 nm 50 nm

0 0

100

200

300

Temperature (K)

Fig. 3 Silicon nanowires for thermoelectrics: (a) cross-sectional SEM image of silicon nanowires and (b) high-resolution transmission electron microscopic (TEM) image of silicon nanowires after the surface treatment. The roughness of silicon nanowires is clearly visible in the TEM image. (c) Experimentally measured thermal conductivities of such roughened silicon nanowires of different diameters as a function of temperatures. Silicon nanowires produced two different techniques, vapor-liquid-solid (VLS) and electroless etching (EE) techniques (Reproduced by the permission of Nature Publishing Group (Hochbaum et al. 2008) and American Institute of Physics (Li et al. 2003))

a quantitative relationship between the nanowire length, diameters and root mean square roughness, and thermal conductivities. Another strategy to reduce thermal conductivity is to introduce defects, in the form of impurities, to affect those short-length phonons (Lee et al. 2012; PerezTaborda et al. 2016). Lee et al. (2012) showed a significant reduction of thermal conductivities of silicon nanowires by alloying with germanium (Ge). Figure 4a, a high-resolution TEM image Si-Ge composite nanowire, is clearly showing the grain boundary between the two components. Figure 4b plots thermal conductivities of Si-Ge composite nanowires (with different compositions), synthesized from bulk

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Fig. 4 Silicon-germanium (Si-Ge) composite nanowires: (a) high-resolution TEM image of Si-Ge composite nanowire. Inset shows electron diffraction pattern obtained from a selected area of Si-Ge nanowire. (b) Thermal conductivities of bulk Si-Ge nanowires at varying Ge content in Si at different temperatures (Reproduced from Dismukes et al. 1964; Li et al. 2003; Kim et al. 2010b). Inset shows the thermal conductivities of Si-Ge bulk nanowires at different Ge content as performed by Lee et al. (2012) (Reproduced by the permission of the American Chemical Society (Lee et al. 2012))

composites. It is clear from experimental data that the effect of Ge content on the overall thermal conductivities is apparently below 100 K temperature. However, σ and S are also sensitive to impurities. Therefore, Lee et al. also measured thermal conductivities as well as σ and S simultaneously from the same nanowires. Impressively, bulk Si-Ge composite nanowires showed a significantly improved ZT, around 0.46 at 450 K. The advances using 2D and 1D nanostructured materials clearly demonstrate a path to engineer thermoelectric materials. However, scalability to large systems is an issue. Bulk nanostructured materials are an alternative and address the issue of scalability. The approach was elegantly demonstrated by Poudel et al. (2008), who introduced bulk alloy BiSbTe as efficient thermoelectric materials. Essentially, nanopowders were prepared by simple ball milling of a block of p-type BiSbTe, followed by hot pressing them to form a bulk ingot with a large number of scattering grain boundaries. The thermal conductivity of the resulting nanostructured ingot was significantly lower than its homogenous counterpart, and, impressively, a ZT value

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of 1.4 at 100  C was achieved. The approach is by no means limited to BiSbTe; systematic reviews document research on materials, establishing the generality of this approach (Minnich et al. 2009; Sootsman et al. 2009). In addition to extensive use of inorganic materials, there have also been a number of reports focusing on alternate materials such as conjugated organic polymers as thermoelectric materials. Immediate advantages of using organic polymers over inorganic materials are their light weight, good mechanical properties, and low production cost (McGrail et al. 2015), which are desirable in realizing the next generation of large-scale flexible electronic devices and also realizing lightweight power sources. Conjugated organic polymers, such as polyaniline (Kaneko et al. 1993), polyacetylene (Yoon et al. 1995; Mateeva et al. 1998), polypyrrole (Kemp et al. 1999), polythiophenes (Hu et al. 2013), etc., show a good electrical conductivity by charge delocalization across the polymer backbone. To enhance thermoelectric performances (ZT) , electrical conductivity of organic polymer needs further improvement. This can be achieved by blending with other polymers (Kim et al. 2013), nanoparticles and nanowires (Moriarty et al. 2013; Kim et al. 2010a, b), inorganic salts (Sun et al. 2011; Bissessur et al. 1993), etc.

3.1.3 Doping Given the ease of nanofabrication with Si, numerous attempts have been made to modify Si nanowires, either by doping with other p-type and n-type materials or altering the surface chemistry, to augment the electrical conductivity and simultaneously to reduce the thermal conductivity (Donadio and Galli 2009; Huang et al. 2007; Balasubramanian et al. 2011; Pokatilov et al. 2005; Pan et al. 2015, 2016). In this regard, Pan et al. (2015) fabricated a silicon nanowire forest via a simple technique, using polystyrene spheres as sacrificial templates, and followed by a gold-assisted silicon etching technique. The fabricated silicon nanowires are coated with germanium (Ge), an n-type dopant, and followed by a thermal annealing step to enhance the interaction between Ge and Si. A SEM image of such a silicon nanowire forest and a high-resolution TEM image of Ge-doped silicon nanowires are shown in Fig. 5a, b. Experimental measurements, as shown in Fig. 5d, clearly indicate a reduction in thermal conductivities of silicon nanowires after doping with Ge and a further reduction after the thermal annealing step. Thermal conductivity and electrical conductivity measurements have also been performed by the same research group (Pan et al. 2016) on free-standing silicon nanowires where nanowires were suspended between a heating and sensing pad. A SEM image of such free-standing silicon nanowires is presented in Fig. 5c. Interestingly, surfaces of free-standing nanowires are modified by different surface modification techniques, such as oxygen plasma, vapor phase HF treatment, and n-type surface charge transfer doping. The experimental results, Fig. 5e, f, showed an enhancement of electrical conductivity and a simultaneous reduction in thermal conductivity of surface-modified silicon nanowires. In addition, the n-type surface charge transfer-doped silicon nanowires had a higher electrical conductivity and a lower thermal conductivity at a specific temperature compared the nanowire modified by the remaining techniques (Pan et al. 2016).

Fig. 5 Doped silicon nanowires: (a) scanning electron microscopic (SEM) image of the silicon nanowire forest, (b) high-resolution TEM image of a germanium-doped silicon nanowire, (c) SEM image of free-standing silicon nanowires, (d) thermal conductivity measurements of pristine silicon nanowire and Ge-doped silicon nanowires, and (e and f) electrical and thermal conductivity measurements of free-standing silicon nanowires with different surface treatments, such as oxygen plasma, vapor HF, and doping with Co(Cp)2 (Reproduced by the permission of the American Institute of Physics (Pan et al. 2015) and American Chemical Society (Pan et al. 2016))

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Generally, the reduction in thermal conductivities due to the surface doping results from the enhancement of the incoherent phonon transport. The effect can also be observed with thin silicon films. For example, Asheghi et al. (2002) reported 80% reduction in thermal conductivity of doped silicon compared to pure crystalline silicon. Dopants, however, also increase the number of electrons (with n-type dopants) and holes (p-type dopants), resulting in a potential decrease in σ and S and, thus, ZT. Note the dopants also increase ke, relatively modest influence on the overall thermal conductivity (Vineis et al. 2010). Therefore, the use of dopants requires a careful trade-off (Vineis et al. 2010).

3.2

Thermal Interface Materials

A thermal interface material (TIM) facilitates heat conduction between two components. TIMs have a plethora of industrial applications where generated waste heat needs to be transferred effectively to heat sinks. In these applications, TIMs provide a conductive path from one material (heat source) to the other (heat sinks). Therefore, thermal resistance and mechanical property, particularly stiffness, are two crucial metrics of materials to be used as a TIM. The thermal resistance of TIMs needs to be low (Marconnet et al. 2013). Stiffness of TIMs, on the other hand, needs to be low to accommodate the mismatch of thermal expansion between hot and cold components (Marconnet et al. 2013). Generally, metals have low thermal resistances, but high stiffness. Polymers, on the contrary, possess low stiffness, but high thermal resistances. Industries have been using different materials like solders (metal alloys), thermal pastes (polymer composites with high thermal conductance), and phase change materials. However, the high thermal resistances offered by those commercially available materials merely satisfy the current industry’s need, particularly for microelectronic industry where the need for miniaturization and high computing speeds lead to the production of huge amount of process heat within a very small area. Nanostructured materials could be a promising future candidate for TIMs due to their extraordinary physical properties, particularly high thermal conductivity and superior mechanical properties: the most exciting candidates are discussed in more detail next.

3.2.1 Carbon Nanotube and Other Nanoparticles in Polymer Matrix The common nanostructured TIMs are fabricated by combining nanoparticles fillers with superior thermal conductivities and polymers with lower stiffness. To this end, carbon nanotubes (CNTs), i.e., sp2-hybridized carbon allotropes, which are formed by a roll-up of single- or multilayer graphene (planar layer of hexagonally arranged carbon atoms), have elicited great interest as filler particles due to their very high conductivity. The experimental investigations on the evaluation of thermal performances of CNT-polymer composite, however, only show moderate increase in the effective thermal conductivity compared to the theoretically predicted values. The discrepancy can be attributed (McNamara et al. 2012) to:

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Fig. 6 Carbon nanotube (CNT) and graphene-based polymer composite thermal interface materials: (a) vertically aligned CNT forest used as a thermal interface material (TIM). Inset shows the high resolution of SEM image of CNT forest; some CNTs are bended at the interface resulting to poorer contacts and high boundary scatterings. (b) SEM image of graphene-based TIM (Reproduced by the permission of Elsevier Masson SAS (Shahil and Balandin 2012b))

1. Random dispersion of CNTs in the polymer matrix; i.e., not all CNTs are participating in the heat conduction event. 2. Inadequate CNT loading and/or poor dispersion quality of CNT in the polymer. 3. High boundary resistance between CNT and polymer. The effective thermal conductivity of the polymer-CNT composite can be enhanced by improving the dispersion quality (Choi et al. 2003; Marconnet et al. 2011). Theoretical studies indicate that vertically aligned CNTs (VACNT) (Balandin 2011; McNamara et al. 2012; Marconnet et al. 2013) stand-alone, as shown in Fig. 6a, may be used as TIM due to higher intrinsic conductivity of CNTs along their axial direction. CNTs were grown directly on metal substrates, and thermal measurements indeed showed higher effective thermal conductivity compared to polymer-CNT composite. However, there are some issues with VACNT that limit their practical exploitation (McNamara et al. 2012): 1. Mean free path of phonons in CNT is 1.5 μm and the area of contact between CNT and substrate only a few nm. Therefore, the ballistic transport of phonon is important along the axial direction of the CNT. However, the measured thermal conductance is much smaller than the actual thermal conductance of CNT. This discrepancy is largely due to the higher CNT boundary resistance. Phonons scatter at the boundary (see Sect. 3.1) and thus increase the boundary resistance. 2. CNTs are mostly grown by chemical vapor deposition (CVD) technique. The resulting CNTs are not exactly the same length, and therefore some CNTs cannot form a good mechanical contact to substrates due to bending at the interface (inset of Fig. 6a).

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3. Inherent acoustic mismatch between CNT and substrates leads to a high phonon scattering. 4. Adhesion between substrates and CNT is weak resulting in a weak heat conduction path. Potential strategies to overcome the issue of poor CNT/substrate contact include chemical modification of substrates (Lin et al. 2008; 2010), which also helps limit phonon’s boundary scattering.

3.2.2 Graphene-Based TIMs Graphene, one-atom-thick sp2-hybridized carbon atoms arranged in a planner sheet, has the higher thermal conductivity than CNT (Shahil and Balandin 2012a, b). Therefore, graphene, as an alternative to CNT, can be used as TIM. In fact, graphene-based polymer composites (Fig. 6b) show better thermal conductivity compared to CNT-polymer composites. This superior behavior is due to the availability of π electrons on graphene surfaces (since graphene is sp2 hybridized) and the planner geometry graphene. The latter facilitates better contacts between graphene and polymer matrix, and thus phonon scattering at the boundary of graphene decreases (Shahil and Balandin 2012a, b). However, the effective thermal conductivity of graphene-polymer composite depends on various parameters, such as surface roughness of graphene, its quality (single or double layer, etc.), etc. (Minnich et al. 2009), which need to be controlled carefully (Shahil and Balandin 2011, 2012a).

4

Heat Convection

In the previous section, the discussion was centered on the heat conduction where the molecular distribution and vibration are important. However, the heat can also be transported through the movement of fluid, i.e., by convection. For example, a hot liquid which is flowing through a pipe heats up the pipe wall. Generally, Reynolds number (Re) – ratio of the inertial force to the viscous force – is used to characterize the fluid flow. In small scales, focused herein (for discussion on large-scale convection, see ▶ Chap. 8, “Single-Phase Convective Heat Transfer: Basic Equations and Solutions”), the viscous forces dominate and Re is small; examples include polymer extrusion processing, electronics cooling, etc. In extrusion processing (Pearson 1985), a polymer is melted and pushed through a metallic channel and an orifice of a given shape. The viscous dissipation is high due to the low flow rate and the high viscosity of the polymer melt. Interestingly, heat convection at low Re is also important in the geophysics, where the flow of the earth’s mantle is studied. The mantle (Birch 1952) is the viscous earth’s interior. The temperature of the mantle can be as high as 500  C. The heat is transported from the earth’s interior to the surface by convection. As the mantle is viscous and it flows slowly, the convective heat transfer resistance is higher than the conductive one (Gebhart 2017; Birch 1952).

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Under the conditions of low Re, the governing physics and equations are amenable to some simplifications, which are discussed in greater details next.

4.1

Governing Equations and Dimensionless Numbers in Heat Convection

For the incompressible flow, in a steady-state process, continuity equation can be written as (Beek et al. 1975) ∇v ¼ 0

(13)

With constant density, the fluid momentum is governed by the Navier-Stokes equation, which equates the rate of change of fluid momentum with net forces on a fluid element and can be written as (Beek et al. 1975) ρ

Dv ¼ ∇p  ∇τ Dt

(14)

D where Dt is called substantial time derivative and is defined as (Beek et al. 1975)

D @ @ ¼ þ Σvi Dt @t @i

(15)

with vi denoting the velocity of the fluid in ith direction. The first term of the lefthand side in Eq. 14 represents the inertial force; the first and the second terms in the right-hand side denote pressure and viscous force, respectively. At steady state, and for Newtonian fluid, Eq. 14 yields (Beek et al. 1975) ρ ðv:∇vÞ ¼ ∇p þ ∇ðμ:∇vÞ

(16)

where μ denotes the fluid viscosity. Neglecting potential energy and kinetic energy, the thermal energy balance equation can be written as (Beek et al. 1975) ρ

DU ¼ ð∇:qÞ  ∇ðp:vÞ  ∇:ðτ0 :vÞ Dt

(17)

U, q and τ0 denote internal energy, amount of heat energy input by conduction, and shear stress of the fluid, respectively. Considering U = U(V, T), at constant pressure and incompressible liquid, the left-hand side of Eq. 17 becomes ρ

DU DT ¼ ρcp Dt Dt

Combining Eqs. 17 and 18 yields

(18)

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(19)

The last term on the left-hand side of Eq. 19 is the energy term associated with viscous dissipation and can be represented via a function of velocity gradients (Beek et al. 1975), μΦv, where Φv denotes the so-called dissipation function. Therefore, Eq. 19 becomes ρcp

DT ¼ k∇2 T  μΦv Dt

(20)

Let us define dimensionless terms T, v, t and xˇ as T t ˇ x T  ¼ ΔT ; v ¼ vvˇ ; t ¼ ðD=ˇ v Þ and x ¼ D

ˇ and D denote the characteristic temperature differential, velocity, and where ΔT, v, length scale, respectively. With the introduction of these nondimensional terms, Eq. 20 becomes (Yazicioglu and Kakaç 2010) DT  1  2  Br  ∇ T þ Φ ¼  Pe Pe Dt

(21)

where the Peclet number (Pe) , the product of Reynolds number (Re) and Prandtl number (Pr) , signifies the relative importance of convective heat transfer in the fluid and heat conduction from the wall to the fluid. At low Re , Pe can be made higher by reducing the characteristics system dimension (D). Brinkman number (Br) is the ratio of viscous dissipation to heat conduction from the wall to the fluid. At low Re, Br becomes relevant as the axial conduction becomes important. The sections following next will focus on application specific use and simplification of the above equations.

4.2

Single-Phase Convection at Microscale

Microscale heat convection has many applications such as cooling of micro-/ nanoelectronics, photovoltaic cells, laser diodes, etc. Newton’s law, q = hΔT where h denotes the heat transfer coefficient of a fluid, describes overall performance of heat convection processes (Kraus et al. 2001b). The heat transfer coefficient h is obtained by solving the Navier-Stokes (momentum) and energy equations, with an assumption of no fluid slippage at solid walls. However, at microscale, the boundary conditions may need modification, depending on the Knudsen number (Kn) (Yazicioglu and Kakaç 2010), which is defined as the ratio of mean free path of fluid molecules λ to the characteristic length scale of the system, e.g., the channel diameter D. Essentially, the Kn is used to characterize the flow regime. If Kn < 0.001, flow can be considered to be continuum where conventional no-slip condition at the wall is maintained. For 0.001 < Kn 10, the flow is dominated by molecular diffusion, and the heat flux and temperature gradient cannot maintain a linear relationship. In fact, the fluid and the solid cannot retain thermodynamic equilibrium at the interface, and the no-slip condition is not valid. Navier-Stokes equations, however, can be applied with the “slip” boundary condition. As an additional feature, as the system dimension becomes smaller, Pe also gets smaller leading to a higher convective heat transfer resistance inside the fluid as compared to the conductive heat transfer resistance. Furthermore, at the small scale, the viscous dissipation becomes significant, and hence Br becomes large. In general, at small scale, flow can be assumed to be laminar (as Re is low), and for fully developed flow, h can be obtained from the correlation (Panigrahi 2015) (see ▶ Electrohydrodynamically Augmented Internal Forced Convection) Nu ¼

hD ¼ 3:657 k

(22)

From Eq. 23, it is clear that h scales inversely as D. Therefore, h is high at small (micro/nano) scales. The decrease in convective heat transfer resistance, however, is at the expense of the higher pressure drop, which increases the fluid pumping power and, therefore, affects the overall efficiency of the process. One particular application of microscale convection and a topic of major scientific in the last two decades is electronics cooling. Salient aspects of this application will be discussed next.

4.2.1 Electronics Cooling The heat generated in microelectronic systems needs to be dissipated effectively and often under severely constrained geometries, due in large part to the steady progress in device miniaturization. This need has led to a broad and active research into thermal management using micro-/nanoscale cooling platforms and led to a new field of research termed “electronics cooling” (Koo et al. 2005). Electronics cooling covers theoretical and experimental investigations on thermal management – which is naturally not just limited to microelectronics industry – and assessment and characterization of thermal reliability of the cooling hardware. Efficient thermal management offers immediate advantages in efficiency, robustness, and power handling capability of microelectronics. There are a number of different approaches to convective electronics cooling. An effective cooling strategy is jet impinging cooling (Kandlikar and Bapat 2017) where high-speed jets issue from nozzles and the microchips are kept at a certain distance away from and perpendicular to the nozzles. A thin boundary layer is formed immediately under the jet, and this boundary layer becomes thicker as the liquid starts to flow radially outward direction. Therefore, due to the variation of the boundary layer thicknesses, the convection coefficient h also varies from the center to the edge of the microchip and, thereby, producing large temperature gradients in the microchip. Due to this, thermal stresses in the chip to the heat sink interface

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increases, and electronic reliability, particularly in hotter regions of the chip, reduces. This is a notable drawback of jet impinging cooling. Spray cooling (Kandlikar and Bapat 2017) is another strategy. This process is based on impinging followed by evaporation of the liquid droplets. As this cooling strategy relies on the liquid-to-vapor phase change process, a large amount of heat can be removed at much lower surface temperatures. This is an immediate advantage. Spray cooling efficiency can be further enhanced by atomizing the fluid into smaller drops, for example, by using pressure-assisted atomizing nozzles. The high pressure requirement in such nozzles, however, poses a mechanical reliability challenge. A third strategy relies on using fluid flow through microchannels for cooling. Microchannel cooling (Tuckerman and Pease 1981; Escher et al. 2010b; Sharma et al. 2013) offers a high heat transfer coefficient due to the smaller channel width, as discussed above. Common coolants used are air and water (Khonsue 2012; Wang and Chi-Chuan 2017). However, water has three orders of magnitude higher density, four times higher specific heat, and an order of magnitude better thermal conductivity compared to air. This facilitates remarkable improvement in heat transfer efficacy and enables designs of compact heat sinks for cooling microprocessors, for example (Alfieri et al. 2010; Escher et al. 2010a). In fact, microchannelbased single-phase cooling has been exploited to remove a large amount of heat, as high as 750 W/cm2 (Sharma et al. 2013). Unfortunately, the associated higher pressure drop limits the use of microchannel cooling. To overcome this issue and to tackle the challenge of cooling 3D chips stacks (see section “Microchannel Geometry: 3D Architecture” below), two important cooling platforms have been introduced, as outlined below. Microchannel Geometry: Manifold Microchannel (MMC) To reduce the pressure drop of the liquid across the microchannel, a manifold microchannel design is adopted where liquid, instead of being delivered at a particular inlet, is delivered across the chip through uniformly spaced nozzles. The nozzles are each part of a manifold sitting on top of the microchannels. Thus the manifold layers distribute the flow, and the flow from the nozzle outlet impinges on to the microchannels underneath. The arrangement is referred as manifold microchannel (MMC) heat sink, as shown in Fig. 7a, and reduces the travel length of the liquid through microchannel and, hence, significantly decreases the pressure drop. A number of studies (Escher et al. 2010a, b; Sharma et al. 2013, 2015a, b) have also been done to optimize manifold and channel geometries in MMC heat sinks to enhance the overall cooling efficiency by reducing the pressure drop. One example of such an optimized cooling strategy is the hotspot-targeted microchannel cooling. This strategy is based on the delivery of cooling liquid to a high heat flux region and throttling the liquid flow in a low heat flux region. With this strategy, much more uniform chip temperature can be attained while consuming low pumping power, thus enhancing the efficacy of the overall cooling process. The effectiveness of MMC heat sinks and significantly better thermal properties of water can be exploited to facilitate the reuse of the waste heat from energyintensive components such as data centers, concentrated photovoltaic systems, etc.

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b

a

Inlet pipe

Inlet manifold

100 μm

Outlet manifold

3 m/s

1.5m/s

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0.1m/s

Outlet pipe

d

c 35 °C

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Tf (°C)

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Fig. 7 Flow and heat transfer in water cooled heat sinks: (a) schematic of manifold microchannel (MMC) heat sink in isometric view and top view. (b) Streamline pattern of the steady flow across a chip with inline micropin array. At Re~170, a stable pair of vortices is formed between the pins in the flow direction. (c) Liquid temperature map on a micropin fin-based heat sink in the pre-vortex flow regime. (d) Transversal temperature profile at three different locations, as indicated by three different arrows. (e) Image sequences of instantaneous temperature map in the post-vortex regime showing gradual destruction of liquid recirculation zones. (f) Transversal temperature profile at two different locations, as indicated by two arrows (Reproduced by the permission of Elsevier Ltd. (Renfer et al. 2013; Sharma et al. 2013) and Springer Verlag (Renfer et al. 2011))

This helps to remarkably improve the overall thermal performance and energy efficiency of these systems. In fact, a recent study showed that the heat recovery efficiency can be as high as 80% using MMC heat sinks with water as a coolant (Kasten et al. 2010; Zimmermann et al. 2012a,b; Tiwari et al. 2012). Interestingly,

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these works used coolant water temperature of ~60  C, i.e., well above the ambient temperature, to cool microprocessor chips. Zimmermann et al. (2012b) performed detailed exergy and feasibility analysis of such a hot water MMC heat sinks. They showed that raising the coolant inlet temperature was an effective way to ensure that even hotter water coming out of the data center was useful in secondary usage such as district heat, adsorption cooling, membrane-based water desalination, etc. The hot water-based cooling strategy opens up a new avenue as it enables us to use efficiently otherwise wasted heat in data centers and offers not only an improved thermal performance of data centers as a whole but also a possible elimination of chiller units used in data centers to precool the air in warm climates. The MMC heat sinks have also been used to cool concentrated photovoltaic (CPV) cells (Zimmermann et al. 2015). Photovoltaic chip efficiency goes down with increase in chip temperature, thus cooling is beneficial. The problem is exacerbated in concentrated photovoltaic chips, where solar light is focused down to a small area to save the expensive chip material by exploiting optical components which are cheaper. For CPVs operating at a high concentration, a high heat flux is produced. Consequently, an efficient heat sink is needed for cooling. Just as in the case of data center cooling mentioned above, Zimmermann et al. (2015) showed that using efficient MMC heat sinks CPVs could be cooled even with hot water. The high inlet cooling temperature only marginally affected the electrical efficiency of the photovoltaic solar cell module. However, crucially, it produced even hotter water at the outlet of the heat sink which could be used in secondary applications such as building heating, thereby boosting the system energy efficiency. Impressively, the strategy enabled a fourfold increase in the overall energy efficiency, from 15% to 60% (Zimmermann et al. 2015). Microchannel Geometry: 3D Architecture Recently, electronics industry is adapting 3D integrated circuit (IC) design to accommodate more functionalities into a small space, lower the overall cost by partitioning large chip into multiple smaller dies, and reduce IC wire length. In 3D ICs, microprocessor cores are stacked on top of each other, through silicon vias (holes) in each core that are filled with conductive materials to connect the cores together. The strategy greatly reduces the data transfer time between cores. However, this compact 3D design brings with it additional challenges. Higher junction temperature and accessibility of cooling fluids to each layer are the two common challenges in 3D IC design. Therefore, a smarter and stronger cooling strategy is needed to address these challenges. Researchers have proposed interlayer integrated chip cooling as a possible strategy (Renfer et al. 2011). In such integrated cooling approach, cooling hardware (e.g., microchannels, micropins, etc.) are embedded at the back of each chip layer. Typically, this comprises of a microcavity with micropins (representing the vias). This arrangement is adopted to seal the electrical interconnects from water. However, the hydrodynamics of the flow of the cooling liquid may be influenced by the embedding micropin array in microcavity. Renfer et al. indeed (2011) showed that with increase in flow Re, there was a clear sign of steady (Fig. 7b) and shedding vortices in the wake of cylinder. This flow transition

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was found to be associated with a fluctuation in the flow field and a sharp transition in the pressure drop trend. The authors also performed visualization of the fluid flow field and its relevant effects on the temperature and pressure distribution of the fluid in the micropin array heat sinks (Renfer et al. 2013). They reported instantaneous temperature maps of the micropin heat sink in pre-vortex and post-vortex shedding stages using the microscale laser-induced fluorescence (μLIF) technique (Renfer et al. 2013), which is a popular technique to study the role of microscale flow on convective heat transfer. The temperature map of a micropin-based heat sink obtained by the μLIF is shown in Fig. 7c–f. Interestingly, in the pre-vortex transition regime, at Re ~350 (calculated based the mean velocity between a pair of pins) a stagnant recirculation zone was formed behind micropins, and a stable microchannel-like flow occurs in between the pins (Fig. 7c). The stagnant recirculation zones had a higher temperature than the zones where microchannellike flow occurred. Figure 7d shows transversal temperature profile at three different locations, extracted from the μLIF maps. The temperature of recirculation zones increases rapidly by 5 K over three pins in the direction of the flow, whereas the temperature only increases by 1 K in the free liquid stream (see Fig. 7d). In addition, the temperature difference between recirculation and microchannel-type flow zones is 10 K. In the post-vortex regime, at Re ~800, the shedding vortices dynamically alter the temperature maps, as shown in image sequences in Fig. 7e (i) to (iv), and leading to advection of liquid hotspots into the main flow stream. Note also that the laminar boundary layer, which is formed in the post transition regime, is destroyed due to the vortex-induced liquid mixing. The complete destruction of liquid recirculation zones and the laminar boundary layer result in a uniform liquid temperature field (Fig. 7f) and nearly a twofold enhancement in heat transfer performances (Renfer et al. 2013).

4.3

Two-Phase Convection at Microscale

In phase change processes, the major amount of heat is transported in the form of the latent heat. Typically, the latent heat is much larger than the sensible heat (i.e., heat transferred due to temperature difference). Therefore, the heat transfer is drastically enhanced compared to the single-phase approach. Due to this superior performance, phase change processes are desired in many energy applications, such as thermal generation of electricity, desalination, different metallurgical processes, electronics cooling, food processing, etc. Although the phase change processes are used widely in industry, the coexistence of two phases and their effects on the thermal transport are not fully understood, despite a few good propositions (Qu and Mudawar 2003; Lee and Mudawar 2005). However, empirical studies have enabled substantial improvement in the performance of two-phase processes, thanks in major part to the advancement of micro-/nanofabrication techniques. Heat pipe is the simplest of heat transfer devices relying on phase change; recent advances have led to microheat pipes for passive thermal management. Additional phase change processes are also employed; among these boiling, condensation, and icing have benefitted

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significantly from developments in novel fabrication technologies. Some relevance phase change devices and their underlying physics are discussed next.

4.3.1 Microheat Pipe A heat pipe is a heat transfer device and used frequently for thermal management in space craft, computers, solar collector array, air conditioning system, etc. It relies on two-phase heat transfer and comprises an evaporation zone, an adiabatic zone, a condensation zone, a liquid (coolant), and a wick structure, as shown in Fig. 8a (see also ▶ Chap. 52, “Heat Pipes and Thermosyphons”). The heat pipe is generally made of high thermal conductivity material, such as copper, aluminum, etc. The internal wall of a standard heat pipe is covered with a porous (wick) structure to facilitate the liquid wicking and to provide the surface area for the heat transfer. The center of the heat pipe remains hollow to facilitate the flow of the evaporating fluid from the evaporation to the condensation zone through the adiabatic zone. The working liquid is chosen depending on the operational temperature of the pipe. After the removal of the air, the heat pipe is partially filled with the saturated coolant liquid and sealed. When the heat is absorbed in one end of the heat pipe, the liquid in the wick of the evaporation zone evaporates. The vapor fills up the center, diffuses along the length of the pipe, and finally reaches the condensation zone. The temperature of the condensation zone is kept slightly below the saturation temperature of the liquid, and therefore, vapor condenses with the release of the latent heat. The transport of the liquid from the condensation to the evaporation zone relies on the surface texture and wettability of the wick and occurs passively without an active control. This is the major advantage of heat pipes and enables compact system designs. Many researchers have investigated on the design of wick structures to facilitate constant supply of the coolant to the heated surface, and routing the vapor phase evolved during the phase change process. Examples include a porous wick structure, e.g., sintered copper (Weibel et al. 2010), different microstructures (Chen et al. 2016), etc., with good wettability toward coolant liquid. Such wick structures, however, provide no control of the evolved vapor, which can saturate the porous materials, particularly at high heat flux, and block further liquid transport. It is even more complicated to control the liquid transport at a small scale (micro- and nanoscale) compared to the relatively large scale (submicron to millimeter). Micro- and nanoporous wick structure provides more surface area per unit volume to the heat transfer. However, this can also lead to a large amount of vapor production in the porous wick structure and block the coolant flow, causing an early evaporation dry out may occur (Semenic et al. 2008; Rice and Faghri 2007; Weibel et al. 2010). Despite these challenges, heat pipes find plethora of applications in small-scale devices in electronics cooling, passive thermal management of electrical devices, solar thermal energy technology to name but a few. Small-scale heat pipes are referred as microheat pipe (MHP). MHPs are capillarity (interfacial force)driven heat pipes where the mean curvature of vapor/liquid interface is comparable to the reciprocal of hydraulic radius of the channel and were first introduced by Cotter in 1984 (Cotter 1984). In contrast to a conventional heat pipe, an MHP has no capillary wick. Rather, the capillary action of the coolant relies on the sharp-angle

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Fig. 8 Different types of heat pipes: schematic of (a) a conventional heat pipe with a capillary wick, (b) a capillary wickless microheat pipe, and (c) a liquid meniscus at a corner in the microheat pipe

corners in a polygonal microchannel. The coolant condenses at the corners in the condensation zone of the MHP, as depicted in Fig. 8b. The liquid meniscus curvature is different between the evaporation and condensation zones, thereby helping to create a pressure differential (due to Laplace pressure) which can drive the liquid coolant. The maximum heat transfer capacity is reached when simultaneously dry out and flooding transpire in the evaporation and condenser zone, respectively. The thermal performance of an MHP is often evaluated on the basis of transient energy and mass balance, which are discussed below.

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4.3.2 Heat Transfer in Solid Wall of the MHP Energy balance equations on the wall of MHP can be written as ρs c p , s

@T ¼ k s ∇2 T þ S @t

(23)

where ρs, cp, s, and ks denote the density, heat capacity, and thermal conductivity of the MHP wall, respectively. The symbol S denotes the heat gained/lost by the wall at any z-coordinate and, for different MHP sections (see Fig. 8b), can be expressed as (Cotter 1984; Carbajal et al. 2006; Liu and Chen 2013) S ¼ hl ðT s  T l Þwb þ qin lo , 0  z  Le

(24)

S ¼ hl ðT s  T l Þwb , Le  z  Le þ La

(25)

S ¼ hl ðT s  T l Þwb  jhfg þ ho ðT s  T c Þ, Le þ La  z  Le þ La þ Lc

(26)

where hl and ho are convective heat transfer coefficient of liquid, respectively, to the evaporation zone and the outer wall of the condenser; Ts , Tl, and Tc denote the temperatures of the solid wall, liquid, and the cooling water used at the condensing outer wall to extract heat, respectively; qin and qout are the heat input in the evaporation and condenser zone, respectively; wb is the wetted perimeter of the MHP grooves; and j and hfg are the condensing mass flux and latent heat of evaporation, respectively. The condensation flux j can be expressed as (Liu and Chen 2013; Carbajal et al. 2006)
j¼

2^ σ 2  σ^




M 2πRu

0:5


pv pl  0:5 0:5 Tv Tl

 (27)

where σ^ , M, Ru, pv , and pl, respectively, denote accommodation factor, molecular weight of the vapor, and universal gas constant, the vapor pressure, and saturate vapor pressure of the liquid at the operating temperature of the MHP.

4.3.3 Capillary Flow in the Evaporation Zone Capillary radius at the liquid/vapor interface can be calculated from the YoungLaplace equation as follows:
Δp ¼ γ lv

1 1 þ R1 R2

 (28)

where Δp , γ lv , R1, and R2, respectively, denote pressure differential across the liquid/vapor interface, liquid/vapor interfacial energy, and radii of curvature in the axial and radial directions. R2 is much larger than R1, and, therefore, Eq. 28 can be approximated as

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Δp  γ lv

1 R1

 (29)

For a constant vapor pressure of the liquid inside the MHP, Eq. 29 can be written in differential variation of pl as d ðpl Þ γ lv dR ¼ 2 dz R1 dz

(30)

This pressure gradient is capillarity induced and helps drives the liquid flow in MHP (Liu and Chen 2013). The mass balance equations for the liquid flow in the MHP can be written as (Cotter 1984; Carbajal et al. 2006; Liu and Chen 2013) @ ð ρ l A l Þ @ ð ρ l Al V l Þ ¼  M l , 0  z  Le @t @z

(31)

@ ðρl Al Þ @ ðρl Al V l Þ ¼ , Le  z  Le þ La @t @z

(32)

@ ð ρ l Al Þ @ ð ρ l A l V l Þ ¼  Mv , Le þ La  z  Le þ La þ Lc @t @z

(33)

where Al is the cross-sectional area for liquid flow, Vl is the liquid velocity, ρl is its density, and Ml and Mv are the masses of the liquid evaporated and condensed per unit length and time. Likewise, transient energy balance in the evaporator, adiabatic, and condenser zone can be written as follows (Cotter 1984; Carbajal et al. 2006; Liu and Chen 2013): ρl cp, l Al

@T l ¼ Al kl ∇2 T l þ ρl cp, l Al ∇T l þ hl ðT s  T l Þwb  Ml j @t

(34)

where kl and cp, l denote the thermal conductivity and specific heat of the coolant liquid. The above mass and energy balance equations are used to evaluate the performance of MHPs. In addition to the capillary forces, the external force like electric field can also be employed to enhance the liquid motion inside the MHP. In those techniques, free space charges are introduced in the liquid coolant either by direct injecting a small amount of mobile charges through an emitter tip (ion injection pump) (Chang and Yeo 2010; Chang and Hung 2017) or by the dissociation of ions upon the application of the electric field (conduction pump) (Chang and Yeo 2010; Chang and Hung 2017). Those ions act as free charges and initiate the electrohydrodynamic flow in the MHP in the presence of the electric field. Alternatively, an electrode is introduced along the axial direction of the MHP to generate polarization forces driving the dielectric coolant liquid into a nonuniform electric field. Therefore, a hydrostatic equilibrium is established in the coolant liquid (induction pump)

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(Jones 1973). Typically, the thermal conductance of induction pump-based MHP is lower than MHPs based on the ion injection or conduction pump mechanisms (Chang and Hung 2017).

4.3.4 Microscale Boiling Boiling occurs when the temperature of a liquid is higher than its saturation temperature. Considering a classic case of a liquid in contact with a solid, depending on the solid-liquid temperature difference (i.e., the degree of superheat), the liquid boiling progresses through different regimes as will be described in brief below. The boiling regimes are depicted in Fig. 9a where heat flux q is plotted as a function of degree of superheat ΔT. When a liquid in contact with a substrate is heated by raising the solid temperature, initially the temperature rise causes natural convection. This is maintained up to a small degree of degree of superheat (ΔT), i.e., the solid temperature rising slightly above the liquid saturation temperature (Ts). Eventually, the first stage of boiling is initiated and is manifested by nucleation and departure of bubbles from the solid surface. This is referred as the onset of nucleate boiling (ONB), and it is characterized by a sharp change in the slope of heat flux (see the boiling curve in Fig. 9a) (see also ▶ Chap. 41, “Nucleate Pool Boiling”). The slope change shows the effectiveness of two-phase heat transfer over single phase, and the change is also marked by a much higher heat transfer coefficient. Therefore, nucleate boiling is sought after in many industrial processes. The bubble nucleation/departure also induces significant fluid motion, especially near to the solid surface. The bubble nucleation intensity depends on the degree of superheat ΔT and the availability of active nucleation sites on the solid surface. As the temperature of the solid surface increases (ΔT increases), more nucleation sites become active, more bubbles nucleate, and coalesces after departing from the solid surface. With progressive rise in ΔT, the boiling become more vigorous and starts producing high density of bubbles that eventually coalesces near to the solid surface to form a thin vapor film. This thin film adds thermal resistance and also starts to obstruct the liquid from coming in contact with the solid surface. This is marked by a decrease in the gradient of the heat flux curve. At sufficient high ΔT, the vapor covers the solid surface completely, preventing the liquid contact with the surface. This is called the dry out or boiling crisis and is marked by a sharp reduction in q, as shown in Fig. 9a. The critical heat flux (CHF) defines the maximum heat flux just before the dry out. CHF is a surfacespecific characteristic, which depends on the surface textures and wettability. Different surface modification techniques are adopted to enhance CHF. The decrease in q reaches a minimum at the Leidenfrost point, beyond which boiling progress with the presence of a stable film on the solid surface. This regime, with much lower heat transfer coefficient compared to the nucleate boiling, is referred as film boiling (see also ▶ Chap. 42, “Transition and Film Boiling”). Literature is rich with many empirical correlations to predict CHF and heat transfer coefficient in nucleate and film boiling regimes (Li and Peterson 2007; Kandlikar 2010). CHF and heat transfer coefficient are, largely, functions of physical properties of the liquid

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a

Free convection

Nucleate

Transition

Film

Isolated Jets and bubbles columns 107

C

q”max

Critical heat flux, q max ” Boiling crisis

106

E

q ”,(W/m2)

P

B

105

D

q”min 104

Equation 9.31 A

ONB

ΔTe,A ΔTe,B 103 1

5

Leidenfrost point, q”min

10

ΔTe,C

ΔTe,D

30 120 ΔT e= T s – T sat(°C)

1000

b

Bubble flow

Slug flow

Slug-annular flow

Annular flow

Fig. 9 Boiling curve and regimes: (a) boiling curve obtained through a typical pool boiling experiments (Reproduced by the permission of John Wiley and Sons (Bergman et al. 2006)) and (b) different fluid regimes in boiling process

(viscosity, density, specific heat, thermal conductivity, and surface tension), bubble departure diameter, liquid flow rate, etc. (Collier and Thome 1994). There are two modes of boiling: pool boiling and convective boiling. 1. In pool boiling, a heated surface is introduced in a pool of liquid (coolant). The motion of the liquid is largely driven by the buoyancy force. The whole liquid,

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except liquid near the solid surface, is below the solid surface temperature, Ts. Pool boiling is also called the subcooled boiling. 2. In convective boiling, the liquid is forced by external means, as, for example, pump, to flow over the heat solid surface. The liquid near to the heated surface is rapidly replenished by the cooler liquid, and therefore, a higher heat transfer coefficient is achieved. Boiling is used widely in industry for enhance heat transfer and thermal management applications across scale. Two most common exploitations in small scale are discussed below. Pool Boiling on Porous Surfaces: Effect of Surface Textures Boiling performance is greatly enhanced on micro- and nanoporous textures that provide a large area for heat transfer and enhance the liquid suction. Therefore, in the past few decades, different engineered micro- and nanoporous surfaces have widely been used to study boiling (Theofanous et al. 2002; Hsu and Chen 2012; Hetsroni et al. 2005). However, these experiments indicate that the design of micro- and nanoporous wick structure needs further improvement to enhance the boiling performance (Attinger et al. 2014). Therefore, there is a clear need to understand the nucleation and the subsequent boiling events on the porous matrices. Typically, surfaces have pits and cavities, which can entrap gas or vapor leading to initiate the heterogeneous nucleation. The Gibbs free energy for creating vapor nuclei on a smooth surface is given by (Collier 1972) Ghet ¼ ϕGhom

(35)

where Ghom denotes the Gibbs free energy required to create a vapor nucleus in the bulk liquid, commonly referred as the homogeneous nucleation, and 2 ϕ ¼ 2þ2 cos θþ4 cos θsin θ , with θ denoting the solid-liquid contact angle. For textured superhydrophobic surfaces, the superheat required for heterogeneous nucleation becomes small as ϕ exponentially goes to zero. Betz et al. (2013) confirmed that indeed the heterogeneous nucleation occurs on textured superhydrophobic surfaces at two orders of magnitude lower superheat compared to smooth surfaces. Therefore, it is clear that surfaces with low wettability toward the coolant liquid favor the nucleation. However, low wetting surfaces are not beneficial in later stage boiling, near to CHF. At CHF, the extended liquid layer on the heated surface ruptures and vaporizes independently. Therefore, a wettable surface is needed to sustain the liquid film on the heater surface, and, by doing so, a higher CHF can be obtained (Theofanous et al. 2002). These contradicting requirements make the fabrication of surfaces for optimal boiling performance challenging. Biphilic surfaces, with both hydrophobic and hydrophilic character, have been proposed for optimum boiling performance (Forrest et al. 2010; Hsu and Chen 2012). Nanotextured surfaces offer a few unique features that can help overcome the surface wettability challenge. Such texturing can help tune the wettability to extremes. For example, it has been shown that a nanotextured superhydrophilic

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surface can not only facilitate the liquid transport to the heated area by its superwettability (Takata et al. 2005) and enhanced capillary transport (Chen et al. 2009) but also provide the enhance surface area (Li and Peterson 2007) for better heat transfer. With the advancement of sophisticated micro- and nanofabrication techniques and their characterization proficiency, different nanotextures (Lu et al. 2011; Yao et al. 2011; Dai et al. 2013; Chen et al. 2009; Li and Peterson 2007) were fabricated, and their potential to significantly enhance boiling heat transfer performance has also been established. For example, a 100% enhanced CHF was achieved with a silicon nanowire forest compared to a smooth silicon surface (Lu et al. 2011). A sustained operation under high temperature and need to optimize the surface thermal resistance are some outstanding challenges which need to overcome before nanotextured surfaces could be widely adopted in practical applications. Additionally, there also is a need for theoretical studies to complement the experimental findings discussed above. Convective Boiling: Microchannel Flow Boiling An enhanced heat transfer performance is achieved by adopting convective boiling, where the liquid is forced by external means (e.g., a pump). Such convective or flow boiling in microscale devise frequently occurs with liquid flowing through a microchannel (see ▶ Chap. 47, “Flow Boiling in Tubes” for macroscale boiling phenomena). The configuration also provides a convenient means to understand the basic principle, thus our discussion will be focused on this. It is pertinent to start by introducing the two dimensionless numbers. First, the Bond number (Cheng et al. 2007) is the ratio of gravitational to surface tension forces and can be expressed as Bo ¼

gðρl  ρv ÞD2 γ lv

(36)

where g and ρv, respectively, denote the acceleration due to gravity and the density of the vapor phase. With small size (D), in microchannels the Bo is small relative to minichannel and macrochannel flow boiling. Therefore, in microchannel flow boiling surface tension effects dominates over the gravity. Second, the boiling number (Wang et al. 2007) is the ratio of heat flux to the mass flux and can be expressed as Boi ¼

q Ghfg

(37)

where G and hfg denote mass flux and latent heat of evaporation, respectively. Boi helps to define the stability of different flow boiling regimes that follow. The flow boiling regimes are classified according to the nature of vapor and liquid phases (Cheng et al. 2007; Wang et al. 2007) and are accordingly termed bubble flow, slug flow, slug-annular flow, and annular flow (see Fig. 9b). These regimes are observed at progressively higher heat fluxes. The liquid acts as the continuous phase

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and vapor as the dispersed phase. With the constant flow of the liquid, and at a low heat flux conditions, small bubbles are dispersed into the liquid continuum. This flow pattern is called bubbly flow (Cheng et al. 2007; Wang et al. 2007). At a slightly higher heat flux, bubbles get bigger and the flow regime turns to slug flow. With increase in heat flux, the bubble generation frequency increases and the bubbles start to coalesce and form a continuous vapor stream. This is referred as the slug-ring (or slug-annular) flow. With even greater heat flux, the vapor starts to occupy the channel core and a thin liquid film surrounds the vapor core. Interestingly, miniature liquid droplets are still entrained in the vapor core. This regime is called annular flow. All the above flow regimes have been observed in microchannel flow boiling. The stability of flow regime depends on the mass and heat flux (Wang et al. 2007; Cheng et al. 2007). When Boi is less than 0.96, a stable flow regime is observed and is characterized by bubble generation at the channel wall, followed by their steady transportation along the channel width. The long oscillation, and unstable flow regime are observed when 0.96 < Boi < 2.14. This is marked by temperature and pressure variations and switching between bubbly and annular flows. Short oscillations with unstable flow regime is observed with Boi > 2.14 (Kandlikar 2010; Wang et al. 2007) and is characterized by short periods of oscillations featuring periodic dry out and rewetting. With further increase in the heat flux increases, the annular flow becomes stable and the vapor quality increases at the outlet of the microchannel leading to poorer heat transfer performance (Qu and Mudawar 2003; Hetsroni et al. 2005). Interestingly, in the annular flow regime, channel wall and inlet liquid temperature increase gradually. The resulting heat transfer performance at different vapor qualities are not well predicted by the existing models (Lee and Mudawar 2005), which clearly reflects a need for deeper theoretical studies on flow boiling in microchannels.

4.3.5

Micro-/Nanotextured Surfaces for Efficient Condensation Heat Transfer Vapor condensation occurs when the environmental temperature is below the saturation temperature of the vapor. During condensation, the latent heat (enthalpy of vaporization) is released. This is used in many processes, such as air conditioning systems, in refrigeration, in generation of electric power, etc. Most of these applications involve condensation on solid surfaces maintained below the saturation temperature. The difference between saturation temperature and the solid surface temperature is referred as degree of subcooling. Depending on the surface texture and surface chemistry, the liquid condenses either in the form of droplets or a film, referred respectively as dropwise and filmwise condensation (see also ▶ Chap. 50, “Film and Dropwise Condensation”). Both droplets and the liquid film (condensate) prevent the contact of the vapor phase with the solid and increase the resistance to the heat transfer. Typically, a higher – even up to an order of magnitude – heat transfer coefficient is achieved in the dropwise condensation compared to the filmwise condensation. Therefore, realizing surfaces that sustain stable dropwise condensation is highly desirable. Regardless of the nature of condensation, similar to the

310

T. Maitra et al.

boiling phenomena, condensation proceeds through two steps- nucleation of the condensate, and the growth of this condensate either to larger drops or a liquid film. In case of dropwise condensation, nucleation, growth and the subsequent departure of the condensate droplets exposes new areas on the solid surface, for cycle to continue. Therefore, a surface offering many nucleation sites and small roll-off angle for droplet departure (i.e. low adhesion between drops and the solid surface) should be ideal to enhance condensation heat transfer. The Gibbs free energy for heterogeneous nucleation of condensate can be expressed by the same Eq. 35 as discussed in the boiling section above. The factor ϕ, however, has the opposite trend than that of boiling; thus, hydrophilic surfaces are better at facilitating nucleation compared to hydrophobic surfaces. However, hydrophilic surfaces promote formation of a liquid condensate film instead of droplets, which affects the efficiency of thermal transport. Droplet adhesion on the other hand is lower on hydrophobic surfaces. Thus, hydrophilicity is needed for nucleation of liquid condensate, whereas hydrophobicity is required to maintain a dropwise condensation (Attinger et al. 2014). Clearly, there is a need for a trade-off to optimize the rate of nucleation, drop adhesion and stabilize dropwise condensation, without worsening the surface thermal resistance. Advancements of micro-/nanofabrication and wettability engineering offer some exciting opportunities and are discussed next. The size of the departing droplet is crucial in the condensation heat transfer. The heat transfer coefficient decreases as the maximum droplet departure size increases (Fevre and Rose 1966; Le Fevre and Rose 1965). A number of studies have focused on passively controlling the drop radius, mobility, and size distribution by engineering the condensing surface and/or tailoring surface chemistry. Firstly, applying the surface energy gradient (Daniel et al. 2001; Macner et al. 2014) is an effective way to promote mobility. In fact, surfaces with patterned hydrophobic and hydrophilic regions can be used to optimize the drop nucleation and mobility. Studies have already established the feasibility of controlling maximum departing droplet diameter and its mobility by tuning the width of hydrophobic and hydrophilic regions (Chatterjee et al. 2013, 2014; Peng et al. 2015). Secondly, on grooved superhydrophobic surfaces, condensate droplets coalesce and can be easily drained through grooves, leaving behind the top portion of surfaces dry due to its superhydrophobic character (Narhe and Beysens 2004; Izumi et al. 2004). Notwithstanding the success of these approaches in demonstrating dropwise condensation, these surface fabrication technologies have been realized only on small samples (surface dimension millimeter to centimeter), and, more significantly, they were evaluated through condensation experiments lasting a few hours or less, which raises questions regarding their industrial exploitation. There also are significant challenges toward scalability of fabrication (Attinger et al. 2014). To this end, robust and rationally conceived surface treatment strategies facilitating stable dropwise condensation are being actively researched. Simple, thick polytetrafluoroethylene (PTFE) coatings have demonstrated stable dropwise condensation without aging for ~1000 h (Rose 1997; Ma et al. 2000). However, the PTFE coatings (especially at a high thickness) introduce substantial thermal resistance, thereby negating the efficiency gained due to dropwise condensation. Applying self-assembled monolayer (SAM) of silanes is a

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way to overcome the thermal resistance issue as the SAMs have lower thickness (~30 A); durability however is an issue in this case. Interestingly, a recent work has shown that a graphene coating can sustain dropwise condensation over 2 weeks on continuous exposure to steam (Preston et al. 2015). Additionally, due to their higher thermal conductivity, the graphene coating also showed a fourfold higher heat transfer coefficient compared to a silane-based coating. Recently, Paxson et al. (2014) synthesized metal-grafted copolymer thin films via initiated chemical vapor deposition (iCVD) technique. Those metal-grafted thin films showed stable dropwise condensation without aging up to ~50 hours, with a good heat transfer performance. As an exciting finding, at low surface subcooling, it has been shown that condensate drops coalescing on nanotextured superhydrophobic surfaces can spontaneously jump from the surface due to a release of excess surface energy (see Fig. 10; Boreyko et al. 2011; Chen et al. 2007; Miljkovic et al. 2013a). The size of maximum departing droplet is reduced and self-regulated by this jumping droplet phenomenon. This results in a stable dropwise condensation and an improvement in heat transfer performance by 25%. Moreover, this spontaneous jumping behavior can also be augmented by application of an external electric field (Miljkovic et al. 2013b). Most of the abovementioned condensation experiments testing surface treatment were performed in a customized closed vessel at low pressure conditions and at low temperature (> Parallel V-Ribs ~100) in such cores tend to promote considerable fluid mixing. The wall waviness also periodically disrupts the boundary layers and provides for an effectively longer flow length. The enhanced convection has been found to be a complex interplay of fin density (and hence inter-fin spacing) and fin height along with the severity of fin waviness (ratio of amplitude and pitch) (Manglik et al. 2005; Vyas et al. 2010). In one estimate (Manglik et al. 2005), the enhanced performance of a set of sinusoidal wavy fins has been shown to provide considerable reduction in the frontal core area for a plate-fin heat exchanger. The use of fins to promote enhanced convection invariably involves the evaluation of their thermal efficiencies as well. This is often addressed by the fin surface area thermal effectiveness (which is based on fin efficiency) in calculating the associated fluid stream thermal resistance in most heat exchanger design analyses. Guidelines for such considerations are outlined in the extended analysis presented by Manglik et al. (2011). These are generally applicable to all plate-fin and tube-fin geometries, as well as to convection with other extended surfaces.

5

Displaced Enhancement Devices

These are typically found in engineering applications in the form of tube inserts, where the geometric features promote transverse mixing in the axial fluid flow. An example is the looped wire insert shown in Fig. 9. Their intended usage is in thermal processing of viscous liquids in the chemical and food processing industry (Oliver and Shoji 1992). The wire-loop inserts have been reported to promote enhancement in both laminar and turbulent flow regimes (Bergles et al. 1991). Bent-strip-type turbulators, manufactured with a variety of geometric features, are also an attractive displaced flow enhancement device (Bergles et al. 1991) and have been used to enhance gas flow convection. In high-temperature applications, such as the flue-gas tubes of a gas-fired water heater, enhancement of combined radiation and convection has been reported (Junkhan et al. 1985). Displaced flow enhancement has also been shown to be effected by angled tabs or protrusions on a heat transfer surface. As shown in Fig. 10, the tabs can take on a variety of wing-like or winglet shapes that are angled away from the axial flow field. Longitudinal vortices are generated from the edges of these tabs that sweep the Fig. 9 A typical looped wire mesh insert for tubular exchangers and mixers used in the chemical process industry

462

R. M. Manglik

Fig. 10 Different types of punched tabs or vortex generators used on plate-fin surfaces for displaced enhancement of forced convection heat transfer

downstream heat transfer surface to produce mixing and enhancement (Fiebig 1995; Fiebig et al. 1991). Such wing-shaped protrusion can be punched out of the fin surface in both plate-fin as well as tube-fin exchangers (Bergles and Manglik 2013). However, no general correlations have been developed for predicting the flow friction factors and heat transfer coefficients.

6

Swirl-Flow Devices

Techniques for producing swirl flows and vortex-induced mixing to enhance heat transfer generally consist of tube inserts, axially twisted noncircular ducts, and tangential fluid entry arrangements in ducts (Manglik and Bergles 2002). Of these, helically twisted metallic strips or ribbons, or twisted-tape inserts, have been extensively used to enhance forced convection heat transfer inside circular tubes. Their typical usage in multi-tube shell-and-tube heat exchangers and geometric features and characteristic descriptors are illustrated in Fig. 11. For predicting thermalhydrodynamic performance of tubes fitted with these inserts, Fanning friction factor and Nusselt number correlations for both laminar and turbulent flows are available (Manglik and Bergles 1993a, b). For laminar flows, the helical swirl generation in the tube fitted with a twistedtape insert has been found to scale with the following dimensionless swirl parameter (Manglik and Bergles 1993a): pffiffiffi

Sw ¼ Res = y

(17a)

Here y is the tape-twist ration (180 helical twist pitch H divided by tube inside diameter Di, as indicated in Fig. 11c), and based on the mass flux G (mass flow rate per unit flow cross-sectional area), the swirl Reynolds number Res and swirl velocity Vs are given by

11

Enhancement of Convective Heat Transfer

463

Fig. 11 Use of twisted-tape inserts in the tubes of a shell-and-tube heat exchanger, and schematic representation of the geometric features a typical twist tape

h i1=2 Res ¼ ðρV s Di =μÞ, V s ¼ ðG=ρÞ 1 þ ðπ=2yÞ2

(17b)

Based on this scaling of the helical swirl flow inside the tube, the isothermal Fanning friction factor is given by fs ¼

 

1=6 15:767 π þ 2  2ðδ=Di Þ 2 1 þ 106 Sw2:55 Res π  4ðδ=Di Þ

(18a)

where δ is the tape thickness and fs is based on the effective swirl velocity and swirlflow length: "
2 #1=2  ΔpDi π , and Ls ¼ L 1 þ 2 2y 2ρV s Ls


fs ¼

(18b)

For the heat transfer in laminar forced convection with twisted tapes inside tubes maintained at a uniform wall temperature, the following mean Nusselt number Num correlation (Manglik and Bergles 1993a) is recommended: 92 28 > > = <



2:5 3:835 6 Num ¼ 4:612ðμb =μw Þ0:14 4 1 þ 0:0951Gz0:894 þ6:413  109 Sw Pr0:391 |fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl} > ; :|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}> fully developed flow

thermal entrance

30:1

7 þ 2:132  1014 ðRea RaÞ2:23 5 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

swirl flows

(19)

free convection

The relative interplay of swirl-induced convections with thermal entrance effects and free convection superimposed on the axial flow are described in the results presented in Fig. 12. Moreover, these correlations have been found to predict very well all the experimental data that are to date available in the literature (Manglik et al. 2001; Manglik and Bergles 2013).

464

a

R. M. Manglik 103

b

Num = φ (Gz, Sw, Pr, Ra, μb/μw)

102 EXPERIMENTAL DATA Marner and Bergles (1978) 7 1.0 x 10 < Ra < 3.0 x 107 Manglik and Bergles (1992) 6 2.0 x 10 < Ra < 4.8 x 106

Num(μb / μw)–0.14/4.612

Ra ~ 0 0.391

Num(μb / μw)–0.14

Sw-Pr 7500 102 4500 3000 1500

10

101

EXPERIMENTAL DATA Manglik and Bergles (1992) Sw-Pr0.391 7125-7875 4275-4725 2850-3150 1425-1575

1

100 101

102

103 Gz

104

105

7

Ra ~ 2

x 10 6

Ra ~ 3

100 102

x 10

SWIRL FLOW, Ra ~ 0

103 Sw . Pr0.391

104

Fig. 12 Influence of swirl flows produced by twisted-tape inserts on laminar forced convection heat transfer inside circular tubes with walls at uniform temperature and comparison of predictions of Eq. 19 with experimental data (Manglik and Bergles 1992; Marner and Bergles 1978)

In the turbulent regime (generally characterized by axial tube-side flows with Re  104), however, the swirl parameter Sw is inapplicable because of the inherent highly mixed nature of the fluid flow. The tape-induced swirl is simply scaled by a power-law type function of the twist ration y (Manglik and Bergles 1993b), and after accounting for tube partitioning effects, the Fanning friction factor correlation is given by f ¼



 1:75   0:0791 π π þ 2  ð2δ=Di Þ 1:25 2:752 1 þ π  ð4δ=Di Þ y1:29 Re0:25 π  ð4δ=Di Þ |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl}

(20)

swirl

tube partitioning

Likewise, for turbulent heat transfer with Re  104, the recommended (Manglik and Bergles 1993b) mean Nusselt number correlation is   0:8   

π π þ 2  ð2δ=Di Þ 0:2 0:769 1þ ϕ Nu ¼ 0:023Re0:8 Pr0:4 π  ð4δ=Di Þ π  ð4δ=Di Þ y |fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} swirl

tube partitioning

(21a) The temperature-dependent property correction factor ϕ in Eq. 21a is given by the following function of viscosity ratio for liquid flows, and temperature ratio for gas flows: ϕ ¼ ðμb =μw Þn or ðT b =T w Þm  0:18 liquid heating n¼ 0:30 liquid cooling

 and m ¼

0:45 gas heating 0:15 gas cooling

(21b)

Again, the predictions of these correlations, Eq. 20 and 21, have been found to agree very well with the experimental data of different investigators available in the

11

Enhancement of Convective Heat Transfer

465

Fig. 13 Axially twisted ducts of oval (elliptical) and rectangular cross sections

literature. Extended discussions on their development, validity, as well as design and applications issues are given in (Manglik et al. 2001; Manglik and Bergles 2002). Swirl flows can also be generated by using noncircular cross-sectional ducts that are helically twisted along their longitudinal or flow axis (Bishara et al. 2009; Manglik and Bergles 2002; Manglik et al. 2012; Patel et al. 2012). Two such examples are the axially twisted tubes with an oval (or elliptical) and a rectangular cross section, as depicted in Fig. 13. Of these the former has been used commercially for enhancing viscous liquid flow heat transfer in external flows over tube bundles (Dzyubenko and Dreitser 1986; Manglik and Bergles 2002) in shell-and-tube heat exchangers in the petrochemical industry. For low Reynolds number flow of air (Pr ~ 0.7–1.0) inside twisted oval tubes maintained at constant surface temperatures, up to 2.0–2.5 times higher Nu can be achieved in comparison with straight oval tubes of same cross-sectional aspect ratio and same pumping power (Bishara et al. 2013). The primary driving mechanism for enhancement is the near-wall swirl generated by the helical curvature of the duct surface (Bishara et al. 2009). Likewise in the case of viscous liquid flows (5  Pr  100) inside twisted rectangular tubes, depending upon their cross-sectional aspect ratio and the severity of longitudinal twist, up to 2.4 to 13 times higher heat transfer rates can be sustained on a constant-pumping-power basis compared to that in a straight duct (Manglik et al. 2012). Moreover, when considering the constraint of fixed heat load and pressure drop, as much as 15% to 50% reduction in surface area can be obtained in a new heat exchanger.

7

Coiled Tubes

Coiled tubes are used in a variety of commercial heat exchangers (Bergles et al. 1991; Manglik 2003; Nandakumar and Masliyah 1986). They not only increase the heat transfer surface area per unit volume of the exchanger but also significantly

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enhance the tube-side flow heat transfer coefficient. The helical curvature of the tube coil induces a centrifugal force on the axial flow inside, and a secondary flow pattern, consisting of two vortices perpendicular to the direction of axial flow, is set up (Collins and Dennis 1975; Mori and Nakayama 1965; Prusa and Yao 1982). This helically swirling flow is referred to as Dean flow, so named in recognition of the early work of Dean (1927, 1928), and its strength and cross-stream fluid-mixing intensity produces enhanced convection heat transfer when compared to that in a straight tube of equal length. The flow characterization and the associated convection heat transfer coefficient in coiled tubes are dependent upon the geometric attributes of the coiled tube, which are identified in Fig. 14. Their dimensionless representation along with that of the fluid flow are given by the Dean number, Helical number, and radius of curvature, which are defined as follows: De ¼ Re

Fig. 14 Schematic representation of curved and coiled tubes and their geometric attributes

pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðd=2RÞ

(22a)

11

Enhancement of Convective Heat Transfer

He ¼ Re

467

h i1=2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðd=2Rc Þ ¼ De 1 þ ðH=2πRÞ2

(22b)

h i Rc ¼ R 1 þ ðH=2πRÞ2

(22c)

It may be noted that when the helicoidal pitch is zero (H = 0), He reduces to De for a simple curved tube. The tube curvature, whether helical or only in the axial plane, essentially acts to impose a centrifugal force on the fluid motion to generate a secondary circulation, and the strength of the vortex increases as well as scales with De. Experimental data and theoretical results for laminar flow heat transfer in curved and coiled circular tubes have been reported in the literature along with a few correlations (Manglik 2003; Kreith et al. 2011). The hydrodynamic behavior of tube-side flow in curved and coiled tubes can be distinguished by three regions: the region of small Dean number, De < 20, in which inertia forces due to secondary flows are negligible; the region of intermediate Dean numbers, 20  De  40, where inertial forces due to secondary flow balance the viscous forces; and the region of large Dean numbers, De > 40, where viscous forces are significant only in the boundary near the tube wall. Based on this characterization, Manlapaz and Churchill (1981, 1980) have devised the following correlations for predicting isothermal friction factors in fully developed helical flows: 20 6B f ¼ f st 4@1  n 8 >

: 0

30:5
2
 d He 7 C 5 , o0:5 A þ 1 þ 6R 88:33 2 1m

0:18 1 þ ð35=HeÞ

(23)

De < 20 20 < De < 40 De > 40

Moreover, Manlapaz and Churchill (1980) have also proposed the following two similar but separate expressions for predicting average Nusselt numbers in fully developed flows in coiled tubes, respectively, with the uniform wall temperature (UWT, subscript T) and uniform heat flux (UHF, subscript H ) conditions: 2( Num, T ¼ 4 3:657 þ 

4:343

2 1 þ 957=Pr He2

)3

 þ 1:158

He ½1 þ ð0:477=PrÞ

3=2

31=3 5

(24) 2( Num, H ¼ 4 4:364 þ 

4:636



2 1 þ 1342=Pr He2

)3



He þ 1:816 ½1 þ ð1:15=PrÞ

3=2

31=3 5

(25)

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The predicted results of Eqs. 23, 24, and 25 agree very well with a fairly large data set from different experimental investigations (Manglik 2003; Nandakumar and Masliyah 1986; Manlapaz and Churchill 1981, 1980). It has been observed that the flow inside coiled tubes remains in the viscous regime for much higher Reynolds number as compared to straight tubes (Taylor 1929; Srinivasan et al. 1968; Manglik 2003). The curvature-induced helical vortices tend to suppress the onset of turbulence and delay transition, and the following correlation given by Srinivasan et al. (1968) for the critical Reynolds number may be considered: h pffiffiffiffiffiffiffiffiffiffiffii Retr ¼ 2100 1 þ 12 d=2R , 10 < ð2R=dÞ < 1

(26)

Moreover, as suggested by Mishra and Gupta (1979), the finite-pitch effects of coiled tubes can be incorporated in Eq. 26 by replacing R with Rc given by Eq. 22c. In the turbulent flow regime (Re > Retr) inside curved or coiled circular tubes, to predict the Fanning friction factor, the following correlation (Mishra and Gupta 1979) is recommended: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

 f ¼ 0:079=Re0:25 þ 0:0075 ðd=2Rc Þ

(27)

This equation, which essentially superposes curvature-induced swirl-flow effects on straight axial flows, is valid for Retr < Re < 105, 6.7 < (2R/d ) < 346, and 0 < (H/ 2R) < 25.4. For the heat transfer in the fully developed turbulent flow inside curved tubes, Mori and Nakayama (1967) have provided the following correlations, respectively, for gases and liquids: 2 (
 )1=5 3
1=10 2 Pr d 41 þ 0:098 Re d 5, for Pr 1 2=3

Re4=5 Nu ¼ 2R 2R 26:2 Pr  0:074

(28a) 2

3

(
 )1=6
 Pr0:4 5=6 d 1=12 4 d 2:5 5, for Pr > 1 Re Nu ¼ 1 þ 0:061 Re 2R 2R 41:0

8

(28b)

Other Techniques and Compound Enhancement

Much of the explorations in using additives for liquids and additive for gases have been confined to single-phase liquid flows and primarily on the drag-reducing effects of the additives on the fluid flow (Manglik 2003; Bergles 1998). The lowering of frictional losses is an indirect enhancement of heat transfer enhancement, particularly when evaluated on the basis of fixed pressure drop or pumping power. For example, with soluble polymeric additives in water, where the solution is rendered

11

Enhancement of Convective Heat Transfer

469

with a shear-thinning rheology, the non-Newtonian effects yield a significant reduction in friction loss along with a modest increase in the heat transfer coefficient (Prusa and Manglik 1994; Manglik and Prusa 1995; Chhabra and Richardson 1999; Manglik and Fang 2002). Some other considerations in the literature have included the injection of gas bubbles where, for example, by injecting air bubbles at the base of a heated vertical wall up to 400% higher free-convection heat transfer coefficients have been reported (Tamari and Nishikawa 1976) in both water and ethylene glycol. In another study (Kenning and Kao 1972), up to 50% increase in the heat transfer in turbulent flow of water was obtained by injecting nitrogen bubbles in the flow stream; in essence this would be considered as quasi two-phase flow. The use of active techniques is somewhat restrictive and difficult because of the additional energy input to drive the enhancement method (Bergles 1998; Bergles and Manglik 2013; Manglik 2003). The external power input required to implement the techniques are often quite large and offset the thermal gains. However, there are instances where the primary “active” energy or driving force is needed to facilitate a thermal process as is the case in the use of mechanical aids. With either surface rotation or surface scraping, which may entail both heat and mass transfer, this technique is commonly used for processing viscous media in the chemical and food industries. For instance, in scraped surface heat exchangers, the heat transfer surface is rejuvenated with unprocessed media, thereby removing the high-thermal resistance processed media as well as effectively reducing surface fouling. A more recent and novel offshoot of this technique, which consists of a reciprocating rod and surface scraper elements to provide a tube-side self-cleaning scheme for shelland-tube exchangers, has been proposed (Bergles and Manglik 2013; Solano et al. 2011). Another active technique involves heat transfer surface vibration, at either low or high frequency in single-phase heat transfer. In one approach to fruitfully implement this technique, piezoelectric actuators have been attached to fins to augment the natural convection heat transfer from air-cooled finned heat sinks (Park and Kim 2010). Likewise, fluid vibration has been considered for both air (e.g., using acoustic fields) and liquids (e.g., using pulsations inducing actuators, and ultrasonic transducers). As an example, ultrasonic vibrations have been applied to a cylinder cooled by degassed subcooled water (Bartoli and Baffigi 2010) to get up to 30% enhancement where the transducer was in close proximity to the heater. However, in both cases of either surface or fluid vibration, equivalent heat transfer gains could be obtained by switching the heat rejection mode to that of simple forced convection (Bergles and Manglik 2013; Manglik 2016). An emerging area of heat transfer enhancement is in the use of compound techniques, where two or more passive and/or active techniques could be used. In principle, by such schemes the overall heat transfer coefficients can be increased above each of the techniques acting alone. A variety of technique combinations have been considered, and their emerging developmental growth is indicated in a recent survey (Manglik and Bergles 2004; Manglik and Jog 2009). For meaningful practical applications, perhaps the most attractive potential is presented in systems where one form of the enhancement method “naturally” preexists. Rotating systems (large

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electric motor and generator rotor windings, and gas-turbine blades, among others) are one such example. Some examples of different passive techniques that have been applied to rotating tubes and ducts in exploratory efforts (Bergles and Manglik 2013; Manglik 2003) of producing compound enhancement include the following: twisted-tape inserts in a tube that rotates around a parallel axis (to simulate the coolant channel of an electric machine), structured roughness or ribs inside rotating U-bend flat or rectangular ducts that simulate internal flow passages of a gas-turbine airfoil, and ribbed roughness and vortex generators that have been used in a two-pass rotating duct. Likewise, in the use of twisted-tape inserts in other tubular enhanced surface to devise compound schemes, single-phase forced convective applications have been explored (Manglik and Bergles 2002). These include adding twisted-tape inserts inside rough tubes or ribbed tubes and internally finned tubes (Zimparov et al. 2012). Other examples of compound enhancement techniques proposed in the literature (Manglik 2003) that hold some promise for practical applications are as follows: • Single-phase mass transfer enhancement in grooved (finned) channel with pulsating fluid flow and heat transfer in an acoustically excited flow field over a rough cylinder • Gas-solid suspension flows in an electric field • Fluidized bed (air-particle suspension) heat transfer with air-flow pulsations and across finned tubes There are, however, some counter-indicative surprises of using compound techniques that have been reported in the literature (Bergles 1998, 1999; Manglik 2003). The computational analysis of forced convection inside coiled tubes with internal fins (Masliyah and Nandakumar 1977) revealed that average Nu was lower than that in plain coiled tubes for the same flow conditions. In one other example (Bergles 2000), the average single-phase heat transfer coefficients in a coiled tube with pulsating flow (fluid vibration) were found to be lower than those in the coiled tube with steady flow.

9

Performance Evaluation Criteria

The enhancement of forced convection heat transfer is invariably accompanied with an increased flow friction loss. The quantification of the performance of different enhancement techniques thus requires the relative evaluation of the thermal heat transfer benefits after balancing the hydrodynamic penalty. For this, consider the tube-side performance of a two-fluid stream heat exchanger that is governed by two dependent variables: the heat transfer rate q and pressure drop Δp or pumping power P. These can be expressed as q ¼ ðUAÞΔT m

(29)

11

Enhancement of Convective Heat Transfer



Δp ¼ 2f ðL=di Þ G2 =ρ

471

and

P ¼ ΔpðGAc =ρÞ

(30)

The primary independent operating variables are the approach or initial temperature _ In the case of tube difference ΔTi [note that ΔTm = ϕ(ΔTi)] and the mass flow rate m. bundles, the heat transfer surface area A, or exchanger size, is given by the tube diameter di and length L and number of tubes N per pass. Also, in Eqs. 29 and 30, U is the overall heat transfer coefficient, f is the Fanning friction factor, G is the mass flux, Ac is the tube-side flow cross-sectional area, and ρ is the fluid density. In the practical use of enhancement techniques, typically the following performance objectives, along with a set of operating constraints or conditions, are considered for optimizing the heat exchanger design: 1. Increase the heat duty of an existing heat exchanger without altering the pumping power (or pressure drop) or flow rate requirements. 2. Reduce the approach temperature difference between the two heat-exchanging fluid streams for a specified heat load and size of exchanger. 3. Reduce the size or heat transfer surface area requirements for a specified heat duty and pressure drop. 4. Reduce the process stream’s pumping power requirements for a given heat load and exchanger surface area. Moreover, in considering and applying these objectives to design a heat exchanger, as recommended by Marner et al. (1983), the results for the enhanced geometries are based on an equivalent “envelope” or “empty” tube diameter. This allows a direct comparison of enhancement data with the “normal” or “smooth tube” performance. It may be noted that in cases of using duct inserts and/or extended surfaces, the hydraulic diameter is different from their “parent” or “empty” flow channel. Objectives (1), (2), and (4) yield savings in operating (or energy) costs, and objective (3) lends to material savings and reduced capital costs. These objective functions and their operating conditions have been described by many different performance evaluation criteria (PEC) (Bergles 1998; Manglik 2003). To understand their context in typical heat exchanger designs, it is instructive to consider some of these PEC for the more common case of applying enhancement techniques to the in-tube flows of conventional shell-and-tube heat exchangers and Eqs. 29 and 30. Twelve (12) different PEC are listed in Table 2 for the relative evaluation of single-phase flow heat transfer inside enhanced and smooth tubes. Note that the envelope or “empty tube” diameter is considered to be the same for both tubes, and the listing represents criteria for comparing the enhanced performance based on the following three broad constraints: 1. FG criteria: where the area of flow cross section (N and di) and tube length L are kept constant. For example, in retrofitting smooth tubes of an existing exchanger with enhanced tubes so as to increase the heat transfer rate q for the same approach temperature ΔTi and mass flow rate m_ or pumping power P.

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Table 2 Performance evaluation criteria (PEC) for forced convection heat transfer inside enhanced tubes of same envelope diameter (di) as the smooth tube Case FG-1a FG-1b FG-2a FG-2b FG-3 FN-1 FN-2 FN-3 VG-1 VG-2a VG-2b VG-3

Geometry N, L N, L N, L N, L N, L N N N – (NL)* (NL)* (NL)*

Fixed m_

P

✓ ✓

ΔTi ✓

✓ ✓ ✓ ✓

✓ ✓ ✓ ✓ ✓ ✓

Q

✓ ✓ ✓

✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓

✓ ✓ ✓ ✓ ✓ ✓ ✓

Objective q" ΔTi # q" ΔTi # P# L# L# P# (NL)* # q" ΔTi # P#

*

The product of N and L

Alternatively, the design would seek to decrease ΔTi or P for fixed q and m_ or P or reduce P for fixed q. 2. FN criteria: where the flow frontal area or cross section (N and di) is kept constant, and the heat exchange length is allowed to vary. This seeks a reduction in either the surface area (A ! L ) or the pumping power P for a fixed heat transfer rate q. 3. VG criteria: where the number of tubes and their length (N and L) are kept constant, but their diameter can change. A heat exchanger is often sized to meet a _ Because the tube-side velocity specified heat transfer rate q for a fixed flow rate m. reduces in such cases so as to accommodate the higher friction losses in the enhanced surface tubes, it becomes necessary to increase the flow area in order to _ This is usually accomplished by using a greater number of maintain constant m. parallel flow circuits so as to avoid the penalty of operating at higher thermal effectiveness inherent in the FG and FN criteria.

For the quantitative evaluation of these PEC, algebraic expression can be obtained that relates the enhanced surface performance (Nu or j and f Re) with that of an equivalent smooth duct. For a specified tube-bundle geometry (N, L, di) in a shell-and-tube heat exchanger, the heat transfer coefficient h and pumping power P can be expressed as   h ¼ cp jG=Pr2=3

(31)



P ¼ fAG3 =2ρ2

(32)

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Enhancement of Convective Heat Transfer

473

Thus, by keeping N, L, and di same, the performance of enhanced tubes can be related to that of equivalent smooth tubes (subscript “o”) as follows: ðhA=ho Ao Þ ðP=Po Þ

1=3

ðA=Ao Þ

2=3

¼

ðj=jo Þ ðf =f o Þ1=3

(33)

If j (or Nu) and f data or correlations for both tubes are available, then the evaluation of the objectives for each PEC in Table 2 is rather straightforward. One of the groupings (hA/hoAo), (P/Po), and (A/Ao) becomes the objective function with the other two set as 1.0 for the corresponding operating constraints, which then also provide the mass flux ratio (G/Go) required to satisfy Eq. 33. Extended details for obtaining the requisite relationships are given by Webb and Bergles (1983), and Manglik (2003). For illustrative applications of these PEC, the reader is directed to several references in the literature, such as (Manglik et al. 2012; Muley and Manglik 2000; Yerra et al. 2006), among others. Acknowledgments Partial support from ARPA-E, US Department of Energy, particularly for the evaluation of convective enhancement in plate-fin and tube-fin flows, is gratefully acknowledged.

References Bartoli C, Baffigi F (2010) Heat transfer enhancement from a circular cylinder to distilled water by ultrasonic waves at different subcooling degrees. In: Proceedings of the international heat transfer conference. ASME, Washinton, DC/New York. Paper IHTC14-22773 Bergles AE (1998) Techniques to enhance heat transfer. In: Rohsenow WM, Hartnett JP, Cho YI (eds) Handbook of heat transfer, 3rd edn. McGraw-Hill, New York. p Ch. 11 Bergles AE (1999) Enhanced heat transfer: endless frontier, or mature and routine? J Enhanc Heat Transf 6(2–4):79–88 Bergles AE (2000) New frontiers in enhanced heat transfer. In: Manglik RM, Ravigururajan TS, Muley A, Papar RA, Kim J (eds) Advances in enhanced heat transfer – 2000. ASME IMECE/ ASME, Orlando/New York, pp 1–8 Bergles AE, Manglik RM (2013) Current progress and new developments in enhanced heat and mass transfer. J Enhanc Heat Transf 20(1):1–15 Bergles AE, Nirmalan V, Junkhan GH, Webb RL (1983) Bibliography on augmentation of convective heat and mass transfer – II. Iowa State University, Ames Bergles AE, Jensen MK, Somerscales EFC, Manglik RM (1991) Literature review of heat transfer enhancement technology for heat exchangers in gas-fired applications. Gas Research Institute, Chicago Bishara F, Jog MA, Manglik RM (2009) Computational simulation of swirl enhanced flow and heat transfer in a twisted oval tube. J Heat Transf 131(8):080902–080901 Bishara F, Jog MA, Manglik RM (2013) Heat transfer enhancement due to swirl effects in oval tubes twisted about their longitudinal Axis. J Enhanc Heat Transf 20(4):289–304 Carnavos TC (1979) Cooling air in turbulent flow with internally finned tubes. Heat Transfer Eng 1(2):41–46 Champagne PR, Bergles AE (2011) Development and testing of a novel, variable-roughness technique to enhance, on demand, heat transfer in a single-phase heat exchanger. J Enhanc Heat Transf 8(5):341–352

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Chang YJ, Wang CC (1997) A generalized heat transfer correlation for louver fin geometry. Int J Heat Mass Transf 40:533 Chhabra RP, Richardson JF (1999) Non-Newtonian flow in the process industries: fundamentals and engineering applications. Butterworth-Heinemann, Oxford, UK Collins WM, Dennis SCR (1975) The steady motion of a viscous fluid in a curved tube. Q J Mech Appl Math 28:133–156 Dean WR (1927) Note on the motion of a fluid in a curved pipe. Philos Mag 4(7):208–233 Dean WR (1928) The stream line motion of fluid in a curved pipe. Philos Mag 5(7):673–695 Dippery DF, Sabersky RH (1963) Heat and momentum transfer in smooth and rough tubes at various Prandtl numbers. Int J Heat Mass Transf 6:329–353 Dzyubenko BV, Dreitser GA (1986) Heat transfer and fluid friction in bundles of twisted tubes. J Eng Phys 50(6):611–617 Fiebig M (1995) Vortex generators for compact heat exchangers. J Enhanc Heat Transf 2(1–2):43–61 Fiebig M, Kallweit P, Mitra N, Tiggelbeck S (1991) Heat transfer enhancement and drag by longitudinal vortex generators in channel flow. Exp Thermal Fluid Sci 4(1):103–114 Filonenko GK (1954) Hydraulic resistance in pipes (in Russian). Teploenergetika 1(4):40–44 Focke WW, Knibbe PG (1986) Flow visualization in parallel-plate ducts with corrugated walls. J Fluid Mech 165:73–77 Fraas AP (1989) Heat exchanger design, 2nd edn. Wiley, New York Fujii M, Seshimo Y, Ueno S, Yamanaka G (1989) Forced air heat sink with new enhanced fins. Heat Transf Jpn Res 18(6):53–65 Gnielinski V (1986) Correlations for the pressure drop in helically coiled tubes. Int Chem Eng 26 (1):36–44 Han J-C, Huang JJ, Lee CP (1993) Augmented heat transfer in square channels with wedge-shaped and delta-shaped turbulence promoters. J Enhanc Heat Transf 1(1):37–52 Huzayyin OA, Jog MA, Manglik RM (2010) Low Reynolds number air-flow heat transfer in trapezoidally corrugated perforated plate-fin ducts. ASHRAE Trans 116(2):339–346 Junkhan GH, Bergles AE, Nirmalan V, Ravigururajan TS (1985) Investigation of turbulators for fire tube boilers. J Heat Transf 107(2):354–360 Kays WM, London AL (1998) Compact heat exchangers, 3rd edn. Krieger Publishing Company, Malabar Kenning DBR, Kao YS (1972) Convective heat transfer to water containing bubbles: enhancement not dependent on thermocapillarity. Int J Heat Mass Transf 15:1709–1718 Kraus AD, Aziz A, Welty J (2001) Extended surface heat transfer. Wiley, New York Kreith F, Manglik RM, Bohn MS (2011) Principles of heat transfer, 7th edn. Cengage Learning, Stamford Lokshin VA, Fomina VN (1978) Correlation of experimental data on finned tube bundles. Teploenergetika 6:36–39 Luo L, Wang C, Wang L, Sundén B, Wang S (2015) Computational investigation of dimple effects on heat transfer and friction factor in a lamilloy cooling structure. J Enhanc Heat Transf 22(2):147–175 Manglik RM (2003) Heat transfer enhancement. In: Bejan A, Kraus AD (eds) Heat transfer handbook. Wiley, Hoboken. p Ch. 14 Manglik RM (2016) Heat transfer enhancement. In: Chhabra RP (ed) CRC handbook of thermal engineering 2e. CRC Press, Boca Raton. p Ch. 4 Manglik RM, Bergles AE (1992) Heat transfer enhancement and pressure drop in viscous liquid flows in isothermal tubes with twisted-tape inserts. Wärme- und Stoffübertragung 27(4):249–257 Manglik RM, Bergles AE (1993a) Heat transfer and pressure drop correlations for twisted-tape inserts in isothermal tubes: part I – laminar flows. J Heat Transf 115(4):881–889 Manglik RM, Bergles AE (1993b) Heat transfer and pressure drop correlations for twisted-tape inserts in isothermal tubes: part II – transition and turbulent flows. J Heat Transf 115(4):890–896

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Manglik RM, Bergles AE (1995) Heat transfer and pressure drop correlations for the rectangular offset-strip-fin compact heat exchanger. Exp Thermal Fluid Sci 10(2):171–180 Manglik RM, Bergles AE (2002) Swirl flow heat transfer and pressure drop with twisted-tape inserts. In: Hartnett JP, Irvine TF, Cho YI, Greene GA (eds) Advances in heat transfer, vol 36. Academic, New York, pp 183–266 Manglik RM, Bergles AE (2004) Enhanced heat and mass transfer in the new millennium: a review of the 2001 literature. J Enhanc Heat Transf 11(2):87–118 Manglik RM, Bergles AE (2013) Characterization of twisted-tape-induced helical swirl flows for enhancement of forced convective heat transfer in single-phase and two-phase flows. J Therm Sci Eng Appl 5(2):021010. (021011–021012). https://doi.org/10.1115/1.4023935 Manglik RM, Fang P (2002) Thermal processing of viscous non-Newtonian fluids in annular ducts: effects of power-law rheology, duct eccentricity, and thermal boundary conditions. Int J Heat Mass Transf 45(4):803–814 Manglik RM, Jog MA (2009) Molecular-to-large-scale heat transfer with multiphase interfaces: current status and new directions. J Heat Transf 131(12):121001. (121001–121011) Manglik RM, Kraus AD (1996) Process, enhanced and multiphase heat transfer. Begell House, New York Manglik RM, Prusa J (1995) Viscous dissipation in non-Newtonian flows: implications for the nusselt number. J Thermophys Heat Transf 9(4):733–742 Manglik RM, Maramraju S, Bergles AE (2001) The scaling and correlation of low Reynolds number swirl flows and friction factors in circular tubes with twisted-tape inserts. J Enhanc Heat Transf 8(6):383–395 Manglik RM, Zhang J, Muley A (2005) Low Reynolds number forced convection in threedimensional wavy-plate-fin compact channels: fin density effects. Int J Heat Mass Transf 48(8):1439–1449 Manglik RM, Huzayyin OA, Jog MA (2011) Fin effects in flow channels of plate-fin compact heat exchanger cores. J Therm Sci Eng Appl 3(4):041004. (041001–041009) Manglik RM, Patel P, Jog MA (2012) Swirl-enhanced forced convection through axially twisted rectangular ducts – part 2, heat transfer. J Enhanc Heat Transf 19(5):437–450 Manglik RM, Bergles AE, Dongaonkar AJ, Rajendran S (2013) Limitations of compiling the global literature on enhanced heat and mass transfer. J Enhanc Heat Transf 20(1):83–92 Manlapaz RL, Churchill SW (1980) Fully developed laminar flow in a helically coiled tube of finite pitch. Chem Eng Commun 7:57–78 Manlapaz RL, Churchill SW (1981) Fully developed laminar convection from a helical coil. Chem Eng Commun 9:185–200 Marner WJ, Bergles AE (1978) Augmentation of Tubeside laminar flow heat transfer by means of twisted-tape inserts, static-mixer inserts, and internally finned tubes. In: International heat transfer conference, heat transfer 1978. Hemisphere, Washington, DC, pp 583–588 Marner WJ, Bergles AE, Chenoweth JM (1983) On the presentation of performance data for enhanced tubes used in shell-and-tube heat exchangers. J Heat Transf 105:358–365 Masliyah JH, Nandakumar K (1977) Fluid flow and heat transfer in internally finned helical coils. Can J Chem Eng 55:27 36 Metwally HM, Manglik RM (2004) Enhanced heat transfer due to curvature-induced lateral vortices in laminar flows in sinusoidal corrugated-plate channels. Int J Heat Mass Transf 47(10–11):2283–2292 Mishra P, Gupta SN (1979) Momentum transfer in curved pipes, 1. Newtonian fluids; 2. Non-Newtonian fluids. Ind Eng Chem Process Des Dev 18:130–142 Mori Y, Nakayama W (1965) Study on forced convective heat transfer in curved pipes (1st report, laminar region). Int J Heat Mass Transf 8:67–82 Mori Y, Nakayama W (1967) Study on forced convective heat transfer in curved pipes (3rd report, theoretical analysis under the condition of uniform wall temperature and practical formulae). Int J Heat Mass Transf 10:681–695

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Muley A, Manglik RM (2000) Enhanced thermal-hydraulic performance optimization of chevron plate heat exchangers. Int J Heat Exch 1(1):3–18 Nandakumar K, Masliyah JH (1986) Swirling flow and heat transfer in coiled and twisted pipes. In: Mujumdar AS, Mashelkar RA (eds) Advances in transport processes, vol IV. Wiley Eastern, New Delhi, pp 49–112 Ngo TL, Kato Y, Nikitin K, Ishizuka T (2007) Heat transfer and pressure drop correlations of microchannel heat exchangers with S-shaped and zigzag fins for carbon dioxide cycles. Exp Thermal Fluid Sci 32(2):560–570 Nikuradse J (1933) Strömungsgesetze in rauhen Rohren. Forsch Arb Ing-Wes, No 361 (English translation as NACA TM 1292, 1965) Nishimura T, Bian Y, Kunitsugu K, Morega AM (2003) Fluid flow and mass transfer in a sinusoidal wavy-walled tube at moderate Reynolds numbers. Heat Transf Asian Res 32(7):650–661 Oliver DR, Shoji Y (1992) Heat transfer enhancement in round tubes using three different tube inserts: non-Newtonian liquids. Chem Eng Sci Des 70(6):558–564 Park H, Kim S (2010) Thermal performance improvement of a heat sink with piezoelectric vibrating fins. In: Proceedings of the international heat transfer conference. ASME, Washington, DC/New York. pp Paper IHTC14-22552 Patel P, Manglik RM, Jog MA (2012) Swirl-enhanced forced convection through axially twisted rectangular ducts – part 1, fluid flow. J Enhanc Heat Transf 19(5):423–436 Prusa J, Manglik RM (1994) Asymptotic and numerical solutions for thermally developing flows of Newtonian and non-Newtonian fluids in circular tubes. Numer Heat Transfer Part A 26 (2):199–217 Prusa J, Yao LS (1982) Numerical solution for fully developed flow in heated curved tubes. J Fluid Mech 123(Oct):503–522 Ravigururajan TS, Bergles AE (1994) Visualization of flow phenomena near enhanced surfaces. J Heat Transf 116(1):54–57 Ravigururajan TS, Bergles AE (1995) Prandtl number influence on heat transfer enhancement in turbulent flow of water at low temperatures. J Heat Transf 117(2):276–282 Ravigururajan TS, Bergles AE (1996) Development and verification of general correlations for pressure drop and heat transfer in single-phase turbulent flow in enhanced tubes. Exp Thermal Fluid Sci 13(1):55–70 Rush TA, Newell TA, Jacobi AM (1999) An experimental study of flow and heat transfer in sinusoidal wavy passages. Int J Heat Mass Transf 42:1541–1553 Saunders EAD (1988) Heat exchangers: selection, design and construction. Longman Scientific & Technical, Harlow Solano JP, Garcia A, Vicente PG, Viedma A (2011) Performance evaluation of a zero-fouling reciprocating scraped-surface heat exchanger. Heat Transf Eng 32(3–4):331–338 Srinivasan PS, Nandapurkar SS, Holland FA (1968) Pressure drop and heat transfer in coils. Chem Eng 218:113–119 Sundén B (1999) Enhancement of convective heat transfer in rib-roughened rectangular ducts. J Enhanc Heat Transf 6(2–4):89–103 Tamari M, Nishikawa K (1976) The stirring effect of bubbles upon the heat transfer to liquids. Heat Transf Jpn Res 5(2):31–44 Taylor GI (1929) The criterion for turbulence in curved pipes. Proc R Soc A 124:243–249 Vyas S, Manglik RM, Jog MA (2010) Visualization and characterization of lateral swirl flow structure in sinusoidal corrugated-plate channels. J Flow Vis Image Process 17(4):281–296 Wang C-C (2000) Technology review – a survey of recent patents of fin-and-tube heat exchangers. J Enhanc Heat Transf 7(5):333–345 Wang C-C, Lee W-S, Sheu W-J, Liaw J-S (2001) Empirical airside correlations of fin-and-tube heat exchangers under dehumidifying conditions. Int J Heat Exch II(2):151–178 Watkinson AP, Miletti DC, Kubanek GR (1975) Heat transfer and pressure drop of internally finned tubes in laminar oil flows. ASME, New York

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Webb RL, Bergles AE (1983) Performance evaluation criteria for selection of heat transfer surface geometries used in low Reynolds number heat exchangers. In: Kakaç S, Shah RK, Bergles AE (eds) Low Reynolds number flow heat exchangers. Hemispherre, Washington, DC, pp 735–752 Webb RL, Kim N-H (2005) Principles of enhanced heat transfer, 2nd edn. Taylor & Francis, Boca Raton Webb RL, Eckert ERG, Goldstein RJ (1971) Heat transfer and friction in tubes with repeated-rib roughness. Int J Heat Mass Transf 14:601–618 Webb RL, Exckert ERG, Goldstein RJ (1972) Generalized heat transfer and friction correlations for tubes with repeated-rib roughness. Int J Heat Mass Transf 15:180–184 Wright LM, Han J-C (2014) Heat transfer enhancement for turbine blade internal cooling. J Enhanc Heat Transf 21(2–3):111–140 Yerra KK, Manglik RM, Jog MA (2006) Optimization of heat transfer enhancement in single-phase tubeside flows with twisted-tape inserts. Int J Heat Exch 8(1):117–138 Zhang J, Kundu J, Manglik RM (2004) Effect of fin waviness and spacing on the lateral vortex structure and laminar heat transfer in wavy-plate-fin cores. Int J Heat Mass Transf 47(8–9):1719–1730 Zimparov V, Petkov VM, Bergles AE (2012) Performance characteristics of deep corrugated tubes with twisted-tape inserts. J Enhanc Heat Transf 19(1):1–11 Žukauskas A (1989) High-performance single-phase heat exchangers. Hemisphere, New York

Electrohydrodynamically Augmented Internal Forced Convection

12

Michal Talmor and Jamal Seyed-Yagoobi

Contents 1 2 3 4

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Transport Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fully Developed Laminar Flow in Circular Tubes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Electrohydrodynamically Driven Internal Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Introduction to Electrohydrodynamic (EHD) Pumping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 EHD Theoretical Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Experimental Studies of EHD Driven Single Phase Flows: Macroscale . . . . . . . . . . . . 4.4 EHD Driven Single Phase Flows: Microscale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 EHD Driven Two Phase Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Example Application: EHD Driven Single Phase Flow Distribution Control . . . . . . . 5 Cross-References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

481 482 483 491 491 492 507 511 514 518 521 521

Abstract

Many thermal devices, such as heat exchangers and heat pipes, utilize forced convection of internal flows as their main mechanism for heat transport. This chapter addresses the fundamentals associated with internal flows by providing the mathematical model for the simplest case of forced convection – a laminar flow in a circular tube. As an example for the application of this fundamental theory, the modern topic of electrohydrodynamically driven dielectric liquids and liquid films for heat transfer in internal flows in macro- and microscales is presented.

M. Talmor (*) · J. Seyed-Yagoobi Multi-Scale Heat Transfer Laboratory, Department of Mechanical Engineering, Worcester Polytechnic Institute, Worcester, MA, USA e-mail: [email protected]; [email protected] # Springer International Publishing AG, part of Springer Nature 2018 F. A. Kulacki (ed.), Handbook of Thermal Science and Engineering, https://doi.org/10.1007/978-3-319-26695-4_7

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Nomenclature Fluid Parameters


 Fluid velocity vector ms
m Mean fluid velocity s Pressure [Pa] Temperature [K] Mean temperature [K]
 Mass flux kgs
 Shear stress mN2 Friction coefficient Reynolds number
 Heat flux density mW2 Nusselt number

v vx , mean P T Tmean m_ τ Cf RE q00 NuD

Electrical Parameters

E E Φ J fg f EHD ρe ne , pe n, p ωe F(ωe) I1 λ ζ ω kw


 Electric field vector mV
 Electric field magnitude mV Electric potential [V]
 Current density vector mC3 s
 Force density of gravity mN3
 EHD body force density mN2
 Net charge density, ne  pe mC3
 -/+ Charge densities mC3 Ionic species densities [m3] Onsager parameter Onsager function Bessel function, first kind, order one Heterocharge layer thickness [m] Zeta potential [V] Angular frequency [s1] Wave number [m1]

Constants

h ρ μ cp k α e σ


 Heat transfer coefficient mW2 K
kg  Fluid mass density m3 Fluid dynamic
viscosity [Pas]  Specific heat KJ
W Thermal conductivity mK h 2i Thermal diffusivity ms
 Electric permittivity mkg3
 Electric conductivity mS

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h 2i

b

Ionic mobility

te n0 neq kD kR g

Charge relaxation time [s] Equilibrium ionic density [m
3] Equilibrium charge density mC3 Dissociation constant [m3s1] Recombination constant [s
1] Gravitational acceleration sm2

m Vs

Domain Parameters

t r, θ, x Ac D r0 ^ n

Time [s] Cylindrical coordinates Channel cross sectional area [m2] Hydraulic Channel Diameter [m] Hydraulic channel radius [m] Normal direction unit vector

Subscripts

Initial condition or surface In the given direction Positive or negative

1

Introduction

Heat transfer to, or from, a confined flow domain is included under the umbrella of forced convection of internal flows. Such confined domains can be tubes, ducts, or other containers. Forced convection of internal flows is present in all areas of science and engineering, from industrial fluid supply lines and heat exchangers to arteries and plant roots. In all of these applications, the flow of a fluid is driven by a variety of forces and opposed by friction against the confines of the liquid volume. The resultant velocity of the fluid affects the transferal of energy to or from the fluid domain, as well as the surface temperature of the confining walls. This chapter describes the fundamental principles associated with forced convection of internal flows as well as the regimes in which such flows can operate. The corresponding mathematical model for a laminar flow inside a tube with a circular cross section is also provided, showing the development of the velocity and temperature profiles typical for such flows. Many applications involving internal flow forced convection rely on traditional mechanical pumping devices for generating pressure and flow. However, many new heat transport systems, using unconventional means for creating pressure and flow,

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are being developed for modern thermal control applications – where miniaturization, low power consumption, and simplicity of manufacturing are key factors. The second part of this chapter provides an overview of a family of such innovative fluid pumping devices known as electrohydrodynamic pumps. These devices use different mechanisms for introducing charges into dielectric working fluids and then utilize electrical body forces to create flow and generate pressure. This overview includes the use of these devices in forced convection heat transfer enhancement of internal flows, the fundamental principles of the operation of these devices, and the mathematical modeling of such systems.

2

Transport Equations

The first transport equation is the equation of conservation of mass, @ρ 1 @ 1 @ @ þ ðρrvr Þ þ ðρvθ Þ þ ðρvx Þ ¼ 0 @t r @r r @θ @x

(1)

Next are the equations of motion (Navier Stokes, conservation of momentum) for a Newtonian fluid, r-direction:   @vr @vr vθ @vr v2θ @vr ρ þ vr þ  þ vx @t @r r @θ r   @x  @P @ 1@ 1 @ 2 vr 2 @vθ @ 2 vr þ ρgr þ μ þ 2 ¼ ðrvr Þ þ 2 2  2 @r @r r @r r @θ r @θ @x (2) θ-direction:   @vθ @vθ vθ @vθ vr vθ @vθ þ vr þ þ þ vx ρ @t @r r @θ r   @x  1 @P @ 1@ 1 @ 2 vθ 2 @vr @ 2 vθ þ ρgθ þ μ þ 2 ¼ ðrvθ Þ þ 2 2 þ 2 r @θ @r r @r r @θ r @θ @x (3) x-direction: ρ

  @vx @vx vθ @vx @vx þ vr þ þ vx @t @r r @θ @x     @P 1@ @vx 1 @ 2 vx @ 2 vx þ ρgx þ μ r ¼ þ 2 2 þ 2 @x r @r r @θ @r @x

The final equation of transport is the thermal energy rate equation,

(4)

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 ρcp

483

     @T @T vθ @T @T 1@ @T 1 @2T @2T þ vr þ þ vx r ¼ k þ 2 2þ 2 @t @r r @θ @x r @r @r r @θ @x (    2  2 ) 2 @vr 1 @vθ @vx þ 2μ þ vr þ þ r @θ @r @x ( 2  2 @vθ 1 @vx @vx @vr þμ þ þ þ @x r @θ @r @x   ) 1 @vr @  vθ 2 þ þr r @θ @r r (5)

These transport equations are used to obtain the velocity and temperature profiles for an internal flow under various operating conditions. Similar equations exist in the Cartesian and spherical coordinates. Numerous textbooks are available which present the derivation of these transport equations in details (e.g., Kays and Crawford 1980; Bird et al. 2007). These textbooks also cover, in much detail, the topic of heat transfer in internal flows presented below. The interested reader is encouraged to refer to such textbooks to get acquainted further with this topic.

3

Fully Developed Laminar Flow in Circular Tubes

To develop the velocity profile for a steady laminar Newtonian fluid flow inside a horizontal circular tube, a uniform velocity profile is assumed at the entrance of the tube. As the fluid flows through the tube, a viscous boundary layer is developed with fluid velocity being zero on the tube wall, as shown in Fig. 1. Eventually, as the fluid moves down the tube, the viscous boundary layer meets itself at the tube center. From this point on, the velocity profiles remain unchanged as long as the fluid properties are not a strong function of temperature. The initial tube length during which the viscous boundary layer grows is referred to as the hydrodynamic entry

Fig. 1 Development of the viscous boundary layer and velocity profile in the hydrodynamic entry region of a tube (Kays and Crawford 1980)

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length, or hydrodynamically developing length. The region beyond this entry length is called the hydrodynamically fully developed region. Assuming an axisymmetric configuration (i.e., no dependency on θ) and vθ = 0, the equations of motion for the hydrodynamic entry length simplify as follows, r-direction       @vr @vr @P @ 1 @ @ 2 vr ρ vr þ ρgr þ μ þ vx ðrvr Þ þ 2 ¼ @r @r r @r @r @x @x

(6)

x-direction       @vx @vx @P 1 @ @vx @ 2 vx þμ r þ vx ρ vr ¼ þ 2 @x r @r @r @x @r @x

(7)

Note that @P/@r , ρgr , @ 2vr/@x2 and @ 2vx/@x2 terms may be neglected in comparison with the magnitudes of the other terms. Similarly to the developing viscous boundary layer, a thermal boundary layer develops as the fluid flows through the tube, as shown in Fig. 2. Note that the hydrodynamic entry length and the tube length during which the thermal boundary layer grows are not necessarily the same, nor are the heights of these different boundary layers the same. Figure 2 also shows the fully developed temperature profile characteristic of two different surface conditions – a specified wall temperature, Ts, and a specified wall heat flux, q00s . The equation of thermal energy rate for the thermally developing region (i.e., the thermal entrance region) of interest reduces to,  ρcp

@T @T þ vx vr @r @x



    1 @ @T @2T r ¼k þ 2 r @r @r @x

(8)

where the last term, representing the change in heat conduction term in the x-direction, may be ignored due to its small magnitude compared to the rest of the

Fig. 2 Development of thermal boundary layer and temperature profile in the thermal entrance region and thermally fully developed region (Bergman et al. 2011)

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terms. Note that in Eq. 8, the viscous dissipation contribution (i.e., the terms in the last two brackets in Eq. 5) are ignored since its magnitude is negligible in typical internal flows. The governing equations provided above are generally solved numerically to obtain the velocity and temperature profiles in order to determine the corresponding convection heat transfer coefficient. In the hydrodynamically fully developed region, as stated above, the axial velocity profile remains unchanged as long as the fluid properties are not a strong function of temperature. Then, under the assumption of an axisymmetric flow, the following conclusion can be reached from the conservation of mass equation, vr ¼ 0

since

@vx ¼0 @x

(9)

This implies that vx is only a function of r , vx = vx(r). Then, Eq. 7 for the hydrodynamically fully developed region simplifies to,   μ d dvx dP r ¼ r dr dx dr

(10)

Since the pressure is independent of r, Eq. 10 can be integrated directly twice with respect to r to yield the desired velocity profile. The boundary conditions to apply are, r¼0:

@vx ¼0 @r

r ¼ r0 :

vx ¼ 0

which results in the following axial velocity profile,    r 2o dP r2  vx ¼ 1 2 dx 4μ ro

(11)

Then, the total mass flow rate through the tube is simply calculated as follows, ð m_ ¼

ρvx dAc

(12)

Ac

The mean velocity, vx, mean, can be determined from the following equation, vx, mean ¼

m_ Ac ρ

(13)

Then, with a constant density for a circular tube of inner radius r0, vx, mean

  r 20 dP  ¼ dx 8μ

(14)

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Substituting Eq. 14 into Eq. 11 gives the axial velocity profile in terms of the mean velocity,  vx ¼ 2vx, mean

r2 1 2 r0

 (15)

The shear stress at the wall surface can be evaluated from the gradient of the velocity profile at the wall. From Eq. 15,   @vx 4μvx, mean ¼ τ0 ¼ μ @r r¼r0 r0

(16)

Similarly, the shear stress within the fluid can be determined by taking the derivative of the velocity profile, τ ¼ 4μvx, mean

r r 20

(17)

Thus, τ r ¼ τ0 r 0

(18)

The wall shear stress can be expressed in dimensionless form in terms of friction coefficient, Cf ¼

τ0 1 2 ρv 2 x, mean

(19)

Considering the absolute value of τw from Eq. 17, Cf ¼

8μ r 0 ρvx, mean

(20)

Note that the friction coefficient is independent of x for the hydrodynamically fully developed region. Equation 20 for a circular tube can also be expressed as, Cf ¼

16 ρvx, mean D , where Re ¼ Re μ

(21)

The energy equation (i.e., Eq. 8) for the thermally fully developed region reduces to, ρcp vx

    @T 1 @ @T @2T ¼k r þ 2 @x r @r @r @x

(22)

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where the last term in the above equation, corresponding to the heat conduction in the x-direction, is negligible for common internal flows. Thus, vx

  @T 1 @ @T ¼α r @x r @r @r

(23)

The above equation will need to be solved in order to obtain the fluid temperature profile. Once the temperature profile is obtained, the convection coefficient can be determined from its definition, q000 ¼ hðT 0  T mean Þ

(24)

where T0 corresponds to the tube wall temperature. The mixed mean fluid temperature at a given tube cross section is calculated as follows, Ð T mean ¼

Ac vx TdAc

Ac vx, mean

(25)

In order to solve Eq. 23 for fluid temperature profile, two separate cases will be considered. Case 1 will correspond to a constant heat flux at the tube wall, and Case 2 will correspond to the tube wall kept at a constant temperature. These two cases will lead to different temperature profiles, thus different convection coefficients. All along, the flow will be assumed to be hydrodynamically fully developed, thus, allowing the use of Eq. 15 in Eqs. 23 and 25. It is important to recognize that in the case of thermally fully developed flow, the term @T/@x varies with x since the heat transfer still takes place. However, it can be shown that the following nondimensional temperature condition is invariant with x,   @ T0  T ¼0 @x T 0  T mean

(26)

The above equation is indicative of the fact that the relative shape of the temperature profile no longer changes in the thermally fully developed region. Taking the derivative of the abovementioned nondimensional temperature at the tube wall with respect to r yields,  

@T=@rjγ¼γ0 @ T0  T

¼ 6¼ f ðxÞ

@r T 0  T mean γ¼γ0 T 0  T mean

(27)

Since the heat flux at the tube wall is defined as, q000 ¼ k

@T

¼ hðT 0  T mean Þ @r γ¼γ0

Then, from Eqs. 27 and 28 it can be readily concluded that,

(28)

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Fig. 3 Illustration of convection coefficient for laminar flow inside a tube

q000 =k h ¼ 6¼ f ðxÞ or h 6¼ f ðxÞ q000 =h k

(29)

This means that in the thermally fully developed region, the convection coefficient, h, has a constant value as long as the thermal conductivity of the fluid, k, is constant, as illustrated in Fig. 3. Differentiating Eq. 26 with respect to x and solving it for @T/@x yields, @T dT 0 T 0  T dT 0 T 0  T dT mean ¼  þ @x dx T 0  T mean dx T 0  T mean dx

(30)

For the case of a constant heat flux at the tube wall, q000 ¼ hðT 0  T mean Þ; T 0  T mean ¼ constant

(31)

dT 0 dT mean  ¼0 dx dx

(32)

or,

Then, from Eq. 30 one can conclude, @T dT 0 dT mean ¼ ¼ @x dx dx

(33)

Furthermore, a simple energy balance along the tube length yields, _ p dT mean πDq00 dx ¼ mc or,

(34)

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Fig. 4 Axial temperature variations for constant heat flux

dT mean ¼ constant dx

(35)

Note that Eq. 35 is valid for both the thermal entrance region and the thermally fully developed region. Based on Eqs. 32 and 35, the fluid axial temperature variations for the case of a constant heat flux at the tube wall are illustrated in Fig. 4. For the case of constant surface temperature, dT 0 ¼0 dx

(36)

@T T 0  T dT mean ¼ @x T 0  T mean dx

(37)

Then, Eq. 30 reduces to,

Figure 5 illustrates the axial temperature variations for the case of constant surface temperature. As stated previously, Eq. 23 will need to be solved for the two cases of constant heat flux and constant surface temperature. Considering the first case of constant heat flux, substituting Eq. 33 into Eq. 23 yields,   1 @ @T dT mean r ¼ vx r @r @r dx

(38)

where dTmean/dx is constant. The corresponding boundary conditions are, r¼0

@T ¼ 0 and @r

r ¼ r0

T ¼ T0

(39)

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Fig. 5 Axial temperature variations for constant surface temperature

Note that T0 is not known in this case and is not constant. However, if the known heat flux boundary conditions at the tube wall is to be applied at this point, then Eq. 38 will not yield a unique solution since both boundary conditions will be of the second kind. Substituting Eq. 15 for vx in Eq. 38 and integrating it twice yields, T ¼ T0 

   2vx, mean dT mean 3 2 r4 r2 r0 þ  16 α dx 16r 20 4

(40)

From the above equation, Tmean can be determined as follows, Tm ¼ T0 

11 2vx, mean dT mean 2 r 96 α dx 0

(41)

Substituting this into Eq. 24 yields, q000 ¼

11 2vx, mean dT mean 2 r h 96 α dx 0

(42)

Note that the unknown Ts no longer appears in the above equation. Finally, substituting Eq. 34 into Eq. 42 with m_ ¼ ρπr 20 vx, mean , the convection coefficient becomes, h¼

48 k 11 D

(43)

where D is the tube diameter. The above equation in dimensionless form becomes, NuD ¼

hD ¼ 4:364 k

(44)

For the case of a constant surface temperature, substituting Eq. 37 into Eq. 23 yields,

12

Electrohydrodynamically Augmented Internal Forced Convection

  1 @ @T vx T 0  T dT mean r ¼ r @r @r α T 0  T mean dx

491

(45)

The corresponding boundary conditions in this case are, r¼0

@T ¼0 @r

(46)

r ¼ r0

T ¼ T0

(47)

Here, T0 is known. Substituting for vx in Eq. 45 from Eq. 15 yields,     1 @ @T 2vx, mean r2 T 0  T dT mean r 1 2 ¼ r @r @r α r 0 T 0  T mean dx

(48)

This is a more complicated equation to solve than the one for the constant heat flux case. One method to solve Eq. 48 is by successive substitution (i.e., trial and error). A good choice for the initial temperature profile could be the temperature profile for the constant heat flux case. After several iterations, the actual temperature profile for constant surface temperature will be obtained. Then, following the same steps as those of the constant heat flux case, the Nusselt number becomes, NuD ¼ 3:658

(49)

As seen from Eqs. 44 and 49, the heat transfer is more effective under a constant heat flux at the wall than the constant wall temperature.

4

Electrohydrodynamically Driven Internal Flows

4.1

Introduction to Electrohydrodynamic (EHD) Pumping

Electrohydrodynamics (EHD) is a field of research studying the interactions between externally applied electrical fields and fluid flow fields. Although the notion that electrical charge can affect fluids has been known for centuries (Pickard 1965), research into utilizing EHD phenomena to generate fluid flow did not begin in earnest until the 1960s (Stuetzer 1959; Pickard 1963a; Melcher 1961). Since then, many innovative applications utilizing EHD have been developed include EHD driven spraying (Hayati et al. 1986), heat transfer enhancement in various configurations (Jones 1979; Seyed-Yagoobi et al. 1989), EHD liquid jet generation (Hanaoka et al. 2002), and EHD driven mixing (Yazdani and Seyed-Yagoobi 2009c). EHD devices have no moving parts and are able to pump traditionally difficult flows such as single and two-phase thin film flows (Brand and SeyedYagoobi 2003; Darabi and Ekula 2003). These devices can therefore be more easily miniaturized than mechanical pumping devices, and recent research has shown that some EHD pumping can operate in microgravity conditions (Patel et al. 2016). Since

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the performance of EHD devices depends primarily on the magnitude of the applied electrical field, these devices can be used in a smart, autonomous control system for real-time flow management. These unique properties of EHD devices make them advantageous for usage in a wide range of applications, including small-scale electronics cooling, flow distribution control for single or two-phase flows, propellant utilization enhancement, and aerospace thermal control. Studying EHD driven internal flows, in terms of flow generation or flow control, requires full understanding of the underlying physical principles, the capabilities, and the limitations of EHD devices. This section aims to provide prospective researchers with the background, approaches and state-of-the-art information needed to explore and utilize EHD driven internal flows.

4.2

EHD Theoretical Background

The model for EHD driven flows couples the familiar Navier-Stokes equations for fluid flow with Maxwell’s equations for electrostatics. This coupling occurs via an additional force density term in the momentum conservation equation. This term describes the sum of EHD body force densities that are applied on a linear dielectric working fluid (Melcher 1981) and is generally written as: f EHD

    1 2 1 @e 2 ¼ ρe E  E ∇e þ ∇ E ρ 2 2 @ρ T

(50)

The first term in the above equation is the Coulomb force density, also known as the electrophoretic force density, which describes the interaction between the applied electrical field, E, and the space charge density, ρe, available in the fluid when free charges are present. The second term is the dielectrophoretic force density, which arises from polarization effects in a working fluid with gradients or discontinuities in its electrical permittivity, e. The final term is the electrostriction force density, which arises from changes in the electrical permittivity of the working fluid as a result of changes in fluid density, and is primarily relevant for compressible flows. Since even in multiphase flows each phase is often considered incompressible for the purpose of modeling, the electrostriction force term can be neglected in the formulation of most EHD driven flows. The dielectrophoretic force is relevant in nonisothermal applications, or when there exists an interface between regions of the fluid domain with significantly different electrical permittivity values, such as a vapor-liquid interface. This force acts along the electric permittivity gradient, unlike the electrophoretic force that acts in the direction of the electric field. This means that the dielectrophoretic force can create undesired flow motion in directions that oppose or reduces the desired effect of the electrophoretic force, as well as introduce flow instabilities. It should be noted that all EHD forces are volumetric forces, which act independently of gravity. In terms of size scaling, as an EHD device is scaled down any adverse effects from gravity would be minimized, and the EHD forces would

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dominate. However, other effects related to miniaturization will need to be taken into consideration as well, such as higher pressure losses, flow instabilities, and any effects on the charge distribution (Ramos 2011). In the case of a bulk fluid where the force being applied is most significant at a boundary location, it is possible to use the Maxwell stress tensor to describe the EHD body force density applied on the surface of a volume enclosing the system of interest (Crowley 1986). This can be done in lieu of determining the charge distribution in the bulk if the electric field arising from these charges is known,     e ρ @e τij ¼ eEi Ej  δij Ek Ek 1  2 e @ρ T

(51)

If a single phase working fluid is assumed to be incompressible and isothermal, the force density will be dominated by the electrophoretic force. Most existing applications of EHD driven internal flows, such as EHD pumps, follow this description due to its simplicity. There exist three recognized types of EHD pumping techniques, which differ by how each introduces free charges into the working fluid – ion drag pumping, induction pumping, and conduction pumping (SeyedYagoobi 2005).

4.2.1 Ion Drag Pumping Also known as injection pumping, this is the most mature of the EHD pumping. An ion drag EHD pump introduces free charges into a working fluid via applying a sufficiently strong direct current (DC) electric field between a sharp emitter electrode and a collector electrode, such that a corona discharge of material from the sharp emitter tip occurs, in the forms of ions of the same polarity as the emitter. The emitted charges travel to the oppositely charged collector electrode, imparting their momentum to the surrounding fluid and generating a strong net flow that can be turbulent regardless of the Reynolds number, depending on the properties of the working fluid (Crowley et al. 1990; Paschkewitz and Pratt 2000). The phenomenon of injected ions discharged from sharp points are able to exert mechanical forces on fluids that has been known for more than a century (Chattock et al. 1901). However, the first model describing the phenomenon in all fluid media was not developed until much later (Stuetzer 1959) and verified experimentally (Pickard 1963b) using transformer oil. This new model showed that significant pressure head could be generated using injected charges into a dielectric liquid, making ion drag pumping a viable theoretical replacement to select mechanical pumps. Figure 6 shows a common configuration in a circular flow channel, with the emitter electrode receiving high voltage while the ring shaped collector electrode mounted into the wall being grounded. It is, however, possible to configure this type of pump with the emitter being grounded and the collector receiving high voltage. Ion drag pumping is the easiest technique to model, using the simplest coupling of the Navier-Stokes and Maxwell equations. In the electrostatics formulation, the electric field is irrotational, meaning that electric field has zero curl and the field can be described as a gradient of a scalar potential, Φ:

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M. Talmor and J. Seyed-Yagoobi +V ring ground electrode

pipe wall

+ + high voltage electrode

e e e

+

flow

+

Fig. 6 EHD ion drag pumping schematic (Al Dini 2007)

E ¼ ∇Φ

(52)

The descriptions of the electric charge density, ρe, and total current density, J, are described via Gauss’ Law and the charge conservation descriptions from the electrostatics formulation:  ∇  eE ¼ ρe (53) ∇Jþ

@ρe ¼0 @t

(54)

Combining Gauss’ Law with the electrostatic potential equation yields the familiar Poisson equation: ∇2 Φ ¼ 

ρe e

(54)

The total current density depends on the different charge transport effects that are present: J ¼ σE þ ρe u þ ρbE þ Dð∇qe Þ

(55)

where σ is the electrical conductivity of the working fluid, u is the flow velocity, b is the ionic mobility of the emitted ions, and D is the diffusion coefficient. The terms shown describe the Ohmic conduction, convection, ionic mobility, and diffusion effects, respectively. The diffusion term is generally negligible for most EHD pumping situation, even at the microscale, and could be omitted. It should be noted that the conduction term for ion drag pumping does not contribute to the overall motion of charges between the emitter and collector electrodes, but represents an additional current path that causes additional power draws in liquid phase, as have been experimentally verified. This term could therefore be neglected for single gas phase flows, but for the sake of generality this term will be kept in the current model description. After some manipulation, the steady state charge conservation equation becomes:

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Electrohydrodynamically Augmented Internal Forced Convection

ðv  b∇ΦÞ  ∇ρe þ

ρe ðbρe þ σ Þ ¼ 0 e

495

(56)

The conservation of momentum equation with the Coulomb force density term is simply: μ f ðv  ∇Þv ¼ f g  ∇P þ ∇2 v þ EHD ρ ρ

(57)

Together with the Poisson equations, these equations form the full EHD ion drag model. In the case of internal flows, it is convenient to use the 1D forms for these equations. For a simple, horizontal, axisymmetric flow channel, in steady state and cylindrical coordinates: d 2 Φ ρe Poisson’ s equation þ ¼0 e dx2   dΦ dρe ρe þ ðbρe þ σ Þ ¼ 0 Charge conservation vx  b dx dx e   1 d duz 1 dP ρe dΦ r þ Momentum conservation ¼ r dr μ dx ρ dx dr

(58) (59)

(60)

These equations, along with simple boundary conditions for the chosen electrode polarity and the initial flow characteristics, allow for theoretical calculations of the pressure generation and flow rate performance of the ion drag pump while accounting for flow inertia and pressure losses. Note that in general, it is not necessary to solve for the charge density in these equations. If such a solution is desired a boundary condition for charge at the emitter will need to be selected, which can be determined from experiments, but otherwise may prove difficult due to the particularities of the corona discharge effects in dielectric fluids. It is easier to solve only for the electric field distribution using a boundary condition of zero electric field magnitude at the emitter electrode (x = 0). This represents the situation where the emitter is an ideal source of ions, capable of generating a sufficiently large amount of charge as to incur the space charge limit, in which the electric field near the emitter is reduced to zero. Such a situation corresponds to the maximum theoretical force achievable by the ion drag pump while accounting for flow inertia and pressure losses, and therefore the maximum theoretical pressure generation. The potential distribution can then be derived from the electric field solution. This set of coupled equations requires a numerical solution. However, as a faster method of estimating the pump’s pressure generation performance, the ideal theoretical pressure generation of an ion drag pump can be calculated by integrating only the electrophoretic force over the space between the electrodes, and neglecting fluid flow effects or pressure losses. The following equations summarize the ideal pressure generation of an ion drag pump, in the general case, and in the space charge limited case that represents the pump’s maximum performance:

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1 1 ΔPgen ¼ eE2x¼L  eE2x¼0 2 2 1 ΔPgen ¼ eE2x¼L 2

General ideal pressure generation Maximum ideal pressure generation

(61) (62)

The primary advantages of the ion drag EHD pumping technique are the simplicity of the design, high pressure and flow rate generation compared to other techniques, and ability to pump single phase gas flows. The main disadvantage of ion drag pumping is the erosion of the emitter electrode over time, and degradation of the fluid’s dielectric properties due to the injection of charges (Seyed-Yagoobi et al. 1984). The erosion effect is of major concern for small scale applications, since it significantly limits the operational lifetime of the ion drag pump. Ion drag also requires higher power consumption than other techniques, due to the strong electric field that must overcome the fluid’s breakdown voltage to generate the corona discharge and the strong ionic currents.

4.2.2 Induction Pumping In this EHD pumping technique, charges are induced in the working fluid as a result of the applied electric field. A positive potential induces negative charges on the liquid side near the electrodes and vice versa. These charges relax through the liquid medium through normal conduction processes but are delayed at a discontinuity or gradient in electric conductivity, leading to a local accumulation of charge. Most commonly, such discontinuities can be found at interfaces between the liquid and vapor phase of a fluid (Melcher 1966), or between two immiscible fluids (Melcher 1981). In addition, since electric conductivity is a function of temperature, a sufficiently strong temperature gradient imposed on a single, liquid phase, fluid can also create regions or layers where a conductivity gradient is significant (Melcher and Firebaugh 1967; Wong and Melcher 1969; Crowley 1980; Seyed-Yagoobi et al. 1989). Unlike ion drag pumping, fluid motion cannot occur under the effect of a direct current electric field. To generate fluid motion, an alternating current (AC) travelling wave electric field must be used. In the formulation of induction pumping, any gradient in electric permittivity is considered negligible so that only the electrophoretic force needs to be considered. To understand the mechanism of induction pumping, it is important to consider the charge relaxation time, te, of the fluid, te ¼

e σ

(63)

The charge relaxation time describes how long the induced charges will retain their charge and how long they will take to relax through the fluid to the locations of conductivity discontinuity (Melcher and Taylor 1969). Charges accumulating at these locations will then be attracted or repelled by the travelling potential wave. Because this relaxation is not instantaneous, the induced charges will accumulate at the discontinuity interface at a certain time lag from when they were initially induced by the applied potential, leading to a phase shift between the original potential travelling wave and the surface charge carried by the travelling wave.

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The direction in which the fluid will move due to this attraction or repulsion is dependent on the direction of the conductivity gradient, as shown in Fig. 7. In this figure, a three-phase travelling wave field is shown applied through a series of identical electrodes embedded in the top or bottom wall of a rectangular channel filled with a layer of the vapor phase over a layer of the liquid phase of some fluid. Since these charges are induced in fluids with different conductivities, and therefore different charge relaxation times, they will relax toward the interface at different rates, affecting the final charge composition on the interface. Figure 7a illustrates the case where the positive potential travelling wave is applied adjacent to the low conductivity fluid, where the charge relaxation time is longer. The induced positive charges on the fluid side will therefore take longer to reach the interface than the induced negative charges on the high conductivity fluid side. The charge composition at the interface across from the positive potential will therefore be negative. As the potential wave progresses, the negative interface charges will be pulled along in the same direction, and the induction pump would be operating in the attraction mode.

Fig. 7 EHD induction pumping schematic, with travelling wave applied on (a) the low conductivity side, (b) the high conductivity side (Al Dini 2007)

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Conversely, Fig. 7b illustrates the case where the positive potential travelling wave is applied adjacent to the high conductivity fluid, where the charge relaxation time is shorter. In this case, the composition of charges at the interface across from the positive potential will be more positive, as the positive induced charges will reach the interface more quickly than their negative counterparts. In this case, the interface charges will be repulsed by the travelling potential wave. This can result either in pumping in the opposite direction from the travelling wave, or increased pumping velocity in the direction of the travelling wave, depending on the time lag between the wave and the charge accumulation at the interface. In either case, the induction pump would be operating in the repulsion mode. In terms of modeling, the main electrostatics and fluid flow equations are the same, but with notable changes in how they are simplified, as well as how boundary conditions are imposed on them. The model presented here follows the assumption that induction pumping can be modeled as an interface phenomenon, with the charges assumed to be concentrated at an interface between phases, fluids, or temperatures. The effects of bulk conduction of charges are not insignificant (Wawzyniak and SeyedYagoobi 2001) but can be neglected to obtain the general trends of induction pumping using a simpler model. The charge transport effects of interest for induction pumping are Ohmic conduction and convection (Wawzyniak and Seyed-Yagoobi 1999a), J ¼ σE þ ρe v

(64)

However, the travelling wave now introduces a time component that does not disappear from the conservation of charge, unlike the ion drag model, ∇Jþ

@ρe ¼0 @t

(65)

Plugging the relevant quantities into the charge conservation equation yields: σ∇2 Φ þ ∇σ  ∇Φ þ v  ∇ρe þ eð@=@tÞ∇2 Φ ¼ 0

(66)

Poisson’s equation and the conservation of momentum equation are the same as in the ion drag model. However, with the introduction of temperature, the conservation of energy equation must also be taken into account. Under the assumption of negligible viscous dissipation, the conservation of energy is:  ∇  ρcp Tv ¼ ∇  k∇T þ E  J (67) where Cp is the specific heat of the fluid and k is its thermal conductivity. Note that the final term in the energy conservation equation is representative of Joule heating due to the application of the electric field, which can also contribute to the gradient in temperature for a small enough pumping device, and which is generally written for alternating electric fields as (Kaviany 2002),

2 E  J ¼ σ E

(68)

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Electrohydrodynamically Augmented Internal Forced Convection

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The temperature boundary conditions for this equation are straightforward. However, the boundary condition for the applied potential is no longer a simple value, but a time dependent one that accounts for the oscillating travelling wave: h i Φðx, tÞ ¼ RE ΦðxÞeiðωtkw xÞ

(69)

Here, i is the imaginary unit number, Φ(x) is the amplitude of the travelling potential wave, ω is the angular frequency of oscillation, and kw is the wave number. Boundary conditions must be specified for the charge density at the interface. Since the charges at the interface are meant to be moving along the z axis, and charge conduction is assumed to occur primarily along the conductivity gradient, any surface conduction along the interface in the z direction can be assumed to be negligible, with the charge motion at the interface only being due to convection effects. J¼σ

@Φ þ qvx @n

(70)

Where n is the direction of the conductivity gradient, perpendicular to the interface. An integration of the charge conservation equation then yields the charge distribution boundary condition: σ

@Φ @q @ ðqvx Þ ¼ þ @^n @t @x

(71)

Lastly, boundary conditions must also be specified for the potential at the interface:   @Φa @Φb  ¼0 @x @x

(72)

^n  ðea ∇Φa  eb ∇Φb Þ ¼ q

(73)

where the subscripts a and b relate to above and below the interface, respectively, and where ^ n is the unit vector for the n direction, such that these two equations define different conditions for different directions. The first condition is simply the continuity of the electric field across the interface, while the second accounts for the free charges delayed at the interface. Note that under the assumption of a fully developed flow, there will be a balance between the electric shear stress, described by the Maxwell stress tensor, and the viscous shear stress due to the flow velocity at the interface. The stable operation of EHD induction pumps requires careful design of the electrode geometry, the operational voltage and frequency ranges, the working fluid properties and any expected loads on the system. Instabilities in operation can appear as sudden drop in pump output or a change in the pumping direction. Early works have addressed the underlying principles for these potential instabilities (Melcher 1981), with general stability criteria developed for attraction and repulsion mode induction pumps of all types (Crowley 1983). Later works developed further stability criteria for

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stratified vapor/liquid induction pumps under external loads (Wawzyniak and SeyedYagoobi 1999b), using different electrode configurations (Brand and Seyed-Yagoobi 2002), and in a vertical configuration (Aldini and Seyed-Yagoobi 2005). The main advantages of induction pumping are its ability to pump different types of fluids, so long as the frequency of the AC travelling wave can be adjusted to match the inverse of the charge relaxation time of the fluid. The ability to pump two-phase flows and layers of immiscible fluids, and the ability to change the pumping direction by adjusting the frequency of the travelling wave, are also unique advantages to this pumping technique. The main disadvantage for this technique is the complex power configurations needed to maintain the electric field wave, using a low frequency AC power source, as well as the temperature gradients. The overall complexity of the system, both in terms of instrumentation of the actual pumping devices and the mathematical modeling, is relatively high as well and prone to many modes of instability that must be designed around. As a testament to this complexity, there is no easy way to perform a quick estimate of the pressure generation performance of an induction pump, since the force calculations cannot be easily simplified. In terms of microgravity operations, it should also be noted that in the case of stratified fluids or fluid phases, where the presence of a distinct interface is the cause for the discontinuity in electric conductivity, the lack of buoyancy forces may create difficulties in maintaining a stable interface and therefore stable pump operation. In single phase fluids under a temperature gradient, the presence or absence of gravity should not be a concern (Seyed-Yagoobi 1990).

4.2.3 Conduction Pumping This technique uses the dissociation of naturally occurring electrolytic impurities within a dielectric working fluid as a source of free charges. In the absence of a strong electric field, under equilibrium conditions, these impurities dissociate into ions and recombine into neutral species at equal reaction rates, as described by,

AB

Dissociation





!

Aþ þ B

(74)

Recombination In this equilibrium state, the working fluid is considered to be electroneutral, with no net charge (Fuoss 1958). However, under a strong direct current electric field, on the order of 105 V/m, an abundance of free ionic charges are generated in the nearelectrode region due to an exponential enhancement of the dissociation rate, while the recombination rate remains unchanged. This effect is known as the Onsager field enhanced dissociation (Onsager 1934). The dissociated positive and negative ions are attracted toward oppositely charged electrodes. Due to a balance between how long the ions can maintain their charge, as described by the charge relaxation time, and the time the ions take to move through the characteristic length of the electrode gap in a viscous medium, as described by the ionic transit time, a new equilibrium forms (Pontiga and Castellanos 1996). In this new steady state, the charges form thin

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501

layers above each electrode of the opposite polarity. These layers are known as the heterocharge layers (Atten and Seyed-Yagoobi 2003). These separate accumulations of free charges generate opposing forces on the fluid. Under the assumption of equal ionic mobilities for both the negative and positive ions (Yazdani and Seyed-Yagoobi 2014), if the high voltage and ground electrodes have the same wetted areas, such that the positive and negative heterocharge layers have the same dimensions and will generate equal forces, only flow circulation will occur between the electrodes. To generate a net flow, EHD conduction pumps use asymmetrical electrode configurations, where the electrode with the larger wetted surface area, and therefore larger heterocharge layer, determines the flow direction (Yazdani and Seyed-Yagoobi 2009b; Kano and Nishina 2013). In most cases, this corresponds to designing the high voltage electrode as having the larger wetted surface area than the ground electrode, to select for the negatively charged heterocharge layer to be more dominant, but the reverse configuration can be used as well. A schematic of a conduction pump with such asymmetrical electrodes is shown in Fig. 8. In this figure, the larger high voltage electrode has a larger negative heterocharge layer than the positive heterocharge layer on the ground electrode, leading to a larger force component and a net flow to the right, as shown. Although this pumping technique also requires strong electric fields in order to activate the field enhanced dissociation, corona discharge and fluid breakdown work against conduction pumping and must be avoided (Yazdani and Yagoobi 2015). EHD conduction pumps therefore operate at voltages below the breakdown limit of the working fluid, and all electrode surfaces are made smooth to ensure that there are no electric field concentrations at sharp features that can undergo corona discharge. In addition, the current consumed in conduction pumping is very small due to the short time the ions maintain their charge before recombination occurs outside the near-electrode region, leading to overall low power consumption for these types of EHD pumps. Numerical simulations of EHD conduction allow for better understanding of the profiles of the heterocharge layers (Yazdani and Seyed-Yagoobi 2009b). Unlike ion drag pumping and induction pumping, various electrode designs can be used with

Fig. 8 EHD conduction pumping schematic (Yang et al. 2017)

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Fig. 9 EHD conduction pump electrode design examples (Pearson and SeyedYagoobi 2009)

conduction pumping. Figure 9 shows some of the existing designs – hollow tube high voltage and flush ring ground electrode, perforated or porous-permeable high voltage electrode, and multitube high voltage electrode. These different electrode designs will have different heterocharge layer profiles and therefore different force distributions in the liquid, which in turn will lead to different pressure generation versus flow rate performance for each electrode design (Pearson and Seyed-Yagoobi 2009). An example of numerical simulations of the heterocharge layer structure for flush electrodes is presented in Fig. 10. The layer profiles are shown for variations in a nondimensional quantity, M0, representing the working fluid properties. Note, however, that the introduction of ion injection into an EHD conduction pump not only weakens the conduction pump’s performance, as the injection current generally opposes the net conduction current (Yazdani and Yagoobi 2015), but also introduces the instabilities associated with ion injection into the system (Pontiga and Castellanos 1994). Therefore, any sharp features must be eliminated from the electrode designs used for EHD conduction pumps, to eliminate ion injection entirely. In terms of modeling, conduction pumping is relatively similar to ion drag pumping. The primary difference is that, unlike ion drag and induction pumping, the motion is not solely dominated by a single polarity of charge carriers. The model for conduction pumping must therefore account for both negative and positive ionic species, ρe ¼ pe  ne

(75)

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Fig. 10 Flush electrodes heterocharge layer structure for different fluid properties (Yazdani and Seyed-Yagoobi 2009b)

Here, ne and pe are the negative and positive charge densities, respectively. These charge densities relate to the initial concentration of neutral species, n0, through the constitutive relation for the balance between dissociation and recombination: k D n0 ¼ k R

n p n2eq e e ¼ k R 2 e2 e

(76)

where kD and kR are the dissociation and recombination rate constants, respectively, and e is the unit charge. Note that these rate constants have different units, s1 for the dissociation and m3 s1 for the recombination in SI units. In the above, neq is the equilibrium charge density of either positive or negative species, since ne and pe are assumed to be the same in that situation. To account for the field enhancement effect, the Onsager function, F(ωe), is used: I 1 ð4ωe Þ , kD ¼ kD0 Fðωe Þ ¼ kD0 2ωe

" ωe ¼

e3 E 16πeðkB T Þ2

# (77)

The Onsager function utilizes the modified Bessel function of the first kind and order one, and the enhanced dissociation rate coefficient, ωe. In the above equation, kB is the Boltzmann constant, and kD0 is the original dissociation constant, which is identical to the recombination constant. As is shown, the Onsager function serves as

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a multiplication factor for the original dissociation constant. In the absence of an electric field the value of F(ω) is 1, and under the effect of the electric field it is greater than 1. As the name of the technique implies, the primary charge transport mechanism is conduction. However, unlike the ordinary Ohmic conduction described for ion drag and induction pumping, the conductivity is now variable, depending on the charge density: σ ¼ ð b ne þ bþ pe Þ

(78)

In the conductivity equation, b and b+ represent the ionic mobilities of the negative and positive ions, respectively. The equation for current density must now be split to account for all the charged species of interest – positive and negative ions. In addition, a flux density equation for the neutral species must be provided: J  ¼ b ρ E þ ρ v  D ∇ρ

(79)

Φ0 ¼ n0 v  D0 ∇n0

(80)

In these equations, ρ, b, and D stand for the charge density, ionic mobility and diffusion coefficient belonging to each charged species, and D0 is the diffusion coefficient for the neutral species, respectively. Note that diffusion effects are not entirely negligible for EHD conduction, since the heterocharge layers’ shapes can be affected by diffusion of charges into the bulk of the working fluid, as well as diffusion of neutral species from the bulk into the near electrode region. Since the charge density is now affected by chemical reaction rates, the charge conservation equations must also be split and made to account for these reactions. In addition, a similar equation for the conservation of concentration of neutral species must also be supplied: ∇  J  ¼ k D n0 e  k R ∇  Φ0 ¼ k R

ne pe e

n p e e  k D n0 e2

(81) (82)

Otherwise the model is similar to ion drag pumping, with the Poisson equation and conservation of momentum equations largely unchanged: ∇2 Φ þ

pe  n e ¼0 e

 μ 1
ðv  ∇Þv ¼ f g  ∇P þ ∇2 v þ ðpe  ne ÞE ρ ρ

(83) (84)

The boundary conditions for the potential and flow are trivial. The charge and concentration density boundary conditions only need to account for incoming and outgoing fluxes at the electrode surfaces and the inlet and outlet of the domain. As a general rule, it is assumed that no diffusion occurs between the electrode surfaces

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and the fluid, and that the charge density of each ionic species is zero on the electrode with the same polarity as the species. In the same fashion as ion drag pumping, solving these coupled equations numerically will account for the full flow inertia effects, including pressure losses and flow velocity effects on the shape of the heterocharge layers. However, a quick estimation of conduction pumping pressure generation is also available here. The simplest model for EHD conduction is the 1D case of parallel plate electrodes (Atten and Seyed-Yagoobi 2003). Neglecting any flow, the charge conservation equations can be solved analytically. This yields a simplified heterocharge layer thickness that depends on the ionic mobility, the charge relaxation time and the nominal magnitude of the electric field: λ ¼ b Eτe

(85)

Under the assumption of equal ionic mobilities, the thickness is expected to be the same over each of the electrodes, such that the magnitude of the net force will depend only on the dimensions of the electrodes. An integration of the electrophoretic force for this 1D model can therefore be performed over this thickness in order to estimate the resulting pressure: ΔP ¼

ðλ

ðpe  ne ÞEdx

(86)

0

Estimating this integral using Poisson’s equation yields the following approximation: ΔP ¼

ðλ 0

eE

@E dx  λeðEÞ2  bσE3 @x

(87)

This relation tends to overestimate the actual magnitude of pressure generation for EHD conduction. Attempts to convert this general relation to account for the lower observed experimental values have been made using altered fluid properties (Mahmoudi et al. 2011a) or accounting for the exact wetted surface areas participating in the formation of the heterocharge layer (Feng and Seyed-Yagoobi 2004b) with some measure of success. The main advantages of EHD conduction pumping are its reliability in long term operations, its low power consumption, and its simple, flexible electrode designs. The main disadvantages of this technique are lower pressure generation capabilities, difficulty pumping two phase flows, and manufacturing concerns due to the need for smooth electrodes in order to avoid charge injection from sharp features.

4.2.4 Size Scaling: Micro- and Nanoscale EHD Theory In dielectric fluids at very small scales, on the order of tens of microns or smaller, effects that are negligible in larger scales become prominent. The most important of these effects is the electric double layer (EDL). This is a unique formation of charge structure in the vicinity of any dielectric surface in contact with a fluid, due to natural

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Fig. 11 Electric double layer and zeta potential (Dutta and Beskok 2001)

Stern Plane Shear Plane

ψ - + + - +

ζ ψo

-

+

-

+

+

-

+

-

+

y’

+

+

+

+ -

+

- + - +

-

+

+

+ EDL

-

+

+ +

-

chemical processes such as adsorption. Once a surface gains a net charge, oppositely charged ionic species from the fluid adhere to the surface under strong electrostatic forces and form what is known as the Stern layer – an immobile layer of ions. Immediately beyond this immobile layer, a diffuse layer of mobile mixed ionic charges is present. The diffuse layer has the same charge sign as the Stern layer, but the electric potential drops off exponentially further into the fluid bulk, such that the bulk is still electroneutral. The characteristic drop off length is the Debye length of the EDL, which is usually on the order of nanometers. Within the diffuse layer there exists a slipping plane, which defines where the diffuse layer is able to shear off from the Stern layer. The potential at this slipping plane, ζ, is known as the zeta potential (Stern 1924). Figure 11 shows a schematic of the EDL and the potential distribution, Ψ, along its length, y0. In this figure, Ψ0 represents the applied potential on the electrode surface at y0 = 0. In the presence of an electric field orthogonal to the EDL surface, with an applied potential that is significantly larger than the zeta potential, the diffuse layer is stripped away. This is the case with most electrode surfaces in EHD pumping. However, the remaining wall surfaces will still retain their EDL. Studies have shown that the presence of the EDL serves as an electroviscous resistance that opposes any pressure driven flow (Gong and Wu 2006). All EHD pumping techniques in small scales would, therefore, need to generate greater forces on the fluid in order to overcome the EDL flow resistance. In addition, an electric field that is locally parallel to a wall surface will incur an electroosmotic flow of the diffuse layer that can further interfere with pumping performance. This is particularly relevant for EHD induction pumping, where the combination of an electroosmotic flows in the AC travelling electric field and the conductivity gradient induced charge accumulation can cause significant instabilities and disruption of the flow (Dutta and Beskok 2001; Chen et al. 2005).

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In low conductivity dielectric fluids, the EDL length can be on the order of microns (Tardu 2004), which means that sufficiently small microscale EHD pumps using such fluids, as is the case for ion drag and conduction pumping, will be affected by the presence of the additional space charge, as well as a potentially nonnegligible zeta potential compared with the applied pumping voltage. Additional potentially adverse effects at small-scales include significant Joule heating relative to the volume of the fluid, which can give rise to potentially unwanted gradients in the fluid’s electrical properties (Cao et al. 2009), and induced buoyancy forces or unexpected electrothermal induction flow (Castellanos et al. 2003). Lastly, localized electrochemical reactions, between the electrode material and the fluid, can be enhanced by the application of an electric field at such small scales, which can cause a more rapid erosion of the electrodes or generate unwanted reaction products in the fluid (Zhakin 1998). In general, Joule Heating is the main cause of low power efficiency in EHD pumping devices in all size scales, but in small scales the effect can become more severe. In terms of modeling for charge transport and heat transfer, diffusion effects become prominent at these small scales (Ramos 2011) and can no longer be neglected from the various models and boundary conditions for the various EHD pumping techniques. The EDL itself is often modeled as a charge distribution boundary condition, but the exact nature of this distribution in the presence of an externally applied electric field is difficult to obtain (Prieve 2004).

4.3

Experimental Studies of EHD Driven Single Phase Flows: Macroscale

This section contains a summary of the seminal experimental works on EHD pumping driven single phase, gas or liquid, internal flows in the macroscale. The majority of these studies are historical references rather than the state of the art, since the advent of miniaturization has made microscale devices a realizable option for flow generation and heat transfer enhancement.

4.3.1 Ion Drag Pumping Early works on ion drag pumping in single liquid phase internal flows at the macroscale used both conducting fluids and nonconductive fluids. In the case of conductive fluids, such as acetone (Krawinkel 1968), flow velocities as high as 7.8 m/s were observed in the 20 mm diameter pump. These high velocities were due to the low viscosity and high dielectric constant. However, these sorts of ion drag pumps offered very low efficiencies and further research focused on low conductivity fluids instead. The first such pump to be used in a thermal control application, for transformer cooling using Shell XSL-916 as the working fluid (Sharbaugh and Walker 1985), showed moderate flow velocities of 10 cm/s in the 60 mm diameter pump. Optimization of the design of the electrodes, using fluid mechanics principles, showed that velocities as high as 33 cm/s could be achieved in a 70 mm diameter pump in a vertical configuration, using dodecylbenzine as the working fluid (Bryan and Seyed-

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Yagoobi 1992). The effect of varying the interelectrode gap, the polarity, the electrode material, the electrode design, and the working fluid for an ion drag pump in a 30 mm horizontal pipe were also studied (Barbini and Coletti 1995). More recently, macroscale ion drag pumps were also proven to be able to pump cryogenic fluids, such as liquid nitrogen, in a 95 mm diameter pump with emitter fin electrodes (Rada et al. 2008). Research on ion drag pumping of a single gas phase began earlier than the liquid phase studies mentioned above. This type of ion drag pumping generates ions via a corona discharge of electrons into a gas, thereby ionizing the gas molecules. This phenomenon is called the ionic wind or corona wind (Robinson 1962). Internal flows affected by the corona wind were first studied in a nuclear reactor, where the gas was ionized due to radiation rather than corona discharge (Berger and Stach 1959), where a heat transfer enhancement of up to 20% was observed (Stach 1962). Most of the later corona wind studies involved external flows over flat plates. Nevertheless, further studies into macroscale corona wind internal flows were conducted on the developing region of air flow through a 25.4 mm diameter circular tube (Molki and Bhamidipati 2004), where a 23% enhancement of the heat transfer coefficient was observed. In addition, secondary circulating flows induced by corona wind were also investigated in macroscale circular tubes (Baghaei Lakeh and Molki 2012), and in macroscale fully developed flows in a 10 mm by 20 mm rectangular channel (Lakeh and Molki 2009), with a reported heat transfer enhancement of 173%. Similarly, more recent work showed secondary flow and overall volumetric flow rate enhancement using multiple stages of corona jets in a 10 mm square vertical duct (Mazumder and Lai 2014). An example for the flow velocities that can be generated using multistage corona wind ion drag pumps, as a function of applied voltage, and location along the channel cross section is given in Fig. 12. This figure shows how emitter electrodes placed in the channel walls act as localized jets, with the highest velocity magnitudes present at the walls and an inverted fully developed profile to what is expected for fully developed internal flows.

4.3.2 Induction Pumping Single phase EHD pumping of internal flows at the macroscale has been experimentally investigated primarily with ion drag pumping and EHD conduction pumping. The focus of single phase induction pumping of such flows has been on small channels, where even small temperature gradients are sufficiently strong to incur the necessary conductivity gradient. Nevertheless, initial studies of induction driven internal flows at the macroscale in horizontal rectangular configurations were conducted using a 22 mm wide channel and Sun 4 as the working fluid (Kervin et al. 1981). Further studies in a vertical, 16 mm diameter examined the effect of different temperature profiles, electric field frequencies and wavelengths, using doped Sun #4 as the working fluid (Seyed-Yagoobi et al. 1989). Additional studies on the same experimental setup looked into the optimal selection of working fluids for single phase induction pumping (Bohinsky and Seyed-Yagoobi 1990), as well as the effects of tilt angle on the generated flow rates and heat transfer enhancement, using n-hexane as the working fluid and different voltage frequencies (Margo and Seyed-Yagoobi 1994). The mass flow rate performance results of the latter study are shown in Fig. 13.

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4.3.3 Conduction Pumping Since EHD conduction is the youngest of the EHD pumping techniques, all studies of EHD conduction driven single phase internal flows are more recent compared to the other pumping techniques. EHD conduction was first proposed as a separate EHD pumping technique after a few experimental studies of induction pumping yielded unexpected results. As an example, in a study of high frequency, high voltage induction pumping initial theories about the existence of heterocharge layers began to emerge (Washabauch et al. 1989), but it was merely assumed to be a side effect of induction pumping. With the introduction of the EHD conduction model and first proper experiments (Atten and Seyed-Yagoobi 1999), further macroscale studies were able to focus on investigating different electrode designs in a 10 mm diameter flow channel (Jeong and Seyed-Yagoobi 2002, 2004) in order to determine the pressure generation performance of EHD conduction pumping. Some of these performance curves are given in Fig. 14. Note that this figure shows performance information for conduction pumps with only one pair of electrodes of any given design. In general, pressure generation of EHD conduction pump increases in a nearly linear fashion with the inclusion of additional electrode pairs (Jeong and Seyed-Yagoobi 2002), as long as there is sufficient spacing between the pairs such that their individual electric fields do not interfere with each other (Yazdani and Seyed-Yagoobi 2009b). These early studies confirmed that EHD conduction pumps generate higher pressures the more asymmetrical the wetted surface area of the high voltage electrode compared to the ground electrode. In addition, it was established that there is a quasi-linear relation between the number of electrode pairs used and the final

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pressure generation values of the EHD conduction pump. In addition to different electrode designs, different working fluids were also tested for static pressure generation in a similar macroscale channel using the same electrode design for all fluids (Ashjaee and Mahmoudi 2005). This study showed the range of electrical conductivities that are most favorable for EHD conduction. More recently, a study involving a combination of rod to rod electrode pairs and parallel mesh electrode pairs was conducted for a 20 mm tall rectangular, serpentine channel (Hanaoka et al. 2009), which showed a linear dependence between the flow rate and applied voltage, and conducted flow pattern visualization using Schlieren photographs. EHD conduction pumps are unable to pump single gas phase flows at any size scale, so no further discussion will be offered for those types of flows and this pumping technique.

4.4

EHD Driven Single Phase Flows: Microscale

This section contains a summary of selected experimental works on single phase EHD pumping at the microscale in internal flow configurations. This summary serves to illustrate the modern-day applications most fitting for each of the EHD pumping technique, as well as the effects of miniaturization of each of these techniques.

4.4.1 Ion Drag Pumping Microscale corona wind ion drag pumps have focused on modeling and experiments of enhancement of external flows rather than internal flows. Therefore, this summary will focus on liquid phase pumps only. The first single liquid phase ion drag micropumps were made of parallel 3 mm square grid, metallized, silicon wafer electrodes. These grids were 30 μm wide and set 350 μm apart, with hole sizes of 70 μm and 140 μm. The 140 μm grid pump achieved a maximum static pressure of 1.2 kPa and a maximum flow rate of 14 mL/min using liquid ethanol as the working fluid (Richter and Sandmaier 1990). The first planar ion drag micropump was comprised of multiple pairs of symmetric parallel line electrodes. The electrodes were a few microns tall, 100 μm wide and spaced 100 μm apart, with a space of 200 μm between each pair. Due to the symmetry and favorable ionic mobilities of the ethyl alcohol working fluid, this pump could generate flow in either direction depending on the polarity of the electrodes (Ahn and Kim 1998). Further studies of microscale ion drag pumps focused on the static pressure performance of different planar electrode geometries in a 1 mm tall channel and using HFE-7100 as the working fluid. These designs included planar parallel line electrodes, planar saw tooth emitters with line ground electrodes at varying interelectrode gaps, and saw tooth emitters with microscale 3D sharp bumps (Darabi et al. 2002). The performance of such designs is shown in Fig. 15, showing that the saw tooth design with the added bumps produced the highest pressures. Thorough studies of the effects of channel depths, interelectrode gap sizes and gap sizes between electrode pairs were also conducted on microscale ion drag pumping in an enclosed packaging configuration where both static pressures and dynamic pressure generation versus flow rates could be measured. These studies

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Fig. 15 Microscale ion drag pumping performance of different planar electrode designs (Darabi et al. 2002)

found that deeper microchannels produce greater flow rates but significantly lower static pressure generation capabilities (Benetis et al. 2003). In terms of manufacturing, the majority of microscale ion drag pumps have been fabricated out of rigid materials and tested inside rigid channels. However, ion drag pumps have sufficiently simple designs that they can be lithographically printed onto flexible, thin substrates – on the order of 10 μm. This technique was demonstrated using multiple pairs of 10 μm wide saw tooth emitter electrodes with a 20 μm interelectrode gap and an 80 μm tall PDMS covered flow channel (Chen et al. 2007). More recently, studies attempting to utilize different nanoscale surface topologies in order to reduce the voltage requirement of microscale ion drag pumps have been conducted. These studies included chemical etching of the emitter electrode surface and deposition of single walled carbon nanotubes (SWCNT) onto a smooth or etched emitter electrode. These studies showed that SWCNT can greatly reduce the required voltage for the initiation of discharge from the emitter electrode, and that the added surface area greatly increases the emitted current, and therefore pressure generation in a 100 μm tall microchannel and using HFE 7100 as the working fluid (Russel et al. 2014). These micropumps only require applied voltages on the order of a few hundred volts or less and have currents in the range of microamperes. However, they have exhibited more severe issues in long term operation than their macroscale counterparts, due to electrochemical effects and fast erosion of the electrode material (Benetis 2005).

4.4.2 Induction Pumping The first microscale induction pump for single phase flows used an array of 10 μm wide electrodes with 10 μm spacing to pump liquid silicone oil using a three-phase

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Fig. 16 Frequency dependence of microscale induction pumping of high conductibity fluids (Fuhr et al. 1994)

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square wave at various frequencies and an externally applied thermal gradient (Bart et al. 1990). This study confirmed that low frequency induction pumping has better performance at the microscale in addition to the macroscale. Further studies of a microscale induction pump relying solely on Joule heating in order to generate the thermal gradient responsible for the conductivity gradient were also conducted in 100 μm microchannels, using planar 30 μm wide electrodes spaced 30 μm apart from each other (Fuhr et al. 1992). As shown in Fig. 16, these studies found that high conductivity working fluids, such as water solutions, could be pumped at such small scales at low frequencies of the applied travelling wave, despite it being impossible in larger scales. Additional works showed that induction pumping of different fluids can yield the same flow velocity if the frequency is tailored to the specific working fluid’s electrical properties (Fuhr et al. 1994). Since the effective thermal gradients for microscale induction pumps do not need to be excessive, even illumination can be used to generate the required property gradients (Green et al. 2001). Studies of the effect of the wall material and wall thickness of planar induction pumps were also conducted, as well as the effect of applying a travelling wave on both top and bottom sides of a 30 μm tall microchannel and 10 μm wide electrodes spaced 10 μm apart (Felten et al. 2006). Other studies of Joule heating induced thermal gradient showed that the induction mechanism can also be used in different pump geometries, such as a T configuration with a high voltage blunt wire tip orthogonal to a grounded horizontal wire (Wu et al. 2007). More recent studies succeeded in gathering PIV measurements of the flow field in a repulsion type Joule heating induction pump with 8 μm wide planar electrodes spaced 16 μm apart, in a 50 μm microchannel and using a high conductivity low concentration KCl solution as the working fluid. Due to the small size of this induction pump, microscale polystyrene particles with a mean diameter of 0.86 μm had to be used. The use of these particles also enabled researchers to verify

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that induction pumping was the dominating flow mechanism, as opposed to other forces like dielectrophoresis, by viewing the particles downstream of the pumping section (Iverson and Garimella 2009).

4.4.3 Conduction Pumping Experimental work in the area of EHD conduction driven microscale internal flows has been fairly limited due to the relative age of this pumping technique. Nevertheless, initial studies of microscale single phase flows were conducted for ten pairs of asymmetrical flush wall electrodes in a 250 μm thick slot channel. The ground electrodes were 51 μm thick and were set 51 μm apart from the 178 μm thick high voltage electrodes. The power consumption for this configuration was on the order of milliwatts, typical for microscale EHD conduction pumps, which produced static pressures of over 200 Pa in ethanol (Pearson and Seyed-Yagoobi 2010; Pearson 2011). Further studies were conducted on planar 120 μm wide ground or positive voltage electrodes versus a negative voltage 21 by 16mm rectangle cover electrode in a 100 μm microchannel using HFE-7100. These showed that EHD conduction pumping can be initiated using negative applied potentials on the electrode with the higher wetted surface area, as long as the required overall field strength is reached via grounding or applying a positive potential to the smaller electrodes (Kano and Nishina 2010). The same concept was adapted into a radial configuration, with a round disc ground electrode 3 cm in diameter and concentric, 25 μm wide high voltage ring electrodes spaced 75 μm from each other, separated by 286 μm. This configuration was tested using three different working fluids and confirmed that lower conductivity fluids still generate the best EHD conduction pumping performance. However, despite there being 75 electrode pairs, the overall pressure generation underperformed the theoretical predictions, leveling off at an asymptotic value at the highest applied voltages due to suspected charge injection (Mahmoudi et al. 2011b). A more traditional configuration of planar asymmetrical 127 μm ground versus 381 μm high voltage electrodes was also studied in a 500 μm microchannel of different channel lengths and electrode pair numbers. These studies confirmed the quasi-linear scaling of EHD conduction pumping performance at the microscale versus the number of electrode pairs (Pearson and Seyed-Yagoobi 2013). This relation is shown in Fig. 17 for HCFC-123. The short design contained 13 electrode pairs, whereas the long one contained 21 pairs. Lastly, a similar design using flush ring electrodes of the same widths, in a circular 1 mm diameter channel was also studied using HCFC-123 in both a terrestrial and microgravity environment, confirming the viability of the EHD conduction mechanism for aerospace applications (Patel et al. 2013). These studies show the flexibility of effective design options for microscale EHD conduction pumps.

4.5

EHD Driven Two Phase Flows

This section contains a summary of selected works on EHD pumping of two-phase flows and their corresponding heat transfer enhancement performance. Many

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Fig. 17 Microscale EHD conduction performance using different active lengths (Pearson and Seyed-Yagoobi 2013)

applications of EHD pumping of two-phase flows focus on enhancement of external flows, pool boiling and thin film evaporation. For these sorts of applications, all three EHD pumping techniques covered thus far have been tested. However, the study of EHD driven enhancement of heat transfer in two-phase internal flows has been limited to induction and conduction pumping.

4.5.1 Macroscale Following the initial, previously described, induction pumping studies of internal two-phase flows that were conducted in the 1960s (Melcher 1966), and the subsequent study of its efficiency (Crowley 1980), most researchers opted to investigate this pumping technique for heat transfer enhancement of external flows. Nevertheless, a comprehensive study on induction pumping of internal, stratified, two-phase flows was conducted using HCFC-123 as the working fluid (Wawzyniak et al. 2000). This study provided detailed velocity profiles, obtained via an LDA system, for different voltages and frequencies applied on a film height of 7.5 mm in a 25 mm tall square channel. The findings of this study showed a maximum value for the velocity below the interface. This showed that induced charges within the fluid bulk must be accounted for as well. Some of these results are shown in Fig. 18. In contrast, several studies on EHD conduction driven two-phase flows have been performed. The first such study investigated the heat transfer performance enhancement of a monogroove heat pipe using EHD conduction pumping (Bryan and SeyedYagoobi 1997). This study showcased the ability of macroscale EHD conduction pumps to enable recovery from dryout conditions, as shown in Fig. 19. In this figure times 1 and 2 define the span between the activation of the EHD pump at its maximum voltage and achieving a steady state, recovered temperature. In the area of stratified thin film flows, a performance characterization of macroscale EHD conduction pumping at different film heights was performed using HCFC-123 as the working fluid (Siddiqui and Seyed-Yagoobi 2009). Further studies

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compared the achievable heat transfer coefficients of a similar conduction pump using different film heights and different temperatures using kerosene (Nourdanesh and Esmaeilzadeh 2013). Another study investigated the effect of adverse and beneficial gravity forces on the heat transfer performance of EHD conduction driven thin film flows in a two-phase system (Pearson and Seyed-Yagoobi 2015). The setup was comprised of a circulating heat pipe, with the EHD conduction pump upstream

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of the evaporator section, and was tilted such that gravity was helping or hindering the supply of fresh liquid to the evaporator. The results showed that removal of heat fluxes of 20 W/cm2 was still possible even with a near 45 tilt of the setup in a macroscale configuration. Enhancement of macroscale in-tube flow boiling was also investigated using EHD conduction pumping in recent years, beginning with performance characterization of a multitube electrode configuration (Jeong and Didion 2007), and continuing to show the effect of voltage and working fluid temperature on the critical heat flux for such a setup (Jeong and Didion 2008), as shown in Fig. 20. Lastly, recent work on EHD conduction enhancement of mesoscale, in-tube flow boiling have yielded sufficient experimental data for HCFC-123 at low mass fluxes for comparisons to be made with commonly used correlations for flow boiling at this size scale (Pearson and Seyed-Yagoobi 2015), as shown in Fig. 21. It should be noted that none of the available correlations adequately address low mass fluxes due to the lack of available data, making this contribution of great importance to the extension of flow boiling correlations to the low mass flux regime.

4.5.2 Microscale Pumping of microscale internal two-phase flows is of immense interest for the electronics cooling industry, due to the high heat flux removal capabilities. Several passive two-phase flow pumping solutions involving capillary action have been investigated at this size scale, but the study of active pumping solutions, such as EHD, has been limited. To date the relevant investigations have involved only EHD

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Fig. 21 Flow boiling heat transfer coefficient vs. quality, with correlation comparison (Patel and Seyed-Yagoobi 2015)

conduction pumping, with induction pumping at this size scale focusing only on induced thermal gradients in single-phase flows as described previously. The first such studies were conducted using a chip-like structure with an embedded heater and flush, printed electrodes on ceramic substrates that could be fitted and pressed together to form a wide, 500 μm tall evaporator section, with an upstream adiabatic pumping section (Pearson and Seyed-Yagoobi 2011). The results of this study showed that for low heat fluxes, and therefore low vapor qualities, EHD conduction pumping is still able to provide an enhancement of up to 30% to the heat transfer when embedded in the evaporator, as shown in Fig. 22. At higher heat fluxes the pressure loads and diminishing fluid volume reduce the effectiveness of the embedded pump. Subsequent studies at this size scale focused on in-tube flow boiling with a microscale EHD conduction pump placed upstream of the evaporator section. These studies characterized the general performance of such a configuration over prolonged periods of time (Patel and Seyed-Yagoobi 2014) and in microgravity conditions (Patel et al. 2013), proving the reliability and robustness of this type of configuration.

4.6

Example Application: EHD Driven Single Phase Flow Distribution Control

Modern day thermal control solutions for high power, high heat applications are expected to have features such as real-time recovery from dryout conditions, smart

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Fig. 22 Effect of applied heat flux on microscale EHD conduction two-phase loop at 750 V (Pearson and SeyedYagoobi 2011)

redirection of flow from areas of low need to areas of high need, and simultaneous autonomous management of multiple heat generating components (Sinnamon 2012). Miniaturization of such thermal control solutions is also expected, to accommodate the rising transistor density in modern electronics and their accompanying high heat fluxes. EHD pumping technology has matured sufficiently for evaluation as a potential tool for such innovative thermal control systems. This section covers the recent advances made in examining EHD conduction pumping capabilities for smart flow distribution control between parallel channels at different size scales. Preliminary studies involving flow distribution control of single phase flows between two parallel channels, using an EHD conduction pump in one of the channels, showed both the flow redirection and maldistribution recovery capabilities of EHD pumping at the macroscale (Feng and Seyed-Yagoobi 2004a). The internal diameter of the channels in this macroscale setup was 10 mm. The EHD conduction pump used was designed for high pressure generation, with a perforated high voltage electrode and flush ring ground electrode, and a maximum static pressure generation of 1,250 Pa. Figure 23 shows that under an externally supplied mass flux of 100 kg/m2 s, equally distributed between the branch channels, activation of the EHD conduction pump in branch 1 at its maximum voltage of 15 kV caused the flow to be completely redirected into branch 1, leaving branch 2 dry. This demonstrates the potential usage of EHD pumps as valves. At higher total mass fluxes the EHD pump was able to redirect some of the flow into its branch, but not to the point of drying branch 2. Figure 24 shows recovery from an initial maldistribution, with an externally supplied mass flux of 200 kg/m2s and an initial difference in mass flux of 30 kg/ m2s between the two branches. Activation of the EHD pump at its maximum voltage was able to recover from the maldistribution and equalize the flow. This study also examined the theoretical response time, τres, of the EHD conduction mechanism, which is critical for evaluating the usability of this technology in actual thermal control systems. This response time is on the order of the charge relaxation time of the working fluid, in milliseconds, and therefore sufficiently short for use in actual applications,

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Fig. 24 Macroscale EHD conduction, maldistribution recovery, G = 200 kg/m2 s, V = 15 kV (Feng and Seyed-Yagoobi 2004a)

e e  τres  2 σ σ Similar studies of flow distribution control of two-phase flows at the macroscale (Feng and Seyed-Yagoobi 2006), where evaporation took place upstream of the branches, showed that for low vapor qualities, on the order of 10% or less, the EHD

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Table 1 Comparison of size scale dependent EHD conduction driven mass flux redistribution at maximum applied voltages (Talmor et al. 2016) Size scale Macroscale Mesoscale Microscale

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conduction was still able to significantly affect the flow distribution between two parallel branches. Subsequent studies of single phase flow distribution control at the mesoscale (Yang et al. 2015) and the microscale (Talmor et al. 2016) used a different EHD conduction pump than the macroscale design but reported similar flow redirection and maldistribution recovery capabilities. Table 1 shows a summary of these capabilities at all size scales. In this table, the mass flux limit refers to the total mass flux supplied to the setup at which activation of the conduction pump at its maximum voltage no longer had any effect on the flow distribution. The data shows that, while the size scale is decreasing by an order of magnitude and pressure losses are expected to increase exponentially, the EHD conduction pumps perform at similar orders of magnitude at each size scale. This application example shows that understanding how EHD devices scale in performance is another critical parameter in the design of EHD driven systems.

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Cross-References

▶ Heat Transfer Media and Their Properties ▶ Process Intensification ▶ Single-Phase Convective Heat Transfer: Basic Equations and Solutions

References Ahn S-H, Kim Y-K (1998) Fabrication and experiment of a planar micro ion drag pump. Sensors Actuators A Phys 70(1–2):1–5. https://doi.org/10.1016/S0924-4247(98)00105-8 Al Dini SAS (2007) Electrohydrodynamic induction and conduction pumping of dielectric liquid film: theoretical and numerical studies. College Station Texas A&M University Aldini SA, Seyed-Yagoobi J (2005) Stability of electrohydrodynamic induction pumping of liquid film in vertical annular configuration. IEEE Trans Ind Appl 41(6):1522–1530 Ashjaee M, Mahmoudi S (2005) Experimental study of electrohydrodynamic pumping through conduction phenomenon using various fluids. In: CEIDP’05. 2005 Annual report conference on electrical insulation and dielectric phenomena, 2005. IEEE, pp 495–498 Atten P, Seyed-Yagoobi J (1999) Electrohydrodynamically induced dielectric liquid flow through pure conduction in point/plane geometry-theory. In: Proceedings of the 1999 I.E. 13th international conference on dielectric liquids (ICDL ‘99). IEEE, Nara, Japan. pp 231–234 Atten P, Seyed-Yagoobi J (2003) Electrohydrodynamically induced dielectric liquid flow through pure conduction in point/plane geometry. IEEE Trans Dielectr Electr Insul 10(1):27–36

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Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Narrow Vertical Plane Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Heat Transfer from Adjacent Narrow Plane Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Horizontal Plane Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Horizontal Two-Sided Circular Bodies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Recessed Heated Horizontal Circular Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Adjacent Horizontal Isothermal Square Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Horizontal Rectangular Surface with a Parallel Adiabatic Covering Surface . . . . . . . 4 Bodies with a Wavy Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Vertical Wavy Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Horizontal Wavy Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Free Convection from Cylindrical Wavy Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Short Vertical Cylinders with an Exposed Upper Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 External Free Convection in Systems Involving a Nanofluid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Cross-References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

530 533 540 547 556 562 566 569 573 574 578 581 585 594 595 596 596

Abstract

Some representative recent basic studies involving external free convective heat transfer in situations that are of practical interest are discussed in this chapter. Most of the results considered have been obtained numerically, and a very brief discussion of the methodology used to generate these results is presented. Attention has here first been given to the heat transfer rate from narrow vertical plane

P. H. Oosthuizen (*) Department of Mechanical and Materials Engineering, Faculty of Engineering and Applied Science, Queen’s University, Kingston, ON, Canada e-mail: [email protected] # Springer International Publishing AG, part of Springer Nature 2018 F. A. Kulacki (ed.), Handbook of Thermal Science and Engineering, https://doi.org/10.1007/978-3-319-26695-4_10

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surfaces with the effect of the width-to-height ratio of the surface on the heat transfer rate in particular being discussed. Heat transfer from horizontally and vertically spaced pairs of narrow plates has also been considered. Consideration has then been given to the heat transfer rate from horizontal heated surfaces of complex shape, to adjacent pairs of horizontal heated surfaces, and to heat transfer from two-sided circular horizontal surfaces. The effect of a covering surface on the heat transfer from a horizontal heated surface has also been considered. The heat transfer rate from bodies with wavy surfaces has been considered next with attention being given to situations involving two-dimensional flow over vertical and horizontal surfaces and to the effect of surface waviness on the heat transfer rate from cylindrical bodies. The heat transfer from relatively short vertical circular and square cylinders with exposed top surfaces is also considered. Lastly, brief attention is given to external free convective heat transfer to nanofluids.

Nomenclature

A Abottom Ac Atop At Atotal AR a cpf cpnf cps Di D d di G Gap g HGap H h k h L Lout l lout

Total surface area Area of bottom surface Area of cylindrical outer surface of cylinder Area of top surface Area of top surface of cylinder Total surface area Aspect ratio Mean side length Specific heat of fluid in which nanoparticles are placed Specific heat of nanofluid Specific heat of nanoparticles Dimensionless diameter of inner adiabatic section of surface Diameter of circular surface Heated surface diameter Diameter of inner adiabatic section of surface Gap between heated surfaces Gap between vertically spaced heated surfaces Gravitational acceleration Dimensionless size of gap between surfaces Height of surface and dimensionless recess depth, h/d Recess depth Thermal conductivity Recess depth l/d Dimensionless side length of rectangular adiabatic covering surface, Lout = lout/w Reference length and cylinder height Side lengths of rectangular adiabatic covering surface

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m Nu Nu0 Nua Nubottom Nuc Num Nur Nurbottom Nurtop Nut Nutop Nutotal n P Pr Q0 Q0 bottom Q0 top q0 q0 q0c q0s q0t q0w R Ra Raa Ram Rar Ra* r rbottom rtop s S T Tf Tw

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Characteristic length scale of surface Nusselt number Reference Nusselt number Mean Nusselt number based on the mean side length, a, of rectangular heated surface Mean Nusselt number averaged over bottom surface Mean Nusselt number averaged over vertical side surface of cylinder Mean Nusselt number based on m Mean Nusselt number based on r Mean Nusselt number based on r averaged over bottom surface Mean Nusselt number based on r averaged over top surface Mean Nusselt number averaged over horizontal top surface of cylinder Mean Nusselt number averaged over top surface Mean Nusselt number averaged over entire surface Coordinate normal to surface or number of surface waves Total perimeter of heated surface Prandtl number Total mean heat transfer rate Total mean heat transfer rate from bottom surface Total mean heat transfer rate from top surface Local heat transfer rate per unit area Mean heat transfer rate per unit area from entire cylinder Mean heat transfer rate per unit area from cylindrical outer surface of cylinder Mean heat transfer rate per unit area from vertical side surface of square cylinder Mean heat transfer rate per unit area from top surface of cylinder Mean heat transfer rate per unit area Dimensionless cylinder radius, r/l Rayleigh number Rayleigh number based on the mean side length, a, of rectangular heated surface Rayleigh number based on m Rayleigh number based on r Heat Flux Rayleigh number Characteristic length scale of surface and the radius of cylinder Characteristic length scale of bottom surface Characteristic length scale of top surface Arm size of +- shaped and I-shaped heated surfaces Dimensionless distance between sides of square heated surfaces Temperature Fluid temperature far from surface Wall temperature

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T0 VGap W Wout w wout

P. H. Oosthuizen

Reference fluid temperature Dimensionless size of gap between surfaces, Gap/h Dimensionless surface width, w/h The dimensionless side lengths of the rectangular surrounding adiabatic surface Wout = wout/w Surface width Side lengths of the rectangular surrounding adiabatic surface

Greek Symbols

α β δ ϕ ν ξ φ μ ρ ρ0

1

Thermal diffusivity Bulk coefficient of thermal expansion Measure of the boundary layer thickness Nanoparticle volumetric fraction Kinematic viscosity 1/(R Ra0.25) or 1/(W Ra0.25) Angle of inclination Viscosity Density Reference density value

Introduction

Convective heat transfer, in general, is the term used to describe heat transfer between a surface and a fluid flowing relative to the surface. Convection of course is one of the three so-called modes of heat transfer, the other modes being conduction and radiation. Free or natural convective heat transfer is convective heat transfer in situations in which the fluid motion occurs purely as a result of the buoyancy forces that arise in the flow as a consequence of the density differences that are generated as a result of the temperature difference in the flow system. These temperature differences must exist in situations in which heat transfer is occurring. Such flows most commonly arise due the fluid density differences in the presence of the gravitational force field. In such a situation, if ρ0 is the fluid density of the undisturbed fluid and ρ is the density of the heated fluid at some point in the flow, then the buoyancy force per unit volume at this point in the flow is (ρ0ρ)g where g is the gravitational acceleration. This force can be vertically upward or vertically downward depending on whether the temperature change causes a decrease or an increase in the density. While the vast majority of free convective flows occur as a result of density changes in the presence of the gravitational force field, similar flows can arise in the presence of other force fields, e.g., in the presence of a centrifugal force field. In the past, a distinction has sometimes been made between the terms natural convection and free convection, the term natural convection then being applied to flows resulting from temperature-induced density changes in the presence of the gravitational field and the term free convection being applied to flows resulting from

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temperature-induced density changes in the presence of any force field including the gravitational field. However, today these two terms tend to be used interchangeably, and no distinction will be made here between the terms free and natural convection although attention will only be given to flows that result from temperature-induced density changes in the presence of the gravitational force field. Convective heat transfer flows are usually classified as being either external or internal. In external flows, the flow is over the outer surface of a body in situations where any other surfaces that exist in the overall flow system are so far from the body that they effectively have no influence on the flow over the body and therefore have no influence on the convective heat transfer rate from the body. In internal flows, the flow is constrained between surrounding surfaces; these surfaces then together determine the nature of the flow and of the convective heat transfer rate from the heated or cooled surfaces. Examples of external and internal free convective flows are shown in Fig. 1. Attention will be limited in this chapter to external free convective flows. In this chapter, attention will be given to some relatively recent studies of various situations involving external free convective flow. While some experimental results will be presented, the discussion in this chapter is mainly concerned with numerically derived results. These results have been obtained by solving the full Reynoldsaveraged Navier-Stokes equations in conjunction, when necessary, with a suitable turbulence model with the full effects of buoyancy forces being accounted for in the turbulence model used, e.g., see Savill (1993), Schmidt and Patankar (1991), Zheng et al. (1998), Plumb and Kennedy (1977), and Albets-Chico et al. (2008). Most of the results presented in this chapter were obtained using the basic k-epsilon turbulence model with standard wall functions. When appropriate, this model was applied in all calculations and in this way used to determine when transition to turbulent flow

Fig. 1 External and internal free convective flows

Adiabatic

Hot

Cold

Adiabatic External Free Convective Flow

Internal Free Convective Flow

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occurred. The results were mainly obtained using commercial CFD codes, the code used in deriving the majority of the results discussed here being ANSYS FLUENT#. In obtaining the solutions, the Boussinesq approximation has been adopted. This approximation assumes that in free convective flows, even though there are temperature variations in the flow, the viscosity, μ, and all other fluid properties except for the density can be assumed to be constant and that the only effect of changes in the density resulting from the temperature changes is the generation of the buoyancy forces. When the Boussinesq approximation is adopted, it is usual to assume that the density change is linearly related to the temperature change, i.e., to assume Δρ = β ΔΤ where the constant β is termed the bulk coefficient of expansion and Δρ is the density change resulting from the temperature change ΔΤ . This approach has been used in obtaining the results discussed here. A number of studies, for example, see Gray and Giorgini (1976), Nikitin and Ryzhak (1981), and Bredmose et al. (2001), have investigated the adequacy of using the Boussinesq approximation and have found that in the majority of situations, unless the temperature differences in the flow are relatively large, the results obtained using the Boussinesq approximation are of acceptable accuracy. The boundary conditions on the solution to the governing equations, since the body over which the flow is occurring is at rest, are basically that: 1. On the surface over which the flow is occurring, the velocity is everywhere zero. 2. In the fluid far from the surface being considered, the temperature is equal to that in the undisturbed fluid and the pressure is equal to the standard undisturbed atmospheric value. 3. At the surface over which the flow is occurring, the thermal conditions are specified. Usually this means that either the temperature distribution or the heat flux distribution over the surface is specified. Since the velocity at the surface is zero, Fourier’s law for conduction applies at the surface. Hence, if q’ is the local specified heat flux per unit area at any point on the surface, then: n¼0:

q0 ¼ k

@T @n

(1)

Here n is the coordinate normal to the surface and n is zero at the surface. In presenting the results discussed in this chapter, the following dimensionless parameters will be used: the Nusselt number, the Rayleigh number, and the Prandtl number. The mean Nusselt number is defined by: Nusselt Number, Nu ¼

q0w l k ðT w  T 0 Þ

(2)

where q0w is the mean heat transfer rate from the surface, l is the reference size of the body being considered, T0 is the temperature in the undisturbed fluid far from the body, and Tw is the mean surface temperature. The Rayleigh number is defined as follows in the case where the surface temperature is specified:

13

Free Convection: External Surface

Rayleigh Number, R a ¼

533

βgl 3 ðT w  T 0 Þ να

(3)

and as follows when the heat flux at the surface is specified: Heat-Flux Rayleigh Number, R a ¼

βgl 4 q0w kνα

(4)

The Prandtl number is a property of the fluid. Almost all of the results presented in this chapter are for the case where the fluid surrounding the body being considered is air and the Prandtl number has been assumed to be 0.74. In this chapter, attention will be given to free convective heat transfer in the following situations that involve external flow: 1. Heat transfer from narrow vertical plates 2. Heat transfer from horizontal heated surfaces of complex shape, from adjacent pairs of horizontal heated surfaces, and from two-sided horizontal surfaces 3. Heat transfer from bodies having a wavy surface 4. Heat transfer from short vertical square and rectangular cylinders with an exposed top surface 5. Heat transfer to nanofluids, a very brief discussion of this topic being presented. The results discussed in this chapter were obtained in new and extended studies of these topic areas. Many reviews of classical studies of external free convective heat transfer situations are available, e.g., see the publications on convective heat transfer by Burmeister (1993), Kaviany (1994), Kakaç and Yener (1995), Bejan (1995), Oosthuizen and Naylor (1999), and Mickle and Marient (2009) and the publications on free convection by Ede (1968), Gebhart (1973), Jaluria (1980), Raithby and Hollands (1985), Gebhart et al. (1988), and Oleg and Pavel (2005),

2

Narrow Vertical Plane Surfaces

Free convective heat transfer from vertical plane surfaces that are wide compared to their height has been quite extensively studied, laminar flow situations having been dealt with, for example, by Sparrow (1955), Sparrow and Gregg (1956, 1958), and Gryzagoridis (1971), and transitional and turbulent flow situations having been considered by Szewczyk (1962), Oosthuizen (1964), Warner and Arpaci (1968), Noto and Matsumoto (1975), Lin and Churchill (1978), and Mahajan and Gebhart (1979). In such a situation because the width of the surface is large compared to its height of the surface, the flow can be assumed to be two dimensional. However, when the width of a vertical plane surface is relatively small compared to its height (see Fig. 2), i.e., when the vertical plane surface is relatively narrow, the mean heat

534

P. H. Oosthuizen

transfer rate from the surface per unit surface area can be significantly greater than it would be from a wide vertical plane surface under the same conditions. The higher transfer rate that exists with a narrow vertical surface compared to that which exists with a wide vertical surface under the same conditions arises from the fact that an inward flow is induced near the edges of the heated surface with the result that the flow is three dimensional near the edges of the plane surface (see Fig. 3). When the surface is wide, the effects of this edge flow are negligible because it covers such a small portion of the total surface area. However, when the surface is narrow, the effects of this inward flow near the edges of the surface are significant and lead to an increase in the mean heat transfer rate. The increase in the heat transfer rate from a narrow surface compared to that for a wide surface is thus said to be due to “edge effects.” Situations in which edge effects are important arise in a number of practical free convective flow situations.

Fig. 2 Wide and narrow vertical plates

g

Wide Vertical Plate Fig. 3 Flow near the edges of a vertical plate

g

Narrow Vertical Plate

g

Flow Near Edge of Plate

Flow Near Edge of Plate

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Free Convection: External Surface

535

Discussions of free convective heat transfer from narrow plane surfaces are presented in Oosthuizen and Paul (2006, 2007a, d) and Kalendar and Oosthuizen (2008). The effect of edge conditions on free convective heat transfer from narrow plane surfaces is discussed in Oosthuizen and Paul (2007b, c, 2010) and Kalendar and Oosthuizen (2011), while free convective heat transfer from narrow plane surfaces that are inclined to the vertical is considered in Kalendar and Oosthuizen (2010c) and Kalendar et al. (2015). The case where there are two vertical narrow plane surfaces that are relatively closely spaced is dealt with in Kalendar et al. (2009, 2015) and Kalendar and Oosthuizen (2009, 2010b). Reviews of studies of free convective heat transfer from narrow plane surfaces are given in Oosthuizen and Kalendar (2013) and Kalendar (2011). The results discussed in this section for the heat transfer rate from narrow vertical plane surfaces have, as mentioned before, all been obtained numerically. Because the edge effects arise as a result of the three-dimensional flow effects near the edges of the surface, the three-dimensional flow equations have to be used in obtaining the results. Most of the available results for free convective heat transfer from narrow vertical plane surfaces have been obtained for the case where the fluid involved is air. The discussion presented here is mainly based on that presented in Oosthuizen and Paul (2007b, c, d). Consideration will first be given to a vertical surface that has a uniform surface temperature, i.e., to the situation shown in Fig. 4. In such a case: Nu ¼ functionðRa, Pr, W Þ

(5)

where Nu is the mean Nusselt number based on the surface height, h, Ra is the Rayleigh number based on h and on the difference between the temperature of the surface and the temperature of the undisturbed fluid far from the surface, Pr is the Prandtl number, and W = w/h (see Fig. 4). Since only results for air are being considered, Pr can be treated as a constant, here being taken to have a value of 0.74, and in this case:

Fig. 4 Vertical isothermal plane surface situation considered

h Isothermal Vertical Heated Surface

W

Adiabatic Surrounding Surface

536

P. H. Oosthuizen

Nu ¼ functionðRa, W Þ

(6)

The results discussed here were obtained for the case where the heated vertical isothermal plane surface is imbedded in a plane adiabatic surrounding surface, as shown in Fig. 4, the heated surface being in the same plane as the surface of the surrounding adiabatic material. The flow was also assumed to be symmetrical about the vertical centerline shown in Fig. 4. Attention will be restricted to the case where the flow over the plane surface is laminar, and results have therefore been obtained for Rayleigh numbers between 102 and 108 for dimensionless surface width, W, values of between 0.2 and 0.6. Typical variations of the mean Nusselt number with Rayleigh number for various values of W are shown in Fig. 5. It will be seen from these results that the mean Nusselt number at a particular Rayleigh number increases as W decreases, this effect increasing in magnitude as the Rayleigh number decreases. This effect is further illustrated by the results presented in Fig. 6 which shows variations of the Nusselt number with W for various values of the Rayleigh number. The results given in Fig. 6 show that at a Rayleigh number of 104, the Nusselt number increases by roughly 25% as the value of W decreases from 0.6 to 0.2, whereas at a Rayleigh number of 107, the Nusselt number increases by less than 2% as the value of W decreases from 0.6 to 0.2. The decrease in the magnitude of the edge effect as the Rayleigh number increases arises from the fact that the extent of the edge effect will be related to the boundary layer thickness which decreases as the Rayleigh number increases. The boundary layer thickness at the top of the surface, since laminar flow is being considered, is proportional to h/Ra0.25. Hence since the extent of the surface flow affected by the edge flow will be dependent on the ratio of the boundary layer thickness to the surface width which will be a function of h/w Ra0.25, i.e., of 1/W Ra0.25, the effect of the surface width on the Nusselt number will be dependent on this parameter.

Fig. 5 Typical variations of mean Nusselt number with Rayleigh number for various W values (Republished with permission of AIAA Inc., from Oosthuizen and Paul (2006); permission conveyed through Copyright Clearance Centre Inc.)

13

Free Convection: External Surface

537

Fig. 6 Typical variations of mean Nusselt number with W for various Rayleigh number values (Republished with permission of AIAA Inc., from Oosthuizen and Paul (2006); permission conveyed through Copyright Clearance Centre Inc.)

Now for a wide plane surface for the Rayleigh number range here being considered and for the Prandtl number being assumed, the mean Nusselt number variation with Rayleigh number is given by: Nu0 ¼ 0:68 þ 0:513Ra0:25

(7)

Here the subscript 0 on the Nusselt number indicates that the value is that for a wide surface. This equation gives results that are in good agreement with the numerical results here being considered for a narrow surface at the highest value of W considered. Using this equation as a guide and recalling that the edge effects depend on 1/W Ra0.25, it has been found that the numerical results for a narrow surface can be approximately represented by: Nu Nu0 7:1 ¼ þ
1:25 Ra0:25 Ra0:25 W Ra0:5

(8)

the second term on the right-hand side of this equation being, as discussed above, a measure of the ratio of the boundary layer thickness to the surface width. This equation indicates that an approximate indication of when edge effects can be neglected for the case where laminar airflow is involved is:


7:1 W Ra

 0:5 1:25

< 0:005, i:e:,

W>

330 Ra0:5

(9)

The variation of the value of W above which edge effects can be ignored with Rayleigh number given by this equation is shown in Fig. 7.

538

P. H. Oosthuizen

Fig. 7 Variation of W value above which edge effects are negligible with Rayleigh number (Republished with permission of AIAA Inc., from Oosthuizen and Paul (2006); permission conveyed through Copyright Clearance Centre Inc.)

The above results were for the case where the narrow vertical plane surface considered was isothermal, i.e., had a uniform temperature. Results have also been obtained for the case where there is a uniform heat flux over the plane surface (see Fig. 8). In this case: Nu ¼ functionðRa , Pr, W Þ

(10)

where Ra* is, as discussed before, the heat flux Rayleigh number based on h and on the uniform heat flux over the surface. Attention will again be restricted to the case where airflow is involved, and Pr has therefore again been treated as constant, its value again being taken as 0.74. In this case: Nu ¼ functionðRa , W Þ

(11)

Typical numerical results for the case of a vertical plane surface which has a uniform surface heat flux are shown in Fig. 9. Now for a wide vertical plane surface with a uniform heat flux for the value of Pr being considered here, the Nusselt number is approximately given by: Nu0 ¼ 0:68 Ra0:25

(12)

Using this equation, it can be shown that the numerical results for a narrow vertical plane surface with a uniform surface heat flux can be effectively represented by the following equation:

13

Free Convection: External Surface

539

Fig. 8 Vertical plane surface with a uniform surface heat flux situation considered Adiabatic Surrounding Surface

Vertical Heated Surface with Uniform Heat Flux

Fig. 9 Typical variations of Nu/Ra*0.2 with W for various heat flux Rayleigh number values (From Oosthuizen and Paul (2007b), Fig. 1, Paper IMECE2007–42712 by permission of ASME)

540

P. H. Oosthuizen

Fig. 10 Variation of W value above which edge effects are negligible with heat flux Rayleigh number (From Oosthuizen and Paul (2007b), Fig. 10, Paper IMECE200742712 by permission of ASME)

Nu 0:75 ¼ 0:68 þ
1:4 Ra0:25 W Ra0:2

(13)

This equation indicates that edge effects can effectively be neglected for the case of a narrow vertical plane surface with a uniform surface heat flux where airflow is involved when: 0:75
1:4 < 0:014, i:e:, W Ra0:2

W>

15:2 Ra0:2

(14)

The variation of the value of W above which edge effects can be ignored with heat flux Rayleigh number given by this equation is shown in Fig. 10. The above discussion was concerned with conditions under which the flow over the vertical plane surface was laminar. Numerical results obtained for conditions under which turbulent flow exists indicate that for geometries that are likely to occur in practical situations, edge effects in almost all cases will be negligible because of the low values of the boundary layer thickness in such flows.

2.1

Heat Transfer from Adjacent Narrow Plane Surfaces

Studies of the interaction of the free convective flows over two adjacent vertical narrow isothermal flat plane surfaces in the laminar flow region are discussed in this section. The discussion presented here is mainly based on that presented in Kalendar and Oosthuizen (2009, 2010b) and Kalendar et al. (2009, 2015). Two cases will be considered. In one case, the surface are horizontally adjacent to each other, i.e., the surfaces are horizontally separated, the situation considered thus being as shown in Fig. 11. In the other case considered, the surfaces are symmetrically arranged one

13

Free Convection: External Surface

541 Horizontally Spaced Vertical Heated Surfaces

Surrounding Adiabatic Surface

h

h

w

w

Fig. 11 Horizontally adjacent vertical narrow plate situation

above the other, i.e., the surfaces are vertically separated, the situation considered in this case thus being as shown in Fig. 12. Attention is here given to the effects of the vertical and horizontal dimensionless gaps between the heated surfaces and to the effects of the dimensionless surfaces width on the mean heat transfer rates from the two heated surfaces for a range of Rayleigh numbers. The concern here is again with surfaces that are relatively narrow, i.e., that have relatively low width, w, to height, h, ratios. As discussed above, when the surface is narrow, the mean heat transfer rate per unit surface area from the surface is greater than that from a wide surface. As discussed above, the increase in the heat transfer rate from narrow surfaces relative to that from wide surface under the same conditions results from the fact that fluid flow is induced inward near the edges of the surface, and the flow near the edge of the surface is three dimensional. When there are two adjacent narrow surfaces, the interaction of the flows over the two surfaces can affect the nature and the magnitude of the edge effects on the heat transfer rate from the surfaces. Thus, when there are two adjacent narrow flat surfaces with a relatively small gap, the interaction of the flows near the edges of the adjacent surfaces alters the nature of the flow compared to that over a single isolated narrow surface, and this can lead to a significant change in the mean heat transfer rate compared to that from a single isolated surface under the same conditions. Situations that can be approximately modeled as two adjacent narrow heated surfaces do occur in practical situations, for example, in some situations involving the cooling of electronic and electrical equipment. The results discussed in this section were numerically derived by solving the full three-dimensional form of the governing equations, the flow being assumed to be

542 Fig. 12 Vertically adjacent vertical narrow plate situation (From Oosthuizen and Kalendar (2013) Fig. 7.2 Vertically adjacent plate flow situation considered, p. 94, # The authors. With permission of Springer)

P. H. Oosthuizen

g

Top Heated Plate

h

v Gap

Bottom Heated Plate

h

w

Surrounding Adiabatic Surface

laminar and steady and the Boussinesq approximation being adopted. Attention has been restricted to the case where the Prandtl number is 0.74, i.e., effectively to the case of airflow. Consideration will first be given to the mean heat transfer rate from two relatively narrow horizontally adjacent isothermal surfaces of the same size that are embedded in a plane adiabatic surface which is in the same plane as the surfaces of the heated surfaces. The discussion presented here is based on the studies described in Kalendar and Oosthuizen (2010b) and Kalendar et al. (2009). The two surfaces are maintained at the same uniform surface temperature and they are aligned with each other as shown in Fig. 11. The surfaces are separated from each other by a relatively small gap. Results have been obtained for a relatively wide range of Rayleigh numbers and dimensionless surface widths and surface separation gap sizes. The mean heat transfer rates from the surfaces has been expressed in terms of a mean Nusselt number, Nu, based on the height of the surfaces, h, and the difference between the uniform surface temperature and the temperature of the undisturbed fluid far from the surfaces. The solution has the following parameters: 1. The Rayleigh number, Ra, based on the heated surface heights, h, and the overall temperature difference between the surface temperature and the undisturbed fluid temperature 2. The dimensionless heated surfaces width, W = w/h 3. The Prandtl number, Pr 4. The dimensionless gap between the two adjacent heated surfaces, HGap = G/h

13

Free Convection: External Surface

543

Results were obtained for Pr = 0.74, Ra values between 103 and 107, W values between 0.15 and 0.6, and HGap values of from 0 to 0.2. Typical variations of Nu/Ra0.25 with the Rayleigh number for various values of the dimensionless gap between the surfaces for dimensionless surface widths, W, of 0.6 and 0.15 are shown in Figs. 13 and 14. The results given in Figs. 13 and 14 show how the Nusselt number increases with decreasing dimensionless surface width and how it decreases with decreasing dimensionless gap size. The results also show that HGap has a higher effect at lower values of Rayleigh numbers. As the dimensionless

Fig. 13 Variation of Nu/ Ra0.25 with Ra for various HGap sizes for W = 0.6 (Based on Kalendar and Oosthuizen (2010b). By the courtesy of JP Journal of Heat and Mass Transfer)

Fig. 14 Variation of Nu/ Ra0.25 with Ra for various HGap sizes for W = 0.15 (Based on Kalendar and Oosthuizen (2010b). By the courtesy of JP Journal of Heat and Mass Transfer)

544

P. H. Oosthuizen

surface width increases and the dimensionless gap decreases to zero, the two adjacent surfaces become a single wide surface, and the Nusselt number becomes closer to the empirical correlation for a wide surface particularly at higher Rayleigh numbers. The effects of W and the dimensionless plate width on the Nusselt number are further illustrated by the results given in Fig. 15. Using the results given in Figs. 13 and 14 and other similar results obtained in the studies on which the present discussion is based, it has been found that the mean heat transfer rate from two horizontally adjacent vertical surfaces is given by: Nu Nu0  Ra0:25 Ra0:25

¼



 3

W H Gap

0:8
HGap  0:39 Ra 0:5þ 2

(15)

where Nu0 is the value of the mean Nusselt number that would exist with a single wide vertical surface under the same conditions, i.e., at the same value of Ra and Pr, which is given for a Prandtl number of 0.74 by Churchill and Chu (1975): Nu0 ¼ 0:68 þ 0:513 Ra0:25

(16)

Equation 15 describes the computed numerical results to an accuracy of better than 5%. The heat transfer from two vertically spaced adjacent isothermal surfaces will next be considered (see Fig. 12). In such a case, the interaction of the flow from the lower surface with that over the upper surface can alter the nature of the flow and the magnitude of the edge effects for the upper surface. Situations that can be approximately modeled as two vertically separated narrow heated surfaces do occur in

Fig. 15 Variation of Nusselt number with HGap for various values of Ra and W (Based on Kalendar and Oosthuizen (2010b). By the courtesy of JP Journal of Heat and Mass Transfer)

13

Free Convection: External Surface

545

practical situations, for example, in some situations involving the cooling of electronic and electrical equipment. The mean heat transfer rates from two vertically separated narrow vertical surfaces of the same size that are embedded in a plane adiabatic surface such that the adiabatic surface is in the same plane as the heated surfaces are considered here. The discussion presented here is based on the study described in Kalendar and Oosthuizen (2009). The two heated surfaces have the same surface temperature and the vertical edges of the top and bottom surfaces are aligned. The heated surfaces are separated from each other by a vertical gap. Results have been obtained for a relatively wide range of Rayleigh numbers, dimensionless surface gaps, and dimensionless surface widths. Attention has again been restricted to results for a Prandtl number of 0.74, this being approximately the value existing in applications that involve the flow of air. The mean heat transfer rates from the heated surfaces have again been expressed in terms of a mean Nusselt number, Nu, based on the height of the surfaces, h, and the difference between the uniform heated surface temperature and the temperature of the undisturbed fluid far from the surfaces. In the situation here being considered, the mean Nusselt numbers for the bottom and top heated surfaces, i.e., Nubottom and Nutop, will be different. The solution has the following parameters: 1. The Rayleigh number, Ra, based on the surface height, h, and the overall temperature difference between the surface temperature and the undisturbed fluid temperature 2. The dimensionless surface width, W = w/h 3. The Prandtl number, Pr 4. The dimensionless gap between the two adjacent heated surfaces, VGap = Gap/h Results have been obtained for Pr = 0.74, Ra values between 103 and 107, W values between 0.2 and 0.6, and VGap values from 0 to 1.5. The effect of the dimensionless vertical gap between two surfaces on the mean Nusselt numbers for the top surface, Nutop, and for the bottom surface, Nubottom, is illustrated by the typical results shown in Figs. 16 and 17. It will be seen from these results that, as is to be expected, the presence of the top surface has essentially no effect on the mean Nusselt number for the bottom surface. It will also be seen that at small dimensionless gap values, the mean Nusselt number for the top surface is much lower than that for the bottom surface but that for dimensionless gaps greater than approximately 1.5, the mean Nusselt numbers for the two heated surfaces become almost the same. The effect of the dimensionless vertical gap between two heated surfaces and of the dimensionless surface width is further illustrated by the results presented in Figs. 18 and 19. These results again show how, in all cases, as a result of the interaction between the flows over the two surfaces, the mean Nusselt number for the top surface is lower than that for the bottom surface but that its value increases toward the bottom plate value with increasing VGap. Using results similar to those discussed above, it has been found that the variation of the mean Nusselt number averaged over the area of the two surfaces for values of VGap between 0.1 and 1.5 can be approximately described by the following equation:

546

P. H. Oosthuizen

Fig. 16 Variations of the mean Nusselt number averaged over the bottom surface with the Rayleigh number for various dimensionless vertical gaps between the plates and for a dimensionless plate width of 0.3 (Based on Oosthuizen and Kalendar (2013) Fig. 7.19 Variation of Nubottom with Rayleigh number for various values of dimensionless gap, VGap, for W = 0.3, p. 108, # The authors. With permission of Springer)

Fig. 17 Variations of the mean Nusselt number averaged over the top surface with the Rayleigh number for various dimensionless vertical gaps between the plates and for a dimensionless plate width of 0.3 (Based on Oosthuizen and Kalendar (2013) Fig. 7.20 Variation of Nutop with Rayleigh number for various values of dimensionless gap, VGap, for W = 0.3, p. 108, # The authors. With permission of Springer)

Nu 1:3 ¼ 0:45 þ "   #0:82 0:25 Ra 0:3þV0:005 Gap W Ra

(17)

This correlation equation describes 80% of the computed results to an accuracy of better than 10%.

13

Free Convection: External Surface

547

Fig. 18 Variations of the mean Nusselt number averaged over the top surface and over the bottom surface with dimensionless vertical gap between the plates for various Rayleigh numbers and for a dimensionless plate width of 0.6 (Based on Oosthuizen and Kalendar (2013) Fig. 7.21 Variation of mean Nusselt number for the top and bottom surfaces with the dimensionless gap between the plates for various Rayleigh numbers for W = 0.6, p. 110, # The authors. With permission of Springer)

Fig. 19 Variations of the mean Nusselt number averaged over the top surface and over the bottom surface with dimensionless vertical gap between the plates for various Rayleigh numbers and for a dimensionless plate width of 0.3 (Based on Oosthuizen and Kalendar (2013) Fig. 7.22 Variation of mean Nusselt number for the top and bottom surfaces with the dimensionless gap between the plates for various Rayleigh numbers for W = 0.3, p. 110, # The authors. With permission of Springer)

3

Horizontal Plane Surfaces

In this section, attention will be given to horizontal heated plane surfaces (see Fig. 20). Studies of heat transfer in this basic situation are described in Husar and Sparrow (1968), Lienhard et al. (1974), Vliet and Ross (1975), Yousef et al. (1982),

548 Fig. 20 Horizontal surface flow situation considered

P. H. Oosthuizen

Adiabatic Surrounding Surface Heated Horizontal Surface

Fig. 21 Type of flow generated with an upwardfacing heated surface and with a downward-facing cooled surface

Flow Pattern

Heated Horizontal Surface

Clausing and Berton (1989), Sahraoui et al. (1990), Oosthuizen (2014d, 2015a, b, c, 2016b), Oosthuizen and Kalendar (2015a, b, 2016), and Kobus and Wedekind (1995, 2001, 2002). Situations in which there is a covering surface over a heated horizontal surface are discussed in Oosthuizen (2014a, b, c, 2016a). Although studies of heat transfer from heated or cooled surfaces that are facing upward and from those that are facing downward have been undertaken, attention in this section will be limited to the case where there is a heated surface facing upward or a cooled surface facing downward. Attention will first be given to the mean heat transfer rate from plane isothermal one-sided surfaces of the type shown in Fig. 21. As shown in Fig. 21, the flow is inward from the sides of the surface and then upward in a centrally positioned plume in the case of a heated surface. The first case that will be considered is that of a plane surface of width w in the x-direction that is long in the z-direction as shown in Fig. 22. In such a situation, the flow can be assumed to be two dimensional in the x-y plane and to be symmetrical about the

13

Free Convection: External Surface

549

x

Adiabatic Side Surface

y

Heated Horizontal Surface

z

Adiabatic Side Surface

Fig. 22 Two-dimensional isothermal horizontal surface flow situation considered

Fig. 23 Variation of the mean Nusselt number for two-dimensional flow over an isothermal plane horizontal surface with Rayleigh numbers, the Nusselt and Rayleigh numbers being based on the width of the surface (From Fig. 4 in Oosthuizen (2016e) Paper IMECE2016-65716 by permission of ASME)

centerline shown in Fig. 22. Numerical solutions for this situation will be considered (Oosthuizen 2016e). The solution has been obtained by solving the full Navier-Stokes equations in conjunction with the full energy equation. The calculated variation of the mean Nusselt number based on the surface width w with Rayleigh number, also based on the surface width w, for the case where the surface is maintained at a uniform temperature, i.e., for the case where the surface is isothermal, and for the case where the heat transfer from the surface is to air, is shown in Fig. 23. Because the heat transfer is to air, the Prandtl number has been treated as a constant and equal to 0.74. The range of Rayleigh numbers considered is such that laminar, transitional, and turbulent flows occur.

550

P. H. Oosthuizen

Attention will next be turned to the heat transfer from plane square and circular isothermal surfaces that are imbedded in a surrounding plane adiabatic surface, these situations being shown in Fig. 24. As shown in Fig. 24, the heated surfaces and the adiabatic surrounding surfaces are in the same plane. In the case of the square surface, the characteristic length used in defining the Nusselt and Rayleigh numbers will be taken as the side length, l, while in the case of the circular surface, the characteristic length will be taken as the diameter, d (see Fig. 24). The results discussed here are based on those presented in Oosthuizen (2014d, 2015b) and Oosthuizen and Kalendar (2015b). Calculated variations of the mean Nusselt number with Rayleigh number for square and circular surfaces are shown in Figs. 25 and 26. A comparison of the results given in Figs. 25 and 26 indicates that the variations of the mean Nusselt number with Rayleigh number are essentially the same for the two surface shapes considered. In many practical situations, heated horizontal surfaces having a more complex shape than those considered above are encountered. To investigate the effect of the

Fig. 24 Plane square and circular isothermal surface situations considered

13

Free Convection: External Surface

551

Fig. 25 Variation of the mean Nusselt number with Rayleigh number for square isothermal horizontal surface (Derived from Fig. 11 in Oosthuizen and Kalendar (2015a). By permission)

Fig. 26 Variation of the mean Nusselt number with Rayleigh number for circular isothermal horizontal surface (From Oosthuizen and Kalendar (2015b) Fig. 5)

surface shape on the variation of the mean Nusselt number with Rayleigh number, some results for heat transfer from I-shaped and from +-shaped heated surfaces will next be considered. The discussion presented here is mainly based on the study described in Oosthuizen (2015c). The shapes considered are shown in Fig. 27. The outside size, w, of the surfaces will be taken as the characteristic length used in defining the Nusselt and Rayleigh numbers. Typical variations of mean Nusselt number with Rayleigh number for the two surface shapes being considered are shown in Figs. 28 and 29. It will be seen from these results that the variations for the two surface shapes are not the same. For simpler shapes and mainly for laminar flows, it has been found (Raithby and Hollands 1985) that if the following

552

P. H. Oosthuizen

Fig. 27 I- and +-shaped horizontal heated surface shapes considered (Reprinted from Oosthuizen (2015c), Fig. 2, #2015, with permission from Begell House, Inc.)

w s

s

w

s

I Shaped Element

s w s

w

+ Shaped Element

Fig. 28 Variation of the mean Nusselt number with Rayleigh number for I-shaped isothermal horizontal surface for two values of the dimensionless arm size (Based on results from Oosthuizen (2015c), #2015, with permission from Begell House, Inc.)

13

Free Convection: External Surface

553

Fig. 29 Variation of the mean Nusselt number with Rayleigh number for +-shaped isothermal horizontal surface for two values of the dimensionless arm size (Based on results from Oosthuizen (2015c), #2015, with permission from Begell House, Inc.)

characteristic length is used in defining the Nusselt and Rayleigh numbers, the results for all surface shapes will be the same: m¼4

A P

(18)

where A is the surface area and P is the total perimeter of the heated surface. Now for the round and square surfaces considered previously: π 2 d , P ¼ π d, so m ¼ d 4 2 Square Surface : A ¼ w , P ¼ 4 w, so m ¼ w

Round Surface : A ¼

(19)

Hence the results given in Figs. 25 and 26 are equivalent to the variations of the mean Nusselt number based on m with the Rayleigh number based on m for these two surface shapes. For the I-shaped surface and the +-shaped surfaces, however: I-Shape : m ¼ 4

ð3ws  2s2 Þ , ð6w  2sÞ

þ -Shape : m ¼ 4

ð2ws  s2 Þ 4w

(20)

The variations of mean Nusselt number with Rayleigh number for these two surface shapes shown in Figs. 28 and 29 have been replotted in terms of Nusselt and Rayleigh numbers based on values of m obtained using Eq. 20, the results being shown in Figs. 30 and 31. A comparison of these results shows that the variations for the two surface shapes are approximately the same and are approximately the same as the variations for square and circular surfaces. Thus, if the characteristic length

554

P. H. Oosthuizen

Fig. 30 Variation of the mean Nusselt number based on the “mean” element size, m, with Rayleigh number also based on the “mean” element size, m, for an I-shaped isothermal horizontal surface for various values of the dimensionless arm size considered (Reprinted from Oosthuizen (2015c), Fig. 13, #2015, with permission from Begell House, Inc.)

Fig. 31 Variation of the mean Nusselt number based on the “mean” element size, m, with Rayleigh number also based on the “mean” element size, m, for a +-shaped isothermal horizontal surface for various values of the dimensionless arm size considered (Reprinted from Oosthuizen (2015c), Fig. 15, #2015, with permission from Begell House, Inc.)

13

Free Convection: External Surface

555

Adiabatic Horizontal Surrounding Surface

Heated Horizontal Rectangular Surface

l

w

Fig. 32 Rectangular horizontal heated surface considered

m is used in defining the Nusselt and Rayleigh numbers, the results for all the surface shapes considered are the same. The variations of Num with Ram for the I- and +shaped surfaces and for the square and circular surfaces are all then adequately represented by the following equation in the laminar flow region: Num ¼ 0:622 Ra0:25 m

(21)

and by the following equation in the turbulent flow region: Num ¼ 0:022 Ra0:375 m

(22)

Results are also available for heat transfer from a horizontal rectangular isothermal plane surface imbedded in an adiabatic plane surrounding surface, the situation considered being shown in Fig. 32. Results have been obtained for the case where the heat transfer is to air for various values of the aspect ratio AR defined as: AR ¼

l w

(23)

where, as indicated in Fig. 32, d and w are the longer and shorter side lengths of the rectangular surface. Results are given in Oosthuizen (2015a) for values of AR between 1 and 6. Now if the characteristic dimension defined in Eq. 18 is introduced, this being given for a rectangular surface: m¼

4wl , i:e:, 2ð w þ d Þ

m 2AR ¼ w 1 þ AR

(24)

and if Nusselt and Rayleigh numbers based on this value of m are used, the results given in Oosthuizen (2015a) for all the values of AR considered are shown in Fig. 33.

556

P. H. Oosthuizen

Fig. 33 Variation of the mean Nusselt number based on the “mean” element size, m, with Rayleigh number also based on the “mean” element size, m, for a rectangular isothermal horizontal surface for various values of the aspect ratio (Reprinted with modification from Oosthuizen (2015a) Paper CHT-15-145, with permission from Begell House, Inc.)

It will again be seen from this figure that the results for all values of AR fall on a single curve and are well represented by Eqs. 21 and 22. The results discussed so far in this section have all been for hot surfaces facing upward or for cold surfaces facing downward. Results for the case of a hot surface facing downward or a cold surface facing upward are also available. If these results are also expressed in terms of the characteristic dimension m defined in Eq. 18, they can be approximately represented for laminar flow conditions by the following equation: Num ¼ 0:27 Ra0:25 m

3.1

(25)

Horizontal Two-Sided Circular Bodies

The results discussed thus far in this section have all been for the case where the horizontal heated body considered is imbedded in an adiabatic surrounding surface, and there is therefore only a single surface from which heat transfer to the surrounding fluid is occurring. In some practical situations, the heated body is fully exposed to the surrounding fluid, i.e., there is heat transfer from the top, bottom, and side surfaces of the body, a typical such situation being shown in Fig. 34. For such a situation involving a circular body, it has been found that for relatively thin bodies, the available experimentally obtained heat transfer rate results in air can be adequately represented by the following equations (Kobus and Wedekind 2001, 2002), the Nusselt and Rayleigh numbers in these equations being based on the diameter of the circular body: For Ra between 3  102 and 104 : Nu ¼ 1:759Ra0:130 For Ra between 104 and 3  107 : Nu ¼ 0:9724Ra0:194

(26)

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Free Convection: External Surface

Fig. 34 Two-sided horizontal circular plate situation

557

Horizontal Circular Two-sided Isothermal Heated Surface

Fig. 35 Two-sided horizontal circular isothermal surface with an inner circular adiabatic section situation considered (From Fig. 1 in Oosthuizen and Kalendar (2016) Paper IMECE201665540 by permission of ASME)

The experimental results used in deriving these empirical equations were obtained with circular bodies having thickness-to-diameter ratios between approximately 0.06 and 0.16. In some situations involving simultaneous free convective heat transfer from the top, bottom, and sides of a horizontal body, the body has a complex shape. To illustrate the type of results obtained in such cases, numerical results (Oosthuizen and Kalendar 2016) for heat transfer from a thin horizontal circular isothermal body that has an adiabatic inner section will be considered. This situation is shown in Figs. 35 and 36. Because the body is thin, there is essentially no heat transfer from the vertical edge of the body. Mean Nusselt number values for the entire body surface, for the top surface, and for the bottom surface, Nutotal, Nutop, and Nubottom, will be considered, these Nusselt numbers being based on the outside diameter of the body and on the difference between the temperature of the heated isothermal section of the body and the temperature of the undisturbed fluid far from the body, i.e.:

558

P. H. Oosthuizen

Fig. 36 Dimensions of two-sided horizontal circular isothermal surface with an inner circular adiabatic section (From Fig. 3 in Oosthuizen and Kalendar (2016) Paper IMECE2016-65540 by permission of ASME)

0

Nutotal ¼

Atotal




Qd  Tw  Tf k

(27a)

0

Nutop ¼

Q d
top  Atop T w  T f k

(27b)

0

Nubottom ¼

Qbottom d
 Abottom T w  T f k

(27c)

where: 0

0

0

Q ¼ Qtop þ Qbottom

(28)

Atotal, Atop, and Abottom are the surface areas of the entire heated area, of the heated area on the upper surface, and of the heated area on the lower surface. Here: Atop ¼ Abottom ,

Atotal ¼ Atop þ Abottom

(29)

These Nusselt numbers will depend on: 1. The Rayleigh number, Ra, based on the outside diameter of the body and on the difference between the temperature of the heated isothermal section of the body and the temperature of the undisturbed fluid far from the body

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Free Convection: External Surface

559

2. The ratio of the diameter of the inner adiabatic section to the outer body diameter, i.e., on Di = di/d 3. The Prandtl number, Pr The fluid surrounding the body has been assumed to be air and the Prandtl number has been treated as a constant and equal to 0.74. Typical variations of the mean Nusselt number averaged over both the upper and lower heated surfaces of the body and of the mean Nusselt numbers for the total surface areas of the body with Rayleigh number for various dimensionless adiabatic section diameter ratios, Di, values are shown in Figs. 37, 38, 39, and 40. In all cases, it will be seen that, as is to be expected for a heated plate, the mean Nusselt number for the top heated surface is larger than the mean Nusselt number for the bottom heated surface. The difference between the two Nusselt number values is greatest for all values of Di when the flow is in the fully turbulent region. It will be seen from Figs. 37, 38, 39, and 40 that the dimensionless adiabatic section diameter ratio, Di, value has a significant influence on the Nusselt number variations. The use of a reference length scale to allow the correlation of these mean Nusselt number-Rayleigh number variations will therefore again be considered. Now, as discussed before, it has often been assumed that for free convective heat transfer from horizontal heated surfaces of various shapes that if a reference surface size, r, defined by: r¼4

A P

(30)

Fig. 37 Variations of the mean Nusselt number for the top and bottom surfaces and for the entire heated surface of the circular plate with Rayleigh number for the case where the dimensionless diameter of the inner adiabatic section is zero, i.e., for the case where there is no inner adiabatic section (From Fig. 4 in Oosthuizen and Kalendar (2016) Paper IMECE2016-65540 by permission of ASME)

560

P. H. Oosthuizen

Fig. 38 Variations of the mean Nusselt number for the top and bottom surfaces and for the entire heated surface of the circular plate with Rayleigh number for the case where the dimensionless diameter of the inner adiabatic section is 0.25 (From Fig. 5 in Oosthuizen and Kalendar (2016) Paper IMECE2016-65540 by permission of ASME)

Fig. 39 Variations of the mean Nusselt number for the top and bottom surfaces and for the entire heated surface of the circular plate with Rayleigh number for the case where the dimensionless diameter of the inner adiabatic section is 0.5 (From Fig. 6 in Oosthuizen and Kalendar (2016) Paper IMECE2016-65540 by permission of ASME)

is introduced and if Nusselt and Rayleigh numbers based on this mean surface size are used, then the variations of Nusselt number with Rayleigh number will be the same for all of the surface shapes. In the above equation, A is the area of the heated surface and P is the total perimeter of the heated surface. In the present situation

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Free Convection: External Surface

561

Fig. 40 Variations of the mean Nusselt number for the top and bottom surfaces and for the entire heated surface of the circular plate with Rayleigh number for the case where the dimensionless diameter of the inner adiabatic section is 0.75 (From Fig. 7 in Oosthuizen and Kalendar (2016) Paper IMECE2016-65540, by permission of ASME)

since a circular surface with an unheated adiabatic inner section is being considered, the values of r for the upper or lower surfaces are given by: r top ¼ r bottom

 ðπ=4Þ d2  d2i ¼ ðd  di Þ ¼4 π ðd þ di Þ

(31)

while for the entire heated surface, since the outer perimeter of the surface applies to both the top and the bottom surfaces:  2  ðπ=4Þ d2  d2i ¼ ðd  d i Þ r¼4 2  π ðd þ d i Þ

(32)

which is the same as that for the top and bottom surfaces. Hence, the value of r is the same for all three heated sections considered, i.e., rtop = rbottom = r. In terms of the reference size, r, the Nusselt and Rayleigh numbers are given by:  r  r r  N ur ¼ N u , N urtop ¼ N utop , N urbottom ¼ N ubottom (33) d d d and Rar ¼ Ra

 r 3 d

(34)

Variations of Nur with Rar for various values of Di for the entire heated surface, for the top heated surface, and for the bottom heated surface are shown in Figs. 45, 46, and

562

P. H. Oosthuizen

Fig. 41 Variations of the mean Nusselt numbers based on the reference size for the entire surface of plate with Rayleigh number based on the reference size for dimensionless inner adiabatic section diameters of 0, 0.25, 0.5, and 0.75 (From Fig. 12 in Oosthuizen and Kalendar (2016) Paper IMECE2016-65540, by permission of ASME)

47, respectively. The results given in these three figures show that for the entire surface, for the top surface, and for the bottom surface, the variations of Nur with Rar in the laminar flow region and in the fully turbulent flow region are essentially the same for all values of Di considered. However, in the transitional flow region, there are differences between the variations of Nur with Rar for the various values of Di considered, these differences resulting mainly from changes in the conditions under which transition occurs with different values of Di. Therefore, the results given in Figs. 41, 42, and 43 can be used to determine the free convective heat transfer rate from a horizontal circular surface of the geometrical type here considered provided that the flow is either laminar or fully turbulent.

3.2

Recessed Heated Horizontal Circular Surface

A numerical study of free convective heat transfer from a horizontal upward-facing circular isothermal heated surface that is imbedded in a large flat adiabatic surrounding surface has been undertaken (Oosthuizen 2016c). The heated surface is recessed by a relatively small amount into the surrounding flat horizontal adiabatic surface, the situation considered being shown in Fig. 44. The surface is at a higher temperature than the surrounding fluid, and attention has been restricted to the case where the surface is facing upward. The range of conditions considered in the present study is such that laminar, transitional, and turbulent flows can occur. The effect of the dimensionless depth that the surface is recessed (see Fig. 45) on the heat transfer rate from the isothermal circular surface will be considered.

13

Free Convection: External Surface

563

Fig. 42 Variations of the mean Nusselt numbers based on the reference size for the top surface of plate with Rayleigh number based on the reference size for dimensionless inner adiabatic section diameters of 0, 0.25, 0.5, and 0.75 (From Fig. 13 in Oosthuizen and Kalendar (2016) Paper IMECE2016-65540, by permission of ASME)

Fig. 43 Variations of the mean Nusselt numbers based on the reference size for the bottom surface of plate with Rayleigh number based on the reference size for dimensionless inner adiabatic section diameters of 0, 0.25, 0.5, and 0.75 (From Fig. 14 in Oosthuizen and Kalendar (2016) Paper IMECE2016-65540 by permission of ASME)

The results discussed here have been obtained numerically. It has been assumed that the flow is steady and axisymmetric about the surface centerline, and the Boussinesq approximation has been used. Attention has been restricted to the case where the heated surfaces are facing upward. Radiant heat transfer effects have been neglected.

564

P. H. Oosthuizen

Fig. 44 Recessed heated horizontal circular surface situation considered (From Fig. 1 in Oosthuizen (2016c). By permission)

Fig. 45 Recess depth of heated horizontal circular surface (From Fig. 2 in Oosthuizen (2016c). By permission)

h Adiabatic

Heated Recessed Circular Element

The basic k-epsilon turbulence model with standard wall functions and with account being taken of buoyancy force effects has been used. This turbulence model is applied under all conditions and is used to predict when transition occurs. The solution has been obtained using the commercial CFD solver ANSYS FLUENT#. The solution has the following governing parameters: • The Rayleigh number, Ra, based on the diameter, d, of the circular surface and on the difference between the temperature of the surface of the heated surface, Tw, and the temperature of the undisturbed fluid well away from the system, Tf, i.e.: Ra ¼


 βgd 3 T w  T f να

(35)

• The dimensionless distance to which the surface is recessed, i.e.: H¼

h d

(36)

• The Prandtl number, Pr Results have only been obtained for a Prandtl number of 0.74, i.e., effectively the value for air. Rayleigh numbers between approximately 105 and 1016 have been considered.

13

Free Convection: External Surface

565 0

The total heat transfer rate from the heated circular surfaces, Q , has been expressed in terms of a mean Nusselt number based on the diameter, d, of the circular surface and on the difference between the temperature of the surface of the heated surface and the temperature of the undisturbed fluid well away from the system, i.e.: 0

Nu ¼

Qd  kA T w  T f


(37)

where A = πd 2/4 is the area of the heated circular surface. Because a fixed value of Pr is being considered, Nu is a function of Ra and of the dimensionless distance to which the surface is recessed, H. Typical variations of the mean Nusselt number with Rayleigh number for various values of the dimensionless distance to which the surface is recessed, H, are shown in Fig. 46. It will be seen from the results given in Fig. 46 that the dimensionless recess depth has a strong influence on the heat transfer rate at low and intermediate values of the Rayleigh number, when the flow is in the laminar and transitional regions. The effect of the dimensionless recess depth on the heat transfer rate in the fully turbulent flow region is however significantly less than in the laminar and transitional flow regions. It will also be noted from the results given in Fig. 46 that at low Rayleigh numbers, the recessing of the surface leads to a decrease in the heat transfer rate, whereas at higher Rayleigh numbers, the recessing of the surface leads to an increase in the heat transfer rate The effect of the dimensionless recess depth is further illustrated by the results presented in Fig. 47 which shows the variations of the mean Nusselt number with dimensionless recess depth for various Rayleigh number values.

Fig. 46 Variation of the mean Nusselt number with Rayleigh number for a recessed circular surface for various values of the dimensionless recess depth (From Fig. 3 in Oosthuizen (2016c). By permission)

566

P. H. Oosthuizen

Fig. 47 Variation of the mean Nusselt number with the dimensionless recess depth for various values of Rayleigh number for a recessed circular surface (From Fig. 4 in Oosthuizen (2016c). By permission)

The results presented in Fig. 47 show how the form of the variation of Nusselt number with the dimensionless recess depth varies very significantly with Rayleigh number. The changes in the form of the Nusselt number variation with dimensionless recess depth is, of course, the result of the changes in the flow pattern over the heated surface.

3.3

Adjacent Horizontal Isothermal Square Surfaces

In the horizontal surface situations considered thus far in this section, attention has been restricted to the heat transfer from a single horizontal surface. However, in some situations that can arise in the free convective air cooling of electrical and electronic devices, the components are mounted horizontally and relatively close together. In such cases, the interaction of the flows over the adjacent components can affect the heat transfer rate from the components. The purpose of the numerical study discussed here was to investigate at what distance between the surfaces of the adjacent components there was likely to be an effect on the heat transfer rate from the surfaces as a result of the interaction of the flows over the surfaces. A very simplified model of real situations has been considered here. The components are represented by two adjacent square plane horizontal isothermal surfaces of the same size embedded in a horizontal plane adiabatic surface as shown in Fig. 48, the arrangement of the heated surfaces being as shown in Fig. 49. The heated surfaces are assumed to be in the same plane as the surrounding adiabatic surface. Attention has been restricted to the case where the heated surfaces are facing upward and where they are isothermal. The heat transfer from the heated surfaces has been assumed to be to air because of the applications being considered. The discussion given here is based on the study described in Oosthuizen and Kalendar (2015a).

13

Free Convection: External Surface

Horizontal Surrounding Adiabatic Surface

567 Horizontal Square Heated Elements

Gap Between Heated Elements

Fig. 48 Adjacent horizontal surfaces flow situation considered (From Fig. 1 in Oosthuizen and Kalendar (2015a). By permission)

Fig. 49 Dimensions used in adjacent horizontal surfaces flow situation considered (From Fig. 2 in Oosthuizen and Kalendar (2015a). By permission)

The solution has the following parameters: • The Rayleigh number, Ra, based on the side length, w, of the square heated surfaces (see Fig. 49) and the difference between the temperature of the heated surfaces, Tw, and the temperature of the undisturbed fluid well away from the system, Tf, i.e.:

568

P. H. Oosthuizen


 βgw3 T w  T f Ra ¼ να

(38)

• The dimensionless distance between the adjacent sides of the two square heated surfaces, i.e., (see Fig. 49) S = s/w • The Prandtl number, Pr As discussed above, results have only been obtained for a Prandtl number of 0.74, i.e., effectively the value for air. Results have been obtained for S values between 0 and 0.7, the case of S = 0 corresponding to the case where the heated surfaces are in contact with each other and effectively form a single rectangular heated surface. For comparison results have also been obtained for the case of an isolated square heated surface, i.e., for the case where the two heated surfaces are so far apart that there is no interaction between the flows over the two square heated surfaces. Rayleigh numbers between approximately 104 and 1012 have been considered. The mean heat transfer rate from the heated surfaces, Q’, has been expressed in terms of a mean Nusselt number based on the side length, w, of the square heated surface and on the difference between the temperature of the heated surface, Tw, and the temperature of the undisturbed fluid well away from the system, Tf, i.e.: 0

Nu ¼

Qw  kA T w  T f


(39)

where A is the surface area of the heated surface, i.e., w2. Because a fixed value of Pr is being considered, Nu is a function of Ra and the dimensionless spacing between the heated surfaces, S. Typical variations of the Nusselt number with the dimensionless heated surface spacing for two Rayleigh number values are shown in Figs. 50 and 51. It will be seen that the form of the variation of the Nusselt number with the dimensionless surface spacing is strongly dependent on the Rayleigh number. It will also be seen from the Fig. 50 Variation of mean Nusselt number with dimensionless element spacing for a Rayleigh number of 106 (From Fig. 6 in Oosthuizen and Kalendar (2015a). By permission)

13

Free Convection: External Surface

569

Fig. 51 Variation of mean Nusselt number with dimensionless element spacing for a Rayleigh number of 1012 (From Fig. 8 in Oosthuizen and Kalendar (2015a). By permission)

results given in Figs. 50 and 51 that the largest percentage difference between the highest and lowest Nusselt number values for a given Rayleigh number is less than 10%. It will also be seen from these figures that the change in the heat transfer rate as a result of the presence of the adjacent surface begins to be significant when the dimensionless surface spacing is less than roughly 0.5. It was also found that while the spacing between the surfaces has a relatively weak effect on the mean heat transfer rate, it has a more significant effect on the local heat transfer rate distribution over the surface of the heated surfaces, and these changes can have effects that are important in some practical situations.

3.4

Horizontal Rectangular Surface with a Parallel Adiabatic Covering Surface

Free convective heat transfer from a horizontal rectangular isothermal surface which is imbedded in a larger rectangular flat adiabatic surface will here be considered. The heated surface is in the same plane as the surface of the surrounding adiabatic surface. A rectangular flat horizontal adiabatic covering surface (a shroud) is mounted parallel to the heated rectangular surface at a relatively short distance from it, i.e., there is a covering surface over the heated rectangular surface and the surrounding adiabatic surface in which it is imbedded. Figure 52 shows the flow situation considered, while the dimensions used in defining the size of the various system surfaces are shown in Figs. 53 and 54. The rectangular heated surface is at a higher temperature than that of the surrounding fluid. The discussion in the section is mainly based on the results obtained in the study described in Oosthuizen (2014a). Attention has been restricted to the case where the rectangular heated surface is facing upward and the adiabatic covering surface is above the heated surface, the situation considered being shown in Fig. 54. The results presented here have been obtained numerically. The main purpose of the study here being discussed was to

570

P. H. Oosthuizen

Fig. 52 Rectangular heated surface with adiabatic covering surface situation considered (From Fig. 1 in Oosthuizen (2014a) Paper IMECE2014-36780 by permission of ASME)

Fig. 53 Dimensions of heated isothermal rectangular element (From Fig. 2 in Oosthuizen (2014a) Paper IMECE2014-36780 by permission of ASME)

Fig. 54 Dimension between upward-facing heated rectangular surface and adiabatic covering surface (From Fig. 3 in Oosthuizen (2014a) Paper IMECE201436780 by permission of ASME)

Upper Rectangular Adiabatic Covering

h

Heated Horizontal Rectangular Element

investigate how the heat transfer rate from the heated rectangular surface varies with the height of the adiabatic covering surface above the heated surface. Previous studies of laminar natural convective heat transfer rates from uncovered (i.e., unshrouded) horizontal heated rectangular plane surfaces have indicated that if the mean side length of the surface, i.e.: a¼

lþw ðAR þ 1Þ ¼w 2 2

(40)

13

Free Convection: External Surface

571

where AR is the aspect ratio of the heated rectangular surface, l/w, is used as a length scale, the variations of the mean Nusselt number based on this length scale with the Rayleigh number also based on this length scale are essentially the same for all aspect ratios. Nusselt and Rayleigh numbers based on this mean length scale will therefore be used here. The solution then in general will depend on the following parameters: • The Rayleigh number, Raa, based on the mean side length, a, of the rectangular heated surface and the difference between the temperature of the undisturbed fluid well away from the system, Tf, and the temperature of the surface of the heated surface Tw, i.e.:
 βga3 T w  T f Raa ¼ να

(41)

• The aspect ratio of the rectangular heated surface, AR = l/w. • The dimensionless side lengths of the rectangular surrounding adiabatic surface and that of the rectangular adiabatic covering surface, Wout = wout/w and Lout = lout/w. • The dimensionless vertical distance of the adiabatic covering surface from the heated surface, H = h/w. It will be noted that the length scale used in defining H is the smaller side length of the rectangular heated surface and not the mean side length of the surface. • The Prandtl number, Pr. Results have only been obtained for a Prandtl number of 0.74 which is effectively the value for air. Results for Wout = 4 and Lout = AR + 3 will be presented here. The same basic characteristics as those displayed by the results presented here were found to exist in results obtained for other values of Wout and Lout. Dimensionless heated surface-to-adiabatic covering surface distances, H, between 0.125 and 1; rectangular heated surface aspect ratios, AR, of 1, 2, and 3; and Rayleigh numbers between approximately 104 and 1012 have been considered in the results presented here. 0 The mean heat transfer rate from the heated surface, Q , has been expressed in terms of a mean Nusselt number based on the mean side length, a, of the rectangular heated surface and on the difference between the temperature of the undisturbed fluid well away from the system, Tf, and the temperature of the heated surface, Tw, i.e.: 0

Qa
 Nua ¼ kwl T w  T f

(42)

where wl is the surface area of the heated rectangular surface. Because fixed values of Pr, Wout, and Lout are being considered here, Nua is a function of Raa, AR, and H.

572

P. H. Oosthuizen

The variations of Nusselt number with Rayleigh number for a fixed heated surface aspect ratio for various values of the dimensionless gap between the heated surface and the covering surface are shown in Figs. 55 and 56 for aspect ratios of 1, and 2, respectively. It will be seen from these figures that at both aspect ratios considered at the two larger dimensionless gaps, i.e., 0.5 and 1, the Nusselt number variation with Rayleigh number is similar at the two aspect ratios considered and is in fact close to that found to exist when there is no covering surface. However, at the two smaller dimensionless gaps considered, i.e., 0.25 and 0.125, the Nusselt number variation with Rayleigh number deviates significantly from that for the no covering surface case at the lower Rayleigh numbers considered, the presence of the covering surface under these conditions producing a significant decrease in the heat transfer rate.

Fig. 55 Variation of Nusselt number with Rayleigh number for heated element aspect ratio of 1 for dimensionless element to covering surface gaps of 1, 0.5, 0.25, and 0.125 (From Fig. 5 in Oosthuizen (2014a) Paper IMECE2014-36780 by permission of ASME)

Fig. 56 Variation of Nusselt number with Rayleigh number for heated element aspect ratio of 2 for dimensionless element to covering surface gaps of 1, 0.5, 0.25, and 0.125 (From Fig. 6 in Oosthuizen (2014a) Paper IMECE2014-36780 by permission of ASME)

13

4

Free Convection: External Surface

573

Bodies with a Wavy Surface

There are many practical situations in which the heat transfer rate by free convection from a surface must be increased, i.e., in which the heat transfer rate must be enhanced. One means of trying to accomplish this enhancement is to use surfaces that have a wavy shape. For example, consider vertical and horizontal plane surfaces. Figure 57 shows situations in which in order to try to increase (or enhance) the heat transfer rate, the plane surface has been replaced by a wavy surface, the size (“height”) of the waves being relatively small compared to the overall size of the surface. The increase in the heat transfer rate produced by using a wavy surface can result from the increase in the surface area that is exposed to the fluid to which the heat is being transferred and, possibly, due to the changes in the flow pattern over the surface resulting from the presence of the surface waves. Many wave shapes have been considered in the past but the most common wave shapes considered remain rectangular, triangular, and sinusoidal waves, these being shown in Fig. 58. The potential increase in the heat transfer rate resulting from the use of a wavy surface depends on the wave shape, the size of the waves, the flow situation considered, and the thermal boundary conditions at the surface. A number of studies of free convective heat transfer from bodies with wavyshaped surfaces have been undertaken. Some of the earlier numerical investigations Adiabatic Top Surface

Vertical Heated Wavy Surface

Adiabatic Bottom Surface Adiabatic Side Surface

Horizontal Heated Wavy Surface

Adiabatic Side Surface

Fig. 57 Vertical heated wavy surface (top) and horizontal heated wavy surface (bottom)

574

P. H. Oosthuizen

Fig. 58 Types of surface wave shapes considered

such as those by Yao (1983) and Moulic and Yao (1989), Tetsu et al. (1973), Molla et al. (2007), and Bhavnami and Bergles (1991) have dealt with laminar flow over a vertical isothermal surface having sinusoidal surface waves. Experimental studies of laminar flow over wavy vertical surfaces are described by Bhavnami and Bergles (1990) and Rahman (2001). A numerical study of heat transfer from a vertical surface having more complex sinusoidal waves is described in Yao (2006), and a study of a similar situation is described in Molla et al. (2007) except that in this study a uniform surface heat flux surface boundary condition was used. A numerical study of heat transfer from a vertical surface having triangular waves for conditions under which laminar, transitional, and turbulent flow exist is described in Oosthuizen (2010). There have also been a number of studies of free convective heat transfer from a horizontal wavy surface (Pretot et al. 2000, 2003; Siddiqa and Hossain 2013; Siddiqa et al. 2015). Numerical studies of heat transfer from a horizontal surface having rectangular waves and triangular waves for conditions under which laminar, transitional, and turbulent flow exist are described in Oosthuizen (2016d, e), respectively. Heat transfer from wavy surfaces is considered by Oosthuizen and Paul (2011), Oosthuizen and Garrett (2001a, 2001b), and Oosthuizen (2011). Studies of free convective heat transfer from cylinders and cones that have wavy surfaces have also been undertaken, for example, see the study described by Oosthuizen and Chow (1986). A review of studies of free convective heat transfer from wavy surfaced bodies is given in Oosthuizen (2016f). Consideration in this section will here be limited to free convective heat transfer from vertical and horizontal isothermal plane surfaces to which surface waves have been added.

4.1

Vertical Wavy Surfaces

Here attention will be given to a vertical isothermal wavy surface, this surface being imbedded in a larger flat adiabatic surrounding surface. Results for surface waves with triangular and sinusoidal shapes will be considered here. The results discussed in this section are those obtained in the numerical studies described in Oosthuizen (2010) and Oosthuizen and Paul (2012). In all cases, the mean heat transfer rate from the surface

13

Free Convection: External Surface

575

expressed in terms of a Nusselt number based on the height of the surface, l, will depend on the Rayleigh number, Ra, also based on the height of the heated surface and the difference between the temperature of the heated surface, Tw, and the temperature of the undisturbed fluid well away from the system, T0, on the number of waves, n, which determines the dimensionless base width W = w/l of the surface waves, on the dimensionless “height,” A = a/l, of the waves, and on the Prandtl number. Only results for a Prandtl number of 0.74, i.e., effectively the value for air, will be considered here. Now, as discussed before, the reason for adding a wavy surface is to increase the heat transfer rate relative to that which exists when there are no surfaces waves, i.e., which exists when the surface is flat, i.e., when the wave “height” is zero. For this reason, even when there is a wavy surface, the heat transfer rate is expressed per unit base area (see Fig. 59). Results for a vertical isothermal surface that has a series of waves having a triangular cross-sectional shape surface will first be considered, the surface considered having the shape shown in Fig. 60. Results will only be presented here for the case where the dimensionless base width of the waves, W, is equal to 0.095, i.e., for the case where there are ten waves. The effect of the dimensionless surface wave “height” on the heat transfer rate is illustrated by the variations of Nu with Rayleigh numbers for various vales of A shown in Figs, 61, 62, and 63. Figure 61 shows results for a non-wavy surface, i.e., for the case where A = 0. The effect of the dimensionless surface wave height on the heat transfer rate is further illustrated by the results shown in Fig. 64, this figure showing the variations of Nusselt number with A for various Rayleigh number values. Now the ratio of the actual surface area per unit depth, AT, to the base surface area per unit depth, AB, is given for the triangular wavy surface shape being considered here by: AT ¼ 21 A

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2 W 2 H þ 2

(43)

The variation of AT/AB with A given by this equation for W = 0.095 is shown in Fig. 65. If the ratio of the Nusselt number values for A greater than 0 to the Nusselt number values at the same Ra value for A = 0 indicated by the results given in Figs. 61, 62, 63, and 64 is compared to the AT/AB values, it will be seen that the increase in the Nusselt number produced by using the surface waves is much smaller than the increase in the surface area, this being particularly true at the smaller values of the Rayleigh Fig. 59 Actual and base (or projected) surface areas

Actual Surface Area

Base (or projected) Surface Area

576

P. H. Oosthuizen

Fig. 60 Vertical isothermal surfaces with triangular waves Adiabatic

Heated Wavy Surface

Adiabatic

Fig. 61 Variation of Nusselt number with Rayleigh number for a non-wavy vertical isothermal surface (From Fig. 4 in Oosthuizen (2010), Paper IMECE2010-38167, by permission of ASME)

13

Free Convection: External Surface

577

Fig. 62 Variation of Nusselt number with Rayleigh number for a vertical isothermal surface with triangular waves with a dimensionless wave height of 0.01191 (From Fig. 5 in Oosthuizen (2010), Paper IMECE2010-38167, by permission of ASME)

Fig. 63 Variation of Nusselt number with Rayleigh number for a vertical isothermal surface with triangular waves with a dimensionless wave height of 0.0476 (From Fig. 7 in Oosthuizen (2010), Paper IMECE2010-38167, by permission of ASME)

number considered when laminar flow exists. Thus, the waves are not very effective in enhancing the heat transfer rate under laminar flow conditions, they being however much more effective at the larger values of the Rayleigh number considered. Results obtained for the heat transfer rate from vertical isothermal surfaces with other wave shapes, e.g., sinusoidal waves, indicate also that it is also only at the larger values of the Rayleigh number that Nu/Nu0 approaches the value of AT/A. However, whereas at the higher Rayleigh numbers considered with a triangular wave shape the fractional increase in the Nusselt number remains less that the

578

P. H. Oosthuizen

Fig. 64 Variation of Nusselt number with dimensionless wave height for various Rayleigh number values (From Fig. 8 in Oosthuizen (2010), Paper IMECE201038167, by permission of ASME)

Fig. 65 Variation of the ratio of the actual surface area to the base area with dimensionless wave height

fractional increase in surface area with a sinusoidal wave shape, the fractional increase in the Nusselt number can somewhat exceed the fractional increase in the surface area.

4.2

Horizontal Wavy Surfaces

Free convective heat transfer from a horizontal upward-facing heated isothermal surface which is imbedded in a larger flat adiabatic surface will here be considered. The heated surface is, in general, covered with a series of equally spaced surface

13

Free Convection: External Surface Adiabatic

579 Isothermal

Adiabatic “Rough’’ Surface

Heated Element

Fig. 66 Flow situation involving a horizontal surface with rectangular surface waves (From Oosthuizen (2016f), Advances in Heat Transfer, Fig. 4, Page 269, Copyright (2016), with permission from Elsevier)

waves. In this section, results for a surface with rectangular-shaped waves will be considered, this situation being that shown in Fig. 66. The distance between the surface waves is equal to the width of the waves in the situation here considered. The heated surface is at a uniform temperature that is higher than that of the surrounding fluid. The flow has been assumed to be two dimensional in the plane shown in Fig. 66. The discussion given in this section is based on some of the results obtained in the numerical study described in Oosthuizen (2016d). With rectangular waves in the situation being considered, the actual total surface area per unit depth of the surface is, if l is the width of the heated surface, given by: AT ¼ l þ 2nh

(44)

But the projected surface area is given by: A¼l

(45)

Therefore, the ratio of the actual surface area to the projected area is given by:  AT l þ 2nh h h=w ¼ 1 þ 2n ¼ 1 þ 2n ¼ l l l=w A

(46)

l ¼ 2n  1 w

(47)

 AT 2n ¼1þ Hw 2n  1 A

(48)

Hence since:

it follows that:

Variations of AT/A with Hw for the cases where there are five waves and where there are ten waves are shown in Fig. 67.

580

P. H. Oosthuizen

Fig. 67 Variation of the ratio of the actual surface area to the base area with dimensionless wave height

Fig. 68 Variation of the ratio of the Nusselt number to the Nusselt number that would exist with a non-wavy surface at the same Rayleigh number with the dimensionless wave height for various Rayleigh number values for the case where there are five surface waves (From Oosthuizen (2016f), Advances in Heat Transfer, Fig. 16, Page 276, Copyright (2016), with permission from Elsevier)

Typical variations of Nu/Nu0 with Hw for various values of Ra for the cases where there are five and ten waves on the surface are shown in Figs. 68 and 69. Here again Nu0 is the Nusselt number that exists when H = 0 at the particular value of Ra being considered. The results given in Figs. 68 and 69 again show that the increase in the heat transfer rate is significantly less than the increase in the surface area produced by using a wavy surface. This is the result of the changes in the nature of the flow over the surface resulting from the introduction of the surface waves. The present results for flow over a surface with rectangular waves therefore indicate that the heat transfer rate

13

Free Convection: External Surface

581

Fig. 69 Variation of the ratio of the Nusselt number to the Nusselt number that would exist with a non-wavy surface at the same Rayleigh number with the dimensionless wave height for various Rayleigh number values for the case where there are ten surface waves (From Oosthuizen (2016f), Advances in Heat Transfer, Fig. 17 Page 276, Copyright (2016), with permission from Elsevier)

increase produced by the surface waves is significantly less than the increase in surface area resulting from the use of the surface waves. Again similar conclusions can be drawn from results obtained with other wave shapes.

4.3

Free Convection from Cylindrical Wavy Surfaces

The discussion given above was concerned with free convective heat transfer from what are basically plane surfaces that have a wavy surface, the size of the waves being comparatively small compared to the overall size of the surface. In this section, a brief discussion of an experimental study of the mean natural convective heat transfer rate from a cylindrical body that has a wavy surface will be presented. The discussion given in this section is based on the results obtained in the experimental study described in Oosthuizen and Chow (1986). The basic situation considered is shown in Fig. 70. Results were obtained in this study for angles of inclination, φ (see Fig. 70), ranging from 0 , when the cylinder is horizontal, to 90 , when the cylinder is vertical. The dimensions used in defining the geometry are shown in Fig. 71. The experimental results obtained in the study here being discussed were generated using the lumped thermal capacity method. The models were made from solid aluminum with a series of thermocouples mounted inside the models to allow their temperature to be determined. In a test, the model was heated and its temperature variation with time then measured as it cooled when placed in stagnant air. The mean heat transfer rate was then determined from the measured rate of temperature change with time by noting that the Biot number was small, i.e., by assuming that the spatial changes in temperature within the model at any instant of time were negligibly small

582

P. H. Oosthuizen

Fig. 70 Inclined cylindrical body with a wavy surface (Reprinted from Oosthuizen (2016f), Advances in Heat Transfer, Fig. 64, Page 308, Copyright (2016), with permission from Elsevier)

Angle of Inclination, j Horizontal

Fig. 71 Dimensions of inclined cylindrical body with a wavy surface (Reprinted with modification from Oosthuizen and Chow, Copyright (1986), with permission from Begell House, Inc.)

h Cylindrical Body

d

compared to the temperature difference between the model and the air to which the heat from the model was being transferred. Tests were undertaken with models having dimensionless surface wave heights H = h/d of 0.333, 0.167, 0.088, and 0. For each of these H values, tests were undertaken with models having dimensionless length-to-diameter ratios, L = l/d, of 3.1, 4.5, 5.8, and 7.9. Tests were therefore undertaken with 16 different model sizes. All of the models used had the same mean diameter, d, of 38 mm. The number of waves on these models ranged from 5 to 48. The estimated uncertainty in the experimentally determined heat transfer rates was estimated to be less than 5%. Because the Prandtl number of the air over the temperature range covered in the tests was effectively constant and, because the temperature range covered in all tests was the same and because the models all had the same mean diameter, the mean Rayleigh number was the same in all tests, its value being 2.2  105. Therefore, the mean Nusselt number depends on the dimensionless “height” of the surface waves, H = h/d; on the dimensionless length of the model, L = l/d; and on the angle of inclination of the model, φ, i.e.: Nu ¼ functionðH, L, φÞ

(49)

Here the mean Nusselt number, Nu, is based on the heat transfer rate per unit mean surface area, i.e., π d l.

13

Free Convection: External Surface

583

Fig. 72 Variation of Nusselt number for a cylindrical body with a wavy surface with angle of inclination for various length-to-diameter ratios for a dimensionless wave height of 0.333 (Reprinted with modification from Oosthuizen and Chow, Copyright (1986), with permission from Begell House, Inc.)

Fig. 73 Variation of Nusselt number for a cylindrical body with a wavy surface with angle of inclination for various length-to-diameter ratios for a dimensionless wave height of 0.167 (Reprinted with modification from Oosthuizen and Chow, Copyright (1986), with permission from Begell House, Inc.)

Variations of mean Nusselt number with angle of inclination for various values of H and L are shown in Figs. 72, 73, 74, and 75. It will be seen from these results that, as is to be expected, the mean heat transfer rate is highest for a given set of conditions when the cylinder is horizontal and lowest when the cylinder is vertical. It will also be seen from these figures that the mean heat transfer rate decreases as the dimensionless cylinder length, L, increases. Finally, it will be seen from these figures that in all cases, the mean heat transfer rate increases as H increases, i.e., in all cases, the presence of the surface waves does increase the mean transfer rate. This can be better illustrated by considering the variation of Nu/Nu0 with the system parameters, Nu

584

P. H. Oosthuizen

Fig. 74 Variation of Nusselt number for a cylindrical body with a wavy surface with angle of inclination for various length-to-diameter ratios for a dimensionless wave height of 0.088 (Reprinted with modification from Oosthuizen and Chow, Copyright (1986), with permission from Begell House, Inc.)

Fig. 75 Variation of Nusselt number for a cylindrical body with angle of inclination for various length-to-diameter ratios for a dimensionless wave height of 0, i.e., for the smooth surface case (Reprinted with modification from Oosthuizen and Chow, Copyright (1986), with permission from Begell House, Inc.)

being the mean Nusselt number for the cylinder with a wavy surface and Nu0 being the value of the Nusselt number existing in the case where H = 0 (a non-wavy surface) for the same conditions. Typical variations of Nu/Nu0 with angle of inclination for various values of H and L are shown in Figs. 76, 77, and 78. These results again show that the fractional increase in the Nusselt number is at most of the same order of magnitude as the fractional increase in the surface area resulting from the introduction of the surface waves.

13

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585

Fig. 76 Variation of the ratio of the Nusselt number to the Nusselt number that would exist with a non-wavy surface at the same length-to-diameter ratio and with the same angle of inclination for various length-to-diameter ratios for a dimensionless wave height of 0.333 (Based on Oosthuizen and Chow, Copyright (1986), with permission from Begell House, Inc.)

Fig. 77 Variation of the ratio of the Nusselt number to the Nusselt number that would exist with a non-wavy surface at the same length-to-diameter ratio and with the same angle of inclination for various length-to-diameter ratios for a dimensionless wave height of 0.167 (Based on Oosthuizen and Chow, Copyright (1986), with permission from Begell House, Inc.)

5

Short Vertical Cylinders with an Exposed Upper Surface

In this section, the free convective heat transfer rate from a relatively short isothermal cylinder mounted on a horizontal adiabatic base will be considered. The cylinder has an “exposed” upper surface that is at the same temperature as the vertical walls of the cylinder. The flow situation considered is therefore shown in Fig. 79. The interest in this situation arises from the fact that it is an approximate model of a number of

586

P. H. Oosthuizen

Fig. 78 Variation of the ratio of the Nusselt number to the Nusselt number that would exist with a non-wavy surface at the same length-to-diameter ratio and with the same angle of inclination for various length-to-diameter ratios for a dimensionless wave height of 0.088 (Based on Oosthuizen and Chow, Copyright (1986), with permission from Begell House, Inc.)

Fig. 79 Short circular cylinder with an exposed top surface situation considered (From Fig. 1 in Oosthuizen (2007) Paper IMECE200742711, by permission of ASME)

Heated Isothermal Surfaces R

h

Adiabatic Base

arrangements that occur in the free convective cooling of some electrical and electronic equipment. A review of heat transfer in such a situation is given in Oosthuizen and Kalendar (2014), Oosthuizen (2013b, c), and Scott and Oosthuizen (2000). Attention will first be given to the heat transfer from a circular cylinder (see Fig. 70), results for this situation being given in Oosthuizen (2007) and Kalendar et al. (2011). The mean heat transfer rate from the entire cylinder, i.e., the heat transfer rate averaged over both the vertical cylindrical surface and the horizontal plane upper surface of the cylinder will be expressed in terms of a mean Nusselt number based on the height, l, of the cylinder and the difference between the temperature of the surface of the cylinder, Tw, and the temperature of the surrounding undisturbed fluid far from the cylinder, T0, i.e., in terms of:

13

Free Convection: External Surface

587

Fig. 80 Variation of the mean Nusselt number for the entire cylinder surface with dimensionless cylinder radius for various Rayleigh number values (From Fig. 4 in Oosthuizen (2007) Paper IMECE2007-42711 by permission of ASME)

Nu ¼

q0 l k ðT w  T 0 Þ

(50)

This Nusselt number will depend on the Rayleigh number based on the height, l, of the cylinder and the temperature difference Tw – T0, on the dimensionless radius of the cylinder R = r/l, and on the Prandtl number. Attention will only be given here to the case where the surrounding fluid is air at near standard atmospheric conditions and the Prandtl number therefore effectively constant and equal to approximately 0.74. Typical variations of Nu with R for various values of Ra are shown in Fig. 80. It will be seen from this figure that at larger values of R, the Nusselt number tends to a constant value, the flow over the vertical cylindrical side wall of the cylinder under these conditions being effectively the same as that over a vertical plane surface. In order to try to develop a correlation equation for the mean heat transfer rate in the situation here being considered, the mean heat transfer rates from the vertical cylindrical side wall of the cylinder and from the horizontal plane upper surface of the cylinder will be separately considered, these mean heat transfer rates being expressed in terms of Nusselt numbers Nuc and Nut, respectively. Typical variations of Nuc, Nut, and Nu with R for various values of Ra are shown in Figs. 81, 82, 83, and 84. Now if q0 , q0c , and q0 t are the mean heat transfer rates for the entire cylinder, for the vertical side wall of the cylinder, and for the horizontal upper surface of the cylinder, then: q0 A ¼ q0c Ac þ q0 t At

(51)

where A, Ac, and At are the surface areas of the entire cylinder surface, the vertical side surface, and the horizontal top surface, respectively. From this, it follows that:

588

P. H. Oosthuizen

Fig. 81 Variations of the mean Nusselt number for the entire cylinder surface, for the vertical cylindrical side surface of the cylinder, and for the top surface of the cylinder with dimensionless cylinder radius for a Rayleigh number of 104 (From Fig. 5 in Oosthuizen (2007) Paper IMECE2007-42711, by permission of ASME)

Fig. 82 Variations of the mean Nusselt number for the entire cylinder surface, for the vertical cylindrical side surface of the cylinder, and for the top surface of the cylinder with dimensionless cylinder radius for a Rayleigh number of 105 (From Fig. 6 in Oosthuizen (2007) Paper IMECE2007-42711, by permission of ASME)

Nu ¼ Nuc

  Ac At þ Nut A A

(52)

Therefore, since Ac = 2πrl = 2πRl2, At = πr2 = πR2l2, and A = Ac + At, it follows from the above equation that:  Nu ¼ Nuc

 2 R þ Nut 2þR 2þR

(53)

13

Free Convection: External Surface

589

Fig. 83 Variations of the mean Nusselt number for the entire cylinder surface, for the vertical cylindrical side surface of the cylinder, and for the top surface of the cylinder with dimensionless cylinder radius for a Rayleigh number of 106 (From Fig. 7 in Oosthuizen (2007) Paper IMECE2007-42711, by permission of ASME)

Fig. 84 Variations of the mean Nusselt number for the entire cylinder surface, for the vertical cylindrical side surface of the cylinder, and for the top surface of the cylinder with dimensionless cylinder radius for a Rayleigh number of 107 (From Fig. 8 in Oosthuizen (2007) Paper IMECE2007-42711, by permission of ASME)

Now, the heat transfer rate from the vertical side surface is equal to the heat transfer rate from a vertical plane surface of the same height as the cylinder plus a term that accounts for the effects of the curvature of the side surface. This latter term will depend on the value of δ/r where δ is a measure of the boundary layer thickness. Therefore, since δ will be dependent on l/Ra0.25, it follows that δ/r will be dependent on:

590

P. H. Oosthuizen

ξ¼

1 R Ra0:25

(54)

Using the results given in Figs. 81, 82, 83, and 84 and similar results for other Rayleigh numbers gives the following correlation equation for the heat transfer rate from the vertical cylindrical surface: Nuc ¼ 0:59 þ 0:28ξ, Ra0:25

(55)

this equation applying for the Prandtl number of air at near standard ambient conditions. The first term on the right-hand side of this equation is the value of the Nusselt number that would exist if curvature effects were negligible, and the second term on the right-hand side represents the contribution of the curvature effects to the overall Nusselt number value. Turning next to the heat transfer rate from the plane horizontal top surface of the cylinder, the length scale that will influence this heat transfer rate will be the cylinder radius, r. Therefore, introducing Rayleigh and Nusselt numbers expressed in terms of r, i.e., Rar and Nur, the results given in Figs. 81, 82, 83, and 84 and other similar results give the following equation for the heat transfer rate from the horizontal upper cylindrical surface: Nur ¼ 0:45 Ra0:25 r

(56)

Nut ¼ 0:45Ra0:25 =R0:25

(57)

from which it follows that:

this equation also applying for the Prandtl number of air at near standard ambient conditions. Equations 53, 55, and 57 allow the heat transfer rate from a short circular cylinder with an exposed top surface to be determined, these equations together giving:  Nu ¼

   2ð0:59 þ 0:28 ξÞ 0:45 R0:75 þ ð 2 þ RÞ ð2 þ R Þ

(58)

This equation gives results that are in good agreement with the numerical results (Oosthuizen 2007) and with available experimental results (Scott and Oosthuizen 2000). The free convective heat transfer from a relatively short isothermal cylinder having a square cross-sectional shape which is mounted on a horizontal adiabatic base will next be considered; see Kalendar and Oosthuizen (2010a, 2013) and Oosthuizen (2008, 2013a). The cylinder has an “exposed” upper surface that is at the same temperature as the vertical walls of the cylinder. The flow situation considered is therefore shown in Fig. 85.

13

Free Convection: External Surface

591

Fig. 85 Short square cylinder with an exposed top surface situation considered (Based on Kalendar and Oosthuizen (2013), Fig. 1)

Again the mean heat transfer rate from the entire cylinder, i.e., the heat transfer rate averaged over both the vertical side surface and the horizontal plane upper surface of the cylinder, will be expressed in terms of a mean Nusselt number based on the height, l, of the cylinder and the difference between the temperature of the surface of the cylinder, Tw, and the temperature of the surrounding undisturbed fluid far from the cylinder, T0, i.e., in terms of: Nu ¼

q0 l k ðT w  T 0 Þ

(59)

This Nusselt number will depend on the Rayleigh number based on the height, l, of the cylinder and the temperature difference Tw – T0, on the dimensionless side length of the cylinder W = w/l , and on the Prandtl number. Attention will again only be given to the case where the surrounding fluid is air at near standard atmospheric conditions and the Prandtl number therefore effectively constant and equal to approximately 0.74. Again in order to try to develop a correlation equation for the mean heat transfer rate for the entire cylinder in the situation here being considered, this mean heat transfer rate for the entire surface and the mean heat transfer rates from the vertical side walls of the cylinder and from the horizontal plane upper surface of the cylinder will be separately considered, these mean heat transfer rates being expressed in terms of Nusselt numbers Nu, Nus, and Nut respectively. Typical variations of Nus, Nut, and Nu with W for four values of Ra are shown in Figs. 86, 87, 88, and 89. Now if q0 , q0s , and q0 t are the mean heat transfer rates for the entire cylinder, for the vertical side walls of the cylinder, and for the horizontal upper surface of the cylinder, then: q0 A ¼ q0s As þ q0 t At

(60)

where A, As, and At are the surface areas of the entire cylinder surface, the vertical side surfaces, and the horizontal top surface, respectively. From this, it follows that:

592

P. H. Oosthuizen

Fig. 86 Variations of the mean Nusselt number for the entire square cylinder surface, for the vertical side surfaces of the cylinder, and for the top surface of the cylinder with dimensionless cylinder width for a Rayleigh number of 104 (From Fig. 4 in Oosthuizen (2008) Paper HT2008-56025, by permission of ASME)

Fig. 87 Variations of the mean Nusselt number for the entire square cylinder surface, for the vertical side surfaces of the cylinder, and for the top surface of the cylinder with dimensionless cylinder width for a Rayleigh number of 105 (From Fig. 5 in Oosthuizen (2008) Paper HT2008-56025, by permission of ASME)

  As At Nu ¼ Nus þ Nut A A

(61)

Therefore, since As = 4wl , At = w2, and A = 4wl + w2, it follows from the above equation that:  Nu ¼ Nus

4 4þW



 þ Nut

W 4þW

(62)

However, if the results given in Figs. 86, 87, 88, and 89 are considered, it will be seen that for W values less than approximately 0.6, the total heat transfer rate is

13

Free Convection: External Surface

593

Fig. 88 Variations of the mean Nusselt number for the entire square cylinder surface, for the vertical side surfaces of the cylinder, and for the top surface of the cylinder with dimensionless cylinder width for a Rayleigh number of 106 (From Fig. 6 in Oosthuizen (2008) Paper HT2008-56025, by permission of ASME)

Fig. 89 Variations of the mean Nusselt number for the entire square cylinder surface, for the vertical side surfaces of the cylinder, and for the top surface of the cylinder with dimensionless cylinder width for a Rayleigh number of 107 (From Fig. 7 in Oosthuizen (2008) Paper HT2008-56025, by permission of ASME)

within 8% of the heat transfer rate from the side surfaces, i.e., under these conditions, the contribution of the heat transfer from the top surface to the total heat transfer rate from the cylinder is negligible. In deriving a correlation equation for the heat transfer rate from a square cylinder for smaller values of W, the heat transfer rate from the upper surface will therefore be neglected. Now, the heat transfer rate from the vertical side surface is equal to the heat transfer rate from a vertical plane surface of the same height as the cylinder plus a term that accounts for the effects of the interaction of the flows over adjoining side surfaces. This latter term will depend on the value of δ/w where δ is a measure of the boundary layer thickness. Therefore, since δ will be dependent on l /Ra0.25, it follows that δ/w will be dependent on:

594

P. H. Oosthuizen

ξ¼

1 W Ra0:25

(63)

Using the results given in Figs. 86, 87, 88, and 89 and similar results for other Rayleigh numbers gives the following correlation equation for the heat transfer rate from a square cylinder for W values less than approximately 0.6: Nuc ¼ 0:42 þ 0:21ξ0:7 Ra0:25

(64)

this equation applying for the Prandtl number of air at near standard ambient conditions. This equation gives results that are in good agreement with the computed heat transfer rates.

6

External Free Convection in Systems Involving a Nanofluid

The use of nanofluids to produce an increase (enhancement) of the heat transfer rate has received extensive attention in recent times, e.g., see Das et al. (2006), Kakaç and Pramuanjaroenkij (2009), Terekhov et al. (2010a, b), Sergis and Hardalupas (2011), and Sarkar (2011). Nanofluids are liquids in which solid nanometer-sized particles are suspended, the suspension being relatively dilute. These nanoparticles typically have a length scale of approximately 1–100 nm and are made from materials which have a high thermal conductivity compared to that of the liquid in which they are suspended. Both metallic and nonmetallic nanoparticle materials are used, typical materials being Al2O3, CuO, Cu, SiO, and TiO2. Typical liquids in which the nanoparticles are suspended are water, oil, and ethylene glycol. Because the nanoparticles are so small, they do not settle out of the liquid in which they are suspended, and when the fluid in which they are suspended is flowing, they basically flow along the same path as that taken by the adjacent fluid particles. The presence of the nanoparticles results in the nanofluid having a much higher thermal conductivity than that of the base liquid, and this tends to increase the convective heat transfer rate compared to that which would exist when the base fluid alone is used. However, it must be realized that the presence of the nanoparticles also causes changes in the other properties of the nanofluid, for example, the viscosity. In numerically investigating nanofluid convective heat transfer, two basic types of model are commonly used. Broadly these models are as follows: Single-Phase Model: Although nanofluids are solid-liquid mixtures, the most widely used numerical approach for studying nanofluid convective heat transfer assumes that, because the solid particles are so small and because their concentration is so low, the nanofluid can be treated as a single-phase fluid with, as discussed above, the solid nanoparticles being assumed to move with the same velocity as the adjacent fluid. The properties of the nanofluid are based on a combination of the properties of the fluid and of the nanoparticles and are dependent on the

13

Free Convection: External Surface

595

concentration of the nanoparticles. For example (Das et al. 2006), it is sometimes assumed that the specific heat of the nanofluid is given by: cpnf ¼ ð1  ϕÞcpf þ ϕ cps

(65)

where cpnf is the specific heat of the nanofluid, cpf is the specific heat of the fluid, cps is the specific heat of the nanoparticles, and ϕ is nanoparticle volumetric fraction. The nanofluid is then treated as a conventional single-phase fluid that obeys the conventional boundary conditions, and the solution is obtained by solving the conventional single-phase flow equations. Two-Phase Model: The single-phase model does not allow accurate account to be taken of the effects of Brownian motion, of the potential gathering of nanoparticles together, and of the effects of slip between the fluid and nanoparticles to be taken into account. Hence, two-phase models have been introduced in which the nanofluid is treated as a two-phase mixture in which the flow of the fluid and the nanoparticles is separately considered. Various other models of convective heat transfer involving nanofluids have also been proposed. While it has received less attention than forced convective heat transfer, free convective heat transfer involving nanofluids has received some attention, e.g., see Wen and Ding (2005, 2006), Pakravan and Yaghoubi (2013), Ahmed and Abd (2013), Haddad et al. (2012), Polidori et al. (2007), Ravipati (2008), and Zaraki et al. (2015). However, most of the available studies of free convection involving nanofluids have been concerned with flows in enclosures. The relatively little attention that has been given to external free convection involving a nanofluid has been mainly concerned with heat transfer from a wide vertical flat surface. The enhancement of free convective heat transfer in situations involving the use of a nanofluid remains controversial. Most numerical studies indicate that the use of the nanofluid does produce an increase in the heat transfer rate compared to that which would exist with a conventional fluid although the increase is often predicted to be relatively modest. However, experimental studies indicate that the use of the nanofluid often results in a decrease in the free convective heat transfer rate compared to that which would exist with a conventional fluid in the same situation. There is thus the need for more numerical and experimental studies of external free convective heat transfer in situations involving a nanofluid.

7

Conclusions

The main conclusions that can be drawn from the material reviewed in this chapter are: 1. With relatively narrow vertical plates, the mean Nusselt number at a particular Rayleigh number increases as the plate width-to-height ratio decreases, this effect increasing in magnitude as the Rayleigh number decreases. This increase in the

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heat transfer rate is associated with the three-dimensional flow that exists near the vertical edges of the plate. Correlation equations for predicting the free convective heat transfer rate from narrow plates were derived. Equations 8 and 13 can be used to predict the heat transfer rate from a vertical narrow plate, and Eqs. 15 and 17 can be used to predict the heat transfer rates from vertical plates that are horizontally adjacent to each other or vertically adjacent to each other. 2. It was shown that the results for the free convective heat transfer rates from horizontal heated surfaces having shapes ranging from simple to complex could all be represented by a single relationship between the Nusselt number and the Rayleigh number provided the Nusselt and Rayleigh number were based on a characteristic length m = 4A/P where A is the surface area of the heated horizontal surface and P is its perimeter. 3. The results presented here show that free convective heat transfer rates can be increased by using a wavy surface. However, this increase in the heat transfer rate, in most cases, is only significantly large at higher Rayleigh number values. The results presented for the heat transfer rate from wavy surfaces also show that the increase in the heat transfer rate at higher Rayleigh numbers is, at most, only slightly larger than the increase in the surface area resulting from the use of the surface waves. The results discussed also indicate that the shape of the surface waves used, i.e., rectangular, triangular, or sinusoidal, has a relatively small effect on the increase in the heat transfer rate resulting from the use of the surface waves. 4. Correlation equations for the free convective heat transfer rate from vertical cylinders having circular and square cross-sectional shapes and an exposed top surface were presented.

8

Cross-References

▶ Enhancement of Convective Heat Transfer ▶ Free Convection: Cavities and Layers ▶ Full-Coverage Effusion Cooling in External Forced Convection: Sparse and Dense Hole Arrays ▶ Numerical Methods for Conduction-Type Phenomena ▶ Single-Phase Convective Heat Transfer: Basic Equations and Solutions ▶ Thermal Transport in Micro- and Nanoscale Systems ▶ Turbulence Effects on Convective Heat Transfer

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Sparrow EM, Abraham JP, Gorman JM (eds) Advances in heat transfer. Elsevier, Oxford, pp 261–317 Oosthuizen PH, Chow K (1986) An experimental study of free convective heat transfer from short cylinders with ‘wavy’ surfaces. In: Proceedings of the 8th international heat transfer conference, San Francisco, 17–22, Aug 1986 Oosthuizen PH, Garrett M (2001a) A numerical study of natural convective heat transfer from an inclined plate with a ‘wavy surface’. In: Proceedings of the 2001 ASME national heat transfer conference, Anaheim, 10–12 June 2001 Oosthuizen PH, M. Garrett (2001b) A numerical study of three-dimensional natural convective heat transfer from a plate with a ‘wavy’ surface. In: Proceedings of the 2001 ASME international. mechanical engineering congress & exposition, New York City, 11–16 Nov 2001 Oosthuizen PH, Kalendar AY (2013) Natural convective heat transfer from narrow plates. In: Kulacki FA (ed) Springer briefs in applied sciences and technology, thermal engineering and applied science. Springer, New York. https://doi.org/10.1007/978-1-4614-5158-7 Oosthuizen PH, Kalendar AY (2014) Natural convective heat transfer from short inclined cylinders. In: Kulacki FA (ed) Springer briefs in applied sciences and technology, thermal engineering and applied science. Springer, New York. https://doi.org/10.1007/978-3-319-02459-2 Oosthuizen PH, Kalendar AY (2015a) A numerical study of natural convective heat transfer from a pair of adjacent horizontal isothermal square elements embedded in an adiabatic surface-effect of element spacing on heat transfer rate. In: Proceedings of the 11th international conference on heat transfer, fluid mechanics and thermodynamics (HEFAT 2015), Skukuza, 20–23 July 2015 Oosthuizen PH, Kalendar AY (2015b) Laminar and turbulent natural convective heat transfer from a horizontal isothermal circular element with an unheated inner circular section. In: Proceedings of the CFD Society of Canada 23rd annual conference, Waterloo, 7–10 June 2015 Oosthuizen PH, Kalendar AY (2016) A numerical study of the simultaneous natural convective heat transfer from the upper and lower surfaces of a thin isothermal horizontal circular plate. In: Proceedings of the 2016 ASME international mechanical engineering congress and exposition (IMECE2016), Phoenix, 11–17 Nov 2016 Oosthuizen PH, Naylor D (1999) Introduction to convective heat transfer analysis. McGraw-Hill, New York Oosthuizen PH, Paul JT (2006) Natural convective heat transfer from a narrow isothermal vertical flat plate. In: Proceedings of the 9th AIAA/ASME joint thermophysics and heat transfer, San Francisco, 5–8 June 2006 Oosthuizen PH, Paul JT (2007a) Natural convective heat transfer from a recessed narrow vertical flat plate with a uniform heat flux at the surface. In: Proceedings of the 5th international conference on heat transfer, fluid mechanics and thermodynamics (HEFAT2007), Sun City, South Africa, 1–4 July 2007 Oosthuizen PH, Paul JT (2007b) Effect of edge conditions on natural convective heat transfer from a narrow vertical flat plate with a uniform surface heat flux. In: Proceedings of the ASME international mechanical engineering congress and exposition (IMECE2007) volume 8: heat transfer, fluid flows, and thermal systems, Parts A and B, Seattle, 11–15 Nov 2007 Oosthuizen PH, Paul JT (2007c) Natural convective heat transfer from a narrow vertical isothermal flat plate with different edge conditions. In: Proceedings of the 15th annual meeting of the computational fluid dynamics Society of Canada, Toronto, 27–31 May 2007 Oosthuizen PH, Paul JT (2007d) Natural convective heat transfer from a narrow isothermal vertical flat plate with a uniform heat flux at the surface. In: Proceedings of the 2007 ASME/JSME thermal engineering summer heat transfer conference (HT2007) Vancouver, 8–12 July 2007 Oosthuizen PH, Paul JT (2010) Natural convective heat transfer from a narrow vertical flat plate with a uniform surface heat flux and with different plate edge conditions. Front Heat Mass Transf. https://doi.org/10.5098/hmt.v1.1.3006 Oosthuizen PH, Paul JT (2011) A numerical study of natural convective heat transfer from an inclined isothermal plate having a square wave surface. In: Proceedings of the ASME 2011

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international mechanical engineering congress and exposition, volume 10: heat and mass transport processes, parts A and B, Denver, 11–17 Nov 2011 Oosthuizen PH, Paul JT (2012) A numerical study of natural convective heat transfer from an inclined isothermal plate with a ‘sinusoidally wavy’ surface. In: Proceedings of the 12th ICHMT international symposium on advances in computational heat transfer (CHT-12), Bath, 1–6 July 2012 Pakravan HA, Yaghoubi M (2013) Analysis of nanoparticles migration on natural convective heat transfer of nanofluids. Int J Therm Sci 68:79–93 Plumb OA, Kennedy LA (1977) Application of a k-e turbulence model to natural convection from a vertical isothermal surface. J Heat Transf 99(1):79–85 Polidori G, Fohanno S, Nguyen CT (2007) A note on heat transfer modelling of Newtonian nanofluids in laminar free convection. Int J Therm Sci 46(8):739–744 Prétot S, Miriel J, Bailly Y, Zeghmati B (2003) Visualization and simulation of the naturalconvection flow above horizontal wavy plates. Num Heat Transf Part A: Appl: Int J Comput Methodol 43(3):307–325 Prétot S, Zeghmati B, Caminat P (2000) Influence of surface roughness on natural convection above a horizontal plate. Adv Eng Softw 31(10):793–801 Rahman SU (2001) Natural convection along vertical wavy surfaces: an experimental study. Chem Eng J 84(3):587–591 Raithby GD, Hollands KGT (1985) Natural convection, Chapter 6. In: Rohsenow WM, Hartnett JP, Ganić EN (eds) Handbook of heat transfer fundamentals, 2nd edn. McGraw-Hill, New York Ravipati D (2008) Free convection along a vertical wavy surface in a nanofluid. MSc (Mechanical Engineering) thesis, Cleveland State University Sahraoui M, Kaviany M, Marshall H (1990) Natural convection from horizontal disks and rings. J Heat Transf 112:110–116 Sarkar J (2011) A critical review on convective heat transfer correlations of nanofluids. Renew Sust Energ Rev 15(6):3271–3277 Savill AM (1993) Evaluating turbulence model predictions of transition. An ERCOFTAC special interest group project. Appl Sci Res 51(1–2):555–562 Schmidt RC, Patankar SV (1991) Simulating boundary layer transition with low-Reynolds-number k-e turbulence models: part 1- an evaluation of prediction characteristics. ASME J Turbomach 113(1):10–17 Scott DA, Oosthuizen PH (2000) An experimental study of three-dimensional mixed convective heat transfer from short vertical cylinders in a horizontal forced flow. In: Proceedings of the IDMME’2000 and Canadian society for mechanical engineering (CSME) Forum 2000, Montreal, 6–19 May 2000 Sergis A, Hardalupas Y (2011) Anomalous heat transfer modes of nanofluids: a review based on statistical analysis. Nanoscale Res Lett 6(1):1–37 Siddiqa S, Hossain MA (2013) Natural convection flow over wavy horizontal surface. Adv Mech Eng. https://doi.org/10.1155/2013/743034 Siddiqa S, Hossain MA, Gorla RSR (2015) Natural convection flow of viscous fluid over triangular wavy horizontal surface. Comput Fluids 106(5):130–134 Sparrow EM (1955) Laminar free convection on a vertical plate with prescribed nonuniform wall heat flux or prescribed nonuniform wall temperature. National Advisory Committee for Aeronautics, Lewis Flight Propulsion Lab. NACA-TN-3508. Available via NASA Technical Notes Reports Server Sparrow EM, Gregg JL (1956) Laminar free convection from a vertical flat plate with uniform surface heat flux. Trans ASME 78:435–440 Sparrow EM, Gregg (1958) Similar solutions for free convection from a nonisothermal vertical plate. Trans ASME 80:379–386 Szewczyk AA (1962) Stability and transition of the free convection layer along a vertical flat plate. Int J Heat Mass Transf 5:903–914

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Free Convection: Cavities and Layers

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Andrey V. Kuznetsov and Ivan A. Kuznetsov

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Some Classical Results of Studies on Heat Transfer in Cavities. Natural Convection Driven by a Temperature Variation in a Fluid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Rectangular Cavity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Three-Dimensional Cavities and Cavities with Shapes Other than Rectangular . . . . 3 Some Emerging Results on Heat Transfer in Cavities. Situations When There Are Other Contributors to the Buoyancy Force, in Addition to the Density Variation Due to a Temperature Change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Double-Diffusive Convection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 MHD Convection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Mixed Convection in Lid-Driven Enclosures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Non-Newtonian Fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Cavities Filled with Nanofluids. Single-Phase Modeling Approach . . . . . . . . . . . . . . . . . . . . . . . 6 Ferrofluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Research on the Onset of Convection Instability in Natural Convection . . . . . . . . . . . . . . . . . . 7.1 Recent Developments in Classical Natural Convection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Emerging Topics: Links between Internal Natural Convection and Bio- and Nanofluidics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Bioconvection: Macroscopic Motion of a Fluid Caused by Many Mesoscale Swimmers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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A. V. Kuznetsov (*) Department of Mechanical and Aerospace Engineering, North Carolina State University, Raleigh, NC, USA e-mail: [email protected] I. A. Kuznetsov Perelman School of Medicine, University of Pennsylvania, Philadelphia, PA, USA Department of Bioengineering, University of Pennsylvania, Philadelphia, PA, USA e-mail: [email protected] # Springer International Publishing AG, part of Springer Nature 2018 F. A. Kulacki (ed.), Handbook of Thermal Science and Engineering, https://doi.org/10.1007/978-3-319-26695-4_9

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Abstract

Natural convection in cavities and layers has many applications. These applications range from heat transfer in refrigeration chambers, solar chimneys, and ceiling cooling systems to passive safety systems for nuclear reactor cooling and thermal energy storage with phase change materials. The goal of this chapter is to give an overview of some classical results on natural convection in cavities and layers as well as to introduce some emerging topics that present recent extensions of natural convection models, such as bio-thermal convection, nanofluidic bioconvection, and bioconvective sedimentation. The latter topics manifest growing interest of the research community in bio-related issues and show some areas in which classical thermofluid researchers can contribute to geophysics, nanofluidics, and biomechanics. Interdisciplinary issues that are relevant to these latter topics range from geophysical applications (such as spring contamination) and microfluidics (such as leftward flow in a nodal cavity that determines left-right symmetry breaking in a developing embryo) to zoological fluid dynamics (rotation of pond snail embryos to improve their oxygen supply) and biomimetic devices (e.g., research on Janus particles).

1

Introduction

Natural convection in cavities and layers has many applications. These applications range from heat transfer in refrigeration chambers (Laguerre and Flick 2004), solar chimneys (Zhai et al. 2011), and ceiling cooling systems (Kim et al. 2015) to passive safety systems for nuclear reactor cooling (Wibisono et al. 2013) and thermal energy storage with phase change materials (Farid et al. 2004; Zalba et al. 2003). The goal of this chapter is to give an overview of some classical results of studies on natural convection in cavities and layers as well as to introduce some emerging topics that present recent extensions of natural convection models, such as biothermal convection, nanofluidic bioconvection, and bioconvective sedimentation. The latter topics manifest growing interest of the research community in bio-related issues and show some areas in which classical thermofluid researchers can contribute to geophysics, nanofluidics, and biomechanics. Interdisciplinary issues that are relevant to these latter topics range from geophysical applications (such as spring contamination, Nebbache et al. 1997) and microfluidics (such as leftward flow in a nodal cavity that determines left-right symmetry breaking in a developing embryo, Kuznetsov et al. 2014) to zoological fluid dynamics (rotation of pond snail embryos to improve their oxygen supply, Nield and Kuznetsov 2013) and biomimetic devices (see, for example, research on Janus particles, DiazMaldonado and Cordova-Figueroa 2015, Michelin and Lauga 2014, and Shklyaev et al. 2014).

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2

605

Some Classical Results of Studies on Heat Transfer in Cavities. Natural Convection Driven by a Temperature Variation in a Fluid

The most common and most well-studied situation of natural convection is when it is caused by a temperature variation in the fluid. The temperature variation, which leads to a density variation and hence a buoyancy force, typically results from asymmetry in boundary conditions (e.g., from different temperatures maintained at the enclosure walls).

2.1

Rectangular Cavity

Cavities are common elements in engineering and geophysical systems; they are also common in biological systems. Situations involving flows in cavities vary from thermal insulation of buildings (Khalifa 2001a) and cooling of microelectronics (Teertstra et al. 1997; Banerjee et al. 2008) to leftward flow in a nodal cavity in a developing mammal embryo (Kuznetsov et al. 2014). In this section, research on heat transfer in various types of cavities is reviewed. Rectangular cavities with height H and width L are considered.

2.1.1

Rectangular Cavities with Adiabatic Horizontal Walls, Heated from the Side Natural convection in rectangular enclosures is well studied. With respect to heat transfer, rectangular enclosures are broadly classified into those heated from the side and those heated from the bottom. For enclosures heated from the side, scaling analysis (see Bejan 2013) reveals that such enclosures can be classified into four types: conduction limit systems, tall systems, high-Rayleigh number systems, and shallow systems. Each of these enclosure types has its own characteristic pattern of convection. One of the most frequently recommended correlations for determining the Nusselt number for enclosures heated from the side is by Berkovsky and Polevikov (1977; Bejan 2013; Bergman et al. 2017). Although there are other correlations, with different ranges of validity and accuracy, the Berkovsky and Polevikov correlation can be viewed as a representative correlation for this class of cavities:
NuL ¼

0:18Ra0:29 L

Pr 0:2 þ Pr

0:29

:

(1)

Equation 1 is recommended for 1  HL  2 , 103  Pr  105, and 0:2 þ Pr . For larger aspect ratios, the recommended correlation is RaL  103 Pr

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NuL ¼ 0:22Ra0:28 L

Pr 0:2 þ Pr

0:28
0:25 H : L

(2)

Equation 2 is recommended for 2  HL  10, Pr  105, and 103  RaL  1010. In Eqs. 1 and 2, the average Nusselt number is defined as: NuL ¼

hL , k

(3)

where h is the average heat transfer coefficient and k is the thermal conductivity of the fluid. The Rayleigh number is defined as: RaL ¼

gβL3 ðT h  T c Þ, aν

(4)

where a is the thermal diffusivity of the fluid, g is the gravity, β is the volumetric thermal expansion coefficient of the fluid, ν is the kinematic viscosity of the fluid, Th is the temperature at the hot vertical wall, and Tc is the temperature at the cold vertical wall. The Rayleigh number characterizes the ratio of thermal driving (buoyancy) and viscous forces in the fluid (it equals to the ratio of these forces times the Prandtl number). Finally, the Prandtl number is defined as: ν Pr ¼ : a

(5)

All properties in Eqs. 3, 4, and 5 must be evaluated at the average temperature, T ¼ ðT c þ T h Þ=2. The correlations given by Eqs. 1 and 2 were validated in Catton (1978) by comparisons with experimental results and numerical solutions. Extensions of the Berkovsky-Polevikov correlations to the case of heterogeneous porous enclosures were investigated in Qiu et al. (2013). Natural convection resonance can occur in rectangular enclosures periodically heated from a side wall by a pulsating heat flux, while the opposite side wall is kept at a uniform (cold) temperature. Such thermal resonances in enclosures filled with a clear fluid or a saturated porous medium were studied in Lage and Bejan (1993) and Antohe and Lage (1994, 1996a, b, 1997). The relationship between pulsating amplitude and frequency was investigated, and the surface averaged heat flux was found to be maximized at the resonant frequency. The situation for an enclosure subjected to a pulsating heating from the top was analyzed theoretically in Lage (1994). In general, heat transfer in this class of cavities is well studied. However, there are still some opportunities for new research, particularly in the areas of transition to turbulence and turbulent convection. A turbulent plume in a rectangular enclosure which was induced by a hot side wall was studied in Caudwell et al. (2016). The plume was found to have an inner layer adjacent to the heated boundary, which is

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dominated by laminar transport. Transition to turbulent flow occurs at the outer layer. In terms of the height, three zones were observed in the enclosure: a lower zone, where the plume is laminar and thin; a middle stratified zone, where the fluid is exchanged with the wall plume; and a top zone, occupied by a warm layer with a strong stratification. Various turbulence models for simulating natural convection in two and three dimensional rectangular enclosures heated from the side have been recently compared in Altac and Ugurlubilek (2016). Kizildag et al. (2014a) used direct numerical simulation (DNS) in order to assess limits of the Oberbeck-Boussinesq approximation for simulating turbulent convection in an enclosure heated from the side. They found that if the temperature difference between the walls does not exceed 30  C, the Oberbeck-Boussinesq approximation estimates heat transfer with an accuracy of 1%. For larger temperature differences the Oberbeck-Boussinesq approximation may affect the way instabilities propagate in the boundary layers along the hot and cold walls. In Kizildag et al. (2014b), the performance of three subgrid-scale models for simulating a turbulent flow in an enclosure of an aspect ratio 5 was investigated by comparing predictions of these three models with results of a DNS solution. Carvalho and de Lemos (2014) used the high Reynolds closure to develop a two-temperature model for turbulent natural convection in a porous rectangular enclosure heated from the side. The existence of macroscopic turbulent structures in turbulent flows in porous media was investigated by extensive DNS modeling in Jin et al. (2015) and Uth et al. (2016).

2.1.2 Rectangular Cavities and Horizontal Layers Heated from Below A commonly used correlation for this class of cavities is reported in Globe and Dropkin (1959): 1=3

NuL ¼ 0:069RaL Pr0:074 ,

(6)

which is valid for 3  105  RaL  7  109. Equation 4 for the Rayleigh number stands, but now Th is the temperature of the lower wall and Tc is the temperature of the upper wall. For convection to occur, RaL must be larger than the critical value of 1708 (Chandrasekhar 1961). Convective circulation in this class of enclosures is characterized by Rayleigh-Bénard convection, in which a representative flow pattern consists of cells with longitudinal rolls (Mishra et al. 1999). (The convection pattern depends on a number of factors, including the Rayleigh number, shape of the container, and lateral boundary conditions.) Recent research in this area is mainly concerned with turbulent convection. Kenjeres and Hanjalic (2000) numerically investigated turbulent thermal convection in enclosures with aspect ratios ranging from 4:1 to 32:1, for Rayleigh numbers ranging from 105 to 1012, using an algebraic model for the turbulent heat flux. A direct numerical simulation (DNS) study of turbulent Rayleigh-Bénard convection in an enclosure heated from below and cooled from above, with effects of thermal radiation accounted for, was conducted in Czarnota and Wagner (2016). Grossmann and Lohse (2000) analyzed scaling laws for the Nusselt number in this

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type of enclosure by decomposing kinetic and thermal energy dissipation rates into their boundary layer and bulk components. Niemela et al. (2000) experimentally studied turbulent convection at very high Rayleigh numbers. In their experiments with cryogenic helium gas, the Rayleigh number changed from 106 to 1017. They found that for Pr ~ 1, a good correlation for the Nusselt number is Nu = 0.124Ra0.309. The effect of the tilt angle has also been extensively studied; various correlations for the average Nusselt number for rectangular enclosures were reviewed and compared in Chang (2014). It should be noted that Rayleigh-Bénard convection patterns, occurring in an enclosure heated from below, are very complex. Contrary to this, enclosures heated from the side are characterized by a much simpler convection pattern with a single roll (Mallinson and de Vahl Davis 1977). Torres et al. (2015) investigated at what critical angles the transition between these two regimes occurs while enclosure’s tilt angle is increased, and addressed physical mechanisms responsible for this transition. Williamson et al. (2016) also studied a bifurcation in the flow structure that occurs as the enclosure’s inclination angle is varied, and the effect of the angle’s variation on the Nusselt number. Convection in a porous enclosure heated from below was studied in many publications, see Zahmatkesh (2008), Kathare et al. (2008), and Su et al. (2015). High Rayleigh number convection in a porous layer heated from below was investigated in Hewitt et al. (2012) for a two-dimensional situation and in Hewitt et al. (2014) for a three-dimensional situation. For convection to start in a porous layer, the Rayleigh number must exceed 4π 2 (Nield and Bejan 2013). The Rayleigh number, based on the layer height, is now defined as RaH ¼ gβKH am ν ðT h  T c Þ, where K is the permeability of the porous medium and am is the effective thermal diffusivity of the porous medium. For Ra < 1300 convection consists of large-scale quasi-periodic convective rolls, while for Ra > 1300 convection transitions to a high-Rayleigh number regime, characterized by different regions. It includes a vertical exchange flow, carried in columns (megaplumes). The columnar megaplumes are fed by mixing of protoplumes, which are entrained into the columns at the lower and upper boundaries (Hewitt et al. 2012, 2014).

2.2

Three-Dimensional Cavities and Cavities with Shapes Other than Rectangular

In many practical situations, the geometry of the cavity is different from a simple rectangle. Heat transfer correlations in three-dimensional enclosures were reviewed in Khalifa (2001b). Correlations for enclosures with localized heating were reviewed in Oztop et al. (2015). Natural convection in a cavity with periodically active heat sources, a geometry motivated by multicore computer processors, was studied in Mahapatra et al. (2015). Natural convection in a rectangular cavity in which regions of the vertical walls were heated/cooled was studied in Nardini et al. (2015); these authors also studied the effect of a horizontal baffle. The effect of the addition of a

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heated region on the bottom wall was studied in Nardini et al. (2016). Choi et al. (2015) investigated natural convection in a rectangular cavity, heated from the bottom, which had a cold circular cylinder located in its center. Correlations for parallelogramic diode cavities were reviewed in Bairi et al. (2014). Correlations that could be applied to solar collector cavities were studied in Singh and Eames (2011). Correlations for triangular enclosures were reviewed in Kamiyo et al. (2010); the effect of the pitch angle was examined. Various radial geometries are of practical significance. Correlations for annular geometries were reviewed in Dawood et al. (2015) and instabilities and bifurcation phenomena in a horizontal annulus were studied in Angeli et al. (2010).

3

Some Emerging Results on Heat Transfer in Cavities. Situations When There Are Other Contributors to the Buoyancy Force, in Addition to the Density Variation Due to a Temperature Change

3.1

Double-Diffusive Convection

Double-diffusive, also called thermosolutal convection, occurs due to simultaneous action of temperature and concentration gradients. It is relevant to a number of industrial applications, including macrosegregation during solidification of binary or multicomponent alloys and the stability of single-crystal solidification (Schneider and Beckermann 1995; Pollock and Murphy 1996; Kuznetsov 1998). One class of problems deals with stability of double-diffusive convection. Ghorayeb and Mojtabi (1997) studied the onset of double-diffusive convection in rectangular cavities with equal and opposing buoyancy forces due to horizontal thermal and concentration gradients. Mamou et al. (1998) investigated the onset of double-diffusive convection in a rectangular porous enclosure. Stationary instability and over stability regimes were studied. Wang and Tan (2011) performed a stability analysis for a Maxwell fluid. Another class of problems deals with a steady-state circulation in an enclosure. Steady-state thermosolutal convection in a square cavity was numerically investigated in Beghein et al. (1992). Double-diffusive convection in a rectangular enclosure driven by opposing horizontal thermal and solutal gradients was studied in Nishimura et al. (1998). Costa (1997) studied double-diffusive convection in a rectangular cavity with given constant values of temperature and concentration maintained at the vertical walls. Double-diffusive convection in a cubic enclosure, induced by opposing temperature and concentration gradients, was studied in Sezai and Mohamad (2000). Different flow patterns occurring in the enclosure at different Rayleigh and Lewis numbers were identified. Wang et al. (2016) performed a numerical investigation of the development of oscillatory double-diffusive convection, concentrating on investigating the influence of Soret and Dufour effects. Transitions between a periodically oscillating flow and chaos were studied. Arbin

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et al. (2016) investigated double-diffusive convection in an open cavity which was heated and salted from the central portion of the vertical wall. A numerical study of double-diffusive convection in a rectangular cavity by using the lattice Boltzmann method (LBM) was performed in Ren and Chan (2016).

3.2

MHD Convection

Effects of the magnetic field on natural convection in an electrically conducting fluid are extensively studied. These effects are due to the Lorenz force (Sheikholeslami et al. 2014b), which is caused by the interaction of the magnetic field and the induced electric current. MHD convection in rectangular cavities was studied in Rudraiah et al. (1995). The situation involving a shallow cavity was investigated in Alchaar et al. (1995a). The effect of the transverse magnetic field was considered in Alchaar et al. (1995b). The effect of a constriction on MHD flow was investigated in Gajbhiye and Eswaran (2015). MHD natural convection in a triangular cavity with a flexible side was investigated in Selimefendigil and Oztop (2016). The effects of an asymmetrically positioned vertical partition on MHD natural convection in an enclosure heated from the side were investigated in Kahveci and Oeztuna (2009). It was found that the heat transfer rate decreases as the distance of the partition from the hot wall is increased. MHD free convection in a rectangular cavity filled with periodically arranged solid blocks was studied in Ashouri et al. (2014). Effects of a sinusoidal temperature variation at the bottom wall on MHD free convection in a square enclosure were investigated in Oztop et al. (2009). Effects of an inclined magnetic field on free convection in a porous rectangular cavity were studied in Grosan et al. (2009). It was found that the Nusselt number at the bottom wall decreases as the direction of the magnetic field changes from horizontal to vertical. MHD natural convection in a square porous enclosure, heated from the side, subjected to a uniform magnetic field, was investigated in Ahmed et al. (2014); the authors studied effects of the magnetic field orientation. A large number of publications are devoted to MHD convection of nanofluids. Sheikholeslami and Ellahi (2015) used the LBM to study three-dimensional MHD convection of a nanofluid in a cavity heated from below, using the Koo-Kleinstreuer correlation to model the thermal conductivity of a nanofluid. The Koo-Kleinstreuer correlation represents the effective thermal conductivity of a nanofluid by the sum of the thermal conductivity of a suspension containing static nanoparticles and thermal conductivity enhancement due to mixing induced by Brownian motion of the nanoparticles (Koo and Kleinstreuer 2004a, b, 2005). Sheikholeslami et al. (2014a) studied MHD free convection of a nanofluid in a semi-annulus enclosure that had a hot inner wall and a cold outer wall while other walls were thermally insulated. It was found that the magnetic field slows down convection due to the effect of the Lorentz force. Despite a significant progress in numerical research involving convection of nanofluids, experimental validation of the obtained results and evaluation of model parameter values remains a challenge.

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3.3

Free Convection: Cavities and Layers

611

Mixed Convection in Lid-Driven Enclosures

Another important class of problems is associated with enclosures in which flow is lid-driven, but the buoyancy force is also significant. This situation results in mixed convection. This area is quite broad and diverse. For example, doublediffusive mixed convection in a lid-driven porous enclosure was investigated in Khanafer and Vafai (2002). Uddin et al. (2015) studied thermosolutal convection in a lid-driven trapezoidal enclosure for the case when the magnetic field is present. Kefayati (2015) numerically studied mixed convection of a non-Newtonian nanofluid in a two-sided lid-driven enclosure. Chatterjee et al. (2014) investigated mixed convection of a nanofluid in a heated lid-driven enclosure with a rotating circular cylinder in the center of the enclosure. Ahmed et al. (2016) investigated how magnetic field affects convection in a lid-driven cavity which has an isothermal heater in its corner. Three-dimensional double-diffusive convection in a lid-driven enclosure was investigated in Ghachem et al. (2016). Another important situation is when the flow occurs due to acoustic streaming induced by vibrations of the cavity’s lid (Wan and Kuznetsov 2004).

4

Non-Newtonian Fluids

Many researchers studied natural convection in cavities filled with non-Newtonian fluids. Some representative papers are listed below. Ozoe and Churchill (1972) studied natural convection in an enclosure filled with the power-law or Ellis fluid. Effects of the enclosure aspect ratio of a rectangular enclosure on Rayleigh-Bénard convection of power-law fluids was studied in Yigit et al. (2015). The case of pseudoplastic fluids was studied in Ohta et al. (2002). Natural convection in a cavity filled with a Carreau-Yasuda fluid was investigated in Alloui and Vasseur (2015). Yigit et al. (2016) investigated natural convection of a power-law fluid in an annulus with a square cross-section; vertical walls of this cross-section were differentially heated.

5

Cavities Filled with Nanofluids. Single-Phase Modeling Approach

The term “nanofluid” was coined by Choi in a famous paper that he presented in 1995 at the ASME Winter Annual Meeting (Choi 1995). It refers to a liquid that contains dispersed submicronic solid particles; a typical size of such nanoparticles is in the range between 1 and 50 nm (Choi 2009). The interest to nanofluids is mostly due to the promise of a significant enhancement of thermal conductivity of the fluid and overall heat transfer by the addition of nanoparticles (Lee et al. 1999; Choi et al. 2001, 2004; Eastman et al. 2001; Maiga et al. 2005; Das et al. 2008). Mechanisms of heat transfer enhancement in nanofluids are widely discussed; see, for example,

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Xuan and Roetzel (2000), Jang and Choi (2004, 2007), Kang et al. (2006), Vadasz et al. (2005), Vadasz (2006), and Wu et al. (2010). There are two common approaches to modeling nanofluids: treating a nanofluid as a single phase fluid with modified (effective) properties (Tiwari and Das 2007) or treating it as a two-phase mixture (Buongiorno 2006). A recent review addressing both approaches to simulating nanofluids can be found in Sheikholeslami and Ganji (2016). Abouali and Ahmadi (2012) reported a numerical study in which they investigated whether special Nusselt number correlations are required for nanofluids that are simulated by a single-phase model. They concluded that it is sufficient to use correlations for clear fluids; it is just necessary to modify the density, specific heat, thermal expansion coefficient, thermal conductivity, and viscosity to account for the effective properties of nanofluids, which are combinations of properties of the base fluid and nanoparticles. These effective properties should be used instead of properties of the base fluid in the correlations for the clear fluid to estimate the Nusselt number for a nanofluid. It should be noted that effective properties of nanofluids are temperature-dependent. Abouali and Ahmadi (2012) used the following correlations for the nanofluid properties. For the nanofluid density, they used a correlation that is based on the mixing theory: ρn ¼ ð1  ϕÞρ0 þ ϕρp ,

(7)

where ϕ is the volume fraction of nanoparticles, and subscripts n, 0, and p refer to nanofluid, base fluid, and nanoparticles, respectively. For the nanofluid-specific heat, Abouali and Ahmadi (2012) used a correlation suggested in Bergman (2009): cp , n ¼

ð1  ϕÞðρcÞ0 þ ϕðρcÞp ð1  ϕÞρ0 þ ϕρp

:

(8)

For the thermal expansion coefficient of a nanofluid, they used a correlation suggested in Khanafer et al. (2003): βn ¼

ð1  ϕÞðρβÞ0 þ ϕðρβÞp : ð1  ϕÞρ0 þ ϕρp

(9)

The thermal conductivity of a nanofluid was determined by a correlation suggested in Corcione (2011):
10
0:03 kn T kp 0:66 ¼ 1 þ 4:4Re0:4 Pr ϕ0:66 , p T fr k0 k0

(10)

where T is the nanofluid temperature and Tfr is the freezing temperature of the base fluid.

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In Eq. 10, the nanoparticle Reynolds number was calculated as: Rep ¼

2ρ0 kb T , πμ20 d p

(11)

where dp is the nanoparticle diameter and kb is the Boltzmann’s constant. The dynamic viscosity of a nanofluid was also estimated by a correlation suggested in Corcione (2011): μn ¼ μ0

1  34:87

1  0:3 dp df

ϕ1:03

:

(12)

In Eq. 12, the equivalent diameter of a molecule of a base fluid was found as: df ¼

6M , πNρ0@293K

(13)

where M is the molecular weight of the base fluid, N is the Avogadro number, and ρ0 @ 293K is the base fluid density at a temperature of 293 K.

6

Ferrofluids

A ferrofluid is a special type of a nanofluid that contains ferromagnetic nanoparticles and thus can be magnetized (Odenbach 2003). A typical size range of nanoparticles in ferrofluids is 10–100 nm. Stability of a heated layer of a ferrofluid in a magnetic field was studied in Rahman and Suslov (2016). Heat transfer enhancement in magnetic ferrofluids was reviewed in Nkurikiyimfura et al. (2013). Ferrofluid convection in a semi-annulus enclosure was studied in Sheikholeslami and Ganji (2014). These authors accounted for both ferrohydrodynamic and magnetohydrodynamic effects and found that the Nusselt number increases with the increase of the Rayleigh number and nanoparticle volume fraction but decreases with the increase of the Hartmann number, a dimensionless parameter that characterizes the ratio of electromagnetic and viscous forces. Kefayati (2014) used the LBM to numerically investigate natural convection in a rectangular enclosure with a linear temperature variation in the vertical direction. Kerosene was assumed to be a base fluid and cobalt was assumed to be a material for nanoparticles. It was observed that heat transfer decreases with the increase of the nanoparticle volume fraction. Free convection of a cobalt-kerosene ferrofluid in a rectangular enclosure heated from below was investigated in Sheikholeslami and Gorji-Bandpy (2014). The LBM was used to study the effect of the Rayleigh number, external magnetic source, nanoparticle size, and nanoparticle volume fraction. Mojumder et al. (2016) compared heat transfer enhancement during natural

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convection in a half-moon-shaped cavity by two different ferrofluids, containing cobalt-kerosene and Fe3O4-water. Heat transfer enhancement by a cobalt-kerosene ferrofluid was found to be superior.

7

Research on the Onset of Convection Instability in Natural Convection

A large body of work is devoted to conditions for the onset of convection instability. A horizontal fluid layer is a paradigmatic geometry for such studies. A classical situation occurs when such a layer is heated from the bottom. This leads to a cold fluid sublayer situated above a hot fluid sublayer. If an increase of temperature results in a decrease of fluid density (which happens for most fluids), this leads to an unstable situation, with a more dense fluid overlaying a less dense fluid. As long as the Rayleigh number is low, the buoyancy force cannot overcome the viscous force, and convection does not develop. The onset of instability occurs at a critical value of Ra ~ 1700, the exact critical value depends on specific boundary conditions (Chandrasekhar 1961). For Rayleigh numbers larger than the critical value, convection produces a flow with a cellular structure (the Rayleigh-Bénard convection). The physical reasons for the existence of the cellular structure were addressed, for example, in Nield (1967). Getling and Brausch (2003) explained the development of the cellular pattern by flow seeking an optimal scale. The cellular structure exists up to Ra ~ 47,000; after that flow becomes irregular and turbulent (Eckert and Drake 1972; Bergman et al. 2017).

7.1 7.1.1

Recent Developments in Classical Natural Convection

Natural Convection Due to Spatially Nonuniform Internal Heating Natural convection due to internal heat generation is related to a number of interesting phenomena. Natural convection caused by electrolytic currents was observed in a laboratory study (Tritton and Zarraga 1967). Cellular convection in the atmosphere of Venus, observed by Mariner 10, was related to internal energy generation caused by sunlight absorption (Tritton 1975). Convection in the earth’s mantle may be related to heat generated due to radioactive decay of isotopes (Moresi et al. 2000). Early theoretical papers on the onset of convection produced by a uniform constant volumetric heat source in a horizontal fluid layer include Sparrow et al. (1964), Roberts (1967), and Kulacki and Goldstein (1975). More recent papers include Shivakumara and Suma (2000), Perekattu and Balaji (2009), and Yadav et al. (2015). Effects of a nonuniform source distribution were studied in Yucel and Bayazitoglu (1979), Liu (1996), Tasaka and Takeda (2005), and Chatterjee et al. (2008), but in each of these papers the source strength varied exponentially in the vertical direction. Straughan (1990) investigated a few special cases involving a

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non-uniform source. A systematic investigation of the effects of a volumetric heat source whose strength varies in the vertical direction has recently been presented in Kuznetsov and Nield (2016). The authors studied two cases where the source strength varied continuously with the height. In the first case, the source strength distribution varied about the mean in a linear fashion, and so was antisymmetric about the mid-layer plane. In the second case, the source strength varied about the mean in a quadratic symmetric manner. Then the authors considered the extreme case of a concentrated source, namely a plane heat source located at various heights. They then briefly considered the general case. Below the problem studied in Kuznetsov and Nield (2016) is briefly described. In the governing equations, presented below, the z-axis is taken in the upward vertical direction, and the fluid layer is unbounded in the x and y directions. The OberbeckBoussinesq approximation is invoked. The equations representing the conservation of mass, conservation of momentum, and conservation of thermal energy are, respectively: ∇  u ¼ 0, ρ

@u þ ρu  ∇u ¼ ∇P þ μ∇2 u  ρ0 βðT  T 0 Þg, @t
 @T þ u  ∇T ¼ k∇2 T þ Q: ðρcÞf @t

(14) (15) (16)

Here the velocity is denoted by u = (u, v, w), t is the time, P is the pressure (excess over hydrostatic), T is the temperature, k is the thermal conductivity of the fluid, (ρc)f is the heat capacity of the fluid, μ is the fluid viscosity, ρ0 is the fluid density at temperature T0, β is the volumetric thermal expansion coefficient of the fluid, g is the gravitational acceleration vector, and Q is the volumetric heat source strength. The layer depth is H. The upper and lower boundaries are assumed to be both rigid and impermeable. Thus, the hydrodynamic boundary conditions are u ¼ 0 at z ¼ 0 and at z ¼ H:

(17)

The heating is assumed to be purely internal, so that when the case of two isothermal boundaries is considered, the two boundaries are held at the same temperature. Hence, it is either assumed that T ¼ T 0 at z ¼ 0 and at z ¼ H

(18)

for two isothermal boundaries, or @T ¼ constant @z

at z ¼ 0, T ¼ T 0 at z ¼ H

(19)

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for a constant heat flux at the lower surface. In the case of volumetric heating, the heat flux is out of the layer and so the constant in Eq. 19 is positive. The volumetric source strength is assumed to be given by Qm f(z/H), where Qm is the mean value over the layer depth, and thus ð1

f ðζ Þdζ ¼ 1:

(20)

0

Kuznetsov and Nield (2016) found that for the linear variation of source strength, the critical Rayleigh number increases almost linearly as the nonuniformity of the source strength variation increases. For the quadratic variation (symmetric about the mid-layer level), the critical Rayleigh number varies little because stabilizing and destabilizing tendencies resulting from the source strength variation almost balance each other. In the case of concentrated heat source, the effect of boundary conditions is significant. In the case of isothermal/isothermal boundaries, the critical Rayleigh number goes through a minimum as the height of the source is raised, with the minimum being attained at about the middle of the layer. In the case of isoflux/ isothermal boundaries, the critical Rayleigh number increases dramatically as the source approaches the upper boundary. This increased stability is due to the fact that the region where the basic temperature gradient is destabilizing becomes confined to a narrow strip. For the exponential variation, the critical Rayleigh number goes through an extremum, which is explained by the variation of the width of the heated region.

7.1.2

Natural Convective Boundary-Layer Flow of a Nanofluid Past a Vertical Plate A model describing thermal processes in a nanofluid that includes the effects of Brownian motion and thermophoresis was introduced by Buongiorno (2006). Buongiorno’s model was applied by Kuznetsov and Nield (2010, 2014) to the classical problem of convective boundary layer flow past a vertical plate (Kuiken 1968, 1969); Bejan 2013). A two-dimensional problem was studied. A coordinate frame in which the x-axis was aligned vertically upwards was utilized. In Kuznetsov and Nield (2010, 2014), a vertical plate was set at y = 0. The Oberbeck-Boussinesq approximation was utilized so the governing equations expressing the conservation of total mass, momentum, thermal energy, and nanoparticles were, respectively: ∇  v ¼ 0,
 @v ρf þ v  ∇v ¼ ∇p þ μ∇2 v @t    þ ϕρp þ ð1  ϕÞ ρf ð1  βðT  T 1 ÞÞ g,

(21)

(22)

14

Free Convection: Cavities and Layers


ðρcÞf

@T þ v  ∇T @t

617

 ¼ k∇2 T þ ðρcÞp ½DB ∇ϕ  ∇T þ ðDT =T 1 Þ∇T  ∇T ,

@ϕ þ v  ∇ϕ ¼ DB ∇2 ϕ þ ðDT =T 1 Þ∇2 T: @t

(23) (24)

In Eqs. 21, 22, 23, and 24, the field variables are the velocity v [where v = (u, v)], the temperature T, and the nanoparticle volume fraction ϕ. Also, ρP is the density of the particles, ρf is the density of the base fluid, and μ, k, and β are the viscosity, thermal conductivity and volumetric expansion coefficient of the nanofluid, respectively. The gravitational acceleration is denoted by g. In Eqs. 23 and 24, the coefficients DB and DT are the Brownian diffusion coefficient and the thermophoretic diffusion coefficient, respectively, each nondimensionalized in terms of the ambient value of the temperature. It is assumed that the temperature does not vary significantly from the ambient temperature, and so DB and DT may each be treated as a constant. It was assumed that at y = 0 the temperature T takes the constant value Tw. The flux of the nanoparticle fraction at y = 0 was taken to be zero. The ambient value of temperature was T1 and the ambient value of the nanoparticle volume fraction was ϕ1; the ambient values were attained at an infinite distance from the wall. Kuznetsov and Nield (2010) employed boundary conditions on the nanoparticle fraction analogous to those on the temperature (case A): u ¼ 0, v ¼ 0, T ¼ T w , ϕ ¼ ϕw at y ¼ 0,

(25a; b; c)

u ¼ v ¼ 0, T ! T 1 , ϕ ! ϕ1 as y ! 1:

(26a; b; c)

Kuznetsov and Nield (2014) used a boundary condition for nanoparticles that was more physically realistic. They no longer assumed that one can control the value of the nanoparticle fraction at the wall, but instead set the nanoparticle flux at the wall to zero (case B): u ¼ v ¼ 0, T ¼ T w , DB

@ϕ DB @T þ ¼ 0 at y ¼ 0, @y T 1 @y

u ¼ v ¼ 0, T ! T 1 , ϕ ! ϕ1 as y ! 1:

(27a; b; c) (28a; b; c)

With the consideration of thermophoresis, Eq. 27c is a statement that the normal flux of nanoparticles is zero at the boundary (Pakravan and Yaghoubi 2013; Sheikhzadeh et al. 2013). The above problem was solved for a steady state situation. Kuznetsov and Nield (2010, 2014) used a similarity transformation and obtained a system of ordinary differential equations. For case A, it was found that the problem is described by five independent dimensionless parameters:

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ν Pr ¼ , α

 ρp  ρf 1 ð ϕw  ϕ 1 Þ NrA ¼ , ρf 1 β ð T w  T 1 Þ ð 1  ϕ 1 Þ NbA ¼ Nt ¼

ðρcÞp DB ðϕw  ϕ1 Þ ðρcÞf α

,

(29) (30)

(31)

ðρcÞp DT ðT w  T 1 Þ , ðρcÞf αT 1

(32)

α : DB

(33)

Le ¼

Here Pr is the Prandtl number, Nr is the buoyancy ratio, Nb is the Brownian motion parameter, Nt is the thermophoresis parameter, and Le the Lewis number. For case B, the problem was also described by five parameters, but parameters Nr and Nb were rescaled as follows:

 ρp  ρf 1 ϕ1 NrB ¼ , ρf 1 β ð T w  T 1 Þ NbB ¼

ðρcÞp DB ϕ1 : ðρcÞf α

(34)

(35)

Equations for other parameters were unchanged. The Nusselt number Nu was defined as: Nu ¼

q00 x : k ðT w  T 1 Þ

(36)

Here q00 is the wall heat flux. The local Rayleigh number Rax was defined by Rax ¼

ð1  ϕ1 ÞβgðT w  T 1 Þx3 , να

(37)

Kuznetsov and Nield (2010) scaled a Nusselt number Nu in terms of Ra1/4 to produce a reduced Nusselt number, Nur = Nu/Rax1/4. For Pr = 10 and Le = 10, case A, Kuznetsov and Nield (2010) reported the following correlation: NurA ¼ 0:465  0:055 NrA  0:256 NbA  0:160 Nt:

(38)

Kuznetsov and Nield (2014) reported the following correlation for Nur for case B: NurB ¼ 0:465  0:001 NrB  0:003 NbB  0:075 Nt:

(39)

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Correlations 38 and 39 are valid for Nr, Nb, and Nt each taking values in the range [0, 0.4]. The prime result is that the reduced Nusselt number is a decreasing function of each of nanofluid numbers Nr, Nb, and Nt. For case B (when the nanoparticle flux on the boundary is taken to be zero), the reduced Nusselt number is almost independent of Nb (Eq. 39). This is different from case A of actively controlled nanofluid particle fraction on the boundary; in case A the effect of Nb is not negligible (Eq. 38). This difference is explained by the fact that the contribution of Brownian motion to the thermal energy equation, which is proportional to the nanoparticle fraction gradient, in case B tends to be zero as the wall is approached.

8

Emerging Topics: Links between Internal Natural Convection and Bio- and Nanofluidics

8.1

Bioconvection: Macroscopic Motion of a Fluid Caused by Many Mesoscale Swimmers

Bioconvection is a phenomenon caused by upswimming of motile microorganisms. Upswimming occurs because motile microorganisms can propel themselves by rotating their flagellar. Examples of motile microorganisms that can induce bioconvection include some species of algae, bacteria Bacillus subtilis and Escherichia coli, and many other single-cell microorganisms. Since microorganisms are heavier than water, their upswimming results in an increase of the fluid density in the upper portion of the fluid layer and, above a certain threshold, causes overturning instability that induces macroscopic motion of the fluid. Thus, from the macroscopic standpoint, bioconvection is quite similar to natural convection, but the reason for the density stratification is not a temperature variation across the layer or cavity, but collective swimming of a large number of self-propelled microorganisms. Bioconvection may be useful in microfluidics applications, for inducing mixing, and enhancing mass transfer (Kuznetsov 2011a). Bioconvection is an interesting physical phenomenon. As noted in Geng and Kuznetsov (2007), while superfluidity and superconductivity are quantum phenomena visible at the macroscale, bioconvection is caused by a mesoscale phenomenon, self-propelled motion of motile microorganisms, which induces macroscopic motion in the fluid. In other words, many similar mesoscale events lead to a phenomenon observable on a macroscale. The first theory of bioconvection was developed in a celebrated paper by S. Childress, M. Levandowsky, and E.A. Spiegel (Childress et al. 1975) and then further developed in a collaborative work of groups of Professors J.O. Kessler and T.J. Pedley as well as their students and colleagues (Pedley and Kessler 1987, 1992; Pedley et al. 1988; Hill et al. 1989; Hillesdon et al. 1995; Ghorai and Singh 2009; Hwang and Pedley 2014). Depending on particular species, the average direction of swimming of motile microorganisms is determined by different factors. For example, gyrotactic microorganisms, such as algae Chlamydomonas, swim generally in the upward direction,

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but with some angle to the vertical, since the vector of their swimming velocity is re-oriented by the fluid vorticity. The mathematical theory describing bioconvection in suspensions of gyrotactic microorganisms was developed in Pedley and Kessler (1987), Pedley et al. (1988), Hill et al. (1989), and Ghorai and Hill (1999, 2000). Oxytactic microorganisms, such as bacteria Bacillus subtilis, swim up the oxygen gradient. Mathematical models for bioconvection in suspensions of oxytactic microorganisms were developed by Hillesdon et al. (1995) and Hillesdon and Pedley (1996).

8.1.1

Bioconvection in Superimposed Fluid and Porous Horizontal Layers The onset of bioconvection in superimposed fluid and porous horizontal layers for the case without throughflow was investigated in Avramenko and Kuznetsov (2004, 2005). The situation with throughflow was studied in Avramenko and Kuznetsov (2006). The results reviewed in this section may be relevant to bioconvection in springs with a porous bottom. The presence of throughflow is relevant to a number of geomechanical problems, such as seawater convection at the ocean crust (Khalili et al. 2003) as well as the ore body formation and mineralization in hydrothermal systems (Zhao et al. 2002; Lin et al. 2003). Kuznetsov and Nield (2015) noted that throughflow tends to confine significant thermal gradients to a thermal boundary layer at that boundary toward which the throughflow is directed. This reduces the effective vertical length and makes the system more stable, so that larger values of the Rayleigh number are needed for the onset of convection. A schematic diagram of the problem studied in Avramenko and Kuznetsov (2006) is displayed in Fig. 1. These authors considered superimposed fluid and porous layers, whose depths are H and Hp, respectively. The vertical throughflow velocity is W0, and the lower boundary of the porous layer and the upper boundary of the fluid layer are permeable to the throughflow but impermeable to the microorganisms (these boundaries can be modeled as porous plates with a very small porosity). The layer geometry is similar to that studied in Khalili et al. (2003). Avramenko and Kuznetsov (2006) considered a dilute suspension of gyrotactic microorganisms (such as algae Chlamydomonas); they assumed that the porous matrix does not absorb microorganisms. They also assumed that on the continuum scale, swimming of microorganisms through the porous layer can be adequately described by a simple gyrotactic balance law. This implies that the pore sizes are significantly larger than the microorganisms; therefore, the local vorticity generated by flow through the pores does not cause the cells to tumble and does not affect their ability to reorient. Darcy law was used to describe the flow in the porous part of the layer. Convection instability in the above system is described by the following equations. Governing equations in the fluid part of the layer are

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Fig. 1 Schematic diagram of bioconvection due to upswimming of gyrotactic microorganisms in superimposed porous and fluid horizontal layers with vertical throughflow


ρ

@U þ ðU  ∇ÞU @t

 ¼ ∇p þ μ∇2 U þ g nθΔρ,

(40)

∇  U ¼ 0,

(41)

@n ¼ divðjÞ, @t

(42)

where the flux of microorganisms is defined as: ^  D∇n: j ¼ nU þ nW c p

(43)

The three terms on the right-hand side of the equation for j represent contributions from the flux of microorganisms due to the macroscopic motion of the fluid, the directional gyrotactic swimming of microorganisms, and a diffusive process that models all random motions of microorganisms, respectively. The governing equations in the porous part of the layer are ρ

@U μ CF ¼ ∇p  U  ρ pffiffiffiffi jUjU þ g nθΔρ, @t K K ∇  U ¼ 0,

(44) (45)

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A. V. Kuznetsov and I. A. Kuznetsov

φ

 @n ¼ div jp , @t

(46)

where ^  Dp ∇n: jp ¼ nU þ nW c, eff p

(47)

Relevant parameters in Eqs. 40, 41, 42, 43, 44, 45, 46, and 47 are defined as follows. CF is the form drag constant; D is the diffusivity of microorganisms in the clear fluid; Dp is the effective diffusivity of microorganisms in the porous medium; g is the gravity vector; K is the permeability of the porous medium; n is the number ^ is density of motile microorganisms; p is the excess pressure (above hydrostatic); p the unit vector indicating the direction of swimming of microorganisms; t is the time; U is the vector of the fluid convection velocity, which over the porous region is understood as filtration (seepage) velocity, (u,v,w); u, v, and w are the x, y, and ^ is the vector of z-velocity components of the fluid velocity vector, respectively; W c p microorganisms average swimming velocity relative to the fluid (Wc is assumed to be constant); x, y, and z are the Cartesian coordinates (z is the vertical coordinate); Δρ is the density difference, ρcell  ρ0; θ is the average volume of a microorganism; μ is the dynamic viscosity, assumed to be approximately the same as that of water; φ is the porosity; ρ0 is the density of water; and the subscript eff denotes the effective value for the porous medium. In a porous medium, the concentrations of cells (unlike the heat) are advected/ convected with the intrinsic velocity, which characterizes velocity of the fluid through the pores, rather than the Darcy filtration velocity, since the cells cannot pass through the solid phase. This explains why porosity φ is involved in the first term on the left-hand side of Eq. 46. An extra factor, φ, has been incorporated into the effective transport coefficients for the porous medium, Wc, eff and Dp, in Eq. 47. Avramenko and Kuznetsov (2006) assumed that W c, ef f ¼ W c :

(48)

The following boundary conditions were utilized. The top and the bottom boundaries of the domain were assumed to be porous and permeable to the vertical throughflow. At the bottom of the layer, the following conditions were imposed: At z ¼ 0 :

^ ¼ 0, jp  k

u ¼ v ¼ 0,

w ¼ W0,

(49)

^ is the vertically-upward unit vector. The first equation in 49 represents the where k condition of no cell flux through the lower surface. At the top of the layer, the following conditions were imposed: At z ¼ H Σ ¼ H þ H p :

^ ¼ 0, jk

u ¼ v ¼ 0,

w ¼ W0:

(50)

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Free Convection: Cavities and Layers

623

At the porous/fluid interface, the following conditions were imposed: At z ¼ H p : njz!Hp þ0 ¼ njz!Hp 0 ,

 @n @n D ¼ Dp , @z z!Hp þ0 @z z!Hp 0
2  @ w αBJ @w p ffiffiffi ffi  wjz!Hp þ0 ¼ wjz!Hp 0 , @z2 K @z z!Hp þ0


  αBJ W 0 H p ρ @w ¼  pffiffiffiffi 1 þ CF , μ @z z!Hp 0 K

(51)

where αBJ is the Beavers-Joseph coefficient (Nield and Bejan 2013). By applying the linear instability analysis and the Galerkin method to Eqs. 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, and 51, Avramenko and Kuznetsov (2006) numerically investigated convective instability in the composite layer displayed in Fig. 1. They established that the effect of increasing the throughflow velocity is to increase the critical Rayleigh number. This means that the vertical throughflow hinders the development of bioconvection and makes the system more stable. Physically, this happens because the additional flow (the vertically upward throughflow) destroys the forming bioconvection patterns that are needed for the bioconvection to start. This is consistent with the effect of throughflow on convection instability in a single fluid layer heated from below when the lower and upper boundaries are of the same type (Nield 1987). Whether a small amount of throughflow can be destabilizing remains to be seen in future research. It was also established that the viscous torque exerted by the fluid on the body of a gyrotactic microorganism, which results in randomizing its swimming direction, promotes bioconvection. This is consistent with the results reported by Childress et al. (1975) who established the fact that an infinite uniform suspension of negatively geotactic microorganisms (no randomization, all microorganisms swim vertically upward) is stable in the absence of cell concentration stratification. The addition of the effect of the randomization of the swimming direction makes the suspension unstable (Pedley et al. 1988).

8.1.2 Bio-thermal Convection The Kuznetsov’s group extended bioconvection theory to the situation when, in addition to motile microorganisms, there is also a temperature gradient, thus developing a theory of bio-thermal convection (also called thermo-bioconvection or thermo-bio-convection in some publications). The theory of bio-thermal convection in a horizontal layer occupied by a clear fluid (here clear fluid means no porous media) was developed in Kuznetsov (2005a, c, 2011b, c, 2013), Avramenko and Kuznetsov (2010a, b) and reviewed in Kuznetsov (2016). The theory of bio-thermal convection in porous layers was developed in Kuznetsov (2006c), Nield and

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A. V. Kuznetsov and I. A. Kuznetsov

Fig. 2 Schematic diagram of bio-thermal convection due to upswimming of a mixture of gyrotactic and oxytactic microorganisms in a horizontal fluid layer

Kuznetsov (2006) and reviewed in Kuznetsov (2008). The area of bio-thermal convection was further developed by many authors, see, for example, Alloui et al. (2006), Sheremet and Pop (2014), Sharma and Kumar (2011), and Taheri and Bilgen (2008). Below some outlines of the theory for clear fluids are presented, for the case where in addition to a vertical temperature gradient two species of motile microorganisms are present, the gyrotactic and oxytactic microorganisms. A horizontal shallow layer of depth H is considered (Fig. 2). The onset of convection can be investigated by using the following governing equations (Kuznetsov 2016). The conservation equation for the total mass is ∇  U ¼ 0,

(52)

where U = (u, v, w) is the fluid velocity. The buoyancy force is due to a nonuniform temperature distribution and the presence of two species of microorganisms. Adopting the Boussinesq approximation, the momentum equation can be written as:
 @U ρf þ U  ∇U ¼ ∇p þ μ∇2 U þ ρf ð1  βðT  T c ÞÞg þ ng θg Δρg g @t þ no θo Δρo g,

(53)

where T is the fluid temperature, Tc is the reference temperature (the temperature of the upper wall), p is the pressure, ng is the concentration of gyrotactic microorganisms, no is the concentration of oxytactic microorganisms, t is the time, ρf is the fluid density at the reference temperature, β is the volumetric thermal expansion coefficient of water, θg is the average volume of a gyrotactic microorganism, θo is the average volume of an oxytactic microorganism, Δρg = ρg  ρf is the density difference between a gyrotactic microorganism and water, Δρo = ρo  ρf is the density difference between an oxytactic microorganism and water, ρf is the density of water at a reference temperature, g is the gravity vector, and μ is the viscosity of the

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Free Convection: Cavities and Layers

625

suspension (assumed to be approximately the same as that of water due to low concentration of microorganisms). The asterisks denote dimensional variables. The thermal energy equation is

ðρcÞf

 @T þ U  ∇T ¼ ∇  ðk∇T Þ, @t

(54)

where (ρc)f is the volumetric heat capacity of water and k is the thermal conductivity of water. Conservations of gyrotactic and oxytactic microorganisms need to be stated separately. The equation postulating conservation of gyrotactic microorganisms is based on the model developed in Pedley et al. (1988) and Hill et al. (1989): @ng ¼ ∇  jg , @t

(55)

^  Dg ∇ng j g ¼ ng U þ ng W g p

(56)

where

is the total flux of gyrotactic microorganisms due to convection, self-propelled swimming, and diffusion. As proposed by Pedley et al. (1988), it is assumed that microorganism’s motion can be split into directional and random components. The random component is then approximated by a diffusion process, where Dg is the ^ is a vector of the gyrotactic diffusivity of gyrotactic microorganisms. In Eq. 56, W g p microorganism’s average swimming velocity relative to the fluid, and Wg is assumed to be constant. In order to postulate conservation of oxytactic microorganisms, the theory developed in Hillesdon et al. (1995) and Hillesdon and Pedley (1996) is utilized, which gives the following equation: @no ¼ ∇  jo , @t

(57)

jo ¼ no U þ no Vo  Do ∇no

(58)

where

is the flux of oxytactic microorganisms. The terms on the right-hand side of Eq. 58 represent the flux of oxytactic microorganisms due to the macroscopic motion of the fluid, the directional swimming of microorganisms up the oxygen gradient, and a diffusive process that models all random motions of oxytactic microorganisms, respectively. The parameter Do is the diffusivity of oxytactic microorganisms. In order to use Eq. 58, an equation for the average directional swimming velocity of an oxytactic microorganism is needed. Based on Hillesdon and Pedley (1996), this quantity is calculated as:

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A. V. Kuznetsov and I. A. Kuznetsov

 ~ ^ C~ ∇C, Vo ¼ bW o H

(59)

where b is the chemotaxis constant [m] and Wo is the maximum swimming speed of an oxytactic microorganism [m/s] (the product bWo is assumed to be constant). The ~ in Eq. 59 is defined as follows: dimensionless oxygen concentration, C, ~ ¼ C  Cmin : C C0  Cmin

(60)

Here C is the dimensional oxygen concentration, C0 is the upper-surface oxygen concentration (the upper surface is assumed to be open to the atmosphere), and Cmin is the minimum oxygen concentration that oxytactic microorganisms need in order to ~ be active. In a shallow layer

C  > 0 throughout the layer thickness, and hence the ~ ^ Heaviside step function, H C , in Eq. 59, is set to unity. The conservation of oxygen is described by the following equation: ~ @C ~ ¼ DS ∇2 C ~  γno , þ U  ∇C @t ΔC

(61)

where DS is the diffusivity of oxygen, γno/ΔC describes the consumption of oxygen by oxytactic microorganisms in the fluid, and ΔC is equal to C0  Cmin. It was assumed that the temperature is constant on the boundaries. At the bottom of the layer, the following boundary conditions were utilized: w ¼ 0,

@w ^ ¼ 0, jo  k ^ ¼ 0, @C ¼ 0 at z ¼ 0, ¼ 0, T ¼ T h , jg  k @z @z

(62)

^ is the vertically upward unit where Th is the temperature at the lower wall and k vector. Physically, at the top of the layer two types of boundary conditions can be utilized. The top surface can be assumed to be either rigid or stress-free. For the rigid upper surface, the boundary conditions are w ¼ 0,

@w ^ ¼ 0, jo  k ^ ¼ 0, C ¼ C0 at z ¼ H: ¼ 0, T ¼ T c , jg  k @z

(63)

For the stress-free upper surface, the second equation in 63 is replaced with the following equation: @2w ¼ 0: @z2

(64)

The model described by Eqs. 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, and 64 was utilized to investigate the onset of convection instability in a horizontal fluid layer. A linear instability analysis and a single-term Galerkin procedure were used (for details see Kuznetsov 2016).

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627

For the case of rigid-rigid boundaries and nonoscillatory instability, this leads to the following equation relating three Rayleigh numbers, Ra, Rbg, and Rbo: Ra þ F1 Rbg þ F2 Rbo ¼

28ð10 þ m2 Þð504 þ 24m2 þ m4 Þ , 27m2

(65)

where m is the dimensionless horizontal wavenumber, Ra is a standard thermal Rayleigh number, Rbg is the bioconvection Rayleigh number for gyrotactic microorganisms, and Rbo is the bioconvection Rayleigh number for oxytactic microorganisms. The three Rayleigh numbers are defined as, respectively: Ra ¼

ρf gβH 3 ðT h  T c Þ , μαf

Rbg ¼

Δρg gθg ng, av H 3 , μDg

Rbo ¼

Δρo gθo no, av H3 , μDo (66)

where ng, av and no, av are the average concentrations of gyrotactic and oxytactic microorganisms, respectively (concentration of gyrotactic and oxytactic microorganisms in a well-stirred suspension). Equation 65 thus establishes convective instability in the layer in response to three factors that contribute to the buoyancy force: the density change due to a temperature variation, an increased concentration of gyrotactic microorganisms in a certain part of the layer, and an increased concentration of oxytactic bacteria in a certain part of the layer. (It should be noted that the fact that microorganisms are swimming does not change the buoyancy force, it is the same as if the microorganisms were simply suspended, which follows from the Newton’s third law.) The analytical expressions for the functions F1 and F2 can be found in Kuznetsov (2016). The neutral stability curve can be obtained by solving Eq. 65 for Ra and then finding the minimum value with respect to m of the right-hand side for given values of Rbg and Rbo. If Rbg and Rbo are both equal to zero, which physically corresponds to the situation when the suspension contains neither gyrotactic nor oxytactic microorganisms, the second and third terms on the left-hand side of Eq. 65 vanish and the righthand side of Eq. 65 takes the minimum value of 1750. This is 2.41% greater than the exact value of 1707.762 for the critical Rayleigh number for the classical RayleighBénard problem (Chandrasekhar 1961); the increased value is due to the utilization of a single-term Galerkin approximation in obtaining Eq. 65. Since microorganisms are heavier than water, Rbg and Rbo must be always nonnegative. This means that the presence of microorganisms always causes a destabilizing effect on the suspension, reducing the critical value of Ra. For the case of rigid-free boundaries and nonoscillatory instability, the result is 2 2 4 ^ 1 Rbg þ F ^ 2 Rbo ¼ 28ð10 þ m Þð4536 þ 432m þ 19m Þ , Ra þ F 507m2

(67)

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A. V. Kuznetsov and I. A. Kuznetsov

^ 1 and F ^ 2 can be found in where the analytical expressions for the functions F Kuznetsov (2016). If Rbg and Rbo are both equal to zero, the right-hand side of Eq. 67 takes the minimum value of 1139. This is 3.48% greater than the exact value of 1100.65 for the critical Rayleigh number for the classical Rayleigh-Bénard problem (Chandrasekhar 1961).

8.1.3 Nanofluid Bioconvection Applications of nanoparticles range widely, from heat transfer enhancement in nanofluids (Buongiorno 2006; Khanafer et al. 2003; Sidik et al. 2015) and drug delivery systems (Abbasi et al. 2015) to modification of surfaces to improve their optical properties (e.g., to improve the efficiency of solar cells, Derkacs et al. 2008, Kuznetsov et al. 2011). There is a significant body of work devoted to investigating convection in enclosures filled with various types of nanofluids. For example, Cho et al. (2016) numerically investigated natural convection of a nanofluid in a cavity with inclined wavy walls. Ternik (2015) studied convection heat transfer in a water-based nanofluid with gold nanoparticles in an enclosure with differently heated side walls. Bouhalleb and Abbassi (2014) computationally investigated natural convection of nanofluids in enclosures with different aspect ratios. There is also significant interest in using nanoparticles in various biotechnological applications, such as detection of pathogenic microorganisms (Lin et al. 2005). Rodriguez-Gonzalez et al. (2010) studied the possibility of application of nanoparticles for deactivation of harmful algae and phytoplankton blooms in sea water and rivers. There is growing interest in fluid systems that include both nanoparticles and motile microorganisms. One potential application is using motile microorganisms to keep nanoparticles suspended and prevent them from agglomeration. This motivates interest in nanofluid bioconvection, the field that was recently developed in Kuznetsov (2010, 2011a, c, 2012a, b, c), and Kuznetsov and Bubnovich (2012). This area was further advanced by many authors, see, for example, Beg et al. (2015), Basir et al. (2016), Ahmed and Mahdy (2016), Khan et al. (2015), Mehryan et al. (2016), Sheremet et al. (2015), and Sk et al. (2016). In what follows, equations governing nanofluid bioconvection are briefly reviewed. Now a horizontal layer with a dilute water-based suspension that contains both gyrotactic and oxytactic microorganisms and nanoparticles is considered (Fig. 3). As in the previous section, the suspension occupies a horizontal shallow layer of depth H. The following equations are still valid: Eq. 52 expressing the total mass conservation, Eq. 55 expressing the conservation of gyrotactic microorganisms, Eq. 57 expressing the conservation of oxytactic microorganisms, and Eq. 61 expressing the conservation of oxygen. The Boussinesq approximation was extended as described in Kuznetsov (2011c); under this approximation, the momentum equation, which now replaces Eq. 53, can be modified as follows:

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Fig. 3 Schematic diagram of nanofluid bio-thermal convection due to upswimming of a mixture of gyrotactic and oxytactic microorganisms in a horizontal nanofluid layer


 @U þ U  ∇U ¼ ∇p þ μ∇2 U ρf @t 

 þ ϕ0 ρp þ ð1  ϕ0 Þρf  ρf βðT  T c Þ þ ρp  ρf ðϕ  ϕ0 Þ þng θg Δρg þ no θo Δρo g: (68) Here ϕ is the nanoparticle volume fraction and ϕ0 is the nanoparticle volume fraction at the lower wall. The thermal energy equation (Eq. 54) is modified as:



 @T DT þ U  ∇T ¼ ∇  ðk∇T Þ þ ðρcÞp DB ∇ϕ  ∇T þ ∇T  ∇T : ðρcÞf @t T

(69)

Here DB and DT are the Brownian and thermophoretic diffusion coefficients, respectively; (ρc)f and (ρc)p are the volumetric heat capacities of the nanofluid and nanoparticles, respectively; and k is the thermal conductivity of the suspension. Equation 69 can be further simplified by assuming that the spatial variation of k is negligible; as an approximation, DT/T can be also replaced by DT/Tc. The requirement that nanoparticles are conserved gives the following equation (Buongiorno 2006; Nield and Kuznetsov 2010):

 @ϕ DT þ U  ∇ϕ ¼ ∇  DB ∇ϕ þ ∇T : @t T

(70)

The two diffusion terms on the right-hand side of Eq. 70 represent Brownian diffusion and thermophoresis, respectively. Equation 70 can be simplified further by assuming that the temperature variation in the fluid is small compared with Tc and also that DT/T in the last term on the right-hand side of Eq. 70 can be replaced with DT/Tc.

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In addition to boundary conditions given by Eqs. 62, 63, and 64, the following boundary conditions are imposed: ϕ ¼ ϕ0 at z ¼ 0

(71)

ϕ ¼ ϕ1 at z ¼ H,

(72)

and

where ϕ1 is the nanoparticle volume fraction at the upper wall. For the nonoscillatory situation and rigid-rigid boundaries, the neutral stability curve is now given by the following equation:

Ra þ ðN A þ LnÞRn þ F1 Rbg þ F2 Rbo ¼

28ð10 þ m2 Þð504 þ 24m2 þ m4 Þ : (73) 27m2

Equation 73 now relates four Rayleigh numbers, Ra, Rn, Rbg, and Rbo. The boundary for nonoscillatory instability is again obtained by solving Eq. 73 for Ra and then finding the minimum with respect to m of the right-hand side of the obtained equation. Equation 73 is an approximate result because it was obtained with the help of the one-term Galerkin approximation; the details can be found in Kuznetsov (2016). The new dimensionless parameters are defined as follows. Rn is the nanoparticle Rayleigh number:

 ρp  ρf ðϕ1  ϕ0 ÞgH 3 Rn ¼ , μαf

(74)

Ln is the nanoparticle Lewis number: αf , DB

(75)

DT ðT h  T c Þ : DB T c ðϕ1  ϕ0 Þ

(76)

Ln ¼ and NA is a modified diffusivity ratio: NA ¼

Equation 73 shows that the critical thermal Rayleigh number is a linear function of Rn with a negative slope, meaning that, for positive values of Rn, nanoparticles produce a destabilizing effect. This is explained as follows. In Eq. 74 Rn is defined such that its positive values correspond to a top-heavy (destabilizing) nanoparticle distribution. Therefore, the increase of Rn produces a destabilizing effect and decreases the critical value of Ra.

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For rigid-free boundaries, the neutral curve is now given by the following equation: ^ 1 Rbg þ F ^ 2 Rbo Ra þ ðN A þ LnÞRn þ F ¼

28ð10 þ m2 Þð4536 þ 432m2 þ 19m4 Þ , 507m2

(77)

^ 1 and F ^ 2 can be found in where the analytical expressions for the functions F Kuznetsov (2016).

8.1.4 Bioconvective Sedimentation If the fluid contains small or large solid particles, natural convection that may exist in a cavity would interact with sedimentation of such particles. Developing tools for controlling natural convection is important to manipulate particle sedimentation and mass transfer. In particular, this would be useful for controlling the rate of particle dissolution for the case where solid particles can be dissolved in the surrounding liquid. In large fluid volumes, controlling convection is relatively easy, but may prove difficult in microvolumes, so bioconvection can provide a useful tool to this end. The importance of mixing in microfluidics is reviewed in Burghelea et al. (2004). This section reviews recently obtained results on bioconvective sedimentation, the area that studies the interactions between sedimentation of solid particles and bioconvection plumes. Relevant papers include Geng and Kuznetsov (2004, 2005, 2006, 2007; Kuznetsov and Geng 2005). The following is a brief review of the theory developed in the above papers for the case where a bioconvection plume interacts with large sedimenting particles that are heavier than water. The two-dimensional model reviewed below was used in Geng and Kuznetsov (2007) to investigate sedimentation of one (Fig. 4a) and two (Fig. 4b) large cylindrical particles in a chamber with a fully developed bioconvection plume. The height of the computational domain is H and its width is L, where L is a typical plume spacing and λ = H/L is the aspect ratio of the chamber. A bioconvection plume caused by gyrotactic microorganisms was modeled using equations given in Ghorai and Hill (1999, 2000): • Momentum equation
 @U ρ0 þ ðU  ∇ÞU ¼ ∇pe þ μ∇2 U þ nm θm Δρm g: @t

(78)

• Continuity equation ∇  U ¼ 0:

(79)

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Fig. 4 Bioconvective sedimentation in a suspension of gyrotactic microorganisms: (a) One large particle, (b) Two large particles

• Conservation of motile microorganisms @ ð nm Þ ^  Dm ∇nm Þ, ¼ divðnm U þ nm W m p @t

(80)

where Dm is the diffusivity of microorganisms (it is assumed that all random motions of microorganisms can be approximated by a diffusive process); g is the gravity ^ is the unit vector vector; nm is the number density of motile microorganisms; p indicating the direction of microorganisms’ swimming (equations for this vector are obtained in Pedley et al. 1988); pe is the excess pressure (above hydrostatic); U is the ^ is the fluid velocity vector; t is the time; (Vx, Vy) is the particle velocity vector; W m p vector of microorganisms’ average swimming velocity (Wm is assumed to be constant); Δρm is the density difference between microorganisms and water, ρm  ρ0; θm is the volume of a microorganism; μ is the dynamic viscosity of the suspension, assumed to be approximately the same as that of water; and ρ0 is the density of water.

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Appling the second Newton’s law to particles, the motion of particles can be described by the following equations: mp

dV x ¼ Fx , dt

mp

dV y ¼ Fy , dt

I

dω ¼ T, dt

(81)


2  d where F = (Fx, Fy) is the total external force on a particle, I ¼ mp =8 is the
 2 polar moment of inertia of a particle, mp ¼ ρp πd =4 is the mass of a cylindrical particle per unit axial length (d is the diameter of a particle), T is the mechanical torque on a particle, and ω is the particle’s angular velocity vector, dx ¼ Vx, dt

dy ¼ Vy, dt

dθ ¼ ω, dt

(82)

where ω is the particle’s angular velocity. Geng and Kuznetsov (2007) assumed that the two particles are of identical mass and size. Particle density is uniform so that the particle geometrical center is also its center of mass. The fluid exerts viscous and pressure forces on the surface of the particles. Since particles are assumed to be symmetric, the viscous friction on the surface of a particle provides the only contribution to the torque: ð
Fx ¼

μ Ω

 @Uτ  pn  ^x ds, @n

 ð
 2  ρp  ρ0 @Uτ Fy ¼  pn  ^y ds þ μ π d =4 g, @n ρp Ω

 ð
@Uτ d  T¼ μ ds, @n 2

(83)

(84)

(85)

Ω

where p is the pressure on the surface of the particle, Uτ is the tangential fluid velocity along the surface of a particle, and Ω is the surface of a particle. The stream function-vorticity method was used to simulate flow in a multiconnected domain that involves moving particles. In order to avoid imposing boundary conditions at moving boundaries of settling particles, Geng and Kuznetsov (2007) used a numerical method which was similar to that used in Liu and Wang (2004). The grid systems were formulated by using the Chimera method (Chattot and Wang 1998; Houzeaux and Codina 2003; Zhang et al. 2008). The Chimera method was implemented by generating subgrids around the moving particles. A global rectangular grid was also generated in the computational domain for resolving the global flow field. Governing equations for the global grid

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and the two subgrids are solved separately. Information was exchanged between the global grid and subgrids by means of the interface (hole and outer) boundaries, back and forth with every iteration. The results obtained in Geng and Kuznetsov (2007) suggest that particle sedimentation changes the shape and location of the bioconvection plume. Settling of a single particle on one side of the plume pushes the bioconvection plume away from the particle. Increasing geometric similarity about the plume center for multiple particle sedimentation (if the particles are released symmetrically on both sides of the plume) decreases the plume displacement, while decreasing this similarity (if the particles are released on the same side of the plume) increases the plume displacement. The particles’ settling is also affected by bioconvection; the particles do not move strictly downward; the plume pushes particles in both vertical and horizontal directions.

8.1.5

Methods of Controlling Bioconvection: Effect of High-Frequency Vertical Vibrations In lab experiments, it is often important to control bioconvection. If the upswimming harvest (Kessler 1986) is used to separate vigorously swimming microorganisms, bioconvection should be suppressed, since it would prevent upswimming microorganisms from concentrating at the top of the chamber. On the other hand, if bioconvection is used to induce mixing in microvolumes of fluids, it should be enhanced. Recent investigations of the effect of vibrations on thermal and thermo-solutal convection in horizontal layers and rectangular enclosures are reported in Bardan and Mojtabi (2000), Bardan et al. (2001, 2004), Cisse et al. (2004), Mojtabi et al. (2004), Benzid et al. (2009), and Saravanan and Sivakumar (2011). Hsu et al. (2016) experimentally studied natural convection in an enclosure with pin fins. They investigated whether mechanical oscillations can enhance heat transfer. The utilization of vertical vibrations to control bioconvection was studied in Kuznetsov (2005b, 2006a, b). The effects of vibrations on bioconvection were further studied in Sharma and Kumar (2012, 2014). Kuznetsov (2006a) investigated the effect of high-frequency low-amplitude vibrations on the stability of a suspension of oxytactic microorganisms. These microorganisms swim up the oxygen concentration gradient (Hillesdon et al. 1995; Hillesdon and Pedley 1996; Metcalfe and Pedley 1998, 2001). It was assumed that vertical vibrations imposed on the system change neither the oxytactic behavior of microorganisms nor their average swimming speed. This assumption was reasonable because the vibrations were assumed to be highfrequency and low-amplitude. Linear stability analysis carried out in Kuznetsov (2006a) shows that highfrequency vertical vibrations produce a stabilizing effect on the suspension of oxytactic microorganisms. This suggests that high-frequency vibrations can be utilized to suppress bioconvection in lab experiments. The stabilizing effect of vertical vibrations was explained as follows. The vibration-induced flow is opposite

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to the flow induced by buoyancy forces (i.e., the flow induced by bioconvection). Therefore, vertical vibrations reduce the overall circulation and stabilize the system.

9

Conclusions

The field of natural convection has grown significantly over the last decade. Traditionally, the field was about establishing correlations for the Nusselt number for various geometries and various types of fluids (including non-Newtonian fluids), for flows driven by temperature variations. The recent expansion of the field includes MHD convection, bioconvection, and nanofluids. These recent advances are dominated by numerical work. Future directions should include more experimental validation of computer models that are being developed, and also careful estimation of parameter values used for the numerical models. Acknowledgment The author acknowledges with gratitude the support of the National Science Foundation (award CBET-1642262) and the Alexander von Humboldt Foundation through the Humboldt Research Award.

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Heat Transfer in Rotating Flows

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Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Methods and Basic Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Similarity Numbers for Rotating Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Experimental Investigations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Governing Equations and Theoretical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Computational Fluid Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Rotating Bodies of Revolution in an Infinite Resting Fluid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Flow and Heat Transfer Regimes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Effect of Prandtl Number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Numerical Investigations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Rotating Spheres . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Enclosed Rotating Disk Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Flow Configurations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Heat Transfer from Enclosed Rotating Disks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Unsteadiness and Numerical Investigations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Rotating Disks Subjected to External Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Perpendicular Jets on Rotating Disks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Rotating Disk Subjected to a Parallel Stream of Air . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Critical Point and Bifurcation Theory: Fundamental Concepts . . . . . . . . . . . . . . . . . . . . . . 5.4 Inclined Rotating Disk in a Stream of Air . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

Convective heat transfer in rotating flows is of great technical and scientific importance. Two kinds of configurations, namely, bodies of revolution spinning in a fluid and rotor-stator disk systems, are considered in this chapter. In many S. aus der Wiesche (*) Department of Mechanical Engineering, Muenster University of Applied Sciences, Steinfurt, Germany e-mail: [email protected] # Springer International Publishing AG, part of Springer Nature 2018 F. A. Kulacki (ed.), Handbook of Thermal Science and Engineering, https://doi.org/10.1007/978-3-319-26695-4_12

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cases, not only centrifugal but also Coriolis force contributions play a significant role, and the boundary layer flow is essentially three dimensional. In this case, the rotating flow and heat transfer cannot be described by a simple change of the reference frame and very complex and unexpected phenomena can be found. A substantial difficulty is given by the fact that the number of input parameters is typically rather large in case of rotating systems subjected to an outer forced flow. Then, not only the rotational Reynolds number and the Prandtl number are important for the resulting heat transfer but also the translational Reynolds number and further input variables like angle of incidence or partial admission factors. In this chapter, experimental, theoretical, and recent numerical methods are reviewed. The following discussion is limited to an incompressible Newtonian fluid. Selected results of current research projects are discussed, too. The phenomena arising from natural convection or heat transfer in a rotating fluid heated from below might be found in ▶ Chap. 16, “Natural Convection in Rotating Flows.” Nomenclature

a a aT Aijk B C Ccr Cf CM d D f F g G G Gr h h hm H K L m mcr n* N Nu

Annular domain parameter Potential flow parameter Thermal diffusivity Tensor (Taylor series for velocity) Angular velocity ratio (Correlation) constant Correlation constant (Landau model) Friction coefficient Moment coefficient Diameter Sphere or nozzle diameter Function Self-similar function Acceleration due to gravity Dimensionless cavity height Self-similar function (azimuthal component) Grashof number Heat transfer coefficient Cavity height Mean heat transfer coefficient Self-similar function Heat transfer correlation constant Characteristic length Correlation exponent Correlation exponent (Landau model) Exponent for temperature distribution function Velocity ratio (potential flow) Nusselt number

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Heat Transfer in Rotating Flows

Num p P Pr qi q_ w r R Rcr Rm Re Reh ReL Reu Reδ Reω Reω,r Ro Sc Sh t T Ta u ui U x xi y z

Mean Nusselt number Pressure Self-similar function (pressure) Prandtl number Heat flux component in i-direction Heat flux Radial coordinate Radius Critical ratio between Reynolds numbers (Landau model) Curvature parameter Reynolds number Reynolds number based on cavity height h Reynolds number based on length scale L Inflow or translational Reynolds number Reynolds number based on boundary layer thickness Rotational Reynolds number Local rotational Reynolds number Rosby number Schmidt number Sherwood number Time Temperature Taylor number Velocity (component) Velocity component in i-direction Characteristic velocity Coordinate Coordinate in i-direction Coordinate Axial coordinate

Greek Symbols

α β βT δ e ζ ζ λ Ʌ μ ν ρ

Angle Angle of attack, incidence Thermal expansion coefficient Boundary layer thickness Gap width Self-similar variable Normal coordinate Thermal conductivity Order parameter (Landau model) Dynamic viscosity Kinematic viscosity Density

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φ Ψ τ ω Ω

S. aus der Wiesche

Azimuthal angle Control parameter (Landau model) Shear stress Angular velocity (disk) Angular velocity (flow)

Subscripts

cr f i j lam mnc nc r tur tr w z 0 1

Critical Film i-direction (i = 1, 2, 3) j-direction ( j = 1, 2, 3) Laminar Mean Natural convection Radial Turbulent Transition Wall Axial Reference, nominal Infinity, ambient, bulk

Mathematical Symbols

ΔT Δui F0 f

1

Temperature difference Laplace operator for velocity component ui Derivative of self-similar function F Normal component of f

Introduction

Flow and heat transfer characteristics of rotating systems are not only of great theoretical interest but are also of major practical importance. In the first half of the twentieth century, practical applications of heat transfer in rotating systems were mostly confined to cooling of conventional rotating machinery, such as simple electric motors. With the advent of the cooled gas turbine as prime mover of modern power plants and aircraft engines, the better understanding of heat transfer in rotating systems became essential for a still growing industrial sector. In addition, many new technical applications involving rotating systems were designed leading to flow and heat transfer phenomena under conditions vastly different from those encountered in conventional rotating machinery. For instance, the possibility of cooling the nose of space vehicles during the reentry flight by a fluid was envisioned in the late 1950s. In

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space vehicles, heat transfer to rotating cryogenic fuels occurs. Rotating heat exchangers are now operating on a variety of different principles in the chemical and automotive industry. New cooling issues have arisen with the introduction of computer hard disks. The thermal design of high-speed gas and oil journal bearings introduce new challenges. It is certainly not an exaggeration to say, that even in case of conventional rotating machinery, the current design and operation issues are mainly governed by heat transfer phenomena from rotating fluids. From a scientific point of view, the importance of flow heat transfer characteristics of rotating flow has been recognized very early by Lord Kelvin (1880) among others. The classical book by Lamb (1916) provides a good review about the early pioneering papers. Probably the most influential paper about the flow over a rotating disk has been written by von Karman (1921). He obtained an exact solution of the Navier-Stokes equations by means of a similarity approach. Later, his pioneering approach was substantially extended by other researchers. A mathematical review of the so-called von Karman swirling flows has been given by Zandbergen and Dijkstra (1987). This classical configuration offers a good starting point for illustrating major phenomena occurring in rotating flow systems. The laminar velocity field induced by a rotating disk placed in an infinite quiescent fluid is shown in Fig. 1. Although the configuration and the flow are axisymmetric, all three velocity components occur. A rather special property of the laminar flow over a rotating disk is its constant boundary layer thickness δ = (v/ω)1/2 with the kinematic viscosity ν and the spinning rate ω of the disk. This defines a natural length scale despite the disk geometry or its dimensions, and it enables the introduction of a dimensionless self-similarity variable ζ = z/δ with the axial coordinate z. Using ζ as new variable, von Karman (1921) obtained his famous exact solution of the Navier-Stokes equation. Mathematical details about this self-similarity solution will be presented later. For the present introduction, some other features of the flow and heat transfer from a rotating disk may be mentioned briefly. The laminar flow is only stable up to a critical value Reω,r,cr of the rotational Reynolds number Reω,r = ωr2/v. For sufficient large local Reynolds numbers Reω,r > Reω,r,cr, a transition from laminar to turbulent flow can be observed. The classical stability analysis has been given by Gregory et al. (1956). They demonstrated the three-dimensional Fig. 1 Laminar velocity field and temperature profile over a heated rotating disk in an infinite quiescent fluid

z T∞ uϕ

T

ϕ Tw

ω

ur

r

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character of the transition that differs from the classical two-dimensional boundary flow past a stationary surface. Even in the fully turbulent flow state, remarkable phenomena can be observed in the vicinity of a rotating surface like the mechanisms promoting sweeps and ejections that play a role in shear-stress production (Littell and Eaton 1994). Many of the above phenomena can be observed in other rotating disk systems. However, the fluid mechanics of an enclosed rotating disk or of parallel corotating disk systems are distinctly different from those of a free rotating disk in an infinite quiescent fluid. In the case of rotor-stator systems, several important types of flow configurations exist which have no analogy in the flow induced by a free rotating disk. This class of rotating disk systems occurs frequently in gas turbine engineering. With the advent of air-cooled gas turbines, the investigation of flow and heat transfer phenomena due to rotor-stator interactions became mandatory for the gas turbine industry. Another substantial extension of the classical von Karman problem is given by a free rotating disk subjected to an outer forced flow. In the case of an axisymmetric configuration, i.e., a forced flow perpendicular to the disk surface, the self-similarity approach offers still a powerful tool for analyzing flow and heat transfer. But in the case of a parallel flow over a rotating disk with finite thickness, an extremely complicated flow field caused by the interaction of the translational and the rotational flow contributions occurs. This is illustrated by means of Fig. 2 where an example of the instantaneous streamline pattern of the flow past a rotating disk subjected to an outer parallel stream is shown. The streamline pattern has been numerically computed from a large-eddysimulation (LES) and it corresponds to a rotating disk with radius 20 cm placed in the test section of a wind tunnel with 0.3 m/s inflow velocity. Due to the finite thickness of the disk, a separation bubble at the leading edge occurs that is clearly visible in Fig. 2. The interaction of the rotating boundary layer over the disk surface with the separated and reattached turbulent flow leads to highly unsteady phenomena. In certain respects, the flow in the vicinity of a rotating surface resembles the flow over a stationary surface if the frame of reference of the observer is moving with the Fig. 2 Flow over a free rotating disk subjected to an outer parallel stream (Results of a LES study conducted by J. Turnow, University of Rostock)

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Heat Transfer in Rotating Flows

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rotating surface. Furthermore, there is an analogy between rotating and stratified fluids as pointed out by Veronis (1970). However, in many cases, substantial additional flow phenomena arise due to the action of centrifugal or Coriolis forces or due to the very special nature of a three-dimensional boundary layer flow. Then, the resulting flow characteristics can be very complex, and according to Kreith (1968), “almost every rotating system reveals novel and unexpected flow characteristics when subjected to a complete stability analysis or when studied experimentally over a sufficiently wide range of all the variables which affect the hydrodynamic phenomena.” Since in rotating systems convective heat transfer and flow are intimately related, they are also very complex, and interesting scientific and practical phenomena occur. It is therefore not surprising that much research has been dedicated to flow and heat transfer in rotating systems. Since the pioneering papers, many articles dealing with flow and heat transfer in rotating systems has been written including excellent reviews and monographs. Dorfman (1963) prepared the first comprehensive collection of the state of knowledge of heat transfer and flow in rotating systems until 1958. Some years later, Kreith (1968) reviewed convection heat transfer in rotating systems while focusing on the progress made after Dorfman. Details of the hydrodynamics and the heat and mass transfer near rotating surfaces were reviewed by Moalem Maron and Cohen (1991). Excellent monographs about rotating disk systems were prepared by Owen and Rogers (1989) and Shevchuk (2009). The complex flow phenomena occurring in case of a rotating free disk subjected to an outer stream of air was considered by aus der Wiesche and Helcig (2016). A review of fluid flow and convective heat transfer within rotating disk cavities was given by Launder et al. (2010) and Harmand et al. (2013). The following chapter assumes that the reader is familiar with the classical literature about boundary layer theory and to some extend with the very basic material contained in the abovementioned reviews. However, some of the basic material will also be repeated, and the main ideas behind the different approaches will be briefly presented. In case of further interests, the reader is invited to consult the cited reviews and the corresponding original papers.

2

Methods and Basic Principles

Prior to a discussion about the various and complex heat transfer phenomena occurring in rotating flows, it is useful to briefly review the available methods and basic principles for investigating rotating flows. In general, experimental, theoretical, and numerical methods can be distinguished. The latter is also known as the “third way” established by computational fluids dynamics (CFD). Since there is a tremendous progress with regard to CFD methods, this approach is rather promising today. However, experimental data and theoretical considerations are still the basic source of understanding. A powerful tool for describing global complex heat transfer phenomena in rotating flows is given by the application of bifurcation theory. This theoretical approach for correlating empirical data will be discussed later in more detail in connection with three-dimensional flow phenomena.

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Similarity Numbers for Rotating Flows

The general behavior of all fluid mechanical systems can be best understood through dimensional analysis. This is the formal procedure whereby the group of involved variables representing the flow and heat transfer configuration is reduced to a smaller number of dimensionless groups (e.g., Taylor 1974). When the number of independent variables is not too large, dimensional analysis enables a clear and transparent picture. In the case of flow and convective heat from rotating disk systems, the following groups are useful based on the main parameters of the considered problem. The most important parameter for rotating flows characterized by a length scale R (e.g., the disk radius) and an angular velocity ω is the rotational Reynolds number Reω ¼

ωR2 : v

(1)

ωr 2 v

(2)

A local rotational Reynolds number Reω,r ¼

based on the radial coordinate r is also frequently used in case of rotating disk systems. An alternative definition of the rotational Reynolds number is based on the boundary layer thickness δ = (v/ω)1/2 and is given as ratio ReL = (L/δ)2 with a suitable length scale L of the configuration under consideration (e.g., a distance between corotating disks). An outer forced flow is described by an inflow or translational Reynolds number u1 R v

Reu ¼

(3)

based on the velocity u1 of the outer uniform forced flow. The thermal fluid properties are generally described by the Prandtl number Pr ¼

v , aT

(4)

with the thermal diffusivity aT of the fluid. The mean heat transfer coefficient hm is useful for the definition of a mean or average Nusselt number Num ¼

hm R λ

with hm ¼

q_ w, m T w, m  T 1

(5)

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Heat Transfer in Rotating Flows

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and a mean wall heat flux q_ w, m . Its value can be obtained by integration over the corresponding surface. In the case of a heated disk with radius R, it calls 1 q_ w, m ¼ 2 πR

2ðπ ð R

q_ w r dr dφ:

(6)

0 0

Instead of a mean Nusselt Number Num, the use of a local Nusselt number Nur ¼

q_ w r λðT w  T 1 Þ

(7)

is also common, employing local values at the radial coordinate r. In the case of concentric cylinders with a gap distance d, the Taylor number Ta ¼

ω2 R d 3 v2

(8)

is a suitable parameter particularly for stability analysis (Taylor 1923). In order to establish similarity between laboratory experiments and large-scale convection phenomena in nature, the Rosby number Ro ¼

U ωR

(9)

with a characteristic (wind) velocity U and the radius R of the rotating system is found to be an appropriate dimensionless group. The Rosby number Ro can be also interpreted as the ratio between the convective acceleration and the Coriolis contribution.

2.2

Experimental Investigations

Heat transfer coefficients or Nusselt numbers Nu are commonly obtained using two different experimental approaches: (i) the use of an electrically heated disk and (ii) the naphthalene sublimation technique. Whereas the first approach directly yields heat transfer quantities, the second approach permits only the determination of mass transfer quantities, and a further data reduction or recalculation is necessary to obtain corresponding Nusselt numbers. Despite the extra calculations and limitations, the naphthalene sublimation technique itself has been used for several decades for the investigation of rotating disk systems. It yields the Sherwood number Sh as a function of the involved Reynolds number Re and the Schmidt number Sc. A statement about the corresponding heat transfer, i.e., the Nusselt number Nu, can be obtained on the basis of the analogy between heat and mass transfer. This assumption leads to the relation

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Pr Nu ¼ Sh  Sc

m :

(10)

Typically, the exponent m in Eq. 10 is assumed to be constant and not a function of Pr and Sc. For air at atmospheric conditions, the values Pr = 0.71 and Sc = 2.28 for naphthalene are appropriate. In line with the methodology by Cho and Rhee (2001), the exponent should be equal to m = 0.4. However, considerable variations of its value ranging from m = 0.31 up to 0.58 can be found in the literature (see, for instance, Shimada et al. 1987 or Cho et al. 2002). As discussed by Shevchuk (2008), it is obvious that an error in the choice of the value of m can lead to poor results for predictions of the Nusselt number. Furthermore, the uniform and reproducible preparation of the surfaces is not free from difficulties in the naphthalene sublimation technique. In spite of these issues, this method should be carefully considered in cases where absolute statements about Nusselt numbers are required. In their review, Harmand et al. (2013), concluded, that at present most of the authors use direct thermal measurements to determine the convective heat transfer. The use of this approach, i.e., an electrically heated disk and the conduction of temperature and heat transfer measurements, circumvents the typical issues arising with the naphthalene technique, but substantial efforts are required to design and manufacture suitable test set-ups. Such a rotating disk apparatus was successfully introduced by Cobb and Saunders (1956). The mean heat transfer coefficients were found directly by measuring the heat input and surface temperature of the disk after steady state had been achieved. The following issues are critical for obtaining highly accurate data with such an apparatus: (i) Uniform or well-defined surface temperature distributions (ii) Elimination of additional heat losses not covered in the data reduction and heat balance calculation (iii) Accurate surface temperature measurements and low uncertainty levels for the data recording system Thermal measurements are frequently performed by means of thermocouples, but a few studies, for instance, Kakade et al. (2009) also use thermochromic liquid crystals (TLC). A challenge of the TLC technique arises from the precise calibration. Today, many authors use infrared thermography for thermal measurements. Usually, the surface under investigation is covered with high emissivity black paint. Precise calibration of the thermography system is crucial, too. In case of free rotating disks with easy optical access to the surface, this method can be very efficient, as demonstrated by Cardone et al. (1997). In the case of enclosed rotating disks or in configurations where reflections can occur, the infrared thermography approach is seriously limited. All of the thermal measurement techniques are faced with the question of the adiabatic wall temperature which is the surface temperature of the system without any heat transfer. In the case of supersonic flow, the deviation between the fluid temperature, the adiabatic temperature, and the measured temperature can be substantial; in typical rotating flow heat transfer measurements

15

Heat Transfer in Rotating Flows

657

reported in the literature, the error introduced by using the ambient fluid temperature as reference is typically negligible. The present chapter focuses on forced convection effects and convective heat transfer due to forced flows. However, for heat transfer experiments with actual disks, the contribution due to natural convection to the observed heat transfer has to be taken into account. Without clarifying the effect of natural convection, experimental data might be seriously misinterpreted. Natural convection plays a dominant role in the case of the very low forced flows that result from very low disk-running speeds or stream velocities. For describing natural convection, the Grashof number Gr ¼

gβT ð2RÞ3 ΔT v2

(11)

defined by a length scale R and gravity g can be employed as a suitable dimensionless group. The temperature difference is given for an isothermal surface by means of ΔT = Tw – T1; the bulk modulus of expansion βT and the physical properties of the fluid is frequently evaluated at the so-called film temperature Tf = (Tw + T1)/2. In the case of significant temperature dependency, the physical properties have to be evaluated more carefully. Since natural convection has a major impact in low forced flow configurations, it is possible to limit the discussion to the laminar forced flow regime in the following. The effect of natural convection can be assessed experimentally by means of measurements of the actual mean Nusselt number Num for low forced flow velocities or corresponding Reynolds numbers Re. For a given disk apparatus with radius R, it is useful to plot the obtained Num/Re1/2 against Gr/Re2 in accordance with the scheme shown in Fig. 3. For a rotating disk in still air, the Reynolds number is given by means of Re = Reω = ωR2/v. For a resting disk subjected to a stream of air, the Reynolds number would be Re = Reu = u1R/v in Fig. 3. For small Grashof numbers Gr in comparison to Re2, the effect of natural convection is of minor importance, and the experimental data should be in reasonable agreement with the theoretical curve of forced laminar flow. For large Grashof numbers Gr or small Reynolds numbers Re, the heat transfer is dominated by natural convection, and the experimental data for the actual Nusselt number Num should be in accordance with a natural convection correlation containing only the Grashof Fig. 3 Effect of natural convection on forced convection of a rotating disk (From aus der Wiesche and Helcig 2016)

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number. For a small perpendicular disk at constant surface temperature, Mabuchi et al. (1971) recommended the Izumi correlation Num, nc ¼ C  Gr1=4 ¼ 0:60 Gr1=4

(12)

for calculating the mean Nusselt number of natural convection. For other configurations and fluids, the reliability of this correlation should be carefully checked using measurements. Typically, the numerical value of the constant C in correlation (12) has to be determined experimentally. In the case of rotating disks subjected to an outer forced uniform flow, a wind tunnel has to be used. A wind tunnel is a tool used in research to study the effects of air moving past solid objects. It consists of a test section with the test object mounted in the center. Air is forced past the object using a powerful fan system. Reviews of the many types of wind tunnels including low-speed, high-speed, intermittent blowdown, and suction tunnels are available in the literature, e.g., Barlow et al. (1999). A detailed description of heat transfer experiments with rotating disks placed in the test section of a wind tunnel can be found elsewhere (aus der Wiesche and Helcig 2016). In many experiments for convective heat transfer effects, the quality of the air stream generated by the wind tunnel is not crucial, and comparably simple facilities can be used. For experiments involving rotating disks with finite thickness, certain flow and heat transfer regime transitions can only be observed in stream of airs with fairly low inflow turbulence over the entire test section. This should be ensured by the selection of the wind tunnel. The vast majority of heat transfer experiments with rotating flows involves dry air as fluid. Experiments as reported by Helcig et al. (2016) with water as working fluid are an exception due to the substantial efforts required for performing corresponding heat transfer experiments. On the other hand, the effect of the Prandtl number on the convective heat transfer can only be assessed by employing other fluids than air or gases. In literature, there is still a substantial lack of experimental research about the effect of the Prandtl number. Since rotating flows are frequently characterized by three-dimensional effects, the well-known results from classical boundary layer theory and essentially two-dimensional flows are only of limited value. This will also be pointed out later.

2.3

Governing Equations and Theoretical Methods

Under the assumption of an incompressible, Newtonian fluid with constant material properties, the governing flow equations for the velocity components ui and pressure p are given by the Navier-Stokes equations that read in the usual Cartesian tensor notation @ui @ui 1 @p μ @ 2 ui þ uj ¼ þ ρ @xi ρ @x2j @t @xj

(13)

15

Heat Transfer in Rotating Flows

659

and the continuity equation @ui ¼ 0: @xi

(14)

Since heat transfer is involved, the energy equation has to be considered which, under the aforementioned assumptions, can be written as a partial differential equation @T @T @2T þ ui ¼ aT 2 @t @xi @xi

(15)

for the temperature field T in the flow domain. In addition to the governing Eqs. 13–15, appropriate boundary conditions have to be provided. For rotating solids, the no-slip condition is usually applied to the velocity field. With regard to heat transfer, prescribed wall heat flux q_ w or wall temperature Tw distributions are common boundary conditions. Since the classic investigation by Dorfman, the assumption of a temperature distribution in accordance to a power law 

T w ¼ T 0 þ C rn

(16)

with constant parameters T0, C, and exponent n* is common for analytical treatments. The isothermal case with constant wall temperature Tw corresponds to n* = 0. For axisymmetric configurations, such as free rotating disks in a resting fluid or flow impingement onto orthogonal disks, the use of a cylindrical coordinate system is convenient. The rotation axis of the disk is typically chosen as z-coordinate axis, while the point z = 0 is located at the disk surface, see Fig. 1. In this case, the governing equations for stationary flow are explicitly given by the following set of partial differential equations: ur

@ur @ur u2φ 1 @p μ  ur  þ Δur  2 , þ uz  ¼ ρ @r ρ @r @z r r

(17)

@uφ @uφ ur uφ μ  uφ  Δuφ  2 , þ uz þ ¼ ρ @r @z r r

(18)

@uz @uz 1 @p μ þ Δuz , þ uz ¼ ρ @z ρ @r @z

(19)

ur

ur

@ur ur @uz þ þ ¼0 @r r @z

(20)

with the operator Δ¼

@2 1 @ @2 þ þ : @r 2 r @r @z2

(21)

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Fig. 4 Flow over a free rotating disk subjected to an orthogonal stream

z

T∞ u∞

T∞

ur,∞

Tw

ω

r

d=2R

In a rotating coordinate system (i.e., a frame fixed to the disk), the governing equations include additional Coriolis force terms. For further details about reformulating the corresponding Navier-Stokes equations in moving or rotating frames, the reader is referred to Shevchuk (2009). For laminar flows over a single rotating disk, exact solutions of the Navier-Stokes equations can be obtained by means of the concept of self-similar solutions. Since the pioneering work by von Karman (1921), this class of exact solutions has been also termed von Karman swirling flows, and a review of its mathematical properties has been given by Zandbergen and Dijkstra (1987). For a free rotating disk subjected to a perpendicular stream of air, as shown in Fig. 4, the following selfsimilar variables are introduced: ur ¼ ða þ ωÞ rFðζ Þ, pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi uz ¼ ða þ ωÞvH ðζ Þ,

(22)

uφ ¼ ða þ ωÞr Gðζ Þ,

(24)

p ¼ ρvωPðζ Þ

(25)

(23)

with the new dimensionless axial coordinate ζ¼z

rffiffiffiffiffiffiffiffiffiffiffiffi aþω : v

(26)

The functions F, G, H, and P obey a set of ordinary differential equations F2  G2 þ F0 H ¼

N 2  B2 ð1 þ N Þ2

þ F00 ,

(27)

15

Heat Transfer in Rotating Flows

661

2FG þ G0 H ¼ G00 ,

(28)

HH0 ¼ P0 þ H00 ,

(29)

2F þ H0 ¼ 0,

(30)

derived from the boundary layer equations. The potential flow variable a is linked to the assumed external flow ζ ! 1 : ur, 1 ¼ a r, uz, 1 ¼ 2a z, uφ, 1 ¼ Ωr

(31)

at infinity and has the dimension of 1/s. The ratio between the disk angular velocity ω and the outer flow rotation Ω is denoted by parameter B = Ω/ω and is assumed to be constant. The nondimensional radial velocity in the potential flow outside the boundary layer is given by N = ur,1/ωr = a/ω. At the disk surface, the boundary conditions are ζ ¼ 0 : F ¼ H ¼ 0, and G ¼ 1:

(32)

In the past, the above set of equations has been solved by means of expansions in power or exponential series and by using the shoot-method. Today, modern mathematical software tools enable user-friendly handling of such sets of ordinary differential equations. The exact solutions provide a reliable database for validation studies of CFD methods. Using self-similar solutions also provides a good method for obtaining approximate analytical solutions for other configurations, but it should be remarked that this approach has been mainly replaced by numerical methods (CFD) in engineering applications. Furthermore, one has to keep in mind, that the similarity equations of von Karman flows do not generally have unique solutions as shown by Zandbergen and Dijkstra (1987). This rather interesting mathematical property was first discovered numerically by Mellor et al. (1968) who considered the flow between a rotating and a stationary disk. For that special configuration, two independently obtained solutions, namely, the so-called Batchelor and Stewartson flows, had been available during the 1950s (see also the later section about rotorstator systems in this chapter). The question about their significance and which solution was the “right” one was disputed in the past. This controversy has been resolved by recognizing the existence of multiple solutions (Kreiss and Parter 1983). In fluid mechanics, boundary layer theory is of fundamental importance, and many problems can only be treated successfully within its framework (e.g., Schlichting 1968). In the case of a rotating disk in still air, the usual boundary layer assumptions are: (i) The axial velocity component uz is by an order of magnitude lower than the other components. (ii) The change of velocity, pressure, and temperature in the normal direction is much larger than in the radial direction. (iii) The static pressure p is constant in the normal direction.

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Under these assumptions, it is possible to replace the full Navier-Stokes equations by the following set of boundary layer equations ur

@ur @ur u2φ 1 @p μ @ 2 ur þ þ uz  ¼ , ρ @r ρ @z2 @r @z r

(33)

@uφ @uφ ur uφ μ @ 2 uφ þ uz þ ¼ , ρ @z2 @r @z r

(34)

ur

@p ¼ 0: @z

(35)

The continuity Eq. 20 remains unchanged. The equation for a stationary thermal boundary layer is ur

@T @T @2T þ uz ¼ aT 2 : @r @z @z

(36)

The above set of equations is closed by the equation 1 du2r,1 u2φ,1 1 dp1  ¼ 2 dr ρ dr r

(37)

that connects the flow variables in the potential flow domain with the boundary layer region. For parallel or inclined disks with finite thickness, wakes and three-dimensional flow separation become essential, as illustrated by means of Fig. 2. While the phenomenon of separation in two-dimensional flow is fairly well understood, the situation in three dimensions is much more complicated and far from being clear. Furthermore, a moving solid boundary wall (i.e., the rotating disk surface) changes the separation condition. In this case, Prandtl’s well-known two-dimensional separation condition  @u ¼0 @z w

(38)

at the wall has to be replaced by the so-called MRS-criterion (Moore 1958; Rott 1956; Sears 1956): @u ¼ 0 at regions with vanishing velocity u ¼ 0, @z

(39)

satisfied typically within the flow domain. The separated flow is usually unsteady, and the laminar boundary layer theory does not provide an efficient approach for calculating the entire flow field. For these types of flows and phenomena occurring in the case of rotating disk systems subjected to an outer stream, the critical point and

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bifurcation theory offers a valuable approach. This will be discussed later in some more details in the corresponding subchapter. The flow over a rotating disk is laminar for sufficiently low local Reynolds numbers Reω,r = ωr2/v. Lingwood’s (1996) stability analysis has shown how the onset of transition over a rotating disk occurs at 502 < Reω,δ2 < 513. The length scale of the Reynolds number Reω,δ2 is defined by the boundary layer momentum thickness δ2. Typically, three-dimensional turbulent boundary layers are formed from initially two-dimensional flow due to action of spanwise pressure gradients or shearing forces. The turbulent boundary layer flow past a rotating disk represents a unique exception because it is three dimensional from its onset. Its underlying structure does not result from a perturbation of an initially two-dimensional flow. Therefore, the flow past a rotating disk offers an ideal starting point for investigations into the underlying structures of three-dimensional turbulent boundary layers, and valuable experimental data have been obtained in this area by Eaton and coworkers (see, for instance, Littell and Eaton 1994; Elkins and Eaton 2000).

2.4

Computational Fluid Mechanics

Computational fluid dynamics, usually abbreviated as CFD, has become a “third” approach in addition to the classic analytical treatment and the experimental investigation of flow and heat transfer phenomena. CFD is a branch of fluid mechanics that uses numerical methods and mathematical algorithms to solve and analyze problems that involve flow phenomena. This approach is especially attractive since powerful computers for performing the calculations are now widely available. The direct numerical simulation (DNS) of turbulent flows resolving the entire range of turbulent length scales is usually still not feasible at high Reynolds numbers for the majority of applications, and appropriate simulation strategies for such flows are still required. In industry and in many research institutions, the Reynolds averaged Navier-Stokes (RANS) equation approach dominates. It is the oldest approach to turbulence modeling. An ensemble version of the governing equations is solved introducing new apparent stresses known as Reynolds stresses. This approach adds a second order tensor of unknowns for which various models can provide different levels of closure. Details about the approach can be found in various books, e.g., Launder and Spalding (1972) or Tennekes and Lumley (1972). RANS simulation can be extremely powerful for calculating time-averaged flow quantities in statistically stationary turbulent flows. On the other hand, many real flows are characterized by major unstationary phenomena such as vortex shedding. In these cases, the classic RANS approach is not sufficient, and substantial efforts are required to extend the RANS formalism. Currently, wide use is made of the so-called transient or unsteady RANS methods (URANS) for complex turbulent flows. URANS models are also offered by many commercial CFD codes. The fundamental problem is that URANS solutions are often not rigorously justified in terms of flow physics, and the significance of the so-computed solutions cannot be checked.

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URANS simulations can be interpreted as loosely resolved large-eddy-simulations (LES), as remarked by Lesieur et al. (2005). LES is a technique in which the smallest scales of the flow are removed through a filtering operation, and their effect is modeled using subgrid scale models. The history of LES began in 1963 with the proposal of the first eddy viscosity model (Smagorinsky 1963). The method requires substantially greater computational resources than RANS methods but is far cheaper than direct numerical simulations (DNS). It is not possible to cover all LES aspects in a single chapter; LES has become a discipline to itself. LES is particularly useful for investigating the spatiotemporal variations in turbulent boundary layer flows, for which the simple empirical correlations or RANS methods do not offer an adequate approach. Today, LES is considered as a powerful tool for investigating flow and heat transfer phenomena involved in rotating disk systems in greater detail, but the choice of the appropriate subgrid model and the influence of the computational mesh always require special attention.

3

Rotating Bodies of Revolution in an Infinite Resting Fluid

The basic configuration of a free rotating disk in an infinite resting fluid, Fig. 1, can directly be extended by two classes of rotating bodies of revolution, as shown in Fig. 5. An axisymmetric body described by the power function  ð2m1Þ=3 x r ðxÞ ¼ L  L

(40)

is shown in Fig. 5a. The dimensions of such bodies are given by the length scale L; x is the distance from the nose, and r is the radius of revolution. The special value m = 2 yields the disk. Another generalization of the disk is given by the cone shown in Fig. 5b. Here, the relation

Fig. 5 Bodies of revolution: (a) body described by a power function and (b) cone

a

z

z

b ω

ω

r

r x

α

x

15

Heat Transfer in Rotating Flows

665

r ðxÞ ¼ x  sin α

(41)

holds. In body-fitted orthogonal coordinates ðx,y,zÞ the laminar boundary layer equations can be reformulated, and a broader class of similarity solutions becomes available (Geis 1956). Further details about this classical laminar flow problem and its solution are provided by Kreith (1968). Even in the case of turbulent flows where the similarity approach fails, major results of the flat disk configuration can be adapted to both configurations, if a transformation to local coordinates is performed. The shapes of spheres and cylinders are not covered by Eq. 40 or Eq. 41; the flow and heat transfer phenomena of rotating spheres and cylinders have to be treated separately.

3.1

Flow and Heat Transfer Regimes

The different flow and convective heat transfer regimes related to spinning bodies of revolution in an infinite resting fluid are best illustrated by a plot of the mean Nusselt number Num against the rotational Reynolds number Reω ¼

ω L2 sin α ν

(42)

for heated cones and disks, Fig. 6. In the case of a disk, the definition (42) reduces to Eq. 1 with the outer disk radius R as length scale L. A close inspection of Fig. 6 shows that three different regimes exist, namely, a laminar, transitional, and fully turbulent regime. The laminar flow and heat transfer regime can be successfully analyzed by means of the similarity approach. Particularly in the case of a flat rotating disk, very accurate information about the velocity and temperature profiles are available (see, for instance, Shevchuk 2009). A frequently employed correlation for the laminar heat transfer from an isothermal disk rotating in still air (Pr = 0.71) is given by Num ¼ 0:33  Re1=2 ω :

(43)

This laminar boundary layer result can also be used for other generalized bodies of revolution if a modified Reynolds number definition, Eq. 42, is employed. For a fully turbulent flow regime, the mean heat transfer can also be covered by the general correlation Num ¼ 0:0157  Re0:8 ω :

(44)

The exponent value of 0.8 is typical for turbulent boundary layers, Schlichting (1968); small deviations exist with regard to the “exact” values of the constants occurring in Eq. 43 and Eq. 44 in the literature. However, it should be remarked that these deviations are typically of minor importance in comparison with experimental

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1000

Nu m= 0.0157 Re0.8 60° Cone Disk

30° Cone

80° Cone

Nu m 100 Nu m= 0.33 Re1/2

20

104

105 Re = Reω = ω L2 sin α / v

106

Fig. 6 Mean Nusselt number Num against rotational Reynolds number Reω for heated cones and disk in still air (Experimental data adapted from Kreith 1968)

uncertainty levels occurring in actual systems. Between the laminar and the fully turbulent flow regimes, a transitional regime can be observed. For rotational Reynolds numbers within this regime, the surface of the rotating body is partially covered by a laminar zone (around the axis of rotation, i.e., the nose) and partially covered by a transitional zone where the laminar flow state becomes unstable. For sufficiently high rotational Reynolds number, an outer turbulent flow zone occurs. With increasing Reynolds number, that turbulent zone becomes larger and finally dominates the mean heat transfer. However, a central laminar and transitional zone still remains. The simultaneous existence of different heat transfer regimes requires an appropriate way to calculate the mean heat transfer coefficient hm. A frequently used approach is given by 0 1 ðL 1B C hm ¼ 2 @hlam x2cr þ 2 htur ðxÞ x dxA (45) L xcr

with the laminar local heat transfer coefficient hlam (a constant value), the local (turbulent) heat transfer coefficient htur, and the critical distance xcr from the vertex of transition onset (at which Reω = Reω,cr). The approach (45) does not employ a local

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heat transfer coefficient for the transitional zone; in practice, the right choice of the value for the critical distance xcr (i.e., Reω,cr) is important for the accuracy of the resulting correlation (45). An alternative way for calculating the mean Nusselt number Num as function of the local Nusselt numbers including the transitional zone between rcr,1 and rcr,2 is given by 0 Num ¼

2B @ R

ð

ð

rcr, 1

ðR

rcr, 2

Nur, lam dr þ 0

Nur, tran dr þ rcr, 1

1 C Nur, tur drA:

(46)

r cr, 2

This equation is valid for the heat transfer from a free rotating disk with radius R. In the case of other bodies of revolution, analogous formulations can be derived. Experimental results for the local Nusselt number and the radial extension of the transition zone are shown in Fig. 7. Inserting the local Nusselt numbers Nur of Fig. 7 into Eq. 46 and performing the integrations yields the qualitative behavior of the mean Nusselt number Num against the rotational Reynolds number Reω as shown in Fig. 6. Especially in the turbulent flow regime, it is necessary to distinguish between the local and the mean Nusselt numbers.

1000

Nu = Nur

100 106

105 Re = Reω r

Fig. 7 Local Nusselt number Nur against local rotational Reynolds number Reω,r for an isothermal rotating disk in still air (Experimental data adapted from Elkins 1997)

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S. aus der Wiesche

Whereas the laminar flow state can be successfully treated by means of the laminar boundary layer theory, the analysis for fully turbulent flow over a rotating surface requires substantial efforts. The classical method is resting on introducing the usual boundary layer simplifications and taking averages of the fluctuating components of velocity and temperature (Reynolds averaging). This leads to modified momentum and energy equations, and a closure problem results, because the new turbulent fluctuation contributions require suitable modeling. With regard to turbulent heat transfer, the question about the value of the turbulent Prandtl number Prt arises. In many cases, the assumption Prt = 1 is made. Fluids with a Prandtl number of unity, Pr = 1, are interesting from an academic point of view, because Reynolds analogy holds. It is then possible to relate the local Nusselt number Nu to the local friction coefficient τyz 1 ρ r 2 ω2 2

(47)

hðxÞx 1 ¼ Cf , x Reω, x λ 2

(48)

Cf , x ¼

by means of Nux ¼

with the local rotational Reynolds number Reω, x ¼ ω x2 sin α=ν in the case of a cone with angle α. For heated surfaces with a temperature distribution in accordance to Eq. 16, the classical Dorfman correlation Nux ¼ 0:0212  ðn þ 2:6Þ0:2 Re0:8 ω, x

(49)

can be obtained for fluids with Pr = 1. After an integration of Eq. 49 over the entire disk or cone surface, the average turbulent Nusselt number is found to be Num ¼

CM ReL : 2π sin 2 α

(50)

The moment coefficient CM can be expressed empirically by CM ¼ 0:15  Re0:2 ð8%Þ: L

(51)

Inserting Eq. 51 into Eq. 50 yields a correlation analogous to Eq. 44 in the case of a rotating flat disk in a resting fluid with Pr = 1.

3.2

Effect of Prandtl Number

The effect of the Prandtl number Pr on the convective laminar and turbulent heat transfer is of extraordinary importance for many practical applications. Typically, an approach

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669

Num ¼ K m ðPrÞ  RenL or Nux ¼ K ðPrÞ  Renω, x

(52)

is preferred with a “constant” K that is in fact a function of the Prandtl number Pr. The value of the exponent n for the rotational Reynolds number depends on the flow regime. The laminar convective heat transfer regime (where the mean and the local constants are identical, i.e., K = Km,) can be obtained by means of the similarity approach, and the effect of the Prandtl number Pr can hence be calculated numerically for a wide range of values. The most reliable information about those theoretical predictions is currently provided by Shevchuk (2009). In addition to the similarity results, the Dorfman correlation K ¼ K m ¼ C  Pr1=2 for Pr  1

(53)

and the Levich correlation K ¼ K m ¼ 0:66  Pr1=3 for Pr ! 1

(54)

can be obtained as asymptotic results. Sparrow and Gregg (1959) conducted an analytical study of the effect of the Prandtl number on the convective heat transfer in the framework of similarity solution methods; they found the two asymptotic correlations K ¼ K m ¼ 0:88447  Pr1=2 for Pr ! 0

(55)

K ¼ K m ¼ 0:62048  Pr1=3 for Pr ! 1:

(56)

and

In the literature, semitheoretical correlations for K as function of the Prandtl number Pr for a large range of Pr are also available for the laminar regime. The simple empirical expression K ¼ Km ¼ 

0:6Pr 0:56 þ 0:26 Pr1=2 þ Pr

2=3

(57)

was suggested by Lin and Lin (1987). A mathematical analysis of the asymptotic matching (Awad 2008) yields almost the same results as the simple expression (57). In Fig. 8, the theoretical predictions and the available experimental data are compared. It should be remarked that reliable heat transfer experiments for Pr significantly larger than unity have been conducted only recently, whereas the Levich correlation (54) was proven much earlier by means of mass transfer data. In the case of turbulent flow, the identity K = Km does not hold any more, and the local and the mean Nusselt numbers and the effect of the Prandtl number on K and

670

S. aus der Wiesche

Fig. 8 Effect of the Prandtl number Pr on the laminar convective heat transfer over a rotating surface (Data in accordance to Helcig and aus der Wiesche 2016)

Km have to be distinguished. For gases with Prandtl number values close to unity, Dorfman (1963) proposed K ¼ 0:0197  ðn þ 2:6Þ0:2 Pr0:6 and K m ¼ K

n þ 2 : n þ 2:6

(58)

Based on an adaption of von Karman’s three layer scheme for turbulent flow over a solid boundary, Kreith (1968) proposed the correlation hðxÞx ¼ Nux, Pr¼1  Nux ¼ λ

Pr pffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ Cf, x =2ð5Pr þ 5 lnð5 Pr þ 1Þ  14Þ

! (59)

with the local friction coefficient Cf, x ¼ 0:0267  Re0:2 ω, x : 2

(60)

The local Nusselt number Nux,Pr=1 for Pr = 1 can be assessed by Eq. 49. The mean Nusselt number Num for the entire disk or cone has to be calculated by integrating the local Nusselt number Nux and considering the local Reynolds number Reω,x. Kreith’s result (59) indicates a nearly vanishing dependency of Pr in the limit case Pr ! 1. For moderate Prandtl number values, recent heat transfer experiments (Helcig and aus der Wiesche 2016) indicate that the effect of the Prandtl number on

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the mean Nusselt number correlates with Pr0.35. This result is in agreement with the classical boundary layer theory for the turbulent flow past a fixed surface. From an experimental point of view it is very challenging to achieve fully turbulent flow states over a rotating disk in fluids with high viscosity.

3.3

Numerical Investigations

Even the basic flow past a free rotating disk exhibits interesting features that cannot be adequately analyzed by the classic RANS approach. Although the flow configuration of free or enclosed rotating disk systems is often axisymmetric, experiments have revealed the existence of large-scale vertical structures in the turbulent regime, see Czarny et al. (2002). Inside the laminar boundary layer near the rotating disk, all velocity components are nonzero. Wu and Squires (2000) performed an LES analysis of the turbulent flow over a rotating disk with the prediction of the three-dimensional turbulent boundary layer and an investigation of the underlying flow structure as primary aims. They considered a Reynolds number of Reω,r = 6.5  105 (corresponding to Reω,δ2 = 2660). Since it was difficult to incorporate the laminar-to-turbulent transition into the calculation, only a segment (in both radial and azimuthal directions) was chosen as the computational domain for the fully turbulent flow. The use of such a reduced domain requires special attention with regard to the boundary conditions. They employed three subgrid models, namely, the dynamic eddy viscosity model of Germano et al. (1991), the dynamic mixed model of Zang et al. (1993), and the dynamic mixed model of Vrenan et al. (1994). The classic Smagorinsky (1963) model was not considered because previous investigations had shown a better performance of the dynamic models. Wu and Squires pointed out that the prediction of complex flows using LES requires special care with regard to the numerical scheme. Upwind methods introduce a dissipative truncation error, that can act as an additional (unphysical) subgrid model. The governing equations including the subgrid models were solved using a semi-implicit fractional step method in cylindrical coordinates. Second-order central differences were used for approximating spatial derivatives on a staggered grid. The numerical scheme was essentially the same as the one used by Akselvoll and Moin (1996). A series of calculations were performed to validate the overall computational approach. This approach is typical for LES analyses because the somewhat empirical input of the sub-grid models and the numerical scheme must be carefully considered before the results from an LES calculation may be used. The effect of the subgrid models and grid refinement were not significant if the resolution was in a range in which large-scale motions were accurately and well resolved. Wu and Squire provided a detailed comparison of LES predictions with experimental measurements of the turbulence quantities by Littell and Eaton (1994), and generally a good agreement was found. The numerical results supported the structural model advanced by Littell and Eaton (1994), where stream wise vortices with the same sign as the stream wise vorticity are mostly responsible for strong sweep events, whereas stream wise vortices having the opposite sign

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promote strong ejections. Near-wall characteristics of the turbulence intensities and turbulent shear stress in the disk flow were presented and compared with available DNS results (Kim et al. (1987) and Spalart (1988)). They concluded that these disk flow results are not markedly different from their counterparts in canonical two-dimensional flows.

3.4

Rotating Spheres

The flow caused by a spinning sphere in a quiescent fluid has found much attention in the literature since the first investigation by Stokes (1845). The spinning sphere acts like a centrifugal fan, and the flow field consists of a flow outwards from the equator and inwards towards the poles. The above treatment is valid for the bodies of revolution, as shown in Fig. 5, and it cannot directly be applied due to the secondary flow features. The transition from laminar to turbulent flow occurs at a critical Reynolds number of order ReD,cr = 50,000. The mean Nusselt number for the laminar flow regime can be correlated by Num ¼

hm D 1=2 ¼ ð0:33  0:43Þ  Pr0:4  Reω, D : λ

(61)

In the case of fully turbulent flow, a correlation Num ¼ 0:066  Pr0:4  Re0:67 ω, D :

(62)

has proven its reliability for spinning spheres in resting fluids. The empirical value of 0.4 for the exponent of the Prandtl number has been determined on the basis of experiments of spheres in air (Pr = 0.71), water (Pr = 4.52), and oil (Pr = 217). The available experimental data for mercury (Pr = 0.024) are not covered by the approach mentioned above, see Kreith (1968).

4

Enclosed Rotating Disk Systems

The flow phenomena of enclosed (also called shrouded) rotating disks (and also of corotating parallel disk systems) are very different from those of free rotating disks in an infinite resting fluid. Particularly in case of a small height between the disks or the surfaces, new types of confined flow configurations can occur which have no analogy to the flow induced by a free disk. The same holds for the corresponding heat transfer regimes. In Fig. 9 two rather important rotor-stator systems involving an enclosed rotating disk are shown. Obviously, many other configurations are available, but even the two basic cases of Fig. 9 exhibit a large number of different flow and heat transfer regimes, that will be discussed later. Prior to an overview of the flow configurations, a review of the characteristic parameters is provided.

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Heat Transfer in Rotating Flows

a

673

b

z

z

h

ω R

ε

ω R

a Fig. 9 Schematically illustration of rotor-stator systems: (a) annular cavity enclosed by an inner hub and an external shroud and (b) cavity with inlet and outflow

A major parameter describing the configuration is given by the dimensionless cavity height, defined by G ¼ h=R:

(63)

In such systems, the Reynolds number Reh = (h/δ)2 expressed by the ratio between the cavity height h and the characteristic length scale δ = (v/ω)1/2 of the flow is frequently used. The relation with the rotational Reynolds number Reω = ωR2/v of Eq. 1 is simply given by Reh = ReωG2. In the case of an annular rotorstator system as sketched in Fig. 9a, the curvature parameter Rm ¼

Rþa Ra

(64)

represents another geometric parameter. The value Rm = 1 corresponds to a cylindrical cavity. In the case of a rotor-stator disk with in- and outlet flow as sketched in Fig. 9b, the strength of the inlet (sink term) also governs the flow field within the cavity.

4.1

Flow Configurations

In the case of an infinite rotor-stator system, i.e., R ! 1, the self-similarity approach offers a good starting point for obtaining analytical solutions, but the major effect of a finite radial extension is that the boundary conditions are not compatible with the classical von Karman (1921) solution (rotating disk in resting fluid) or the Bödewadt (1940) solution (rotating fluid over a resting disk). However, some qualitative resemblance far from the end walls still exists in the laminar flow regime. In the literature, the flow structure illustrated by the normalized velocity profile in Fig. 10a are referred to as Batchelor flow, since Batchelor (1951) solved the system of NavierStokes equations relating to the stationary axisymmetric flow between two infinite disks.

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S. aus der Wiesche

The Batchelor flow structure is characterized by two unmerged boundary layers in the vicinity of the disks, which are separated by a nonviscous core region. It can be interpreted as the extension of the Ekman-Bödewadt flow over a single disk. Prior to Prandtl’s boundary layer theory, Ekman (1902) proposed a rather similar picture of the atmospheric flow field, and he also found the skewing effect of the mean velocity vector over a rotating surface. Stewartson (1953) obtained a quite different analytical solution for the same configuration that had been considered by Batchelor. The Stewartson flow without a nonviscous core region is illustrated by the velocity profiles shown in Fig. 10b. Both analytical solutions are mathematically correct, and historically there was an intense controversy about their significance. In terms of mathematics, the controversy was resolved by the discovery of multiple solutions of the corresponding system of differential equations. Brady and Durlofsky (1987) suggested that the base flow is completely determined by the radial boundary conditions. For sufficiently large Reh, a Batchelor flow structure results within the rotor-stator system, whereas the Stewartson solution appears in cavities with through-flow or in open cavities without a shroud. The first comprehensive study of enclosed rotor-stator systems was carried out by Daily and Nece (1960). With regard to technical applications, the discussion was confined to systems with small cavity heights, i.e., G Rcr).

686

S. aus der Wiesche

a

Num

u∞

0°

SP

β

βtr

b

Num

u∞ 0°

SP

β

βtr

c

Num

Reω > Rcr Reu

u∞ 0°

βtr

βtr,ω

β

ω

Fig. 17 Location of the stagnation point SP and illustration of the transitions observed for the mean heat transfer from an inclined rotating disk (From aus der Wiesche and Helcig 2016)

In terms of bifurcation theory, the inclined rotating disk subjected to an external flow exhibits both a supercritical bifurcation (Landau model Eq. 67 for the parallel disk) as well as a subcritical bifurcation (transition with respect to inclination). Based on the critical point and bifurcation theory, it is therefore possible to obtain suitable heat transfer correlations for inclined rotating disks (aus der Wiesche and Helcig 2016). Despite their idealized origin, the performance of these correlations is satisfying (deviations not larger than 10%) at least in case of a low inflow turbulence level. If high inflow turbulence levels or rough surfaces are involved, the resulting

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flow and heat transfer depends on further influence parameters not covered by the critical point concept. The main value of the critical point and bifurcation concept is its capability to qualitatively structure the data, which is particularly important with regard to the large number of parameters involved in heat transfer from rotating surfaces, see Eq. 71. Acknowledgments The author gratefully acknowledges the Deutsche Forschungsgemeinschaft DFG for its strong financial support of his research projects devoted to heat transfer in rotating disk systems. The efforts and contributions of the many students involved in the author’s research projects are appreciated. In particular, the work of Christian Helcig is acknowledged. Among the many scientists and workers on the field, the author would like to deeply acknowledge the fruitful discussions with Igor V. Shevchuk. The editorial assistance of the staff at Springer and the interests of the editor of the book, Professor Kulacki, are also gratefully appreciated.

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Natural Convection in Rotating Flows

16

Peter Vadasz

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Natural Convection in Fluids Subject to Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Taylor–Proudman Columns and Geostrophic Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Natural Convection in a Rotating Fluid Layer Heated from Below . . . . . . . . . . . . . . . . . 3 Modeling of Flow and Heat Transfer in Porous Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Modeling of Flow in Porous Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Modeling of Heat Transfer in Porous Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Natural Convection and Buoyancy in Porous Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Classification of Convective Flows in Porous Media Subject to Rotation . . . . . . . . . . 4 Fundamentals of Flow in Rotating Porous Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Taylor–Proudman Columns and Geostrophic Flow in Rotating Porous Media . . . . . 5 Natural Convection in Porous Media due to Thermal Buoyancy of Centrifugal Body Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 General Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Temperature Gradients Perpendicular to the Centrifugal Body Force . . . . . . . . . . . . . . . 5.3 Temperature Gradients Collinear with the Centrifugal Body Force . . . . . . . . . . . . . . . . . 6 Coriolis Effects on Natural Convection in Porous Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Coriolis Effect on Natural Convection in Porous Media due to Thermal Buoyancy of Centrifugal Body Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Coriolis Effect on Natural Convection due to Thermal Buoyancy of Gravity Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Other Effects of Rotation on Flow and Natural Convection in Porous Media . . . . . . . . . . . . 7.1 Natural Convection in Porous Media due to Thermal Buoyancy of Combined Centrifugal and Gravity Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

693 696 696 699 702 709 709 713 716 722 724 724 728 728 729 732 740 740 745 748 748

P. Vadasz (*) Department of Mechanical Engineering, Northern Arizona University, Flagstaff, AZ, USA e-mail: [email protected] # Springer International Publishing AG, part of Springer Nature 2018 F. A. Kulacki (ed.), Handbook of Thermal Science and Engineering, https://doi.org/10.1007/978-3-319-26695-4_11

691

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7.2 Onset of Convection due to Thermohaline (Binary Mixture) Buoyancy of Gravity Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Finite Heat Transfer Between the Phases and Temperature Modulation . . . . . . . . . . . . 7.4 Anisotropic Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5 Applications to Nanofluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6 Applications to Solidification of Binary Alloys . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Cross-References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

751 752 753 754 754 755 755

Abstract

The effect of rotation is shown to have a significant impact on the natural convection in pure fluids as well as in porous media. In isothermal systems, this effect is limited to the effect of the Coriolis acceleration on the flow. It is shown that Taylor–Proudman columns and geostrophic flows exist in both pure fluids as well as porous media subject to rotation. Results of linear stability analysis for natural convection in a rotating fluid layer heated from below are presented, identifying the unique features corresponding to this problem as compared to the same problem without rotation. In nonisothermal porous systems, the effect of rotation is expected in natural convection. Then the rotation may affect the flow through two distinct mechanisms, namely thermal buoyancy caused by centrifugal forces and the Coriolis force (or a combination of both). Since natural convection may be driven also by the gravity force, and the orientation of the buoyancy force with respect to the imposed thermal gradient has a distinctive impact on the resulting convection, a significant number of combinations of different cases arise in the investigation of the rotation effects in nonisothermal porous systems. Results pertaining to some of these cases are presented.

Nomenclature

X


t
V
ω
T
p
ρ
ρo μo αo βT
^e x ,^e y ,^e z

Position vector from the origin located on the axis of rotation to any point in the flow domain in Cartesian coordinates, equals x
^e x þ y
^e y þ z
^e z Time Velocity vector, equals u
^e x þ v
^e y þ w
^e z Angular velocity of rotation Temperature Pressure Fluid density A constant reference value of the density Dynamic viscosity (assumed constant) Thermal diffusivity, equals ko/ρocp Thermal expansion coefficient assumed constant, equals 1/ρo(@ρ
/@T
) Unit vectors in the x
, y
, and z
directions, respectively

16

1

Natural Convection in Rotating Flows

693

Introduction

This chapter focuses on natural convection in pure fluids as well as porous media subject to rotation. The first part of the chapter deals with rotation in natural convection in pure fluids, while the second part deals with natural convection in porous media subject to rotation. The fundamental equations and assumptions applicable to single-phase convective heat transfer are presented by Acharya (2017). A review of the effects of rotation on heat transfer in general is presented in Wiesche (2017). The investigation of natural convection subject to rotation was traditionally motivated by its large variety of applications in geophysics and astrophysics. Geophysical applications consist of circulation in the ocean (Marshall and Schott 1999), convection in the earth’s core, and mantle convection (Glatzmaier et al. 1999), while astrophysical applications include among others solar convection (Miesch 2000) and flows in major planets (Glatzmaier et al. 1994). Similarly, a wide range of industrial applications for the effects of rotation on convection heat transfer is available that includes rotating machinery and other industrial processes involving rotation and heat transfer. A typical example is the air-gap convection in rotating electrical machines (Howey et al. 2012). Others involve turbomachinery, e.g., the first rows of blades in a high pressure turbine of a gas turbine need to be cooled down to prevent material damage. The cooling process occurring in the interior of the blade that is attached to the rotating shaft is an example of the combined effect of heat convection subject to rotation. The study of flow in rotating porous media is motivated by its theoretical significance and practical applications in geophysics and engineering. Flows in porous geological formations subject to earth rotation and the flow of magma in the earth mantle close to the earth crust (Fowler 1990) represent examples of geophysical applications. Among the engineering applications of rotating flow in porous media, one can find the food process industry, chemical process industry, centrifugal filtration processes, and rotating machinery. More specifically, packedbed mechanically agitated vessels are used in the food processing and chemical engineering industries in batch processes. The packed bed consists of solid particles or fibers of material, which form the solid matrix while fluid flows through the pores. As the solid matrix rotates, due to the mechanical agitation, a rotating frame of reference is a necessity when investigating these flows. The role of the flow of fluid through these beds can vary from drying processes to extraction of soluble components from the solid particles. The molasses in centrifugal crystal separation processes in the sugar milling industry and the extraction of sodium alginate from kelp are just two examples of such processes. Another important application of rotating flows in porous media is in the design of a multipore distributor in a gas-solid fluidized bed. A multipore distributor is a device, which is constructed from foraminous materials, wire compacts, filter cloth, compressed fibers, sintered metal, or similar (Whitehead 1985). Research results (Davidson and Harrison 1963) showed that the porous distributor allowed a more even expansion of the bed than the other

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distributors and its design affected the behavior of the bed over most of its height. An even distribution of the gas is necessary to avoid instability in the fluidized bed, which can break down proper fluidization. A commonly used solution to avoid maldistribution of gas and bed instability is cyclic interchange fluidization (CIF) (Kvasha 1985), where the distributor is rotating at constant angular velocities which vary between 20 and 2500 rpm, depending on the size of the bed (the higher its diameter, the lower the angular velocity). Some examples of applications of the cyclic interchange fluidization are the highly exothermic synthesis of alkylchlorsilanes polymer filling of composites, treatment of finely dispersed solids, drying of paste-like polymers, and permanganate of potash and iodine (Kvasha 1985). Therefore, evaluating the flow field through a porous rotating distributor becomes a design necessity. Modeling of flow and heat transfer in porous media is also applied for the design of heat pipes using porous wicks and includes the effects of boiling in unsaturated porous media, surface tension driven flow with heat transfer, and condensation in unsaturated porous media. Plumb (1991) presented a comprehensive review of the heat transfer in unsaturated porous media flow with particular applications to the heat pipe technology. Again, when the heat pipe is used for cooling devices, which are subject to rotation the corresponding centrifugal, and Coriolis effects become relevant as well. The macrolevel porous media approach is gaining an increased level of interest in solving practical fluid flow problems, which are too difficult to solve by using a traditional microlevel approach. As such Direct Chill (DC), casting models apply the Darcy law to predict the heat transfer, fluid flow, and ultimately the thermal stresses in the solidified metal. Such a model was applied by Katgerman (1994) to analyze the heat transfer phenomena during continuous casting of aluminum alloys. When centrifugal casting processes are being considered, rotation effects become relevant to the problem. The porous media approach is also used in processing of composite materials. Güçeri (1994) states that “most of the studies of resin transfer molding (RTM) processes and structural reaction injection molding (SRIM) treat the flow domain as an anisotropic porous medium and preform a Darcy flow analysis utilizing a continuum model.” The electrocatalysis of the oxygen reduction reaction (ORR) in alkaline media on ultrathin porous coating rotating electrodes is another application of transport phenomena in rotating porous media and its application was discussed by Lima and Ticianelli (2004) in their study that used experimental methods. Additional applications of the porous media approach are discussed by Nield and Bejan (2013), and Bejan (2013) in comprehensive reviews of the fundamentals of heat convection in porous media. Bejan (2013) mentions among the applications of heat transfer in porous media the process of cooling of winding structures in high-power density electric machines. When this applies to the rotor of an electric machine, say generator (or motor), rotation effects become relevant as well. Mohanty (1994) presented a study of natural and mixed convection in rod arrays motivated by safety-related thermalhydraulic modeling of nuclear reactors with particular attention to the rod-bundle geometry. The author concluded that “bundle average experimental friction factor values in forced convection are better explained through a porous medium model” and “the porous medium parameters so derived also yield quantitative corroboration

16

Natural Convection in Rotating Flows

695

of the flow through vertical bundles induced solely by buoyancy.” The porous media approach was also successfully applied to simulate complex transport phenomena in mass and heat exchangers (Robertson and Jacobs 1994) and in cooling of electronic equipment (Vadasz 1991). Additional applications of the porous media approach are the flow of liquids in biological tissues like the human brain, the cardiovascular flow of blood in the human heart or other physiological processes, pebble-bed nuclear reactors, and cooling of turbine blades in the high pressure and hot portion of a turboexpander. Regarding the last application, such a cooling process enables the expander inlet gas temperature to increase beyond the allowed metal temperature, bringing a significant contribution to the cost-effectiveness of the expander. The cooling process occurs by injecting air through channels in the internal part of the blade. As long as the geometry of the channels is not too complicated, the traditional heat transfer approach can be applied to evaluate the cooling performances. However, for complicated channel geometry, the porous medium approach will prove again the most effective way of simulating the phenomenon. With the emerging utilization of the porous medium approach in nontraditional fields, including some applications in which the solid matrix is subjected to rotation (like physiological processes in human body subject to rotating trajectories, cooling of electronic equipment in a rotating radar, cooling of turbomachinery blades, or cooling of rotors of electric machines) a thorough understanding of the flow in a rotating porous medium becomes essential. Its results can then be used in the more established industrial applications like food processing, chemical engineering, or centrifugal processes, as well as to the less traditional applications of the porous medium approach. Reviews of the fundamentals of flow and heat transfer in rotating porous media were presented by Vadasz (1994a, 1997, 1998a, 2000, 2002a, b). No reported results were found on isothermal flow in rotating porous media prior to 1994. Pioneering research results on natural convection in rotating porous media were reported by Rudraiah et al. (1986), Patil and Vaidyanathan (1983), Jou and Liaw (1987a, b), and Palm and Tyvand (1984). Nield (1991a) while presenting a comprehensive review of the stability of convective flows in porous media found that the effect of rotation on convection in a porous medium attracted limited interest. The main reason behind the limited interest for this type of flow is first the fact that isothermal flow in homogeneous porous media following Darcy law is irrotational (Bear 1991); hence, the effect of rotation on this flow is insignificant. However, for a heterogeneous medium with spatially dependent permeability or for natural convection in a nonisothermal homogeneous porous medium, the flow is not irrotational anymore; hence the effects of rotation become significant. In some applications, these effects can be small, e.g., when the porous media Ekman number is high. Nevertheless, the effect of rotation is of interest even then, as it may generate secondary flows in planes perpendicular to the main flow direction. Even when these secondary flows are weak, it is essential to understand their source, as they might be detectable in experiments. To back up this claim, it is sufficient to look at the corresponding rotating flows in pure fluids (nonporous domains). There, the Coriolis effect and secondary motion in planes perpendicular to the main flow direction are

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controlled by the Ekman number. Experiments (Hart 1971; Johnston et al. 1972; Lezius and Johnston 1976) showed that this secondary motion is detectable even for very low or very high Ekman numbers, although the details of this motion may vary considerably according to pertaining conditions. It is therefore expected to obtain secondary motion when a solid phase is present (a porous matrix) in a similar geometric configuration, although its details cannot be a priori predicted based on physical intuition only. This creates a strong motivation to investigate the effect of rotation on the fluid flow in isothermal heterogeneous porous media. For high angular velocity of rotation or extremely high permeability, conditions pertaining to some engineering applications, the Ekman number can become of unit order of magnitude or lower, and then the effect of rotation becomes even more significant. The same motivation applies for investigating the effect of rotation on natural convection in porous media. A significant effort leading to a large volume of publications on this topic was evident during the past 30 years, producing a good fundamental understanding of the related phenomena. The methodology adopted in this chapter consists of a presentation of the dimensionless equations governing the flow and transport phenomena in a rotating frame of reference. Then each class of problems is analyzed and solved. Conclusions regarding the effect of rotation are then drawn and discussed.

2

Natural Convection in Fluids Subject to Rotation

2.1

Governing Equations

The equations governing the fluid flow and heat transfer for natural convection when the fluid domain is subject to rotation are being derived from the Navier-Stokes and Energy equations, i.e., the continuity, momentum, and energy equations. Subject to the Boussinesq approximation (Boussinesq 1903), which indicates that the density can be regarded constant in all terms of the governing equations except the buoyancy terms in the momentum equation, these equations take the form ∇
 V
¼ 0
ρo

@V
þ ðV
 ∇
ÞV
þ 2ω
 V
þ ω
 ω
 X
@t


¼ ∇
p
þ μo ∇2
V
þ ρ
g @T
þ V
 ∇
T
¼ αo ∇2
T
@t


(1) 

(2) (3)

and an equation of state relating the density to the temperature is added assuming a linear relationship between the two, i.e.,

16

Natural Convection in Rotating Flows

ρ
¼ ρo ½1  βT
ðT  T o Þ

697

(4)

where X
¼ x
^e x þ y
^e y þ z
^e z is the position vector from the origin located on the axis of rotation to any point in the flow domain in Cartesian coordinates, t
is time, V
¼ u
^e x þ v
^e y þ w
^e z is the velocity vector, ω
is the angular velocity of rotation, T
is the temperature, p
is the pressure, and ρ
is the fluid density, while ρo is a constant reference value of the density, μo is the dynamic viscosity assumed constant, αo = ko/ρocp is the thermal diffusivity, and βT
= 1/ρo(@ρ
/ @T
) is the thermal expansion coefficient assumed constant. The symbols ^e x , ^e y , and ^e z represent unit vectors in the x
, y
, and z
directions, respectively. Only constant angular velocities of rotation are being considered, and therefore ω ¼ ωo ^e ω , where ^e ω is a unit vector in the direction of the angular velocity of rotation. Also g ¼ go^e g where go = 9.81 . . . [m/s2] is the numerical value of the acceleration due to gravity, and ^e g is a unit vector in the direction of the acceleration due to gravity. The nabla operator ∇
is defined in the form ∇
 ð@  = @ x
Þ^e x þ ð@  = @ y
Þ^e y þ ð@  = @ z
Þ^ez , and the Laplacian operator is ∇2
 @ 2  = @ x2
þ @ 2  = @ y2
þ @ 2  = @ z2
. Equations (1), (2), (3), and (4) can be transformed into a dimensionless form by using as scales the reference value of density ρo, a characteristic velocity uc, a characteristic length lc, and a characteristic time that can be defined in terms of the latter two in the form tc = lc/uc, a characteristic pressure pc = 2ωolcρouc, and a characteristic temperature difference ΔTc. The specific choice of ρo, uc, lc, and ΔTc will be made in each instance when dealing with a specific problem. For now all one needs to remember is that they are constant values. Using these scales the dimensionless position vector, the dimensionless time, the dimensionless velocity, the dimensionless pressure, the dimensionless density, the dimensionless temperature, and the dimensionless ∇ and ∇2 operators (in Cartesian coordinates) become X¼

X
x
y z
uc t
¼ ^e x þ
^e y þ ^e z ¼ x^e x þ y^e y þ z^e z , t ¼ lc lc lc lc lc

V
ρ p
T  To , ρ¼
, p¼ , T¼ uc ρo 2ωo lc ρo uc ΔT c   @ @ @ @ @ @ ^e x þ ^e y þ ^e z ¼ ^e x þ ^e y þ ^e z ∇ ¼ lc ∇
¼ lc @x
@y
@z
@x @y @z  2  @2 @2 @2 @2 @ @2 þ þ ∇2 ¼ l2c ∇2
¼ l2c ¼ 2þ 2þ 2 2 2 2 @x @y @z @x
@y
@z
V¼

(5a) (5b) (5c) (5d)

Consequently the dimensionless form of Eqs. (1), (2), (3), and (4) becomes ∇V¼0

(6)

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P. Vadasz


 @V þ ðV  ∇ÞV þ ^e ω  V þ Ce^e ω  ^e ω  X Ro @t ¼ ∇p þ Ek∇2 V þ

1 ρ^e g Fr 2ω

(7)

@T 1 2 þ V  ∇T ¼ ∇T @t Pe

(8)

ρ ¼ 1  βT T

(9)

where the following dimensionless groups emerged νo uc ωo l c Ek ¼ , Ro ¼ , Ce ¼ , Fr ω ¼ 2ωo lc 2uc 2ωo l2c

sffiffiffiffiffiffiffiffiffiffiffiffi 2ωo uc uc lc , , Pe ¼ go αo

(10)

βT ¼ βT
ΔT c In Eqs. (7), (8), and (9), Ek represents the Ekman number, Ro is the Rossby number, Ce is the centrifugal number, Frω is the rotation Froude number, Pe is Peclet number, and βT is the dimensionless thermal expansion coefficient, where νo = μo/ρo is the kinematic viscosity. For natural convection, a typical choice of characteristic velocity is uc = αo/lc. With this choice, the characteristic time becomes tc ¼ l2c =αo leading to the dimensionless time variable t ¼ αo t
=l2c . Consequently, the dimensionless velocity and pressure become V = V
lc/αo and p = p
/2ωoρoαo, respectively. Then Peclet number becomes one, i.e., Pe = 1, and therefore the dimensionless energy equation becomes @T þ V  ∇T ¼ ∇2 T @t

(11)

There are two additional terms in the momentum equation that are due to the contribution of rotation. The velocity is measured relative to the rotating frame of reference, a necessity dictated by the fact that for geophysical applications such as atmospheric or ocean flows it is only normal to measure the velocity relative to the rotating earth on which our instruments are located. For rotating machinery, the same is applicable as one would be interested in velocities relative to the moving rotor. Therefore, in addition to the centripetal acceleration term Ce^e ω  ^e ω  X, Eq. (7) includes a Coriolis acceleration term in the form ^e ω  V. In addition, three different dimensionless groups that are typical for the effects of rotation emerged in the form of Ekman number that replaces the reciprocal of Reynolds number as a coefficient to the Newtonian shear stress term (viscous force), the Rossby number that represents the ratio between the inertial and Coriolis accelerations, and the Centrifugal number that represents the ratio between the centrifugal and the inertial accelerations.

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Natural Convection in Rotating Flows

699

The following identities will be used in the next derivations: 1 ^e ω  ^e ω  X ¼  ∇½ð^e ω  XÞ  ð^e ω  XÞ 2   ^e g ¼ ∇ ^e g  X

(12) (13)

The proof of these identities is provided in Vadasz (2016). Replacing the terms ð^e ω  ^e ω  XÞ and ^e g in Eq. (7) with their equivalent forms from Eqs. (12) and (13), and grouping all terms under a common gradient together with the pressure term yields


 @V 1 þ ðV  ∇ÞV þ ^e ω  V ¼ ∇pr þ Ek∇2 V þ 2 ðρ  1Þ^e g Ro @t Fr ω

(14)

where these terms grouped together form the reduced pressure pr that is defined by pr ¼ p þ Ce^e ω  ^e ω  X 

 1  ^e g  X 2 Fr ω

(15)

Substituting into Eq. (14), the equation of state (9) leads to Ro


 @V β þ ðV  ∇ÞV þ ^e ω  V ¼ ∇pr þ Ek∇2 V  T2 T ^e g @t Fr ω

(16)

Dividing Eq. (16) by Ro Pr, where Pr = νo/αo is the Prandtl number, and accounting for the selected characteristic velocity of uc = αo/lc yields Ek/(RoPr) = 1 and βT βT ΔT c go l3c ¼ ¼ Ra νo αo Fr 2ω

(17)

which is the Rayleigh number, transforming Eq. (16) into
 1 @V 1 þ ðV  ∇ÞV þ ^e ω  V ¼ ∇~ p r þ ∇2 V  Ra T ^e g Pr @t Ro

(18)

where the reduced pressure was rescaled in the form p~r ¼ pr =Ro Pr Equations (6), (11), and (18) are the equations governing natural convection subject to rotation.

2.2

Taylor–Proudman Columns and Geostrophic Flow

Spectacular effects of rotation on the fluid flow can be obtained in the limit of steady isothermal flows (βT = 0) at high rotation rates, such that Ek  1 and Ro  1.

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Such high rotation rates are not significantly high to comply with Ek  1, even earth’s rotation of 7.3  105[s1] is sufficiently high to produce a very small Ekman number, but require very slow flows at large length scales in order to comply with Ro  1. When these conditions are satisfied one may observe from Eq. (16) that the buoyancy term that includes as coefficient βT =Fr 2ω becomes zero, and the terms that include as coefficients Ro and Ek become very small and therefore negligible. The resulting equation at the leading order of magnitude, after selecting to align the direction of the angular velocity of rotation with the vertical direction, i.e., ^e ω ¼ ^e z , is then ^e z  V ¼ ∇pr

(19)

showing that in these limit of high rotation rates and slow steady isothermal flows the Coriolis acceleration is balanced by pressure forces. Taking the “curl” of Eq. (19) yields ∇  ð^e z  VÞ ¼ 0

(20)

Evaluating the “curl” operator on the cross product of the left-hand side of Eq. (20) gives ð^e z  ∇ÞV ¼ 0

(21)

Equation (21) represents the Taylor–Proudman theorem (Chandrasekhar 1981; Greenspan 1980) for rotating flows and can be presented in the following simplified form @V ¼0 @z

(22)

The conclusion expressed by Eq. (22) is that V = V(x, y), i.e., it cannot be a function of z, where z is the direction of the angular velocity vector. This means that all velocity components can vary only in the plane perpendicular to the angular velocity vector. The consequence of this result can be demonstrated by considering a particular example (see Greenspan 1980). Figure 1 shows a closed cylindrical container filled with a fluid. The topography of the bottom surface of the container is slightly changed by fixing securely a small solid object. The container rotates with a fixed angular velocity ωo. Any forced horizontal flow in the container is expected to adjust to its bottom topography. However, since Eq. (22) applies for each component of V it applies in particular to w, i.e., @w/@z = 0. But the impermeability conditions at the top and bottom solid boundaries require V  ^e n ¼ 0 (where ^e n is a unit vector normal to the solid boundaries) at z = h(r, θ) and at the top z = 1, where h(r, θ) represents the bottom topography in polar coordinates (r, θ). The combination of these boundary conditions with the requirement that @w/@z = 0 yields w = 0 anywhere in the field. Hence, a flow over the object as described qualitatively in Fig. 2a becomes impossible as it introduces a vertical component of velocity. Therefore, the resulting flow may adjust around the object as presented qualitatively in Fig. 2b. However, since this flow pattern u(x, y),

16

Natural Convection in Rotating Flows

701

Fig. 1 A closed cylindrical container filled with a fluid. A solid object fixed at the bottom and a qualitative description of a Taylor–Proudman column

Fig. 2 (a) An impossible type of flow over the object. (b) The flow adjusts around the object (as seen from above) and extends at all heights creating a column above the object, which behaves like a solid body

v(x, y) is also independent of z, it extends over the whole height of the container resulting in a fluid column above the object, which rotates as a solid body. This demonstrates the appearance of a Taylor–Proudman column, as presented qualitatively in Fig. 1. Experimental results demonstrating the Taylor–Proudman column are presented by Greenspan (1980). A further significant consequence of Eq. (22) is represented by a geostrophic type of flow. Taking the z-component of Eq. (22) yields @w/@z = 0, and the continuity equation (6) becomes two dimensional @u @v þ ¼0 @x @y

(23)

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P. Vadasz

Therefore the flow at high rotation rates has a tendency toward two-dimensionality and a stream function, ψ, can be introduced for the flow in the x  y plane u¼

@ψ ; @y

v¼

@ψ @x

(24)

which satisfies identically the continuity equation (23). Substituting u and v with their stream function representation given by Eq. (24) into Eq. (19) yields @ψ @pr ¼ @x @x

(25)

@ψ @pr ¼ @y @y

(26)

As both the reduced pressure and the stream function can be related to an arbitrary reference value, the conclusion from Eqs. (25) and (26) is that the stream function and the pressure are the same in the limit of high rotation rates (Ek  1 , Ro  1). This type of geostrophic flow means that isobars represent streamlines at the leading order, for Ek  1 , Ro  1. This is the reason that weather predictions focus on two-dimensional horizontal sections describing pressure lines (isobars) that correspond to the streamlines of the flow structure at the leading order.

2.3

Natural Convection in a Rotating Fluid Layer Heated from Below

When introducing the selected choice of characteristic velocity (see text prior to Eq. (11)) uc = αo/lc into Eq. (18), it produces the Taylor number, Ta, defined in the form Ta ¼

1 ðPr RoÞ2

¼

 2 2ωo l2c νo

(27)

leading to
 1 @V þ ðV  ∇ÞV þ Ta1=2 ^e ω  V ¼ ∇~ p r þ ∇2 V  Ra T ^e g Pr @t

(28)

For the problem of a rotating horizontal layer of fluid heated from below the angular velocity of rotation is aligned with the vertical axis, i.e., ^e ω ¼ ^e z and the acceleration due to gravity is opposing the vertical z direction, i.e., ^e g ¼ ^e z. Subject to this alignment of angular velocity of rotation and acceleration due to gravity equation (28) becomes

16

Natural Convection in Rotating Flows

703


 1 @V þ ðV  ∇ÞV þ Ta1=2 ^e z  V ¼ ∇~ p r þ ∇2 V þ Ra T ^e z Pr @t

(29)

Chandrasekhar (1953) presented a pioneering study of linear stability of a rotating layer of fluid heated from below that was extended in the seminal book (Chandrasekhar 1981). The summary presented here is mainly inspired and follows the latter. The method applied introduces a basic motionless conduction solution Vb, Tb to the governing equations (6), (11), and (29) in the form Vb ¼ 0;

T b ¼ 1  z,

  p~r, b ¼ Ra z  z2 =2 þ C1

(30)

where C1 is a constant of integration. One may observe that the basic solution (30) satisfies the governing equations (6), (11), and (29) by a simple substitution. Then one introduces small perturbations eV1(t, x, y, z), eT1(t, x, y, z) around this basic solution such that the complete solution V ¼ Vb þ eV1 ðt, x, y, zÞ ¼ eV1 ðt, x, y, zÞ

(31)

T ¼ T b þ eT 1 ðt, x, y, zÞ ¼ 1  z þ eT 1 ðt, x, y, zÞ

(32)

where e  1 is a small constant introduced only to indicate that the perturbations are small. Substituting Eqs. (31) and (32) into Eqs. (6), (11), and (29) and neglecting terms of powers of e that are larger than 1, such as e2 , e3 , . . . yields the linearized equations in the form ∇  V1 ¼ 0

(33)

1 @V1 þ Ta1=2 ^e z  V1 ¼ ∇~ p r, 1 þ ∇2 V1 þ Ra T 1 ^e z Pr @t

(34)

@T 1 @T b @T b @T b þ u1 þ v1 þ w1 ¼ ∇2 T 1 @t @x @y @z

(35)

where in Eq. (35) V1 was explicitly replaced with its components, i.e., V1 ¼ u1^e x þ v1^e y þ w1^e z . Substituting Tb = 1  z from Eq. (30) into Eq. (35) and presenting explicitly the components of Eq. (34) yields @T 1  w1 ¼ ∇2 T 1 @t

(36)

@ p~r,1 1 @u1  ∇2 u1  Ta1=2 v1 ¼  Pr @t @x

(37)

@ p~r,1 1 @v1  ∇2 v1 þ Ta1=2 u1 ¼  Pr @t @y

(38)

704

P. Vadasz

@ p~r,1 1 @w1  ∇2 w1 ¼  þ Ra T 1 ^e z Pr @t @z

(39)

In the solution of the linear equations (33), (34), and (36), one needs to use the vorticity equation, which is obtained by applying the curl operator (∇) on Eq. (34) an operation that also eliminates the reduced pressure terms, leading to   1 @ζ 1 @V1 @T 1 @T 1 ^e x  ^e y  Ta1=2 ¼ ∇2 ζ 1 þ Ra Pr @t @z @y @x

(40)

where ζ 1 ¼ ∇  V1 ¼ ð@w1 =@y  @v1 =@xÞ^e x þ ð@u1 =@z  @w1 =@xÞ^e y þ ð@v1 =@x  @u1 =@yÞ^e z

(41)

is the vorticity perturbation. It is particularly noteworthy that the vertical component in Eq. (40) is independent of temperature. Applying the curl operator (∇) again on Eq. (40) and using the property of V1 being solenoidal (i.e., ∇  V1 = 0 from Eq. (33)) yields



2  1 @ @ζ @ T1 @2 T1 ^e x þ ^e y  ∇2H T 1 ^e z ¼ 0  ∇2 ∇2 V1 þ Ta1=2 1 þ Ra Pr @t @z @x@z @y@z

(42)

where ∇2H T 1 ¼ @ 2 T 1 = @ x2 þ @ 2 T 1 = @ y2 is the horizontal Laplacian. The coupling between Eqs. (36), (41), and (42) can be removed by eliminating V1 and ζ1. The latter can be accomplished by using the z component of Eqs. (40) and (42) to eliminate first ζz, 1 and then use the resulting equation together with Eq. (36) to eliminate w1 leading to an equation for T1 only. Similarly one may obtain uncoupled equations for w1 as well as the horizontal velocity components. The resulting equation for T1 is "  2   2   # @ 1 @ @ @ 1 @ 2 2 2 2 2 ∇ ∇ ∇  ∇ ∇2H ∇ þ Ta  Ra @t Pr @t @t Pr @t @z2 T1 ¼ 0 Expanding the solution into normal modes in the x and y directions, i.e.,  

T 1 ¼ θðzÞexp i κ x x þ κy y þ σt þ c:c:

(43) (44)

where c.c. stands for the complex conjugate terms, and substituting it into Eq. (43) produces an ordinary differential equation for θ(z). In addition, the following boundary conditions have to be satisfied for the case of a free top and bottom boundaries

16

Natural Convection in Rotating Flows

z ¼ 0, 1 : T 1 ¼ 0 ) θ ¼ 0 &

705

w1 ¼ 0 |{z} )

Eq:ð36Þ

)

@2T1 @2θ ¼ 0 ) ¼0 @z2 @z2

@4θ @6θ ¼ 6 ¼0 @z4 @z

(45)

A solution of the form θðzÞ ¼ A sin ðn π zÞ

(46)

satisfies the boundary conditions (45) for n = 1 , 2 , 3 , . . ., and upon substitution into Eqs. (44) and (43) one obtains a characteristic algebraic equation. The perturbation growth rate is represented by the parameter σ that generally is a complex constant, i.e., σ = σ r + i σ i. One needs to analyze separately the two possible convection solutions, i.e., stationary convection (steady state) and oscillatory convection (standing or travelling waves). For the stationary convection, the imaginary part of σ is zero by definition, i.e., σ i = 0. Then at marginal stability, i.e., at the point where the basic motionless solution looses stability and a steady convection solution takes over the value of σ r = 0. Therefore for marginal stability at stationary convection σ = 0 and the characteristic equation takes the form 

n2 π 2 þ κ 2

3

þ n2 π 2 Ta  κ2 Ra ¼ 0

(47)

 1=2 is the horizontal wave number. Equation (47) can be where κ ¼ κ2x þ κ 2y presented in the form Rast ¼

3

ðn2 π 2 þ κ2 Þ þ n2 π 2 Ta κ2

(48)

where the superscript st stands for “stationary convection.” From Eq. (48), it can be observed that the smallest values of Rast can be obtained when n = 1 producing neutral curves on the Ra  κ 2 plane as presented in Fig. 3. For the oscillatory (or overstable) convection, the imaginary part of σ is nonzero, i.e., σ i 6¼ 0, implying that the solution oscillates with a frequency σ i. Using the same solution for θ(z) as presented in Eq. (46) for the stationary convection leads to the following characteristic equation for the lowest mode n = 1 

     π 2 þ κ2 þ Pr σ π 2 þ κ2 þ σ π 2 þ κ2 þ π 2 Ta ¼ Ra2 π 2 þ κ2 þ σ

(49)

Then, at marginal stability, i.e., at the point where the basic motionless solution looses stability and an oscillatory convection solution takes over, the value of

706

P. Vadasz

σ r = 0. The solution perturbation T1 has therefore the form  for the temperature

T 1 ¼ θðzÞeiσi t exp i κ x x þ κy y þ c:c:, and the characteristic equation (49) becomes 

i h  2   π 2 þ κ2 þ i Pr σ i π 2 þ κ2 þ i σ i π 2 þ κ2 þ π 2 Ta   ¼ κ2 Raos π 2 þ κ2 þ i σ i

(50)

where the superscript os stands for “oscillatory convection.” It is convenient to rescale the parameters in Eqs. (48) and (58) in the form

α¼

κ2 σi Ra Ta ; e σi ¼ 2 ; R ¼ 4 ; χ ¼ 4 π π π2 π

(51)

Then Eq. (48) applicable for the lowest mode n = 1 and Eq. (50) take the form Rst ¼

ð α þ 1Þ 3 þ χ α


 ðα þ 1 þ i Pr e σ iÞ χ R ¼ ð α þ 1Þ ð α þ 1 þ i e σ iÞ þ α σ iÞ ðα þ 1 þ i e os

(52) (53)

Equation (53) is a complex equation and the aim is at deriving explicit relationships between the scaled Rayleigh number R, and the oscillations frequency σ i, versus the scaled square wave number α, for different values of the parameters Pr and the scaled Taylor number χ. Toward this end one needs to express the complex equation (53) by separating the imaginary part from the real part and solving for R and e σ 2i , a process that yields ð1  Pr Þχ  ð α þ 1Þ 2 ð1 þ Pr Þðα þ 1Þ " # 1 Pr 2 χ 3 os R ¼ 2ð1 þ Pr Þ ðα þ 1Þ þ α ð1 þ Pr Þ2 e σ 2i ¼

(54)

(55)

From Eq. (54) it can be observed that a frequency of oscillatory convection exists (i.e., e σ 2i > 0) only for Prandtl numbers smaller than 1, i.e., Pr < 1. Actually there are values of α for which e σ 2i 0 even when Pr < 1. Oscillatory convection is therefore possible only if Pr < 1 and χ>

ð1  Pr Þ ð α þ 1Þ 3 ð1 þ Pr Þ

(56)

where χ = Ta/π 4. For a given value of χ (i.e., Ta), oscillatory convection is possible σ 2i ¼ 0, i.e., only for wave numbers α < α
, where α
is the value of α associated with e

16

Natural Convection in Rotating Flows

707

Fig. 3 Characteristic curves representing the marginal stability with respect to both stationary as well as oscillatory convection for Ta = 104 and different values of Prandtl number, Pr

ð α
þ 1Þ 3 ¼

ð1 þ Pr Þ χ ð1  Pr Þ

(57)

At this value of e σ i ¼ 0, the oscillatory Rayleigh number Ros at marginal stability is evaluated from Eq. (53) in the form Ros ¼

ðα
þ 1Þ3 þ χ α


(58)

and this is precisely the value of the stationary Rayleigh number Rst at marginal stability as presented in Eq. (52). For α > α
, oscillatory convection is not possible for the given values of χ and Pr. Then the onset of stationary convection is the only possibility. The characteristic curves representing the marginal stability with respect to both stationary as well as oscillatory convection for Ta = 104 and different values of Prandtl number, Pr, are being presented graphically in Fig. 3. It can be observed that the oscillatory convection neutral curves (dotted lines) branch off from the stationary convection neutral curve (continuous line) at the point where α = α
and e σ i ¼ 0. The frequency of oscillations at marginal stability for Ta = 104 and different values of Prandtl number, Pr, are being presented in Fig. 4. Chandrasekhar (1981) presented a more detailed analysis as well as solutions for different types of boundary conditions on the top and bottom of the horizontal layer, such as free-rigid and rigid-rigid. In addition, he presented variational principles for both stationary as well as oscillatory convection. Veronis (1959, 1966) expanded the linear stability analysis using a stability criterion established by Malkus and Veronis (1958) and introduced a finite amplitude analysis to the same problem by using an asymptotic expansion of the dependent variables. He showed that “the boundary of a steady convection cell is distorted by the rotation in such a way that the wavelength of the cell measured along the

708

P. Vadasz

Fig. 4 Frequency of oscillations at marginal stability for Ta = 104 and different values of Prandtl number, Pr

distorted boundary is equal to the wavelength of the non-rotating cell.” He traced back this conservation of the cellular wavelength to the “constancy of the horizontal vorticity in the rotating and non-rotating systems.” A finite amplitude solution to the problem eventually leads to a solution in the form h i T 1 ¼ A1 eiðκx xþtÞ þ B1 eiðκxtÞ þ A
1 eiðκxþtÞ þ B
1 eiðκxtÞ sin ðπzÞ (59) where the amplitudes A1(τo, τ, X) and B1(τo, τ, X) describe modulations of the waves on the slow time (τo = e t0 , τ = e2t0) and space (X = ex) scales for a Hopf bifurcation. This form of the solution represents travelling waves. The special cases of a pure left travelling wave (B1 = 0) or a pure right travelling wave (A1 = 0) and of standing waves (A1 =  B1 or A1 ¼ B
1 ) can be recovered. This general solution will eventually provide two coupled equations for the complex amplitudes. Experimental studies undertaken by Fultz and Nakagawa (1955) and Nakagawa and Frenzen (1955) confirmed the theoretical predictions. Rossby (1969) also presented experimental results that for water (Pr 7, so oscillatory convection cannot occur according to the theory) confirm the theoretical predictions for Taylor numbers less than 5  104 but subcritical convection occurs beyond this value of Taylor number. For mercury (Pr 0.025 < 1 and oscillatory convection is possible), “excellent agreement between the theory and experiment” was found when Taylor number was greater than 105 but “for Taylor numbers less than 1.8  104 the agreement between the linear theory and experiment was poor.” More recent experimental results are being presented by Zhong et al. (1993). Numerical studies including transition to turbulence were presented among others by Julien et al. (1996). Numerical methods were applied also to the problem of supercritical

16

Natural Convection in Rotating Flows

709

convection in rotating spherical shells by Christensen (2002). Direct numerical simulations as well as experimental measurements were performed by Stevens et al. (2013) to investigate the heat transfer and flow structure in rotating Rayleigh–Benard convection. The convection heat transfer drops as a consequence of oscillatory convection. Scaling behavior in Rayleigh–Benard convection with and without rotation was analyzed numerically by King et al. (2013).

3

Modeling of Flow and Heat Transfer in Porous Media

3.1

Modeling of Flow in Porous Media

While the same symbol V
¼ u
^e x þ v
^e y þ w
^e z that was used in the previous section applicable for pure fluids is used in the present section that deals with porous media the meaning of the two is distinct. In pure fluids this is the fluid’s velocity, but in porous media this is a filtration velocity that averages the fluid velocity over a Representative Elementary Volume (REV). Similarly identical symbols are used for some dimensionless groups in porous media as in pure fluids although their definition differs. The convention applied in this chapter is that the symbols defined in specific sections that deal with pure fluids are applicable to pure fluids alone, and those defined in specific sections that deal with porous media are applicable to porous media alone. A common equation to all models of flow in porous media is the continuity equation representing the mass balance of the fluid. Regardless of what kind of dynamic model is being used, the continuity equation is to be used in conjunction with the latter. The continuity equation for a nondeformable porous medium (the porosity is constant, ϕ = const.) is presented in the form ϕ

@ρ
þ V
 ∇
ρ
þ ρ
∇
 V
¼ 0 @t


(60)

where ϕ is the porosity of the porous medium defined as the ratio between the volume of effective pores to the total volume of the porous matrix (voids + solid). The definition of effective pores is applicable to pores that are not completely isolated from fluid flow, i.e., interconnected pores. In Eq. (60) ρ
is the density of the fluid, V
¼ u
^e x þ v
^e y þ w
^e z is the filtration velocity vector and u
, v
, w
are its components in the x
, y
, and z
direction, respectively, in a Cartesian coordinate system. When the flow is definitely incompressible the density is identically constant, i.e., ρ
= ρo = const. and Eq. (60) above becomes ∇
 V
¼ 0

(61)

Even when the flow is only approximately incompressible Eq. (61) can be used as a very accurate approximation. An example of such an application that is widely

710

P. Vadasz

used in this chapter refers to the Boussinesq approximation (Boussinesq 1903) that is acceptable in reference to buoyancy driven flows and indicates that the density can be regarded as constant in all terms of the governing equations except the buoyancy terms in the momentum equation. Therefore, subject to Boussinesq approximation (Boussinesq 1903), Eq. (61) will be used in this chapter to represent a good approximation of the continuity equation (60). When relaxation of the latter approximation is applied in particular circumstances one shall revert to the accurate form of the continuity equation (60). Dimensionless Form of the Continuity Equation The two forms of the continuity equation presented above can be presented in a dimensionless form by using a reference value of density say ρo, a characteristic velocity uc, a characteristic length lc, and a characteristic time that can be defined in terms of the latter two in the form tc = lc/ucMf (The coefficient Mf represents the ratio between the effective heat capacity of the fluid and effective heat capacity of the porous medium, a coefficient that will be defined later. The choice of including this coefficient in the characteristic time is done for later convenience in notation.) By definition, any characteristic or reference values are constant. The distinction between the two is made only for physical reasons. A reference value typically applies to material properties, which act as system parameters in the equations and may or may not change, while a characteristic value applies to the dependent and independent variables that naturally always change. The specific choice of ρo, uc, and lc will be made in each instance when one deals with a specific problem. For now all one need to remember is that they are constant values. Then by introducing the dimensionless position vector, the dimensionless filtration velocity, the dimensionless density, and the dimensionless ∇ operator (in Cartesian coordinates) as X¼

X
x
y z
¼ ^e x þ
^e y þ ^e z ¼ x^e x þ y^e y þ z^e z , lc lc lc lc

V
ρ ,ρ ¼
, uc ρo   @ @ @ @ @ @ ^e x þ ^e y þ ^e z ¼ ^e x þ ^e y þ ^e z ∇ ¼ lc ∇
¼ lc @x
@y
@z
@x @y @z V¼

(62a) (62b) (62c)

it leads to the dimensionless form of Eq. (1) ϕ

@ρ þ V  ∇ρ þ ρ∇  V ¼ 0 @t

(63)

and the dimensionless form of Eq. (2) ∇V¼0

(64)

16

Natural Convection in Rotating Flows

711

3.1.1 Darcy Model The most fundamental model of flow in porous media is the Darcy model. It represents the momentum balance for the fluid and can be presented in the form V
¼ 

K
∇
p
 ρ
g
^e g μ


(65)

where K
is the permeability of the porous matrix (carrying units of m2 or darcy, 1 darcy unit = 9.869233  1013 m2), μ
is the dynamic viscosity of the fluid (carrying units of Pa  s = N  s/m2), g
is the acceleration due to gravity (carrying units of m/s2), ρ
is the density of the fluid (carrying units of kg/m3), and ^e g is a unit vector in the gravity acceleration direction. The permeability of the porous medium is a property of the matrix and is independent of the fluid flowing through it. It can be regarded as an effective cross sectional area perpendicular to the direction of the flow and hence represents a measure of how permeable the matrix is. The Darcy equation accounts for the lumped effect of the friction forces between the fluid and the solid matrix on the filtration velocity, in addition to the gravitational forces. The forces between the fluid particles within a Representative Elementary Volume (REV) are being assumed negligible and therefore neglected. The latter are of the order of magnitude of Darcy number (Da ¼ K
=l2c ), i.e., O(Da), which for typical porous media has very small values. Dimensionless Form of the Darcy Equation The dimensionless form of the Darcy equation, assuming constant values of permeability and dynamic viscosity, can be obtained by introducing a characteristic pressure difference Δpc and by using the same characteristic and reference values as used for the continuity equation in the previous section. Again, except for the fact that the characteristic pressure difference that is introduced, Δpc, is constant there is no need to specify anything else about Δpc. This will be done later when dealing with specific problems. Therefore, the dimensionless form of the Darcy equation is V ¼ N p ∇p þ Frρ^e g

(66)

where two dimensionless groups, a pressure number Np and a porous media Froude number Fr, emerged in the form Np ¼

K
Δpc g K
, Fr ¼
μ
uc l c ν
uc

(67)

where ν
= μ
/ρo is the kinematic viscosity of the fluid.

3.1.2

Darcy Equation in a Rotating Frame of Reference and Other Extended Forms Darcy equation can be extended to include additional body forces, such as centrifugal and Coriolis effects in the case of a rotating porous matrix, or an additional term

712

P. Vadasz

that accounts for time dynamics of the evolving values of the filtration velocity, i.e., a time derivative term. For the former, rotation effects introduce the centrifugal and Coriolis terms as follows: V
¼ 


 K
2ρ ∇
p
 ρ
g
^e g þ ρ
ω
 ðω
 X
Þ þ
ω
 V
μ
ϕ

(68)

where ω
is the angular velocity of rotation (carrying units of s1), ϕ is porosity, and X
is the position vector measured from the axis of rotation. The third term in the brackets represents the centrifugal force while the fourth term represents the Coriolis acceleration. Another extension to the Darcy equation is applicable when fast transients or high frequency effects are of interest. Then, the time resolution obtained by assuming a very fast reaction of Darcy flow to changes and therefore the quasisteady approximation which is inherent in the formulation of the Darcy law is not sufficient and a time derivative of the filtration velocity needs to be incorporated leading to ρ
@V
μ þ
V
¼ ∇
p
þ ρ
g
^e g ϕ @t
K


(69)

Combining Eq. (68) with Eq. (69) yields the Darcy equation in a rotating frame of reference extended to include fast transients and high frequency effects. ρ
@V
μ 2ρ þ
V
¼ ∇
p
þ ρ
g
^e g  ρ
ω
 ðω
 X
Þ 
ω
 V
(70) ϕ @t
K
ϕ Dimensionless Forms of the Extensions to Darcy Equation Assuming a constant angular velocity of rotation, ω
= const., Eq. (68) can be transformed into the following dimensionless form V ¼ N p ∇p þ Fr ρ^e g  Cn ρ^e ω  ð^e ω  XÞ 

ρ ^e ω  V EkΔ

(71)

where two additional dimensionless groups emerged, a centrifugal number Cn and a porous media Ekman number EkΔ defined in the form Cn ¼

K
ω2
lc ϕν
, EkΔ ¼ ν
uc 2ω
K


(72)

and where ν
= μ
/ρo is the kinematic viscosity of the fluid, and Eq. (70) is transformed into the dimensionless form DaReMf @V ρ ^e ω  V þ V ¼ N p ∇p þ Fr ρ^e g  Cn ρ^e ω  ð^e ω  XÞ  ρ @t EkΔ ϕ (73)

16

Natural Convection in Rotating Flows

713

where the additional dimensionless groups that emerged are the Darcy number Da, and the Reynolds number Re Da ¼

K
uc l c , Re ¼ ν
l2c

(74)

and where the definition of Mf will follow later. Note that despite the fact that it is the porous media filtration velocity that emerged in the definition of the Reynolds number, the corresponding length scale is a macrolevel length scale, not the pore size. However the Reynolds number appears in Eq. (73) in a product combination with the Darcy number, bringing therefore the pore-scale effects into account too.

3.2

Modeling of Heat Transfer in Porous Media

Similarly as in fluid flow, the equation governing the heat transfer in porous media is the energy equation averaged over the phases of the porous medium. Fourier law is assumed for the conduction heat fluxes in each phase.

3.2.1 Finite Heat Transfer Between the Phases Assuming different temperatures associated with the different phases, i.e., Ts
for the solid phase temperature and Tf
for the fluid phase temperature one obtains the following equations of energy balance ð1  ϕÞρs
cs
ϕρf
cp, f


  @T s
¼ ð1  ϕÞk~s
∇2
T s
þ h
T f
 T s
@t


  @T f
þ ϕρp, f
cp
V
 ∇
T f
¼ ϕk~f
∇2
T f
 h
T f
 T s
@t


(75) (76)

where ρs
, cs
, and k~s
are the density, specific heat, and thermal conductivity of the solid phase material, and ρf
, cp, f
, and k~f
are the density, the constant pressure specific heat, and the thermal conductivity of the fluid phase material. The following notation is useful γ s
¼ ð1  φÞρs
cs
, γ f
¼ φ ρf
cp, f


(77)

representing the solid phase and fluid phase effective heat capacities, respectively, and ks
¼ ð1  φÞk~s
, kf
¼ φ k~f


(78)

representing the effective thermal conductivities of the solid and fluid phases, respectively. The term h
(Tf
 Ts
) represents the rate of heat generation per unit volume in the solid phase within the REV due to the heat transferred from the fluid

714

P. Vadasz

over the fluid-solid interface. The coefficient h
> 0, carrying units of W m3 K1, is a macrolevel integral heat transfer coefficient for the heat conduction at the fluidsolid interface (averaged over the REV) that is assumed independent of the phases’ temperatures and independent of time. Note that this coefficient is conceptually distinct from the convection heat transfer coefficient and is anticipated to depend on the thermal conductivities of both phases as well as on the surface area to volume ratio (specific area) of the medium. Subject to the notation (77) and (78), Eqs. (75) and (76) become   @T s
¼ ks
∇2
T s
þ h
T f
 T s
@t


(79)

  @T f
þ γ f
V
 ∇
T f
¼ kf
∇2
T f
 h
T f
 T s
@t


(80)

γ s
γf


3.2.2 Thermal Equilibrium Between the Phases When the temperature difference between the solid and fluid phases is very small, conditions applicable to most circumstances when for example the thermal conductivity of the solid phase is not extremely small, then a sensible approximation would be to set the temperatures of the phases equal to each other, i.e., T
= Ts
Tf
. These conditions are identified as Local Thermal Equilibrium (LTE or Lotheq), while the former case of distinct phase temperatures is identified as Local Thermal Non Equilibrium (LTNE) or Lack of Local Thermal Equilibrium (LaLotheq). Then the two Eqs. (79) and (80) can be combined by adding them to yield γ e


@T
þ γ f
V
 ∇
T
¼ ke
∇2
T
@t


(81)

where γ e
= γ s
+ γ f
and ke
= ks
+ kf
are the effective heat capacity and effective thermal conductivity of the porous medium, and the interface heat transfer term disappeared as the amount of heat gained by the solid as it transferred from the fluid is equal to the amount of heat lost by the fluid as it transferred to the solid via the fluid-solid interface at each point in space. Dividing Eq. (81) by γ e
yields @T
~ e
∇2
T
þ Mf V
 ∇
T
¼ α @t


(82)

where ~ e
¼ α

γf
ke
, Mf ¼ γ e
γ e


(83)

are an effective thermal diffusivity of the porous medium, and the heat capacity ratio, i.e., the ratio between the effective heat capacity of the fluid phase and the effective heat capacity of the porous medium, respectively.

16

Natural Convection in Rotating Flows

715

Dimensionless Forms of the Energy Equation for Local Thermal Equilibrium The derivation of the dimensionless form of the energy equation (82) follows the definition of the dimensionless temperature T¼

ðT
 T o Þ ΔT c

(84)

where To is a reference value of temperature and ΔTc is a characteristic temperature difference to be explicitly defined for every specific problem. Introducing the ~ e
=Mf ¼ ke
=γ f
the definition of the adjusted effective thermal diffusivity αe
¼ α dimensionless energy equation becomes @T 1 2 þ V  ∇T ¼ ∇ T @t Pe

(85)

where Peclet number emerged as an additional dimensionless group, defined in the form Pe ¼

uc l c ¼ PrRe αe


(86)

where the Prandtl number Pr is defined by Pr ¼

ν
αe


(87)

3.2.3 Equation of State A relationship between the density, temperature, and pressure (and solute concentration when the fluid is a solution of soluble substances, e.g., salt in water, alcohol in water, etc.) is needed in order to complete the model formulation. If the fluid is compressible such as a gas, in certain circumstances one may use as an approximation the ideal gas equation ρ
¼

1 p
Rg
T


(88)

where Rg
is the specific gas constant (i.e., the universal gas constant divided by the molecular mass of the gas). For liquids and also for gases (even when Eq. (88) applies), one may use a linear approximation relating the density to temperature and pressure, when temperature and pressure differences are not excessively large. Then

ρ
¼ ρo 1  βT
ðT
 T o Þ þ βp
ðp
 po Þ

(89)

where ρo is a reference value of density obtained when the temperature is To and the pressure is po. The coefficients βT
and βp
are the thermal expansion coefficient

716

P. Vadasz

(carrying units of K1) and the pressure compression coefficient (carrying units of Pa1), respectively, defined in the form βT
¼  βp
¼

1 @ρ
1 @ρ
 ρ
@T
ρo @T


(90)

1 @ρ
1 @ρ
ρ
@p
ρo @p


(91)

It can be observed that by using the ideal gas equation (88) with Eqs. (90) and (91) it leads to the following thermal expansion and pressure compression coefficients: βT
= 1/T
and βp
= 1/p
, respectively. Dimensionless Forms of Equation of State The dimensionless form of the linear approximation of the equation of state can be obtained by using the definition of the dimensionless temperature from Eq. (84) and the dimensionless pressure in the form p = ( p
 po)/Δpc. Then Eq. (89) becomes

ρ ¼ 1  β T T þ βp p

(92)

where ρ = ρ
/ρo is the dimensionless density, and βT = βT
ΔTc and βp = βp
Δpc are the dimensionless thermal expansion and pressure compression coefficients, respectively. Typically, for most fluid flows βp  βT, especially for incompressible flows, i.e., flows of liquids. Therefore a common approximation of the dimensionless equation of state would be ρ ¼ ½1  β T T 

(93)

When the fluid phase is a solution of a soluble substance in a liquid, such as salt in water, then the density would depend also on the concentration S. Then an extension of the equation of state (93) would be required in the form ρ ¼ ½1  βT T þ βS S

(94)

where S = (S
 So)/ΔSc, and βS = βS
ΔSc is the dimensionless compression coefficient due to solute concentration, and S
is the solute concentration. In such cases, an additional transport equation for mass (solute) transfer accounting for the molecular diffusion and the convection of the solute within the fluid phase would be required too.

3.3

Natural Convection and Buoyancy in Porous Media

3.3.1 Homogeneous Porous Media Natural convection is the effect of flow and convection heat transfer due to the existence of density gradients in a body force field (such as gravity or centrifugal

16

Natural Convection in Rotating Flows

717

force field). As density depends on temperature as demonstrated in the derivation of the equation of state, temperature gradients may create natural convective flows when a body force field is present. What characterizes natural convection is the lack of a known value of characteristic filtration velocity that can be applied upfront in a problem. No characteristic velocity can be specified because the latter is dictated by the temperature gradients and their resulting buoyancy rather than being known upfront. Therefore a sensible choice of uc would be uc = αe
/lc. With this choice of uc the Froude number Fr, the pressure number Np, and the centrifugal dimensionless group Cn in Eqs. (71) and (73) become Fr ¼

g
K
lc K
Δpc ω2 l2 K
, Np ¼ , Cn ¼
c ν
αe
μ
αe
ν
αe


(95)

Without loss of generality for the same reason as for the filtration velocity one can chose the characteristic pressure difference to be such that the pressure number Np becomes unity, i.e., Δpc = μ
αe
/K
leading to Np = 1. Also the Reynolds number in Eq. (73) renders into the reciprocal Prandtl number Re ¼

αe
1 ¼ Pr ν


(96)

and the Peclet number in Eqs. (86) and (85) equals one by definition Pe ¼

uc lc αe
lc ¼ ¼1 αe
lc αe


(97)

One may define the effective Prandtl number in terms of the effective thermal ~ e
(see Eq. (83)) diffusivity α Pr e ¼

ν
Pr ¼ ~ e
Mf α

(98)

Then the coefficient to the time derivative term in Eq. (73) becomes DaReMf DaMf Da 1 ¼ ¼ ¼ ϕPr e Va ϕ ϕPr

(99)

a new dimensionless group that Straughan (2001) named the Vadasz number (Va), or the Vadasz coefficient named by Straughan (2008) (see also Sheu 2006; Govender 2010). An extremely useful approximation that is usually applied to natural convection problems is the Boussinesq approximation (Boussinesq 1903). Boussinesq approximation states that the density can be considered constant, i.e., ρ
= ρo = const. (and consequently its dimensionless value is ρ = 1) everywhere except in the terms associated with body forces, where the variations of density are to be accounted for. According to this approximation the equation of state (93) will be substituted instead

718

P. Vadasz

of ρ in the gravity and centrifugal terms in Eqs. (71) and (73), but ρ = 1 will be substituted everywhere else. Introducing these results from Eqs. (95), (96), (97), (98), and (99) with Np = 1 and applying the Boussinesq approximation (Boussinesq 1903) one obtains the following equations for natural convection to replace Eqs. (64), (71), (73), and (85) ∇V¼0 V ¼ ∇p þ Fr ρ^e g  Cn ρ^e ω  ð^e ω  XÞ 

(100) 1 ^e ω  V EkΔ

1 @V 1 ^e ω  V þ V ¼ ∇p þ Fr ρ^e g  Cn ρ^e ω  ð^e ω  XÞ  Va @t EkΔ @T þ V  ∇T ¼ ∇2 T @t

(101) (102) (103)

It becomes appealing in the application of the porous media models presented in the previous sections for problems of natural convection where buoyancy effects are to be investigated to introduce some identities that are extremely useful in what follows. These identities are   ^e g ¼ ∇ ^e g  X
 1 ^e ω  ð^e ω  XÞ ¼ ∇ ð^e ω  XÞ  ð^e ω  XÞ 2 ^e ω  ð^e ω  XÞ ¼ ð^e ω  XÞ^e ω  X

(104) (105) (106)

and their proof is provided in Vadasz (2016). Substituting these identities into Eqs. (101) and (102) and presenting the body force terms by using the artificial representation ρ = ρ  1 + 1 that will prove useful shortly, produces     V ¼ ∇p þ Fr ðρ  1Þ∇ ^e g  X þ Fr∇
^e g  X   1 1 ^e ω  V Cnðρ  1Þ½ð^e ω  XÞ^e ω  X þ Cn∇ ð^e ω  XÞ  ð^e ω  XÞ  2 EkΔ (107)     1 @V þ V ¼  ∇p þ Fr ðρ  1Þ∇ ^e g  X þ Fr∇ ^e g  X Va @t  Cnðρ  1Þ½ð^e ω  XÞ^e ω  X
 1 1 ^e ω  V þ Cn∇ ð^e ω  XÞ  ð^e ω  XÞ  2 EkΔ

(108)

Introducing (ρ  1) =  βTT from Eq. (93) and grouping all gradient terms under a common gradient operator yields

16

Natural Convection in Rotating Flows

719




   Cn V ¼  ∇ p  Fr ^e g  X  ð^e ω  XÞ  ð^e ω  XÞ þ Fr βT T ^e g 2 (109)  Cn βT T ½ð^e ω  XÞ^e ω  X 1 ^e ω  V  EkΔ
   Cn 1 @V þ V ¼ ∇ p  Fr ^e g  X  ð^e ω  XÞ  ð^e ω  XÞ þ Fr βT T ^e g  Va @t 2 1 ^e ω  V Cn βT T ½ð^e ω  XÞ^e ω  X  EkΔ (110) The term under the common gradient operator is a reduced pressure pr, defined as   Cn pr ¼ p  Fr ^e g  X  ð^e ω  XÞ  ð^e ω  XÞ 2

(111)

The product of βT by Fr and Cn yields two new dimensionless groups in the form of the gravity-related Rayleigh number and the centrifugal Rayleigh number, respectively in the form Rag ¼ Fr βT ¼

βT
ΔT c g
K
lc ν
αe


(112)

Raω ¼ Cn βT ¼

βT
ΔT c ω2
l2c K
ν
αe


(113)

Equations (111), (112), and (113) transform Eqs. (109) and (110) into the following form V ¼ ∇pr þ Rag T ^e g  Raω T ½ð^e ω  XÞ^e ω  X 

1 ^e ω  V EkΔ

1 @V 1 ^e ω  V þ V ¼ ∇pr þ Rag T ^e g  Raω T ½ð^e ω  XÞ^e ω  X  Va @t EkΔ

(114) (115)

The particular case when ^e g ¼ ^e z and ^e ω ¼ ^e z will be selected to analyze later. Subject to this orientation of the gravity and angular velocity of rotation Eqs. (114) and (115) take the form   1 ^e z  V V ¼ ∇pr þ Rag T ^e z  Raω T x^e x þ y^e y  EkΔ   1 @V 1 ^e z  V þ V ¼ ∇pr þ Rag T ^e z  Raω T x^e x þ y^e y  Va @t EkΔ

(116) (117)

720

P. Vadasz

  The vector r ¼ x^e x þ y^e y in Eqs. (116) and (117) represents the perpendicular radius vector from the axis of rotation to any point in the flow domain. Three dimensionless groups emerged in Eq. (116) when fast transients or high frequencies are not of interest. These dimensionless groups control the significance of the different phenomena. Therefore, the value of Ekman number (EkΔ) controls the significance of the Coriolis effect and the ratio between the gravity related Rayleigh number (Rag) and the centrifugal Rayleigh number (Raω) controls the significance of gravity with respect to centrifugal forces as far as natural convection is concerned. This ratio is Rag =Raω ¼ g
=ω2
lc . When fast transients or high frequencies are of interest Eq. (117) is to be considered. In such a case one additional dimensionless group emerged, the Va number representing the ratio between two characteristic frequencies, i.e., the fluid flow frequency ων
= ϕν
/K
and the ~ e
=l2c , i.e., Va ¼ ων
=ωα
¼ ϕν
l2c =K
α ~ e
¼ ϕ thermal diffusion frequency ωα
¼ α Pr e =Da, or alternatively the ratio between two time scales, i.e., the thermal diffusion time scale τα
¼ l2c =~ α e
, and the fluid flow time scale τν
= K
/ϕν
, i.e., Va ¼ τα
= ~ e
¼ ϕPr e =Da. In addition to such cases, Eq. (117) should be used τν
¼ ϕν
l2c =K
α also when the Prandtl number is of the order of magnitude of Darcy number, i.e., Pre = O(Da), i.e., a very small number (as Da  1 in most porous media). Such small values of the Prandtl number are typical for liquid metals. In such cases too, the time derivative term in Eq. (117) should be retained. Equations (100), (114) or (115), and (103) include three distinct mechanisms of coupling. Two of these couplings are linear. The first is the coupling between the pressure terms and the filtration velocity components, i.e., a coupling between Eq. (100) and the three components of Eq. (114) or (115). The second linear coupling is due to the Coriolis acceleration acting on the filtration velocity components in the plane perpendicular to the direction of the angular velocity. The third coupling is nonlinear as it involves the gravity or centrifugal buoyancy terms in Eq. (114) or (115) and the energy equation (103). As V in Eq. (114) or (115) is dependent on temperature (T ) due to the buoyancy terms, it follows that the convective term V  ∇T in the energy equation (103) causes the nonlinearity. Therefore the coupling between the temperature, T, and the filtration velocity, V, is the only source of nonlinearity in the Darcy’s formulation of flow and heat transfer in rotating porous media Eq. (114), or its extension Eq. (115).

3.3.2 Heterogeneous Porous Media In heterogeneous porous media, the permeability and thermal conductivity can be dependent on the space variables, i.e., K
(x
, y
, z
) and possibly ke
(x
, y
, z
). In this chapter only cases when the permeability is allowed to vary in space are being considered. Then, a reference value of permeability Ko is being used in all previous definitions and the dimensionless permeability function is then defined in the form K (x, y, z) = K
(x
, y
, z
)/Ko. Subsequently, following the same derivations one obtains the following set of equations for the heterogeneous porous media ∇V¼0

(118)

16

Natural Convection in Rotating Flows

721


 1 ^e ω  V V ¼ K ∇pr þ Rag T ^e g  Raω T ½ð^e ω  XÞ^e ω  X  (119) EkΔ
 K @V 1 ^e ω  V þ V ¼ K ∇pr þ Rag T ^e g  Raω T ½ð^e ω  XÞ^e ω  X  Va @t EkΔ (120) @T þ V  ∇T ¼ ∇2 T @t

(121)

Decoupling the equations is difficult without losing generality although the linear couplings can be resolved. Resolving the Coriolis related coupling is particularly useful. In doing so a Cartesian coordinate system is used and without loss of generality one can choose ^e ω ¼ ^e z . A further choice is made, which reduces the generality of the problem, namely that the gravity acceleration is collinear with the z axis and directed downwards, i.e., ^e g ¼ ^e z . Some generality is lost, as problems where the direction of rotation and gravity are not collinear will not be represented by the resulting equations. Nevertheless, it is of interest to demonstrate this particular choice of system as it represents a significant number of practical cases. Subject to these choices (^e ω ¼ ^e z and ^e g ¼ ^e z ) Eq. (119) can be expressed by extending Eq. (116) to heterogeneous media in the following form (with the r dropped from pr and the Δ dropped from EkΔ for convenience)

 



1 þ K Ek1^e z  V ¼ K ∇p þ Raω T x^e x þ y^e y  Rag T^e z

(122)

An important observation can be made by presenting Eq. (122) explicitly in terms of the three scalar components
 @p þ Raω Tx u  K Ek v ¼ K @x
 @p þ Raω Ty K Ek1 u þ v ¼ K @y
 @p þ Rag T w ¼ K @z 1

(123) (124) (125)

where u, v, and w are the corresponding x, y, and z components of the filtration velocity vector V. It is now observed that the first two Eqs. (123) and (124) for the horizontal components of V are “Coriolis coupled,” while the third one is not. Since this type of coupling is linear it is possible to decouple them to obtain VH ¼ E  ½∇H p þ Raω T XH  ~H

(126)

722

P. Vadasz

where VH ¼ u^e x þ v^e y is the horizontal filtration velocity, XH ¼ x^e x þ y^e y is the horizontal position vector, ∇H  @ = @ x^e x þ @ = @ y^e y is the horizontal gradient operator, and EH is the following tensor operator ~
K 1

EH ¼ 1 2 2 Ek K ~ 1 þ Ek K

Ek1 K 1

 (127)

The definition of the tensor operator E is extended to include the z component of ~H V, thus leading to V ¼ E  ½∇p þ B T  ~

(128)

where B ¼ Raω x^e x þ Raω y^e y  Rag^e z is the buoyancy coefficient vector and the tensor operator E is defined in the form ~ 2 1 K 4 Ek1 K

E¼ ~ 1 þ Ek2 K 2 0

Ek1 K 1 0

3 0 5 0   2 2 1 þ Ek K

(129)

It is interesting to observe from Eqs. (128) and (129) that the Coriolis effect due to rotation is equivalent to a particular form of an anisotropic porous medium. The anisotropy is represented by the tensor operator E. It should be pointed out that the ~ analogy between the Coriolis effect and anisotropy was identified by Palm and Tyvand (1984) for the problem of gravity-driven thermal convection in a rotating porous layer at marginal stability. It has been shown here and by Vadasz (1997, 2000) that this analogy is more general and applies also for centrifugally driven convection and for isothermal flows in rotating heterogeneous porous media (Raω = Rag = 0 ! B = 0 in Eq. (128)). Subsequently, Auriault et al. (2000, 2002) have obtained the same conclusion by using a method of multiple scale expansions. Although for a significantly high number of practical instances, Darcy model in a rotating frame of reference is sufficient for representing the effects of rotation, and non-Darcy models have been used as well. Their relevance and limitations are subject to professional discourse (e.g., Nield 1983, 1991b, 1995; Vafai and Kim 1990). Nevertheless there are circumstances where there is sufficient consensus that Darcy model falls short in representing accurately the fluid flow in porous media. Details on non-Darcy models can be found in Vadasz (2016).

3.4

Classification of Convective Flows in Porous Media Subject to Rotation

Natural convective flows in porous media subject to rotation can be dealt with by classifying them first into two major categories:

16

Natural Convection in Rotating Flows

723

(a) Convective flows in homogeneous porous media subject to rotation (b) Convective flows in heterogeneous porous media subject to rotation This chapter is concerned with the first category. The second category is not covered, simply because there are no reported research results on natural convection in rotating heterogeneous porous media. It is evident from Eq. (119) subject to isothermal conditions, i.e., Rag = Raω = 0, that heterogeneity of the medium is an important condition for the effects of rotation to be significant. This can be observed, for example, from the basic Darcy law, which under homogeneous conditions (i.e., K = const. = 1) yields irrotational types of flows, i.e., the vorticity ∇  V = 0. The heterogeneity of the medium, K  K(X), introduces a nonvanishing vorticity, i.e., ∇  V 6¼ 0. Nonisothermal flows, as a result of natural convection, allow also a nonvanishing vorticity. Convective flows in rotating porous media are further classified into three categories and each one of them can be separated into three cases. In order to justify this classification one proceeds first by defining the phenomenon of natural convection. Natural convection is the phenomenon of fluid flow driven by density variations in a fluid subject to body forces. Therefore, there are two necessary conditions to be met in order to obtain natural convective flow; (i) the existence of density variations within the fluid and (ii) the fluid must be subjected to body forces. As density is in general a function of pressure, temperature, and solute concentration (in the case of a binary mixture), i.e., ρ  ρ( p, T, S) and its variation with respect to pressure is much smaller than that with respect to temperature or concentration (i.e., βp  βT, leading to Eq. (93) ρ = 1  βTT for example). Hence the convection can be driven either by temperature variations or by variations in solute concentration or by both. In the two latter cases, Eq. (93) should be extended to include S in the form of Eq. (94) ρ = 1  βTT + βSS, the solute transport equation should be included in the model, and Eq. (119) should be extended as well to account for the solute effect on density. Gravity, centrifugal forces, and electromagnetic forces (in the case of liquid metals subject to an electric field) are only examples of body forces which represent the second necessary condition for natural convection to occur. Some of these body forces are constant, such as gravity for example, while others can vary linearly with the perpendicular distance from the axis of rotation like the centrifugal force. The lack of body forces, for example under microgravity conditions in the outer space, prevents occurrence of convection. The two conditions mentioned above are indeed necessary for natural convection to occur; however they are not sufficient. The relative orientation of the density gradient with respect to the body force is an important factor for providing the sufficient conditions for convection to occur. This is shown graphically in Fig. 5 for the particular case of thermal convection, where B represents the body force and ∇ρ =  βT∇T is the direction of the density gradient. Given this basic introduction on the causes for the setup of convection, the following classification of convective flows is introduced: (i) Convection due to thermal buoyancy (ii) Convection due to thermo-solutal buoyancy

724

P. Vadasz

Fig. 5 The effect of the relative orientation of the temperature gradient with respect to the body force on the setup of natural convection

A third category related to convection due to solutal buoyancy alone could have been introduced but it would not bring any significant contribution to (i) above as the results obtained for thermal buoyancy alone can be easily converted to the third case by analogy. Three separate cases for each one of the above categories can be considered, depending on the driving body force, i.e., convection driven by gravity, convection driven by the centrifugal force, and convection driven by both gravity and centrifugal force. In each of these cases, the effect of Coriolis acceleration on the natural convective flow is another rotation effect of interest in the present chapter.

4

Fundamentals of Flow in Rotating Porous Media

4.1

Taylor–Proudman Columns and Geostrophic Flow in Rotating Porous Media

To present the topic of Taylor–Proudman columns, Eq. (119) is considered in the limit of steady isothermal flows (Rag = 0 & Raω = 0) multiplied by [Ek/K] for ^e ω ¼ ^e z and rescale the pressure in the form p~r ¼ Ek pr to yield


Ek þ ^e z  K

 V ¼ ∇~ pr

(130)

Given typical values of viscosity, porosity, and permeability, one can evaluate the range of variation of Ekman number in some engineering applications. There, the angular velocity may vary from 10 to 10,000 rpm leading to Ekman numbers in the range from Ek = 1 to Ek = 103. The latter value is very small, pertaining to the conditions considered in this section. Therefore, in the limit of Ek!0, say Ek = 0, Eq. (130) takes the simplified form

16

Natural Convection in Rotating Flows

725

^e z  V ¼ ∇~ pr

(131)

and the effect of permeability variations disappears. Taking the “curl” of Eq. (131) leads to ∇  ð^e z  VÞ ¼ 0

(132)

Evaluating the “curl” operator on the cross product of the left-hand side of Eq. (132) gives ð^e z  ∇ÞV ¼ 0

(133)

Equation (133) is identical to the Taylor–Proudman condition for pure fluids (nonporous domains); it thus represents the proof of the Taylor–Proudman theorem in porous media and can be presented in the following simplified form @V ¼0 @z

(134)

The conclusion expressed by Eq. (134) is that V = V(x, y), i.e., it cannot be a function of z, where z is the direction of the angular velocity vector. This means that all filtration velocity components can vary only in the plane perpendicular to the angular velocity vector. The consequence of this result can be demonstrated by considering a particular example that was presented by Vadasz (1994b) (see Greenspan (1980) for the corresponding example in pure fluids). Figure 6 shows a closed cylindrical container filled with a fluid saturated porous medium. The topography of the bottom surface of the container is slightly changed by fixing securely a small solid object. The container rotates with a fixed angular velocity ω
. Any forced horizontal flow in the container is Fig. 6 A closed cylindrical container filled with a fluid saturated porous medium. A solid object fixed at the bottom and a qualitative description of a Taylor–Proudman column in porous media

z w*

r

726

P. Vadasz

a

b

y

r

z

x

Impossible type of flow Fig. 7 (a) An impossible type of flow over the object. (b) The flow adjusts around the object (as seen from above) and extends at all heights creating a column above the object, which behaves like a solid body

expected to adjust to its bottom topography. However, since Eq. (134) applies for each component of V, it applies in particular to w, i.e., @w/@z = 0. But the impermeability conditions at the top and bottom solid boundaries require V  ^e n ¼ 0 at z = h(r, θ) and at z = 1, where h(r, θ) represents the bottom topography in polar coordinates (r, θ). The combination of these boundary conditions with the requirement that @w/@z = 0 yields w = 0 anywhere in the field. Hence, a flow over the object as described qualitatively in Fig. 7a becomes impossible as it introduces a vertical component of velocity. Therefore, the resulting flow may adjust around the object as presented qualitatively in Fig. 7b. However, since this flow pattern is also independent of z, it extends over the whole height of the container resulting in a fluid column above the object, which rotates as a solid body. This will demonstrate a Taylor–Proudman column in porous media, as presented qualitatively in Fig. 6. It should be pointed out that the column in porous media should be expected in the macroscopic sense, i.e., the diameter of the column (and naturally the diameter of the obstacle) must have macroscopic dimensions which are much greater than the microscopic characteristic size of the solid phase, or any other equivalent pore scale size. Experimental confirmation of these theoretical results is necessary. However, because of the inherent difficulty of visualizing the flow, i.e., the Taylor–Proudman column through a granular porous matrix, an indirect method was adopted. The rationale behind the proposed method was to place a porous layer in-between two pure fluid layers inside a cylinder (see Fig. 8a). A disturbance in the form of a small fixed obstacle was created on the bottom wall in the fluid layer located below the porous layer causing the appearance of a Taylor–Proudman column in this layer. As the fundamental property of a Taylor–Proudman column is that it does not allow for variations of velocity in the vertical direction, the proposed experimental setup should allow us to detect Taylor–Proudman columns in the top undisturbed fluid layer located above the porous layer, if the column exists in the porous layer as a result of the disturbance in the bottom fluid layer. This means that a small object located at the bottom of the

16

Natural Convection in Rotating Flows

727

Fig. 8 (a) A closed cylindrical container divided into a porous layer and two fluid layers above and below the porous layer. A solid object is fixed at the bottom surface of the container. (b) Qualitative description of the anticipated Taylor–Proudman column as observed in the preliminary runs of the experiment

lower fluid layer should create a Taylor–Proudman column, which extends upward through the porous layer into the upper fluid layer. It is because of this expectation that the top fluid layer will “feel” through the buffer porous layer the small object located at the bottom of the lower fluid layer, which some colleagues of the author named the effect resulting from the experiment “The Princess and the Pea.” Of course another possibility exists which is consistent with Eq. (134). This could for example imply that the filtration velocity is zero in a rotating porous layer, meaning that the whole fluid in a rotating porous layer rotates as a solid body instantaneously (almost). The experimental apparatus consists of a record player turntable adjusted to allow variable angular velocity in the designed range and to provide a better dynamic balance when the cylindrical container is placed securely on the rotating plate. Preliminary results confirm the appearance of Taylor–Proudman columns in the top fluid layer as shown qualitatively in Figs. 8b and 9. A further significant consequence of Eq. (134) is represented by a geostrophic type of flow. Taking the z-component of Eq. (134) yields @w/@z = 0, and the continuity equation (118) becomes two dimensional @u @v þ ¼0 @x @y

(136)

Therefore the flow at high rotation rates has a tendency toward two-dimensionality and a stream function, ψ, can be introduced for the flow in the x  y plane

728

P. Vadasz

Fig. 9 A Taylor–Proudman column as observed in the top fluid layer in the preliminary runs of the experiment

u¼

@ψ ; @y

v¼

@ψ @x

(136)

which satisfies identically the continuity equation (135). Substituting u and v with their stream function representation given by Eq. (136) into Eq. (131) yields @ψ @ p~r ¼ @x @x

(137)

@ψ @ p~r ¼ @y @y

(138)

As both the pressure and the stream function can be related to an arbitrary reference value, the conclusion from Eqs. (137) and (138) is that the stream function and the pressure are the same in the limit of high rotation rates (Ek!0). This type of geostrophic flow means that isobars represent streamlines at the leading order, for Ek!0.

5

Natural Convection in Porous Media due to Thermal Buoyancy of Centrifugal Body Forces

5.1

General Background

Considering Darcy’s regime for a homogeneous porous medium (K = 1) subject to a centrifugal body force and neglecting gravity (Rag/Raω  1) Eqs. (118), (119), and (121) with K = 1 and Rag = 0 represent the mathematical model for this case. The objective in the first instance is to establish the convective flow under small rotation rates, then Ek 1, and as a first approximation the Coriolis effect can be neglected, i.e., Ek!1. Following these conditions, the governing equations when heat generation is absent (i.e., Q_ ¼ 0) become (using identity (106) to revert back)

16

Natural Convection in Rotating Flows

729

∇V¼0

(139)

V ¼ ½∇pr  Raω T ^e ω  ð^e ω  XÞ

(140)

@T þ V  ∇T ¼ ∇2 T @t

(141)

The three cases corresponding to the relative orientation of the temperature gradient with respect to the centrifugal body force as presented in Fig. 5 are considered. Case 5(a) in Fig. 5 corresponds to a temperature gradient, which is perpendicular to the direction of the centrifugal body force and leads to unconditional convection. The solution representing this convection pattern is the objective of the investigation. Cases 5(b) and 5(c) in Fig. 5 corresponding to temperature gradients collinear with the centrifugal body force represent a stability problem. The objective is then to establish the stability condition as well as the convection pattern when this stability condition is not satisfied.

5.2

Temperature Gradients Perpendicular to the Centrifugal Body Force

An example of a case when the imposed temperature is perpendicular to the centrifugal body force is a rectangular porous domain rotating about the vertical axis, heated from above and cooled from below. For this case the centrifugal buoyancy term in Eq. (140) becomes Raω T x^e x leading to V ¼ ∇pr  Raω T x^e x

(142)

An analytical two-dimensional solution to this problem (see Fig. 10) for a small aspect ratio of the domain was presented by Vadasz (1992). The solution to the nonlinear set of partial differential equations was obtained through an asymptotic expansion of the dependent variables in terms of a small parameter representing the aspect ratio of the domain. The convection in the core region far from the sidewalls was the objective of the investigation. To first order accuracy, the heat transfer coefficient represented by the Nusselt number was evaluated in the form
  2 Raω Nu ¼  1 þ þO H 24

(143)

where Nu is the Nusselt number and the length scale used in the definition of Raω, Eq. (113), was lc = H
. A different approach was used by Vadasz (1994c) to solve a similar problem without the restriction of a small aspect ratio. A direct extraction and substitution of

730

P. Vadasz

w*2 x*

z w*

TH*

z=1 ¶T =0 ¶x

¶T =0 ¶x

TC * x=0

x

L*

H* z=0

x=L

Fig. 10 A rotating rectangular porous domain heated from above, cooled from below, and insulated on its sidewalls (Courtesy Elsevier Science Ltd.)

the dependent variables was found to be useful for decoupling the nonlinear partial differential equations, resulting in a set of independent nonlinear ordinary differential equations, which was solved analytically. To obtain an analytical solution to the nonlinear convection problem one assumes that the vertical component of the filtration velocity, w, and the temperature T are independent of x, i.e., @w/@x = @T/ @x = 0 8x  (0, L ), being functions of z only. It is this assumption that will subsequently restrict the validity domain of the results to moderate values of Raω (practically Raω < 5). Subject to the assumptions of two-dimensional flow v = 0 and @()/@y = 0 and that w and T are independent of x the governing equations take the form @u dw þ ¼0 @x dz u¼

(144)

@pr  Raω x T @x

(145)

@pr @z

(146)

w¼

d2 T dT ¼0 w 2 dz dz

(147)

The method of solution consists of extracting T from Eq. (145) and expressing it explicitly in terms of u , @pr/@x and x. This expression of T is then introduced into Eq. (147) and the derivative @/@x is applied to the result. Then, substituting the continuity equation (144) in the form @u/@x = dw/dz and Eq. (146) into the results yields a nonlinear ordinary differential equation for w in the form

16

Natural Convection in Rotating Flows

731

d3 w d2 w  w ¼0 dz3 dz2

(148)

An interesting observation regarding Eq. (148) is the fact that it is identical to the Blasius equation for boundary layer flows of pure fluids (nonporous domains) over a flat plate. To observe this, one simply has to substitute w(z) = f(z)/2 to obtain 000 2f + f f 00 = 0, which is the Blasius equation. Unfortunately, no further analogy to the boundary layer flow in pure fluids exists, mainly because of the quite different boundary conditions and because the derivatives [d()/dz] and the flow (w) are in the same direction. The solutions for the temperature T and the horizontal component of the filtration velocity u are related to the solution of the ordinary differential equation 000

φ0 φ  φ00 þ φ φ0 ¼ 0 2

2

(149)

where ()0 stands for d()/dz and u ¼ x φð z Þ T ðzÞ ¼ 

(150)

1 ½ P þ φð z Þ  Raω

(151)

where P is a constant defined by P ¼ Raω

ð1

T ðzÞdz

(152)

0

The relationship (152) is a result of imposing a condition of no net flow through Ð1 any vertical cross-section in the domain stating that 0 udz ¼ 0. The following boundary conditions are required to the solution of Eq. (148) for w: w = 0 at z = 0 and z = 1 representing the impermeability condition at the solid boundaries and T = 0 at z = 0 and T = 1 at z = 1. Since @u/@x = φ according to Eq. (150), then following the continuity equation (144) φ = dw/dz and the temperature boundary conditions can be converted into conditions in terms of w by using Eq. (151), leading to the following complete set of boundary conditions for w: z¼0: z¼1:

w¼0

w¼0

and

and

dw ¼P dz

(153)

dw ¼ P þ Raω dz

(154)

Equations (153) and (154) represent four boundary conditions, while only three are necessary to solve the third order Eq. (148). The reason for the fourth condition comes from the introduction of the constant P, whose value remains to be determined. Hence, the additional two boundary conditions are expressed in terms of the unknown constant P and the solution subject to these four conditions will determine

732

P. Vadasz

y=0 y = 0.518 y = 1.036 y=0

y=0

y = 1.295 y = 0.777 y = 0.259

Fig. 11 Graphical description of the resulting flow field; five streamlines equally spaced between their minimum value ψ min = 0 at the rigid boundaries and their maximum value ψ max = 1.554. The values in the figure correspond to every other streamline (Courtesy Elsevier Science Ltd.)

the value of P as well. A method similar to Blasius’s method of solution was applied to solve Eq. (148). Therefore, w(z) was expressed as a finite power series and the objective of the solution was to determine the power series coefficients. Once the solution for w(z) and P was obtained, u and T were evaluated by using φ(z) = dw/dz and Eqs. (150) and (151). Then, for presentation purposes a stream function ψ was introduced to plot the results (u = @ψ/@z, w = @ψ/@x). An example of the flow field represented by the streamlines is presented in Fig. 11 for Raω = 4 and for an aspect ratio of 3 (excluding a narrow region next to the sidewall at x = L). Outside this narrow region next to x = L, the streamlines remain open on the right-hand side. They are expected to close in the end region. Nevertheless, the streamlines close on the left-hand side, throughout the domain. The reason for this is the centrifugal acceleration, which causes u to vary linearly with x, thus creating (due to the continuity equation) a nonvanishing vertical component of the filtration velocity w at all values of x. The local Nusselt number Nu, representing the local vertical heat flux, was evaluated as well by using the definition Nu = |@T/@z|z=0 and using the solution for T. A comparison between the heat flux results obtained from this solution and the results obtained by Vadasz (1992) using an asymptotic method was presented graphically by Vadasz (1994c). The two results compare well as long as Raω is very small. However, for increasing values of Raω, the deviation from the linear relationship pertaining to the first-order asymptotic solution (Nu = 1 + Raω/24, according to Vadasz 1992) was evident.

5.3

Temperature Gradients Collinear with the Centrifugal Body Force

The problem of stability of free convection in a rotating porous layer when the temperature gradient is collinear with the centrifugal body force was treated by Vadasz (1994d, 1996a) for a narrow layer adjacent to the axis of rotation (Vadasz 1994d) and distant from the axis of rotation (Vadasz 1996a), respectively. The problem formulation corresponding to the latter case is presented in Fig. 12. In order to include explicitly the dimensionless offset distance from the axis of rotation

16

Natural Convection in Rotating Flows

733

z'

z'

z

w*

2 w* x'*

z w2 * x'

T=1

T=0

z=H

y'

x'

x

x

x0

0

y

x'

x

W

Fig. 12 A rotating fluid saturated porous layer distant from the axis of rotation and subject to different temperatures at the sidewalls

x0, and to keep the coordinate system linked to the porous layer, Eq. (142) was presented in the form V ¼ ∇pr  ½Raωo þ Raω xT ^e x

(155)

Two centrifugal Rayleigh numbers appear in Eq. (155); the first one, Raωo ¼ βT
ΔT c ω2
x0
L
K o =αe
ν
, represents the contribution of the offset distance from the rotation axis to the centrifugal buoyancy, while the second, Raω ¼ βT
ΔT c ω2
L2
K o =αe
ν
, represents the contribution of the horizontal location within the porous layer to the centrifugal buoyancy. The ratio between the two centrifugal Rayleigh numbers is η¼

Raω 1 ¼ Raωo x0

(156)

and can be introduced as a parameter in the equations transforming Eq. (155) in the form V ¼ ∇pr  Raωo ½1 þ ηxT ^e x

(157)

From Eq. (157), it is observed that when the porous layer is far away from the axis of rotation then η  1 (x0 1) and the contribution of the term η x is not significant, while for a layer close enough to the rotation axis η 1 (x0  1) and the

734

P. Vadasz

contribution of the first term becomes insignificant. In the first case the only controlling parameter is Raωo while in the latter case the only controlling parameter is Raω = η Raωo. The flow boundary conditions are V  ^e n ¼ 0 on the boundaries, where ^e n is a unit vector normal to the boundary. These conditions stipulate that all boundaries are rigid and therefore nonpermeable to fluid flow. The thermal boundary conditions are: T = 0 at x = 0, T = 1 at x = 1 and ∇T  ^e n ¼ 0 on all other walls representing the insulation condition on these walls. The governing equations accept a basic motionless conduction solution in the form    

½Vb , T b , pb  ¼ 0, x, Raωo x2 =2 þ ηx3 =3 þ C

(158)

The objective of the investigation was to establish the condition when the motionless solution (158) is not stable and consequently a resulting convection pattern appears. Therefore a linear stability analysis was employed, representing the solution as a sum of the basic solution (158) and small perturbations in the form ½V, T, p ¼ ½Vb þ V0 , T b þ T 0 , pb þ p0 

(159)

where ()0 stands for perturbed values. Solving the resulting linearized system for the perturbations by assuming a normal modes expansion in the y and z directions, and θ(x) in the x direction, i.e., T0 = Aκθ(x) exp[σt + i(κ yy + κzz)], and using the Galerkin method to solve for θ(x) one obtains at marginal stability, i.e., for σ = 0, a homogeneous set of linear algebraic equations. This homogeneous linear system accepts a nonzero solution only for particular values of Raωo such that its determinant vanishes. The solution of this system was evaluated up to order 7 for different values of η, representing the offset distance from the axis of rotation. However, useful information was obtained by considering the approximation at order 2. At this order, the system reduces to two equations which lead to the characteristic values of Raωo in the form h i β ð1 þ αÞ2 þ ð4 þ αÞ2   R0, c ¼ 2α β2  γ 2 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi h i2   β2 ð1 þ αÞ2 þ ð4 þ αÞ2  4 β2  γ 2 ð1 þ αÞ2 ð4 þ αÞ2   (160)  2α β2  γ 2 where the following notation was used Ro ¼

Raωo Raω κ2 η 256η2 2 ; γ ; R ¼ ; α ¼ ; β ¼ 1 þ ¼ 2 π2 π2 π2 81π 4

(161)

and κ is the wavenumber such that κ2 ¼ κ 2y þ κ2z while the subscript c in Eq. (160) represents characteristic (neutral) values (values for which σ = 0). A singularity in

Natural Convection in Rotating Flows

a

4

7

6

6

5

5

4

2

3 1

-2

-1

0 log 10 η

1

2

8

7

R cr

R o,cr R cr

0

b

8

3

R o,cr

735

Ra ω /p 2

16

Raω ,cr Ra + ω 0,2 cr = 1 7.81π 2 4π

Unstable

4 3

2

2

1

1

0

0

Stable

0

0.5

1

1.5

2

2.5

Raωο /π

3

3.5

4

2

Fig. 13 (a) The variation of the critical values of the centrifugal Rayleigh numbers as a function of η; (b) The stability map on the Raω  Raωo plane showing the division of the plane into stable and unstable zones

the solution for Ro,c, corresponding to the existence of a single root for Ro,c, appears when β2 = γ 2. This singularity persists at higher orders as well. Resolving for the value of η when this singularity occurs shows that it corresponds to negative η values implying that the location of the rotation axis falls within the boundaries of the porous domain (or to the right side of the hot wall – a case of little interest due to its inherent unconditional stability). This particular case will be discussed later in this section. The critical values of the centrifugal Rayleigh number as obtained from the solution up to order 7 are presented graphically in Fig. 13a in terms of both Ro, cr and Rcr as a function of the offset parameter η. The results presented in Fig. 13 are particularly useful in order to indicate the stability criterion for all positive values of η. It can be observed from the figure that as the value of η becomes small, i.e., for a porous layer far away from the axis of rotation, the critical centrifugal Rayleigh number approaches a limit value of 4π 2. This corresponds to the critical Rayleigh number in a porous layer subject to gravity and heated from below. For high values of η, it is appropriate to use the other centrifugal Rayleigh number R, instead of Ro, by introducing the relationship R = ηRo (see Eqs. (156) and (161)) in order to establish and present the stability criterion. It is observed from Fig. 13a that as the value of η becomes large, i.e., for a porous layer close to the axis of rotation, the critical centrifugal Rayleigh number approaches a limit value of 7.81π 2. This corresponds to the critical Rayleigh number for the problem of a rotating layer adjacent to the axis of rotation as presented by Vadasz (1994d). The stability map on the Raω  Raωo plane is presented in Fig. 13b, showing that the plane is divided between the stable and unstable zones by the straight line (Raω, cr/7.81π 2) + (Raωo, cr/4π 2) = 1. The results for the convective flow field are presented graphically in Fig. 14 following Vadasz (1996a), where it was concluded that the effect of the variation of the centrifugal acceleration within the porous layer is definitely felt when the box is close to the axis of rotation, corresponding to an eccentric shift of the convection

736

P. Vadasz

Fig. 14 The convective flow field at marginal stability for three different values of x0; ten stream lines equally divided between ψ min and ψ max. At x0 = 1010 : ψ min =  1.378 ; ψ max = 1.378, at x0 = 0.02 : ψ min =  1.374; ψ max = 1.374 and at x0 = 50 : ψ min =  1.319; ψ max = 1.319

cells toward the sidewall at x = 1. However, when the layer is located far away from the axis of rotation (e.g., x0 = 50), the convection cells are concentric and symmetric with respect to x = 1/2 as expected for a porous layer subject to gravity and heated from below (here “below” means the location where x = 1). Although the linear stability analysis is sufficient for obtaining the stability condition of the motionless solution and the corresponding eigenfunctions describing qualitatively the convective flow, it cannot provide information regarding the values of the convection amplitudes, nor regarding the average rate of heat transfer. To obtain this additional information, Vadasz and Olek (1998) analyzed and provided a solution to the nonlinear equations by using Adomian’s decomposition method to solve a system of ordinary differential equations for the evolution of the amplitudes. This system of equations was obtained by using the first three relevant Galerkin modes for the stream function and the temperature in the form ψ ¼ 2θ

πz pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2γ ðR  1Þ XðtÞ sin ðπxÞ sin H

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi πz ðR  1Þ 2γ ðR  1Þ Z ðtÞ sin ð2πxÞ Y ðtÞ sin ðπxÞ cos T ¼xþ þ H πR πR 2

(162) (163)

16

Natural Convection in Rotating Flows

737

  where γ ¼ H 2 = H2 þ 1 , θ = (H2 + 1)/H, H being the layer’s aspect ratio, R = ξ/ π 2θ2, ξ = Raωo + Raω/2, and X , Y , Z the possibly time dependent amplitudes of convection. In this model, it was considered of interest including the time derivative term in Darcy’s equation in the form (1/Va)@V/@t, where Va = ϕ Pre/Da; and Da, Pre are the Darcy and the effective Prandtl numbers, respectively, defined as Da ¼ K o =L2
and Pr e ¼ ν
=~ α e
(see Eq. (117) with Rag = 0 and Ek!1). The reason for including the time derivative term in the Darcy equation was the fact that one anticipates oscillatory and possibly chaotic solutions that some might have very high frequencies. Then, the following equations were obtained for the time evolution of the amplitudes X , Y , Z X_ ¼ αðY  XÞ

(164)

Y_ ¼ R X  Y  ðR  1ÞX Z

(165)

Z_ ¼ 4γ ðX Y  Z Þ

(166)

where α ¼ γVa=π 2, and R is a rescaled Rayleigh number introduced above following Eq. (163). The results obtained are presented in Fig. 15 in the form of projection of trajectories data points onto the Y  X and Z  X planes. Different transitions as the value of R varies are presented and they relate to the convective fixed point which is a stable simple node in Fig. 15a, a stable spiral in Fig. 15b, c, and loses stability via an inverse Hopf bifurcation in Fig. 15d, where the trajectory describes a limit cycle, moving toward a chaotic solution presented in Fig. 15e, f. A further transition from chaos to a periodic solution was obtained at a value of R slightly above 100, which persists over a wide range of R values. This periodic solution is presented in Fig. 15g, h for R = 250. Previously in this section (see Eq. (160)), a singularity in the solution was identified and associated with negative values of the offset distance from the axis of rotation. It is this resulting singularity and its consequences, which were investigated by Vadasz (1996b) and are the objective of the following presentation. As this occurs at negative values of the offset distance from the axis of rotation, it implies that the location of the rotation axis falls within the boundaries of the porous domain, as presented in Fig. 16. This particular axis location causes positive values of the centrifugal acceleration on the right side of the rotation axis and negative values on its left side. The rotation axis location implies that the value of x0 is not positive. It is therefore convenient to explicitly introduce this fact in the problem formulation specifying explicitly that x0 = |x0|. As a result Eq. (157) can be expressed in the form V ¼ ∇pr  Raω ½x  jx0 jT ^e x

(167)

The solution for this case is similar to the previous case leading to the same characteristic equation for Rc at order 2 as obtained previously in Eq. (160) for Ro,c, with the only difference appearing in the different definition of β and γ as follows

738

P. Vadasz

Fig. 15 Different transitions in natural convection in a rotating porous layer (Courtesy Elsevier Science Ltd.)

16

Natural Convection in Rotating Flows

739

z

z'

z'

z 2 w*

x'*

*

w2

2 w* x'*

*

x' *

z=H

w2

x' *

T=0

T=1

*

y x

x

y' 0

x'

W

x

x0

x'

Fig. 16 A rotating porous layer having the rotation axis within its boundaries and subject to different temperatures at the sidewalls (Courtesy Elsevier Science Ltd.)

  1  jx0 j ; β¼ 2

γ2 ¼

256 81π 4

(168)

The singularity is obtained when β2 = γ 2, corresponding to β = γ or β = γ. Since β is uniquely related to the offset distance |x0| and γ = 16/9π 2 is a constant, one can relate the singularity to specific values of |x0|. At order 2, this corresponds to |x0| = 0.3199 and |x0| = 0.680. It was shown by Vadasz (1996b) that the second value |x0| = 0.680 is the only one, which has physical consequences. This value corresponds to a transition beyond which, i.e., for |x0|  0.68, no positive roots of Rc exist. It therefore implies an unconditional stability of the basic motionless solution for all values of R if |x0|  0.68. The transitional value of |x0| was investigated at higher orders showing |x0|  0.765 at order 3 and the value increases with increasing the order. The indications are that as the order increases the transition value of |x0| tends toward the limit value of 1. The results for the critical values of the centrifugal Rayleigh number expressed in terms of Rcr versus |x0| are presented graphically by Vadasz (1996b), who concluded that increasing the value of |x0| has a stabilizing effect. The results for the convective flow field as obtained by Vadasz (1996b) are presented in Figs. 17, 18, and 19 for different values of |x0|. Keeping in mind that to

740

P. Vadasz

Fig. 17 The convective flow field at marginal stability for three different values of |x0|; ten streamlines equally divided between ψ min and ψ max (Courtesy Elsevier Science Ltd.)

a

b

z'

c

z'

ω

ω

*

ω

*

*

x'

x'

x' | x0 | = 0

z'

| x0 | = 0.3

| x0 | = 0.5

the right of the rotation axis, the centrifugal acceleration has a destabilizing effect while to its left a stabilizing effect is expected; the results presented in Fig. 17b, c reaffirm this expectation showing an eccentric shift of the convection cells toward the right side of the rotation axis. When the rotation axis is moved further toward the hot wall, say at |x0| = 0.6 as presented in Fig. 18a, weak convection cells appear even to the left of the rotation axis. This weak convection becomes stronger as |x0| increases, as observed in Fig. 18b for |x0| = 0.7, and the formation of boundary layers associated with the primary convection cells is observed to the right of the rotation axis. These boundary layers become more significant for |x0| = 0.8 as represented by sharp streamlines gradients in Fig. 19a. When |x0| = 0.9, Fig. 19b shows that the boundary layers of the primary convection are well established and the whole domain is filled with weaker secondary, tertiary, and further convection cells. The results for the isotherms corresponding to values of |x0| = 0 , 0.5 , 0.6 , and 0.7 are presented in Fig. 20 where the effect of moving the axis of rotation within the porous layer, on the temperature is evident.

6

Coriolis Effects on Natural Convection in Porous Media

6.1

Coriolis Effect on Natural Convection in Porous Media due to Thermal Buoyancy of Centrifugal Body Forces

In a previous section, centrifugally driven natural convection was discussed under conditions of small rotation rates, i.e., Ek 1. Then, as a first approximation the Coriolis effect was neglected. It is the objective of the present section to show the

16

Natural Convection in Rotating Flows

741

z'

z' w

w

*

*

x'

x'

| x0 | = 0.6

| x0 | = 0.7

Fig. 18 The convective flow field at marginal stability for two different values of |x0|; ten streamlines equally divided between ψ min and ψ max (Courtesy Elsevier Science Ltd.)

a z

z'

w*

b z

z'

x'

x' | x 0 | = 0.8

w*

| x 0 | = 0.9

Fig. 19 The convective flow field at marginal stability for two different values of |x0|; ten streamlines equally divided between ψ min and ψ max (Courtesy Elsevier Science Ltd.)

742

P. Vadasz

T=1

x'

x' | x0| = 0

d

z'

w*

w*

T=1

w*

z'

T=0

c

T=0

w*

z'

T=1

b

T=1

T=0

z'

T=0

a

x'

x' | x0 | = 0.6

| x 0 | = 0.5

| x0 | = 0.7

Fig. 20 The convective temperature field at marginal stability for four different values of |x0|; ten isotherms equally divided between Tmin = 0 and Tmax = 1 (Courtesy Elsevier Science Ltd.) Fig. 21 Three-dimensional convection in a long rotating porous box (Courtesy: The ASME)

w* z* w

*

2

x *

HOT y *

L*

COLD

H* H* x*

effect of the Coriolis acceleration on natural convection even when this effect is small, i.e., Ek 1. A long rotating porous box is considered as presented in Fig. 21. The possibility of internal heat generation is included but the case without heat generation, i.e., when the box is heated from above and cooled from below is dealt with separately. This case was solved by Vadasz (1993) by using an asymptotic expansion in terms of two small parameters representing the reciprocal Ekman number in porous media and the aspect ratio of the domain. Equations (123), (124), and (125) were used with Rag = 0 , K = 1 and neglecting the component of the centrifugal acceleration in the y direction (a small aspect ratio). Then an expansion of the form

16

Natural Convection in Rotating Flows

z

w*

743

z

z

z

z

z

1

1

1

1

1

1

0.5

0.5

0.5

0.5

0.5

0.5

0

0

0

0

0

0

0

0

u00 / xRaw

x

0

u00 / xRaw

a

u00 / xRaw

b

c

x

0

0

0

1

T00

T00

T00

d

e

f

Fig. 22 The leading order convection for the core region: (a) filtration velocity, with heat generation, perfectly conducting top and bottom walls; (b) filtration velocity, with heat generation, perfectly conducting top wall, insulated bottom wall; (c) filtration velocity, no heat generation, heating from above; (d) temperature, as in (a); (e) temperature, as in (b); (f) temperature, as in (c)

½V, T, pr  ¼

1 X 1 X

H m Ekn ½Vmn , T mn , pmn 

(169)

m¼0 n¼0

is introduced in the equations, where H is the aspect ratio, i.e., H = H
/L
. To leading order the zero powers of the aspect ratio Hm and Ekn are used, which yields v00 ¼ w00 ¼ 0; T 00 ¼ z; u00 ¼ 

Raω x ½2z  1 2

(170)

This solution holds for the core region of the box and is presented in Fig. 22c, f. To orders 1 in Ekn and 0 in Hm, the Coriolis effect is first detected in a plane perpendicular to the leading order natural convection flow leading to the following analytical solution for the stream function ψ 01 in the y  z plane ψ 01 ¼ 

1 X 1 16Raω x X sin ½ð2i  1Þπy sin ½ð2j  1Þπz h i 4 2 2 π i¼1 j¼1 ð2i  1Þð2j  1Þ ð2i  1Þ þ ð2j  1Þ

(171)

and the corresponding temperature solution is T 01 ¼ 

1 X 1 16Raω x X cos ½ð2i  1Þπy sin ½ð2j  1Þπz h i2 π5 i¼1 j¼1 ð2j  1Þ ð2i  1Þ2 þ ð2j  1Þ2

(172)

744

P. Vadasz

a

T= 0

b

z

T= 1

z

z y

c

T= 0

y

T= 0

@T =0 @z

y

T= 0

Fig. 23 The flow and temperature field in the y  z plane; (a) with heat generation, perfectly conducting top and bottom walls; (b) with heat generation, perfectly conducting top wall, insulated bottom wall; (c) no heat generation, heating from above

From the solutions, it was concluded that the Coriolis effect on natural convection is controlled by the combined dimensionless group σ¼

Raω 2βT
ΔT c ω3
L
H
K 2o ¼ Ek αe
ν2
ϕ

(173)

The flow and temperature fields in the plane y  z, perpendicular to the leading order natural convection plane as evaluated through the analytical solution, are presented in Fig. 23c in the form of streamlines and isotherms. The single vortex in this plane is a consequence of the monotonic variation of the natural convection flow field in the core region, i.e., by the sign of @u00/@z. By extending this argument to evaluate the natural convection flow and temperature field in a similar box to that in Fig. 21 but including internal heat generation and subject to different top and bottom boundary conditions, the following cases were considered by Vadasz (1995): (i) A uniform rate of internal heat generation and perfectly conducting top and bottom walls, i.e., Q = 1 , z = 0 : T = 0, and z = 1 : T = 0 (ii) A uniform rate of internal heat generation, perfectly conducting top wall and adiabatic bottom wall, i.e., Q = 1 , z = 0 : @T/@z = 0 and z = 1 : T = 0 As a result a basic flow u00, at the leading order, was evaluated in the form (a) For boundary conditions – set (i) u00


 Raω x 2 1 z zþ ¼ 2 6

(174)

(b) For boundary conditions – set (ii) u00


 Raω x 2 1 z  ¼ 2 3

(175)

16

Natural Convection in Rotating Flows

745

The graphical description of the leading order core solutions in terms of u00/Raωx is presented in Fig. 22a, b and the solutions for T00 in Fig. 22d, e. Figure 22a, d correspond to the boundary conditions – set (i), while Fig. 22b, e correspond to the boundary conditions – set (ii). Because of the change of sign of the gradient @u00/@z in case (i) (see Fig. 22a) a double vortex secondary flow as presented in Fig. 23a was obtained in the y  z plane. Since the gradient @u00/@z for case (ii) does not change sign a single vortex secondary flow is the result obtained in the y  z plane for this case, as shown in Fig. 23b. A comparison of the flow direction of this vortex with the direction of the vortex resulting from the analytical solution for the case without heat generation while heating from above (see Fig. 23c), shows again the effect of the basic flow vertical gradient @u00/@z on the secondary flow, i.e., a positive gradient is associated with an counterclockwise flow and a negative gradient favors a clockwise flow. The resulting effect of these secondary flows on the temperature is presented by the dashed lines in Fig. 23, representing the isotherms.

6.2

Coriolis Effect on Natural Convection due to Thermal Buoyancy of Gravity Forces

The problem of a rotating porous layer subject to gravity and heated from below (see Fig. 24) was originally investigated by Friedrich (1983) and by Patil and Vaidyanathan (1983). Both studies considered a non-Darcy model, which is probably subject to the limitations as shown by Nield (1991b). Friedrich (1983) focused on the effect of Prandtl number on the convective flow resulting from a linear stability analysis as well as a nonlinear numerical solution, while Patil and Vaidyanathan (1983) dealt with the influence of variable viscosity on the stability condition. The latter concluded that variable viscosity has a destabilizing effect

Fig. 24 A rotating fluid saturated porous layer heated from below (Courtesy: Cambridge University Press)

746

P. Vadasz

while rotation has a stabilizing effect. Although the non-Darcy model considered included the time derivative in the momentum equation, the possibility of convection setting-in as an oscillatory instability was not explicitly investigated by Patil and Vaidyanathan (1983). It should be pointed out that it was shown in Sect. 2 that for a pure fluid (nonporous domain) convection sets in as oscillatory instability for a certain range of Prandtl number values (Chandrasekhar 1981). This possibility was explored by Friedrich (1983), which presents stability curves for both monotonic and oscillatory instability. Jou and Liaw (1987b) investigated a similar problem of gravity-driven thermal convection in a rotating porous layer subject to transient heating from below. By using a non-Darcy model, they established the stability conditions for the marginal state without considering the possibility of oscillatory convection. An important analogy was discovered by Palm and Tyvand (1984) who showed, by using a Darcy model, that the onset of gravity driven convection in a rotating porous layer is equivalent to the case of an anisotropic porous medium. The critical Rayleigh number was found to be h i2 Rag,cr ¼ π 2 ð1 þ TaÞ1=2 þ 1

(176)

where Ta is the Taylor number defined here as  Ta ¼

2ω
K o ϕ ν


2 (177)

and the corresponding critical wave number is π(1 + Ta)1/4. The porosity is missing in Palm and Tyvand (1984) definition of Ta. Nield (1999) has pointed out that these authors and others have omitted the porosity from the Coriolis term. This result, Eq. (176) (amended to include the correct definition of Ta), was confirmed by Vadasz (1998b) for a Darcy model extended to include the time derivative term (see Eq. (115) with Raω = 0), while performing linear stability as well as a weak nonlinear analyses of the problem to provide differences as well as similarities with the corresponding problem in pure fluids (nonporous domains). As such, Vadasz (1998b) found that, in contrast to the problem in pure fluids, overstable convection in porous media at marginal stability is not limited to a particular domain of Prandtl number values (in pure fluids the necessary condition is Pr < 1). Moreover, it was also established by Vadasz (1998b) that in the porous media problem, the critical wave number in the plane containing the streamlines for stationary convection is not identical to the critical wave number associated with convection without rotation and is therefore not independent of rotation, a result which is quite distinct from the corresponding pure-fluids problem. Nevertheless, it was evident that in porous media, just as in the case of pure fluids subject to rotation and heated from below, the viscosity at high rotation rates has a destabilizing effect on the onset of stationary convection, i.e., the higher the viscosity the less stable is the fluid. An example of stability curves for overstable convection is presented in Fig. 25 for

16

Natural Convection in Rotating Flows

747

Ta = 5

20

(st) Rc

γ =0. γ =0.2

(ov)

Ra c /π

2

18

γ =0.4

16

γ =0.6 γ =0.8

14 12 10 8 6

0

0.5

1

κ /π

1.5

2

2.5

Fig. 25 Stability curves for overstable gravity driven convection in a rotating porous layer heated from below (γ = Va/π 2, R = Rag/π 2) (Courtesy: Cambridge University Press)

Ta = 5, where κ is the wave number. The upper bound of these stability curves is represented by a stability curve corresponding to stationary convection at the same particular value of the Taylor number, while the lower bound was found to be independent of the value of Taylor number and corresponds to the stability curve for overstable convection associated with Va = 0. Two conditions have to be fulfilled for overstable convection to set in at marginal stability, i.e., (i) the value of Rayleigh number has to be higher than the critical Rayleigh number associated with overstable convection, and (ii) the critical Rayleigh number associated with overstable convection has to be smaller than the critical Rayleigh number associated with stationary convection. The stability map obtained by Vadasz (1998b) is presented in Fig. 26, which shows that the Ta  γ (γ = Va/π 2) plane is divided by a continuous curve (almost a straight line) into two zones, one for which convection sets in as stationary, and the other where overstable convection is preferred. The dotted curve represents the case when the necessary condition (i) above is fulfilled but condition (ii) is not. Weak nonlinear stationary as well as oscillatory solutions were derived, identifying the domain of parameter values consistent with supercritical pitchfork (in the stationary case) and Hopf (in the oscillatory case) bifurcations. Unfortunately due to a typo affecting the sign of one of the nonlinear terms in the weak nonlinear analysis, the direction of the bifurcations presented might be incorrect. The identification of the tricritical point corresponding to the transition from supercritical to subcritical bifurcations was presented on the γ  Ta parameter plane. The possibility of a codimension-2

748

P. Vadasz 35

σ i,2 cr = 0

30

(ov)

(st)

R cr = R cr

25

Stationary Convection

g

20 15

Overstable Convection

10 5 0

0

40

80

120

160

200

Ta Fig. 26 Stability map for gravity-driven convection in a rotating porous layer heated from below (γ = Va/π 2, R = Rag/π 2) (Courtesy: Cambridge University Press)

bifurcation, which is anticipated at the intersection between the stationary and overstable solutions, although identified as being of significant interest for further study, was not investigated by Vadasz (1998b).

7

Other Effects of Rotation on Flow and Natural Convection in Porous Media

7.1

Natural Convection in Porous Media due to Thermal Buoyancy of Combined Centrifugal and Gravity Forces

Section 5 dealt with the onset of natural convection due to thermal buoyancy caused by centrifugal body forces. Consequently the effect of gravity was neglected, i.e., Rag = 0. For this assumption to be valid the following condition has to be satisfied: Rag =Raω ¼ g
=ω2
L
 1. It is of interest to investigate the case when Rag ~ Raω and both centrifugal and gravity forces are of the same order of magnitude. Vadasz and Govender (1998, 2001) present investigations of cases corresponding to the ones presented in Sect. 5.3, including the effect of gravity in a direction perpendicular to the centrifugal force. Figure 12 is still applicable to the present problem with the slight modification of drawing the gravity acceleration g
in the negative z direction. The notation remains the same and Eq. (157) becomes

16

Natural Convection in Rotating Flows

749

V ¼ ∇pr  Raωo ½1 þ η x T^e x þ Rag T^e z

(178)

where η = 1/x0 = Raω/Raωo represents the reciprocal of the offset distance from the axis of rotation. The approach being the same as in Sect. 4.3, the solution is expressed as a sum of a basic solution and small perturbations as presented in Eq. (159). However, because of the presence of the gravity component in Eq. (178), a motionless conduction solution is not possible any more. Therefore, the basic solution far from the top and bottom walls is obtained in the form 

 1 ub ¼ vb ¼ 0; wb ¼ Rag x  ; T b ¼ x; 2
 1 1 1 2 þ η x þ const: pr,b ¼ Rag z  Raωo x 2 2 3

(179)

Substituting this basic solution into the governing equations and linearizing the result by neglecting terms that include products of perturbations, which are small, yield a set of partial differential equations for the perturbations. Assuming a normal modes expansion in the y and z directions in the form  

T 0 ¼ Aκ θðxÞexp σt þ i κ y y þ κ z z

(180)

where κy and κ z are the wave numbers in y and z directions respectively, i.e., κ 2 ¼ κ 2y þ κ2z , and using the Galerkin method, the following set of linear algebraic equations is obtained at marginal stability (i.e., for σ = 0) ( M h  i X 2 2 m2 π 2 þ κ2  κ2 Raωo ð2 þ ηÞ δml m¼1

) # 

8 m l κz Rag 2  2 16m l κ2 Raωo 2 2  i  þ 2 2 2 π l þ m þ 2 κ δmþl,2p1 am ¼ 0 π l  m2 π 2 l2  m2 "

(181) pffiffiffiffiffiffiffi for l = 1 , 2 , 3 , . . . , M and i ¼ 1. In Eq. (181), δml is the Kronecker delta function and the index p can take arbitrary integer values, since it stands only for setting the second index in the Kronecker delta function to be an odd integer. A particular case of interest is the configuration when the layer is placed far away from the axis of rotation, i.e., when the length of the layer L
is much smaller than the offset distance from the rotation axis x0
. Therefore for x0 = (x0
/L
)!1 or η!0, the contribution of the term ηx in Eq. (178) is not significant. Substitution of this limit into Eq. (181) and solving the system at the second order, i.e., M = 2, yields a quadratic equation for the characteristic values of Raωo. This equation has no real solutions for values of α ¼ κ2z =π 2 beyond a transitional value αtr = (27π 3/16 Rag)2.

750

P. Vadasz

This value was evaluated at higher orders too, showing that for M = 10 the transitional value varies very little with Rag, beyond a certain Rag value around 50π. The critical values of Raωo were evaluated for different values of Rg (=Rag/π) and the corresponding two-dimensional convection solutions in terms of streamlines are presented graphically for the odd modes in Fig. 27a, showing the perturbation solutions in the x  z plane as skewed convection cells when compared with the case without gravity. The corresponding convection solutions for the even modes are presented in Fig. 27b, where it is evident that the centrifugal

z

a

z

z

z'

T=0

x'

x

x Rg=0

x

Rg=5

Rg=20

z

z

z

x

Rg=10

b z'

T=1

T=0

T=1

T=0

T=1

T=0

ωc

T=1

z

x

x' Rg=5

T=1

T=0

T=1

T=0

T=1

ωc

T=0

Fig. 27 The convective flow field (streamlines) at marginal stability for different values of Rg (=Rag/π): (a) the odd modes; (b) the even modes

x Rg=10

x Rg=20

16

Natural Convection in Rotating Flows

751

effect is felt predominantly in the central region of the layer, while the downward and upward basic gravity-driven convection persists along the left and right boundaries, respectively, although not in straight lines. Beyond the transition value of α, the basic gravity driven convective flow (Eq. (179)) is unconditionally stable. These results were shown to have an analogy with the problem of gravitydriven convection in a nonrotating, inclined porous layer (Govender and Vadasz 1995). Qualitative experimental confirmation of these results was presented by Vadasz and Heerah (1998) by using a thermosensitive liquid-crystal tracer in a rotating Hele-Shaw cell. When the layer is placed at an arbitrary finite distance from the axis of rotation, no real solutions exist for the characteristic values of Raωo corresponding to any values of γ other than γ = κzRag = 0. In the presence of gravity, Rag 6¼ 0 and γ = 0 can be satisfied only if κz = 0. Therefore the presence of gravity in this case has no other role but to exclude the vertical modes of convection. The critical centrifugal Rayleigh numbers and the corresponding critical wave numbers are the same as in the corresponding case without gravity as presented in Sect. 5.3. However, the eigenfunctions representing the convection pattern are different as they exclude the vertical modes replacing them with a corresponding horizontal mode in the y direction. Therefore, a cellular convection in the x  y plane is superimposed to the basic convection in the x  z plane.

7.2

Onset of Convection due to Thermohaline (Binary Mixture) Buoyancy of Gravity Forces

Limited research results are available on natural convection in a rotating porous medium due to thermohaline buoyancy caused by gravity forces. Chakrabarti and Gupta (1981) investigated a non-Darcy model, which includes the Brinkman term as well as a nonlinear convective term in the momentum equation (in the form (V  ∇) V). Therefore the model’s validity is subject to the limitations pointed out by Nield (1991b). Both linear and nonlinear analyses were performed, and overstability was particularly investigated. Overstability is affected in this case by both the presence of a salinity gradient and by the Coriolis effect. Apart from the thermal and solutal Rayleigh numbers and the Taylor number, two additional parameters affect the stability. These are the Prandtl number Pr = ν
/αe
, and the Darcy number Da ¼ K o =H 2
, where H
is the layer’s height. The authors found that, in the range of values of the parameters, which were considered, the linear stability results favor setting-in of convection through a mechanism of overstability. The results for nonlinear steady convection show that the system becomes unstable to finite amplitude steady disturbances before it becomes unstable to disturbances of infinitesimal amplitude. Thus the porous layer may exhibit subcritical instability in the presence of rotation. These results are surprising at least in the sense of their absolute generality and the authors mention that further confirmation is needed in order to increase the degree of confidence in these results.

752

P. Vadasz

A similar problem was investigated by Rudraiah et al. (1986) while focusing on the effect of rotation on linear and nonlinear double-diffusive convection in a sparsely packed porous medium. A non-Darcy model identical to the one used by Chakrabarti and Gupta (1981) was adopted by Rudraiah et al. (1986); however, the authors spelled out explicitly that the model validity is limited to high porosity and high permeability which makes it closer to the behavior of a pure fluid system (nonporous domain). It is probably for this reason that the authors preferred to use the nonporous medium definitions for Rayleigh and Taylor numbers which differ by a factor of Da and Da2, respectively, from the corresponding definitions for porous media. It is because of these definitions that the authors concluded that for small values of Da number, the effect of rotation is negligible for values of Ta < 106. This means that rotation has a significant effect for large rotation rates, i.e., Ta > 106. If the porous media Taylor number had been used instead, i.e., the proper porous media scales, then one could have significant effects of rotation at porous media Taylor numbers as small as Ta > 10. Hence, the results presented by Rudraiah et al. (1986) are useful provided Da = O(1), which is applicable for high permeability (or sparsely packed) porous layers. Marginal stability as well as overstability was investigated and the results show different possibilities of existence of neutral curves by both mechanisms, i.e., monotonic as well as oscillatory instability. In this regard, the results appear more comprehensive in the study by Rudraiah et al. (1986) than in Chakrabarti and Gupta (1981). The finite amplitude analysis was performed by using a severely truncated representation of a Fourier series for the dependent variables. As a result, a seventh-order Lorenz model of double diffusive convection in a porous medium in the presence of rotation was obtained. From the study of steady, finite amplitude analysis the authors found that subcritical instabilities are possible, depending on the parameter values. The effect of the parameters on the heat and mass transport was investigated as well, and results presenting this effect are discussed in Rudraiah et al. (1986).

7.3

Finite Heat Transfer Between the Phases and Temperature Modulation

Lack of local thermal equilibrium (LaLotheq) or local thermal non-equilibrium (LTNE) implies distinct temperature values between the solid and fluid phases within the same REV. Malashetty et al. (2007) presented the linear stability and the onset of convection in a porous layer heated from below and subject to rotation, accounting for the Coriolis effect as in Vadasz (1998b) but allowing for distinct temperature values between the solid and fluid phases, i.e., lack of local thermal equilibrium (LaLotheq), or local thermal nonequilibrium (LTNE). The nonlinear part of the analysis was undertaken by using a truncated mode spectral system, such as the one used by Vadasz and Olek (1998) but adapted for the LaLotheq conditions. The effect of finite heat transfer between the phases leading to lack of local thermal equilibrium was investigated also by Govender and Vadasz (2007) while investigating also the effect of mechanical and thermal anisotropy on the stability of a rotating

16

Natural Convection in Rotating Flows

753

porous layer heated from below and subject to gravity. The topic of anisotropic effects is discussed in the next section. Bhadauria (2008) investigated the effect of temperature modulation on the onset of thermal instability in a horizontal fluidsaturated porous layer heated from below and subject to uniform rotation. An extended Darcy model, which includes the time derivative term, has been considered, and a time-dependent periodic temperature field was applied to modulate the surfaces’ temperature. A perturbation procedure based on small amplitude of the imposed temperature modulation was used to study the combined effect of rotation, permeability, and temperature modulation on the stability of the fluid saturated porous layer. The correction of the critical Rayleigh number was calculated as a function of amplitude and frequency of modulation, the porous media Taylor number, and the Vadasz number. It was found that both rotation and permeability suppress the onset of thermal instability. Furthermore, the author concluded that temperature modulation could either promote or retard the onset of convection.

7.4

Anisotropic Effects

The effect of anisotropy on the stability of convection in a rotating porous layer subject to centrifugal body forces was investigated by Govender (2006). The Darcy model extended to include anisotropic effects, and rotation was used to describe the momentum balance and a modified energy equation that included the effects of thermal anisotropy was used to account for the heat transfer. The linear stability theory was used to evaluate the critical Rayleigh number for the onset of convection in the presence of thermal and mechanical anisotropy. It was found that the convection was stabilized when the thermal anisotropy ratio (which is a function of the thermal and mechanical anisotropy parameters) increased in magnitude. Malashetty and Swamy (2007), and Govender and Vadasz (2007), investigated the Coriolis effect on natural convection in a rotating anisotropic fluid-saturated porous layer heated from below and subject to gravity as the body force. Malashetty and Swamy (2007) assumed local thermal equilibrium while Govender and Vadasz (2007) dealt with lack of local thermal equilibrium (LaLotheq), or local thermal non-equilibrium (LTNE). Malashetty and Swamy (2007) used the linear stability theory as well as a nonlinear spectral method. The linear theory was based on the usual normal mode technique and the nonlinear theory on a truncated Galerkin analysis. The Darcy model extended to include a time derivative and the Coriolis terms with an anisotropic permeability was used to describe the flow through the porous media. A modified energy equation including the thermal anisotropy was used. The effect of rotation, mechanical, and thermal anisotropy parameters and the Prandtl number on the stationary and overstable convection was discussed. It was found that the effect of mechanical anisotropy is to prefer the onset of oscillatory convection instead of the stationary one. It was also found (just as in Vadasz 1998b) that the existence of overstable motions in case of rotating porous media is not restricted to a particular range of Prandtl number as compared to the pure viscous fluid case. The steady finite

754

P. Vadasz

amplitude analysis was performed using the truncated Galerkin modes to find the Nusselt number. The effect of various parameters on heat transfer was investigated. Govender and Vadasz (2007) analyzed the stability of a horizontal rotating fluid saturated porous layer exhibiting both thermal and mechanical anisotropy, subject to lack of local thermal equilibrium (LaLotheq), or local thermal nonequilibrium (LTNE). All of the results were presented as a function of the scaled interphase heat transfer coefficient. The results of the linear stability theory have revealed that increasing the conductivity ratio and the mechanical anisotropy has a destabilizing effect, while increasing the fluid and solid thermal conductivity ratios is stabilizing. In general it was found that rotation has a stabilizing effect in a porous layer exhibiting mechanical or thermal (or both mechanical and thermal) anisotropy.

7.5

Applications to Nanofluids

An interesting recent application is related to nanofluids. A nanofluid is a suspension of nanoparticles or nanotubes in a liquid. When the liquid is saturating a porous matrix one deals with nanofluids in porous media. Rana and Agarwal (2015) investigated the natural convection in a rotating porous layer saturated by a nanofluid and a binary mixture. This implies that the nanoparticles are suspended in a binary mixture, e.g., in a water and salt solution. Therefore double-diffusive convection is anticipated. The model used for the nanofluid incorporates the effects of Brownian motion and thermophoresis, while the Darcy model is used for the porous medium. The neutral and critical Rayleigh numbers for stationary and oscillatory convection have been obtained in terms of various dimensionless parameters. The authors concluded that the principle of exchange of stabilities is applicable in the present problem, while more amount of heat is required in the nanofluid case for convection to set in. Agarwal et al. (2011) considered the convection in a rotating anisotropic porous layer saturated by a nanofluid. The model used for nanofluid combines the effect of Brownian motion along with thermophoresis, while for a porous medium the Darcy model has been used. Using linear stability analysis, the expression for the critical Rayleigh number has been obtained in terms of various dimensionless parameters. Agarwal et al. (2011) indicate that bottom-heavy and top-heavy arrangements of nanoparticles tend to prefer oscillatory and stationary modes of convection, respectively.

7.6

Applications to Solidification of Binary Alloys

During solidification of binary alloys, the solidification front between the solid and the liquid phases is not a sharp front but rather a mushy layer combining liquid and solid phases each one being interconnected. It is not surprising therefore that the treatment of this mushy layer follows all the rules applicable to a porous medium. Natural convection due to thermal as well as concentration gradients occurs in the mushy layer resulting in possible creation of freckles that might affect the quality of

16

Natural Convection in Rotating Flows

755

the cast. When such a process occurs in a system that is subject to rotation, centrifugal buoyancy as well as Coriolis effects is relevant and essential to be included in any model of this process. Govender and Vadasz (2002b) investigated such a system via a weak nonlinear analysis for moderate Stefan numbers applicable to stationary convection in a rotating mushy layer. Consequently, Govender and Vadasz (2002a) investigated a similar system via a weak nonlinear analysis for moderate Stefan numbers applicable to oscillatory convection in a rotating mushy layer. A near-eutectic approximation and large far-field temperature were employed in both papers in order to decouple the mushy layer from the overlying liquid melt. The parameter regimes in terms of Taylor number for example where the bifurcation is subcritical or supercritical were identified. In the case of oscillatory convection, increasing the Taylor number leads to a supercritical bifurcation.

8

Cross-References

▶ Heat Transfer in Rotating Flows ▶ Single-Phase Convective Heat Transfer: Basic Equations and Solutions

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Fultz D, Nakagawa Y (1955) Experiments on over-stable thermal convection in mercury. Proc R Soc Lond A 231(1185):211–225 Glatzmaier GA, Coe RS, Hongre L, Roberts PH (1994) Convection driven zonal flows and vortices in the major planets. Chaos 4:123–134 Glatzmaier GA, Coe RS, Hongre L, Roberts PH (1999) The role of the Earth’s mantle in controlling the frequency of geomagnetic reversals. Nature 401:885–890 Govender S (2006) On the effect of anisotropy on the stability of convection in rotating porous media. Transp Porous Media 64(4):413–422 Govender S (2010) Vadasz number influence on vibration in a rotating porous layer placed far away from the axis of rotation. J Heat Transf 132:112601/1–112601/4 Govender S, Vadasz P (1995) Centrifugal and gravity driven convection in rotating porous media – an analogy with the inclined porous layer. ASME-HTD 309:93–98 Govender S, Vadasz P (2002a) Weak non-linear analysis of moderate Stefan number oscillatory convection in rotating mushy layers. Transp Porous Media 48(3):353–372 Govender S, Vadasz P (2002b) Weak non-linear analysis of moderate Stefan number stationary convection in rotating mushy layers. Transp Porous Media 49(3):247–263 Govender S, Vadasz P (2007) The effect of mechanical and thermal anisotropy on the stability of gravity driven convection in rotating porous media in the presence of thermal non-equilibrium. Transp Porous Media 69(1):55–66 Greenspan HP (1980) The theory of rotating fluids. Cambridge University Press, Cambridge, pp 5–18 Güçeri SI (1994) Fluid flow problems in processing composites materials. In: Proceedings of the 10th international heat transfer conference, vol 1, Brighton, pp 419–432 Hart JE (1971) Instability and secondary motion in a rotating channel flow. J Fluid Mech 45:341–351 Howey DA, Childs PRN, Holmes S (2012) Air-gap convection in rotating electrical machines. IEEE Trans Ind Electron 59(3):1367–1375 Johnston JP, Haleen RM, Lezius DK (1972) Effects of spanwise rotation on the structure of two-dimensional fully developed turbulent channel flow. J Fluid Mech 56:533–557 Jou JJ, Liaw JS (1987a) Transient thermal convection in a rotating porous medium confined between two rigid boundaries. Int Commun Heat Mass Transf 14:147–153 Jou JJ, Liaw JS (1987b) Thermal convection in a porous medium subject to transient heating and rotation. Int J Heat Mass Transf 30:208–211 Julien K, Legg S, McWilliams J, Werne J (1996) Rapidly rotating turbulent Rayleigh-Benard convection. J Fluid Mech 322:243–273 Katgerman L (1994) Heat transfer phenomena during continuous casting of aluminum alloys. In: Proceedings of the 10th international heat transfer conference, SK-12, Brighton, pp 179–187 King EM, Stellmach S, Buffett B (2013) Scaling behaviour in Rayleigh-Benard convection with and without rotation. J Fluid Mech 717:449–471 Kvasha VB (1985) Multiple-spouted gas-fluidized beds and cyclic fluidization: operation and stability. In: Davidson JF, Clift R, Harrison D (eds) Fluidization, 2nd edn. Academic, London, pp 675–701 Lezius DK, Johnston JP (1976) Roll-cell instabilities in a rotating laminar and turbulent channel flow. J Fluid Mech 77:153–175 Lima FHB, Ticianelli EA (2004) Oxygen electrocatalysis on ultra-thin porous coating rotating ring/ disk platinum and platinum–cobalt electrodes in alkaline media. Electrochim Acta 49:4091–4099 Malashetty MS, Swamy M (2007) The effect of rotation on the onset of convection in a horizontal anisotropic porous layer. Int J Therm Sci 46(10):1023–1032 Malashetty MS, Swamy M, Kulkarni S (2007) Thermal convection in a rotating porous layer using a thermal nonequilibrium model. Phys Fluids 19:1–16. Art. 054102 Malkus WVR, Veronis G (1958) Finite amplitude cellular convection. J Fluid Mech 4(3):225–260 Marshall J, Schott F (1999) Open-ocean convection: observations, theory, and models. Rev Geophys 691:1–64

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Vadasz P (1994b) On Taylor-Proudman columns and geostrophic flow in rotating porous media. SAIMechE R&D J 10(3):53–57 Vadasz P (1994c) Centrifugally generated free convection in a rotating porous box. Int J Heat Mass Transf 37(16):2399–2404 Vadasz P (1994d) Stability of free convection in a narrow porous layer subject to rotation. Int Commun Heat Mass Transf 21(6):881–890 Vadasz P (1995) Coriolis effect on free convection in a rotating porous box subject to uniform heat generation. Int J Heat Mass Transf 38(11):2011–2018 Vadasz P (1996a) Stability of free convection in a rotating porous layer distant from the axis of rotation. Transp Porous Media 23:153–173 Vadasz P (1996b) Convection and stability in a rotating porous layer with alternating direction of the centrifugal body force. Int J Heat Mass Transf 39(8):1639–1647 Vadasz P (1997) Flow in rotating porous media. In: du Plessis P (ed), Rahman M (series ed) Fluid transport in porous media. Advances in fluid mechanics, vol 13. Computational Mechanics Publications, Southampton, pp 161–214 Vadasz P (1998a) Free convection in rotating porous media. In: Ingham DB, Pop I (eds) Transport phenomena in porous media. Elsevier, Oxford, pp 285–312 Vadasz P (1998b) Coriolis effect on gravity-driven convection in a rotating porous layer heated from below. J Fluid Mech 376:351–375 Vadasz P (2000) Fluid flow and thermal convection in rotating porous media. In: Vadfai K (ed) Handbook of porous media. Marcel Dekker, New York/Basel, pp 395–439 Vadasz P (2002a) Heat transfer and fluid flow in rotating porous media. In: Hassanizadeh SM, Schotting RJ, Gray WG, Pinder GF (eds) Computational methods in water resources, vol 1. Development in water science, vol 47. Elsevier, Amsterdam, pp 469–476 Vadasz P (2002b) Thermal convection in rotating porous media. In: Trends in heat, mass & momentum transfer, vol 8. Research Trends, Trivandrum, pp 25–58 Vadasz P (2016) Fluid flow and heat transfer in rotating porous media. Springer briefs in applied science and engineering, Kulacki FA (series ed). Springer, Cham/Heidelberg/New York/ Dordrecht/London Vadasz P, Govender S (1998) Two-dimensional convection induced by gravity and centrifugal forces in a rotating porous layer far away from the axis of rotation. Int J Rotating Mach 4(2):73–90 Vadasz P, Govender S (2001) Stability and stationary convection induced by gravity and centrifugal forces in a rotating porous layer distant from the axis of rotation. Int J Eng Sci 39(6):715–732 Vadasz P, Heerah A (1998) Experimental confirmation and analytical results of centrifugally-driven free convection in rotating porous media. J Porous Media 1(3):261–272 Vadasz P, Olek S (1998) Transitions and chaos for free convection in a rotating porous layer. Int J Heat Mass Transf 41(11):1417–1435 Vafai K, Kim SJ (1990) Analysis of surface enhancement by a porous substrate. J Heat Transf 112:700–706 Veronis G (1959) Cellular convection with finite amplitude in a rotating fluid. J Fluid Mech 5 (3):401–4735 Veronis G (1966) Motions at subcritical values of the Rayleigh number in a rotating fluid. J Fluid Mech 24(3):545–554 Whitehead AB (1985) Distributor characteristics and bed properties. In: Davidson JF, Clift R, Harrison D (eds) Fluidization, 2nd edn. Academic, London, pp 173–199 Wiesche S (2017) Heat transfer in rotating flows. In: Kulacki FA (ed) Handbook of thermal science and engineering. Springer Zhong F, Ecke RE, Steinberg V (1993) Rotating Rayleigh-Benard convection: asymmetric modes and vortex states. J Fluid Mech 249:135–159

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Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Refractive Index Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Optical Configurations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Data Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Color Schlieren Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Tomography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Experimentally Recorded Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Scattering Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Liquid Crystal Thermography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Infrared Thermography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Cross-References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

This chapter introduces optical techniques for the measurement of temperature and species concentration in heat and mass transfer processes occurring in fluids. Measurement is initiated by a light source and changes in the attributes of light emerging from the test cell are recorded by a suitable detector. The fluid medium is taken to be transparent to the passage of light. The light intensity distribution and contrast generated during the measurement can arise either due to variations in the refractive index of the fluid or by scattering from preselected optically active particles. Among the refractive index methods, interferometry, schlieren (monochrome and color), and shadowgraph are discussed. Liquid crystal

P. K. Panigrahi (*) · K. Muralidhar Department of Mechanical Engineering, Indian Institute of Technology Kanpur, Kanpur, India e-mail: [email protected]; [email protected] # Springer International Publishing AG, part of Springer Nature 2018 F. A. Kulacki (ed.), Handbook of Thermal Science and Engineering, https://doi.org/10.1007/978-3-319-26695-4_15

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thermography is presented as an example of a scattering technique. Infrared measurements are also included for completeness. Specific advantages of optical methods are that they are nonintrusive, image a flow cross section, and are inertiafree. A continuous sequence of optical images can be recorded in a computer using CCD cameras, thus extending measurements to the time domain. Optical images can be analyzed and methods of extracting quantitative data from such images are discussed. Several applications and representative images recorded on a laboratory scale are also presented. Nomenclature

a0-a1-a2-a3 ak A-B-C C d D f h H I k L n

Nu Pe Pr Q R-G-B Re Ri t T V x-y-z

Constants appearing in the Lorentz-Lorenz formula Diameter of the focal spot (m) Constants appearing in the Cauchy formula Species concentration in a solution (kg/m3) Diameter of the cylinder (m) Distance of the exit plane of the apparatus from the screen (m) Focal length of the decollimating lens/mirror (m) Local heat transfer coefficient (W/m2-K) Hue distribution in a color image (units of angle) Light intensity distribution on the screen, grayscale value Thermal conductivity (W/mK) Length of the apparatus in the direction of propagation of light, m Refractive index of the medium; na (or n0) for the ambient; nfor the average in the viewing direction, nλ for the wavelength dependent refractive index Nusselt number Peclet number Prandtl number Flow rate, lit/min Color scales of red, green, and blue in an image Reynolds number based on the cylinder diameter and upstream conditions; Rex for the local Reynolds number Richardson number based on cylinder diameter, upstream velocity, and overall temperature difference Time (s) Temperature; Tw for the wall temperature (K) Voltage; Vexc for excitation voltage (V) Cartesian coordinates with z along the viewing direction and x-y, the cross-sectional plane

Greek Symbols

α δ Δ

Thermal diffusivity, m2/s Angular deflection of the light beam, radians Laplacian operator

17

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ΔCe, ΔTe, λ ρ θ

1

761

Change in concentration per fringe shift, kg/m3 Change in temperature per fringe shift, K Wavelength of light, nm Material density, kg/m3 Deflection angle (θy in the y-direction), radians

Introduction

Transport of energy, mass, and momentum occur frequently in several applications. Modeling these phenomena continues to be complex and comparison against experiments is a continuing challenge. Optical methods were initially developed as a visualizing tool for transport phenomena. These techniques have evolved and are now considered as powerful tools in heat and mass transfer measurements (Lauterborn and Vogel 1984; Merzkirch 1987; Mayinger 1994; Goldstein 1996; Lehner and Mewes 1999). As a consequence, optical techniques find extensive use in temperature and species concentration measurements, combustion diagnostics, and process monitoring related to physical and engineering sciences. Optical measurements were initially based on visible light, followed by lasers. These have expanded over time to include a wider electromagnetic spectrum. Apart from a radiation source, optical methods require a camera for recording images. In an unsteady measurement, an image sequence is collected, invariably through a computer. Several features of optical techniques make them quite effective and convenient. Such methods are nonintrusive and perform whole-field imaging with small time lag. In view of these advantages, optical methods are best suited to treat unsteady scalar and vector fields in three-dimensional geometries. Other applications uniquely suited for optical measurements include determination of wall heat fluxes from wall temperatures and the estimation of material properties on multiple scales. Optical methods are classified as (i) transmission techniques and (ii) scattering techniques (Mayinger 1994). In (i), the medium is transparent and imaging relies on changes in refractive index, which is, in turn, uniquely related to density. Examples of transmission techniques are interferometry, schlieren, and shadowgraph (Settles 2001; Muralidhar 2001; Panigrahi and Muralidhar 2012). When light (wavelength λ) falls on a particle of a given diameter d, the scattered radiation will experience changes in intensity, directionality, wavelength, phase, and polarization. The property that changes significantly forms the basis of the measurement technique. The relevant quantity here is the ratio of the radiation wavelength and the particle diameter. Laser Doppler velocimetry (LDV), particle tracking velocimetry (PTV), and particle image velocimetry (PIV), are examples that exploit changes initiated by scattering for the measurement of fluid velocity (Goldstein 1996; Tropea et al. 2007). Temperature and species concentration can be measured using liquid crystal thermography, laser-induced fluorescence, and Mie scattering.

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Refractive Index Techniques

Refractive index based techniques are applicable when the fluid medium is transparent to visible light. The light intensity distribution emerging from the test cell and the contrast generated depend on changes in refractive index in the region being studied. Since these techniques rely on the change in refractive index, interferometry, schlieren (monochrome and color), and shadowgraph are interrelated. Interferometry depends on changes in the optical path length but can be limited by beam bending (Anderson and Milton 1989). Changes in the light beam direction, in turn, form the basis of schlieren and shadowgraph techniques. Using multiwavelength lasers and suitable cameras, distribution of several dependent variables such as temperature and solute concentration can be simultaneously determined. Optical images can be viewed as path-integrated data of thermal and species concentration in the viewing direction. As a result, three- dimensional variation of the variable of interest can be reconstructed by algorithms such as tomography (Herman 1986; Natterer 1986). Seen together, optical measurements pave way for time-dependent three-dimensional measurements of temperature and concentration in engineering applications. In transparent media, light interaction with the material depends on the refractive index. The usefulness of refractive index relies on the fact that, for isotropic transparent media, it is a unique function of material density. Temperature and species concentration influence the density of the material. Hence, inhomogeneities in the refractive index distribution will necessarily carry data related to the heat and mass transfer phenomena. The anisotropic nature of refractive index has been utilized in complex fluids and rheology using polarized light (Schmidt et al. 2002).

2.1

Optical Configurations

Interferometry, schlieren, and shadowgraph generate optical signals on the basis of changes in refractive index of the transparent medium with temperature and species concentration (Panigrahi and Muralidhar 2012). The layouts of the three optical techniques discussed in the present chapter are shown in Fig. 1. For measurements, a continuous wave, highly coherent, low power helium-neon laser serves as the light source. A monochrome CCD camera of adequate spatial resolution records the optical images of the convective field. The camera may be interfaced with a computer through an A/D card. The working principle of a Mach-Zehnder interferometer can be inferred from Fig. 1a. In a laboratory scale arrangement, it has two mirrors and two 50% beam splitters of, say, 150 mm diameter. The mirrors have a 99.9% silver coating with silicon dioxide layer for protection. The interferometer should be mounted on pneumatic legs so that the instrument is isolated from external vibrations.

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Fig. 1 Optical configurations of (a) Mach-Zehnder interferometer, (b) laser schlieren, and (c) shadowgraph (Panigrahi and Muralidhar 2012)

Experiments can be carried out in the infinite and the wedge fringe settings. In the former, the initial optical path difference between the test and the reference beams is zero. This is a condition of constructive interference when a bright patch is formed on the screen. When the test beam passes through a region of density disturbance, a set of fringes appear on the screen. At each fringe, the depth-averaged density is a constant. In an unsteady problem, fringes move with time. In the wedge fringe

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setting, the optics is first misaligned so that a set of straight fringes is formed – horizontal or vertical. When the test beam passes through a region of thermal or concentration disturbance, fringes are displaced in proportion to the change in temperature (or concentration) from the reference. The fringes recorded in the wedge fringe setting, thus, yield the spatial profiles of these quantities. Closely related to Fig. 1a is dual wavelength interferometry (Lehner and Mewes 1999) for joint measurement of multiple variables and phase shifting interferometry (Min et al. 2010) for controlling resolution. Among various arrangements possible, a Z-type schlieren system is shown in Fig. 1b. On a laboratory scale, it can have concave mirrors of 1.30 m focal length and 200 mm diameter. Relatively large focal lengths make the schlieren technique sensitive to the temperature and concentration gradients. The knife-edge is placed at the focus of the second concave mirror. It is positioned to cut off a part of the light falling on it. A schlieren apparatus is a single light beam arrangement and does not have a reference. In the absence of a density disturbance, the illumination on the screen is gradually reduced by the knife edge. The intensity values initially in an experiment may be chosen to be less than 20, for example, on a gray scale of 0–255. The knife-edge is commonly set perpendicular to the density gradients in the experimental apparatus. Further developments in schlieren instrumentation have led to the color (rainbow-) schlieren (Al-Ammar et al. 1998) that improves the appearance of the image in terms of color and coherent gradient sensing (Atcheson et al. 2009) that simplifies analysis. Shadowgraph is also a single light beam diagnostic technique that simplifies the schlieren by eliminating the knife-edge and the decollimating optics. A laboratoryscale shadowgraph arrangement is shown in Fig. 1c. The position of the screen alters the light intensity distribution in shadowgraph images and has an important role in data analysis (Lewis et al. 1987; Rasenat et al. 1989; Schopf et al. 1996). The screen position may be chosen by the user to improve image contrast while selectively extracting relevant features of the flow field. In schlieren and shadowgraph, the light beam passing through the region of high density gradient may interfere with the portion passing through a uniform region, giving rise to newer methods of measurement such as schlieren-interferometry (Gayhart and Prescott 1949). Such effects may be avoided in an exclusive schlieren or shadowgraph arrangement while fixing the camera position. Moving images in a convective field can be inverted to yield fluid velocity data, giving rise to techniques such as schlieren-PIV. Information of temperature and concentration in an interferogram is localized at the fringes formed. In contrast, schlieren and shadowgraph images carry information in the form of an intensity distribution over the image. Data is thus available at the pixel-level of the image. There are drawbacks to be considered here, including errors linked to noise arising from scattering within the fluid and the optical windows apart from the possibility of device saturation. The first error may be less significant because the measurement integrates the field variable in the direction of propagation of the light beam, an operation that tends to damp noisy profiles. Device saturation can be minimized by working with reduced laser intensity.

17

2.2

Visualization of Convective Heat Transfer

765

Data Analysis

Refractive index techniques for a transparent material rely on the following unique relationship between refractive index n and density ρ (Mayinger 1994; Goldstein 1996): n2
1 ¼ constant ρð n2 þ 2Þ

(1)

Equation 1 is referred as the Lorentz-Lorenz formula. In gases, n ~ 1 and Eq. 1 simplifies to n
1 ¼ constant ρ

(2)

Hence, Eq. 2 shows that in gases, the derivative dn/dρ is constant and a material property. In liquids, if the overall changes in density are small, the derivative is practically constant. For moderate changes in temperature, density and temperature T are linearly related. It follows that dn/dT is also constant, depending on the choice of the medium. Thus, changes in temperature will manifest as those in refractive index. This discussion carries over to mass transfer, where changes in density arise from a solutal concentration field. In the above formulation, pressure changes in gas can be accommodated via an equation of state, for example, the ideal gas law. The Lorentz-Lorenz formula is a special case of a more general dependence of refractive index on density. The general expression is written in the form of a virial expansion in the dependent variables (ρ, T, and C) as follows: n2
1 ¼ a0 þ a1 ρ þ a2 T þ a3 C ρð n2 þ 2Þ

(2a)

The explicit dependence of refractive index on temperature and concentration via coefficients a2 and a3 is usually small for most fluids of interest. Hence, it is appropriate to generalize this equation as n2
1 ¼ a0 þ a1 ρ ρð n2 þ 2Þ

(2b)

The correction has negligible effect in gases. When the second term is included, the derivative dn/dρ may have 5–10% maximum error in liquids. Though a monochromatic source is anticipated in the above discussions, the dependence of refractive index on wavelength λ is also of interest. It is expressed by the Cauchy’s formula: nλ ¼ A þ

B C þ λ2 λ4

(2c)

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The dependence is known to be stronger in liquids and solids as compared to gases. See Schiebener et al. 1990 for a discussion related to refractive indices of water and steam. The governing equations for the three optical methods are summarized below (Muralidhar 2014). Interferometry: Start with the passage of a laser through a test section of length L in the viewing direction as shown in Fig. 1a. The interferometer is in the infinite fringe setting. For a light source of wavelength λ the change in concentration and temperature required per fringe shift ΔCe (or ΔTe) is given as (Mayinger 1994; Goldstein 1996; Panigrahi and Muralidhar 2012): ΔCe ¼

λ=L λ=L ; ΔT e ¼ dn=dC dn=dT

(3)

The fringe positions are determined from the interferogram using image processing tools. In the wedge fringe setting, the fringe displacement from the initial location is proportional to the change in temperature (or concentration) with respect to the portion of the fluid region where fringes are straight and undisturbed. The above results ignore beam bending in a variable refractive index field. Hence, they require the approximation that the light travels in a straight line with beam deflection and displacement effects being small. Schlieren: Image formation in such a system relies entirely on beam bending (Fig. 1b). The measurement relies on deflection of the light beam in a variable refractive index field from regions of low refractive index toward regions that have a higher refractive index. Quantitative information from a schlieren image is extracted from an expression for the cumulative angle of refraction of the light beam emerging from the experimental apparatus. This angle is distributed over the cross-sectional x-y plane when the direction of propagation of the light beam is the z-coordinate. Using Snell’s law, the total angular deflection δ normal to the knife edge can be expressed as (Settles 2001): 1 δ¼ na

ðL n 0

@lnðnÞ dz @y

(4)

Here n(x, y, z) is the local refractive index within the experimental chamber and na refers to the ambient. Beam deflection is the driver for modulating light intensity over the screen of the camera. The distortion in the intensity field ΔI beyond the knife edge recorded on the screen relative to the background intensity distribution Ik can be related to the refractive index field directly as: ΔI f ¼ Ik ak  na

ðL 0

@n dz @y

(5)

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767

Here na, the refractive index of the ambient is practically unity, ak is the size of the focal spot at the knife edge, with f, the focal length of the decollimating lens or mirror. This equation confirms that the schlieren technique depends on the derivative of refractive index (normal to the knife edge) integrated over the path of the light beam. In terms of the ray-averaged refractive index, the governing equation for the schlieren process is derived from Eq. 5 as ΔI f @n L ¼ Ik ak  na @y

(6)

The above equation relies on the approximation that changes in the light intensity occur only due to beam deflection, instead of a displacement in the x-y plane (Kumar et al. 2007). In Eq. 6, the contribution of additional refraction of light at the optical windows that confine the experimental set-up should be accounted for. Shadowgraph: The shadowgraph measurement depends on the change in the light intensity arising from beam deflection as well as beam displacement relative to its original path (Fig. 1c). Complete shadowgraph analysis is possible only by tracing the path of a bundle of rays through the physical region. With linearizing approximations such as small displacement of the light ray, a second order PDE is derived for the refractive index distribution. The second derivative referred here is related to the intensity contrast over the shadowgraph image. With D as the distance of the screen from the exit plane of the test cell and Δ as the Laplace operator defined in the x-y plane, the governing equation for shadowgraph measurement is written as (Schopf et al. 1996): ΔI ¼ L  D Δfln nðx, yÞg Ik

(7)

Here, Ik is the final intensity distribution and ΔI is the difference in intensity value at a pixel location relative to the initial value. The ratio of these two intensities defines contrast and hence the sensitivity of the measurement. Equations 6 and 7 have to be suitably integrated with boundary conditions to determine the local refractive index followed by concentration (or temperature) field using Eqs. 1 and 2. Integration of the above-referred Poisson Eq. 7 can be performed by the method of finite differences. When the approximations enforced in the derivation of the schlieren and shadowgraph equations are not valid, optical techniques can only be used for field visualization (Merzkirch 1987).

2.3

Color Schlieren Technique

The primary measurement in a schlieren arrangement that relates to the heat and mass transfer process is the angle of deflection of the light rays emerging from the test apparatus. The knife edge affords a method of measuring the component of the angle

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normal to its edge. Other possibilities include using a color filter composed of a grid of colors, with angle being determined by the highlighted color. This route requires a white light source along with a color CCD camera. Achromatic optical elements are essential in such a measurement system. Analysis can be carried out in terms of color changes or equivalently, the hue. Images recorded in this arrangement are in color. Also called rainbow schlieren (or, color schlieren deflectometry), it can serve the function of flow visualization better than the monochrome version of Fig. 1b. The schematic diagram of the color schlieren technique with the test cell in place is shown in Fig. 2a. A cold white light lamp along with a pinhole serves as a point light source. The source and the pinhole are connected with a fiber optic cable. The pinhole is placed at the focal plane of a convex lens producing a collimated beam whose diameter is matched with the experimental test cell. The parallel beam of light falls on the second convex lens of sufficiently large focal length so that small angles of beam deflection can be detected. The second lens decollimates the light beam, focusing it on to the calibrated color filter. A 1-D color filter is schematically shown in Fig. 2b. The changes in the color scale of the filter are set in the direction along which density gradients are to be observed. The color CCD camera of adequate spatial resolution, bit resolution of color, and frame rate is connected to a computer for image acquisition and analysis. A 24-bit A/D frame grabber card records the images formed on the filter plane. a

Color CCD Camera

x y

Color filter Convex lens

Test cell

Convex lens

Pin-hole

Fiber optic cable

Original beam Deflected beam

White light source

Computer

b

Fig. 2 (a) Schematic diagram of the color schlieren technique and (b) example of a 1-D rainbow filter

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2.3.1 Data Analysis Let the z-axis be the direction of the undisturbed light ray while x-y is the crosssectional plane in the test cavity. The angular deflection of the light beam measured at the exit plane of the test section over a length L in the y-direction and given a symbol θy is determined as: 1 θy ¼ n0

ðL z¼0

@n dz @y

(8)

Here n0 is the refractive index of air surrounding the test section. Schlieren records the cumulative refraction of the light beam over the length of the test cell. Hence, the image contains the path-integrated refractive index gradient data. The deflection of the light beam leads to a variation in hue over the color filter and hence the recorded image. For a deflection angle θy, the beam displacement Δay at the position of the color filter will be given by Δay ¼ f 2 θy

(9)

The sign in Eq. 9 is determined by the change in hue in the filter plane. One can relate the change in hue with beam displacement through the calibration curve of the color filter. From the RGB tristimulus values distributed over the pixels of the image, hue distribution can be obtained using its definition  pffiffiffi  3 ð G
BÞ H ¼ arctan 2R
G
B

(10)

The refractive index gradient is then obtained from Eqs. 8 and 9 as Δay ¼

2.4

f 2 @n L n0 @y

(11)

Tomography

Light passes through a three-dimensional region during refractive index based measurements. However, the image is recorded on the screen or a plane. The planar light intensity data is obtained as an integration of the refractive index (and hence, density, temperature, and concentration) along the viewing direction. If the density field prevails over a two-dimensional cross section, the recorded intensity data is fully representative of it. In many applications, optical imaging is recorded on a plane while the measured variable itself is distributed in three dimensions. Mathematically, projecting a three-dimensional field analytically onto a plane is simple and direct. Recovering three-dimensional information from planar data is a common

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Tc

Th

0d

eg

ree

90 degree

–4.9

0

+4.9

One Roll

z/h

0

x/h

25

(a) 0° Projection

8 (b) 60

0 (c) 90

s/h

z/h

Fig. 3 (continued)

25

27.5

8 (d) 150

27.5

s/h

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771

requirement in optical imaging but is difficult and is referred as an inverse problem. Tomography is an analytical technique that is available for reconstruction of the three-dimensional data from its planar projections (Herman 1986; Natterer 1986; Mayinger 1994). Implementation aspects of tomography for refractive index based techniques can be found in Srivastava et al. (2012). POD tools can be used for analysis of unsteady phenomena (Srivastava et al. 2009; Sirovich 1987).

2.5

Experimentally Recorded Images

Images arising from the use of refractive index methods in laboratory-scale experiments are presented in the following sections.

2.5.1 Rayleigh-Benard Convection in a Rectangular Cavity Optical imaging of buoyancy-driven convection in a horizontal differentially heated cavity, rectangular in plan has been studied earlier by Mishra (1998) and also discussed by Muralidhar (2001). The lower surface is warmer than the top. The overall arrangement is the familiar Rayleigh-Benard configuration. The fluid medium considered is air. For temperature differences and cavity height considered, Rayleigh number is 1.39  104. As a result, three- dimensional fluid motion, both steady and unsteady is to be expected, with a comparable temperature distribution. The flow pattern is sketched in Fig. 3 (top row). The circulatory motion of the fluid particles creates a roll-type of motion. The roll patterns of the flow field are seen as fringe displacement in the interferograms (Fig. 3, second row). An intermediate step that assists conversion of fringe patterns to temperature is fringe thinning. It replaces the fringe bands by a skeleton, using image processing techniques. The corresponding interferograms in the form of a fringe skeleton and seen from four viewing directions are also shown in the figure. The schematic drawing at the top indicates how a roll is created within the cavity. Here, a parcel of warm fluid rises as a buoyant plume at a certain location, spreads on its way upward, reaches the cold top, and descends downward at a second location by gravity. Points where the fluid rises with those where the fluid descends together form a cellular pattern within the cavity. The geometry of the pattern can be understood by recording interferograms from various directions. The sample interferogram shown in the figure shows fringe displacement upward and can be correlated with the rising buoyant plume. Conversely, fringe displacement downward shows the descent of the cold fluid. The entire thermal field can be reconstructed from the projection data for temperature by tomography, referred in Sect. 2.4. ä Fig. 3 Buoyancy-driven convection in a differentially heated cavity. The roll patterns of the flow field are seen as fringe displacement in the interferograms. The flow pattern is three- dimensional and sketched at the top of the figure. The corresponding interferograms seen from four viewing directions (0, 60, 90, and 150 ) are shown below in the form of thinned fringes (Mishra 1998)

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0°

45°

90°

135°

0°

45°

90°

135°

Fig. 4 (left) Buoyancy driven convection in an octagonal cavity half-filled with 50 cSt silicone oil, the rest being air. The lower surface is heated while the top surface is cooled. The figure shows interferometric projections when the thermal field is viewed in various directions – 0, 45, 90, and 135 . (right) Long-time interferograms formed in an octagonal cavity containing silicone oil (50 cSt) floating over water as recorded from four different angles; overall temperature difference is 1.8 K, the top plate being cooler than the one at the base (Punjabi 2002)

2.5.2 Convection in a Two-Fluid Layer System Figure 4 shows long-time interferograms formed in an octagonal cavity containing silicone oil (50 cSt) with air above (left) and floating over water (right). The configuration is equivalent to Rayleigh-Benard convection with a two-fluid system. The height ratio of the fluid layers is unity. The nature of coupling of the fluid layers at the interface, mechanical versus thermal, is convection of interest. In mechanical coupling, the motion of the fluid in one layer drives the other by viscous stresses at the interface. In thermal coupling, the fluid layers independently experience fluid convection as appropriate for a pair of temperature differences. Fluid circulation in the two forms of coupling can be alike (thermal) or opposed (mechanical), giving rise to several interesting possibilities. Interferometric projections recorded from four different angles are shown in Fig. 4 (Punjabi 2002). The flow is driven by an overall temperature difference. The temperature difference across the cavity is 1.8 K, the top plate being cooler than the one at the base. Thus, cold/dense fluid set over warm/light fluid generates buoyancy forces and drives fluid convection. Water and oil are immiscible and independent convection patterns are set up in each part of the cavity. Since the fluid conductivities of water and silicone oil are comparable, a finite temperature difference is available across each fluid layer and detectable fluid convection is to be seen in each half. In air, the temperature difference available is small and no convection is visible. Of special interest is the energy and momentum transfer at the oil-water interface. When water is set in motion, it can drive the oil layer beneath even if buoyancy is absent.

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The relevant mechanism is momentum transfer and the fluid layers are mechanically coupled. The other possibility, namely, thermal coupling, is that each fluid layer experiences buoyant convection whose strength scales with the temperature difference appropriate for each of them. In the present experiment, interferometric images show that the nature of coupling in air-oil and water-oil systems is thermal in origin. In addition, the images of each pair of fluids look different in each view angle, indicating the onset of a three-dimensional temperature fields.

2.5.3 Crystal Growth Buoyancy-driven and forced convection patterns around a potassium-di-hydrogenphosphate (KDP) crystal growing from its aqueous solution have been visualized using refractive index techniques (Verma and Schlicta 2008). The growth process is initiated in a thermally controlled apparatus by inserting the KDP seed into its supersaturated solution. Slow cooling of the solution follows over a period of several hours (Fig. 5). Excess salt from the solution deposits on the crystal and changes the concentration field around it. The associated change in solution density creates a 7 8 5 2 4

Water In

Laser beam

6 29°C

1 9 3 Water out

(1) Growing crystal (2) Growth chamber (3) Outer chamber (4) Heating element (5) Thermocouple (6) Optical window (7) Seed holder (8) Covering lid (9) Temperature controller unit

b

Water In

Water Out Pump

Fig. 5 Schematic diagram of a crystal growth apparatus from an aqueous solution

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a

b

First visible fringe during the growth phase Initial dissolution

c

f

24 hours

d

36 hours

24 hours

g

e

60 hours

36 hours

h

Initial dissolution

60 hours

Fig. 6 Time sequence of the evolution of interferograms around the growing crystal. (a–d) Infinite fringe setting; (e–h) Wedge fringe setting. The initial crystal size in the infinite fringe setting is greater than in the wedge fringe setting. The opposing fringe curvatures above and below the crystal in (h) show a lighter and a denser solution formed by stratification (Srivastava et al. 2012)

natural convection patterns in the growth cell. Forced convection is initiated when the crystal is given rotation. Convection currents in terms of solute concentration (or its gradients) have been recorded around the growing crystal as a function of time. Optical techniques employed for visualization of the convective field are laser interferometry (Fig. 6), schlieren (Fig. 7), and shadowgraph (Fig. 8). Images recorded by these techniques may be compared with respect to the ease of instrumentation, quality of images, and the possibility of quantitative analysis. The three optical techniques show clearly the spatial distribution of concentration differences in the vicinity of the growing crystal. With the passage of time, the solution in the growth cell develops stable density stratification, leading to a practically stagnant solution. Vigorous convection can be seen when the crystal is given rotation about its own axis (Fig. 7, right). From the viewpoint of image clarity and simplicity of

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Fig. 7 (left) Evolution of schlieren images around the growing crystal from an aqueous solution. In (h) the original photograph as recorded by the camera is shown while (g) is a close-up. (right) Schlieren patterns around a KDP crystal growing from its aqueous solution when the crystal is given rotation about its own axis (Srivastava et al. 2012)

analysis, schlieren is felt to be more suitable for monitoring convection in crystal growth when compared to interferometry and shadowgraph (Srivastava et al. 2012).

2.5.4 Protein Crystal Growth by a Hanging Drop Technique The application of color schlieren for studying growth of Lysozyme crystals is discussed by Gupta et al. (2013) and is summarized in the following discussion. A drop of Lysozyme solution in water is placed on the cover slip and inverted on the

776 Fig. 8 Evolution of shadowgraph images around the growing crystal from an aqueous solution. A bright streak of light indicates the separation of the light solution from the heavy. The streak is seen to move downward in (d–h), till it stabilizes just around the crystal (Srivastava et al. 2012)

P. K. Panigrahi and K. Muralidhar

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Fig. 9 (a) Protein crystal growth chamber in the hanging drop configuration and (b) color schlieren images of transient evolution of the mass diffusive field in the reservoir solution during the crystal growth process. Seven drops of protein solution are placed on the underside of the top surface. Each drop is of 10 μl volume; the reservoir volume is 50 ml (left) and 75 ml (right)

reservoir (Fig. 9a). Differing salt concentrations are maintained between the drop and the reservoir. The concentration of NaCl in the reservoir solution is higher than that of the drop, namely, the protein solution. According to Raoult’s law, the vapor pressure of a solution containing a nonvolatile solute such as NaCl is equal to the vapor pressure of the pure solvent – water, at that temperature multiplied by the

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solvent mole fraction. In other words, the vapor pressure of a dilute solution is higher compared to the concentrated. Hence, the vapor pressure of the drop is higher when compared to the reservoir resulting in water evaporating from the drop and condensing over the reservoir. The process of evaporation from the drop supersaturates the solution with protein that crystallizes out in time. Water evaporation increases salt concentration within the drop and the differences in vapor pressure between the drop and the reservoir may diminish to zero. Additionally, the drop size decreases and the moisture transported across the air gap decreases with time. Color schlieren images for reservoir volumes of 50 and 75 ml are shown in Fig. 9b. The corresponding reservoir heights are in proportion to the volumes. A total of seven drops have been placed over the cover slip, each with a volume of 10 μl in a circular arrangement so that three of them are visible in a single view. The drop composition (protein p: reservoir r) is 7:3 in these experiments. During crystal growth, moisture from the drop diffuses through the intervening air gap and condenses over the reservoir. Fresh water diffuses through the reservoir but the system is stably stratified in density and convection currents are absent. At later times, water condensation over the reservoir ceases and the salt diffuses back toward the reservoir surface. The initial phase of drop evaporation and water condensation takes place over a time frame of 3–5 h. The reverse migration of salt lasts a few hours, with an asymptotic tail of nearly 24 h. Color schlieren images have been recorded over a 24 h period. Since the volume of water in a drop is quite small, the overall change in average salt concentration in the reservoir is also quite small. Hence, the initial and final schlieren images look practically identical. Additionally, the change in refractive index of air due to the presence of moisture is not large enough to create a spread of colors. Crystals grown over a 24 h period have been examined under a microscope and discussed by Gupta et al. (2013). Changes in the color distribution within the reservoir are visible in the schlieren images of Fig. 9b. The change in color is initiated near the interface where fresh water condensation takes place. The dark band seen in the images from the start is due to light interactions with the gas-liquid interface, particularly the meniscus at the optical windows. The deformation of the color streak below the interface arises from the placement of three drops closer to the center. At later times, when evaporation has ceased, the condensate layer becomes horizontal (t = 3 h, Fig. 9b). Outside the condensate layer, slight changes occur in the color distribution of the reservoir but are not prominently seen.

2.5.5 Buoyancy-Affected Wake of a Circular Cylinder Experiments on a heated cylinder performed in a vertical flow facility (Fig. 10a) are discussed here. The facility resembles an open-circuit wind tunnel. The test cell is made of Plexiglas and consists of a honeycomb structure, settling chamber, antiturbulence wire screens, contraction section, test section, and outflow section. The contraction section guides the flow from the settling chamber to the test section with an area contraction ratio of 4:1. The cross section of the test section is 0.4  0.4 m2

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Fig. 10 (a) Schematic diagram of the experimental setup for study of flow around a heated cylinder using schlieren interferometry and (b) the heating arrangement for the heated cylinder

square section with an overall length of 0.95 m. The test section is provided with a removable side panel for ease of installation of the cylinder. A centrifugal blower is used to maintain steady air flow in the test section. The suction side of the blower is connected to the outflow section of the test cell through a flexible PVC pipe. The blower is mounted on a suitably dampened base with rubber sheet padding for vibration isolation. The speed of the blower motor is regulated by a frequencybased inverter drive. The inner details of a heated cylinder with its heating arrangement for temperature control are shown in Fig. 10b. Both cylinders are internally heated by passing

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P. K. Panigrahi and K. Muralidhar

direct current through a nichrome wire, positioned along the axis of the tube and throughout its length. The wire is insulated from the copper body by packing powder mica in the gap. Mica has good thermal conductance to heat the cylinder walls from the nichrome wire but poor electrical conductance, thus providing electrical insulation between the cylinder body and the nichrome wire. The power to the heater is supplied by a regulated DC power supply. To reduce end heat losses to the wall of the test section, two small Teflon plugs are attached to the ends of the cylinder. Two pointed metallic holders wrapped with Teflon insulation tape support the cylinder at each plug through small openings in the tunnel. Temperature resistant sleeve is used to cover nichrome wire coming out of the cylinder ends. The changes in the organized wake structures with respect to their shape, size, and time-dependent movement is readily perceived from the instantaneous schlieren images before the wake degenerates into a steady plume. The wake characteristics depend on the Reynolds number Re and the mixed convection parameter, namely, Ri, the Richardson number (Sohankar et al. 1999; Luo et al. 2007). Figure 11a presents instantaneous images at a cylinder temperature of 40  C (Re = 114; Ri = 0.052), ambient temperature being 25  C. The heated wake zones, i.e., the bright zones behind the cylinder extend to a downstream distance of about x/d = 6.0. The near field region (x/d < 1) close to the cylinder shows very small variation with time and is practically stationary during the complete vortex shedding cycle. The far field region shows a stronger time-dependence. Thus, the base region of the wake shows very low levels of velocity and temperature fluctuations. The growth of the heated shear layer on both sides of the cylinder takes place asymmetrically at different phases of the vortex shedding cycle. The instability of the growing shear layers results in the shedding of two alternate rows of vortices from the opposite side of the cylinder. Figure 11b shows the time-sequence of schlieren interferograms behind the circular cylinder at a surface temperature of 75  C (Ri = 0.140). Regular vortex shedding with a higher number of fringes is seen. This indicates that vortex is not an isothermal packet of fluid; rather it is the recirculation bubble with a temperature distribution within. With increasing Richardson number, the increase in the number of fringes is a consequence of a higher overall temperature difference on one hand and large localized temperature gradients, on the other. These gradients arise from the structure of a shear layer, with thermal gradients correlating with velocity gradients at the limit of near-unity Prandtl number. The schlieren-interferograms at a higher surface temperature of 82  C (Ri = 0.157) are presented in Fig. 11c. The vortex shedding structure is completely altered at this Richardson number. The alternate shedding pattern observed at lower Richardson numbers is now replaced by a plume. The thin plume in the far field region is aligned with the cylinder mid-plane. The interferograms do not oscillate any longer and the wake can be termed as steady. When the cylinders are heated to high temperatures, vortex shedding is suppressed. This result is demonstrated by schlieren visualization. At a Reynolds number of 114, vortex shedding is suppressed at a Richardson number of 0.157, the

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Fig. 11 Instantaneous schlieren images for a circular cylinder separated by a time interval of one eighth of the time period of vortex shedding at various heating levels. Reynolds number Re = 114. Richardson numbers (a) Ri = 0.052, (b) Ri = 0.14, and (c) Ri = 0.157

corresponding Strouhal number becoming zero. Critical Richardson number as a function of Reynolds number is discussed by the authors in a previous study (Singh et al. 2007; Kakade et al. 2010).

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Scattering Techniques

The term scattering is used to indicate light-matter interaction, specifically, light interaction with particulates. Such particles may be coated on a surface or dispersed in the fluid medium. In the former, we arrive at an instantaneous surface temperature distributed over the surface. In the second approach, temperatures and velocities of the scattering particles can be detected. Both approaches are exemplified by liquid crystal thermography (LCT), described below with applications (Smith et al. 2001). Closely related to LCT is infrared thermography (IRT) which relies on background emissions from surfaces. Principle of operation of IRT and selected examples are also discussed in the following sections.

3.1

Liquid Crystal Thermography

Determination of convective heat transfer from a surface requires measurement of the wall temperature. Traditional techniques employing sensors such as thermocouples can measure temperature at a single location. A large number of sensors are required for complete surface temperature measurement but is limited by the space constraint. Convective heat transfer measurement using a limited number of sensors leads to erroneous results for systems with hot spots where localized peaks of heat transfer are formed. Liquid crystal thermography (LCT) is an ideal technique that can provide temperature data over the whole region of the test surface. Liquid crystals can also be used to identify the hot spots when local heat fluxes are high, as in electronic components. Liquid crystal is a unique substance, which exists between the solid and the isotropic-liquid phase of some organic compounds. It scatters incident light selectively, and thus exhibits the specific property of a solid crystal. Each liquid crystal compound possesses a helical structure with a characteristic pitch. The pitch dimensions are in the range of the wavelength of visible light. The pitch length can be altered by changing external stimulus typically, temperature and shear stress. The event temperature and clearing point temperatures are used to describe the properties of TLCs. The lowest temperature, where liquid crystals scatter visible light is called the event temperature. Below the event temperature, liquid crystals will be in the solid state and will appear transparent. At a temperature above the clearing point temperature, it will enter the pure liquid state and will revert back to being transparent. The difference between the two temperatures is called the bandwidth. The reflected color spectrum of liquid crystals varies continuously from the longer wavelengths (i.e., red) corresponding to event temperature to shorter wavelengths (i.e., blue) corresponding to clearing point temperature. Liquid crystals transmit a significant amount of the incident light with virtually no modification. Therefore, they are viewed against a nonreflecting background. This prevents the transmitted light from getting reflected without adversely affecting the interpretation of selectively scattered light from the liquid crystals. The narrow-band formulation of liquid

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crystals has a bandwidth below 1 or 2  C while wide-band formulations have bandwidths between 5  C and 30  C. The highest spatial resolution achievable with liquid crystal thermography is limited by the type of liquid crystal formulation and resolving capacity of the camera and lenses. Accuracy and temperature resolution of a measurement made with liquid crystals is related to the accuracy and consistency used in calibrating the colortemperature response of the liquid crystals. Inconsistency in lighting-viewing arrangements present when performing the color-calibration process and interpreting the actual color response can contribute significantly to measurement errors. The speed with which optical properties of liquid crystals change with surface temperature is an important parameter for transient experiments. Liquid crystals have a typical response time in the range of 5–150 ms (Ireland and Jones 1987; Moffat 1990; Kobayashi et al. 1998). A sufficiently bright and stable white light source without the infrared (IR) and ultraviolet (UV) radiation from the output spectrum is preferred for liquid crystal thermography. Any IR energy present in the incident light spectrum will cause unwanted radiant heating of the test surface. Exposure to UV radiation can cause rapid deterioration of liquid crystal surface and cause the surface to produce an unreliable performance in terms of the color-temperature response. The performance of several engineering devices is limited by proper dissipation of heat. Examples are gas turbine blade cooling, nuclear reactors, and high heat flux electronics. Several active and passive control schemes have been used for enhancement of heat transfer. The flow field in these applications is three-dimensional leading to uneven temperature distribution and hot spots. Therefore, local heat transfer coefficient calculation using surface temperature measurement at very high spatial resolution is required for proper design and analysis of these systems. Liquid crystal thermography is an ideal tool for local and instantaneous measurement of heat transfer coefficient with high spatial resolution. In the following sections, two applications of LCT for the measurement of heat transfer coefficient are discussed. The first is a surface mounted rib serving as a passive control device for heat transfer enhancement and the second is a synthetic jet used as an active control strategy.

3.1.1 Surface Mounted Rib Turbulator The extent to which a solid rib of square cross section can enhance heat transfer rates for flow over a flat surface is of interest. The sketch of the experimental setup of flow over a flat plate with a surface-mounted rib is shown in Fig. 12. The experimental facility comprises a flow circuit, the heating section, the A/D data acquisition system and an image acquisition system. Air is sucked into the test section through a honeycomb section, five screens for reducing turbulence and a 9:1 contraction cone. The test section is followed by a flow straightener to minimize the influence of blower noise in the test section. The speed of the blower is set by a speed controller leading to air velocity changes within 0.08%. Free stream turbulence is less than 0.5% at the entrance of the test cell for the velocity range used. The test channel is 3300 mm long with an aspect ratio of 1.8:1 (298  160 mm2 in the vertical

Fig. 12 An experimental facility for measurement of heat transfer coefficient for flow over a flat plate with a surface mounted rib using liquid crystal thermography

784 P. K. Panigrahi and K. Muralidhar

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plane) and is made of Perspex sheet of 12 mm thickness. The use of Perspex serves as thermal insulation and provides optical accessibility for the liquid crystal measurements. The hydraulic diameter of the smooth channel is 0.208 m and the rib height to hydraulic diameter ratio is set to be 0.0624. Thermocouples are used to measure the air temperature entering and leaving the test section. Thermocouples are also mounted at selected locations below the LCT sheet for calibration. The heat transfer section fits into the lower portion of the test section. A single aluminum plate (680  298  3 mm3) is heated by six stainless steel foil heaters (thickness = 0.045 mm) connected in series and cemented on the top of a Bakelite plate (thickness, 25.4 mm). The use of an aluminum plate ensures the smoothening of lateral temperature nonuniformity, caused by the use of foil heaters of definite width. Foil heaters are electrically connected to a regulated DC power source. To minimize conductive heat losses to the ambient, the lower surface of the Bakelite sheet is insulated using a 2 mm air gap. The air cavity, between the Bakelite plates serves as an effective insulator between the foil heaters and the ambient. The heat transfer surface is instrumented with thirteen calibrated thermocouples of the chromelalumel (K-) type, along the centerline and spanwise directions of the heated aluminum plate. The junction bead (0.2 mm diameter, time constant 20 ms) of the thermocouple is mounted in the counter-sunk holes of the aluminum test surface using thermally conductive epoxy from underneath. Conduction loss to the lower side of the heating section is estimated by using the thermocouples mounted across the Bakelite plate. Ribs are made of aluminum and are highly polished to minimize radiation heat transfer. K-type thermocouples are connected to the National Instrument’s data acquisition card (NI-4351) for data acquisition. The image acquisition system consists of a high-resolution 3-CCD video camera (SONY XC-003P) with 16 mm focal length lens (VCL-16WM), a 24-bit true color frame grabber board (Imaging Technology Inc.), and a high speed PC. Experimental Procedure and Data Analysis For transient experiment, the test section is heated to a spatially uniform temperature. This is followed by step cooling due to the initiation of the mainstream flow in the flow apparatus. The velocity response at the inlet of the test section measured by the pitot probe indicates the transient period to be 3–5 s, for the blower RPM to stabilize. A sequence of LCT images of the cooling surface is collected after flow has reached steady state. The LCs display the temperature of the thin aluminum plate, mounted over the upper-Bakelite sheet. High thermal conductivity and small thickness of the aluminum plate enables the approximation that the changes in the color of the liquid crystals indicate those in surface temperature of the Bakelite sheet. The inverse-transient technique for the determination of the heat transfer coefficient ate the surface of the aluminum sheet employs the semi-infinite solid approximation through the Bakelite sheet below. The criterion for the validity of semipffiffiffiffi infinite solid assumption is that the material thickness should be greater than 4 αt where α is thermal diffusivity and t is the total time. A simple calculation for Bakelite

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of 25 mm thickness reveals the maximum penetration time to be around 340 s. This is much larger than the duration of a typical transient experiment of 80 s for which LCT images are recorded. For a semi-infinite solid, the solution of the 1-D transient heat conduction equation with convective boundary conditions at the surface is given as:  pffiffiffiffi  2  T w, i
T w ð t Þ h αt h αt ¼ 1
exp erfc 2 T w, i
T b k k

(12)

Here, Tw(t) is the surface temperature at time t, Tb is the bulk fluid temperature inside the channel, Tw,i is the initial surface temperature, h is the convective heat transfer coefficient, α is thermal diffusivity and k is thermal conductivity of the Bakelite sheet. The transient nondimensional temperature variation of the test surface is curve fitted by using least squares with the local heat transfer coefficient as a parameter. Details of the experimental apparatus and data analysis can be found in Tariq et al. (2003, 2004). Calibration The color-temperature relationship of the liquid crystal sheet should be known prior to its use for temperature measurement. Calibration is carried out to develop the intrinsic color-temperature response of the liquid crystals. Color image of the test surface is acquired after bringing the test surface and the liquid crystal sheet to its event temperature. An average color value is computed and stored against the recorded temperature of the test surface. This process is repeated at higher temperatures until the clearing point temperature is reached. Figure 13 shows the calibration curve of the liquid crystal sheet (R35C5W) used in the present work. It is preferable Fig. 13 Calibration data of a liquid crystal sheet (Hallcrest, R35C5W), relating huesaturation-intensity with temperature

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Fig. 14 Transient liquid crystal images of the flat plate with surface mounted rib at different time instants (t1 < t2 < t3 < t4) during plate cooling. Flow direction is from left to right

to replace color intensities R, G, and B with hue (H), saturation (S), and intensity (I). Figure 13 shows that hue varies monotonically with temperature and can be used in measurements to represent color data.

LCT Images and Heat Transfer Coefficient The color images recorded in a transient experiment, with one solid rib mounted on the test surface is shown in Fig. 14. Flow direction is from left to right. In these color images, red indicates the coldest region followed by green, yellow, and blue respectively for higher temperatures. Initially the plate is heated up to the clearing temperature and the correspondingly, the LC sheet appears uniformly blue. The heater is then turned off while the flow is initiated over the surface. On forced cooling, gradual changes in temperature occur with time, as depicted in the images. The effect of the reattaching boundary layer causing a high heat transfer rate, i.e., a low-temperature region downstream of the rib is clearly evident. Streaks of yellow/ red color indicating zones of high heat transfer are revealed within the reattachment region. Comparatively, high temperature near the rib is attributed to the low heat transfer zone due to stagnant flow in the recirculation zone. As a first step, LCT images can be used as a visualization tool to predict qualitatively the regions of low and high wall heat fluxes. These images are subsequently digitized frame by frame into a color scale. Mapping color to hue and then to temperature yields the temperature history of the test surface. Figure 15 compares the surface heat transfer coefficient distribution retrieved from LCT images for a smooth surface, smooth surface with one surface mounted rib and two surface mounted ribs. The heat transfer coefficient of the smooth surface

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Nu

12 21 36 18 60 72 81 96 108 120

100 No rib One rib Two rib

80

Nuavg

60

40

20

0

5

10

15

20

25

30

35

40

45

x/e

Fig. 15 Nusselt number distribution for a plane wall (top), single rib (middle), and two ribs (bottom) at Re = 20,900

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reduces in the streamwise direction due to the growth of the boundary-layer. In a single-rib configuration, the heat transfer coefficient increases at the reattachment point. The high heat transfer region for the surface mounted rib in Fig. 15 corresponds to large temperature drop observed during transient cooling (Fig. 14). The streamwise extent of high heat transfer observed in the reattachment region for the single surface mounted rib increases in the two-ribs configuration. The average heat transfer coefficient for two ribs is higher than that of the single rib.

3.1.2 Synthetic Jet Figure 16a shows the experimental facility which comprises of a flow circuit, the heating section, the imaging system, and the A/D data acquisition system for temperature measurement. The goal of the study is to examine heat transfer enhancement when a synthetic jet is used with channel flow. The surface temperature distribution is recorded with the liquid crystal imaging system. The flow circuit is a two-pass square channel with an arrangement for visualization using liquid crystal

Fig. 16 (a) Schematic drawing of the experimental arrangement and (b) synthetic jet actuator

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thermography. Air is sucked into the test section through a honeycomb section, antiturbulence screens, and a nozzle with 3:1 contraction ratio. The test section is 8 8 cm2 in cross section and 100 cm long. It is made of Plexiglas, with the top surface made of glass for clear visualization. The test section is surrounded by a Plexiglas enclosure of size 43  40  145 cm3. Filament heaters placed inside the enclosure are connected to a variable power supply for heating the enclosure around the test cell. Fans are mounted inside the enclosure for uniform mixing of hot air. A blower is used for generating flow inside the test cell and the motor RPM is varied with the help of a speed controller (Kirloskar Electric). The blower is connected to the test cell via a three-way diverter valve. This valve is used to initiate transient cooling in the test cell by sudden introduction of flow. Thermocouples are used to measure the air temperature entering and leaving the test section. The imaging system used for LCT is identical to the one used for experiments with a surfacemounted ribbed duct, Sect. 3.1.1. Synthetic Jet Actuator Piezoelectric-based synthetic jet actuator (Fig. 16b) is used for the modification of the flow field. The actuator consists of an aluminum cantilever (50  20  0.4 mm3) and a piezoelectric element (SP-5H material measuring 40  19.75  0.4 mm3) bonded to the cantilever near its support. The piezo-element is made of lead zirconate titanate and the electrical contacts to this element are provided by soldering the electrical leads to the terminals. The cantilever is set into forced oscillations by a piezo-element. The deformation of the element can be controlled by adjusting the excitation voltage amplitude. The cantilever is enclosed from all sides with Plexiglas sheets forming a cavity. The top surface of the cavity has a circular orifice. Orifice diameter of 0.5 mm and a height of 1.8 mm have been used. The piezoelectric element oscillates when supplied with a voltage signal from the function generator (HP 33120A) via an amplifier unit (Spranktronics). The highest amplitude of oscillation is obtained by exciting the piezo element at the resonant frequency of the cantilever, i.e., 951 Hz. Nusselt Number Measurements The experimental procedure and assumptions for data analysis are validated for flow in a straight undisturbed channel by comparing the heat transfer coefficient from LCT measurements with correlations. Figure 15 compares the heat transfer coefficient obtained from LCT measurements with the flat plate correlation given below (Holman 1997): Nu ¼

 x 3=4 
1=3 hx 0 ¼ 0:332Pr1=3 Rex 1=2 1
k x

(13)

Here, x is the streamwise distance from the leading edge where the velocity boundary layer starts, and xo is the streamwise distance from the leading edge to the location where heating starts. The equation is valid for Rex < 5  105 and

Visualization of Convective Heat Transfer

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7

LCT measurements Correlation

6

5

2

Fig. 17 Comparison of the heat transfer coefficient distribution along the normalized streamwise distance (x/D) inside the smooth channel from LCT measurement and comparison with that of the flat plate correlation at Reynolds number based on the hydraulic diameter of the channel equal to 5500, where D is the diameter of the synthetic jet orifice

h, W/m K

17

4

3

2

0

20

40

60

x/D

80

100

0.6 < Pr < 50 under constant wall temperature conditions. Figure 17 shows that the heat transfer coefficients determined from the present experiments match the correlation within 5%. The small difference can be attributed to the uncertainty in measurements of the streamwise distance (x) and the difficulty in determination of the origin for channel flow. The heat transfer coefficient measurements of the two-pass square channel have been carried out at different actuation conditions of the synthetic jet. The actuation voltage amplitude is varied as 20, 30, and 55 V. The modulation frequency of the actuator is selected as 10 Hz and 50 Hz. Figure 18 shows the contours of the surface heat transfer coefficient for various operating conditions of the synthetic jet (Qayoum et al. 2010a, b). An increase in the heat transfer coefficient is consistently seen in the wake region of the synthetic jet. The highest increase in the heat transfer coefficient is observed at the 55 V excitation. In view of the larger vortices created at lower frequencies, the increase in heat transfer coefficient is correspondingly higher. Specifically, 10 Hz modulation yields a larger heat transfer coefficient when compared to 50 Hz. The influence of the synthetic jet is significant in the near field region when compared to the far field where the vortices have weakened in strength. Table 1 compares the average heat transfer coefficient for different actuation conditions. It shows a maximum 44% increase in heat transfer coefficient compared to the base flow for an actuation voltage of 55 V. Higher heat transfer enhancement of 22.8% is observed at lower modulation frequency of 10 Hz compared to 16.1% for 50 Hz at a constant actuation amplitude of 30 V.

3.1.3 Heat Transfer through a Single Drop Heat transfer coefficients arising in dropwise condensation are large. The related temperature difference driving heat transfer is small making experimental

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c 80

60

60

60

x/D

x/D

b 80

x/D

a 80

40

40

40

20

20

20

Base

20 V

d 80

e 80

60

60

30 V f

80

h 12 10

60

x/D

x/D

x/D

8

40

40

6 40

4 2

20

20

55 V

20

30 V – 10 Hz

1

30 V – 50 Hz

Fig. 18 Contours of heat transfer coefficient for a synthetic jet in cross-flow with a laminar boundary layer at various excitation conditions: (a) base, (b) Vexc = 20 V, (c) Vexc = 30 V, (d) Vexc = 55 V, (e) Vexc = 30 V (fAM = 10 Hz), and (f) Vexc = 30 V (fAM = 50 Hz)

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Table 1 Percentage increase in heat transfer coefficient with respect to the base flow for a synthetic jet in cross-flow with a laminar boundary layer Actuation condition Vexc (V) Base flow 20 30 55 30 30

fAM(Hz) 0 0 0 0 10 50

h (W/m2-K) 11 12.1 13 14.85 13.75 12.96

% increase – 10 18.21 35.01 24.96 17.83

determination of heat transfer coefficient a challenge. The stochastic and unsteady form of droplet distribution in the ensemble contributes to additional intricacy in the analysis and interpretation of data. In the present application, temperature distribution during dropwise condensation has been measured over a polyethylene substrate using liquid crystal thermography (LCT). In the experiments, pendant drops form on the underside of the liquid crystal sheet. In view of the spatial resolution afforded by the LCT sheet, temperature variation at the base of the drops, as small as 0.4 mm, have been resolved. The signature of the drop in the form of its contour is also visible in the LCT images. The drop size distribution formed on the substrate has also been imaged. Static contact angles of water on polyethylene were independently measured and for comparison, drop shapes were estimated via a mathematical model. Using a one-dimensional heat transfer model, heat flux profiles through individual droplets are computed. Temperature profiles derived from this approach combined with drop sizes recorded from direct visualization are sufficient for understanding mechanisms arising in dropwise condensation. Results show that the estimated heat flux as a function of drop diameter matches published data for large drop sizes. It fails for small drops where the limiting factor is the thermal resistance of the LCT sheet. To a first approximation, however, drop size can be correlated to the local heat flux. Hence the average heat flux over a surface can be obtained entirely from the drop-size distribution. Experimental Procedure The condensing chamber with a small quantity of liquid inventory placed in the annular space of the apparatus is shown in Fig. 19. It is first evacuated by a turbomechanical combo vacuum pump. A fraction of water flashes to vapor form and fills the chamber. Evacuation continues for several minutes and complete removal of noncondensable gases in the chamber is ascertained. The absolute pressure in the chamber is practically equal to the saturation pressure of water at the chamber temperature. Any operating temperature-pressure combination can be obtained by a suitable combination of coolant water temperature and the heat input by the circular heaters in the chamber placed below.

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Fig. 19 Experimental setup developed to study dropwise condensation on the underside of a substrate. (a) Photograph showing the details of the condensing chamber and (b) exploded view of the condensing chamber. (c) Camera view A from below shows the condensing droplets while camera view B is the RGB image of the liquid crystal sheet (Bansal et al. 2009)

Condensation in the form of pendant drops commences soon after evacuation is initiated. Hence, liquid crystal thermographs are hard to record in the initial phase of nucleation and droplet growth. Nucleated drops grow in size by direct condensation of the vapor followed by coalescence between adjacent drops. When the weight of the drop exceeds the restraining force, it falls into the reservoir. A vacant space is thus created where fresh condensation is initiated. In this manner, a dynamic steady state of evaporation and condensation is established in the apparatus. The relevant data, namely, LCT images and condensations patterns of drops have been acquired in the present work after quasi-steady state under dynamic conditions is arrived at. Experiments show that drop growth at this stage is primarily dominated by coalescence. Condensation occurs at virgin areas vacated by grown drops falling off the substrate. Heat Transfer through Individual Drops Figure 20a shows the liquid crystal thermograph recorded during dropwise condensation under a single droplet of diameter 2.96 mm. The hue distribution over the base is shown in Fig. 20b. Figure 20c shows the variation of heat throughput through the mid-plane of the single drop as identified in Fig. 20a. Examples of other isolated droplets recorded during the experiment are shown in Fig. 20d. Figure 21a and b are an adjacent droplet pair and also show the hue distribution over the base. Using a one-dimensional model of heat transfer in a semi-infinite solid, instantaneous heat

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Fig. 20 (a) and (b) LCT image and the RGB enhanced version of an isolated pendant droplet during dropwise condensation with superimposed hue contours. (c) Heat transfer rate at a plane passing through the middle of the droplet as a function of position. (d) Examples of hue profiles of three other isolated droplets of smaller sizes (Bansal et al. 2009)

transfer rates are shown in Fig. 21c and d through planes passing through these individual drops. The following qualitative features can be recorded from these data. (a) Iso-hue lines indicating isotherms can be seen on the base area of the condensing pendant droplets. For a given temperature difference between the glass plate and the condensing vapor, droplet thermal resistance due to changing height is clearly manifested as temperature distribution over the base of the drop. Smaller drops have a lower thermal resistance per unit area compared to larger drops. (b) Maximum heat transfer rate occurs at the three-phase contact line. Here, the thickness of the droplet is the lowest, corresponding to the region of least thermal resistance. (c) Circumscribing the contact line, the adsorbed liquid film increases the thermal resistance to heat transfer. (d) The center of the drop, where its height is a maximum, offers the greatest heat transfer resistance.

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Fig. 21 (a) and (b) Original LCT image and RGB enhanced image of two adjacent pendant droplets during dropwise condensation process with hue contours superimposed. (c) and (d) Local heat transfer rate through plane#1 (Drop-A) and plane#2 (Drop-B) as a function of position (Bansal et al. 2009)

Using a one-dimensional model of heat conduction, heat transfer rates through individual droplets as a function of drop diameter and degree of subcooling have been computed and presented in Fig. 22. Among the two bands, one corresponds to results obtained by neglecting the thermal resistance of the polyethylene substrate and the second, when this thermal resistance is included. The degree of subcooling is varied from 0.4  C to 2  C. When the thermal resistance due to the polyethylene sheet is not considered, heat flux passing through droplets drastically increases with decreasing droplet diameter. Specifically, the greatest heat flux passes through the smallest diameter droplets. The inclusion of the thermal resistance due to the polyethylene sheet substantially decreases heat transfer. The deterioration is felt strongly for smaller-sized droplets. Thus, for the present data set, low thermal conductivity polyethylene sheet causes a large reduction in heat transfer through the smaller droplets.

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Fig. 22 Average heat flux over the base of a drop as a function of the drop diameter. Experimentally determined heat transfer rates (shaded squares) are determined from LCT data by combining it with the one-dimensional heat transfer model. The simulations shown are (i) without and (ii) with the thickness of the PE substrate taken into account. Although the highest heat flux passes through the small drops, the thermal resistance due to small droplets diminishes with decreasing diameter, and the resistance due to the PE foil becomes increasingly significant (Bansal et al. 2009)

3.2

Infrared Thermography

At temperatures higher than 0 K, the absolute zero temperature, each body emits thermal radiation. The intensity of this radiation depends on wavelength and the body temperature. The basic component of an infrared system is an infrared camera. The atmosphere has two bands of good transmission in the infrared range (i.e., shortwave band between 2 and 5 μm and long-wave band between 8 and 14 μm). Hence, most detectors and infrared (IR) cameras are divided in a natural way into short-wave (SW) and long-wave (LW) devices. Another classification follows from the detector type: there are cameras with cooled detectors, containing a refrigerator (cooling) unit, and noncooled detectors, operating at the ambient temperature. Noise equivalent temperature difference (NETD) is used as a parameter for characterization of IR camera. It is defined as the difference between the temperature of the observed object and the ambient temperature that generates a signal level equal to the noise level. Measurement cameras have lower NETD value (less than 100 mK) compared to that of imaging cameras (more than 200 mK). The temperature measurement by IR camera is dependent on atmospheric temperature, emissivity of the object, distance of the object from the detector. Proper mathematical model needs to be used for interpretation of the IR measurement.

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3.2.1 Emissivity Estimation and Correction Emissivity of the surface influences the IR measurements and is one of the key input parameters. Thus, it is important to have its correct estimation before any measurement. There are various methods available for the evaluation of surface emissivity. Orlove (1982) proposed a scheme to measure the emissivity where a piece of known high emissivity material with good thermal conductivity is pasted over a portion of the object whose emissivity has to be determined. The object is heated at least 50  C above the ambient temperature and after sufficiently long time when steady-state surface temperature is ensured, temperature of the small area of known emissivity is read from the IR camera. Then, this point is moved to the area where the object emissivity has to be determined. Keeping other parameters unchanged, the object emissivity in the camera software is tuned such that the reading of the temperature of this point matches with the temperature of the known emissivity area. Another commonly adopted method for evaluation of emissivity consists of determining the object temperature by a contact method, for example, accurate thermocouples. The emissivity parameter in the IR camera should be tuned until the same reading of temperature is obtained. The value of emissivity at which the two temperatures match is used as the emissivity of the surface. To obtain an average value of emissivity for the complete zone of measurement, the above steps have to be repeated for many different locations. Measurement of heat transfer coefficient inside a tube under magnetic actuation of ferrofluid flow is described in the following section. 3.2.2

Heat Transfer Augmentation Using Magnetic Actuation of Ferrofluid Figure 23 shows the schematic drawing of the setup developed for carrying out convective heat transfer measurements of laminar ferrofluid flow through a stainless steel tube in the presence of magnetic field produced by pairs of permanent magnets. The stainless steel (SS) tube has dimension of 2 mm ID  2.6 mm OD and the ferrofluid flow rates ranges from 10–40 mL/min. The SS tube is kept over a PMMA platform and connected to the flow delivery system through flexible tubing. Cylindrical NdFeB permanent magnets are used to generate the magnetic field inside the SS tube. For generating varying magnetic field strengths inside the tube, the permanent magnets are positioned with a gap of 1.5 and 3 mm from the SS tube. These are arranged in two different configurations: (a) single inline (Fig. 23b) and (b) double inline with respect to the SS tube (Fig. 23c). A constant heat flux boundary condition is provided across the tube using a DC power supply. The power is delivered to the tube through the copper electrode blocks providing direct resistive heating to the tube. A FLIR ThermoVision SC4000 MWIR infrared camera is used for the measurement of temperature of the heated surface of the SS tube. The IR camera has an operational spectral band of 3–5 μm, 14-bit signal digitization, spatial resolution of 320 (H)  256 (V) pixel2, and a noise equivalent temperature difference (NETD) of less than 0.02 K at 30  C. Before conducting experiments, the calibration of the IR camera is carried out by comparing the surface temperatures of a heated stainless

Visualization of Convective Heat Transfer

Fig. 23 (a) Schematic drawing of the experimental setup for heat transfer augmentation, (b) single-inline arrangement of magnets, and (c) double-inline arrangement of magnets

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Fig. 24 IRT measurement over the heated surface of the SS tube: (a) without magnetic field and (b) with magnetic field applied to the ferrofluid flow

steel tube against thermocouple measurements. Excellent comparison between the IR camera reading and thermocouple reading is observed when emissivity of the carbon-black is set equal to 0.964. Figure 24a shows the streamwise temperature distribution of the heated surface of the SS tube for the case of ferrofluid flow in the absence of magnetic field for flow rates of Q = 20 and 40 mL/min. The temperature distribution for the same heated surface of SS tube for single-inline and double-inline arrangement of magnets is reported in Fig. 24b. For both the flow rates of the ferrofluid (20 and 40 mL/min), temperature for the single-inline arrangement is found to be higher compared to the double-inline arrangement of magnets. Figure 25 shows the variation of Nusselt number (Nu) versus dimensionless distance (Z*) for single-inline and double-inline arrangement of magnets (Asfer et al. 2016). Results are reported at flow rates Q = 40 mL/min for two different gaps of the magnets, 1.5 and 3 mm from the tube surface. Nusselt number for the single-inline arrangement is lower compared to the double-inline arrangement of magnets. For double-inline arrangement, ferrofluid is strongly attracted toward the magnets, resulting in higher momentum and energy transport in the transverse direction leading to higher heat transfer from the heated SS surface to the ferrofluid as compared to single-inline arrangement. At a flow rate of Q = 40 mL/min, Nusselt number is higher for both the single-inline and double-inline arrangement of magnets compared to the experiment with no magnetic field. This is because of higher inertia of the ferrofluid compared to the magnetic force, which results in less aggregation of nanoparticles at the wall of the tube compared to lower flow rates of the ferrofluid through the tube. The changes in the convective heat transfer characteristics during flow of a ferrofluid in the presence of magnetic field are attributed to several factors. These include: (a) ratio of magnetic force to inertia force acting on the ferrofluid, (b) interaction of main flow with the aggregates of

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Fig. 25 Comparison of Nusselt number (Nu) versus dimensionless distance (z*) for single-inline and double-inline arrangement of magnets for gap size: (a) 1.5 mm and (b) 3 mm. The ferrofluid flow rate is set equal to Q = 40 mL/min (Pe = 2496) through the SS tube

IONPs at the wall adjacent to each magnet, and (c) enhancement in local thermal conductivity due to the formation of chain- like clusters of IONPs within the ferrofluid in the presence of a magnetic field. The extent to which the changes take place are quantified using infrared thermography.

4

Closure

Images of fluid flow, heat transfer, and species transport can be recorded using several refractive index and scattering-based techniques. The measurement is over a cross section and follows transients without delay. The recorded data can be analyzed to recover additional details such as three dimensionality and unsteadiness as well as surface-level fluxes. Such an approach will require knowledge of the natural laws that govern transport as well as principles of image formation from radiation sources. Refractive index techniques have been used in the literature for measurement of temperature and concentration in the body of the fluid medium. Scattering techniques are more commonly seen in surface temperature measurements.

5

Cross-References

▶ Electrohydrodynamically Augmented Internal Forced Convection ▶ Enhancement of Convective Heat Transfer ▶ Film and Dropwise Condensation ▶ Free Convection: External Surface ▶ Inverse Problems in Radiative Transfer ▶ Turbulence Effects on Convective Heat Transfer

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References Al-Ammar K, Agrawal AK, Gollahalli SR, Griffin D (1998) Application of rainbow schlieren deflectometry for concentration measurements in an axisymmetric helium jet. Exp Fluids 25:89 Anderson RC, Milton JE (1989) A large aperture inexpensive interferometer for routine flow measurements, in: Instrumentation in aerospace simulation facilities, ICIASF’89 record, International congress on, IEEE, pp 394–399 Asfer M, Mehta B, Arun K, Khandekar S, Panigrahi PK (2016) Effect of magnetic field on laminar convective heat transfer characteristics of ferrofluid flowing through a circular stainless steel tube. Int J Heat Fluid Flow 59:74–86 Atcheson B, Heidrich W, Ihrke I (2009) An evaluation of optical flow algorithms for background oriented schlieren imaging. Exp Fluids 46:467 Bansal GD, Khandekar S, Muralidhar K (2009) Measurement of heat transfer during dropwise condensation of water on polyethylene. Nanoscale Microscale Thermophys Eng 13(3):184–201 Gayhart EL, Prescott R (1949) Interference phenomenon in schlieren system. J Opt Soc Am 39:546–550 Goldstein RJ (ed) (1996) Fluid mechanics measurements, 2nd edn. Taylor and Francis, New York Gupta AS, Gupta R, Panigrahi PK, Muralidhar K (2013) Imaging transport phenomena during lysozyme protein crystal growth by the hanging drop technique. J Cryst Growth 372:19–33 Herman GT (1986) Image reconstruction from projections. Academic, New York Holman JP (1997) Heat transfer. McGraw Hill, New York Ireland PT, Jones TV (1987) Response time of a surface thermometer employing encapsulated thermochromic liquid crystals. J Physics E 20:1195–1199 Kakade A, Singh SK, Panigrahi PK, Muralidhar K (2010) Schlieren investigation of the square cylinder wake: joint influence of buoyancy and orientation. Phys Fluids 22(054107):118 Kobayashi T, Saga T, Doeg-Hee D (1998) Time response characteristics of microencapsulated liquid-crystal particles. Heat Tran Jpn Res 27:390–398 Kumar R, Kaura SK, Sharma AK, Chhachhia DP, Agarwal AK (2007) Knife-edge diffraction pattern as an interference phenomenon: an experimental reality. Opt Laser Technol 39:256–261 Lauterborn W, Vogel A (1984) Modern optical techniques in fluid mechanics. Annu Rev Fluid Mech 16:223 Lehner M, Mewes D (1999) Applied optical measurements. Springer, Berlin Lewis RW, Teets RE, Sell JA, Seder TA (1987) Temperature measurements in a laser-heated gas by quantitative shadowgraphy. Appl Opt 26(17):3695 Luo SC, Chew YT, Ng YT (2003) Characteristics of square cylinder wake transition flow. Phys Fluids 15(9):2549–2559 Mayinger F (ed) (1994) Optical measurements: techniques and applications. Springer, Berlin Merzkirch WF (1987) Flow visualization, second edn. Academic, New York Min J, Yao B, Gao P, Guo R, Zheng J, Ye T (2010) Parallel phase-shifting interferometry based on Michelson-like architecture. Appl Opt 49:6612 Mishra D (1998) Experimental study of Rayleigh-Benard convection using interferometric tomography. Doctoral dissertation, Indian Institute of Technology Kanpur, India Moffat RJ (1990) Some experimental methods for heat transfer studies. Exp Thermal Fluid Sci 3:14–32 Muralidhar K (2001) Temperature field measurement in buoyancy-driven flows using interferometric tomography. Ann Rev Heat Tran 12:265 Muralidhar K (2014) Imaging unsteady three dimensional transport phenomena. Pramana J Phys 82(1):3 Natterer F (1986) The mathematics of computerized tomography. Wiley, New York Orlove GL (1982) Practical thermal measurement techniques. Proc SPIE 371:72–81 Panigrahi PK, Muralidhar K: (i) Schlieren and shadowgraph methods in heat and mass transfer (Springer briefs in thermal engineering and applied science, New York, Aug, 2012); (ii) Imaging

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heat and mass transfer processes – visualization and analysis (Springer briefs in thermal engineering and applied science, New York, Oct, 2012) Punjabi S (2002) Interferometric study of convection in superposed gas-liquid layers. Doctoral dissertation, Indian Institute of Technology Kanpur, India Qayoum A, Gupta V, Panigrahi PK, Muralidhar K (2010a) Influence of amplitude modulation on piezoelectric synthetic jet actuator. Sens Actuators: A Phys 162:36–50 Qayoum A, Gupta V, Panigrahi PK, Muralidhar K (2010b) Perturbation of a laminar boundary layer by a synthetic jet for heat transfer enhancement. Int J Heat Mass Transf 53:5035–5057 Rasenat S, Hartung G, Winkler BL, Rehberg I (1989) The shadowgraph method in convection experiments. Exp Fluids 7:412 Schiebener P, Straub J, Sengers JMHL, Gallagher JS (1990) Refractive index of water and steam as function of wavelength, temperature, and density. J Phys Chem Ref Data 19(3):677–717 Schmidt G, Nakatani AI, Tan CC (2002) Rheology and flow-birefringence from viscoelastic polymer-clay solutions. Rheol Acta 41:45 Schopf W, Patterson JC, Brooker AMH (1996) Evaluation of the shadowgraph method for the convective flow in a side-heated cavity. Exp Fluids 21:331 Settles GS (2001) Schlieren and shadowgraph techniques. Springer, Berlin Singh SK, Panigrahi PK, Muralidhar K (2007) Effect of buoyancy on the wakes of circular and square cylinders: a Schlieren-interferometric study. Exp Fluids 43(1):101–123 Sirovich L (1987) Turbulence and the dynamics of coherent structures. Part 1: coherent structures. Q App Math 45(3):561 Smith CR, Sabatino DR, Praisner TJ (2001) Temperature sensing with thermochromic liquid crystals. Exp Fluids 30:190–201 Sohankar A, Norberg C, Davidson L (1999) Simulation of three-dimensional flow around a square cylinder at moderate Reynolds numbers. Phys Fluids 11(2):288–306 Srivastava A, Singh D, Muralidhar K (2009) Reconstruction of time-dependent concentration gradients around a KDP crystal growing from its aqueous solution. J Cryst Growth 311:1166 Srivastava A, Muralidhar K, Panigrahi PK (2012) Optical imaging and three dimensional reconstruction of the concentration field around a crystal growing from aqueous solution: a review. Prog Cryst Growth Charact Mater 58:209 Tariq AK, Singh K, Panigrahi PK (2003) Flow and heat transfer in a rectangular duct with single-rib and two-ribs mounted on the bottom surface. J Enhanc Heat Tran 10(2):171–198 Tariq A, Panigrahi PK, Muralidhar K (2004) Flow and heat transfer in the wake of a surfacemounted rib with a slit. Exp Fluids 37:701–719 Tropea C, Yarin AL, Foss JF (eds) (2007) Springer handbook of experimental fluid mechanics. Springer, Berlin Verma S, Schlicta PJ (2008) Imaging techniques for mapping solution parameters, growth rate, and surface features during the growth of crystals from solution. Prog Cryst Growth Charact Mater 54:1

Part III Single-Phase Heat Transfer in Porous and Particulate Media

Applications of Flow-Induced Vibration in Porous Media

18

Khalil Khanafer, Mohamed Gaith, and Abdalla AlAmiri

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Fluid-Structure Interaction Analysis of Non-Darcian Effects on Natural Convection in a Porous Enclosure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Numerical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Solution Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Fluid-Structure Interactions in a Tissue During Hyperthermia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Mathematical Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Cross-References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

808 809 811 813 813 817 820 823 825 826 826

Abstract

This chapter reviews the applications of flow-induced vibration using fluidstructure interaction in porous media. Two examples are discussed, namely, fluid structure interaction analysis of non-Darcian effects on natural convection in a porous enclosure and the analysis of pulsatile blood flow and heat transfer in living tissues during thermal therapy. The transport equations are solved for various pertinent parameters using a finite element formulation based on the

K. Khanafer (*) · M. Gaith Department of Mechanical Engineering, Australian College of Kuwait, Kuwait City, Kuwait e-mail: [email protected]; [email protected] A. AlAmiri Mechanical Engineering Department, United Arab Emirates University, Al-Ain, UAE e-mail: [email protected] # Springer International Publishing AG, part of Springer Nature 2018 F. A. Kulacki (ed.), Handbook of Thermal Science and Engineering, https://doi.org/10.1007/978-3-319-26695-4_37

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Galerkin method of weighted residuals. The fluid domain is described by an Arbitrary Lagrangian–Eulerian (ALE) formulation that is fully coupled to the structure domain. Different flow models for porous media such as Darcy’s law model and Darcy-Forchheimer model are considered in this investigation. Comparisons of isotherms, streamlines, average Nusselt number, and vibration characteristics are presented. The results presented in this review show that the Rayleigh number and the elasticity of the flexible wall have a profound effect on vibration characteristics, the shape of the flexible wall, and consequently on the heat transfer enhancement within the enclosure. For hyperthermia application, larger vessels and flexible arterial wall models exhibited higher variation of the temperature within the treated tumor owning to the enhanced mixing in the vicinity of the bottom wall.

1

Introduction

Transport phenomena through porous media have received considerable attention by many authors due to an increasing need for a better understanding of the associated transport processes. This interest stems from its importance in numerous practical applications which can be modeled or can be approximated as transport through porous media such as thermal insulation, packed bed heat exchangers, drying technology, catalytic reactors, petroleum industries, geothermal systems, and electronic cooling. Vafai and Tien (1981, 1982) presented an in-depth analysis of the generalized transport through porous media. They established a set of governing equations using the local volume-averaging technique. In their work, the concept of momentum boundary layer and introduction of proper averaging volume for interpreting the results within a momentum boundary layer were presented. The effects of presence of a solid boundary and inertial forces on the transient mass transfer in porous media were studied thoroughly by Vafai and Tien (1982) with particular emphasis on mass transfer through a porous medium near an impermeable boundary. Some aspects of transport in porous media were also discussed in monographs by Nield and Bejan (1995), Vafai (2000, 2005), Hadim and Vafai (2000), and Vafai and Hadim (2000). Major advances were made in modeling fluid flow, heat, and mass transfer through a porous medium including clarification of several important physical phenomena. For example, non-Darcy effects on momentum, energy, and mass transport in porous media were studied in detail for different geometrical configurations and boundary conditions. Many of the research works in porous media for the past couple of decades used what is now commonly known as the Brinkman–Forchheimer-extended Darcy or the generalized model. Significant advances were also been accomplished in applying porous media theory in modeling biomedical applications. Examples include computational biology, tissue replacement production, drug delivery, advanced medical imaging, porous scaffolds for tissue engineering and effective tissue replacement to alleviate organ shortages, and transport in biological tissues (Yang and Vafai 2006; Ai and Vafai 2006). Porous

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media theory can also be utilized in biosensing systems (Khaled et al. 2003; Yang et al. 2003; Khanafer et al. 2004; Khaled and Vafai 2004a, b). Fluid-structure interaction (FSI) problems, as well as many other multifield problems, have received a great consideration in recent years and their importance is still continuously increasing. The main reason for this attention is that they are of great relevance in all fields of engineering as well as in the applied sciences. Hence, the development and application of respective modeling and simulation approaches have gained great attention over the past decades. Fluid-structure interaction (FSI) is a multiphysics coupling between the laws that describe fluid dynamics and structural mechanics. This phenomenon is characterized by interactions – which can be stable or oscillatory – between a deformable or moving structure and a surrounding or internal fluid flow. When a fluid flow encounters a structure, stresses and strains are exerted on the solid object – forces that can lead to deformations. These deformations can be quite large or very small, depending on the pressure and velocity of the flow and the material properties of the actual structure. In this review, two applications, namely, heat transfer augmentation and hyperthermia in cancer treatment are analyzed as related to the advances of fluid-structure interaction in porous medium. The flow-induced vibration characteristics will be determined and discussed in this review.

2

Fluid-Structure Interaction Analysis of Non-Darcian Effects on Natural Convection in a Porous Enclosure

It is worth noting that the vast majority of the previous studies on natural convection flow and heat transfer in geometries filled with a porous medium have been limited to both regular and irregular rigid walls (Khanafer et al. 2009a; Misirlioglu et al. 2005, 2006; Lauriat and Prasad 1989; Nithiarasu et al. 1997; Moya et al. 1987; Baytas and Pop 1999; Das and Mahmud 2003; Adjlout et al. 2002; Mahmud et al. 2002; Dalal and Das 2006; Kumar and Shalini 2003; Kumar 2000; Murthy et al. 1997). Khanafer et al. (2009) analyzed numerically natural convection heat transfer inside a cavity with a sinusoidal vertical wavy wall filled with a porous medium. The vertical walls were assumed isothermal while the top and bottom horizontal walls were assumed insulated. The transport equations were solved using the finite element formulation based on the Galerkin method of weighted residuals. The importance of non-Darcian effects on convective flow and heat transfer in a wavy porous cavity was analyzed in their work. Moreover, different flow models for porous media such as Brinkman-extended Darcy, Forchheimer-extended Darcy, and the generalized flow models were considered. Their results illustrated that the amplitude of the wavy surface and the number of undulations affected heat transfer characteristics inside the cavity. Misirlioglu et al. (2006) studied numerically natural convection inside an inclined wavy cavity filled with a porous medium using Darcy model. Finite element method was used to discretize the governing differential equations with nonstaggered variable arrangement. For high Rayleigh numbers, their results showed that flow and thermal structures were found to be highly

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dependent on surface waviness for inclination angles less than 45 . Kumar and Shalini (2003) analyzed numerically non-Darcy free convection induced by a vertical wavy surface in a thermally stratified porous medium using Forchheimer based non-Darcian formulation. Their results showed that the local heat flux has a complex oscillatory pereiodic pattern in the non-Darcian case and is similar to that observed for the vertical wavy surface. Due to problems associated with machining wavy surfaces, Khanafer (2013) suggested the application of a flexible surface with known elasticity to enhance heat transfer characteristics as the fluid motion will cause the deformation of the flexible wall. Therefore natural convective flow and heat transfer inside a differentially heated enclosure filled with a non-Darcian fluid-saturated porous medium was used to test the hypothesis owing to its fundamental nature. Fluidstructure interaction (FSI) has received a great attention in recent years due to its significance in many applications such as biomechanics, automotive, turbomachinery, wind engineering, MEMS, and aerospace (AlAmiri and Khanafer 2011; Khanafer et al. 2010). AlAmiri and Khanafer (2011) analyzed numerically steady laminar mixed convection heat transfer in a lid-driven cavity with a flexible bottom surface using a fully coupled fluid-structure interaction analysis. The results of that investigation revealed that the elasticity of the bottom wall surface was found to have a profound impact on the shape of the bottom wall and consequently on the heat transfer characteristics within the cavity. Moreover, the effect of increasing Grashof number was also found to significantly impact the flexible wall shape and displacement, especially at low Reynolds number. There are a number of studies cited in the literature related to the effect of moving boundaries on the flow and heat transfer characteristics (Khaled and Vafai 2002, 2003b; Nakamura et al. 2000, 2001). These studies involved boundaries that move with a prescribed motion such as displacement, pressure, or velocity. For example, Khaled and Vafai (2002) studied the effects of both external squeezing and internal pressure pulsations on flow and heat transfer inside nonisothermal and incompressible thin films supported by soft seals. The lower plate of the thin film was fixed while the vertical motion of the upper plate was assumed to have sinusoidal behavior. Their results showed that an increase in the fixation number resulted in more cooling and a decrease in the average temperature values. Also, it was found that an increase in the squeezing number decreased the turbulence level at the upper plate. Less attention was given to natural convection flow and heat transfer in a differentially heated cavity filled with a non-Darcian saturated porous medium while incorporating flexible walls. For example, Khanafer (2013) investigated numerically non-Darcian effects on natural convective flow and heat transfer in a square enclosure filled with a porous medium using fluid-structure interaction (FSI) model. The results of that investigation showed that Rayleigh number and the elasticity of the flexible wall had a profound effect on the shape of the flexible wall and consequently on the heat transfer enhancement within the enclosure.

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2.1

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Numerical Analysis

The problem under consideration is schematically shown in Fig. 1. It was assumed in the investigation by Khanafer (2013) that the left vertical wall was maintained at a high temperature TH while the right flexible vertical wall was maintained at a low temperature TC while the horizontal walls were assumed insulated. The working fluid was assumed incompressible Newtonian fluid with a Prandtl number of 1. The porous medium was viewed as a continuum with the solid and fluid phases in local thermal equilibrium. Thermophysical properties were assumed constant except for the density variation in the body force term (Boussinesq approximation). Furthermore viscous heat dissipation in the fluid was assumed to be negligible in comparison to convection and conduction heat transfer effects. In the porous medium, the Darcy-Forchheimer model was assumed valid where the viscous drag was assumed neglected in the governing equations. It was also assumed in that study that the porosity and the permeability of the porous medium remained constant when deforming the porous medium (Khanafer 2013). An Arbitrary Lagrangian–Eulerian formulation (ALE) was employed to describe the fluid motion in the FSI model. When using the ALE technique in engineering simulations, the computational mesh inside the domains can move arbitrarily to optimize the shapes of elements, while the mesh on the boundaries and interfaces of the domains can move along with materials to precisely track the boundaries and interfaces of a multi-material system. By incorporating the above assumptions, the system of the governing equations can be expressed in general vectorial form as: Continuity ∇u¼0

Fig. 1 Schematic diagram of the problem under investigation

y

(1)

∂T = 0 ∂y

h Flexible plate

H TH

Tc

x ∂T = 0 ∂y

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Momentum ρFku  wkðu  wÞ pffiffiffi þ μ κ1  ðu  wÞ þ ∇p ¼ f Bf κ

(2)

ðu  wÞ  ∇T ¼ αeff ∇2 T

(3)

Energy

where κ is the permeability tensor of the fluid, αeff = keff/(ρfcp) is the effective thermal diffusivity, f Bf the body force per unit volume, T the temperature, u the fluid velocity vector, w the moving coordinate velocity, and (u  w) the relative velocity of the fluid with respect to the moving coordinate velocity. The porous medium was assumed isotropic and therefore the permeability was constant. The governing equation for the solid domain of the FSI model can be described by the following elastodynamics equation: þ f Bs ρs d€s ¼ ∇  σtotal s

(4)

where d€s represents the local acceleration of the solid region ( d€s ¼ w_ ), f Bs the externally applied body force vector at time t, ρs the density of the left vertical wall, and σs the solid stress tensor. The physical properties of the flexible wall were assumed to be constant and homogeneous. The default values chosen for the flexible wall were as follows: a density ρs = 600 kg/m3 and a Poisson’s ratio ϑ = 0.45. The geometric function F in the momentum equation is based on the experimental results of Ergun (1952) and may be written as a function of porosity as (Khanafer and Vafai 2001; Amiri and Vafai 1994, 1998): 1:75 F ¼ pffiffiffiffiffiffiffiffiffiffiffi 150e3

(5)

The boundary conditions used in this investigation are given by: ψ ¼ v ¼ 0,

@T ¼ 0, @y

ψ ¼ u ¼ 0, T ¼ T H , ψ ¼ 0, T ¼ T C ,

y ¼ 0, H : 0  x  H

(6a)

x ¼ 0, 0  y  H

(6b)

x ¼ H, 0  y  H

(6c)

where ψ is the stream function. The final set of boundary conditions is applied to the FSI interfaces such that the conditions of displacement compatibility and traction equilibrium between fluid mesh and solid mesh must be satisfied. These conditions can be expressed mathematically as follows (AlAmiri and Khanafer 2011; Khanafer et al. 2010; Khanafer 2013):

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Displacement compatibility df ¼ ds

(7)

This condition also indicates that the moving mesh velocity in ALE formulation is identical with the solid velocity: w ¼ d_ s . Traction equilibrium n  σf ¼ n  σtotal s

(8)

where df and ds are the displacements, σf and σtotal the tractions of the fluid and s solid, respectively. The microscope fluid stress must be added to structural model as additional internal stress. Therefore, the total Cauchy stress in solid models can be written as ¼ σs  pI σtotal s

(9)

The physical quantity of interest in this investigation is the average Nusselt number along the vertical wall defined by:  Nu ¼

keff

@T @x



ðH ) Nu ¼ Nudy

x¼0

(10)

0

where keff is the effective thermal conductivity of the porous medium given by: keff ¼ ekf þ ð1  eÞks

2.2

(11)

Solution Method

A finite element formulation based on the Galerkin method was utilized in the present investigation to solve the governing equations of fluid-structure interaction model in a square cavity filled with a saturated porous medium using ADINA software (Adina v 9.03) subject to the boundary conditions described above. A variable grid-size system was employed to capture the rapid changes in the dependent variables especially in the vicinity of the walls where the major gradients occur inside the boundary layer. As such, a nonuniform mesh of 120  120 nodes was used for the solution of the Navier–Stokes equations. Further, the right vertical flexible wall was modeled using isoparametric beam elements (Khanafer 2013).

2.3

Results and Discussion

The characteristics of the flow and heat transfer fields within the porous cavity were examined by analyzing the effects of the Rayleigh number, elasticity of the flexible

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wall, porosity, and the effective thermal conductivity of the porous medium. The field variables were presented by outlaying the nondimensional streamlines, isotherms, vibration characteristics, and average Nusselt number in the subsequent sections. In this chapter, the parametric domains of the dimensionless groups considered are: Rayleigh number, 103  Ra 105, and nondimensional elasticity, 107  E* 109. The effective thermal conductivity of the porous medium and the Prandtl number are assumed a value of one in this review. Darcy number (Da) of 0.01 and porosity of 0.9 were used (Khanafer 2013). In addition, the maximum and minimum recirculation intensity levels and dimensionless temperature bounds were documented for the presented streamline results to reflect on the flow intensity levels.

2.3.1

Comparison of Streamlines, Isotherms, and Average Nusselt Number Between FSI and Rigid Models Figures 2, 3, and 4 present streamlines, isotherms, and average Nusselt number for both the FSI model and the rigid model at various Rayleigh numbers. The results show an enhancement in heat transfer when a flexible right vertical wall is used. It can be seen from Figs. 2 and 3 that the shape and the degree of flexible wall penetration into the fluid domain are found to depend considerably on the employed Rayleigh number value. At high Rayleigh number (Ra = 104), the flexible wall bulges inward significantly and the wall approaches asymmetric parabolic shape. Figure 4 demonstrates a comparison of the average Nusselt number between FSI and rigid models using Darcy-Forchheimer formulation. Flexible vertical wall exhibits higher Nusselt number than rigid wall model for various Rayleigh numbers. It is worth noting that the maximum relative difference in Nusselt number prediction between both models is 13.6% and occurs at Ra = 104. As the Rayleigh number increases, the average Nusselt number difference between both models increases. This is associated with higher penetration of the flexible wall into the fluid domain. As such, the parabolic shape of the flexible wall augments heat transfer as the motion of the cold fluid accelerates along the flexible wall. This suggests that heat transfer augmentation can be achieved using flexible walls rather than incorporating, for example, irregular geometries. The vibration characteristics of the flow-induced vibration model are also investigated in this investigation (Khanafer et al. 2016). Fourier analysis was performed for the point with maximum penetration inside the cavity for different Rayleigh numbers as shown in Table 1. This table illustrates that as the Rayleigh number increases, the frequency amplitude as well as the natural frequency increases. This increase is more profound at high Rayleigh numbers. 2.3.2

Effect of Varying the Elasticity of the Flexible Wall on Isotherms, Streamlines, and Average Nusselt Numbers Figure 5 demonstrates the effect of varying the elasticity of the flexible wall on the isotherms, streamlines, and the average Nusselt number. It can be seen from Fig. 5 that the elasticity of the flexible wall has a significant effect on the shape of the

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Rigid Ψ(−0.697,0)

FSI Ra=103

Ψ(−0.665,0.0052)

-0.0500 -0.1500 -0.2500 -0.3500 -0.4500 -0.5500 -0.6500

Ψ(−4.05,0.0006)

815

Ra=104

-0.300 -0.900 -1.500 -2.100 -2.700 -3.300 -3.900

-0.0450 -0.1350 -0.2250 -0.3150 -0.4050 -0.4950 -0.5850

Ψ(−3.85,0.167)

0.000 -0.600 -1.200 -1.800 -2.400 -3.000 -3.600

Fig. 2 Comparison of the streamlines between rigid and FSI models for various Rayleigh numbers using Darcy-Forchheimer formulation (ε = 0.9, Da = 0.01, Pr = 1, keff = 1, E* = 107)

wall. As such, the flexible wall bulges significantly inward at a small value of the elasticity. As the elasticity of flexible wall increases, its stiffness increases and consequently decreases thermal activities within the cavity. This is clearly seen in Fig. 6 which shows the impact of elasticity on the average Nusselt number. Figure 6 illustrates that the average Nusselt number decreases and asymptotically approaches the value of the rigid model case as the elasticity of the flexible wall increases. It is worth noting that the relative difference in the average Nusselt number predictions between both models was found to be 15.3% and 2.05% at E* = 107 and 109, respectively. Table 2 shows the effect of varying the elasticity of the wall on the frequency amplitude. This table demonstrates that as the elasticity of the wall increases, the frequency amplitude decreases significantly when the wall becomes very rigid (E* = 109).

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Rigid θ(0,1)

FSI Ra=103

θ(0,1)

0.975 0.825 0.675 0.525 0.375 0.225 0.075

0.975 0.825 0.675 0.525 0.375 0.225 0.075

θ(0,1)

Ra=104

0.975 0.825 0.675 0.525 0.375 0.225 0.075

θ(0,1)

0.975 0.825 0.675 0.525 0.375 0.225 0.075

Fig. 3 Comparison of the isotherms between rigid and FSI models for various Rayleigh numbers using Darcy-Forchheimer formulation (ε = 0.9, Da = 0.01, Pr = 1, keff = 1, E* = 107)

Fig. 4 Comparison of the average Nusselt number between rigid and FSI models for various Rayleigh numbers using Darcy-Forchheimer formulation (ε = 0.9, Da = 0.01, Pr = 1, keff = 1, E* = 107)

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Table 1 Effect of varying Rayleigh number on the frequency amplitude and natural frequency using Fourier analysis Rayleigh number 103 5  103 104

θ(0,1)

Frequency amplitude 0.11 0.19 0.26

E*=107

Natural frequency 10 23.75 28.75

Ψ(−3.85,0.167)

0.975 0.825 0.675 0.525 0.375 0.225 0.075

θ(0,1)

E*=109

0.000 -0.600 -1.200 -1.800 -2.400 -3.000 -3.600

Ψ(−3.96,0.159)

0.975 0.825 0.675 0.525 0.375 0.225 0.075

0.000 -0.600 -1.200 -1.800 -2.400 -3.000 -3.600

Fig. 5 Effect of varying the elasticity of the flexible wall on the isotherms and streamlines (Ra = 104, ε = 0.9, Da = 0.01, keff = 1)

3

Fluid-Structure Interactions in a Tissue During Hyperthermia

Thermal therapy is a relatively new cancer treatment which is receiving significant attention amongst radiation therapists in recent years. Its concept is to slightly raise the temperature of body tissues up to 45
C to kill tumor cells for therapeutic purposes without affecting the surrounding healthy tissues. This treatment, also

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Fig. 6 Effect of varying the elasticity of the flexible wall on the average Nusselt number (Ra = 104, ε = 0.9, Da = 0.01, Pr = 1, keff = 1)

Table 2 Effect of varying the elasticity of the wall on the frequency amplitude using Fourier analysis Nondimensional elasticity 107 108 109

Frequency amplitude 0.26 0.12 0.054

known as hyperthermia and temperature regulation, is of paramount interest in this medical treatment. In essence, tumor cells have difficulty dissipating heat and, hence, hyperthermia would cause such cells to undergo a systemic cell death in direct response to applied heat, while healthy tissues can more easily maintain a normal temperature. In addition, maintaining a uniform temperature distribution during the thermal therapy is critical as the application of temperature values above 45
C may destroy the surrounding healthy tissues. Many investigations were launched to model and analyze the flow motion and thermal response in a blood vessel under hyperthermia treatment (Xuan and Roetzel 1997, 1998; Khaled and Vafai 2003, 2006, 2009; Khanafer et al. 2007; Nakayama and Kuwahara 2008; Mahjoob and Vafai 2009). It is worth noting that it was found that the temperature variation within the tissues depends on many parameters such as tissue thermal conductivity, heat transfer resulting from blood flow, heat generation by tissue metabolism, and the heating source’s power deposition pattern characteristics during the thermal treatment process (Khanafer et al. 2007; Shih et al. 2012). The transport of energy in biological tissues, which is usually expressed by the bioheat equation, is in fact a complicated process. Therefore many researchers have developed mathematical models of bioheat transfer as an extended or modified

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version of the original work of Pennes (1948), as reported by Charny (1989) and Arkin et al. (1994). Shih et al. (2012) investigated numerically the coupled effects of the pulsatile blood flow in blood vessels and the thermal relaxation time in living tissues on temperature distributions using wave bioheat transfer equation (WBTE) for various conditions of heating speed, thermal relaxation time, and pulsation frequency of blood flow. Their results showed that the thermal behavior was found to be very sensitive to the heating speed and insensitive to the pulsation frequency. Furthermore, Craciunescu and Clegg (2001) introduced a simple model of a straight rigid blood vessel with unsteady periodic flow to study the effect of blood velocity pulsations on bioheat transfer without considering the surrounding tissue. Their results showed that the reversed flow enhances as the Womersley number, which is a dimensionless expression of the ratio of the pulsatile flow frequency to the viscous effects, becomes larger in magnitude, thus resulting in a smaller temperature difference between forward and reverse flows. Although Pennes’s bioheat equation is considered to be a useful model to predict temperature distribution in the human body due to its simplicity, it is still considered questionable. On the contrary, Khanafer and Vafai (2006) and Khaled and Vafai (2003) pointed out that the theory of porous media is more suitable for modeling heat transfer in biological tissues. This is due to the fact that it contains fewer assumptions as compared to different bioheat transfer equations, which incorporates several empirical parameters that are considered valid only in restricted domains. Moreover, Khanafer et al. (2007) conducted a numerical study to determine the influence of pulsatile laminar flow and heating protocol on temperature distribution in a single blood vessel with a tumor tissue receiving hyperthermia treatment. The wall of the tissue was modeled as a homogenous porous medium. Their results indicated that the choice of motion waveform significantly influences the findings concerning temperature distribution and heat transfer rate during hyperthermia treatment. Moreover, the authors illustrated that large vessels have a profound effect on the heat transfer characteristics. Meanwhile, Vafai and co-workers (Ai and Vafai 2006; Yang and Vafai 2006, 2008; Khakpour and Vafai 2008) developed a new fundamental four-layer model for the description of the mass transport in the arterial wall coupled with the mass transport in the arterial lumen. In essence, the endothelium, intima, internal elastic lamina (IEL), and media layers were all treated as macroscopically homogeneous porous media and mathematically modeled using proper types of the volumeaveraged porous media equations. Recently, Iasiello et al. (2016) investigated analytically low-density lipoprotein transport through an arterial wall under hyperthermia and hypertension conditions. A four-layer model was used to characterize the arterial wall. Transport governing equations were obtained as a combination between Staverman–Kedem–Katchalsky membrane equations and volume-averaged porous media equations. Temperature and solute transport fields were coupled by means of Ludwig-Soret effect. Results were in excellent agreement with numerical and analytical literature data under isothermal conditions and with numerical literature data for the hyperthermia case. Effects of hypertension combined with hyperthermia were also analyzed in that work.

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It is worth noting that most of the earlier studies treated the arterial wall as a solid nonelastic medium, which does not represent the real physiological condition. In the cardiovascular system, the fluid blood and the arterial wall structure constitute an intrinsically coupled system where the blood flow is under constant interaction with arterial walls. The interactions between blood flow and wall deformation can involve a wide range of fluid-mechanical phenomena (Khanafer et al. 2009). When blood flows through the lumen, the forces associated with the flow may deform the flexible arterial walls and consequently alter the properties of the wall which in turn affect the flow structure in the lumen as well as the thermal transport during hyperthermia treatment. This phenomenon is classified under “fluid-structure interaction” and is of primary interest in modeling blood flow because of the arterial wall remodeling process and the subsequent altered flow pattern in pathological conditions. Khanafer and Berguer (2009) numerically analyzed turbulent pulsatile flow and wall mechanics in an axisymmetric three-layered wall model of a descending aorta. A fullycoupled fluid-structure interaction (FSI) analysis was utilized in that investigation. Their findings showed that peak wall stress and maximum shear stress are highest in the media layer which has the highest elastic value. Recently, Alamiri et al. (2014) outlined a robust model that coupled the lumen with the arterial wall under pulsatile flow condition. In fact, their work was the first attempt, up to the author’s knowledge, to fully incorporate the fluid-structure interaction effect during hyperthermia treatment. In addition, the flow and temperature characteristics due to the wall deformation were analyzed under prescribed physiological conditions, different sizes of blood vessels, and three different heating schemes.

3.1

Mathematical Formulation

This review is concerned with modeling the transient behavior of blood flow in a human vessel while retaining the flexible behavior of the surrounding arterial wall. In essence, the problem configuration deals with the internal flow of a liquid motion in elastic tube driven by a pulsating-force function. At a macroscopic level, the blood vessel consists of a lumen and an arterial wall, which is a complex multilayer structure that is subject to deformation under blood pressure. In the current investigation, however, the arterial wall will be viewed as a single homogenous, isotropic porous layer. Furthermore, the system under investigation is a two-dimensional straight and circular tube of an inner radius Ri, a uniform wall thickness b, and finite axial length L as depicted in Fig. 7. Moreover, the blood flow will be treated as an incompressible laminar Newtonian flow, which is a valid assumption for the experienced velocity magnitudes as reported by Xu et al. (1992). In reference to the work of Prosi et al. (2005), the pertinent blood physiological properties used were the density ρf = 1050 kg/m3, dynamic viscosity μf = 0.00345 Pa.s, thermal conductivity kf = 0.51 W/m.K, and specific heat cf = 3.78 kJ/kg.K. Meanwhile, the porous layer is characterized by Khanafer et al. (2010) to have a porosity of e = 0.258 and a permeability of K = 2  1012 m2. Furthermore, the blood vessel was assigned the following numerical values for the

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porous tissue

T = Ttumor or q = q’’

T = Tin

T = Tin b = 1 mm

L1 = 50 mm u(t)

Ltumor = 20 mm

L2 = 80 mm

r

P(t)

Ri = 8 mm

x

L = 150 mm

Fig. 7 Schematic of the physical model and coordinate system

inner radius Ri = 8 mm, arterial wall thickness b = 1 mm, and the axial length L = 150 mm. These values are in accordance with typical sizes of human blood vessels as reported by Khanafer et al. (2007) and Khanafer and Berguer (2009). Further, a tumor of length ‘ = 20 mm was considered to exist at a prescribed axial location as shown in Fig. 7. In the lumen part, the clear flow can be described by the Navier–Stokes equation. Accordingly the governing equations for conservation of mass, momentum, and energy can be written as following. Continuity ∇u¼0

(12)

Momentum ρf

@u þ ρf ðu  wÞ  ∇u ¼ ∇  σf þ ρf f Bf @t

(13)

Energy @T þ u  ∇T ¼ αf ∇2 T @t

(14)

where u is the fluid velocity vector, ρf the fluid density, w the moving coordinate velocity, f Bf the body force per unit volume, (u  w) the relative velocity of the fluid with respect to the moving coordinate velocity, and σf the fluid stress tensor. Owning to the low encountered velocity magnitudes, the Darcy-Forchheimer model is invoked in this investigation where the viscous drag is assumed negligible in the governing equations. Also, it is assumed that the porosity and the permeability of the porous medium remain constant when deformation of the porous tissue takes place (Chung and Vafai 2012). Meanwhile, the arterial wall was considered saturated

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with a blood fluid that is in local thermodynamic equilibrium with the solid structure of the porous medium. In the present review, the physical properties of the flexible arterial wall and the tumor were assumed constant. What is more, simulation of a deformable arterial wall under a pulsating blood flow was characterized by physiological values for the arterial wall density, Poisson’s ratio and Young’s modulus. The two aforementioned terms are fixed, respectively, as wall density ρs = 2200 kg/m3 and Poisson’s ratio ϑ = 0.45. Further, the employed tumor was assumed solid with the following physical properties (AlAmiri 2013): ktumor = 0.511 W/m.K, ρtumor = 2000 kg/m3, and specific heat ctumor = 3.6 kJ/kg.K. As the tumor is typically considered stiffer than the vessel itself, the elastic properties, namely the Young modulus, of the arterial wall and the tumor tissue were taken as 2 MPa and 6 MPa, respectively. The inlet velocity to the vessel (shown in Fig. 8) was presumed to follow a fullydeveloped pulsating pattern as provided by Khanafer et al. (2007): h i uz ðr, tÞ ¼ 2um ðtÞ 1  ðr=Ri Þ2

(15)

Meanwhile, a time-dependant mean velocity, um ðtÞ, was employed to describe the inlet velocity as presented by Khanafer et al. (2007). It should be emphasized that such a velocity profile is a more accurate description of the pulsating blood flow than a steady version of the fully-developed flow given by Eq. 15. In addition, the inlet blood temperature was assigned a value of Tin = 37
C, while the declared tumor was subjected to three different heating schemes, namely, a constant temperature scheme with Ttumor = 54
C, a uniform heat flux scheme with q}tumor ¼ 7500 W=m2, and a step-wise heat flux dosage of 7500 W/m2 with a time interval of 1.5 s. The latter imposed condition was a more realistic condition as utmost care must be exercised in order not to harm the healthy tissues (Khanafer et al. 2007). Finally, the exit boundary condition for the pressure is applied as given by Khanafer and Berguer (2009). Fig. 8 Velocity and pressure waveforms employed in the current investigation

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823

The associated boundary conditions for the problem under consideration involve a defined inlet axial velocity and a zero transverse velocity at the inlet, and zero gradient at the symmetry line and a normal traction at the outlet. All this can be expressed mathematically as x¼0:

u ¼ uðtÞ, v ¼ 0, T ¼ T in @v @T ¼ ¼0 p ¼ pðtÞ, @x @x

x¼L: r¼0:

@u @T ¼v¼ ¼0 @r @r

(16) (17)

The heat flux equality and the temperature at the lumen-arterial wall interface (r = Ri) for the rigid wall case will be sustained upon employing the following conditions: T lumen wall  ¼ T arterial @T    kf @T ¼ k eff @r lumen @r arterial  kf @T  ¼ ktumorf @T  @r lumen

wall

(18)

@r tumor wall

Finally, several parameters were normalized as follows: R¼

r u 2um Ri , Re ¼ , U¼ Ri um ðtÞ νf

(19)

where Re is the time-averaged Reynolds number.

3.2

Results and Discussion

The variation of the local heat flux and the bottom surface temperature along the axial length of the vessel was examined for a prescribed tumor top wall temperature of 54
C while employing FSI model. This particular temperature was chosen in order to regulate the temperature within the tumor layer to the range used in hyperthermia treatment, i.e., 41–45
C. In this regard, the effect of the vessel size can be represented by manipulating the prescribed Reynolds numbers at peak flow condition, which is identified in Fig. 8. Hence, the Reynolds number impact on the local heat flux and temperature variations along the bottom surface of the tumor is examined. Figure 9 illustrates that large vessels (Re = 300) exhibits higher local heat flux distribution than small vessels (Re = 100). This indicates that the dissipation of energy from heated tissues during hyperthermia treatment, which is carried out by convection and conduction heat transfer, is greater for larger vessels. Consequently, the increase in Reynolds number renders lower bottom surface temperature of the tumor as shown in Fig. 10.

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Fig. 9 Comparison of the local heat flux distribution along the tumor surface at peak flow condition for different Reynolds numbers (ltumor = 20 mm)

Fig. 10 Comparison of the temperature variation along the bottom surface of the tumor at peak flow condition for different Reynolds numbers

This relatively higher cooling effect attained at Re = 300 may cause some parts of the tumor not reaching the desired treatment temperature, especially if the Reynolds numbers were further increased. Figures 11 and 12 show, respectively, the local heat flux distribution and the bottom wall temperature of the tumor along the axial length of the vessel. Further, the results are presented for both FSI and stationary rigid wall models. The results manifest that FSI model exhibits higher local heat flux magnitudes and bottom surface temperature variations along the tumor axial length as compared to rigid model. It is evident that the flexible wall model undergoes further cooling effect owning to improved mixing and mass/heat transport in the vicinity of bottom wall of the vessel. Such observations require revision of rigid arterial wall assumption as currently employed in the literature. This is an interesting observation as all parts of the tumor should be heated to the desired rise in temperature during hyperthermia therapy. Moreover, the peak temperature plays an important role in thermal treatments giving that the

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825

Fig. 11 Comparison of the local heat flux distribution along the tumor surface at peak flow condition between flexible and rigid wall models using Re = 300 (ltumor = 20 mm)

Fig. 12 Comparison of the temperature variation along the bottom surface of the tumor at peak flow condition between flexible and rigid models using Re = 300

maximum temperature magnitude directly dominates the thermal dose levels. However, prolonged heating time may induce areas of overheating (beyond the therapeutic regions) that could damage normal tissues.

4

Conclusions

Natural convective flow and heat transfer in a cavity filled with a saturated porous medium was investigated numerically using fluid-structure interaction for various pertinent parameters such as Rayleigh number, elasticity of the flexible wall, effective thermal conductivity of porous medium, and porosity. The presented results in this review showed that Rayleigh number and elasticity of the flexible wall had a profound effect on the shape and the penetration of the flexible wall and consequently on the heat transfer enhancement. Moreover, average Nusselt number was

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found to increase with an increase in the effective thermal conductivity and porosity of the porous medium. It was also noted that Darcy’s model experienced a sinusoidal shape of the flexible wall at low Rayleigh number and asymmetric parabolic shape at high Rayleigh numbers. However, Darcy-Forchheimer model exhibited asymmetric parabolic shape for the studied range of Rayleigh number. The second example presented in this review dealt with modeling and simulation of hyperthermia treatment of tumor in blood vessels. The arterial wall was modeled using volume-averaged porous media theory. The results showed that large blood vessels exhibited significant cooling effect on the temperature distribution and must be accounted for when planning for hyperthermia treatment. What is more, the elasticity of the tumor was found to influence the temperature variation along the bottom surface of the tumor. The results presented in this review may assist physicians in positively identifying individuals who exhibit the development of vascular diseases and begin appropriate treatment long before symptoms or clinical signs tend to appear.

5

Cross-References

▶ Applications of Flow-Induced Vibration in Porous Media ▶ Heat Exchanger Fundamentals: Analysis and Theory of Design

References Adjlout L, Imine O, Azzi A, Belkadi M (2002) Laminar natural convection in an inclined cavity with a wavy-wall. Int J Heat Mass Transf 45:2141–2152 Ai L, Vafai K (2006) A coupling model for macromolecule transport in a stenosed arterial wall. Int J Heat Mass Transf 49:1568–1591 AlAmiri A (2013) Fluid-structure interaction analysis of pulsatile blood flow and heat transfer in living tissues during thermal therapy. J Fluids Eng 135:041103–004110 Alamiri A, Khanafer K (2011) Fluid-structure interaction analysis of mixed convection heat transfer in a lid-driven cavity with a flexible bottom wall. Int J Heat Mass Transf 54:3826–3836 AlAmiri A, Khanafer K, Vafai K (2014) Fluid-structure interactions in a tissue during hyperthermia. Numer Heat Transf 66:1–16 Amiri A, Vafai K (1994) Analysis of dispersion effects nonthermal equilibrium non-Darcian variable porosity incompressible-flow through porous-media. Int J Heat Mass Transf 37:939–954 Amiri A, Vafai K (1998) Transient analysis of incompressible flow through a packed bed. Int J Heat Mass Transf 41:4259–4279 Arkin H, Xu LX, Holmes KR (1994) Recent developments in modeling heat transfer in blood perfused tissues. IEEE Trans Biomed Eng 41:97–107 Baytas AC, Pop I (1999) Free convection in oblique enclosures filled with a porous medium. Int J Heat Mass Transf 42:1047–1057

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Charny CK, Levin RL (1989) Bioheat transfer in a branching countercurrent network during hyperthermia. ASME J Biomech Eng 111:263–270 Chung S, Vafai K (2012) Effect of the fluid-structure interactions on low-density lipoprotein within a multi-layered arterial wall. J Biomech 45:371–381 Craciunescu OI, Clegg ST (2001) Pulsatile blood flow effects on temperature distribution and heat transfer in rigid vessels. ASME J Biomech Eng 123:500–505 Dalal A, Das MK (2006) Natural convection in a cavity with a wavy wall heated from below and uniformly cooled from the top and both sides. J Heat Transf 128:717–725 Das PK, Mahmud S (2003) Numerical investigation of natural convection inside a wavy enclosure. Int J Therm Sci 42:397–406 Ergun S (1952) Fluid flow through packed columns. Chem Eng Prog 48:89–94 Hadim H, Vafai K (2000) Overview of current computational studies of heat transfer in porous media and their applications-forced convection and multiphase transport. Adv Numer Heat Transfer 2:291–330. Taylor and Francis, NY Iasiello M, Vafai K, Andreozzi A, Bianco N (2016) Low-density lipoprotein transport through an arterial wall under hyperthermia and hypertension conditions – an analytical solution. J Biomech 49:193–204 Khakpour M, Vafai K (2008) A critical assessment of arterial transport models. Int J Heat Mass Transf 51:807–822 Khaled A-RA, Vafai K (2002) Flow and heat transfer inside thin films supported by soft seals in the presence of internal and external pressure pulsations. Int J Heat Mass Transf 45:5107–5115 Khaled A-RA, Vafai K (2003a) The role of porous media in modeling flow and heat transfer in biological tissues. Int J Heat Mass Transf 46:4989–5003 Khaled A-RA, Vafai K (2003b) Analysis of flow and heat transfer inside oscillatory squeezed thin films subject to a varying clearance. Int J Heat Mass Transf 46:631–641 Khaled ARA, Vafai K (2004a) Optimization modeling of analyte adhesion over an inclined microcantilever-based biosensor. J Micromech Microeng 14:1220–1229 Khaled ARA, Vafai K (2004b) Analysis of oscillatory flow disturbances and thermal characteristics inside fluidic cells due to fluid leakage and wall slip conditions. J Biomech 3:721–729 Khaled ARA, Vafai K, Yang M, Zhang X, Ozkan CS (2003) Analysis, control and augmentation of microcantilever deflections in bio-sensing systems. J Sens Actuators B 94:103–115 Khanafer K (2013) Fluid-structure interaction analysis of non-Darcian effects on natural convection in a porous enclosure. Int J Heat Mass Transf 58:382–394 Khanafer K, Berguer R (2009) Fluid–structure interaction analysis of turbulent pulsatile flow within a layered aortic wall as related to aortic dissection. J Biomech 42:2642–2648 Khanafer K, Vafai K (2001) Isothermal surface production and regulation for high heat flux applications utilizing porous inserts. Int J Heat Mass Transf 44:2933–2947 Khanafer K, Vafai K (2006) The role of porous media in biomedical engineering as related to magnetic resonance imaging and drug delivery. Heat Mass Transf 42:939–953 Khanafer K, Vafai K (2009) Synthesis of mathematical models representing bioheat transport, Chapter 1. In: Advances in numerical heat transfer, vol III. CRC Press, New York, pp 1–28 Khanafer K, Khaled ARA, Vafai K (2004) Spatial optimization of an array of aligned microcantilever biosensors. J Micromech Microeng 14:1328–1336 Khanafer K, Bull JL, Pop I, Berguer R (2007) Influence of pulsative blood flow and heating scheme on the temperature distribution during hyperthermia treatment. Int J Heat Mass Transf 50:4883–4890 Khanafer K, Al-Azmi B, Marafie A, Pop I (2009a) Non-Darcian effects on natural convection heat transfer in a wavy porous enclosure. Int J Heat Mass Transf 52:1887–1896 Khanafer K, Bull JL, Berguer R (2009b) Fluid–structure interaction of turbulent pulsatile flow within a flexible wall axisymmetric aortic aneurysm model. Eur J Mech B-Fluids 28:88–102

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Khanafer K, Alamiri A, Pop I (2010) Fluid-structure interaction analysis of flow and heat transfer characteristics around a flexible microcantilever in a fluidic cell. Int J Heat Mass Transf 53:1646–1653 Khanafer K, Vafai K, Gaith M (2016) Fluid–structure interaction analysis of flow and heat transfer characteristics around a flexible microcantilever in a fluidic cell. Int Commun Heat Mass Transf 75:315–322 Kumar BVR (2000) A study of free convection induced by a vertical wavy surface with heat flux in a porous enclosure. Numer Heat Transfer A: Appl 37:493–510 Kumar BVR, Shalini RKBV (2003) Free convection in a non-Darcian wavy porous enclosure. Int J Eng Sci 41:1827–1848 Lauriat G, Prasad V (1989) Non-Darcian effects on natural convection in a vertical porous enclosure. Int J Heat Mass Transf 32:2135–2148 Mahjoob S, Vafai K (2009) Analytical characterization of heat transfer through biological media incorporating hyperthermia treatment. Int J Heat Mass Transf 52:1608–1618 Mahmud S, Das PK, Hyder N, Islam AK (2002) Free convection in an enclosure with vertical wavy walls. Int J Therm Sci 41:440–446 Misirlioglu A, Baytas AC, Pop I (2005) Free convection in a wavy cavity filled with a porous medium. Int J Heat Mass Transf 48:1840–1850 Misirlioglu A, Baytas AC, Pop I (2006) Natural convection inside an inclined wavy enclosure filled with a porous medium. Transport Porous Med 64:229–246 Moya SL, Ramos E, Sen M (1987) Numerical study of natural convection in a tilted rectangular porous material. Int J Heat Mass Transf 30:741–756 Murthy PVSN, Kumar BVR, Singh P (1997) Natural convection heat transfer from a horizontal wavy surface in a porous enclosure. Numer Heat Transf A Appl 31:207–221 Nakamura M, Nakamura T, Tanaka T (2000) A computational study of viscous flow in a transversely oscillating channel. JSME Int J Ser C 434:837–844 Nakamura M, Sugawara M, Kozuka M (2001) Heat transfer characteristics in a two-dimensional channel with oscillating wall. Heat Trans Asian Res 304:280–292 Nakayama A, Kuwahara F (2008) A general bioheat transfer model based on the theory of porous media. Int J Heat Mass Transf 51:3190–3199 Nield DA, Bejan A (1995) Convection in porous media, 2nd edn. Springer, Berlin/ Heidelberg/New York Nithiarasu P, Seetharamu KN, Sundararajan T (1997) Natural convective heat transfer in a fluid saturated variable porosity medium. Int J Heat Mass Transf 40:3955–3967 Pennes HH (1948) Analysis of tissue and arterial blood temperature in the resting human forearm. J Appl Physiol 1:93–122 Prosi M, Zunino P, Perktold K, Quarteroni A (2005) Mathematical and numerical models for transfer of low-density lipoprotein through the arterial walls: a new methodology for the model set up with applications to the study of disturbed laminar flow. J Biomech 38:903–917 Shih TC, Horng TL, Huang HW, Ju KC, Huang TC, Chen P (2012) Numerical analysis of coupled effects of pulsatile blood flow and thermal relaxation time during thermal therapy. Int J Heat Mass Transf 55:3763–3773 Vafai K (2000) Handbook of porous media, 1st edn. Marcel Dekker, New York Vafai K (2005) Handbook of porous media, 2nd edn. Taylor and Francis Group, New York Vafai K, Hadim H (2000) Overview of current computational studies of heat transfer in porous media and their applications – natural convection and mixed convection. Adv Numer Heat Transfer 2:331–371. Taylor and Francis, NY Vafai K, Tien CL (1981) Boundary and inertia effects on flow and heat transfer in porous media. Int J Heat Mass Transf 24:195–203 Vafai K, Tien CL (1982) Boundary and inertia effects on convective mass transfer in porous media. Int J Heat Mass Transf 25:1183–1190

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Xu XY, Collins MW, Jones CJH (1992) Flow studies in canine artery bifurcations using a numerical simulation method. ASME J Biomech Eng 114:504–511 Xuan YM, Roetzel W (1997) Bioheat equation of the human thermal system. Chem Eng Technol 20:268–276 Xuan YM, Roetzel W (1998) Transfer response of the human limb to an external stimulus. Int J Heat Mass Transf 41:229–239 Yang N, Vafai K (2006) Modeling of low-density lipoprotein (LDL) transport in the artery – effects of hypertension. Int J Heat Mass Transf 49:850–867 Yang N, Vafai K (2008) Low density lipoprotein (LDL) transport in an artery – a simplified analytical solution. Int J Heat Mass Transf 51:497–505 Yang M, Zhang X, Vafai K, Ozkan C (2003) High sensitivity piezoresistive cantilever design and optimization for analyte-receptor binding. J Micromech Microeng 13:864–872

Imaging the Mechanical Properties of Porous Biological Tissue

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John J. Pitre Jr. and Joseph L. Bull

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Biot Theory of Poroelasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Constitutive Relationships . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Physical Interpretation of the Poroelastic Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Ultrasound Elastography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Ultrasound Poroelastography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Background and Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Early Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 In Vivo Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Magnetic Resonance Poroelastography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Background and Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Laboratory Studies of MR Poroelastography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Compression-Sensitive MR Elastography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Measured Properties of Poroelastic Materials and Tissues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Cross-References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

832 834 834 836 839 841 846 846 848 849 850 850 851 852 853 855 855 855

Abstract

Of the roughly 42 l of water contained in a normal 70 kg person, approximately a quarter makes up what is known as the interstitial fluid. This fluid permeates a dense porous network of proteins called the extracellular matrix. The nonlinear mechanical behavior of many tissues is, at least partly, a result of this porous structure. That is, many tissues are poroelastic. Abnormalities in the poroelastic properties of tissue are often indicative of disease states ranging from renal failure J. J. Pitre Jr. (*) · J. L. Bull Biomedical Engineering Department, Tulane University, New Orleans, LA, USA e-mail: [email protected]; [email protected] # Springer International Publishing AG, part of Springer Nature 2018 F. A. Kulacki (ed.), Handbook of Thermal Science and Engineering, https://doi.org/10.1007/978-3-319-26695-4_38

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to traumatic brain injury to cancer. As such, it is of broad clinical interest to develop methods for measuring these abnormalities to help guide diagnosis and clinical decision-making. This chapter describes methods for imaging the poroelastic properties of biological tissue using ultrasound and magnetic resonance (MR). These poroelastography methods are derived from earlier work which focused on imaging linearly elastic properties of tissue, ignoring the porous nature of the tissue structure. Incorporating poroelastic tissue models into elasticity imaging makes it possible to image not only tissue stiffness but also fluid content, permeability, and internal pressure. This chapter will introduce the theory of poroelasticity, as described by both the Biot and KLM biphasic models, and discuss its role in the development of both quasi-static ultrasound poroelastography and time-harmonic MR poroelastography. For each method, the current state-of-the-art from both a technical perspective and a clinical perspective will be reviewed, offering insight into the continuing development of both technologies. The authors hope to leave the reader with a better understanding of the challenges faced by these methods as well as the role that advances in porous media science can play in improving medical imaging.

1

Introduction

In many ways, medical imaging is a qualitative problem. The goal is often to give clinicians the ability to “peek inside” the body and identify features that might be indicative of a disease state. For example, a magnetic resonance (MR) or computed tomography (CT) image can show the boundaries of a tumor, allowing oncologists to make clinical decisions for surgery or to check the progress of treatment. An ultrasound image can allow an obstetrician to monitor fetal development during pregnancy or give a cardiologist a real-time view of the beating heart. These types of imaging rely on tissue contrast, and the source of that contrast is related to the nature of the imaging modality. Ultrasound contrast derives from differences in the speed of sound between various tissues. MR contrast derives from aligning and perturbing the spins of hydrogen nuclei, which are plentiful because of the human body’s high water content. Taking advantage of these properties has led to a revolution in modern medicine – giving clinicians accurate pictures of nearly any pathology imaginable. What then do modern medical imaging methods leave to be desired? As mentioned before, these methods lead to largely qualitative images. Clinical decisionmaking still requires a highly trained expert to interpret the images. This makes many imaging methods largely impractical for point-of-care applications or rural medicine, where access to a physician’s time is a valuable commodity. Furthermore, these medical images can sometimes only tell if there is a problem but not how bad that problem might be. Many pathologies still require a (sometimes highly invasive) biopsy. Imagine an imaging system that could allow an untrained eye to identify a tumor and predict the metastatic potential. Consider the usefulness of a point-of-care system that measures the exact volume of excess fluid in swollen tissues, allowing nephrologists to improve the quality of life of hundreds of thousands of dialysis

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patients who suffer from peripheral edema. These are just two examples of some of the work being done to advance new quantitative imaging methods to the clinic. How does porous media fit into this scheme? There exists certain class of quantitative imaging methods called elastography that seeks to image the mechanical properties of tissue. All elastography methods rely on some choice of a material model for the tissue to be imaged. As a simple assumption, one might assume that tissue behaves as a linearly elastic solid, in accordance with Hooke’s law. While this works fairly well in practice for a number of applications, it does have its limitations. First and foremost is that biological tissue is not a homogeneous solid. Rather, it can best be described as a porous matrix of proteins that houses the various cells of the body (Fig. 1). This extracellular matrix is saturated with the interstitial fluid, a solution of water and the various dissolved ions necessary for physiological function. It follows that when a region of tissue is compressed, one should not expect a linearly elastic response but rather a poroelastic response. As an example, suppose a compressive stress is applied to a tissue. In response, the porous extracellular matrix deforms elastically and pressurizes the interstitial fluid. The increase in internal fluid pressure drives fluid flow through the porous matrix (according to Darcy’s law). As more fluid leaves the compressed region, less fluid remains to support the extracellular matrix. This leads to the tissue exhibiting a slow additional deformation, frequently called a creep response, which approaches a constant value once all the mobilizable fluid is driven from the tissue. Poroelastography methods seek to image the mechanical properties that lead to this poroelastic response – the elastic coefficients of the extracellular matrix, the fluid volume contained in it, and the porosity and permeability of the tissue. It has been hypothesized that these measurements may be useful in a diverse number of clinical settings. Poroelastography remains a nascent field, however, and much work remains to develop more robust methods and translate them to the clinic. It is in this developmental stage that porous media scientists can greatly aid the field. The following sections will outline in more detail the poroelastic nature of tissue, the Fig. 1 Schematic of soft tissue structure. Biological tissue is composed of a porous matrix of proteins and cells. This extracellular matrix is saturated with water and ions needed for physiological functions. This pore fluid (shown here in light blue) can move freely though the porous matrix leading to the poroelastic response of tissue under compression

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theory of linear poroelasticity, the early poroelastography work that has been done using both ultrasound and MR, and the next steps in the development of poroelastography in which porous media scientists can play a role.

2

Biot Theory of Poroelasticity

The general theory of poroelasticity has its roots in soil mechanics. Soil undergoes a process called consolidation whereby the material slowly deforms in response to an applied load. Terzaghi (1923) was the first to propose a mechanism to explain this phenomenon, basing it on assumptions of one-dimensional deformation and a constant load. It was Biot (1941) who outlined the first general theory in the threedimensional case. Consider, as Biot did, an isotropic material composed of a porous matrix saturated by a pore fluid. It is helpful to require that pores of the matrix are small enough compared to the macroscopic behavior that the material can be considered homogeneous. In addition, it is reasonable to limit the analysis to assume small strains, linearity of the stress-strain relations, and reversibility of the equilibrium stress-strain relationship.

2.1

Constitutive Relationships

The derivation begins by defining a variable ζ called the variation in fluid content that describes the increment of water volume per unit volume of the porous matrix. The fluid pressure p acts on the pore fluid. In the case that ζ ¼ 0 , the material by definition will behave as a perfectly elastic solid – that is, there is no pore fluid and therefore no pore pressure ðp ¼ 0 Þ: One would therefore expect the material to behave according to Hooke’s law. Writing Hooke’s law in terms of Young’s modulus E and Poisson’s ratio ν, the following constitutive relationship applies between the stress tensor σ ij and the strain tensor eij: eij ¼


 1 σ ij  ν σ kk  σ ij , E

(1)

where summation is implied over repeated indices. For a nonzero pressure, isotropy requires that the pressure cannot produce any shearing strain. It must also act equally on the three principal directions. By the assumption of small strain, it is reasonable to assume a linear relationship between the strain and the pore pressure. Letting H0 denote some physical constant and δij denote the Kronecker delta, the stress-strain-pressure relationship for a poroelastic material can then be written as eij ¼


 p 1 σ ij  ν σ kk  σ ij þ 0 δij : E H

(2)

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The volumetric strain ϵ  ekk can be written by summing the principal strains to give ϵ ¼ e11 þ e22 þ e33 ¼

1 3p ð1  2νÞσ kk þ 0 : E H

(3)

Solving Eq. 3 for σ kk and defining H 0 ¼ 3 H (for convenience) yields σ kk ¼

E  p ϵ : 1  2ν H

(4)

This now makes it possible to determine the stress tensor for a poroelastic material. Combining Eqs. 2 and 4 and rearranging terms to solve for the stress yields  σ ij ¼ 2G eij þ

 νϵ δij  αpδij : 1  2ν

(5)

Here, two new constants have been defined – the shear modulus G and the BiotWillis coefficient α. The shear modulus is not independent from the parameters E and ν. Rather, it can be written as a combination of the two – specifically, G¼

E : 2ð 1 þ ν Þ

(6)

The Biot-Willis coefficient α, as will be shown later, is a measure of the ratio of the fluid volume change to the total volume change of the porous matrix. It can be written in terms of the shear modulus and Poisson’s ratio or alternatively in terms of the bulk modulus K and the poroelastic constant H as α¼

2ð1 þ νÞG K ¼ : 3ð1  2νÞH H

(7)

The derivation now returns to the variation of fluid content ζ. As the simplest possible assumption, ζ has the general linear form ζ ¼ a1 σ 11 þ a2 σ 22 þ a3 σ 33 þ a4 σ 12 þ a5 σ 13 þ a6 σ 23 þ a7 p:

(8)

As before, the isotropy assumption is used to eliminate shear terms (a 4 ¼ a 5 ¼ a 6 ¼ 0 ). Since all three principal stresses must have equivalent effects on the fluid content, this leaves the following expression for the variation of fluid content: ζ¼

1 1 p, 0 σ kk þ R H1

(9)

where H01 and R are physical poroelastic constants. Assuming that the material has potential energy – this follows naturally if the stress-strain relationship is reversible – it can be shown that H 01 ¼ H0 ¼ 3 H (Biot 1941). Combining Eqs. 4 and 9, the fluid

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content can be expressed as a function of the volumetric strain and the pore pressure. Some algebraic manipulations yield ζ ¼ αϵ þ

1 p, M

(10)

where 1 =M ¼ 1 =R  α=H is sometimes called the Biot modulus.

2.2

Physical Interpretation of the Poroelastic Parameters

It is worth mentioning now that two unique poroelastic constants have not been fully defined: H and R, which enter the constitutive equations through the Biot-Willis coefficient α and the Biot modulus 1/M. In general, these represent physical constants that must be measured by experiment. In the case of an ideal porous solid, however, expressions can be derived for these in terms of more familiar material properties such as porosity, compressibility, and bulk modulus. An ideal porous solid is a material composed of a homogeneous, isotropic solid matrix containing a fully connected pore space. Such materials exhibit a very useful property. Consider applying a known confining pressure P0 to the material while holding the fluid pore pressure p at an equal magnitude – a useful thought experiment Detournay and Cheng (1993) termed a “Π-loading”. Biot referred to this as an unjacketed test, and it is sometimes also referred to as an undrained test. Under a Π-loading, the solid matrix and pore space of an ideal porous solid deform with a uniform volumetric strain. This means that the porosity of an ideal porous solid is constant under a Π-loading, making the analysis much simpler. The first goal is to derive a simple physical interpretation of the Biot-Willis coefficient α. This is done by applying a Π-loading with an all-around confining pressure P0 to an ideal porous material. In other words, the three principle stresses all equal P0 (where the negative denotes compression). It follows that the sum of the principle stresses can be written σ kk ¼ σ 11 þ σ 22 þ σ 33 ¼ 3P0 :

(11)

An equivalent form of Eq. 4, expressed in terms of the bulk modulus rather than the Young’s modulus and Poisson’s ratio, can be written  p σ kk ¼ 3K ϵ  : H

(12)

By virtue of the chosen Π-loading, some substitutions can be made into Eq. 12. First, the pore pressure p = P0, and second, σ kk = 3P0 from Eq. 11. Finally, for a Π-loading, the volumetric strain is uniform – that is, the volumetric strain in the bulk porous material is equal to the volumetric strain in the solid phase alone. These substitutions yield

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Imaging the Mechanical Properties of Porous Biological Tissue

P0 3P ¼ 3K ϵ s  : H

837



0

(13)

Focusing on the solid components of the porous material only, the definition of the bulk modulus can be used to describe the volumetric strain in the solid  ϵ s  

ΔV V

¼

s

P0 , Ks

(14)

where V denotes volume and Ks is the bulk modulus of a homogeneous (non-porous) volume of the solid matrix material. Combining Eqs. 7, 13, and 14 and rearranging, it can be shown that K : Ks

α¼1

(15)

This means that α can be thought of as a measure of the relative compressibility of the solid matrix material itself and that of the porous material as a whole. In the case of an incompressible solid phase, Ks approaches infinity and α = 1. An expression must also be found for the Biot modulus 1/M. From Eq. 10, this constant relates the variation of fluid content to the volumetric strain and pore pressure. In fact, the Biot modulus can be defined very precisely as the change in fluid content with respect to pressure under conditions of constant volumetric strain. If the pore space of the porous solid is saturated, then the pore volume Vp and fluid volume Vf are equal. The same holds for any changes in those two volumes. This allows the following decomposition: ð1Þ

ð2Þ

ΔV f ΔV p ΔV f ΔV f ¼ ¼ þ , Vp Vf Vf Vf

(16)

ð1Þ

where ΔV f is associated with volume changes due to dilatation of the pore fluid and ð2Þ

ΔV f is due to fluid exchange between the pore space and the environment. By the definition of compressibility, χ f of the pore fluid ð1Þ

ΔV f Vf

¼ χ f p:

(17)

The variation in fluid content ζ describes the increment of water volume per unit volume of the porous matrix. That is, ð2Þ

ζ

ΔV f V

:

(18)

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This is very close to the form seen in Eq. 16. Using the porosity of the material ϕ, defined as the ratio of pore volume to total volume, Eq. 18 can be written in terms of the total fluid volume (again noting that for a saturated pore space Vp = Vf) ð2Þ

ζ¼

V f ΔV f  V Vf

ð 2Þ

¼ϕ

ΔV f Vf

:

(19)

Substituting Eqs. 17 and 19 into Eq. 16, noting that the left hand side of the equation is equal to the volumetric strain in the pore space, yields ϵ p ¼ χ f p þ

ζ : ϕ

(20)

Equation 10 can be used to eliminate ζ from Eq. 20. With some mild rearranging, this yields ϕϵ p ¼ ϕχ f p þ αϵ þ

1 p: M

(21)

As for the Biot-Willis coefficient, a Π-loading can again be applied to the ideal porous material. For a Π-loading, the following hold: (1) p equals a uniform confining pressure P0 and (2) ϵ p = ϵ = ϵ s. Applying these conditions to Eq. 21, yields ϕϵ s ¼ ϕχ f P0 þ αϵ s þ

1 0 P, M

(22)

which can then be simplified further using Eq. 14: P0 ϕ ¼ Ks



1 1 ϕχ f  α þ P0 : Ks M

(23)

From here, one can eliminate P0 and solve for the Biot modulus 1 1 ¼ ϕχ f þ ðα  ϕÞ , M Ks

(24)

or, equivalently, making use of the definition of α (Eq. 15), 1 1α ¼ ϕχ f þ ðα  ϕÞ : M K

(25)

The poroelastic parameters now have clear physical interpretations, fully expressible in terms of the mechanical properties of the constituent materials (solid, fluid, and bulk matrix) and their internal structure (porosity). This will be particularly beneficial to poroelastography studies, as will be discussed later, since it allows

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greater flexibility in making material assumptions. For example, consider the case where the bulk porous material is much more compressible than an equivalent block of the non-porous solid (Ks  K ), which yields the case α = 1 (from Eq. 15). Another reasonable assumption might be that the pore fluid has the same properties as water. In many applications ranging from soil dynamics to poroelastography, this is indeed the case. The constitutive equations under these assumptions then involve only bulk material properties (G , ν , K , ϕ) and one poroelastic parameter 1/M that can be calculated rather than measured.

2.3

Governing Equations

A full constitutive model for a poroelastic material has now been derived. This means that the relationship between stresses, strains, and pressures can be described. However, the system of equations is not closed and cannot be solved. This requires two equations: (1) an equilibrium equation for the stress and (2) an evolution equation for the pore pressure. The equilibrium equation is typically found by assuming that the deformation of the poroelastic material can be described as quasi-static. That is, deformation to an equilibrium state occurs very quickly, particularly when compared to the timescale over which the pore fluid pressure changes. This allows one to enforce an equilibrium condition. The equilibrium condition is identical to that for an elastic solid, in accordance with Cauchy’s momentum equation. Neglecting gravitational effects, the divergence of the total stress tensor must equal zero: σ ij, j ¼ 0:

(26)

Applying this to Eq. 5, this leads to the equilibrium pressure-strain relationship  2G eij, j þ

 ν ϵ , i  αp, i ¼ 0, 1  2ν

(27)

or equivalently using the bulk modulus K instead of Poisson’s ratio ν  2 2Geij, j þ K  G ϵ , i  αp, i ¼ 0: 3

(28)

This equation has three components, one for each of the three spatial dimensions and relates the fluid pore pressure to the displacement field ui, since eij = (ui,j + uj,i)/2. There are four unknowns – the three displacements ui and the pore pressure p. Closing the system of equations requires an equation for the evolution of the pore pressure. The pressure evolution is a time-dependent equation, but one that varies on a much slower time scale than the deformation. It is in this sense that the timedependent behavior – for example, in the consolidation of soil or the creep response

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of biological tissue – enters the problem even though the deformation is assumed to be quasi-static. Continuity requires that the variation of fluid content is dependent on the volumetric flux qi of fluid into and out of the porous matrix. Mathematically, this relationship can be written as @ζ þ qi;i ¼ 0: @t

(29)

The volumetric flux of fluid moving through a porous matrix is described by Darcy’s law. If κ is the permeability of the porous matrix (with units of m2) and μ is the viscosity of the pore fluid, then Darcy’s law states that the volumetric flux is proportional to the pressure gradient κ qi ¼  p, i : μ

(30)

Combining Eqs. 10, 29, and 30 gives the pressure evolution equation 1 @p κ @ϵ  p ¼ α : M @t μ , ii @t

(31)

Equations 28 and 31 are sufficient to solve for all the unknown variables – three equations for the displacement components u, v, and w, and one equation for the pressure p – for some deformation problem. Table 1 gives a summary of the variables, parameters, and governing equations for poroelastic deformation. As an example application, one may be interested in determining the expected displacement and pressure in a region of biological tissue placed under a known external load. Knowing the properties of the tissue (i.e., K , G , α , κ , ϕ , χ f , μ), one can determine the displacement field. Alternatively, it is often the case that one is Table 1 Summary of the poroelasticity parameters and equations Material parameters

Poroelastic parameters

K G Ks κ ϕ χf μ α ¼ 1  KKs

Matrix bulk modulus Matrix shear modulus Solid bulk modulus Matrix permeability Matrix porosity Fluid compressibility Fluid viscosity Biot-Willis coefficient

¼ ϕχ f þ ðα  ϕÞ 1α K   2Geij;j þ K  23 G ϵ ;i  αp;i ¼ 0

Biot modulus

1 M

Governing equations

1 @p M @t

Unknown variables



u, v, w p

κ μ p;ii

¼ α

@ϵ @t

Equilibrium equations Pressure evolution equation Displacement field Pore fluid pressure

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interested in the inverse problem. That is, suppose the displacement can be measured at every point in the tissue (using ultrasound, for example). Then the inverse problem can be stated as finding the material properties that produce the measured displacement under a known loading. This problem forms the core of poroelastography methods, which seek to measure tissue mechanical properties in vivo.

3

Ultrasound Elastography

Elastography refers to any medical imaging method that seeks to measure the elastic properties of biological tissue. Research in this field began in 1991, when Ophir and colleagues introduced ultrasound elastography. In their paper, they outlined the basic premise by which many elastography studies still work today (Fig. 2). Briefly, an ultrasound transducer is placed so that a region of interest is visible and an initial image is captured. A known axial compressive stress is then applied to the tissue, inducing a deformation, and a second ultrasound image is acquired. These two ultrasound images and the magnitude of the applied stress represent the entirety of the data collection. Assuming the compression occurs in the axial direction only, Hooke’s law for linear elasticity states that the stress σ in the tissue must be proportional to the strain e, σ ¼ Ee:

(32)

The proportionality constant E is known as the Young’s modulus, and it is a measure of the elasticity of a material. Since the applied stress is known and uniform, only the internal strain of the tissue is needed to estimate the Young’s modulus, that is, E¼

σ applied : emeasured

(33)

Internal strain measurements typically rely on estimating the internal displacement at each point in the tissue between the first and second ultrasound images. The strain is related to the displacement by definition. If the displacement vector ui = (u1, u2, u3), then the strain tensor is defined eij ¼

 1 ui , j þ uj , i : 2

(34)

Since it is assumed that only axial strains are present, Eq. 34 can be simplified by only considering the third component. If z represents the axial direction, this leaves the relationship emeasured ¼ e33 ¼

@u3 : @z

(35)

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0.00 −0.15 −0.30 −0.45 −0.60 −0.75 −0.90 −1.05

−1.20

Fig. 2 (continued)

5.0 4.5 4.0 3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.0

80 72 64 56 48 40 32 24 16 8 0

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The task now becomes to reliably estimate the displacement at each point in the ultrasound images. Before discussing how to measure internal displacement from the acquired data, it is useful to describe how an ultrasound image is created. First, a pressure transducer emits a high frequency sound wave into the tissue. The wave travels through the tissue, interacting with the many microscale features present. Each of these features creates a scattered sound wave that returns to the transducer, which records the sound waves as an electrical signal. This recorded signal is called the raw radiofrequency (RF) signal, and it forms the basis of the ultrasound image. Applying enveloping and compression to this RF signal yields the familiar ultrasound brightness-mode (B-mode) image (Fig. 3a). In this sense, an ultrasound image represents the reflectivity of the tissue at each location. The direct reflection of sound waves from each microscopic tissue feature is not the only phenomenon present in the image. Figure 3a shows that the ultrasound B-mode image has a “grainy” character that one might be tempted to describe as random noise. This phenomenon is called ultrasonic speckle, and it is by no means random. Rather, speckle is the result of the interference patterns of sound waves 5.0 4.5

3.5 3.0 2.5 2.0 1.5

Axial Strain (%)

4.0

1.0 0.5 0.0

Fig. 3 (a) An ultrasound B-mode image of a tissue-mimicking phantom constructed by embedding a cylinder of firm tofu inside a block of soft tofu. Pre- and postcompression images were acquired while applying a 2% compressive strain. Displacement estimation was performed using a speckle tracking algorithm. (b) A smoothing spline and gradient operator were then used to compute an axial strain elastogram. Note that the background strain is roughly 2%, while the stiffer inclusion has a smaller strain, as expected ä Fig. 2 Typical steps in an ultrasound elastography measurement. (1) Pre- and postcompression images are acquired of a region of interest under a constant strain. (2) Speckle tracking algorithms are then used to estimate the internal displacement field in the image. This works by finding the best match of small sub-images, or kernels, between the pre- and postcompression images. (3) Using the displacement estimates, an axial strain or Young’s modulus elastogram can be computed

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scattering throughout the tissue. As the sound scatters, the hills and valleys of the waves interact constructively and destructively creating small bright and dark spots, respectively. The key point here though is that this speckle pattern is intrinsically linked to the unique microscale geometry of the tissue. It follows that motion of the tissue induces a near identical motion of the speckle pattern. Speckle can therefore be used as a type of identifying “tissue marker” to track how discrete regions move between two ultrasound images. Speckle tracking algorithms estimate motion between two ultrasound images by searching for matching blocks in the two images, a precompression image and a post compression image. These images may use either the raw RF or processed B-mode data. The simplest algorithms utilize a brute force search, though cleverer search strategies have been developed. The basic speckle tracking algorithm proceeds as follows. First, the precompression image is divided into a grid of sub-images, each centered at the point xi = (xi, yi). For each sub-image, a search region is defined in the post compression image. The normalized cross-correlation between the sub-image and the search region is then computed, typically using the Fast Fourier Transform. The maximum of the normalized cross-correlation corresponds to the  point xi ¼ xi , yi at which the sub-image best matches the search region. The best displacement estimate then is given by the difference in positions between the preand post compression images: ui ¼ xi  xi :

(36)

The partial derivative from Eq. 35 can then be approximated using finite differences to obtain a strain measurement (Fig. 3b), and the Young’s modulus estimate calculated as E¼

σ applied  Δz u3 ðz þ ΔzÞ  u3 ðzÞ

(37)

where Δz is the axial spacing between adjacent grid points in the speckle tracking grid. Note that this finite difference scheme can amplify noise in the displacement measurements, and the fact that it appears in the denominator can exacerbate this effect. A great deal of elastography research has followed this principle for estimating the Young’s modulus of tissue. Numerous analyses and improvements have been made in an attempt to optimize the method. The early work by Ophir et al. (1991) was expanded upon by O’Donnell et al. (1994) who characterized the effects of finite strains (those greater than 2%). Their proposed methods were verified by Skovoroda et al. (1994) who compared experimental elastography measurements of gelatin phantoms to theoretical predictions of a linearly elastic model. Following these early studies, a great deal of work was done to seek optimal methods for strain estimation and speckle tracking. One important development was the strain filter, a theoretical framework used to estimate the internal tissue strain required to obtain the most accurate estimates of displacement estimates (Varghese and Ophir 1997a). This

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framework is particularly useful for estimating the expected signal-to-noise ratio of an elastography measurement for a given ultrasound system. Other studies sought to develop more accurate speckle tracking and displacement estimate algorithms. These include fixed and adaptive temporal stretching (Varghese et al. 1996, Varghese and Ophir 1997b, Alam and Ophir 1997, Alam et al. 1998), short time correlation (Lubinski et al. 1999), and the use of dynamic programming (Petrank et al. 2009). These methods can greatly increase the accuracy of the speckle tracking displacement estimates, allowing accurate measurements even at much higher strains where decorrelation of the speckle pattern becomes problematic. Up to this point, two critical assumptions present in many elastography methods have been overlooked. The first is that the stress is uniform throughout the tissue and that it is equal to the applied stress at the boundary. In general, this is not true. The presence of lateral strains in the compressed tissue points to a slightly more complex relationship between the mechanical properties, stress, and strain. It becomes important to consider not only the Young’s modulus but also the Poisson’s ratio of the tissue being imaged. This analysis requires a more nuanced approach. One must consider the general form of the equilibrium equations for linear elastic solids σ ij;j ¼ 0, σ ij ¼

(38)


 E νδij ekk þ ð1  2νÞeij , ð1 þ νÞð1  2νÞ

(39)

 1 ui, j þ uj;i : 2

(40)

eij ¼

Typically, studies simplify this approach by assuming that the tissue is nearly incompressible, for example, assigning it an assumed Poisson’s ratio of 0.495. In addition, since most ultrasound systems can only produce a two-dimensional image, one might assume a two-dimensional plane strain approximation. Even with simplifying assumptions, the problem remains difficult: • Given a set of known displacements ui(x, y) and known boundary forces, determine the Young’s modulus field E(x, y) that satisfies Eqs. 38, 39, and 40. The above is known as an inverse elasticity problem. Posing the elastography problem in this way allows for more complicated physics and even more complex material models to be employed. Furthermore, the inverse problem formulation reduces artifacts and performs better than the strain-based formulation in cases of high elasticity contrast (Doyley et al. 2005). Provided the solution to the inverse problem can be computed (Barbone and Bamber 2002), this approach to generating an elastogram becomes very appealing. Skovoroda et al. (1994, 1995, 1999) posed the inverse elastography problem using a displacement-pressure formulation, solving it for both small and large deformations using a gradient descent method. In contrast, Kallel and Bertrand (1996) solved the

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elastography inverse problem using a modified Gauss-Newton method. This approach has become much more common in model-based elastography. Using both ideal displacement data and simulated RF ultrasound data, they were able to reconstruct Young’s modulus images for materials with hard cylindrical inclusions. Doyley et al. (2000) provided experimental validation of this technique using a similar inverse problem solver, the Levenberg-Marquardt method. Both of these studies faced a similar disadvantage. The modified Gauss-Newton method and Levenberg-Marquardt algorithm both require an estimation of the system Jacobian, or sensitivity, matrix. This matrix can be very computationally expensive to compute since a single forward elasticity problem must be solved for each pixel in the image. Oberai et al. (2003, 2004) derived and tested an adjoint-based method for computing the Jacobian matrix that greatly reduces computational costs. More recent work has focused on implementing more accurate tissue models into the inverse elastography framework. In particular, numerous studies have sought to measure the properties of tissue assuming it behaves as a more complex hyperelastic material (Samani and Plewes 2004, Goenezen et al. 2011, 2012).

4

Ultrasound Poroelastography

4.1

Background and Theory

Under the linear elasticity assumption, ultrasound elastography methods are able to measure the mechanical properties of the tissue, particularly the Young’s modulus. In reality though, biological tissue displays a more complex stress-strain relationship that results from the transport of interstitial fluid through the porous extracellular matrix. In short, tissue can be better described as being poroelastic. This more accurate assumption presents itself as an interesting avenue for improvement upon previous elastography methods. The idea of imaging the poroelastic nature of tissue began to take shape in 1998 when Konofagou and Ophir introduced a method for imaging lateral strains and Poisson’s ratio using ultrasound. Lateral strains had previously presented a problem for elastography studies. Ultrasound resolution in the axial direction is determined by the pulse length. For a typical ultrasound system, this will be on the order of 0.5–1 mm (Hoskins et al. 2010). In contrast, resolution in the lateral direction is determined by the beam width and typically is on the order of 1–5 mm (Hoskins et al. 2010). This difference in resolution also makes speckle tracking measurements of tissue displacement much less accurate in the lateral direction. By using an adaptive lateral speckle tracking method, Konofagou and Ophir (1998) were able to accurately measure lateral strains. This allowed them to compute the lateral-to-axial strain ratio, giving a measurement of the effective Poisson’s ratio of the tissue. Konofagou et al. (2001) later termed this method poroelastography when they combined their measurement method of the lateral-to-axial strain ratio with a theoretical solution for the deformation of a biphasic material.

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The initial theoretical basis for poroelastography was an analytical solution to the Kuei-Lai-Mow (KLM) biphasic model of tissue (Armstrong et al. 1984, Kuei et al. 1978, Mow and Kuei 1980). The KLM model can be derived by considering the conservation of mass, momentum, and energy for a binary mixture of fluid and solid constituents. The model was originally used to describe the stress-strain response of articular cartilage where it successfully described the creep and stress relaxation responses of cartilage under sustained compression (Mow and Kuei 1980). Armstrong et al. (1984) later derived an analytical solution to the KLM equations for a uniform unconfined cylinder with slip boundary conditions subjected to an axial compressive stress. This solution (Eq. 41) dictates that the average lateral-to-axial strain ratio (the average effective Poisson’s ratio) in the cylinder should decay in time to that of the drained matrix. Additional insight into this behavior was provided by Berry et al. (2006a) who derived an expression for the local (rather than average) strain ratio (Eq. 42). The following equations summarize these two results. They depend on the following material properties: HA, the aggregate modulus (itself a function of the matrix Young’s modulus and Poisson’s ratio); νs, the Poisson’s ratio of the drained matrix; κ, the permeability of the matrix; and a, the radius of the cylinder. More details about these equations can be found in Armstrong et al. (1984) and Berry et al. (2006a), respectively.  2 α HA κt exp  n 2 a

1 X u n o ða, tÞ ¼ νs þ ð1  2νs Þð1  νs Þ ae0 α2n ð1  νs Þ2  ð1  2νs Þ n¼1

(41)

  8 9 <   J 1 αn r = 1  2νs αn r J ð α Þ 1 n J 0 ðαn Þ   αnar þ J0 a αn ; 1  νs : 1 X err a ðr, tÞ ¼ νs þ 1 e0 n¼1 J 0 ðαn Þ  αn J 1 ðαn Þ ð1  ν s Þ  2 α H A κt exp  n 2 a (42) 

The typical model for poroelastography studies has proceeded as follows (Konofagou et al. 2001; Righetti et al. 2004, 2005a, b; Berry et al. 2006a, b). First, displacement data are collected according to a standard elastography protocol. The tissue to be imaged is compressed, and pre- and postcompression images are collected. Ultrasound speckle tracking provides an estimate of the internal displacement field in the imaged region of interest, both in the axial and lateral directions. The lateral-to-axial strain ratio is then computed using finite differences, and this value is inserted into the left hand side of Eq. 41 or 42. A curve fitting procedure can then be used at each pixel in the poroelastogram to estimate the poroelastic parameters HA, νs, and/or κ.

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Early Studies

Initial poroelastography studies made heavy use of simulated ultrasound RF data and tissue mimicking materials. One can simulate RF data by convolving the point spread function of a simulated ultrasound scanner with the positions of randomly distributed scatterers with Gaussian distributed strengths. Academic and commercial software packages exist to perform these simulations (for example, Field-II and FOCUS). Konofagou et al. (2001) used the solution of a finite element model to displace the scatterers between simulated ultrasound frames. The finite element model used the Biot model of poroelasticity outlined in the previous sections. Note that the Biot model is equivalent to the KLM model under common simplifying assumptions for soft tissue (Simon 1992). The measured lateral-to-axial strain ratio showed good agreement with the computational true value, but the image was very noisy. Righetti et al. (2004) followed a similar procedure but also tested the method experimentally using tofu as a tissue mimicking phantom. Their study exhibited similar results – namely, the Poisson’s ratio image generated (though accurate) was highly noisy, especially for the experimental case. They also noted the difficulty in applying corrections to the lateral measured strains in the presence of contrast in both the Poisson’s ratio and Young’s modulus. Despite these drawbacks, the study produced an important extension to earlier work. Using experimentally measured values of the Young’s modulus and permeability, the researchers used Eq. 41 to solve for the Poisson’s ratio of the drained matrix. They observed an expected decay in the mean Poisson’s ratio, but the true value was difficult to obtain since the permeability could only be accurately estimated experimentally to an order of magnitude. In a similar study, Rightetti et al. (2005a) generated two new varieties of poroelastogram using two types of tofu phantom as well as porcine tissue. The first, the Poisson’s ratio time constant elastogram, was generated by fitting an exponential curve of the form ν = ν0(1 + exp(t/τ)) to the effective Poisson’s ratio at each pixel in a time series of poroelastograms, where the constant ν0 was estimated from the experimental data directly at time t = 0. The time constant τ for each pixel then gives a measure for the degree to which the tissue deviates from a purely elastic behavior (since for a purely elastic material, the time constant should approach zero). The second type of poroelastogram outlined in the study, the permeability poroelastogram, relies on a priori estimates of the drained Poisson’s ratio and the Young’s modulus. With both of these measurements, the effective Poisson’s ratio at each pixel in the time series of poroelastograms can be fit to Armstrong’s biphasic solution to the KLM model (Eq. 41). This provides an estimate for the permeability κ at each pixel. In practice, such a procedure is difficult since it requires accurate estimates of both the Poisson’s ratio and Young’s modulus. Berry et al. (2006a) improved upon Konofagou’s work and Righetti’s work by deriving a local, rather than average, expression for the lateral-to-axial strain

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ratio (Eq. 42). This approach, which focused more heavily on the underlying theory of poroelastic deformation, centered around a curve fitting algorithm for determining the poroelastic constants. They also provided experimental validation (Berry et al. 2006b) for homogeneous tofu cylinders with slip boundary conditions. While this work corrected some of the earlier theoretical flaws of earlier work, it also relies on the assumption of material homogeneity throughout the region of interest. In general, this may not hold in the body, and in some cases, such as tumors, it should be expected that inhomogeneity exists.

4.3

In Vivo Studies

To date, two clinical ultrasound poroelastography studies have been performed (Righetti et al. 2007b; Berry et al. 2008). The first compared Poisson’s ratio time constant elastograms for patients with and without lymphedema. Ultrasound measurements were obtained from the thigh or forearm subcutis of seven females and one male with clinically diagnosed lymphedema. Similar measurements were obtained from five female subjects without lymphedema. All subjects were asked to lie in a supine position while a constant 2–5% strain was applied, corresponding to a stress relaxation experiment. Effective Poisson’s ratio time constant elastograms were then used to differentiate the two patient populations. While the images themselves were difficult to interpret, the mean value of the time constant for each patient provided a useful metric for differentiating the presence of lymphedema. For subjects in the non-edematous group, the mean Poisson’s ratio time constant was on the order of 225 s. In contrast, patients with lymphedema exhibited time constants on the order of 55 s with a distribution heavily skewed towards lower values. Although staging the degree of lymphedema was not addressed in this study and indeed remains an open problem, the ability to differentiate edematous and non-edematous subjects represents an important step forward for clinical translation and marks effective Poisson’s ratio poroelastography as a feasible method for edema measurements. The second clinical poroelastography study, from Berry et al. (2008), examined the differences in the poroelastic response measured from patients with chronic unilateral lymphedema. The study examined six female subjects who exhibited at least a 20% fluid volume difference between the ipsilateral and contralateral arms, as measured using optoelectronic volumetry. A constant strain was applied to the measured arm for 500 s (stress relaxation), and ultrasound images were collected during and following a ramp-like compression. Axial strain images were then computed, and strains were compared at various depths along the image centerline. While the researchers did not analyze any poroelastic constants in this study, they were able to observe poroelastic behavior in both the ipsilateral (edematous) and contralateral (non-edematous) arms for each patient. Specifically, they observed

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that the axial strain in both arms exhibited temporal and spatial dependence, as is typical for poroelastic materials under compression. Moreover, they reported that these effects seemed to be more pronounced in the ipsilateral (edematous) arm. Because of the small sample size and single measurements obtained per patient, more definite conclusions could not be made regarding significant differences or reproducibility. They also noted the need to reduce patient motion and to reduce experimental variance. Nevertheless, the fact that this study observed poroelastic effects in all of their clinical measurements gives ample motivation to continuing research into poroelastic imaging methods.

5

Magnetic Resonance Poroelastography

5.1

Background and Theory

Magnetic resonance (MR) elastography is very similar to ultrasound elastography, differing primarily in the imaging modality and the form of linear elasticity equations used to estimate tissue mechanical properties. As with ultrasound elastography, the first step of MR elastography is to acquire images of tissue subject to some kind of mechanical stimulus. In MR elastography, a harmonic excitation is used, and time-harmonic forms of the linear elasticity equations are used to reconstruct tissue properties. Just as ultrasound poroelastography offers an improvement in describing tissue mechanical properties when compared to standard ultrasound elastography, MR poroelastography offers an improvement over standard MR elastography. Purely elastic MR elastography does not accurately estimate the properties of a poroelastic medium (Perriñez et al. 2009). In addition, the ability to estimate the internal fluid pressure field further differentiates MR poroelastography from purely elastic methods. MR poroelastography studies almost exclusively make use of an inverse problem formulation for estimating the mechanical properties. First, phase contrast MR imaging is used to estimate motion in a volume of tissue. This motion is typically induced using an external low-amplitude, low-frequency vibration source (10–1000 Hz), though in some cases, physiological motion has been used. Once the internal motion field is measured, an iterative procedure such as the Gauss-Newton method is used to solve the inverse problem – that is, to find the poroelastic parameter distribution that makes the result of the forward problem most closely match the measured displacement. The poroelasticity forward problem for MR poroelastography uses the timeharmonic form of Biot’s equations (Eqs. 43, 44, 45, 46, and 47). These equations are written in terms of the complex-valued frequency-dependent displacements ^u ðxi , ωÞ and pressures ^ p ðxi , ωÞ. As before, the equations depend on a number of poroelastic parameters including the total bulk modulus K, solid bulk modulus Ks, shear modulus G, bulk density ρ, fluid viscosity μ, fluid compressibility χ f, porosity ϕ, permeability κ,

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and Biot-Willis coefficient α. Also of interest are the apparent mass density ρa (a measure of the work done by relative motion) and the fluid density ρf. 

G^ u i, j



þ ;j



   1 K þ G ^u k;k  ðα  βÞ^p ;i ¼ ω2 ρ  βρf ^u i 3 ,i ω2 ρ f ð α  β Þ ω2 ρ f ϕ 2 ^p þ ^p ;ii ¼ 0 ^u i;i þ βR β β¼

ωϕ2 ρf κ   iϕμ þ ωκ ρa þ ϕρf

R¼

ϕ2 K s  1 α þ ϕK χ f  Ks α¼1

5.2

K Ks

(43)

(44)

(45)

(46)

(47)

Laboratory Studies of MR Poroelastography

As mentioned previously, one of the pivotal initial studies in MR poroelastography was the derivation of a finite element formulation for the time-harmonic Biot equations by Perriñez et al. (2009). This study highlighted the importance of the poroelastic assumption in estimating tissue properties and laid the foundation for future studies that would use the same methods. In 2010, Perriñez et al. expanded on this initial work, using the method to estimate the poroelastic properties of simulated inclusion phantoms as well as tissue-mimicking tofu-gelatin inclusion phantoms (Perriñez et al. 2010a). This study used multiple scan planes in all three dimensions to form a full estimate of the 3D displacement field in the phantom. The inverse problem was solved by decomposing the full domain into overlapping subzones. Each of these subzones was minimized individually and global iterations were applied to link the subzone calculations. The reconstructions were performed on 16 CPUs of a Linux cluster with average runtimes of 3–5 h. This demonstrates the computational complexity of the three-dimensional poroelastic inverse problem, but the results help to highlight the usefulness of MR poroelastography. The poroelastic reconstructions far outperformed linearly elastic reconstructions, providing superior estimates of both the shear modulus and Poisson’s ratio. In the poroelastic reconstructions, higher levels of noise were associated with larger errors. In addition, poor estimates of the tissue permeability (assumed to be known a priori) resulted in larger errors as well. A subsequent study further

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examined the sensitivity of the algorithm to poor a priori estimates of the tissue porosity, permeability, and apparent density (Perriñez et al. 2010b). They found that the a priori permeability was the most influential factor and recommended that its spatial distribution be estimated algorithmically along with the other poroelastic parameters.

5.3

Compression-Sensitive MR Elastography

In the clinic and the laboratory, a series of studies by Hirsch et al. (2013a, b, 2014) sought to develop clinical measures of tissue compressibility based on the divergence of the measured displacement field, |div(u)|, which was estimated using phase contrast imaging. Since soft tissue can typically be assumed to act as a biphasic mixture of incompressible solid and fluid constituents, any measured volumetric strain must arise from the porous structure (which would allow movement of the fluid phase through the solid as the pores collapse). For this reason, compression-sensitive MR elastography can provide useful measurements related to porous tissue properties even though it is not always used in a poroelastography context. In the first study, they measured volumetric strain in the brains of eight healthy male subjects while applying either no excitation, a 25 Hz harmonic excitation, or abdominal muscle contraction (Hirsch et al. 2013a). They demonstrated the clinical feasibility of their compression-sensitive MR elastography method by showing measurements of |div(u)| throughout the cardiac cycle for both no excitation and harmonic excitation. In addition, they showed that mean |div(u)| increases significantly following abdominal muscle contraction. Using the biphasic poroelastic model, they then estimated a compression modulus and shear modulus for each subject. The shear modulus showed no change with abdominal muscle contraction, but the compression modulus showed a strong tendency to decrease. They were not able to demonstrate statistical significance for this measurement, but the compression modulus was much lower than what would be expected for a linearly elastic tissue, thus highlighting the importance of the poroelastic assumption. The second study provided more insight into the sensitivity of the volumetric strain measurement (Hirsch et al. 2013b). They constructed compressible phantoms by mixing water, agarose, and ultrasound gel with sodium hydrogen carbonate and citric acid. This formed a suspension of CO2 bubbles that set in place with the gel, creating a porous solid. They then characterized changes in |div(u)| for different vibration amplitudes and phantom densities (corresponding to different porosities). Although the study did not use a poroelastic model to describe the measurements, it did provide important insight into a magnetic resonance measurement of a poroelastic property. The third study investigated compression-sensitive MR elastography of the liver as a potential biomarker for intrahepatic blood pressure abnormalities. They studied 13 patients with portal hypertension, each with different values of hepatic vein pressure gradient, before and after intervention with a transjugular intrahepatic portosystemic shunt. Patients showed a significant 21  13% increase in the relative

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change of volumetric strain. Although the study did not provide a general method for directly measuring intrahepatic pressure, it did show that compression-sensitive MR elastography could provide a measurement that is sensitive to clinical parameters.

6

Measured Properties of Poroelastic Materials and Tissues

Poroelastography is still a nascent field, and as such, most of the work has taken place within the laboratory as opposed to the clinic. Tofu has frequently been used as a tissue mimicking phantom because of its porous structure and time-dependent response to applied loads. Most poroelastography studies, whether ultrasound-based or MR-based, have sought to compare the imaged properties of tofu to those measured using standard mechanical creep and stress-relaxation tests. The goal has typically been to estimate the time constant of the temporal response (Table 2) or the shear modulus (Table 3). Studies have reported a wide range of values for these parameters. This may in part be due to the variability inherent in studying a material like tofu whose properties may vary by type (soft, firm, extra-firm), manufacturer, age, and storage conditions. These variables make it difficult to reasonably evaluate comparisons across studies. Clinical pilot studies have also shown a wide range in variation when measuring the poroelastic properties of tissue. Often, comparisons between imaging studies and mechanical measurements are difficult because tissue structure and loading vary greatly between the in vivo case (typically employed for imaging studies) and case of excised tissue (typically used for mechanical tests). Peripheral edema imaging represents a special case where a more direct comparison can be made. Both imaging tests and mechanical tests in this setting apply a compressive stress to the surface of the skin and measure the resulting deformation. The behavior is then described by fitting an exponential model to the measured axial strain or EPR, yielding a time constant. This allows comparisons of the temporal response between patients with edema and those without it. The literature in this area remains sparse, however, and the conclusions of existing studies vary. Table 4 summarizes some key results of one poroelastic imaging study (Righetti et al. 2007b) and three studies utilizing mechanical tests. Two of the Table 2 Mechanical properties of tofu–time constants

a

Type Unspecified (TT1 in study) Unspecified (TT1 in study)

Axial strain TC (s) NA 290 (creep)

Unspecified (TT1 in study) Unspecified (TT2 in study) Unspecified (TT2 in study)

270 (relaxation) NA 56 (creep)

Unspecified (TT2 in study) Extra firm Extra firm

67 (relaxation) 7–10 (creep) 19.68 (relaxation)

Denotes a mechanical, rather than imaging test

EPR TC (s) 339 (relaxation) 240 (creep), 340 (relaxation) NA 50 (relaxation) 32 (creep), 49 (relaxation) NA NA NA

Study Righetti et al. (2005a) Righetti et al. (2007a) Righetti et al. (2007a)a Righetti et al. (2005a) Righetti et al. (2007a) Righetti et al. (2007a)a Pitre et al. (2016) Belmont et al. (2013)a

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Table 3 Mechanical properties of tofu–shear modulus Type Silken Soft Firm Extra firm Extra firm Unspecified (fully hydrated) Unspecified (partially dehydrated) Unspecified (TT1 in study) Unspecified (TT2 in study)

Imaged 6–7 (MR) 4.5–4.8 (MR) 5.1–10.6 (MR) 7.1–14 (MR) 5.4 (MR) 0.64 (USN)a

Mechanical quasistatic NA 0.89 2 2.3 NA NA

Mechanical harmonic NA 6.4 16 22 NA NA

Study Perriñez et al. (2010b) Perriñez et al. (2010a) Perriñez et al. (2010a) Perriñez et al. (2010a) Perriñez et al. (2009) Righetti et al. (2004)

0.69 (USN)a

NA

NA

Righetti et al. (2004)

NA

0.67a

NA

Righetti et al. (2007a)

NA

1.2a

NA

Righetti et al. (2007a)

Denotes that the shear modulus result μ was computed from the Young’s Modulus E and Poisson’s ratio ν determined by the study according to μ = (1/2)E/(1 + ν)

a

Table 4 Material properties of edematous tissue–time constants

Normal

Edematous

Time constant (s) 225

Type EPR

0.39

AS

Subcutis, contralateral (normal) arm, unilateral edema Subcutis, contralateral (normal) arm, unilateral edema Subcutis, arm and leg, lymphedema (USN) Subcutis, lower leg, edema patients

71

AS

135

AS

25

EPR

1.4–2.8

AS

Subcutis, ipsilateral (edematous) arm, unilateral edema Subcutis, ipsilateral (edematous) arm, unilateral edema

230

AS

30

AS

Tissue Subcutis, arm and leg, normal patients (USN) Subcutis, lower leg, normal patients

Study Righetti et al. (2007b) Mridha and Odman (1986) Bates et al. (1994) Lindahl (1995) Righetti et al. (2007b) Mridha and Odman (1986) Bates et al. (1994) Lindahl (1995)

mechanical studies suggest that edematous tissues exhibit longer time constants, but their time constant measurements differ from each other by two orders of magnitude (Mridha and Odman 1986; Bates et al. 1994). In contrast, one mechanical study and the previously mentioned imaging study suggest the opposite – namely, that time constants are lower in edematous tissues (Lindahl 1995). These studies demonstrated similar time constant measurements and similar differences between edematous and

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non-edematous tissues. Because of the discrepancies in the literature, more study is clearly required to produce a satisfactory conclusion. A study that combines both ultrasound and mechanical measurements on a large cohort of patients for varying degrees of edema should be able to more accurately answer these questions.

7

Conclusion

Poroelastography provides a unique avenue for porous media scientists to contribute to the advancement of medical imaging technology. The development of poroelastography methods has the potential to improve clinical decision-making and patient care in a number of settings. While future clinical work is needed to fully understand the clinical usefulness of poroelastography, technical advances are still needed to improve image quality. Improved algorithms for solving the poroelastic inverse problem more accurately and more efficiently will be especially important as the field moves forward. To this end, porous media scientists can play a crucial role in implementing efficient methods for solving the poroelastic forward problem, developing new strategies for efficiently computing the system Jacobian matrix for the poroelastic inverse problem, and determining the optimal loading conditions for clinical measurements. These advances will enable clinicians to study poroelastic responses of tissue in many disease states and will hopefully lead to new quantitative metrics that can guide decision-making and improve patient care.

8

Cross-References

▶ Applications of Flow-Induced Vibration in Porous Media ▶ Heat Transfer In Vivo: Phenomena and Models ▶ Inverse Problems in Radiative Transfer ▶ Thermal Properties of Porcine and Human Biological Systems

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Berry GP, Bamber JC, Armstrong CG, Miller NR, Barbone PE (2006a) Towards an acoustic modelbased poroelastic imaging method: I. Theoretical foundation. Ultrasound Med Biol 32(4):547–567 Berry GP, Bamber JC, Miller NR, Barbone PE, Bush NL, Armstrong CG (2006b) Towards an acoustic model-based poroelastic imaging method: II. experimental investigation. Ultrasound Med Biol 32(12):1869–1885 Berry GP, Bamber JC, Mortimer PS, Bush NL, Miller NR, Barbone PE (2008) The spatio-temporal strain response of oedematous and nonoedematous tissue to sustained compression in vivo. Ultrasound Med Biol 34(4):617–629 Biot MA (1941) General theory of three-dimensional consolidation. J Appl Phys 12(2):155–164 Detournay E, Cheng AH-D (1993) Fundamentals of poroelasticity, Chapter 5. In: Fairhurst C (ed) Comprehensive rock engineering: principles, practice and projects, vol. II, analysis and design method. Pergamon Press, Oxford, pp 113–171 Doyley MM, Meaney PM, Bamber JC (2000) Evaluation of an iterative reconstruction method for quantitative elastography. Phys Med Biol 45(6):1521–1540 Doyley MM, Srinivasan S, Pendergrass SA, Wu Z, Ophir J (2005) Comparative evaluation of strain-based and model-based modulus elastography. Ultrasound Med Biol 31(6):787–802 Field-II [Software]. J. A. Jensen, Technical University of Denmark. Available from http://field-ii.dk. Accessed 28 Oct 2016 FOCUS: Fast Object-Oriented C++ Ultrasound Simulator [Software]. Michigan State University. Available from http://www.egr.msu.edu/~fultras-web/. Accessed 28 Oct 2016 Goenezen S, Barbone PE, Oberai AA (2011) Solution of the nonlinear elasticity imaging inverse problem: the incompressible case. Comput Methods Appl M 200(13–16):1406–1420 Goenezen S, Dord J-F, Sink Z, Barbone PE, Jiang J, Hall TJ, Oberai AA (2012) Linear and nonlinear elastic modulus imaging: an application to breast cancer diagnosis. IEEE T Med Imaging 31(8):1628–1637 Hirsch S, Klatt D, Freimann F, Scheel M, Braun J, Sack I (2013a) In vivo measurement of volumetric strain in the human brain induced by arterial pulsation and harmonic waves. Magn Reson Med 70(3):671–682 Hirsch S, Beyer F, Guo J, Papazoglou S, Tzschaetzsch H, Braun J, Sack I (2013b) Compressionsensitive magnetic resonance elastography. Phys Med Biol 58(15):5287–5299 Hirsch S, Guo J, Reiter R, Schott E, Büning C, Somasundaram R, Braun J, Sack I, Kroencke TJ (2014) Towards compression-sensitive magnetic resonance elastography of the liver: Sensitivity of harmonic volumetric strain to portal hypertension. J Magn Reson Im 39(2):298–306 Hoskins P, Martin K, Thrush A (2010) Diagnostic ultrasound: physics and equipment, 2nd edn. Cambridge University Press, New York, p 147 Kallel F, Bertrand M (1996) Tissue elasticity reconstruction using linear perturbation method. IEEE T Med Imaging 15(3):299–313 Konofagou EE, Ophir J (1998) A new elastographic method for estimation and imaging of lateral displacements, lateral strains, corrected axial strains, and Poisson’s ratios in tissues. Ultrasound Med Biol 24(8):1183–1199 Konofagou EE, Harrigan TP, Ophir J, Krouskop TA (2001) Poroelasticity: imaging the poroelastic properties of tissues. Ultrasound Med Biol 27(10):1387–1397 Kuei SC, Lai WM, Mow VC (1978) A biphasic rheological model of articular cartilage. In: Eberhardt RC, Burstein AH (eds) Advances in bioengineering. American Society of Mechanical Engineers, New York Lindahl O (1995) The evaluation of a biexponential model for description of intercompartmental fluid shifts in compressed oedematous tissue. Physiol Meas 16:17–28 Lubinski MA, Emilianov SY, O’Donnell M (1999) Speckle tracking methods for ultrasonic elasticity imaging using short-time correlation. IEEE T Ultrason Ferroelectr 46(1):82–96 Mow VC, Kuei SC (1980) Biphasic creep and stress relaxation of articular cartilage in compression: theory and experiments. J Biomech Eng 102(Feb):73–84 Mridha M, Odman S (1986) Noninvasive method for the assessment of subcutaneous oedema. Med Biol Eng Comput 24(4):393–398

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Nanoparticles and Metal Foam in Thermal Control and Storage by Phase Change Materials

20

Bernardo Buonomo, Davide Ercole, Oronzio Manca, and Sergio Nardini

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Models and Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Results and Discussion for Main Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Cross-References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

861 864 874 880 881 882

Abstract

The solid-liquid phase change process realizes a buffer system to adsorb and then release heat loads. This property can be used in thermal control and in thermal energy storage. In the first case, it allows to have an assigned temperature range where the system works and to reject high heat loads, mainly when they are intermittent. In the second case, it is used to obtain a constant supply compared to a continuous variation of the consumption demand which leads to waste of excess energy. The phase change materials (PCMs) are materials employed for solidliquid phase change process. They present many advantages such as stability and high stored energy density. Nevertheless, the main drawback of these materials is the small value of the thermal conductivity, and it necessitates a long time for melting and implicates a broad difference of temperature in the system between

B. Buonomo (*) · D. Ercole · O. Manca · S. Nardini Dipartimento di Ingegneria Industriale e dell’Informazione, Università degli Studi della Campania “Luigi Vanvitelli”, Aversa (CE), Italy e-mail: [email protected]; [email protected]; oronzio. [email protected]; [email protected] # Springer International Publishing AG, part of Springer Nature 2018 F. A. Kulacki (ed.), Handbook of Thermal Science and Engineering, https://doi.org/10.1007/978-3-319-26695-4_39

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the solid zone and liquid zone. To overcome this drawback, improvement techniques are implemented to optimize the system like the inserting of metal foam or the addition of highly conductive nanoparticles. The new material created with nanoparticles in the base PCM is called nano-PCM. In the present chapter, the study of these systems will be analyzed numerically after a review of current literature. The governing equation models will be described in the cases of base PCM, nano-PCM, PCM, and nano-PCM in metal foam. Some results related to the main applications of the different systems will be provided in order to underline their advantages and disadvantages. Nomenclature

a Amush b c CF df dp E

Coefficient term (Eq. 33) Mushy zone costant (kgm
3 s
1) Parameter term (Eq. 33) Specific heat (Jkg
1 K
1) Inertial drag factor Ligament diameter (m) Pore diameter (m)
1 Specific enthalpy
(Jkg )

Fo

Fourier number

!

Gravity acceleration (ms
2) Sensible enthalpy (J) Local heat transfer coefficient (Wm
2 K
1) Enthalpy (J) Latent heat content (Jkg
1) Latent Heat of material (Jkg
1) Thermal conductivity (Wm
1 K
1) Permeability (m2)


g h hsf H ΔH HL k K Nui,d p Pr q Ra Red Ste !

S T Tsolidus Tliquidus !

V,V

kt2 ρcp L2R

Interstitial Nusselt number

hsf d f k

Relative pressure (Pa) μc  Prandtl number k p
 t
Δt Performance parameter Et
E HL
2  ρ cp gβΔTL3R Rayleigh number μk
 ρVd Ligament Reynolds number μ f
 c ΔT Stefan number pHL Source term (Eq. 2) Temperature (K) Solidus temperature (K) Liquidus temperature (K) Velocity, velocity vector (ms
1)

20

Nanoparticles and Metal Foam in Thermal Control and Storage by Phase Change. . . 861

Greek Symbols

αsf β γ ε η μ ξ ρ ω ψ

Surface Area density (m
1) Liquid fraction Thermal expansion coefficient (K
1) Porosity Dimensionless coordinate (x/LR) Viscosity of material (kgm
1 s
1) Dimensionless coordinate (y/LR) Density (kgm
3) Pore size (Pore per Inch (PPI)) Volume fraction of nano-PCM

Subscripts

0 eff m nanoparticles nanopcm pcm t 1-t

1

Operating Effective Metal foam Nanoparticle Nano-PCM Phase change material Time Time step

Introduction

One of the big issues of the thermal energy, in general, is to match the supply with the demand. Hence, it is required to build a thermal energy storage system (TESS) to prevent the demand picks or the lacks in supply, especially for the solar energy source. Moreover, for the thermal control, it is useful that the thermal storage system works at constant temperature, in order to avoid high temperatures that they could damage the downstream system. The use of phase change material (PCM) to store energy at constant temperature is the best choice of a thermal energy storage system for building. Principally, there are two types of thermal energy storage systems (TESS): sensible heat thermal energy storage system (SHTESS), which stores thermal energy by increasing the temperature of the material contained in the system (Gil et al. 2010), and latent heat thermal energy storage system (LHTESS), which uses the phase change materials (PCMs) to store thermal energy at nearly constant temperature; in fact, during the phase change process, the stored thermal energy leads to change phase and the temperature remains constant (Zhou et al. 2012). The thermal property responsible for the storing during the phase change is the latent heat. The LHTESS is better because it presents many advantages like the low range of the working temperature and high energy density. About the type of PCM employed, there are three types of phase change processes: solid-solid phase change process, solid-liquid phase change process, and liquid-gas phase change process

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(Cabeza 2014). In the solid-solid phase change process, the heat is stored during the crystallization and the material changes its own crystal lattice. The principal aspects are the low variation of volume during the phase change process and the small value of the latent heat. In the solid-liquid phase change process, the material melts during the storing of the heat and solidifies during the releasing of the heat. In this transformation, the latent heat in general is higher than the solid-solid transformation but the variation of volume is not low. The liquid-gas phase change process presents the highest value of latent heat but the volume variation is unacceptable because it leads to problems to the design of the storage system. Therefore, the solid-liquid phase change process is more suitable to store thermal energy because it represents the best compromise among these phase transformations. In this chapter, the PCMs considered are subjected to the solid-liquid transformation. There is not a unique classification of the solid-liquid PCMs, but generally the main subdivision is organic and inorganic materials, as hydrated salts (Khudhair and Mohammed 2004). The principal organic material is the paraffin, and it is widely used in the literature because it is stable, noncorrosive, nontoxic, and the phase change process is reversible (Abhat 1983). These characteristics are not sufficient to design a LHTESS because the worse disadvantage is the small value of thermal conductivity, leading to a long time for melting with respect to a working cycle and a huge temperature gradients in the system. Therefore, it is necessary to integrate the base PCM with another material like metal foam (Zhao et al. 2010) or injection of nanoparticles into PCM (Khodadadi and Hosseinizadeh 2007). In particular, the metal foams enhance the effective thermal conductivity of the whole system because they have a high value of conductivity and their structure is made to obtain a large area for the heat exchange between the PCM and the metal foam. Furthermore, the injection of nanoparticles improves the thermal conductivity of the PCM, thanks to the higher value of thermal conductivity of the nanoparticles. An in-depth research activity has been realized on the PCM for the thermal storage and thermal control. In particular, according to Agyenim et al. (2010), the first pioneering study on the PCM was accomplished in 1940s, but it was only after the energetic crisis that the scientific community began to research the use of PCM in thermal energy applications. Liu et al. (2013) numerically investigated the PCM and metal foams using the enthalpy porosity theory for the melting simulation and the local thermal nonequilibrium model for the energy equation. They found that the melting rate improved with the presence of the metal foam. An experimental study on the effective thermal conductivity was accomplished by Xiao et al. (2014) using open-cell metal foams impregnated with pure paraffin for latent heat storage. The metal foams employed are made of copper and nickel with high value of porosity and at assigned pore density. The results show that the effective thermal conductivity for the copper metal foam is 13, 31, and 44 times larger than that of pure paraffin for the porosities equal to 96.95%, 92.31%, and 88.89%, respectively. For the nickel foam the improvement of thermal conductivity is about three, four, and five times larger than that of pure paraffin for 97.45%, 94.24%, and 90.61%, respectively. An experimental investigation was made by Zhou and Zhao (2011) to analyze the heat transfer of PCM in metal foam for charging and discharging processes. They conclude that the heat transfer is

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Nanoparticles and Metal Foam in Thermal Control and Storage by Phase Change. . . 863

improved significantly and the melting time is lower of a quarter respects to the pure paraffin. Chen et al. (2014) compared an experimental and a numerical study on the melting of the PCM with metal foam showing a good agreement between the numerical model and the experimental sample. A numerical study by Chen et al. (2010) was performed on an integrated solar collector using paraffin and metal foam. The enthalpy porosity method is used for the simulation of the melting and the local thermal nonequilibrium (LTNE) model is employed for the heat transfer between PCM and metal foam; hence, there is no thermal equilibrium between the paraffin and metal foam. The results show that the heat transfer is improved and the temperature field of paraffin with aluminum foam is more uniform with respect to that of the pure paraffin. An analogous investigation was carried out by Nithyanandam and Pitchumani (2014) where a LTES system for concentrating solar power system was studied both charging and discharging process of the system at different arrangements of heat pipes and the parameters of metal foam. An experimental study is accomplished to analyze the heat transfer in a copper foam with the paraffin wax RT58 as PCM by Zhao et al. (2010) in order to evaluate the improvement of metal foams for low-temperature thermal energy storage systems. Moreover, a numerical simulation is made in two-dimensional domain using the LTNE model in order to compare the experimental data with the numerical results. About the rectangular geometry, a numerical study is carried out by Krishnan et al. (2005) to evaluate the phase change of the PCM in a metal foam using the LTNE assumption. Yang et al. (2013) studied a three-dimensional model of a thermal energy storage system with PCM in metal foam and fins. Forchheimer-Darcy law was employed to simulate the porous metal foam and the local thermal equilibrium (LTE) model was used to evaluate the temperature field. A graphite foam with magnesium chloride as PCM was numerically investigated by Zhao et al. (2014) for a high-temperature LHTES system in a three-dimensional model. The equivalent heat capacity method was used for the phase change process, and an energy efficiency parameter was defined to assess the performance of the system. The results showed that the graphite foam improves the heat transfer rate and the efficiency parameter from 68% to 97% without and with foam, respectively. A numerical study on the pore size and the effect of porosity on the behavior of the PCM in metal foam was accomplished by Sundarram and Li. (2014) with a 3D model and LTNE assumption. A heat sink based on PCM was numerically and experimentally investigated for application in thermal management of electronic devices by Kandasamy et al. (2008). The PCM-based heat sink was placed on a quad flat package electronic device (QFP), and the results were developed and then compared with those of a pure heat sink without PCM. These results showed that the cooling performances are increased when the PCM is present only for high input power levels. Kalbasi and Salimpour (2015) investigated numerically a 2D model of a PCM-based heat sink for the thermal control of an electronic device to achieve the best design for optimization of the cooling system. A similar study was accomplished by Jaworski (2012) on thermal performance of a heat spreader with PCM, where the presence of PCM inside the fins provides a lower thermal resistance with respect to PCM-based heat sink reported in the work of Kandasamy et al. (2008).

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A pioneering numerical study on nano-PCM was proposed by Khodadadi and Hosseinizadeh (2007). They argued that the nanoparticles in the PCM improve the thermal conductivity of the system and this new improved technique can be very promising for the future applications. Tasnim et al. (2015) analytically investigated the melting process for a nano-PCM inside the porous metal foam enclosed in a rectangular box. The nano-PCM is assumed to behave like a nanofluid and the thermal field was calculated by the single-phase model, and for the porous media the Darcy model was employed. The scale analysis was employed to obtain relationships among nondimensional parameters, and a numerical simulation was made to verify the correctness of the scale analysis. The results showed that the phase change of the nano-PCM passes through four steps, since the phenomenon of heat transfer initially is by conduction and lastly is by convection. Sebti et al. (2013) numerically studied the melting process of a nano-PCM in a 2D square cavity assuming the single-phase model for the nano-PCM and the phase change process with the enthalpy-porosity theory. They concluded that the adding of the nanoparticles to the base PCM improves the heat transfer, thanks to the increase of the thermal conductivity of the nano-PCM. Nevertheless, an increase of the viscosity is visible leading to a decrease of the velocity. Harikrishnan et al. (2014) experimentally studied the behavior of a hybrid nano-PCM with paraffin wax, as base PCM, and nanoparticles at 1.0% in mass fraction, with 50% CuO and 50% TiO2. They compared the stability, thermal conductivity, and viscosity between the pure PCM and the nano-PCM. They concluded that the nanoparticles delay the decomposition of the paraffin, and therefore the nano-PCM is more stable than the pure PCM. Moreover, the heating and cooling rates of the nano-PCM are faster than paraffin; in particular, the melting time is reduced by 29.8% and the cooling time by 27.7%. Hossain et al.’s (2015) is the first study related to the use of the nano-PCM inside a metal foam. They numerically studied the effect of the volume fraction of the nanoparticles and the porosity of the metal foam on the evolution of temperature, melt fraction, and the heat transfer rate. They observed that the melting front is more affected by the metal foam rather than the nanoparticles; in particular, the lower porosities increase the melting rate and even the higher volume fractions. Wu et al. (2012) numerically investigated the melting process for a nano-PCM of Cu/paraffin at 1% of volume fraction. The results showed that the melting time of the paraffin is reduced by 13.1%. In Table 1 is reported some information on the different studies previously reviewed.

2

Models and Governing Equations

In this paragraph are described the models employed in the previous reviewed works. There are mainly four models to describe the melting of the PCM or nanoPCM: the enthalpy-porosity method, the temperature-based method, the equivalent heat capacity method, and the thermal lattice Boltzmann method. Briefly the temperature-based method was used by Tasnim et al. (2015) and by Hossain et al. (2015) for the nano-PCM and Zhao et al. (2010) for PCM. In these cases, there is a

Copper

Not specified (579.15–580.15 K)

–

Copper

Dimensionless

–

Stainless steel

Dimensionless

Nithyanandam and Pitchumani (2014) Zhao et al. (2010)

Aluminum

Krishnan et al. (2005) Yang et al. (2013)

Temperaturebased Enthalpy porosity Enthalpy porosity

–

Aluminum

–

–

–

Thermal lattice Boltzmann Equivalent heat capacity Enthalpy porosity

–

Copper

Paraffin (333–343 K) 56% Li2CO3 + 44% Na2CO3 Paraffin RT58

Zhou and Zhao (2011) Chen et al. (2014)

Chen et al. (2010)

–

–

Copper nickel

Paraffin (60–62  C) Paraffin RT27/ CaCl2•6H20 salt Paraffin (55–60  C)

Xiao et al. (2014)

Nanoparticle –

Metal foam Aluminum

PCM Paraffin RT58

References Liu et al. (2013)

Theoretical method for melting Enthalpyporosity –

Table 1 Studies on thermal control and energy storage by PCM and nano-PCM in metal foams

LTE

LTE/LTNE

LTNE

LTE

LTNE

LTE

–

–

Theoretical method for porous media LTNE

–

–

–

–

–

–

–

–

Theoretical method for the nanoparticle –

(continued)

Numerical

Experimental/ numerical Numerical

Numerical

Numerical

Numerical/ experimental

Experimental

Experimental

Experimental or numerical Numerical

20 Nanoparticles and Metal Foam in Thermal Control and Storage by Phase Change. . . 865

–

Aluminum

–

Paraffin (60–62  C)

Not specified (273.15 K)

Cyclohexane

Paraffin (329–333 K)

Harikrishnan et al. (2014)

Khodadadi and Hosseinizadeh (2007) Hossain et al. (2015) Wu et al. (2012)

–

–

Paraffin (18  C)

Copper oxide Copper

Copper oxide Copper oxide 50% CuO + 50% TiO2 Copper

–

–

Lauric acid

Aluminum

–

–

Cyclohexane

–

–-

Tasnim et al. (2015) Sebti et al. (2013)

–

Nanoparticle –

Aluminum

Sundarram and Li (2014) Kandasamy et al. (2008) Kalbasi and Salimpour (2015) Jaworski (2012)

Metal foam Graphite

Paraffin (321–335 K) Paraffin (53–57  C) Paraffin RT27

PCM MgCl2

References Zhao et al. (2014)

Table 1 (continued)

Temperaturebased –

Enthalpy porosity

Theoretical method for melting Equivalent heat capacity Equivalent heat capacity Enthalpy porosity Enthalpy porosity Enthalpy porosity Temperaturebased Enthalpy porosity –

–

LTE

–

–

–

LTE

–

–

–

LTE

Theoretical method for porous media –

Single phase

Single phase

Single phase

–

Single phase

Single phase

–

–

–

–

Theoretical method for the nanoparticle –

Numerical/ experimental

Numerical

Numerical

Experimental

Numerical

Numerical

Numerical

Numerical

Numerical

Numerical

Experimental or numerical –

866 B. Buonomo et al.

20

Nanoparticles and Metal Foam in Thermal Control and Storage by Phase Change. . . 867

clear difference between the solid region and liquid region, and a liquid-solid interface is explicitly traced. There are two energy equations, respectively, for the liquid and solid part of the domain plus an additional equation to balance the energy transfer between the liquid and solid domain at the interface. Moreover, the position of the melting front is related to the temperature function by applying the energy conservation law at the interface zone. The equivalent heat capacity method modifies the value of the heat capacity in the energy equations in base of the latent heat of fusion and the melting temperature. The heat capacity is represented as a Heaviside step along the temperature, and the viscosity is function of the temperature to simulate the phase change from solid to fluid phase. The lattice Boltzmann method is used to simulate the behavior of the PCM in a work by Chen et al. (2014). In this case, the Navier-Stokes equations are not present and fluid is considered as a set of virtual particles whose interactions and collisions among themselves allow to simulate the viscosity and other properties of the fluid. Finally, the most important method to simulate the phase change is the enthalpy-porosity method. This method was introduced by Voller and Prakash (1987). In this method, the solid-liquid interface is not explicitly traced but a mushy region is defined and is described as a “pseudo” porous zone. In this zone a new parameter is introduced, called liquid fraction, β, defined as the fraction of liquid form in a cell of the mushy zone (Al-abidi et al. 2013). The similitude of the mushy zone with a porous media is justified by the fact that when the liquid fraction tends to zero, the mixed region is solidifying and the velocity tends to decrease. The value of the liquid fraction is zero when the zone is fully solid, 1 when it is fully liquid, and between 0 and 1 in the mixed region: 8 β¼0 for T < T SOLIDUS > < T
T SOLIDUS β¼ for T SOLIDUS < T < T LIQUIDUS T LIQUIDUS
T SOLIDUS > : β¼1 for T > T LIQUIDUS

(1)

where T is the local temperature and Tsolidus and Tliquidus are, respectively, the temperature above which the PCM is liquid and below which the PCM is solid. The region of melting lies in a range of temperature between Tliquidus and Tsolidus. Moreover, an additional term depending on the liquid fraction is added to the momentum equation to take account of the presence of the solid part in the mushy zone. For the phase change problem, the energy equation is written in term of enthalpy:
!  @ ðρH Þ þ ∇  ρ V H ¼ ∇  ðk∇T Þ þ S @t

(2)

where H is the mixture enthalpy of the material, defined as the sum of the sensible enthalpy, h, and the latent heat content ΔH: H ¼ h þ ΔH

(3)

h ¼ cT

(4)

ΔH ¼ βH L

(5)

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where c is the mixture specific heat of the material, T is the temperature, HL is the latent heat of the material, ρ is the density, V is the fluid velocity of the material in the liquid phase, k is the mixture thermal conductivity, and S is a source term due to other phenomena. To simulate the heat transfer between the PCM and the metal foam, two models are considered. The first model is the LTE where the local temperature is the same between the PCM and the metal foam. The second one is the local thermal nonequilibrium (LTNE) where PCM and metal foam do not have the same local temperature and two energy equations should be written, one for the PCM and the other one for the metal foam (Liu et al. 2013). About the addition of the nanoparticles in the PCM, there are mainly two models to simulate the interaction between the PCM and the nanoparticle: single-phase and dual-phase models. In the single-phase model, the domain is assumed continuous and there is no difference between the base fluid and the nanoparticles, and the thermal properties of the nano-PCM are calculated as the weighted average between the properties of the nanoparticles and the properties of the PCM. In the dual-phase model, there is a difference between the fluid and the nanoparticles and a possible slip velocity is not equal to zero due to several factors such as friction between the fluid and particles, gravity, Brownian forces, and sedimentation. This method is more accurate but the computational costs are very higher. It seems that, in literature, there are no papers about the nano-PCM employing the dual-phase in governing equations. In the following, the governing equations are described using the single-phase model. The governing equations are written for pure PCM or Nano-PCM without metal foam using the enthalpy-porosity method and for pure PCM or nano-PCM with metal foam in both models (LTE and LTNE). The metal foam is assumed isotropic and homogenous, and it is described as an open porous media which follows the Darcy-Forchheimer law. The density of the PCM or nano-PCM follows the Boussinesq approximation and the other properties are assumed constant. Under these assumptions, the equations of continuity, momentum, and energy are: 1. Pure PCM or nano-PCM without metal foam • Continuity equation !

∇ V ¼ 0

(6)


! ! ! ! @ρV ! þ ∇  ρV V ¼
∇ p þμ∇2 V þ S @t

(7)

• Momentum equation !

• Energy equation
! 
!  @ρH þ ∇  ρ V H ¼ ∇  k ∇T @t

(8)

20

Nanoparticles and Metal Foam in Thermal Control and Storage by Phase Change. . . 869

In the momentum equation, with the Boussinesq approximation, the density of the PCM of nano-PCM is given as: ρ ¼ ρ0 ½ 1
γ ð T
T 0 Þ 

(9)

where μ is the viscosity of the PCM or nano-PCM, p is the relative pressure and S is a source term given as (Nithyanandam and Pitchumani 2014): !

!

S ¼ ρ g γ ðT
T 0 Þ þ 

ð1
βÞ2 3

β þ 0:001

!

3 Amush V

(10)

The first term is the Boussinesq approximation term, which simulates the presence of the buoyancy forces, and g is the gravity, T0 is the reference temperature and γ is the thermal expansion coefficient of the PCM or nano-PCM. The second term is the Karman-Koseny term, which models the presence of the solid part in the mushy region, where Amush is the mushy zone constant which represents the damping of the velocity to zero during the solidification. The energy equation uses the enthalpy of the material according to enthalpy-porosity method and the thermal conductivity is considered constant. 2. Pure PCM or nano-PCM with metal foam • Continuity equation !

∇ V ¼ 0

(11)


! ! ! ! @ρV ! þ ∇  ρV V ¼
∇ p þμ∇2 V þ S @t

(12)

• Momentum equation !

• Energy equation • LTE ρc

  @T ! ! @T þ V  ∇T ¼ keff ∇2 T
eρpcm H L @t @t

(13)

• LTNE

! ! @T pcm eðρcÞpcm þ ðρcÞpcm V  ∇T pcm @t   @T ¼ kpcm, eff ∇2 T pcm þ hsf αsf T pcm
T m
eρpcm H L @t

(14)

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ð1
eÞðρcÞm

  @T m ¼ km,eff ∇2 T m þ hsf αsf T m
T pcm @t

(15)

In the momentum equation the source term S includes the following terms (Nithyanandam and Pitchumani 2014): ! ð1
βÞ2 μ ! CF ! ! ! V þ pffiffiffiffi ρ V V S ¼ ρ g γ ðT
T 0 Þ þ  A V þ  mush 3 K K β3 þ 0:001

!

(16)

Governing equations of the considered models are summarized in Table 2. The first and second terms are the same as those of the previous equation. The third term is the Darcy term where K is the permeability of the porous media and the last term is the Forchheimer extension term, where CF is inertial or drag factor. The permeability and drag factor values can be calculated using the relations presented by Calmidi and Mahajan (2000): K ¼ 0:00073ð1
eÞ


0:224

CF ¼ 0:00212ð1
eÞ

 
1:11 df d2p dp

(17)

 
1:63 df dp

(18)


0:132

where e is the porosity of the metal foam, defined as the ratio between the void volume inside the metal foam to the total volume, df is the ligament diameter and dp is the pore diameter of the foam. These parameters are related to in a work by Calmidi and Mahajan (2000): df ¼ 1:18 dp

rffiffiffiffiffiffiffiffiffiffiffi  1
e 1 3π 1
e
ð1
eÞ=0:04 dp ¼

0:0224 ω

(19) (20)

where ω is the pore density of the metal foam which represents the number of pores across 1 in. In the energy equations, the product ρc is evaluated as the weighted average of the densities of metal foam and PCM: ρc ¼ ð1
eÞρm cm þ eρpcm cpcm

(21)

where ρpcm and cpcm are, respectively, the density and the specific heat of the PCM or nano-PCM while ρm and cm are, respectively, the density and specific heat of the metal foam. keff is the effective thermal conductivity calculated by: keff ¼ ð1
eÞkm þ ekpcm

(22)

Energy

Boussinesq approximation Source term in Eqs. 10 and 16

Momentum

Equation Continuity

ðβ3 þ0:001Þ

ð1
βÞ2 3

(10)

Amush V

!

(8)

3

ð1
βÞ2

! μ ! CF ! ! 3 Amush V þ V þ pffiffiffiffi ρ V V K K β þ 0:001

(9)

Pure PCM or nano-PCM with metal foam LTNE

! ! @ T pcm þ ðρcÞpcm V  ∇T pcm ¼ eðρcÞpcm @t   @T 2 kpcm, eff ∇ T pcm þ hsf αsf T pcm
T m
eρpcm HL @t (14)   ð1
eÞðρcÞm @@Ttm ¼ km, eff ∇2 T m þ hsf αsf T m
T pcm (15)

(13)

Pure PCM or nano-PCM with metal foam LTE
! !  ρc @@ Tt þ V  ∇T ¼ keff ∇2 T
eρpcm H L @@ Tt

!

S ¼ ρ g γ ðT
T 0 Þ þ 

!

ρ = ρ0[1
γ(T
T0)]

(7) and (12)

(6) and (11)

Pure PCM or nano-PCM with metal foam

∇ V ¼ 0
! ! ! ! ! þ ∇  ρV V ¼
∇ p þμ∇2 V þ S

PCM or nano-PCM without metal foam
! 
!  @ ρH @ t þ ∇  ρV H ¼ ∇  k ∇T

!

S ¼ ρ g γ ðT
T 0 Þ þ

!

!

@ ρV @t

!

PCM or nano-PCM without metal foam

Table 2 Governing equations for pure PCM and nano-PCM without and with metal foam

(16)

20 Nanoparticles and Metal Foam in Thermal Control and Storage by Phase Change. . . 871

872

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where km and kpcm are, respectively, the thermal conductivities of metal foam and PCM or nano-PCM. HL is the latent heat of the PCM or nano-PCM. In the energy equations for LNTE model, kpcm,eff and km,eff are, respectively, the effective thermal conductivity of the PCM and metal foam, and the explanation of these parameters is found in Boomsma and Poulikakos (2001) and are not reported for brevity; αsf is the surface area density that represents the whole contact area between the metal foam and PCM and hsf is the local heat transfer coefficient. Tm is the temperature of the metal foam while Tpcm is the PCM temperature or nano-PCM temperature. In this model, Tpcm and Tm are not equal and the heat is transferred between the metal foam and the PCM by means of the product hsf αsf. To evaluate the products αsf and hsf, the relations of Calmidi and Mahajan (2000) are used: αsf ¼ 

3πdf 0:59d p


 ð1
eÞ
2 1
e 0:04

(23)

and for hsf:  k  f 0:37 0:76Re0:4 Pr , 1  Red  40 d pcm df
 k  f 0:5 0:37 0:52Red Pr pcm , 40  Red  1000 hsf ¼ > d f > >   >
 k > > f > 0:37 : 0:26Re0:6 , 1000  Red  2  105 d Pr pcm df 8 > > > > > > >
> 0:76Re0:4 ,1  Red  40 Pr > d pcm > df > > >  

df > >
  >  k > > f 0:37 > : 0:26Re0:6 , 1000  Red  2  105 d Pr pcm df (24)

Reynolds number referred to ligament diameter

Red ¼

ρVd f μ

(25)

the ratio between the volume of the nanoparticles and the total volume of the nano-PCM: ψ¼

V NANOPARTICLES V NANOPCM

(29)

The viscosity is calculated by Brinkman (1949): μNANOPCM ¼

μPCM ð1
ψ Þ2:5

(30)

and the thermal conductivity is calculated from the Maxwell equation (1873): kNANOPCM ¼ kPCM

kNANOPARTICLES þ 2kPCM
2ψ ðkPCM
kNANOPARTICLES Þ kNANOPARTICLES þ 2kPCM þ ψ ðkPCM
kNANOPARTICLES Þ

(31)

where μNANOPCM is the viscosity of the nano-PCM and kNANOPCM is the thermal conductivity of the nano-PCM. The values do not change significantly at varying of the volume fraction. The latent heat of the nano-PCM is evaluated using Khodadadi and Hosseinizadeh (2007) equation:

874

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Table 4 Evaluation of nano-PCM properties, as reported by Sebti et al. (2013) Property Volume fraction of the nanoPCM Density Specific heat Thermal expansion coefficient Viscosity

Equation (29) ψ ¼ V NANOPARTICLES V NANOPCM ρNANOPCM = (1
ψ)ρPCM + ψρNANOPARTICLES (26) (ρc)NANOPCM = (1
ψ)(ρc)PCM + ψ(ρc)NANOPARTICLES (27) (ργ)NANOPCM = (1
ψ)(ργ)PCM (28) μPCM (30) μNANOPCM ¼ ð1
ψ Þ2:5

Thermal conductivity

þ2kPCM
2ψ ðkPCM
kNANOPARTICLES Þ kNANOPCM ¼ kPCM kkNANOPARTICLES NANOPARTICLES þ2k PCM þψ ðk PCM
k NANOPARTICLES Þ (31)

Latent heat of the nano-PCM

ðH L ÞNANOPCM ¼

ðH L ÞNANOPCM ¼

ð1
ψ Þ ðρH L ÞPCM ρNANOPCM

(32)

ð1
ψ Þ ðρH L ÞPCM ρNANOPCM

(32)

The equations related to the evaluation of thermophysical properties of nanoPCM are summarized in Table 4. The melting temperature is the same for the base PCM and the nano-PCM. The temperature field is resolved iterating the energy equation with the updated liquid fraction up to the convergence of the energy equation. This method is developed by Voller and Prakash (1987). In this method, a fully implicit time scheme with upwind difference approach is used, for example, about the energy Eq. 2, the method of Pantankar (1980) gives: t
1 atP htP ¼ aH hH þ aL hL þ aN hN þ aS hS þ at
1 P hP þ b

(33)

where a represents the coefficient terms, subscripts indicate the nodal value and they are represented in Fig. 1, while t-1 represents the previous time step. The parameter b discretizes the eventual source term S in Eq. 2. The finite equations are solved by the SIMPLE algorithm. A complete explanation of the numerical algorithm is given by Voller and Prakash (1987). A grid independence test is performed for each model, conducting various simulations using different mesh sizes. The test results are compared in order to establish the best mesh size that represents a good compromise between the computational cost and the accuracy. These tests are showed in Table 5. The relative error is the difference between the chosen mesh and the finest mesh.

3

Results and Discussion for Main Contributions

In this section various numerical results for different geometry, boundary condition, and other characteristics are compared in order to understand the benefits of the PCM. Independent grid tests have been made on every model to check the quality of

20

Nanoparticles and Metal Foam in Thermal Control and Storage by Phase Change. . . 875

a

300000

Total energy (J)

250000 200000 150000 Porosity=0.95,Tin=373 K,Vin=10 m/s Porosity=0.90,Tin=373 K,Vin=10 m/s Porosity=0.85,Tin=373 K,Vin=10 m/s Porosity=0.95,Tin=353 K,Vin=10 m/s Porosity=0.95,Tin=373 K,Vin=8 m/s

100000 50000 0

3000

2000 t (s)

1000

b

4000

Liquid fraction (-)

1.0 0.8 0.6 0.4 Porosity=0.95,Tin=373 K,Vin=10 m/s Porosity=0.90,Tin=373 K,Vin=10 m/s Porosity=0.85,Tin=373 K,Vin=10 m/s Porosity=0.95,Tin=353 K,Vin=10 m/s Porosity=0.95,Tin=373 K,Vin=8 m/s

0.2 0.0 0

1000

2000 t (s)

c 1.0

4000

PPI=10 PPI=30 PPI=60

0.8 Liquid fraction (-)

3000

Porosity=0.95

0.6

0.95 0.90

0.4

0.85

0.2

0.75

0.80

850

900

950

1000

0.0 0

250 500 750 1000 1250 1500 1750 2000 2250 Time (s)

Fig. 1 Total energy (a) and liquid fraction (b) at varying porosities or inlet conditions; (c) liquid fraction at varying PPI (Reprinted from Liu et al. 2013, Copyright # 2013, with permission from Elsevier Ltd.)

876

B. Buonomo et al.

Table 5 Grid independence tests for each model Model Liu et al. (2013)

Krishnan et al. (2005) Yang et al. (2013)

Tasnim et al. (2015) Sebti et al. (2013)

Hossain et al. (2015)

Grid sizes 20,000 elements 30,000 elements 50,000 elements 48  48 102  102 186  186 7.1414 * 105, elements 1.2648 * 106, elements 1.5235 * 106 elements Not reported 41  41, 51  51, 71  71, 91  91 101  101 Not reported

Physical quantity Melting time

Chosen mesh 30,000 elements

Relative error (%) 3.9

Melting time

102  102

2

Melting time

1.2648 * 106, elements

50 the convection phenomenon is clear because the isothermal lines are inclined. Moreover, the volume fraction of the nanoPCM does not affect the evolution of melting for low value of Ra (minus of 12.5). It also studied the effect of the nanoparticle on the fluid motion; in particular, there is no variation in the streamline pattern when the nanoparticles are added to the PCM,

880

B. Buonomo et al. 10 y =0 y =0.025

8

y =0.05 T –T melting (°C)

Fig. 6 Variation of the temperature along a central line in a square domain at assigned time for different volume fractions Φ (Reprinted from Sebti et al. 2013, Copyright # 2013, with permission of Springer)

6

4

2

0

0

0.002

0.004 0.006 x (m)

0.008

0.010

but the stream function values decrease, because there is a reduction of the liquid PCM velocity due to the increase of the viscosity and the density of the nano-PCM. This leads to an increase of the resistance to convection and also a decrement of the melting process. Sebti et al. (2013) studied the effect of the volume fraction of the nanoparticles on the melting rate. In Fig. 6 the variation of the temperature for various volume fractions of nanoparticles along the central line of a square domain at an assigned time is displayed. It can be seen that the addition of the nanoparticles enhances the heat transfer, thanks to the increase of the thermal conductivity of the nano-PCM. Nevertheless, the presence of the nanoparticles increases the viscosity and therefore the velocity convection is lower. For example, the nano-PCM at 5% of volume fraction has a melting time 15% lower respect to the pure PCM. Finally, the study of Hossain et al. (2015) compares the benefits of the metal foam with the nanoparticles, varying the porosity of the metal foam or the volume fraction of the nanoparticles. They argue that the change of the volume fraction has a small influence compared to the variation of the porosity. Results of different models are summarized in Table 6.

4

Conclusions

In general, the PCMs have many benefits in various applications such as the thermal storage or thermal control, but it is necessary to combine the phase change materials with other materials in order to eliminate their drawbacks, especially the low value of the thermal conductivity. The metal foams and the nanoparticles are two improvements that can optimize the PCM-based systems. The viewed literature has shown

20

Nanoparticles and Metal Foam in Thermal Control and Storage by Phase Change. . . 881

Table 6 Studies on thermal control and energy storage by PCM and nano-PCM in metal foams References Liu et al. (2013)

Geometry Cylindrical

Enhancement Metal foam

Krishnan et al. (2005)

Rectangular

Metal foam

Yang et al. (2013)

3D parallelepiped

Metal foams Fins

Tasnim et al. (2015)

Rectangular

Nanoparticles

Sebti et al. (2013) Hossain et al. (2015)

Rectangular Rectangular

Nanoparticles Nanoparticles Metal foam

Operating parameters Porosity Pore size Inlet HTF velocity Rayleigh number Stefan number Porosity Pore size Rayleigh number Volume fraction Rayleigh number Volume fraction Volume fraction Porosity

that the improvements depend on the characteristics of the metal foam and the nanoparticles. In particular, it is possible to affirm the following conclusions: The porosity of the metal foam has an influence on the melting rate and on the stored energy; when the value of the porosity is low then the melting rate is faster and the stored energy is lower. This can be explained by the Eq. 22, where the low porosity implies higher effective thermal conductivity but there is a reduction of the PCM inside the domain and therefore the stored energy is smaller. The increase of the pore density of the metal foam accelerates the melting process, due to the increase of the value of the surface area between the PCM and metal foams, because higher pore density means smaller pore diameter and this leads to higher value of the surface area density, as it can be noted in the Eq. 20. In this way, there is more area to transfer the heat between the foam and the PCM and therefore the melting rate is faster. The volume fraction of the nanoparticles influences positively the melting rate but at the same time reduces the velocity of the liquid PCM, reducing the convection phenomenon. This can be seen only for high value of Rayleigh number. Finally, it is important to remark the Rayleigh number effect on the melting process of the PCM, because for low value the conduction phenomenon is predominant, whereas for high value (Ra > 50) the convection begins to be significant.

5

Cross-References

▶ Compact Heat Exchangers ▶ Design of Thermal Systems ▶ Energy Efficiency and Advanced Heat Recovery Technologies ▶ Enhancement of Convective Heat Transfer ▶ Free Convection: Cavities and Layers ▶ Heat Transfer Media and Their Properties ▶ Macroscopic Heat Conduction Formulation

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▶ Numerical Methods for Conduction-Type Phenomena ▶ Phase Change Materials ▶ Thermal Transport in Micro- and Nanoscale Systems ▶ Thermophysical Properties Measurement and Identification

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Maxwell JC (1873) Treatise on electricity and magnetism. Clarendon, Oxford Nithyanandam K, Pitchumani R (2014) Computational studies on metal foam and heat pipe enhanced latent thermal energy storage. J Heat Transf 136:051503 Pantankar SV (1980) Numerical heat transfer and fluid flow. Hemisphere, Washingtonm, DC Sebti SS, Mastiani M, Mirzaei H, Dadvand A, Kashani S, Hosseini SA (2013) Numerical study of melting of nano-enhanced phase change material in a square cavity. JZUS-A 14(5):307–316 Sundarram SS, Li W (2014) The effect of pore size and porosity on thermal management performance of phase change material infiltrated microcellular metal foams. Appl Therm Eng 64:147–154 Tasnim SH, Hossain R, Mahmud S, Dutta A (2015) Convection effect on the melting process of nano-PCM inside porous enclosure. Int J Heat Mass Transf 85:206–210 Voller VR, Prakash C (1987) A fixed grid numerical modelling methodology for convectiondiffusion mushy region phase-change problems. Int J Heat Mass Transf 30:1709–1719 Wu S, Wang H, Xiao S, Zhu D (2012) Numerical simulation on thermal energy storage behavior of Cu/paraffin nanofluids PCMs. In: International conference on advances in computational modeling and simulation, vol 31, pp 240–244 Xiao X, Zhang P, Li M (2014) Effective thermal conductivity of open-cell metal foams impregnated with pure paraffin for latent heat storage. Int J Therm Sci 81:84–105 Yang J, Du X, Yang L, Yang Y (2013) Numerical analysis on the thermal behavior of high temperature latent heat thermal energy storage system. Sol Energy 98:543–552 Zhao CY, Lu W, Tian Y (2010) Heat transfer enhancement for thermal energy storage using metal foams embedded within phase change materials (PCMs). Sol Energy 84:1402–1412 Zhao W, France DM, Yu W, Kim T, Singh D (2014) Phase Change Material with graphite foam for applications in high-temperature latent heat storage systems of concentrated solar power plants. Renew Energy 69:134–146 Zhou D, Zhao CY (2011) Experimental investigations on heat transfer in phase change materials (PCMs) embedded in porous materials. Appl Therm Eng 31:970–977 Zhou D, Zhao CY, Tian Y (2012) Review on thermal energy storage with phase change materials (PCMs) in building applications. Appl Energy 92:593–605

Modeling of Heat and Moisture Transfer in Porous Textile Medium Subject to External Wind: Improving Clothing Design

21

Nesreen Ghaddar and Kamel Ghali

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Review of Fabric Heat, Air, and Water Vapor Transport Models . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Fabric Physical Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Diffusive and Convective Fabric Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Mathematical Formulation of Thin Fabric Model for Clothing Ventilation Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Integration of Thin Fabric Model with Segmental Clothed Human Thermal Model . . . . . . 4.1 Clothed Cylinder Model of Independent Body Segments . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Bio-Heat Model Integration and Overall Clothing Ventilation . . . . . . . . . . . . . . . . . . . . . . 4.3 Connected Clothed Cylinders Model to Improve Clothing Ventilation Predictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Effect of Walking on Ventilation and the Clothed Swinging Arm Model . . . . . . . . . . . 5 Closing Remarks and Future Trends . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Cross-References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

888 888 889 892 894 897 899 904 906 911 913 914 914

Abstract

This chapter covers convective modeling approaches of heat and moisture transfer in textile materials coupled with human thermal response models. Fabrics are highly porous and relatively thin materials consisting mainly of solid fiber, adsorbed water vapor, and gaseous mixture of water vapor and air in the void space. Fabric ventilation is induced by external wind or body motion which causes the air to penetrate the fabric and transfer heat and water vapor away from the human skin to the environment.

N. Ghaddar (*) · K. Ghali Department of Mechanical Engineering, Faculty of Engineering and Architecture, American University of Beirut, Beirut, Lebanon e-mail: [email protected]; [email protected] # Springer International Publishing AG, part of Springer Nature 2018 F. A. Kulacki (ed.), Handbook of Thermal Science and Engineering, https://doi.org/10.1007/978-3-319-26695-4_40

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Convective fabric models are developed to predict local and overall clothing ensemble ventilation rates. This modeling approach is combined with segmental bio-heat model to predict human local and overall comfort in hot humid environment. The integration of clothing ventilation models with “cylindrical” segments of the clothed human body is presented showing examples of how segmental and inter-segmental ventilation, sensible heat loss, and moisture transport through clothing are used to assess the whole body comfort. Nomenclature

Af Ai Ao Ca Cf Cin Cp Cv D ef g had H0 ci H0 co hc(f-1) hc(o-air) hc(skin-air) hfg H0 mi H0 mo hm(f-1) hm(o-air) hm(skin-air) im ka

Area of the fabric (m2) Inner node area in contact with the outer node (m2) Outer-node exposed surface area to air flow (m2) Gas concentration in the micro-climate measurement location (m3 Ar /m3 air) Fiber specific heat (J/kg K) Gas concentration in the distribution system (m3 Ar /m3 air) Specific heat of air at constant pressure (J/kg
K) Specific heat of air at constant volume (J/kg
K) Water vapor diffusion coefficient in air (m2/s) Fabric thickness (m) Gravitational acceleration (m/s2) Heat of adsorption (J/kg) Normalized conduction heat transfer coefficient between inner node and outer node (W/m2
K) Normalized convection heat transfer coefficient between outer node and air flowing through fabric (W/m2
K) Heat transport coefficient from the fabric to the environment (W/m2
K) Heat transport coefficient from the fabric to the trapped air layer (W/m2
K) Heat transport coefficient from the skin to the trapped air layer (W/m2
K) Heat of vaporization of water (J/kg) Normalized diffusion mass transfer coefficient between inner node and outer node (kg/m2
kPa
s) Normalized mass transport coefficient between outer node and air void in the fabric (kg/m2
kPa
s) Mass transfer coefficient between the fabric and the environment (kg/m2
kPa
s) Mass transfer coefficient between the fabric and the air (kg/m2
kPa
s) Mass transfer coefficient between the skin and the air layer (kg/m2
kPa
s) Permeability index Thermal conductivity of air (W/m
K)

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Modeling of Heat and Moisture Transfer in Porous Textile Medium Subject to. . .

Linearized radiative heat transfer (W/m2
K) Fabric length in z direction (m) Mass flow rate of air in radial direction (kg/m2
s) Mass flow rate of air in angular direction(kg/m2
s) Mass flow rate of air in axial direction (kg/m2
s) Total ventilation rate (kg/m2
s) Pressure of the microclimate air (kPa) Pressure at the external surface of the fabric (kPa) Temperature of the microclimate air ( C) Ambient temperature ( C) Skin temperature ( C) Temperature of the fabric outer layer ( C) Temperature of the fabric void layer ( C) Heat loss (W/m2) Total regain in fabric (kg of adsorbed H2O/kg fiber) Fabric dry resistance (m2
K/W unless specified in the equation per mm of thickness) Fabric evaporative resistance (m2
kPa/W) The dynamic resistance for ventilation through the fabric (m2
K/W) Fabric cylinder radius (m) Segment cylinder radius (m) Liquid movement velocity (m/s) Velocities of the environment cross wind (m/s) Humidity ratio (kg of water/kg of air) Air layer thickness (m) Coordinate in vertical direction (m)

hr L m_ aY m_ aθ m_ aZ m_ a Pa Ps Ta Tamb Tskin To Tv Q R RD RE Rdynamic Rf Rs V V1 w Y z

Greek Symbols

α β μ θ ε ρa

Fabric air permeability (m3/m2
s) The volumetric thermal expansion ( C1) Viscosity of air (N
s/m2) Angular coordinate Porosity of fabric Density of air (kg/m3)

Subscripts

a fabric o void skin 1

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Conditions of air in the annulus Fabric Fabric outer node Fabric void node Conditions at the skin surface Environment condition

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Introduction

Modeling transient mechanisms of coupled heat and moisture transfer into textile materials has been a topic of interest for many decades since clothing is a crucial parameter in determining human thermal comfort. In fact, the thermal comfort status of people is largely affected by the way the clothing mediates the flow of heat and moisture from the human skin to the environment through the mediating microclimate air layer between trapped between skin and fabric. Fabrics are highly porous and relatively thin materials consisting mainly of solid fiber, adsorbed water vapor, and gaseous mixture of water vapor and air in the void space. The porosity of most fabrics ranges from 50% to 95% depending on the fiber fineness, the tightness of the twist in the yarns, and the yarn count. Water vapor transport may take place by molecular diffusion due to gradients in the partial vapor pressure and by bulk transport due to convective air flow. Models of clothing varied in complexity depending on fabric hygroscopic properties and applicability of the assumption of thermal and vapor concentration equilibrium between the solid fiber and its entrapped air voids. This chapter will review modeling approaches of clothing material and associated microclimate, focusing on convective heat and moisture transfer due to material ventilation induced by relative wind penetrating the fibrous medium. Convective fabric and clothed cylinder models have been used to predict local and overall clothing ensemble ventilation rates. This latter modeling approach has a wider applicability since it can be combined with segmental bio-heat models to predict human local and overall comfort in hot humid environment. The integration of clothing ventilation model with “cylindrical” segments of the clothed human body is presented showing examples of how segmental and inter-segmental ventilation, sensible heat loss, and moisture transport through clothing are used to assess the whole body comfort and hence improve clothing design. The chapter will end with a recommendations’ section of best interventions for improving clothing performance in hot humid climate.

2

Review of Fabric Heat, Air, and Water Vapor Transport Models

Mathematical modeling of heat and mass transfer in porous media has been widely studied in the literature. Fabrics used in clothing systems, in particular, are highly porous thin material that allowed simplified modeling of heat and mass transfer by diffusion and convection. Thermal modeling of fabrics is important in evaluating clothed human thermal response and hence human thermal comfort. The heat exchange between human body and the environment is significantly affected by the way the clothing layers mediates the flow of heat and moisture from the human skin to the environment by diffusion in fabric and air layers and by ventilation induced by relative wind resulting in air penetration through the fabric to microclimate air trapped between skin and fabric. Models that consider heat and vapor transport through the clothing system when exposed to different environmental

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conditions and external wind will be considered in this section after discussing the most important fabric physical parameters that are used in these models.

2.1

Fabric Physical Parameters

Fabrics are highly porous and relatively thin materials made from fibers that have been spun into yarns that are used in weaving the fabric structure. The fabric system consists of the solid fiber matrix and void space. Water vapor can exist in both the adsorbed state in the solid fiber and in void air between the solid fibers, and liquid water can be present in the interstices of the fabric when subjected to wet conditions. Water vapor transport may take place by molecular diffusion due to gradients in vapor pressure and by bulk transport due to convective air flow. Liquid transport may occur by molecular diffusion caused by concentration gradients and by capillary forces as well as gravity. Energy transport occurs by conduction and by convective flows of all phases that are able to move. Predicting the heat and mass transfer in fabric system is a complex task and cannot be accomplished without the input information of the transfer properties. The following physical properties are important when considering modeling of heat and moisture transfer in fabrics:

2.1.1 Air Permeability (a) It can also be used to provide an indication of the breathability of fabrics. It is highly correlated to the yarn weave structure and the size of Micro pores inside the yarn and macropores between the yarns. It is usually determined by subjecting the fabric to a prescribed pressure difference and measuring the air flow rate passing perpendicularly to a fabric of a known cross-sectional area, ASTM D737-96 (2012). It is generally expressed in SI units as cm3/s/cm2. 2.1.2 Porosity The fabric porosity “ef” is generally defined as the air ratio of the air volume in the fabric to the total volume of the fabric structure. It is expressed with Eq. 1 in terms of the corresponding densities:
 ρfabric ef ¼ 1  ρfiber

(1)

where ρfabric and ρfiber are the densities of the fabrics (g/cm3) and of the fiber (g/cm3), respectively. Fabric thickness: Fabric thickness is highly dependent on yarn diameter and the fabric weave and usually expressed in mm. Determination of thickness of fabric samples is usually determined using a precision thickness gauge. The fabric sample is placed horizontally on a circular anvil without any tension and a standard load of 3.4 lbs./sq. in is pressed on the fabric specimen, ASTM D-39-49 (1965).

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2.1.3 Wetting and Wickability of Fabrics Textile researchers distinguish between two phenomena related to liquid transport in fabrics, the wettability and the wickability. Each term addresses a different application of a fabric that is brought in contact with liquid. Harnett and Mehta (1984) gives the following definitions “Wickability is the ability to sustain capillary flow” whereas wettability “describes the initial behavior of a fabric, yarn, or fiber when brought into contact with water.” While wetting and wicking refer to what appear to be separate phenomena, they can be described by a single process. In wetting, the fabric is initially dry but the liquid must still move through the fabric in order to spread and be soaked up by the fabric. In wicking, a continuous liquid phase is present. There is no sharp distinction between wetting and wicking. At some point during wetting, the fabric becomes sufficiently saturated that the process becomes wicking. Fabrics are normally weaved as a homogenous medium in their structure (porosity, permeability) and thin in their thickness. Moisture movement in fabrics is generally slow and if the flow does not affect the pore fabric size, then the moisture transport can be described by Darcy law and liquid velocity V is then given by V¼

α dPc μ dx

(2)

Where α is the permeability which describes the ease with which water liquid flows through the porous media, μ is the viscosity of the advancing liquid, and Pc is capillary pressure which is the driving force for the liquid movement. Ghali et al. (1994) used a variation of the “long column” to measure the capillary pressure of long strips of fabrics suspended vertically above a container of water with the bottom end immersed. Ghali et al. (1995) expressed the capillary pressure function of the fabric degree of saturation. As for the permeability, they used the siphon to measure the permeability function of saturation, and they proposed a new transient measuring technique for determining permeability.

2.1.4 Dry and Evaporative Resistance of Fabrics Steady heat and mass transport from clothing systems were based on dry and evaporative resistances of the fabric. The basic energy balance for dry heat flow is given by QD ¼ A

T skin  T env RD

(3)

where QD (W/m2) is the dry heat transfer and takes place across the clothing layer between the skin and the environment, A (m2) is the clothing surface area, Tskin is the skin temperature, Tenv is the environment temperature, and RD is the total resistance of the fabric layer. The dry resistance of the fabric is dependent upon the amount of still air entrapped in the interstices between the fibers and yarns since the

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conductivity of air is much lower than that of fiber materials (Fourt and Hollies 1970). The solid fibers arrangement and their volume in the fabric influence the fabric insulation more than the fiber itself (Rees 1941). Any fabric characteristics that would increase the amount of still air in the fabric would also increase its dry resistance. Thermal resistance of the fabric is usually negatively correlated with fabric density. The dry heat resistance for indoor worn fabrics is reported by McCullough et al. (1985, 1989) as follows: RD ¼ 0:015  ef

(4)

where RD is the dry resistance of the fabric in m2
K/mm
W and ef is the fabric thickness in mm. Similarly the evaporative heat flow QE is given by QE ¼ A

Pskin  Penv RE

(5)

where RE is the total evaporative resistance of the clothing layer, Pskin is the water vapor pressure at the skin, and Penv is the water vapor pressure in the environment. Similar to dry heat transfer, vapor transfer in fabrics depends on the physical properties of the entrapped air medium and on the arrangement of the solid fibers. The solid fibers not only absorb/desorb moisture but they also represent an obstacle for the vapor molecules on their way through the fabric (Chatterjee 1985). Therefore, the vapor resistance of fabrics is expected to be larger than that of equally thick air layer and is expressed as an equivalent thickness of still air that would give the same resistance to vapor transfer as that of the actual fabric. This equivalent air thickness was found by McCullough et al. (1989) to increase linearly with the fabric thickness for low-density fabrics and to some extent for dense fabric materials. The dry and evaporative resistances are also related through the permeability index, im, which was first proposed by Woodcock (1962). The relationship is expressed by im ¼ ðRD =RE ÞLR

(6)

where RE is the evaporative resistance of the fabric in m2
kPa/W and LR is the Lewis ratio equals approximately 16.65 K/kPa at typical indoor conditions. The heat transfer through a textile layer is a complex combination of conduction, radiation, and convection. However, it is rather common to use the total thermal resistance of textile medium instead of evaluating thermal processes. The thermal resistance for the clothing system is obtained by summation of each individual fabric and air layer of the clothing system. Thermal and evaporative resistances for each individual fabric sample can be measured using dry and sweating hot plate in accordance to ASTM D 1518 (1985) and ISO 11092 (1993), respectively.

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2.1.5

Dynamic Dry and Evaporative Resistances of Fabric Subject to Normal Flow Ventilation causes a dynamic change of clothing insulation due to wind penetration through fabric or ensemble openings, wearer displacement due to motion causing a wind effect, and relative motion of clothed body parts (limbs) with respect to their clothing cover. The fabric static dry and evaporative resistances are corrected for normal flow rate through the fabric denoted as m_ aY . The normal airflow passing through thin fabric at constant fabric permeability is given by (Lotens 1993): m_ aY ¼

αρa ð Pa  P1 Þ ΔPm

(7)

where m_ aY is the flow rate through fabric normal to the fabric surface, α is the fabric air permeability in m3/m2
s, ΔPm = 0.1245 kPa from standard tests on fabrics’ air permeability (ASTM D737-75 1983), Pa is the air pressure in the microclimate trapped air layer between the human skin and the fabric (kPa), and P1 is the outside environment air pressure (kPa). The dynamic dry resistance of the clothing is given by (Havenith et al. 1990, 2000; Ghali et al. 2009): 1  1 þ m_ aY Cpa RD

RD, dynamic ¼


(8)

where RD,dynamic is the dynamic dry resistance for ventilation through the fabric and Cpa is the penetrating air specific heat. The dynamic resistance is a parameter to correct for ventilation effect on clothing static resistance at no wind. Similarly, the dynamic evaporative fabric resistance is given by RE, dynamic ¼ 0

1

1

(9)

B1 C 1 B þ
C @ RE ρa RH 2 O T 1 A m_ aY had where RH2 O is the water vapor gas constant, ρa is the air density, T1 is the penetrating air temperature, and had is the fabric water-vapor heat of adsorption.

2.2

Diffusive and Convective Fabric Models

The first clothing model that describes the mechanism of transient diffusion of heat and moisture transfer into an assembly of hygroscopic textile materials was introduced and analysed by Henry (1939). He developed a set of two differential coupled

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governing equations for the mass and heat transfer in a small flat piece of clothing material. Henry’s analysis was based on a simplified analytical solution. In order to describe the complicated process of adsorption behavior in textile materials, Nordon and David (1967) presented a model in terms of experimentally adjustable parameters appropriate for a first stage of rapid moisture sorption followed by a second stage of slow moisture sorption. However, their model did not take into account the physical mechanisms of the sorption process. For this reason, Li and Holcombe (1992) introduced a new two-stage adsorption model to better describe the coupled heat and moisture transport in fabrics. Moreover, Farnworth (1986) developed a numerical model that took into account the condensation and adsorption in a multilayered clothing system. Jones and Ogawa (1993) developed another new unsteady-state thermal model for the whole clothing system. The whole clothing system model was based on simplified expressions of Henry’s model. Li and Holcombe (1992) presented a transient mathematical clothing model that describes the dynamic heat and moisture transport behavior of clothing and human body. The abovementioned models assumed instantaneous equilibrium between the local relative humidity of the penetrating air and the moisture content of the fiber. However, the hypothesis of local equilibrium was shown to be invalid during periods of rapid transient heating or cooling in porous media as reported by Minkowycz et al. (1999). In the absence of local thermal equilibrium, the solid and fluid should be treated as two different constituents. Under vigorous clothed human movement of a relatively thin porous textile material or with clothing exposure to external wind, the air will pass quickly between the fibers, invalidating the local thermal equilibrium assumption. The ventilation rate was affected mainly by the walking velocity as described by Lotens (1993) who derived empirically the steady clothing ventilation rate as function of the air permeability of the fabric and the effective wind velocity. Ghali et al. (2002a) developed a two-node fabric absorption model using empirically predicted transfer coefficients of a cotton fibrous medium to account for the state of nonthermal equilibrium between solid fiber and air passing through the fabric void space. They extended their model of the fiber to include modeling of the microclimate air in the voids of the fabric and predict the change of air temperature across the fiber (Ghali et al. 2002b, c). They also considered the periodic movement of clothed limbs on renewal of trapped air between clothing and skin by flowing through the fabric void space and clothing apertures. The advantage of the non-equilibrium thin fabric models is that they can be combined with ventilation models estimating the renewal rate of the microclimate air in the gap between skin and clothing. This facilitates the integration with human thermal bio-heat models. In the next sections, the thin fabric model is derived and it is then combined with the microclimate air clothed cylinder ventilation model for effective use in many applications related to human thermal comfort. The combined model is integrated with bio-heat models to predict clothed human thermal response and associated comfort as a function of clothing, activity level, and environmental parameters.

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Mathematical Formulation of Thin Fabric Model for Clothing Ventilation Applications

The fabric model presented here is based on Ghali et al. (2002b) while using a lumped layer of two fabric nodes and an air void node to represent the fibrous medium. The model is simple and is applicable to highly permeable thin fabrics. Lumped parameters have commonly been used in models of thin permeable fabrics (Farnworth 1986; Jones and Ogawa 1993). The three-node model lumps the fabric into an outer node, inner node, and an air void node. The fabric outer node represents the exposed surface of the yarns, which is in direct contact with the penetrating air in the void space (air void node) between the yarns. The fabric inner node represents the inner portion of the “solid” yarn, which is surrounded by the fabric outer node. The outer node exchanges heat and moisture transfer with the flowing air in the air void node and with the inner node, while the inner node exchanges heat and moisture by diffusion only with the outer node. The air flowing through the fabric void spaces does not spend sufficient time to be in thermal equilibrium with the fabric inner and outer nodes. The moisture uptake in the fabric occurs first by the convection effect from the air in the void node to the yarn surface (outer node), followed by sorption/diffusion to the yarn interior (inner node). The fabric model is best represented by a flow of air around cylinders in cross flow, where the air voids are connected between the cylinders (yarns) as shown in Fig. 1. The fabric is represented by a large number of these three-node modules in cross flow depending on the fabric effective porosity. The fabric area is L  W and the fabric thickness is ef. The airflow is assumed normal to the fabric plane. In the derivation of the water vapor mass balances in the outer node, inner node, and the air void node, the water vapor is assumed very dilute compared to the air and the bulk velocity of the mixture is very

Air Flow through Fabric Void

ef W

Inner Nodes Outer Nodes

Air Flow

Fig. 1 Schematic of lumped fabric three-node model

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close to the velocity of the air. The second main assumption is that the volume changes of fibers due to sorption process are small enough to be neglected. Effective heat and mass transfer coefficients, reported by Ghali et al. (2002a, b), Hco and Hmo for the outer node of the fabric, and the heat and mass diffusion coefficients Hci and Hmi for the inner nodes of the fabric, were used in the model in normalized form as follows: H 0mo ¼ H mo

Ao , Af

H 0co ¼ Hco

Ao , Af

H 0mi ¼ H mi

Ai , Af

H 0ci ¼ H ci

Ai Af

(10)

where Af is the overall fabric surface area, Ao is the outer-node exposed surface area to air flow and Ai is the inner node area in contact with the outer node. The values of these normalized convection coefficients for cotton fabric are given by (Ghaddar et al. 2005) H 0Co ¼ 495:72m_ aY  1:85693 H 0Co ¼ 2:0

W=m2
K, W=m2
K,

m_ aY > 0:00777kg=m2
s (11a) m_ aY  0:00777kg=m2
s (11b)

H 0mo ¼ 3:408103 m_ Y  1:2766  105 kg=m2
kPa
s, m_ aY > 0:00777 kg=m2
s (11c)

H 0mo ¼ 1:3714  105

kg=m2
kPa
s, m_ aY  0:00777kg=m2
s (11d)

where the normal air flow rate is as defined in Eq. 7 as a function of the fabric air permeability. The inner node transport coefficients to be used in the fabric model are as reported by Ghali et al. (2002a) at H0 ci = 1.574 W/m2
K, and H0 mi = 7.58  106 kg/m2
kPa
s. The time-dependent mass and energy balances are for the outer and inner nodes of the fabric yarn and of the air void node in terms of the heat and mass transport coefficients between the penetrating air and the outer node and between inner and outer node. In the derivation of the water vapor mass balances in the fabric and void space nodes, the water vapor is assumed dilute compared to the air and the bulk velocity of the mixture is very close to the velocity of the air. This assumption simplifies the mass balances by ignoring the effect of counter transfer of the air and assuming constant total pressure of the system. According to ASHRAE Handbook of Fundamentals (2005), no appreciable error is introduced when diffusion of a dilute gas through an air layer is done. The derivation included a term to correct for bulk motion of the fluid and its value is typically between 1.00 and 1.05 for conditions of the ventilating air. The water vapor mass balance in the air void node is given in Eqs. 12a and b when air flow enters the fabric void from the environment space to the microclimate layer (Pa < P1) and when air flow enters the fabric void space from the microclimate layer to the environment (Pa > P1), respectively, as

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N. Ghaddar and K. Ghali

   @ ρ ðwa  wvoid Þ 0 ρ ef wvoid ef ¼ m_ ay wp  wvoid þ Hmo ½Po  Pa  þ D a @t a ef =2 þD

ρa ðw1  wvoid Þ ef =2

where

 wp ¼

w1 wa

(12a)

Pa < P1 Pa  P1

(12b)

where ef is the fiber porosity. The last two terms in equations are the mass diffusion terms within the fabric in angular and axial directions. The outer fiber node and the inner fiber node mass balances are expressed in terms of the fabric regain in Eqs. 13a and b, respectively  dRo 1  0 ¼ Hmo ðPvoid  Po Þ þ H 0mi ðPi  Po Þ ργef dt

(13a)

dRi H 0mi ¼ ½ Po  Pi  dt ρð1  γ Þtf

(13b)

where Ro is the regain of the outer node (the mass of moisture adsorbed by fiber outer node divided by dry mass of the fiber outer node), Ri is the regain of the inner node, and H0 mo and H0 mi are the mass transfer coefficients between the outer node and the penetrating air and the outer node and the inner node, respectively. The parameter γ is the fraction of mass that is in the outer node and it depends on the fabric type and the fabric porosity. The total fabric regain R (kg of adsorbed H2O/kg dry fiber) is given by R ¼ γRo þ ð1  γ ÞRi

(14)

According to the model of Ghali et al. (2002a), the value of γ is equal to 0.6. The energy balance for the air vapor mixture in the air void node is given by ef

    @ ρ ef Cv T void þ hfg wvoid ¼ m_ ay ½H e  þ m_ ay Cp T void þ wvoid hfg @t a T a  T void T 1  T void 0 þ ka þ Hco ½T o  T void  þ ka ef =2 ef =2 þ Dhfg

ρa ðwa  wvoid Þ ρ ðw1  wvoid Þ þ Dhfg a ef =2 ef =2

(15a)

where He is given by  He ¼

Cp T 1 þ w1 hfg Cp T a þ wa hfg

for for

Pa < P1 Pa  P1

(15b)

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The heat transfer coefficient between the outer node and the penetrating air in the voids is H0 co and ka is the thermal conductivity of air. The last four terms of the energy balance are heat diffusion term in axial and angular directions. These terms are negligible when only normal flow through the fabric is present. The energy balance on the outer nodes gives

dT o dRo H0 H0  had ρf ð 1  γ Þ C f ¼ co ½T void  T o   ci ½T o  T i  dt dt ef ef þ

hr hr ðT skin  T o Þ þ ðT 1  T o Þ 2ef 2ef

(16)

where H0 ci is the heat diffusion coefficient between the outer node and the inner node, hr. is the linearized radiative heat exchange coefficient, and had is the enthalpy of the water adsorption state. The density of the adsorbed phase of water is similar to that of liquid water. The high density results in the enthalpy and internal energy of the adsorbed phases being very nearly the same. Therefore, the internal energy, uad, can be replaced with the enthalpy of the adsorbed water. Data on had, as a function of relative humidity, is obtained from the work of Morton and Hearle (1975). The energy balance on the inner node gives

dT i dRi H0  had ρf γ Cf ¼ ci ½T o  T i  dt dt ef

(17)

The above-coupled differential Eqs. 12a, b, 13a, b, 14, 15a, b, 16 and 17 describe the time-dependent convective mass and heat transfer from the skin-adjacent air layer through the fabric induced by the sinusoidal motion of the fabric. To solve the equations for the fabric transient thermal response, the fabric void microscopic transport coefficients, namely, H0 mo, H0 co, and the inner node diffusion coefficients H0 mi and H0 ci, and the internal convection coefficients from the skin to the air layer hm(skin-a) and h(skin-a) must be known.

4

Integration of Thin Fabric Model with Segmental Clothed Human Thermal Model

One of the important applications of the thin fabric model is clothing ventilation for prediction of clothed human heat loss when subject to external wind or relative wind due to walking. Predicting segmental clothing ventilation is important for improving garment design by modifying design parameters to enhance or decrease local ventilation rate depending on the type of clothing including protective clothing. This is why researchers have been interested in developing mathematical models from the first principles to predict local clothing ventilation.

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Modeling approach has mainly been based on representing the human body as a number of independent cylindrical segments covered with garment material in cross air flow (Ismail et al. 2014; Othmani et al. 2008). The clothed human body can be constructed a number of connected or disconnected clothed cylinders. These segments or cylinders are typically the clothed two arms, the clothed two legs, and the clothed trunk as shown in Fig. 2. Each segment consists of two concentric vertical cylinders for standing or walking person. The outer cylinder representing the fabric covers the inner heated cylinder representing the human skin (Ghaddar et al. 2010). The human skin temperature can be obtained from integration with a bio-heat model that simulates the human thermal response based on physiology and heat and mass transfer mechanisms that take place at the skin and through respiration (Ghaddar et al. 2008; Ismail et al. 2014). Mathematical modeling of clothed limbs and trunk would also require consideration of the air annulus between the human skin and the clothing. Clothing apertures are most likely to be at the top for the trunk at the neck and at the bottom for the limbs. The bio-heat model and its clothed segments including limbs are coupled to the clothed cylinder ventilation model in order to determine the associated boundary condition of skin temperatures to be used as inner cylinder thermal condition as mentioned earlier. This section starts with the clothed heated cylinder ventilation Fig. 2 The clothed human body segments

Head

Clothed arms Clothed trunk

Clothed legs

Feet

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model followed by some results showing how the model can be used to predict the clothed human overall clothing ventilation and compare these predictions with published empirical findings.

4.1

Clothed Cylinder Model of Independent Body Segments

There are several challenges to modeling local ventilation based on clothed cylinder model due to the presence of clothing apertures and the microclimate air layer size. Clothing ventilation is essentially related to the flow characteristics in the air layer between skin and clothes, the clothing permeability (Lotens 1993), and on the open clothing apertures (Ghaddar et al. 2005, 2010). The flow characteristics in the trapped air layer are the results of two phenomena: natural convection associated with warm body skin and forced flow induced by both wind that penetrates permeable clothing and by flow through the opening. The physical configuration of the present study shown in Fig. 3 consists of two concentric cylinders of radius Rs and Rf and height L. A microclimate air annulus of thickness Y = Rf  Rs is trapped between the inner solid cylinder maintained at temperature Tskin and the outer porous cylinder represented by an isotropic fabric layer of permeability and thickness ef. The top end of the annulus is closed and adiabatic, while the bottom end is open to the environment at temperature Tamb. The configuration is placed in perpendicular to an air flow at V1. Some air penetrates

Fig. 3 Schematic of clothed cylinder with open aperture at (a) the bottom end and (b) top end

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through the porous fabric into the air layer and is mixed with the incoming air from the bottom aperture at environment temperature and is then driven upward by the presence of the pressure gradient and the natural convection. The annulus trapped air thickness Y is small compared to the length L and the inner radius cylinder Rs which permits assuming Poiseuille flow in axial and angular directions even for the mixed buoyant upward flow. The outer clothing cylinder is assumed to consist of a layer made of fibers containing air voids. Air penetrates fabric through pores and entered to the microclimate space between skin and clothing. Water vapor is assumed to diffuse through air void space to be absorbed or desorbed by fibers depending on the type of the fabric. The skin is assumed exchanging heat and moisture with the microclimate air layer and radiation heat transfer with the outer node. Thus skin is considered as an interface so that there is no storage term (transient term) to be considered. Convection takes place in the void space between penetrating air and the fabric. The flow characteristics in the air layer are the results of two phenomena: natural convection associated with warm body skin and forced-flow induced by wind effect that causes penetration of air through clothing and induced flow through the opening driven by pressure difference between ambient air and air layer (Sobera et al. 2003; Kind et al. 1991; Leong and Lai 2006). The mass and energy balances in the microclimate air layer are derived in what follows for the cylindrical geometry. Since both ventilation through fiber and natural convection flow derived in axial direction by the open aperture, the mass conservation equation and the energy balance are coupled in the air layer and are given by @ ðY m_ az Þ @ ðY m_ aθ Þ þ  m_ aY ¼ 0 @z Rf @θ

(18)

where m_ az is the mass flux in the vertical z-direction in kg/m2
s, m_ aθ is the mass flux in the angular θ-direction, and m_ aY is the radial infiltrating air flow rate through the fabric as defined in Eq. 7 previously. The gap width Y and the slope @Y/(Rf@θ) can be assumed small for clothed cylinder and the annulus channel length πRf is much larger width (πRf >> Y ) which justify the assumption of 1-D fully developed quasi-parallel flow in the angular direction within the annulus. This is a commonly used model in hydrodynamic lubrication theory for journal bearings (Massey 1989). So, assuming Poiseuille flow in the θ-direction, the angular air mass flow rate per unit area is expressed in terms of the annular pressure as m_ aθ ¼ 

Y 2 dPa 12Rf v dθ

(19)

The upward mass flow rate is expressed in terms of pressure and driving temperature gradient of the air and temperature difference between the inner wall and the

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infiltrating air at fabric void temperature Tvoid. The upward mass flow rate per unit area due to forced flow is given by m_ az ¼ 

Y 2 dPa 12v dz

P a ¼ Pa þ ρa gz

(20) (21)

while the induced buoyant mass flow rate term is m_ an ¼ ρ

gβY 2 ðT a  T void Þ: 12v

(22)

The radial, angular, and upward mass flow rates of Eqs. 19, 20, 21, and 22 are substituted in Eq. 18 to get the mass conservation equation in pressure and temperature form as
3 
 αρa ðPa  Ps Þ @ Y @Pa @ Y 3 @Pa ρgβY 3 þ 2 þ þ ΔPm @z 12v @z 12v Rf @θ 12v @θ 

dðT a  T void Þ ¼0 dz

(23)

The steady state dry energy balance on the air spacing annulus is a balance of the dry convective heat transfer from the surface of the inner cylinder; the heat flow to the air associated with mass fluxes from the radial, angular, and upward directions; the heat diffusion from void air of the thin fabric to the air layer; and the angular conduction of heat in the air layer. The dry energy balance of the air layer is then given by     hc ðskinairÞ ðT skin  T a Þ þ hco ðT o  T a Þ þ max m_ ay , 0 cp T void  max m_ ay , 0 cp T a

   

 @ Y:qm_ az @ Y:qm_ aθ ka @ @T a @ @T a Y Y þ  2 ¼  ka @z @z Rf @θ @θ @z Rf @θ

(24) where heat flow per gap width in z and θ directions are given by qm_ az ¼ m_ az
cp
T

a

qm_ aθ ¼ m_ aθ
cp
T a

(25)

where hc(skin-air) is the convection coefficient from the skin to the air, hc(o-air) is the convection coefficient from the fabric to the air, To is the fabric outer node temperature, ka is the thermal conductivity of air in the trapped air annulus, and cp is the air specific heat. The values of the convective coefficients are provided in Table 1.

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Table 1 Summary of convection heat transfer coefficients used in the model Convection coefficient hcskin  air

Value or equation (W/m2
K) 10.5

hco

9.6

hc(o  air)

hcoae ¼ a:V 1 b  0:33 where a ¼ 4:8 2:Rf and b ¼ 0:5

hr

3.7

References Mohanty and Dubey (1996) Mohanty and Dubey (1996) Danielson (1993) Song (2007)

4.1.1 Boundary Conditions Modeling mixed convection in small-thickness vertical annulus with open bottom lower boundary, connecting to the environment, has relatively more complex boundary condition to tackle at the opening than mixed convection with open top boundary. The complexity arises from the change in the direction of the flow (upward or downward) depending on angular position in the open annulus which requires special treatment of the boundary conditions at the opening (Patankar 1980). When the space between the two concentric cylinders is small enough, the velocity of the inlet or exit natural convection flow is only axial (Anil Lal and Reji 2009; Alarabi et al. 1987; Anil Lal and Kumar 2012). In this situation, the boundary conditions are influenced only by the pressure and temperature conditions. Researchers identified two different types of pressure boundary condition that can be used for bottom openings. The first type is when the static pressure is equal to zero gauge pressure in both the inlet and the exit flow (Chan and Tien 1985). The second type is when the static pressure at the exit and the stagnation pressure at the inlet are equal to zero gauge pressure (Anil Lal and Reji 2009; Zamora and Hemandez 2011; Evangellos et al. 2007). Anil Lal and Kumar (2012) compared the two types and found that the second type of boundary condition is more reasonable and adequate to simulate developing natural convection flow between parallel surfaces. For thermal boundary conditions at the inlet and outlet of an annulus, researches have assumed that the inlet fluid temperature is considered uniform and ambient. However, the gradient of the temperature of the fluid leaving the annulus is set equal to zero (Anil Lal and Reji 2009; Alarabi et al. 1987; Anil Lal and Kumar 2012). These conditions are applicable to the natural convection inflow and outflow only through one end of the annulus top or bottom. In the presented model, the second type of pressure boundary condition is used for open aperture at the lower end of the cylinder while Bernoulli’s equation is applied between P1 in the far environment to the opening at z = 0, using CD, the loss coefficient, at the aperture of the domain dependent on area ratio of the aperture to the air layer thickness Y. The associated boundary conditions for the air flow and pressure are summarized as follows:

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(a) Aperture is z = 0 for the lower opening: m_ az ðz ¼ 0, θÞ ¼ CD ½ 2ρa ðPa  P1 Þ1=2 ; Pa ðz ¼ 0, θÞ ¼ P1 

½m_ az ðz ¼ 0, θÞ 2 2ρa (26a)

At the closed end: m_ az ðz ¼ L, θÞ ¼ 0 &

 dPa ðz ¼ L, θÞ ¼0 dz

(26b)

(b) Aperture is at z = L with opening at the top: m_ az ðz ¼ L, θÞ ¼ CD ½ 2ρa ðPa  P1 Þ1=2 ; Pa ðz ¼ L, θÞ ¼ P1 

½m_ az ðz ¼ L, θÞ 2 2ρa (26c)

At the closed end whether at top or bottom level: m_ az ðθÞ ¼ 0

&

dPa ð θÞ ¼0 dz

(26d)

At the angular symmetry line: m_ aθ ðz, θ ¼ 0 or π Þ ¼ 0

(26e)

Considering the inner cylinder as isothermal at Tskin and the environment temperature at T1, the associated thermal boundary condition use zero temperature gradient when the air is leaving or when end is closed and assumes ambient temperature when the flow is entering.

4.1.2 Microclimate Ventilation Rate and Sensible Heat Loss Of interest is to calculate the total ventilation rate of the fabric, based on the renewal rate induced by external wind penetrating the porous fabric and enhanced by buoyancy. The total ventilation rate is calculated as the positive flow of air into the air annulus integrated per unit area of the clothed surface as follows: The segmental sensible heat loss Qs from the skin to the microclimate air layer used in the clothed cylinder model and bio-heat model is given by 

m_ a kg=s
m

2



1 ¼ πL

ðL ðπ 0 0

 max 0, m_ ay Þ dθ dz

(27a)

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ðð Qs ðW Þ ¼

 hc skinair Rs ðT skin ðθ, zÞ  T a ðθ, zÞÞ

þ hr Rs ðT skin ðθ, zÞ  T o ðθ, zÞÞdθ dz

4.2

(27b)

Bio-Heat Model Integration and Overall Clothing Ventilation

The bio-heat model (Ghali et al. 2011) is adopted in this chapter, and it divides the body into 17 segments: head, chest, back, pelvis, buttocks, lower arms, upper arms, hands, thighs, calves, and feet. The model equations simulate the heat and moisture transport from the human skin through the clothing system using a modified Gagge’s model (Gagge 1973). Each body segment body is represented by Gagge’s two-node model (Gagge 1973), where two energy balance equations were developed for the core node and the skin node. The sensible heat exchange from the skin to the air layer is expressed in terms of the convection heat loss at the skin surface and the radiation exchange between the skin and the fabric. The skin temperature obtained from the human body model is an input boundary condition to the clothed cylinder model. For the latent heat transfer, the boundary condition is more complex. If liquid is present on the skin surface then the skin boundary condition is the saturation pressure at the skin temperature Psk = P(Tsk). The vapor pressure at the skin surface is determined by a balance between the diffusion of vapor through the skin, the sweat secreted, and the transport of moisture away from the skin as reported by Jones and Ogawa (1993). In the analysis of this chapter, it is assumed that no liquid exists at the skin surface. The bio-heat model is coupled through the skin temperature to the proposed mixed convection model of the clothed trunk and limb segments with open apertures to estimate local ventilation rates. Local ventilation rate is an important parameter because of its effect on the clothing resistance. Therefore, the bio-heat model incorporates as input a dynamic resistance that takes into account the correction on the dry resistance due to the local ventilation as given in Eqs. 8 and 9. The most comprehensive data on dynamic clothing dry resistance originates from the work of Havenith et al. (1990) describing the changes in clothing insulation values due to motion of the wearer and wind. Ismail et al. (2014) has reported results of the coupling of bio-heat model, and clothing cylinder models and validating the model with Havenith et al. (1990) published experimental data on total clothing ventilation, vapor resistance, and permeability index for different human postures and wind velocities. Ismail et al. (2014) selected three clothing ensembles to validate the segmentally constructed ventilation and bio-heat model as follows: • Ensemble A (permeable case): workpants (cotton, Φ = 0.66, α = 2.121 m/s); poly-shirt (50% polyester, 50% cotton, porosity = 0.85, α = 5.465 m/s); and sweater (cotton, acrylic, porosity = 0.82, α = 4.13 m/s). • Ensemble B (semi-permeable): workpants (cotton); poly-shirt (50% polyester, 50% cotton); sweater (cotton, acrylic); coverall (cotton, porosity = 0.5, α = 2.161 m/s).

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• Ensemble C (impermeable): workpants (cotton); poly-shirt (50% polyester, 50% cotton); sweater (cotton, acrylic); rain coverall (nylon, porosity = 0.1, α = 2.3.103 m/s). The bio-heat model of Ghali et al. (2011) was coupled through the skin boundary condition (temperature) to the clothed cylinders model. The metabolic rate was set for a standing person. The ambient conditions (temperature and humidity) and external wind velocity were also required as input to the bio-heat model. The resulting segmental ventilation rate obtained from the clothed cylinder model was entered to the bio-heat model by means of the dynamic resistance for the ventilation through fabric to predict the skin temperature. The new skin temperature was used again in the connected cylinder model to predict segmental ventilation. The alternating solution between the two models is repeated until converging skin temperature and segmental ventilation rate were obtained between the bio-heat model and the connected cylinders model. Figure 4 presents the comparison of Ismail et al. (2014) between model predictions and experimental results of ventilation. Note that all the values predicted fell in the range of the standard deviation of the experimental study. Thus, good agreement was found between the parametric study predictions and experimental measurements of ventilation using the tracer gas. The model showed that when the wind speed increased, the opening in the bottom did not show any advantage in the drawing of the ambient air. This is due to the diminishing effect of the natural convection for wind speeds above 2.0 m/s. The effect of permeability was shown to enhance ventilation. A 36% increase in ventilation was obtained by changing from semipermeable to permeable ensemble when only natural convection was present (no wind). However, when V1 was larger than 2.0 m/s (forced convection), the 250.0 Model Predicted Values Wind Speed at 0 m/s

Ventilation Rate (I/min)

200.0

Wind Speed at 0.7 m/s Wind Speed at 4.0 m/s

150.0 Experimentally Published Values

100.0

50.0

0.0 A

B

C

Fig. 4 Comparison between overall clothing ensemble ventilation predicted by constructed clothed human model and those of experimental results

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increase in ventilation by changing from semipermeable to permeable ensemble was 24%. For higher permeability ensemble, the natural convection enhancement effect on ventilation rate is more considerable than that encountered at low permeability ensemble. Thus, the increase in permeability is indispensable to improve the ventilation in no wind condition because the natural convection enhancement effect on the flow within the air layer induces flow from ambient air due to pressure difference created between the ambient air and air within the air gap.

4.3

Connected Clothed Cylinders Model to Improve Clothing Ventilation Predictions

The ventilation model constructed from the clothed cylinder model was improved by considering the effect of clothing connection at the shoulder between the arms and the trunk. The mixed convection connected clothed cylinders model of microclimate air of the human upper body was recently developed by Ismail et al. (2015) using the physically connected clothed cylinders model of the clothed human upper body as shown in Fig. 5. The upper body connection through the shoulders was incorporated as connection element and pathway for the microclimate air flow between clothed arms and trunk. One large cylinder represented the upper shoulder level of the connected upper part of the two arms and the trunk, while three independent cylinders were extended downward from the large cylinder. Each of the arm and the trunk was formed by two co-axial annuli of different inner and outer radii. The Fig. 5 Physical configuration of the upper part of the human body

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inner solid cylinder represented the heated skin and the outer cylinder represented the permeable fabric. The two cylinders were separated by a microclimate air annulus where the flow and heat characteristics were modeled. Ismail et al. (2015) derived and solved the coupled pressure equation of mass and momentum for narrow annular flow as well as the water vapor transport and energy transport equations for the microclimate air layer. They also coupled the microclimate air balance equations to the three-node fabric model described in this chapter representing the thin fabric. The interconnected cylinder model was described in the work of Ismail et al. (2015) and was validated by CFD modeling of the microclimate air geometry and by comparison to published experimental results on clothing ventilation of Ke et al. (2014) for a permeable jacket (α = 0.135 m/s) at different wind velocities of V1  0.1 m/s (no wind), V1 = 0.6 m/s, and V1 = 0.9 m/s. The experiment was carried out on a standing shop manikin in an air-conditioned chamber at 20  C ambient temperature and 40 10% relative humidity. The dimensions used in the published experiment 3 (garment S1) are the same used in the model. Table 2 presents the total ventilation values predicted by the connected model as compared with the published experimental values at different wind speeds for the same permeable jacket when the connection effect is included and when it is excluded. It is observed that the connected model is closer in accuracy to the published experiment (with an error not exceeding 12%) than the unconnected one (with an error exceeding 15%) at relatively high wind speeds (V1  0.6 m/s). However, it shows approximately the same result at low wind speeds (V1  0.1 m/s). Figure 6a shows the inter-segmental ventilation using a permeable jacket (α = 0.135 m/s) at different wind speeds. It was shown that when the velocity increases, the inter-segmental ventilation increased. At low wind speeds (V1  0.1 m/s), it was found that the inter-segmental ventilation was no longer significant. In this case, the trunk and the arm can be modeled as independent segments. However, at high wind speeds (V1  0.9 m/s), the inter-segmental ventilation exceeds 5 l/min. Another important factor that affects the inter-segmental ventilation is the clothing permeability. In general, the jacket permeability allows air to enter to the segmental microclimate air layers and the interconnection allows the air exchange between the segments. In order to study the impact of permeability on the inter-segmental ventilation, different jacket permeability are investigated at a wind speed of 1 m/s. Figure 6b illustrated this impact. It showed that at relatively high permeability (α = 0.135 m/s), the inter-segmental ventilation was significant and reaches 5 l/min. However, at lower permeability (α = 0.05 m/s), the connection impact vanished and the air exchange between the trunk and the arm was no longer important. The third important parameter that affects the inter-segmental ventilation is the opening at the bottom end of the lower arm and at the neck. Figure 6c illustrated this effect by showing the inter-segmental ventilation rates at different apertures. At V1 = 1 m/s and for a permeable jacket (α = 0.135 m/s), it was found that the intersegmental ventilation became more significant when the bottom opening of the arm was open than when it was closed. This is because the pressure at the arm microclimate air layer increases when the bottom end of the arm is closed. This pressure increase caused lower air exchange leaving the trunk to the arm. However, this was not the case

Model connected (Ismail et al. 2015) Model unconnected (Ismail et al. 2014)

V1 (m/s) Experiment ventilation (Ke et al. 2013) Ventilation (l/m) 40.65

Relative error (%) 12.16 11.02

Ventilation (l/m) 32.12

32.54

37.79

0.6 m/s 44.5 (l/min)

0.1 m/s 36.57 (l/min)

15.07

Relative error (%) 8.65

55.09

Ventilation (l/m) 60.01

0.9 m/s 58.5 (l/min)

5.82

Relative error (%) 2.58

Table 2 Comparison of published experimental values (Ke et al. (2013) and predicted model ventilation values for unconnected and connected clothing model (Ismail et al. 2014, 2015) at different wind speeds

908 N. Ghaddar and K. Ghali

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Fig. 6 Estimation of the inter-segmental ventilation obtained from the ventilation model as a function of (a) wind speed (b) permeability (c) apertures

when the top end of the trunk was closed. No significant variation in the intersegmental ventilation in both cases: open top end or closed top end. Ismail et al. (2016) performed experiments on thermal manikin with connected sleeve and trunk and unconnected clothed segments to assess the effect on associated

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dry heat loss at environmental conditions of 24 0.5  C and relative humidity of 40 3% and wind velocity of 1.2 m/s. Figure 7 shows a comparison between their analytical and the experimental segmental ventilations for both cases of closed and open connection between clothed arm and clothed trunk of the jacket of 0.09 m/s permeability. Good agreement was reported between the analytical and experimental local ventilation rates. Table 3 shows a comparison between the analytical prediction and experimental data of segmental ventilation and heat losses in both cases: open and closed connection for an ensemble with permeability of 0.09 m/s. Good agreement was shown between them. Closing the connection affected the segmental ventilation and heat losses for both arm and trunk. The heat losses and ventilation from the trunk decreased when the connection is closed. This is compensated by an increase of the arm ventilation and heat losses. This was due to the flow of the heated air from the trunk to the arm when the connection was opened which allowed fresh air to enter the trunk microclimate. This fresh air increased the segmental ventilation of the trunk by 12% and the heat losses by 5.46%. In the other hand, the heated air that left the trunk and entered the arm reduced the arm ventilation by 3% and reduced the heat losses by 6.68%. As discussed earlier, the inter-segmental ventilation causes the trunk ventilation to increase by allowing some fresh air mass flow rate to penetrate instead of the heated mass flow rate (inter-segmental ventilation). Indeed, the trunk ventilation is

35

30

analytical experimental

Ventilation (I/min)

25

20

15

10

5

0 arm

trunk

inter-segmental

Fig. 7 Comparison between the analytical and experimental ventilations for the permeable jacket at α = 0.09 m/s

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Table 3 Comparison between the segmental ventilation and heat losses between open and closed connection (Ismail et al. 2016) Clothed segment Arm

Type of connection Open connection Closed connection

Trunk

Open connection Closed connection

Segment Arm

Type of connection Open connection Closed connection

Trunk

Open connection Closed connection

Ventilation (l/min) Model Experiment Model Experiment Model Experiment Model Experiment Heat losses (W/m2) Model Experiment Model Experiment Model Experiment Model Experiment

11.26 10.56 12.034 10.856 31.20 30.88 28.63 27.5 52.15 53.3 56.705 57.12 64.914 65.4 59.126 61.536

essential in providing the human thermal comfort (Zhang 2003). The trunk is reported as the most influential segment because of its highest segmental heat loss (Zhang and Zhao 2007). Ismail et al.’s (2016) predictive model of clothing ventilation accounted for inter-segmental air exchanges between clothed arms and trunk. They found that the inter-segmental ventilation was substantial at relatively high wind speed and high clothing air permeability and should not be neglected. Moreover, they reported that accounting for the inter-segmental ventilation resulted in more accurate predictions of the overall clothing ventilation where the relative error between the predicted and published experimental overall clothing ventilation data was reduced from 15% to 8% at relatively high wind speed and air permeability.

4.4

Effect of Walking on Ventilation and the Clothed Swinging Arm Model

Clothing microclimate ventilation is modeled for the condition when the human is subject to external wind that penetrates through the fabric or ensemble openings. These predictive models are presented in the previous subsections using connected clothing cylinder models integrated with bio-heat models with the assumption that both the limbs and trunk are subject to the same wind conditions and may not be applicable to the more complex physical situation due to wearer displacement

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causing relative motion of clothed body parts (limbs) with respect to their clothing cover. During walking, ventilation is not uniform over the clothed human body. The trunk local ventilation is different than ventilation induced by swinging motion of limbs at small or large swinging amplitudes and frequencies (Ghali et al. 2002c). Different body parts during walking are subject to different mechanisms of wind, swinging motion, and combined wind and swinging motion that stimulate ventilation at different rates. When different segments have different ventilation rates, local thermal comfort evaluated by skin wittedness and comfort sensation becomes important in assessing whole body comfort. The standard ISO-7730 (2005) requires determination of clothing insulation of active persons or persons exposed to significant wind. Data on dynamic insulation have limited use since they are applicable for the specific ensembles, activity levels, and experimental conditions used in generating them. Most of the research on clothing ventilation of walking humans was empirical and few studies dealt with the mechanism of microclimate ventilation by wind and motion through modeling. Previous work of Ghali et al. (2009) and Ghaddar et al. (2008) addressed the effect of the changing gap width induced by an oscillating body part (cylinder) within a fixed single clothing cover at uniform external environment pressure with close and open clothing apertures. A clothed swinging arm model was developed by Ghaddar et al. (2008). The developed model was capable of estimating the renewal flow rates for general limb motion configuration taking into consideration the periodic motion of the limbs and clothing, their geometric interaction at skin-fabric contact or no contact, open or closed clothing apertures, and in presence of wind or no wind. However, the model assumed the clothed swinging arm as an independent segment that is not connected with the trunk as shown in Fig. 8. They used the swinging clothed arm model to determine the dynamic clothed limb resistance in a follow-up work by Ghali et al. (2009) to predict the mean steady periodic heat loss from the clothed cylinder using Lotens’ simple heat resistance network model and validated their model with experiments (Lotens 1993). They reported that the use of lumped dry and evaporative heat resistance network that accounted for the effect of ventilation through clothing fabric and clothing opening resulted in relatively good predictions of heat loss from the a clothed swinging solid cylinder in the presence of wind. The use of accurate ventilation rates is an important input to the simplified thermal model using dynamic resistance. The ventilation through clothing by wind and motion was also reported to double or triple heat loss as compared to the case of still cylinder in quiet air. The clothed swinging limb model is still lacking the interconnection effect during walking between clothed trunk and arms and to determine whether interconnection air follows enhances trunk or limb ventilation in similar way to static clothed cylinder interconnection. This remains subject of future research to have a versatile and complete ventilation model of clothed human. Such a model can be used to estimate accurately human heat losses and assess human thermal comfort under different climate conditions and activity level.

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Fig. 8 Representation of the human limb motion inside the clothing cylinder

fmax

V∞

5

Closing Remarks and Future Trends

Ventilation is an important design parameter for summer weather clothing of active people to increase heat loss from the human body. Based on research work done for modeling and predicting clothing ventilation, enhancement of microclimate exchanges between different body segments is desirable in garment design. Walking ventilates moving arms and legs. Clothed trunk ventilation can be enhanced with larger microclimate air exchange permitted between body segments to channel the airflow from the relatively high-ventilated limbs microclimate to the less-ventilated trunk microclimate. Ventilation of the microclimate air layer is an attractive solution for design of versatile clothing for active people that could result in enhancing effectiveness of moisture removal away from the skin. Ventilation rates can be altered where needed by the proper choice of fabric permeability, arrangement of clothing layers, reducing internal microclimate flow resistance for internal air exchanges between connected clothed body segments, and aperture positions, size, and operational flexibility of controlling the extent of microclimate air direct connection with the environment.

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Extensive research is still needed to understand the relationship between segmental ventilation and local discomfort during walking and when subject to wind for optimizing active wear designs. Inter-segmental air exchanges modeling is target of future work to complete the clothed human thermal model based on physiology and first principles. Moreover, development of models that predict liquid moisture transport in fabric is also needed to allow extending the clothed human model to high activity rate where moisture wicks the fabric and moves to outer layers depending on gradients of temperature between skin and environment. Such models would be useful for predicting the human thermal stress state and endurance at high activity level.

6

Cross-References

▶ Analytical Methods in Heat Transfer ▶ Applications of Flow-Induced Vibration in Porous Media ▶ Electrohydrodynamically Augmented Internal Forced Convection ▶ Full-Coverage Effusion Cooling in External Forced Convection: Sparse and Dense Hole Arrays ▶ Heat Transfer Media and Their Properties ▶ Macroscopic Heat Conduction Formulation ▶ Single-Phase Convective Heat Transfer: Basic Equations and Solutions ▶ Thermophysical Properties Measurement and Identification

References Alarabi M, El-Shaarawi MAI, Khamis K (1987) Natural convection in uniformly heated vertical annuli. Int J Heat Mass Tran 30(7):1381–1389 American Society for Testing and Materials (1965) ASTM D-39-49. ASTM D39-65 Method of test for construction characteristics of woven fabrics American Society for Testing and Materials (1983) ASTM D737-75, Standard test method for air permeability of textile fabrics, (IBR) approved 1983 American Society for Testing and Materials (1985) ASTM D1518: Standard test method for thermal resistance of batting systems using a hot plate American Society for Testing and Materials, ASTM D737-96 (2012) Air permeability. Standard test method for air permeability of textile fabrics, ASTM Organization American Society of Heating, Refrigerating and Air-Conditioning Engineers (2005) ASHRAE handbook of fundamentals. ASHRAE, Atlanta Anil Lal S, Kumar A (2012) Numerical prediction of natural convection in a vertical annulus closed at top and opened at bottom. Heat Tran Eng 33(15):70–83 Anil Lal S, Reji C (2009) Numerical prediction of natural convection in vented cavities using restricted domain approach. Int J Heat Mass Transf 52:724–734 Chan YL, Tien CL (1985) A numerical study of two dimensional laminar natural convection in shallow open cavities. Int J Heat Mass Transf 28:603–612 Chatterjee PK (1985) Absorbency. Elsevier Science Publishing Company, Amsterdam

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Danielson U (1993) Convection coefficients in clothing air layers. Doctoral thesis, The Royal Institute of technology, Stockholm Evangellos B, Vrachopoulos M, Koukou M, Margaris D, Filios A, Mavrommatis S (2007) Study of the natural convection phenomena inside a wall solar chimney with one wall adiabatic and one wall under a heat flux. App Therm Eng 27:226–234 Farnworth B (1986) A numerical model of combined diffusion of heat and water vapor through clothing. Text Res J 56:653–655 Fourt L, Hollies NRS (1970) Clothing: comfort and function. Martin Dekker, New York Gagge AP (1973) A two node model of human temperature regulation in FORTRAN. In: Parker JF, West VR (eds) Bioastronautics data, 2nd edn. NASA, Washington, DC Ghaddar N, Ghali K, Harathani J (2005) Modulated air layer heat and moisture transport by ventilation and diffusion from clothing with open aperture. ASME Heat Transf J 127(3):287–297 Ghaddar N, Ghali K, Jreije B (2008) Ventilation of wind-permeable clothed cylinder subject to periodic swinging motion: modeling and experimentation. J Heat Transf 130:1107–2020 Ghaddar N, Ghali K, Othmani M, Holmer I, Kuklane K (2010) Experimental and theoretical study of ventilation and heat loss from clothed vertical isothermally-heated cylinder in uniform flow field. J Appl Mech 77(3):1–8 Ghali K, Ghaddar N, Bizri M (2011) The influence of wind on outdoor thermal comfort in the city of Beirut: a theoretical and field study. Int J HVAC R res 17(5):813–828 Ghali K, Ghaddar N, Jones B (2002a) Empirical evaluation of convective heat and moisture transport coefficients in porous cotton medium. J Heat Transf 124(3):530–537 Ghali K, Ghaddar N, Jones B (2002b) Multi-layer three-node model of convective transport within cotton fibrous medium. J Porous Media 5(1):17–31 Ghali K, Ghaddar N, Jones B (2002c) Modeling of heat and moisture transport by periodic ventilation of thin cotton fibrous media. Int J Heat Mass Transf 45(18):3703–3714 Ghali K, Jones B, Tracy J (1994) Experimental techniques for measuring parameters describing wetting and wicking in fabrics. Text Res J 64:106–111 Ghali K, Jones B, Tracy J (1995) Modeling heat and mass transfer in fabrics. Int J Heat Mass Transf 38:13–21 Ghali K, Othmani M, Jreije B, Ghaddar N (2009) Simplified heat transport model of wind permeable clothed cylinder subject to swinging motion. Text Res J 79:1043–1055 Harnett PR, Mehta PN (1984) A survey and comparison of laboratory test methods for measuring wicking. Text Res J 54(7):471–478 Havenith G, Heus R, Lotens WA (1990) Resultant clothing insulation: a function of body movement, posture, wind clothing fit and ensemble thickness. Ergonomics 33(1):67–84 Havenith G, Holmér I, Parsons KC, Den Hartog E, Malchaire J (2000) Calculation of dynamic heat and vapor resistance. Environ Ergon 10:125–128 Henry HPS (1939) Diffusion in absorbing media. Proc R Soc 171A:215 Ismail N, Ghaddar N, Ghali K (2014) Predicting segmental and overall ventilation of ensembles using an integrated bio-heat and clothed cylinder ventilation models. Text Res J 84:2198–2213 Ismail N, Ghaddar N, Ghali K (2016) Theoretical and experimental estimation of inter-segmental clothing ventilation and impact on human segmental heat losses. In: Proceedings of ASME IMECE2015-50255E, Houston, Nov 2015 Ismail N, Ghaddar N, Ghali K (2016) Effect of inter-segmental air exchanges on local and overall clothing ventilation. Text Res J 86(4):423–439 ISO 11092 (EN31092) (1993) Textiles-physiological effects - Measurement of thermal and watervapor resistance under steady-state conditions (sweating guarded-hotplate test) ISO 7730 (2005) Ergonomics of the thermal environment - Analytical determination and interpretation of thermal comfort using calculation of the PMV and PPD indices and local thermal comfort criteria Jones BW, Ogawa Y (1993) Transient interaction between the human and the thermal environment. ASHRAE Trans 98(1):189–195

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Ke Y, Havenith G, Li J, Li X (2013) A new experimental study of influence of fabric permeability, clothing sizes, openings and wind on regional ventilation rates. Fibers Polym 14:1906–1911 Ke Y, Havenith G, Zhang X, Li X, Li J (2014) Effects of wind and clothing apertures on local clothing ventilation rates and thermal insulation. Text Res J 84:941–952 Kind RJ, Jenkins JM, Seddigh F (1991) Experimental investigation of heat transfer through windpermeable clothing. Cold Reg Sci Technol 20:39–49 Leong JC, Lai FC (2006) Natural convection in a concentric annulus with a porous sleeve. Int J Heat Mass Transf 49:3016–3027 Li Y, Holcombe BV (1992) A two-stage sorption model of the coupled diffusion of moisture and heat in wool fabrics. Text Res J 62(4):211–217 Lotens W (1993) Heat transfer from humans wearing clothing, doctoral thesis. TNO Institute for Perception, Soesterberg Massey BS (1989) Chapter 6. In: Mechanics of fluids, 6th edn. Springer, New York McCullough EA, Jones BW, Huck J (1985) A comprehensive data base for estimating clothing insulation. ASHRAE Trans 91:29–47 McCullough EA, Jones BW, Tamura T (1989) A data base for determining the evaporative resistance of clothing. ASHRAE Trans 95(2):316–328 Minkowycz WJ, Haji-Shikh A, Vafai K (1999) On departure from local thermal equilibrium in porous media due to a rapidly changing heat source: the sparrow number. Int J Heat Mass Transf 42:3373–3385 Mohanty AK, Dubey MR (1996) Buoyancy induced flow and heat transfer through a vertical annulus. Int J Heat Mass Transf 39(10):2087–2093 Morton WE, Hearle LW (1975) Physical properties of textile fibers. Heinemann, London Nordon P, David HG (1967) Coupled diffusion of moisture and heat in hygroscopic textile materials. Int J Heat Mass Transf 10(7):853–866 Othmani M, Ghaddar N, Ghali K (2008) An angular multi-segmented human bio-heat model to assess local segment comfort in transient and asymmetric radiative environment. Int J Heat Mass Transf 51(23–24):5522–5533 Patankar SV (1980) Numerical heat transfer and heat flow. Hemisphere Publishing Corporation, McGraw Hill Book Company, New York Rees WH (1941) The transmission of heat through textile fabrics. J Text Inst 32:149–165 Sobera MP, Kleijn CR, Brasser P, Van den Akker HEA (2003) Convective heat and mass transfer to a cylinder sheathed by a porous layer. AICHE J 49:3018–3028 Song G (2007) Clothing air gap layers and thermal protective performance in single layer garment. J Ind Text 36(3):193–204 Woodcock A (1962) Moisture transfer in textile systems, part I. Text Res J 32:628–633 Zamora B, Hernandez J (2011) Influence of upstream conduction on the thermally optimum spacing of isothermal, natural convection-cooled vertical plate arrays. Int Comm Heat Mass Transf 28 (2):201–210 Zhang H (2003) Human thermal sensation and comfort in transient and non-uniform thermal environments. PhD thesis, University of California, Berkeley Zhang Y, Zhao R (2007) Effect of local exposure on human responses. Build Environ 42:2737–2745

Part IV Thermal Radiation Heat Transfer

A Prelude to the Fundamentals and Applications of Radiation Transfer

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M. Pinar Mengüç

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Fundamental Concepts and Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 The Planck and Wien Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Solid Angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Radiative Transfer Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Solution of the Radiative Transfer Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Applications of Radiative Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Near-Field Radiative Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Cross-References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

920 920 922 924 926 928 928 929 930 930 930

Abstract

Radiation transfer is one of the three pillars of heat transfer processes, along with conduction and convection. It is essential for any high-temperature industrial process taking place in combustion chambers, flames and fires, manufacturing processes, energy harvesting systems, atmospheric processes, large-scale and local thermal management problems, as well as in high-resolution thermal sensing and control applications. This chapter provides an introduction to the fundamental principles of radiation transfer, including the Planck law, Wien law, radiative intensity, solid angle, and radiative transfer equation. It is a prelude to the eight related chapters in this handbook, and provides an overview that leads to detailed discussion of the radiative transfer equation and its solution, the radiative properties of particles and gases, applications in combustion chambers, inverse

M. P. Mengüç (*) Cekmeköy Campus, Özyegin University, Çekmeköy - Istanbul, Turkey e-mail: [email protected] # Springer International Publishing AG, part of Springer Nature 2018 F. A. Kulacki (ed.), Handbook of Thermal Science and Engineering, https://doi.org/10.1007/978-3-319-26695-4_55

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problems, near-field radiation transfer, and advances in surface optical and radiative properties.

1

Introduction

The origin of radiation transfer can be traced back to the beginning of the universe. Our sun has been shining on Earth and providing light and heat for more than 4.5 billion years. During all these years, all kinds of plants, reptiles, birds, mammals and fish have adapted to the sun and its energy. This adaptation goes down to the level of cells and sub-cellular structures, and all governed with the fundamentals of radiative transfer in its most general form. Of course, the light-matter interactions are even more fundamental in nature and include all atomic and sub-atomic level physics. And the humans have been playing with fire, and with its light and heat, for over 200,000 years since they have started controlling it in Olduvai Gorge, in Tanzania. All kinds of technological developments since then have benefitted from the understanding of radiation transfer from flames and fires, within combustion systems and industrial processes at high temperatures. The subject of radiation transfer has been discussed by many researchers over the years. Theory and applications were considered from the classical point of view by Planck (1906), Chandrasekhar (1960), Sparrow and Cess (1966), Hottel and Sarofim (1967), Brewster (1992), Viskanta and Mengüç (1987), Viskanta (2005), Modest (2013), and Howell et al. (2016). In addition, Chen (2005) and Zhang (2007) outlined the principles of near-field radiation transfer (NFRT) and nanoscale conduction heat transfer. A rigorous analysis of radiative transfer is based on Maxwell’s equations for electromagnetic (EM) wave propagation and scattering. The interaction of EM waves with particles and surfaces is an important aspect of the physics behind such an analysis. Detailed discussion of EM waves and particle–surface interactions are found in the texts by Van de Hulst (1981), Bohren and Huffman (1983), and Mishchenko (2014). EM-wave-based analysis is also required for NFRT, as discussed by Chen (2005), Zhang (2007), and more recently by Basu (2016). This chapter outlines the fundamental principles of radiation transfer. Different but related concepts are discussed in subsequent chapters to help the reader put them in perspective. References are listed at the end of the chapter for specific cases.

2

Fundamental Concepts and Equations

All objects radiate EM wave energy to their surroundings. At the same time, all objects receive energy emitted by other objects around them. A fraction of the incident energy on an object is absorbed and the rest is either transmitted through the object’s boundaries or reflected back. The balance between the absorbed and emitted radiation, along with the other modes of energy transfer mechanisms, determines the thermal equilibrium state of an object, specifically by considering conduction, convection, and

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radiation transfer through the system boundaries and taking into account the heat source and mass and work flow distributions. Temperature is the measure of the internal energy of an object. This energy balance and the correlation between internal energy and temperature are manifested by the zeroth and the first law of thermodynamics; the latter is also known as the law of conservation of energy. If the net energy received is more than the energy lost, the internal energy of a body increases; otherwise, it decreases. Once energy is emitted, the object loses part of its energy; in other words, it cools off. However, it gets warmer by absorbing electromagnetic energy emitted by other objects. When EM radiation travels from one object to another, it can be scattered by all molecules, particles, cracks, or any inhomogeneity between them, which makes the problem quite complex as the absorption, emission, and scattering mechanisms should be evaluated at each wavelength (or frequency) of the EM wave spectrum. The polarization of EM waves may also be important because it affects the underlying physics, depending on the length scale of the system under consideration. Radiation transfer enters into the conservation of energy equation as a source term. This equation, as written in terms of the temperature, T, of a control volume is expressed as follows: ρc

@T _ dV þ ∇ðk∇T ÞdV ¼ qdV @t

(1)

Detailed discussion of the equations for the conservation of mass, momentum, and energy are provided in other sections of this handbook. The radiation source term in Eq. 1 is denoted as q._ Note that there may be more than one type of volumetric source, because they can be the result of different physical or chemical phenomena; thermal radiation transfer is only one of the sources. The radiation source term q_ is equivalent to divergence of the radiative flux in a volume element dV. This means that for the calculations we first have to determine the radiative flux profile in the medium of interest. In its most general sense, this profile is calculated in threedimensional space and in transient fashion by solving the integro-differential and spectral radiative transfer equation (RTE). Note that such directional and spectral behavior is unique to radiation transfer, which makes it more complicated than conduction or convection heat transfer modes. Obtaining the radiative flux profile in a medium starts with predicting all underlying physical phenomena within the medium and at its boundaries, in directional, spectral, and transient manners. The distribution of radiative flux depends on the temperature profile in the medium and on the boundaries. This makes the problem even more complex, because the temperature profile can be determined from Eq. 1 only if the radiative source term is available. Therefore, the problem requires an iterative solution of the energy and radiative transfer equations until the temperature profile converges. Beyond this obvious numerical complication, additional complexity arises because of the information required for the spectral and directional properties of surfaces (plus data on their composition and structural variation), of gases and their interactions, and of particles (plus data on their size, shape, structure, and complex indices of refraction).

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A discussion of fundamental equations for radiative emission and absorption should start with the definition of a perfect emitter or absorber. The key point to be emphasized here is the spectral variation in thermal emission. Because of the dependency on wavelength (or frequency), the physics of radiation transfer is rich and complex, and allows intriguing design and process ideas. In other words, one can control the amount of radiation heat transfer to an object by modifying the geometry of the object and its surroundings, as well as its spectral radiative properties and those of its boundaries. To tackle these ideas, the fundamental expressions for blackbody emission should be introduced, including solid angle, radiation intensity, and the RTE. Here, we summarize these fundamental expressions. ▶ Chap. 23, “Radiative Transfer Equation and Solutions” by Zhao and Liu details the most important expressions required for prediction of thermal radiation heat transfer.

2.1

The Planck and Wien Laws

A perfect (ideal) blackbody emits the maximum radiative energy at all wavelengths. This emission from a surface to all directions is expressed by the Planck law, as given by Eq. 2: 2πhc2
0   hc0 5 2 n λ exp 1 nkB λT 

Eλb ðT Þ ¼

(2)

where Eλb is the radiative energy emitted by a surface (of an object) to all directions, per area, at a given wavelength λ; h is the Planck constant (h = 6.626070  1034 m2kg/s); kB is the Boltzmann constant (kB = 1.380648  1023 J/K); and T is temperature in Kelvin. There are two additional constants in this expression: c0, the speed of light in a vacuum (c0 = 2.99792458  108 m/s) and n, the index of refraction of the medium in which the wave is propagating. For a vacuum, n = 1; for air it is very close to 1 and varies with the wavelength λ, which is omitted for the sake of clarity. The subscript “b” refers to “blackbody” to identify the ideal and maximum possible emission. The spectral behavior of the Planck law for different temperatures is depicted in Fig. 1. With increasing temperature, the peak wavelength of emission shifts to lower wavelengths, and the corresponding emissive power increases. If the blackbody emissive power from an object is integrated at all angles within the entire spectrum, one can obtain a relatively simple expression proportional to the fourth power of temperature, T 4 (T being in Kelvin), which is known as the Stefan–Boltzmann Law, where σ is the Stefan–Boltzmann constant (σ = 5.67  108 W/m2K4): ð all λ

Eλb dλ ¼ σT 4

(3)

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The Planck blackbody equation given by Eq. 2 can be normalized with T5 and plotted against λT to consolidate all the T-dependent curves shown in Fig. 1 to a single curve. The peak of this normalized curve, as shown in Fig. 2, is important and known as the Wien law: ðλT Þmax ¼ 2893:6 μm-K

(4)

This equation helps to determine the spectral behavior of radiation emission in an intuitive way in heat transfer analyses at any given T. For example, an object at room temperature, roughly at 20 C (293 K), has maximum emission at a wavelength of about λ=10 μm. On the other hand, the sun, at T=5867 K, has a peak emission at about λ=550 nm. The peak emission of solar light corresponds to the median of the visible radiation spectrum (at green), suggesting that the eyes of many organisms may have evolved to sense the maximum amount of sunlight. Note that the wavelength obtained from the Wien law is also referred to as the characteristic wavelength of emission. This value represents the demarcation between far- and near-field radiation transfers, as discussed in Sect. 6. In many heat transfer calculations, the Stefan–Boltzmann law, as given in Eq. 3, is used to account for radiation transfer. Yet, by doing so, the spectral richness of radiation transfer phenomenon is neglected. In addition to the spectral nature of radiation transfer, we wish to emphasize once again its directional variation, which

108 10

T = 5762 K T = 5000 K T = 3000 K T = 2000 K T = 1000 K T = 300 K

7

106 105 104 103 102 101 100 10–1 10–1

100

101

Fig. 1 Planck blackbody emission as a function of wavelength at different temperatures

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1.4

× 10–11

1.2 1 0.8 0.6 0.4 0.2 0

103

104

Fig. 2 Normalized Planck blackbody function Eλb/Τ5. The peak value is determined from the Wien law, as given in Eq. 4

is unparalleled compared with conduction and convection heat transfer. The spectral and directional behaviors of radiation transfer are discussed in the following sections.

2.2

Solid Angle

We all know that a beam of sunlight streaming off a dense cloud line is more noticeable than the background. Under these conditions, the directional variation of light is obvious just as the directional radiative heat coming from a fireplace. In an enclosure, the radiation from a hot surface is naturally more intense than from the rest of the enclosure. To account for that disparity, it is necessary to formulate the problem of directional variation with clear understanding. To do that, instead of using hemispherical emissive power, it is preferable to define the so-called radiation intensity (Howell et al. 2016). To understand this fundamental quantity, consider radiative energy emanating from a surface, per unit area and per unit time, and propagating in a given direction within a small directional “pencil.” For mathematical derivation, it is preferable to define the pencil of light or the “solid angle,” based on the geometry given in Fig. 3. Here, a small area on the sphere with radius r around the emitting surface is defined in terms of incremental azimuthal (ϕ) and zenith (θ) angles and related to the solid angle:

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Fig. 3 Geometry for definition of the solid angle

= =

dΩ ¼

rdθr sin θdϕ r2

(5a)

The integration of this solid angle in all directions over the hemisphere gives the total angle, 2π. Integration of all angles would yield 4π: ð 4π

dΩ ¼

ð π=2 ð 2π θ¼0

ϕ¼0

sin θdθdϕ ¼ 2π

(5b)

The radiation intensity is defined as the energy leaving a surface at location S, per unit area dA, in the direction of Ω within an incremental solid angle dΩ, within a small wavelength interval dλ at the wavelength λ, per unit time interval dt, as: I λ ðS, Ω, tÞ ¼ lim

dEλ ðS, Ω, tÞ dAdλdΩdt

(6)

If the surface is a blackbody, then Iλ is replaced with Iλb, and Eλ with Eλb. Using this formulation, the problem of radiative exchange between different volume and surface elements in an enclosure can be tackled. As an example, imagine the radiative energy leaving the surface element A1 and reaching A2, as seen in Fig. 4. To determine the net energy going from A1 to A2, a small area element dA1 and an incremental solid angle between dA1 and A2 are considered. After that, dA1 can be scanned over the entire area of A1. Note that this is purely a geometric problem. This geometric analysis is known as the shape factor or configuration factor analysis, which is explained in text books by Sparrow and Cess (1966), Hottel and Sarofim (1967), Modest (2013), and Howell et al. (2016). A compendium of most exchange factors is provided by Howell and Mengüç (2011); the factors are also listed on the website www.thermalradiation.net by the same authors (Mengüç and Howell 2016).

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( + , Ω) = ( , Ω) + ( , Ω)

( ,Ω )

Ω

C A

B

D ( , Ω)

Fig. 4 Geometry for the derivation of the radiative transfer equation: A gain of energy resulting from emission, B loss of energy along the line-of-sight via absorption, C gain of energy resulting from scattering from other directions into the direction of the beam, D loss of energy resulting from scattering to other directions

Note that the shape factor analysis can be carried out if the properties of the radiating surfaces are available. If the problem requires a wavelength-dependent analysis, then the calculations should be carried out at each wavelength interval. The results for flux and the divergence of radiative flux are then integrated over the entire wavelength spectrum. It is also important to realize that the shape factor analysis is used if a medium is non-participating (i.e., completely transparent). If a medium is absorbing, emitting, and scattering, then it is called participating. Any type of medium, including foams, paints, combustion gases, and particles, that is nontransparent within a spectral window of importance falls into this category. The exchange of energy in a participating medium is more complex because of the change in direction of EM waves as a result of any inhomogeneity along their path. This change of direction is called scattering and encompasses reflection, refraction, and diffraction of beams. Under these conditions, we need to solve the RTE, which is derived phenomenologically in Sect. 3. ▶ Chap. 23, “Radiative Transfer Equation and Solutions” gives details.

3

Radiative Transfer Equation

The conservation of radiative energy for a beam of light along a given direction in a participating medium can be written using the nomenclature and geometry shown in Fig. 4. The resulting expression, in terms of spectral radiative intensity, is called the

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radiative transfer equation (RTE). As the beam propagates, it loses some of its energy along the line-of-sight via absorption (B in Fig. 4) and scattering to other directions (D). Furthermore, as the beam propagates through the medium, it gains energy as a result of emission (A) and scattering from all other directions into the direction of the beam (C). These radiative gains and losses can be added, giving the time-dependent RTE: @I λ ðr, Ω, tÞ @I λ ðr, Ω, tÞ þ ¼ κλ I λb ðr, tÞ  κ λ I λ ðr, Ω, tÞ  σ Sλ I λ ðr, Ω, tÞ c@t @S ð 1 þ σ S, λ I λ ðr, Ωi , tÞΦλ ðΩi , ΩÞdΩi 4π

(7)

Ωi ¼4π

Here, κλ is the spectral absorption coefficient and σ λ is the spectral scattering coefficient (in units of m1). The sum of these two gives the spectral extinction coefficient, βλ: β λ ¼ κ λ þ σ s, λ

(8)

In Eq. (7), Φλ(Ωi, Ω) is the scattering phase function, which gives the probability of the radiative energy being redirected from any given direction Ωi to the direction of propagation Ω. Therefore, all possible directions must be accounted for in the analysis, which is the reason for the integral term in the RTE. The radiative properties coefficients and the scattering phase function can be determined starting from the Maxwell equations, which describe the propagation of EM waves in a given medium. For particulate matter, the methodologies for determining these properties have been discussed by Van de Hulst (1981), Bohren and Huffman (1983), Mishchenko (2014), and Howell et al. (2016). These calculations for particle radiative properties require information on the shape, size, and structural details as well as the spectral complex index of refraction data. The details and requirements for different techniques are outlined by Vaillon in ▶ Chap. 27, “Radiative Properties of Particles.” For radiative heat transfer calculations, scattering in gaseous media is not important and only the absorption coefficient is required. There are different types of gases that need to be considered for these calculations, the most important of them being water vapor (H2O), carbon dioxide (CO2), and carbon monoxide (CO). Their presence in the atmosphere, along with nitride oxides (NO, NO2) and methane (CH4), determines for the radiative balance of the Earth and climate change concerns. On the other hand, in combustion chambers, the spectral properties of H2O, CO2, and CO are needed at high temperatures and pressures. The spectral absorption coefficients of these gases show quite complex behavior and their calculation is by no means trivial. Details of the spectral radiative properties of different gases and the corresponding methodologies are available in Modest (2013) and Howell et al. (2016), as well as in ▶ Chap. 26, “Radiative Properties of Gases” by Solovjov et al.

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Solution of the Radiative Transfer Equation

The RTE given in Eq. 7 forms the backbone of radiative transfer calculations. Because of the in-scattering term, the RTE is an integro-differential equation and its solution must first be obtained for every direction, before being integrated over all directions (i.e., over 4π). As one can imagine, this integration and the solution of the RTE is not a trivial task. Depending on the application, the problem can be simplified by considering the radiative intensity to be uniform over a given solid angle interval; however, the applicability of any approximation should be carefully evaluated. Over the years, several different publications have been devoted to this subject, as outlined by Chandrasekhar (1960), Hottel and Sarofim (1967), Viskanta and Mengüç (1987), Modest (2013), and Howell et al. (2016). Discussion of the RTE and many solutions developed for different cases are given in ▶ Chap. 23, “Radiative Transfer Equation and Solutions” by Zhao and Liu. Among all these techniques, the most versatile, without question, is the Monte Carlo technique. A detailed analysis is available for fundamentals and applications in ▶ Chap. 29, “Monte Carlo Methods for Radiative Transfer” by Ertürk and Howell.

5

Applications of Radiative Transfer

Radiative transfer is important in both atmospheric and thermal sciences as well as in industrial systems. Atmospheric radiation transfer is not covered in this handbook, but its analysis can be found in texts such as Thomas and Stamnes (2002) and Coakley and Yang (2016). A detailed discussion of radiative transfer for thermal sciences is given by Howell et al. (2016). For fires, flames and combustion systems radiation is the most dominant mode of heat transfer, as outlined by Coelho in ▶ Chap. 28, “Radiative Transfer in Combustion Systems.” The applications to combustion and industrial systems require the solution of the RTE coupled with the energy equation in complex geometries and with significant variations in temporal and spatial distribution of the properties within the medium and on the surfaces. The properties of combustion gases and particles need to be accounted for spectrally, and they may change as a result of interaction of chemical species–turbulence–radiation within the medium. It is impossible to cover all these complexities here; however, this chapter, coupled with ▶ Chap. 23, “Radiative Transfer Equation and Solutions” by Zhao and Liu, provides fundamental ideas and detailed references for further discussion. Understanding the details of radiation transfer mechanism is required to tailor a furnace or combustion chamber to perform under optimum conditions. For this, an inverse solution of the RTE can be employed to design an enclosure. The problem is not straightforward, but ill-conditioned (i.e., it may have more than one optimum solution), given that the problem is mathematically very challenging and very important. In ▶ Chap. 30, “Inverse Problems in Radiative Transfer,” Daun outlines inverse problems in depth. He first discusses the inverse design criteria and then

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follows up with inverse parameter estimation problems for the determination of radiative properties in carefully conducted experiments.

6

Near-Field Radiative Transfer

Radiative transfer takes place between any two objects at different temperatures, regardless of the distance and the medium between them. As discussed above, the Planck law can predict the spectral emission from a surface at a given temperature. The solution of the RTE can be obtained following many available approaches. Examples are given in ▶ Chap. 23, “Radiative Transfer Equation and Solutions” by Zhao and Liu and ▶ 29, “Monte Carlo Methods for Radiative Transfer” by Erturk and Howell, as well as in texts by Modest (2013) and Howell et al. (2016). If the objects are very close to each other, with distances below the characteristic wavelength (as determined from the Wien law given by Eq. 4), then an additional energy transfer mechanism takes place as a result of the coupled lattice and electron vibrations from two adjacent objects. This stems from the electromagnetic modes caused by evanescent waves generated by total internal reflection and surface polaritons, in addition to the emission by any heat source and manifested as propagating modes (Polder and Van Hove 1971; Chen 2005; Zhang 2007; Basu 2016). The thermal aspect of excitation of optical resonances, or polaritons, and the evanescent wave coupling is what we call Near-Field Radiative Transfer (NFRT). NFRT can be explained within the framework of fluctuational electrodynamics, starting with analysis of Maxwell’s equations and the fluctuating current sources representing thermal emission (Rytov 1953; Rytov et al. 1989). Under these conditions, radiative heat transfer may exceed the blackbody limit between two bulk materials. Emission and tunneling of evanescent modes, volumetric thermal emission, and coherence effects of the NFRT based on fluctuational electrodynamics were outlined first by Rytov (Rytov 1953), (Rytov et al. 1989). In ▶ Chap. 24, “NearField Thermal Radiation,” Francoeur discusses these concepts and the theories for predicting NFRT between close bodies. Understanding the NRFT between close structures has matured in parallel with advances in nanoscale measurement and manufacturing techniques. These developments contributed to the materials revolution of the twenty-first century. With these advances, engineering design of tools and processes can now be enhanced with the design of materials used for these purposes. In that sense, the design of radiative and optical properties of surfaces has also become a reality. Zhao and Zhang outline these studies as related to radiation transfer in ▶ Chap. 25, “Design of Optical and Radiative Properties of Surfaces.” The chapter summarizes how micro/nanostructured surfaces can interact with EM waves by excitation of optical resonances (polaritons) that can modify the polarization-dependent directional and spectral radiative properties. In addition, recent computational advances allow the consideration of different shapes and structures of nano-particles and nano-inclusions on or within the medium, and provide a predictive capability for the spectral optical properties of advanced functional surfaces (Didari and Mengüç 2017). These

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developments are crucial for the future design of energy harvesting systems, advanced photodetectors, large-scale and local thermal management solutions, and high-resolution thermal sensing.

7

Remarks

The radiative transfer section of this handbook describes state-of-the art developments in radiative transfer in thermal systems. The present chapter is an introduction to the concepts discussed in subsequent chapters, providing a coherent framework for the reader. Specific discussions to guide the reader to new ideas and applications are available in ▶ Chaps. 23, “Radiative Transfer Equation and Solutions” by Zhao and Liu, ▶ 27, “Radiative Properties of Particles” by Vaillon, ▶ 26, “Radiative Properties of Gases” by Solovjov et al., ▶ 29, “Monte Carlo Methods for Radiative Transfer” by Erturk and Howell, ▶ 28, “Radiative Transfer in Combustion Systems” by Coelho, ▶ 30, “Inverse Problems in Radiative Transfer” by Daun, ▶ 24, “NearField Thermal Radiation” by Francoeur, and ▶ 25, “Design of Optical and Radiative Properties of Surfaces” by Zhao and Zhang. These developments are crucial for the future design of combustion systems; manufacturing processes for glass, steel, and aluminum; solar energy harvesting systems; advanced heating and cooling devices; large-scale and local thermal management solutions; and high-resolution thermal diagnostic and sensing applications, among others. The discussion of these concepts is obviously extensive and cannot be exhausted in finite number of pages. The coverage here and the following chapters simply provide a starting point for any research or application related to radiative heat transfer.

8

Cross-References

▶ Design of Optical and Radiative Properties of Surfaces ▶ Inverse Problems in Radiative Transfer ▶ Monte Carlo Methods for Radiative Transfer ▶ Near-Field Thermal Radiation ▶ Radiative Properties of Gases ▶ Radiative Properties of Particles ▶ Radiative Transfer Equation and Solutions ▶ Radiative Transfer in Combustion Systems

References Basu S (2016) Near-field radiative heat transfer across nanometer vacuum gaps: fundamentals and applications. Elsevier, Amsterdam Bohren CF, Huffman DR (1983) Absorption and scattering of light by small particles. Wiley, New York

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Brewster MQ (1992) Thermal radiative transfer and properties. Wiley Interscience, Hoboken Chandrasekhar S (1960) Radiative transfer. Dover, New York Chen G (2005) Nanoscale energy transfer and conversion. Oxford University Press, Oxford Coakley JA Jr, Yang P (2016) Atmospheric radiation: a primer with illustrative solutions. Wiley, Hoboken Didari A, Mengüç, MP (2017) J Quant Spectrosc Radiat Transf 197 (August):95–105 Hottel HC, Sarofim AF (1967) Radiative transfer. McGraw Hill, New York Howell JR, Mengüç MP (2011) The JQSRT web-based configuration factor catalog: a listing of relations for common geometries. J Quant Spectrosc Radiat Transf 112(5):910–912 Howell JR, Mengüç MP, Siegel R (2016) Thermal radiation heat transfer, 6th edn. CRC, Boca Raton Mishchenko MI (2014) Electromagnetic scattering by particles and particle groups: an introduction. Cambridge University Press, Cambridge Mengüç MP, Howell JR (2016) http://www.thermalradiation.net. Accessed 15 Dec 2016 Modest MF (2013) Radiative heat transfer, 3rd edn. Academic Press, Oxford Planck M (1906) Vorlesungen Uber die Theorie der Warmestrahlung. Barth, Leipzig Polder D, Van Hove M (1971) Theory of radiative heat transfer between closely spaced bodies. Phys Rev B 4(10):3303–3314 Rytov SM (1953) Theory of electrical fluctuation and thermal radiation. Academy of Science of USSR Publishing, Moscow Rytov SM, Kravtsov YA, Tatarskii VI (1989) Principles of statistical radiophysics 3: elements of random fields. Springer, New York Sparrow EM, Cess RD (1966) Radiation heat transfer. Brooks/Cole, Belmont Thomas GE, Stamnes K (2002) Radiative transfer in the atmosphere and ocean. Cambridge University Press, Cambridge Van de Hulst HC (1981) Light scattering by small particles. Dover, New York Viskanta R, Mengüç MP (1987) Radiation heat transfer in combustion systems. Prog Energy Combust Sci 13(2):97–160 Viskanta R (2005) Radiative transfer of combustion systems: fundamentals and applications. Begell House, New York Zhang Z (2007) Nano/microscale heat transfer. McGraw Hill, New York

Radiative Transfer Equation and Solutions

23

Junming M. Zhao and Linhua H. Liu

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Radiative Transfer Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 The Classical Radiative Transfer Equation (RTE) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 The Second-Order Form of RTE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 The Radiative Transfer Equation in Refractive Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Solution Techniques of the Radiative Transfer Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Spherical Harmonics Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Discrete-Ordinate Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Finite Volume Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Finite Element Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Solution Methods for RTE in Refractive Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Numerical Errors and Accuracy Improvement Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Origin of Numerical Errors in DOM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Error from Differencing Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Scattering Term Discretization Error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Error from Heat Flux Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Cross-References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

934 936 936 947 950 955 955 957 961 963 966 969 969 970 971 972 973 974 975

Abstract

Radiative transfer equation (RTE) is the governing equation of radiation propagation in participating media, which plays a central role in the analysis of radiative transfer in gases, semitransparent liquids and solids, porous materials, and particulate media, and is important in many scientific and engineering disciplines. There are different forms of RTEs that are suitable for different

J. M. Zhao (*) · L. H. Liu School of Energy Science and Engineering, Harbin Institute of Technology, Harbin, China e-mail: [email protected]; [email protected] # Springer International Publishing AG, part of Springer Nature 2018 F. A. Kulacki (ed.), Handbook of Thermal Science and Engineering, https://doi.org/10.1007/978-3-319-26695-4_56

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applications, including the RTE under different coordinate systems, the transformed RTE having good numerical properties, the RTE for refractive media, etc. This chapter gives a comprehensive overview and introduction of the different forms of RTEs. Furthermore, several fundamental numerical methods for solving RTEs are introduced with the focus on the deterministic methods, such as the spherical harmonics method, discrete-ordinate method, finite volume method, and finite element method. The understanding of the numerical errors for solving the RTEs, including their origin and effects on numerical results, and the related accuracy improvement strategies are reviewed and discussed.

1

Introduction

Radiative transfer equation is the governing equation of radiation propagation in participating media, which describes the general balance of radiative energy transport in the participating media taking into account the interactions of attenuation and augmentation by absorption, scattering, and emission processes (Howell et al. 2011; Modest 2013). The equations of radiative transfer play a central role in the analysis of radiative transfer in gases, semitransparent liquids and solids, porous materials, and particulate media, which are important in many scientific and engineering disciplines, such as combustion systems (Viskanta and Mengüç 1987; Modest and Haworth 2016), rockets (Simmons 2000), atmospheric radiation (Liou 2002), remote sensing, astrophysics, noncontact temperature field measurement (Zhou et al. 2005), optical tomography (Klose et al. 2002), photo-bioreactors (Pilon et al. 2011; Berberoglu et al. 2007), and solar energy harvesting (Benoit et al. 2016; Agrafiotis et al. 2007; Mahian et al. 2013). The classical equation of radiative transfer is a first-order integral-differential equation describing radiative energy transport in media with uniform refractive index, i.e., light beam propagates through straight lines in the media. It has been widely applied to radiative transfer analysis in scientific and engineering problems and demonstrated to be a reliable theory for engineering applications. There are many variant forms of radiative transfer equations. For example, in order for convenience of solution for specific problems, the equations of radiative transfer are usually formulated under different coordinate systems and shown in different forms, such as Lagrange form in ray-path coordinate and Eulerian forms in common orthogonal coordinate systems, Cartesian coordinate system and cylindrical coordinate system, etc. Furthermore, the traditional form of radiative transfer equations, namely, the first-order integral-differential equation, can be transformed to secondorder forms to improve stability for numerical solution, such as the even-parity formulation of radiative transfer equation (Song and Park 1992) and the secondorder radiative transfer equations (Zhao et al. 2013; Zhao and Liu 2007a). However, due to the structural characteristics of a material or a possible temperature/pressure dependency, the refractive index of a medium may be a function of spatial position. Some examples of participating media with gradient refractive index distribution are

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earth’s (or other planets’) atmosphere, the ocean water, the hot air/gas of a flame, and artificial materials, such as graded index lens, graded index optical fiber, etc. In such cases, the classical equation of radiative transfer has to be extended to take into account the effect of curved ray path, resulting in the equation of radiative transfer in refractive media (Liu and Tan 2006). Radiative transfer in graded index media has attracted the interest of many researchers; some recent works include Refs. Asllanaj and Fumeron (2010), Wu and Hou (2012), Zhang et al. (2012), Hou et al. (2015), Chai et al. (2015), and Huang et al. (2016), to name a few. Numerical simulation is crucial to analyze radiative transfer in real applications, since analytical solutions exist only for a few simple cases due to the mathematical complexity of radiative transfer equation and the complex configuration of the problems. However, numerical simulation of radiative transfer in participating media is usually time consuming and requires considerable effort due to the complexity and the high dimensionality of radiative transfer process, which contains additional dimensions of one frequency and two angular dimensions besides the common three spatial dimensions. Hence efficient and accurate numerical methods are very important for most practical applications. Many efforts have been devoted to devise effective methods for the analysis of radiative transfer in participating media. Until recently, many numerical methods have been developed for the solution of radiative transfer equation. Generally, the methods can be classified into two groups, the first group is based on stochastic simulation, which includes various implementation of Monte Carlo methods (MCM) (Howell 1968; Farmer and Howell 1994; Siegel and Howell 2002) and the DESOR method (Zhou and Cheng 2004), and the second group is the deterministic methods, such as spherical harmonics method (or PN approximation) (Mengüç and Viskanta 1985; Larsen et al. 2002), discrete-ordinate methods (DOM) (Carlson and Lathrop 1965; Fiveland 1988; Coelho 2002a), finite volume method (FVM) (Raithby and Chui 1990; Chai and Lee 1994; Murthy and Mathur 1998; Asllanaj and Fumeron 2010), finite element method (FEM) (Liu et al. 2008), radiation element method (Maruyama 1993), spectral element method (Zhao and Liu 2006), spectral methods (Li et al. 2008), and meshless methods (Sadat 2006; Liu and Tan 2007), to name a few. A review of numerical methods for solving the RTE refers to the textbook by Modest (2013). As being approximate methods, all numerical methods suffer several kinds of numerical errors. The MC method suffers from statistic errors. The DOM, FVM, and FEM suffer from space and angular discretization errors. The significance of numerical errors is problem dependent. It will add unphysical features to the solution to make the solution difficult to be interpreted and may sometimes totally spoil the solution. Hence to know the origin and characteristics of numerical errors is important, which can help to interpret the results of numerical simulation and to design strategies to reduce or eliminate the errors. It has been known for decades that DOM method suffers two kinds of numerical errors, i.e., false scattering and ray effects, and several strategies have been proposed to reduce these errors (Chai et al. 1993). Since the FVM can be considered a DOM with a special angular quadrature scheme, FEM and many other methods are based on the discrete-ordinate equations. Thus the

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false scattering and ray effects are two general kinds of numerical errors, which need to be thoroughly understood. Recently, Hunter and Guo (2015) gave a comprehensive analysis on the numerical errors on solution of RTE. In this chapter, the classical radiative transfer equation and several variant forms of radiative transfer equation, different solution techniques for the radiative transfer equations, numerical errors on the solution of radiative transfer equation, and the related improvement strategies are presented and discussed. The chapter is organized as follows. Firstly, the classical radiative transfer equation and variant forms of radiative transfer equation are presented in Sect. 2. Then, the different solution techniques for the radiative transfer equation are introduced in Sect. 3. Finally, the numerical errors on the solution of radiative transfer equation and the related improvement strategies are presented in Sect. 4.

2

Radiative Transfer Equation

In this section, the governing equations of radiative transfer, including the classic radiative transfer equation, the radiative transfer equation in refractive media, and the different variant forms of the radiative transfer equations, are introduced.

2.1

The Classical Radiative Transfer Equation (RTE)

The classical equation of radiative transfer describes the balance radiative energy transport in absorbing, emitting, and scattering media with uniform refractive index distribution. Generally, the radiative power of a light beam in the medium is a function of wavelength λ (μm), transfer direction Ω, and spatial location r, which is described using the physical quantity of radiative intensity Iλ(r, Ω). It has unit W/(m2μm sr), denoting the transferred radiative power per unit cross-section area along the transfer direction, per wavelength, and per solid angle. The RTE is a governing equation of radiative intensity Iλ(r, Ω). In the following, the RTE in different coordinate system, the energy relations, and the numerical property of RTE are presented.

2.1.1 Ray-Path Coordinate System Formulation Ray-path coordinate is the natural coordinate system for light transfer. Here the RTE is formulated in the one-dimensional Lagrangian ray-path coordinate at first. The Lagrangian form of RTE is the physical clearest, in the simplest mathematical form, and considered to be the most general formulation such that the RTE under other different coordinate systems can be derived just by expressing the stream operator under the system. A control volume in cylinder shape along the ray path between s and s + ds is considered as shown in Fig. 1. A light beam enters the left surface at s and exits at s + ds. The end surface of the cylinder is perpendicular to the ray transfer direction s with an area of dA. The radiative intensity along the ray-path direction can be

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937

I(s+ ds,s)

I(s,s)

s

Emitted photons

Incident photons

Transmitted photons

ds

Absorbed photons

Scattered photons

Fig. 1 Schematic of light transport in participating medium. The photon beam is attenuated by absorption and out-scattering and augmented by emission and in-scattering processes

expressed as Iλ(s, s). At the same location, the radiative intensity of any other direction Ω can be expressed as Iλ(s, Ω). When the light beam (photons) moves from location s to s + ds, the balance of spectral radiative power Qλ (W) can be expressed as follows:

 ΔQλ ¼ ΔQλ, abs þ ΔQλ, out-scatt þ ΔQλ, emit þ ΔQλ, in-scatt |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} Attenuation

(1)

Augmentation

where ΔQλ denotes the variation of spectral radiative power along the differential ray-path ds, which can be calculated by definition as ΔQλ = [Iλ(s + ds, s)  Iλ(s, s)] dAdΩ, dΩ is a differential solid angle, and the four terms at right-hand side indicate the contributions of the basic interaction mechanisms in participating media, namely, absorption (ΔQλ, abs), scattering (ΔQλ, out - scatt and ΔQλ, in - scatt), and emission processes (ΔQλ, emit). Absorption process transfers the radiative power to kinetic energy of heat carriers (e.g., electrons and phonons), which only attenuates the radiative power. Thermal emission process only augments the radiative power. As for the scattering process, it can both attenuate and augment the radiative power depending on whether it is the scattering of the current light beam Iλ(s, s) to other directions, i.e., the out-scattering process, or the scattering of light beam of other direction Iλ(s, Ω) to current transfer direction s, i.e., the in-scattering process. The attenuated radiative power in the control volume is proportional to the incident radiative power, which for the processes of absorption and out-scattering can be established respectively as ΔQλ, abs ¼ κa, λ I λ ðs, sÞdsdAdΩ,

ΔQλ, out-scatt ¼ κs, λ I λ ðs, sÞdsdAdΩ

(2)

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J. M. Zhao and L. H. Liu

where κa,λ (m1) and κ s,λ (m1) are the spectral absorption coefficient and scattering coefficient, respectively. The emitted radiative power in the control volume is established based on the black body radiative intensity as ΔQλ, abs ¼ κa, λ I b, λ ½T ðsÞdsdAdΩ

(3)

in which Kirchhoff’s law of thermal radiation is applied, κa,λds can be viewed as the emissivity of the layer of medium with thickness ds, and Ib, λ is the black body spectral radiative intensity. For a black body in a transparent medium with refractive index nλ, Ib, λ is calculated from (Modest 2013) I b, λ ¼ n2λ I 0b, λ ¼

2hc2 n2λ λ5 ðehc=λkB T  1Þ

(4)

where I 0b, λ is the radiative intensity of black body in vacuum, c = 2.998  108 (m/s) is vacuum light speed, h = 6.626  1034 (J s) is the Planck’s constant, and kB = 1.3807  1023 (J/K) is the Boltzmann’s constant. Note that the wavelength λ denotes vacuum wavelength throughout this text. The scattering process generally changes the direction of incident photons. This angular redistribution of incident photons by scattering process is described by the scattering phase function, which expresses the ratio of radiative power scattered to each direction per solid angle. By definition, the scattering phase function Φλ(cosΘ) (sr1) must satisfy scattering energy conservation, which is often called normalization relation and written as 1 4π

ð Φλ ð cos ΘÞdΩ ¼ 1

(5)

4π 0

where cosΘ = Ω  Ω is the cosine of the angle between incident (Ω0) and scattering direction (Ω). For a light beam with radiative intensity Iλ(Ω0, s) incident on a differential control volume with volume dV, the total scattered radiative power by the scatterers in the control volume is κ s,λIλ(Ω0, s)dVdΩ0 according to Eq. (2), where dΩ0 is a differential solid angle related to the incident beam of direction Ω0. Then the scattered power from an arbitrary incident direction Ω0 to the current transfer 1 direction s is κs, λ I λ ðs, Ω0 Þ 4π Φλ ðΩ0  sÞdVdΩ0 dΩ . The total in-scattering radiative power augmentation from all directions can then be calculated by integration as ΔQλ, in-scatt ¼

κ s, λ 4π

ð

I λ ðs, Ω0 ÞΦλ ðΩ0  sÞdΩ0 dsdAdΩ

(6)

4π

By substitution of Eqs. (3), (2), and (6) into Eq. (1), the Lagrangian form of RTE can be obtained as

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Radiative Transfer Equation and Solutions

939

dI λ ðs, sÞ κ s, λ þ βλ I λ ðs, sÞ ¼ κa, λ I b, λ ½T ðsÞ þ ds 4π

ð

I λ ðs, Ω0 ÞΦλ ðΩ0  sÞdΩ0

(7)

4π

where βλ = (κ a,λ + κ s,λ) is the extinction coefficient. If the ray coordinate is not moved with beam propagation, namely, Eulerian frame is used. The radiative intensity will be function of time t, and the fixed ray coordinate s can be expressed as Iλ(s, t, s). In this case, the Lagrangian stream operator d/ds can be expanded as d @ @ dt @ nλ @ ¼ þ ¼ þ ds @s @t ds @s c @t

(8)

where c is the light speed in vacuum. Using Eq. (8), the RTE can be expressed in Eulerian form as nλ @I λ ðs, t, sÞ @I λ ðs, t, sÞ þ þ βλ I λ ðs, t, sÞ @t @s c ð κ s, λ ¼ κa, λ I b, λ ½T ðsÞ þ I λ ðs, t, Ω0 ÞΦλ ðΩ0  sÞdΩ0 4π

(9)

4π

Equations (7) and (9) are the basic form of RTEs in uniform refractive index media. As can be seen, for steady-state radiative transfer, Eqs. (7) and (9) are the same. Equation (9) is specially useful for transient radiative transfer analysis. The RTEs in other coordinate systems can be derived by simply expressing the stream operator in the coordinate system. In the following, only steady-state RTE is considered unless otherwise mentioned.

2.1.2 Cartesian Coordinate System Formulation In Cartesian coordinate system, radiative intensity is expressed as Iλ(s(x, y, z), Ω); hence, dI dx @I dy @I dz @I ¼ þ þ ds ds @x ds @y ds @z

(10)

Considering ds as the arc length along a curve, the coordinate transformation coefficients dx/ds, dy/ds, and dz/ds are the direction cosines of the transport direction Ω = μi + ηj + ξk. As such, Eq. (10) can be written as dI @I @I @I ¼ μ þ η þ ξ ¼ Ω  ∇I ds @x @y @z The RTE in Cartesian coordinate system can then be written as

(11)

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J. M. Zhao and L. H. Liu

Fig. 2 1D and 2D Cartesian system with variable defined to formulate the RTE. (a) 1D and (b) 2D

z

a

q

b q

O z

O

ϕ

y

x

κ s, λ Ω  ∇I λ ðr, ΩÞ þ βλ I λ ðr, ΩÞ ¼ κ a, λ I b, λ ½T ðrÞ þ 4π ð 0 0  I λ ðr, Ω ÞΦλ ðΩ  ΩÞdΩ0

(12)

4π

where r = xi + yj + zk is the spatial location vector. For 1D and 2D cases, the equation can be simplified by employing the symmetries of radiative intensity distribution, i.e., the axisymmetric around z-axis for 1D and mirror symmetry about z-axis for the 2D case as shown in Fig. 2. The 1D RTE can be written as dI λ ðz, ξÞ ξ þ βλ I λ ðz, ξÞ ¼ κa, λ I b, λ ½T ðzÞ þ dz 1 where ξ = cos θ, Φ1, λ ðξ0 , ξÞ ¼ 2π

2Ðπ

ð1

I ðz, ξ0 ÞΦ1, λ ðξ0 , ξÞdξ0

(13)

1

ΦðΩ0  ΩÞdφ is the 1D scattering phase func-

0

tion. The scattering phase function can be expanded in Legendre polynomials Pm as ΦðΩ0  ΩÞ ¼ ΦðcosΘÞ ¼ 1 þ

M X

Am Pm ðcosΘÞ

(14)

m¼1

where Am is the m-th order expansion coefficients. Then the 1D scattering phase function can be expressed as (Modest 2013) Φ1, λ ðξ0 , ξÞ ¼ 1 þ

M X

Am Pm ðξ0 ÞPm ðξÞ

(15)

m¼1


 For the 2D case, the mirror symmetry indicates I ðΩ, rÞ ¼ I Ω, r , where Ωm ¼ ½μ, η,  ξ; hence, the RTE can be written as (Zhao et al. 2013) @I λ @I λ κ s, λ þη þ βλ I λ ¼ κ a, λ I b, λ þ μ @x @y 4π

ð 2π

I λ ðr, Ω0 ÞΦ2, λ ðΩ0  ΩÞdΩ0

(16)

23

Radiative Transfer Equation and Solutions

a

941

b

z

ez

ez ϕ e Ψ

eΨ

q

r

z

Ω

ϕ

er

er y

y Ψ

x

Ψ

ρ

Ω

q

r

ρ

x

Fig. 3 Definition of the cylindrical coordinate system. (a) Type I and (b) Type II

  where Φ2, λ ðΩ0  ΩÞ ¼ Φλ ðΩ0  ΩÞ þ Φλ Ω0  Ω is the 2D scattering phase function, and the angular quadrature is only over half solid angular space at θ  [0, π/2].

2.1.3 RTE in Other Coordinate Systems Besides the Cartesian coordinate system, cylindrical and spherical coordinate systems are the other two commonly used orthogonal coordinate systems. Here the RTE in these coordinate systems is presented. Currently, two types of cylindrical coordinate system (ρ-Ψ -z-θ-φ) were proposed for radiative transfer analysis in literatures, whose definitions are shown in Fig. 3. The Type I cylindrical coordinate system is the traditional one (Modest 2013), and the Type II cylindrical coordinate system is a relatively new one proposed recently (Zhang et al. 2010). The difference between these two systems lies in the definition of local angular variables, i.e., the zenith angle θ and azimuthal angle φ. By definition, the optical plane of reflection or refraction at the cylindrical interfaces coincides with the iso-surface of azimuthal angle φ in the Type II system; hence, it facilitates the treatment of reflection/ refraction at the cylindrical interfaces/boundaries. For the Type I cylindrical coordinate system (Fig. 3a), the stream operator d/ds can be expanded as (Modest 2013) d dρ @ dΨ @ dz @ dθ @ dφ @ ¼ þ þ þ þ ds ds @ρ ds @Ψ ds @z ds @θ ds @φ η @ ¼ Ω  ∇I  ρ @φ

(17)

where Ω = μeρ + ηeΨ + ξez is the local direction vector of the beam; μ = sin θ cos φ; η = sin θ sin φ; ξ = cos θ; eρ, eΨ, and ez are the unit coordinate vector; and ∇I = eρ@/@ρ + eΨρ1@/@Ψ + ez@/@z is the gradient operator in the Type I cylindrical coordinate system. Hence the RTE in the Type I cylindrical coordinate system can be written as

942

J. M. Zhao and L. H. Liu

η @I λ κ s, λ þ β λ I λ ¼ κ a, λ I b, λ þ Ω  ∇I λ  ρ @φ 4π

ð

I λ ðr, Ω0 ÞΦλ ðΩ0  ΩÞdΩ0

(18)

4π

This is in nonconservative form, and it can be further rewritten in conservative form as e I λ  1 @ηI λ þ βλ I λ ¼ κa, λ I b, λ þ κs, λ Ω∇ I ρ @φ 4π

ð

I λ ðr, Ω0 ÞΦλ ðΩ0  ΩÞdΩ0

(19)

4π

e I ðÞ ¼ eρ ρ1 @ ðρÞ= @ρ þ eΨ ρ1 @ ðÞ= @Ψ þ ez ρ1 @ ðÞ= @z is a modwhere ∇ ified gradient operator in the Type I cylindrical coordinate system. For the Type II cylindrical coordinates system (Fig. 3b), the stream operator can be expanded as (Zhang et al. 2010; Zhao et al. 2012b) d cos φ @ sin φ cos φ @ ¼ Ω  ∇II  μ þξ ds ρ @θ ρ @φ

(20)

where Ω = μeΨ + ηez + ξeρ is the local direction vector of the beam and the gradient operator is given as ∇II = eΨρ1@/@Ψ + ez@/@z + eρ@/@ρ. The RTE in the Type II cylindrical coordinate system can thus be written as cos φ @I λ sin φ cos φ @I λ þξ þ βλ I λ ρ @θ ð ρ @φ κ s, λ ¼ κ a, λ I b, λ þ I λ ðr, Ω0 ÞΦλ ðΩ0  ΩÞdΩ0 4π

Ω  ∇II I λ  μ

(21)

4π

This is in nonconservative form, and it can be further rewritten in conservative form as 1 @  2  1 @ μ Iλ þ ½ξ sin φ cos φI λ  þ βλ I λ ρ sin θ @θ ρ @φ ð κ s, λ ¼ κ a, λ I b, λ þ I λ ðr, Ω0 ÞΦλ ðΩ0  ΩÞdΩ0 4π

e Iλ  Ω∇ II

(22)

4π

e II ðÞ ¼ eΨ ρ1 @ ðÞ= @Ψ þ ez @ ðÞ= @z þ eρ ρ1 @ ðρÞ= @ρ is a modified where ∇ gradient operator. For the spherical coordinate system (Θ-Ψ-ρ-θ-φ) defined in Fig. 4, the stream operator can be expanded as (Liu et al. 2008) d dΘ @ dΨ @ dρ @ dθ @ dφ @ ¼ þ þ þ þ ds ds @Θ ds @Ψ ds @ρ ds @θ ds @φ η cot Θ @ sin θ @  ¼Ω∇ ρ @φ ρ @θ

(23)

23

Radiative Transfer Equation and Solutions

Fig. 4 Definition of the spherical coordinate system

943

z

er eΨ

q

Θ

r

Ω

ϕ ρ

O

eΘ y

Ψ x

where Ω = μeΘ + ηeΨ + ξeρ is the local direction vector of the beam and the gradient operator is defined as ∇ = eΘρ1@/@Θ + eΨ(ρ sin Θ)1@/@Ψ + eρ@/@ρ. The RTE in the spherical coordinate system can be obtained in the nonconservative form as sin θ @I λ η cot Θ @I λ  þ βλ I λ ρ @θ ð ρ @φ κ s, λ ¼ κ a, λ I b, λ þ I λ ðr, Ω0 ÞΦλ ðΩ0  ΩÞdΩ0 4π

Ω  ∇I λ 

(24)

4π

and in the conservative form as
 1 @ 1  ξ2 I λ cot Θ @ηI λ  þ βλ I λ ρ sin θ ρ @φ @θ ð κ s, λ ¼ κ a, λ I b, λ þ I λ ðr, Ω0 ÞΦλ ðΩ0  ΩÞdΩ0 4π

e Iλ  Ω∇

(25)

4π

e ðÞ ¼ eΘ ðρ sin ΘÞ1 @ ð sin Θ Þ= @ Θ þ eΨ ðρ sin ΘÞ1 @ = @Ψ þ eρ ρ2 @ where∇ 2 ðρ Þ= @ρ is a modified gradient operator.

2.1.4 Overall Energy Conservation If the radiative intensity is known, then any other derived quantities such as radiative heat flux vector, incident radiation, radiation energy density, volumetric radiation source term (or divergence of radiative heat flux vector), absorbed radiative power per unit volume, etc. can be readily calculated. The radiative heat flux vector qλ (W/(m2μm)) and incident radiation G (W/(m2μm)) are calculated based on radiative intensity as

944

J. M. Zhao and L. H. Liu

ð qλ ðrÞ ¼

ð I λ ðr, ΩÞΩdΩ,

Gλ ðrÞ ¼

4π

I λ ðr, ΩÞdΩ

(26)

4π

The total radiative heat flux vector q and total incident radiation G can be calculated by a spectral integration with wavelength to qλ and Gλ, respectively. The net spectral radiative heat flux onto a surface element qs , λ (W/(m2μm)) can thus be calculated from qs, λ = qλ  nw. The spectral radiation energy density uλ (J/(m3μm)) is nλ uλ ð r Þ ¼ c

ð I λ ðr, ΩÞdΩ ¼ 4π

nλ Gλ ðrÞ c

(27)

The absorbed spectral radiative power per unit volume wλ (W/(m3μm)) can be calculated from w λ ðrÞ ¼ κ a G λ ðrÞ

(28)

The balance equation of spectral radiative heat flux can be obtained by integration of the RTE in Cartesian coordinates in Eq. (12) over entire solid angle, that is, ∇  qλ ¼ 4πκ a, λ I b, λ  κa, λ Gλ

(29)

The left-hand side of Eq. (29), namely, ∇  qλ, stands for the outflow radiation power per unit volume, which is thus can be understood as a volumetric radiation source term. The first term and the second term of the right-hand side are the emitted radiation power and the absorbed radiation power per unit volume, respectively. Balance equation of total radiative heat flux can be obtained as 1 ð

∇q¼


 κa, λ 4πI b, λ  Gλ dλ

(30)

0

which can be simplified for gray medium (κa, λ = κ a and nλ = n are constant) as
 ∇  q ¼ κa ð4πI b  GÞ ¼ κa 4n2 σT 4  G

(31)

where σ = 5.67  108 (W/(m2K4)) is the Stefan-Boltzmann constant. If temperature field is to be determined, and only radiation heat transfer is considered, the equation of overall energy conservation can be written as ρCv 000


 @T 000 ¼ Qrad ¼ κ a G  4n2 σT 4 @t

(32)

where Qrad ¼ ∇  q (W/m3) denotes the equivalent radiative heat source, ρ (Kg/m3) is the density, and Cv (J/(Kg K)) is the specific heat capacity of the medium. At equilibrium, the temperature field can be determined as

23

Radiative Transfer Equation and Solutions



G 4n2 σ

T¼

945

14 (33)

If combined mode heat transfer of conduction and radiation is considered, the governing equation of overall energy conservation can be written as @T 000 ¼ ∇  ðk∇T Þ þ Qrad @t
 ¼ ∇  ðk∇T Þ þ κa G  4n2 σT 4 ρCv

(34)

where k (W/(m K)) is the heat conductivity of medium. If further the convection heat transfer is also considered, the overall energy conservation equation can be written as

@T 000 000 ρCv þ u  ∇T ¼ ∇  ðk∇T Þ þ Qrad þ Qf @t
 000 ¼ ∇  ðk∇T Þ þ κa G  4n2 σT 4 þ Qf

(35)

000

where u (m/s) is the fluid velocity vector and Qf denotes the volumetric heat source other than thermal radiation, such as from viscous friction, chemical reaction, etc.

2.1.5 Boundary Conditions for RTE The inflow radiative intensity at the boundary walls must be set before the solution of the RTE. Generally speaking, three processes contributed to the emanated radiative intensity at the boundary walls, namely, the emission, reflection, and transmission. For a diffuse emitting and reflecting opaque wall, the boundary condition can be written as 1  ew, λ I λ ðrw , ΩÞ ¼ ew, λ I b, λ ½T ðrw Þ þ π

ð nw Ω0 >0

I λ ðrw , Ω0 Þj nw  Ω0 j dΩ0 , nw  Ω< 0 (36)

where ew λ is the wall spectral emissivity and nw is the normal vector of the wall pointing outside the enclosure. The boundary condition for diffuse wall (Eq. (36)) can also be written as (Modest 2013) I λ ðrw , ΩÞ ¼ I b, λ ½T ðrw Þ þ

1  e w, λ q  nw , nw  Ω < 0 πew, λ

(37)

For real opaque rough surfaces, there will be significant part of radiation reflection at the specular direction. As such, the wall reflection can be separated into a diffuse reflection part and a specular reflection part. In this case, the boundary condition can be written as

946

J. M. Zhao and L. H. Liu

I λ ðrw , ΩÞ ¼ ew, λ I b, λ ½T ðrw Þ þ

ρdw, λ π

ð nw Ω0 >0

I λ ðrw , Ω0 Þj nw  Ω0 j dΩ0

þ ρsw, λ I λ ðrw , Ωs Þ

(38)

where ρdw, λ and ρsw, λ are the diffuse and specular reflectivity, respectively, and Ωs is the corresponding incident direction of specular reflection, which can be determined as Ωs = Ω  2(Ω  nw)nw. If an external radiative source, such as a laser, a lamp, or a solar beam, is irradiated to a medium with semitransparent walls, the boundary condition can be written as I λ ðrw , ΩÞ ¼ ew, λ I b, λ ½T ðrw Þ þ þρsw, λ I λ ðrw , Ωs Þ

ρdw, λ π

ð 0

nw Ω >0

I λ ðrw , Ω0 Þj nw  Ω0 j dΩ0

(39)

þ τw, λ I ext ðrw , ΩÞ

where τw,λ is spectral transmittance of the semitransparent wall and Iext(rw, Ω) is the radiative intensity of the external source.

2.1.6 Numerical Properties of the Classical RTE The classical RTE (Eq. (12)) can be written shortly as Ω  ∇I þ β I ¼ S

(40)

where S is the source term accounting for thermal emission and in-scattering contribution. The wavelength subscript is omitted for brevity. The first term of the left-hand side of Eq. (40) can be seen as a convection term with a convection velocity of Ω, namely, μ, η, and ξ, which are taken as the velocity in x-, y-, and z- directions, respectively. Hence the RTE can be considered as a special kind of convectiondiffusion equation without the diffusion term (Chai et al. 2000b). The convectiondominated property is a source of numerical instability, which may cause unphysical numerical results, and shows strong ray effects (Chai et al. 2000a). In order for illustrating the numerical stability of the first-order RTE, Fig. 5a shows the example results solved by a finite element method discretization of the first-order RTE for a 1D case with a Gaussian-shaped emissive source at different optical thickness excerpted from Ref. Zhao and Liu (2007a). The solved radiative intensity distribution shows significant unphysical oscillations, which is a good demonstration of the instability issue caused by the convection-dominated characteristics of the RTE. Following the theoretical framework presented in Ref. Zhao et al. (2013), the solution error can be predicted in frequency domain as shown in Fig. 5b. The frequency range of the reduced frequency ϖ ¼ Δsϖ=2π is plotted in [0, 0.5], where ϖ denotes the angular frequency in Fourier analysis and Δs is the grid spacing. This is based on the fact that the maximum frequency (or shortest

23

a

Radiative Transfer Equation and Solutions

947

b

0.08

Exact FEM-RTE

0.06

103

tΔ= 0.01 tΔ= 0.1 tΔ= 1 tΔ= 10

tL= 0.1

I(z)

101 0.04

1.0

|EI|

0.02

10-1 10

0.00 0.0

0.2

0.4

0.6

0.8

1.0

10-3

z/L

0.0

0.1

0.2

ϖ

0.3

0.4

0.5

Fig. 5 Example results illustrating the numerical stability of the first-order RTE. (a) Intensity distribution solved by finite element method for the Gaussian-shaped emissive source problem, (b) theoretical frequency domain relative error of the RTE discretized using central difference scheme

wavelength) of a harmonic that can propagate on a uniform grid of spacing Δs is π/Δs (or wavelength 2Δs), namely, ϖ = 0.5. It can be seen that the relative error of intensity jEIj increases with ϖ for different grid optical thickness and the maximum relative error occurs at ϖ = 0.5 with a huge relative error greater than 300 for τΔ= 0.01. Hence significant error can be observed at around ϖ = 0.5, interpreting the observed high-frequency unphysical oscillations in Fig. 5a. The solution errors (especially at the high frequency) of the results obtained by the RTE reduces significantly with the increasing of grid optical thickness τΔ, indicating the solution error will decrease for problem with larger extinction coefficient on a specified grid, interpreting the observed decreasing of unphysical oscillations with increasing optical thickness in Fig. 5a.

2.2

The Second-Order Form of RTE

The classical form of the RTE under different coordinate systems has been presented in the previous section. The convection-dominated characteristics of the RTE may cause unphysical oscillation in the numerical results as discussed in the previous section. This type of instability occurs in many numerical methods, such as finite difference methods, finite element methods, and meshless methods, if no special stability treatment is taken (Chai et al. 2000b). It has been demonstrated that the classical RTE can be transformed to a new equation with a naturally introduced second-order diffusion term to circumvent the stability issue. In this section, several second-order form of RTEs is presented, including the even-parity formulation of RTE (EPRTE), the second-order radiative transfer equation (SORTE), and its variants.

948

J. M. Zhao and L. H. Liu

2.2.1 The Even-Parity Formulation The EPRTE is the first attempt that transforms the classic RTE into an equation with the second-order diffusion term and hence eliminates the convection-dominated property. The EPRTE was initially proposed in the field of neutron transport and has been used for decades. Song and Park (1992) initially applied the EPRTE in heat transfer field. It has been applied to DOM (Cheong and Song 1997) and FEM (Fiveland and Jessee 1995) discretization. In this approach, new variables, i.e., even- and odd-parity intensities, are defined as function of radiative intensity both at forward direction and backward direction, ψ E ðr, ΩÞ ¼


 1 I ðr, ΩÞ þ I r,  Ω 2

(41)

ψ O ðr, ΩÞ ¼


 1 I ðr, ΩÞ  I r,  Ω 2

(42)

By adding and subtracting the RTE (Eq. (12)) for forward direction Ω and backward direction Ω, respectively, it yields the governing equations of ψ E(r, Ω) and ψ O(r, Ω) for isotropic scattering media as Ω  ∇ψ O þ βψ E ¼ κ a I b þ

κs 2π

ð

ψ E ðr, Ω0 ÞdΩ0

(43)

2π

Ω  ∇ψ E þ βψ O ¼ 0

(44)

These two equations can be decoupled. From Eq. (44), ψ O =  β1Ω  ∇ψ E, which is substituted into the first term of Eq. (43) to obtain a second-order diffusiontype equation of ψ E, namely, the EPRTE,
 κs Ω  ∇ β1 Ω  ∇ψ E þ βψ E ¼ κa I b þ 2π

ð

ψ E ðr, Ω0 ÞdΩ0

(45)

2π

It is noted that the angular integration for the scattering term is only over half solid angular space. Since this is a second-order partial differential equation, boundary condition both at the inflow and outflow boundaries should be prescribed. Following similar approach, by adding and subtracting of the boundary condition (Eq.(36)) for forward direction Ω and backward direction Ω, the boundary condition for ψ E at inflow (nw  Ω < 0) and outflow (nw  Ω > 0) boundary is obtained as ψ E ðrw , ΩÞ  β1 Ω  ∇ψ ð E   1  ew ¼ ew I b ðrw Þ þ ψ E ðΩ, rw Þ þ β1 Ω  ∇ψ E j nw  Ω0 j dΩ0 π nw Ω0 >0 and

(46)

23

Radiative Transfer Equation and Solutions

949

ψ E ðrw , ΩÞ þ β1 Ω  ∇ψ ð E   1  ew ¼ ew I b ðrw Þ þ ψ E ðΩ, rw Þ  β1 Ω  ∇ψ E j nw  Ω0 j dΩ0 π nw Ω0 >0

(47)

respectively.

2.2.2 The Second-Order RTEs The SORTE (Zhao and Liu 2007a) and its variant (Zhao et al. 2013) proposed recently are diffusion-type equations similar to the heat conduction equation in anisotropic medium. Its governing variable is the radiative intensity, as compared to the EPRTE, of which the governing variable is the even parity of radiative intensity. These two approaches share similar stability due to the same basic underlying principle. The using of radiative intensity as solution variable is more convenient and easier to be applied to complex radiative transfer problems for the SORTEs, such as anisotropic scattering. The SORTE is rather easy to be derived based on the RTE. From Eq. (7), it is rearranged to have I ¼ β

1

dI ω þ ð1  ωÞI b þ ds 4π

ð

I ðs, Ω0 Þ ΦðΩ0  ΩÞdΩ0

(48)

4π

where ω = κs/β is the single scattering albedo. Substituting this relation back into the first term of the RTE, the SORTE is then obtained,  d dS 1 dI β  þ βI ¼ βS  ds ds ds

(49)

where S is the source function defined as S ¼ ð1  ωÞI b þ

ω 4π

ð

I ðΩ0 , sÞΦðΩ0  ΩÞdΩ0 4π

Following the approach in Sect. 2.1.2, the SORTE can be written in Cartesian coordinate as   Ω  ∇ β1 Ω  ∇I þ βI ¼ βS  Ω  ∇S

(50)


 ∇  K  ∇I ¼ βðI  SÞ þ Ω  ∇S

(51)

and rewritten as

where K ¼ β1 ΩΩ, which is similar to the tensorial heat conductivity for anisotropic medium, and the terms at the right-hand side can be viewed as effective heat source. Similar to the EPRTE, the boundary condition for the SORTE should be prescribed both at the inflow and outflow boundaries. Since the governing variable is

950

J. M. Zhao and L. H. Liu

radiative intensity, the boundary condition is straightforward, which is given at the inflow and outflow boundary as (Zhao and Liu 2007a) I ðrw , ΩÞ ¼ ew I b ðrw Þ þ

1  ew π

ð nw Ω0 >0

I ðrw , Ω0 Þj nw  Ω0 j dΩ0 , nw  Ω < 0

Ω  ∇I ðrw , ΩÞ þ βI ðrw , ΩÞ ¼ βSðrw , ΩÞ, nw  Ω > 0

(52) (53)

Note that the inflow boundary condition for SORTE is the same for that of the RTE and the outflow boundary condition is just the RTE itself. Following the similar principle, the modified SORTE (MSORTE) was proposed (Zhao et al. 2013), in which no β1 coefficient appears; hence, it is better in dealing with inhomogeneous media where some locations have very small/zero extinction coefficient. The MOSRTE is obtained by applying the stream operator d/ds once to the RTE, which can be written in ray-path coordinate as d2 I dβI dβS ¼ þ ds2 ds ds

(54)

and in Cartesian coordinate system as ðΩ  ∇Þ2 I þ Ω  ∇ðβI Þ ¼ Ω  ∇ðβSÞ

(55)

The boundary conditions for the MSORTE are the same as that for the SORTE.

2.2.3 Numerical Properties of the Second-Order RTE The second-order form of RTE contains a second-order diffusion term, which circumvents the convection-dominated characteristics of the RTE, and hence is numerical stable. The numerical properties of the RTE, SORTE, and MSORTE have been studied theoretically using Fourier analysis (Zhao et al. 2013), which confirms the stability of the second-order forms of RTE. Figure 6 gives a comparison of the predicted relative solution error in frequency domain for the first-order and second-order form of RTEs. As can be seen, at high frequency (ϖ closes to 0.5, which is the frequency of the unphysical oscillations), the relative error for the central difference discretization of the second-order form of RTEs is far less than (two orders of magnitude) that of the RTE, proving the numerical stability of the secondorder form of RTEs.

2.3

The Radiative Transfer Equation in Refractive Media

Due to the structural characteristics of a material or a possible temperature, pressure, and composition dependency, the refractive index of a media may be a function of spatial position. In this case, the ray goes along a curved path determined by the Fermat principle rather than along the straight lines. The formulation of the RTEs

23

Radiative Transfer Equation and Solutions

Fig. 6 The frequency domain distribution of solution error for the central difference discretization based on the RTE, the MSORTE, and the SORTE at different grid optical thickness (Zhao et al. 2013)

103

951

tΔ= 0.01

1

0.1

10

RTE

100 10-3 |EI|

10-1

SORTE

10-2 10-3 10-1

MSORTE

10-2 10-3

0.0

0.1

0.2

ϖ

0.3

0.4

0.5

presented in the previous section implicitly assumed straight-line ray path. The effect of ray curvature or gradient index refraction has to be taken into account to formulate the radiative transfer equation in refractive media. Hence the RTE in uniform index media cannot be applied to gradient index media. The radiative heat transfer in semitransparent media with graded index is of significant importance in thermooptical systems, atmospheric radiation, ocean optics, etc. and has evoked the wide interest of many researchers (Zhu et al. 2011; Asllanaj and Fumeron 2010; Sun and Li 2009; Liu 2006; Xia et al. 2002; Ben Abdallah and Le Dez 2000a, b). In this section, the radiative transfer equation in gradient refractive index media and its formulation under different coordinate system are presented.

2.3.1 Ray-Path Coordinate System Formulation When a light beam propagates in gradient index media, its direction will gradually change due to the effect of gradient of refractive index, besides the attenuation and augmentation effect caused by absorption, scattering, and emission processes, as shown in Fig. 7. The governing equation of radiative transfer in gradient index media can be considered as an extension of the RTE to take into account the effect of gradient of refractive index. The variation of radiative intensity along the curved ray path can be attributed to two mechanisms: the first is due to the variation of refractive index and the second is from any other processes as discussed in participating media of uniform refractive index distribution. The total variation of radiative intensity along the curved ray path can be written as dI λ ðs, nðsÞ, sÞ ¼

@I λ ðs, nðsÞ, sÞ @I λ ðs, nðsÞ, sÞ ds þ dn @s @n

(56)

952

J. M. Zhao and L. H. Liu

I(s, s)

I(s, s + ds) Emitted photons

s

s O

ds

Absorbed photons

Scattered photons

Fig. 7 Schematic of light transport in gradient index participating medium and variable definition in ray-path coordinate

where the first term on the right-hand side stands for the variation caused by the common processes of absorption, scattering, and emission, which can be expressed based on the RTE in uniform refractive index media as @I λ ðs, nðsÞ, sÞ κ s, λ ¼ βλ I λ ðs, sÞ þ κa, λ I b, λ ½T ðsÞ þ @s 4π ð  I λ ðs, Ω0 ÞΦλ ðΩ0  sÞdΩ0

(57)

4π

The second term stands for the variation only caused by variation of refractive index, which is needed to be explicitly calculated. The Clausius invariant relation for transparent gradient index media gives d[Iλ/n2] = 0, from which the second term in Eq. (56) can be obtained as @I λ Iλ dn ¼ 2 dn @n n

(58)

Substituting Eq. (58) into Eq. (56), then the radiative transfer equation in gradient index medium (GRTE) in Lagrange form along the ray coordinate can be obtained as  d I λ ðs, sÞ κ s, λ n þ βλ I λ ðs, sÞ ¼ κa, λ I b, λ ½T ðsÞ þ 2 ds n 4π ð  I λ ðs, Ω0 ÞΦλ ðΩ0  sÞdΩ0 2

(59)

4π

By expanding the Lagrangian stream operator in Eulerian frame (Eq. (8)), the transient GRTE is obtained,

23

Radiative Transfer Equation and Solutions

953

 nλ @I λ ðs, t, sÞ 2 @ I λ ðs, t, sÞ þn þ βλ I λ ðs, t, sÞ @t @s n2 c ð κ s, λ ¼ κa, λ I b, λ ½T ðsÞ þ I λ ðs, t, Ω0 ÞΦλ ðΩ0  sÞdΩ0 4π

(60)

4π

Note that the GRTE does not contain information about the curved ray path. To solve the equation, the ray equation (Born and Wolf 1970) must be solved, that is, d ðnsÞ ¼ ∇n ds

(61)

The wavelength subscript will be omitted without loss of generality. More detailed derivation of radiative transfer equation in gradient refractive index media, including light polarization, refers to Ref. Zhao et al. 2012b.

2.3.2 Cartesian Coordinate System Formulation To formulation the GRTE in Cartesian coordinate system, the stream operator needs to be explicitly expressed in this system. Assuming the radiative intensity is expressed as I(r, Ω) = I(x, y, z, θ, φ), then the stream operator can be expanded as d dx @ dy @ dz @ dθ @ dφ @ ¼ þ þ þ þ ds ds @x ds @y ds @z ds @θ ds @φ dθ @ dφ @ þ ¼Ω∇þ ds @θ ds @φ

(62)

To obtain the explicit formulation of the two angular derivatives, i.e., dθ/ds and dφ/ds, the ray equation (Eq. (61)) must be applied. Following the derivation in (Liu 2006), they can be written as

 dφ 1 ∇n dθ 1 ∇n ¼ s1  ¼ ðξΩ  kÞ  , ds sin θ n ds sin θ n

(63)

where s1 is an auxiliary vector defined as s1 =  sin φi + cos φj. Substituting Eqs. (62) and (63) into the GRTE in ray coordinate (Eq.(59)) and after some manipulations, the final conservative form of the GRTE in Cartesian coordinate system can be obtained as 1 @ ∇n ½I ðr, ΩÞðξΩ  kÞ  sin θ @θ n 1 @ ∇n þ ðκa þ κ s ÞI ðr, ΩÞ þ ½I ðr, ΩÞ s1   sin θ @φ n ð κs ¼ κa I b ðrÞ þ I ðr, Ω0 Þ ΦðΩ, Ω0 Þ dΩ0 4π 4π

Ω  ∇I ðr, ΩÞ þ

(64)

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J. M. Zhao and L. H. Liu

The complete derivation refers to the work of Liu (2006). As compared to the RTE in Cartesian coordinate system (Eq. (12)), it is seen that there are two additional terms related to the gradient of refractive index (the second and third term) that appears in Eq. (64), which are also terms about derivatives of angular variable θ and φ and are usually called angular redistribution terms in literatures. It is angular redistribution terms that account for the effect of gradient refractive index distribution.

2.3.3 Formulation in Other Coordinate Systems The formulation of GRTE in cylindrical and spherical coordinate system is presented here. Two types of the cylindrical coordinates system are considered as shown in Fig. 3. Following similar procedure outlined in Sect. 2.3.2, the GRTE in the Type I cylindrical coordinate system (ρ-Ψ -z-θ-φ) can be derived and written in conservative form as (Liu et al. 2006) e II  Ω∇





 1 @ηI 1 @ ∇n @ ∇n þ s1  ðξΩ  ez Þ  I þ I ρ @φ sin θ @θ n @φ n

þ ðκa þ κ s ÞI ðr, ΩÞ ð κs I ðr, Ω0 Þ ΦðΩ, Ω0 Þ dΩ0 ¼ κa I b ðrÞ þ 4π 4π

(65)

where Ω = μeρ + ηeΨ + ξez is the local direction vector of the beam; μ = sin θ cos φ; e I ðÞ ¼ eρ η = sin θ sin φ; ξ = cos θ; eρ, eΨ and ez are the unit coordinate vector; ∇ 1 1 1 ρ @ ðρ Þ= @ ρ þ eΨ ρ @ ðÞ= @ Ψ þ ez ρ @ ðÞ= @ z is a modified gradient operator in the Type I cylindrical coordinate system, and the auxiliary vector is defined as s1 =  eρ sin φ + eΨ cos φ. Similarly, the GRTE in the Type II cylindrical coordinate system (Ψ -z-ρ-θ-φ) can be obtained and written in conservative form as (Zhang et al. 2010; Zhao et al. 2012b)   e II I  1 @ μ2 I þ 1 @ ½ξ sin φ cos φI  Ω∇ ρ sin θ @θ ρ @φ





 1 @ ∇n @ ∇n þ s1  ðξΩ  ez Þ  I þ I sin θ @θ n @φ n þ ðκa þ κ s ÞI ðr, ΩÞ ð κs I ðr, Ω0 Þ ΦðΩ, Ω0 Þ dΩ0 ¼ κ a I b ðrÞ þ 4π 4π

(66)

where Ω = μeΨ + ηez + ξeρ is the local direction vector of the beam, the auxiliary e II ðÞ ¼ eΨ ρ1 @ ðÞ= @ Ψ þ ez @ ðÞ= @ vector s1 =  sin φeΨ + cos φez, and ∇ 1 z þ eρ ρ @ ðρ Þ= @ ρ is a modified gradient operator in the Type II cylindrical coordinate system.

23

Radiative Transfer Equation and Solutions

955

The definition of the spherical coordinate system (Θ-Ψ-ρ-θ-φ) is shown in Fig. 4. Following the procedure outlined in Sect. 2.3.2, the GRTE in the spherical coordinate system can be obtained and written in conservative form as follows (Liu et al. 2006). 1 @sin 2 θI cot Θ @ηI  ρ sin θ @θ ρ @φ

  


 ∇n 1 @ @ ∇n ξΩ  eρ  s1  þ I þ I sin θ @θ n @φ n

eI Ω∇

þ ðκ a þ κs ÞI ðr, ΩÞ ð κs I ðr, Ω0 Þ ΦðΩ, Ω0 Þ dΩ0 ¼ κa I b ðrÞ þ 4π 4π

(67)

where Ω = μeΘ + ηeΨ + ξeρ is the local direction vector of the beam, s1 =  sin e is defined as ∇ e ðÞ φeΘ + cos φeΨ is an auxiliary vector the gradient operator, and ∇ 1 1 2 2 ¼ eΘ ðρ sin ΘÞ @ ð sin Θ Þ= @ Θ þ eΨ ðρ sin ΘÞ @ = @ Ψ þ eρ ρ @ ðρ Þ= @ ρ.

3

Solution Techniques of the Radiative Transfer Equation

In this section, four fundamental deterministic methods for radiative transfer in participating media are introduced, including the spherical harmonics method, the DOM, FVM, and FEM. The spherical harmonics method is highly efficient for complex multidimensional problems. The DOM, FVM, and FEM are versatile, with good accuracy and convenient to be coupled with conduction and convection solvers. Note that several important aspects of numerical solution of radiative transfer equation are not covered in this chapter, such as the spectral models for non-gray media, problems with collimated irradiation, and transient radiative transfer.

3.1

Spherical Harmonics Method

Spherical harmonics method also known as PN approximation is one basic type of method to solve radiative transfer. Especially, the lower-order approximations, such as P1 and P3 approximation, have achieved broad range of applications (Mengüç and Viskanta 1985; Mengüç and Iyer 1988). It is considered to suffer less from the ray effects, which can significantly deteriorate the accuracy of discrete-ordinate method. In the spherical harmonics method, the angular dependence of radiative intensity is expanded as a series of spherical harmonics, and the expansion coefficients are finally formulated into a set of partial differential equations to be solved. In this approach, the radiative intensity is approximated as

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J. M. Zhao and L. H. Liu

I ðr, ΩÞ ¼

1 X

m Im l ðrÞY l ðΩÞ

(68)

l¼0

where Y m l ðΩÞ are spherical harmonics, which are orthogonal functions in solid angular space, and I m l ðrÞ are the corresponding expansion coefficient. The governing equations for I m l ðrÞ can be obtained by substituting Eq. (68) into the RTE (Eq. (12)), and then do weighted angular integration with different orders of spherical harmonics. It is usually very cumbersome to obtain the equations of I m l ðrÞ, especially for higher-order approximation for multidimensional problems. Here, the P1 approximation is presented. For the P1 approximation, the expansion in Eq. (68) is truncated for l > 1. Four terms are retained in the series, and the radiative intensity can be shortly written as I ðr, ΩÞ ¼ aðrÞ þ bðrÞ  Ω

(69)

Substitute Eq. (69) into the RTE (Eq. (12)) to obtain Ω  ∇aðrÞ þ ðΩΩÞ : ∇bðrÞ þ β½aðrÞ þ bðrÞ  Ω ¼ κa I b þ κs aðrÞ þ κs gbðrÞ  Ω Note that the last term is obtained using the following relation ð ð 2 sin θ cos φ 3 0 0 0 Ω ΦðΩ  ΩÞdΩ ¼ 4 sin θ sin φ 5Φð cos θÞ sin θdθdφ ¼ 4πgΩ cos θ 4π

(70)

(71)

4π

where g is by definition the asymmetry factor of the scattering phase function. Equation (71) is in general not limited to the linear anisotropic scattering phase function. Integrate Eq. (70) for the zeroth and first moment to obtain the following two equations. ∇  bðrÞ ¼ 3κ a ½I b  aðrÞ

(72)

1 ∇aðrÞ β  κs g

(73)

bð r Þ ¼ 

The P1 approximation equation can be obtained by substituting Eq. (73) into Eq. (72). The physical meaning of a(r) and b(r) can be made clear by substituting Eq. (69) into the definition formula of radiative heat flux and incident radiation ð rÞ 3 (Eq. (26)), which yields aðrÞ ¼ G4π and bðrÞ ¼ 4π q. Finally, the P1 approximation equations are summarized below.  G equation :

∇

1 ∇G ¼ κa ð4πI b  GÞ 3ð β  κ s gÞ

(74)

23

Radiative Transfer Equation and Solutions

q equation : I equation :

957

q¼

I ðr, ΩÞ ¼

1 ∇G 3ðβ  κs gÞ

1 ½GðrÞ þ 3qðrÞ  Ω 4π

(75) (76)

By using the I equation, the boundary condition for diffuse emission and reflection boundary can be determined as 2e 2 nw  ∇G þ G ¼ 4πI bw e 3ð β  κ s gÞ

(77)

At radiative equilibrium, namely, ∇  q = 0 and G = 4πIb, the q-equation (Eq. (75)) indicates q¼

4π ∇I b 3ð β  κ s gÞ

(78)

This is the same as the Rosseland approximation (or diffusion approximation). As can be seen, only one equation is needed to be solved (G equation) for radiative heat transfer; hence, it is highly efficient to be used to analyze engineering radiative transfer problems. Even though the P1 approximation can give reasonable results for optically thick media, it may produce significant errors for optically thin media, in case the approximation Eq. (69) fails. The accuracy of P1 approximation can be improved by using higher-order spherical harmonics, such as P3 approximation. There are also a variant of PN approximation, called the simplified PN approximation (SPN) (Larsen et al. 2002), which can generate equations consistent with P1 approximation at low order and can be relatively easy to be extended to higher order.

3.2

Discrete-Ordinate Method

The discrete-ordinate method for the solution of radiative transfer was first proposed by Chandrasekhar (1960). It was then introduced to solve neutron transport, such as the work of Carlson and Lathrop (1965). Fiveland (1984) and Truelove (1988) applied the method to solve general radiative heat transfer problems. A recent review of the DOM and FVM was given by Coelho (2014). In the following, the basic principle of the method is presented. The solution of radiative transfer equation requires discretization of both angular and spatial domains. The idea of DOM is to represent the angular space by a discretized set of directions, and only radiative intensity at these discrete directions is solved. Each direction is associated with a quadrature weight. Both the directions and the weight are chosen carefully to ensure accuracy of angular integration, which is important for discretizing the in-scattering term and calculating the radiative heat flux. After the angular discretization is

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J. M. Zhao and L. H. Liu

finished, the original integral-differential form of RTE becomes a set of coupled partial differential equations, which can then be discretized and solved by traditional techniques for solving partial differential equations.

3.2.1 Angular Discretization The angular space is discretized as a set of discrete directions, Ωm = μmi + ηmj + ξmk; then the RTE (Eq. (12)) can be written into a set of partial differential equations, namely, the discrete-ordinate equations, as Ωm  ∇I m ðrÞ þ β I m ðrÞ ¼ κa I b ðrÞ þ

M κs X I m0 ðrÞ ΦðΩm0  Ωm Þ wm0 , 4π m0 ¼1

(79)

m ¼ 1, . . . , M where wm is the weight of direction Ωm for angular quadrature and M is the total number of discrete discretions. For the opaque and diffuse boundary, the boundary conditions for each discrete-ordinate equation are written as I w ðΩm Þ ¼ ew I bw þ

1  ew π

X

I w ðΩm0 Þj nw  Ωm0 j wm0 , Ωm  nw < 0

(80)

nw Ωm0 >0

By definition, the radiative heat flux and incident radiation are determined as qðrÞ ¼

M X

I m ðrÞΩm wm , GðrÞ ¼

m¼1

M X

I m ðrÞ wm

(81)

m¼1

The definition of the angular discretization includes the selection of the set discrete directions (ordinates) and the design of the related angular quadrature, which is critical about the accuracy of the method. There are several criteria proposed on the selection of angular discretization and the weights (Fiveland 1984; Carlson and Lathrop 1965): (1) The symmetry criterion, namely, the discrete set of directions and weights, should be the same after the rotation of π/2 about each principle axis (x-,y- and z-). (2) The full space moment preserving criterion, i.e., the angular quadrature defined based on the selected directions, should satisfy the zeroth, first, and second moments integrated over 4π, namely, ð dΩ ¼ 4π ¼ 4π

ð ΩdΩ ¼ 0 ¼ 4π

M X

wm

(82)

Ωm w m

(83)

m¼1 M X m¼1

23

Radiative Transfer Equation and Solutions

ð ΩΩdΩ ¼ 4π

959

M X 4π δ¼ Ωm Ωm w m 3 m¼1

(84)

where 0 denotes the zero vector and δ is the unit tensor. (3) The half space moment preserving criterion, i.e., the defined angular quadrature, should preserve the first moment integration over 2π, requiring ð μdΩ ¼ π ¼

X

μm wm

μm >0

μ>0 ð

ηdΩ ¼ π ¼

X

ηm wm

ηm >0

η>0 ð

ξdΩ ¼ π ¼

X

(85)

ξm w m

ξm >0

ξ>0

The most well-known family of the discrete-ordinate set is the SN sets, initially proposed for simulation of neutron transport (Carlson and Lathrop 1965), which are also tabulated in the classic textbooks (Howell et al. 2011; Modest 2013). The angular discretization using the SN discrete-ordinate set is usually called SN-approximation. For the SN-approximation, such as S2, S4, or S6, N means the number of discrete direction cosines used for each principal direction. Total number of directions for the SN-approximation is M = N(N + 2). Several other discrete-ordinate sets were also proposed, such as the TN sets by Thurgood et al. (1995). A recent review of the angular discretization schemes in DOM was given by Koch and Becker (2004).

3.2.2 Spatial Discretization After angular discretization, the resulting discrete-ordinate equations (Eq. (79)) for each direction Ωm can then be discretized by common methods for solving partial differential equations, such as finite difference method and FVM. Note that upwinding scheme is required to obtain reliable results considering the numerical property of the RTE discussed in Sect. 2.1.6; an alternative is to use the second-order form of RTE for discretization. Here, the FVM is used to discretize Eq. (79) to obtain the final algebraic equation. The advantage of FVM is that it is easy to be applied to unstructured mesh to solve problems with complex geometry. Figure 8 shows the 2D grid used to define the FVM discretization scheme. The unknown radiative intensities are stored at the center of each grid cell. By integrating Eq. (79) over the control volume P and using the Gaussian divergence theorem to the first term, it yields
Δx μm I m,

e

 μ m I m,

 w


þ Δy ηm I m,

n

 η m I m,

 s

þ βP I m, P ΔxΔy ¼ Sm, P ΔxΔy

(86)

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J. M. Zhao and L. H. Liu

Δx

Fig. 8 Grid used to define the FVM discretization

N

n W

Δy

w

P

Ωm

e

E

s S

y x where the subscript of capitalized letters denotes the value at cell center (P) and subscript of small letters (e, w, n and s) denotes values at the center of faces as shown in Fig. 8; Sm , P is the source term defined as Sm, P ¼ Sm ðrP Þ ¼ κ a I b ðrP Þ þ

M κs X I m0 ðrP Þ ΦðΩm0  Ωm Þ wm0 4π m0 ¼1

(87)

Since radiative intensities are stored at cell center only, interpolation of radiative intensity at faces to that at cell center is required to obtain the final algebraic equations. Using interpolation and supposing uniform grid, the face values can be interpolated using neighboring cell values as
 I m, n ¼ αy I m, P þ 1  αy I m, N , I m, e ¼ αx I m, P þ ð1  αx ÞI m, E
 I m, s ¼ αy I m, S þ 1  αy I m, P , I m, w ¼ αx I m, W þ ð1  αx ÞI m, P

(88a) (88b)

where αx and αy are the interpolation parameters for x and y directions, respectively. Substituting Eq. (88) into Eq. (86), the final discretization can be written as aP I m, P ¼ aN I m, N þ aS I m, S þ aW I m, W þ aE I m, E þ b

(89)

where the discrete coefficients are determined as
 aN ¼ ηm Δy αy  1 , aE ¼ μm Δxðαx  1Þ, aS ¼ ηm Δyαy , aW ¼ μm Δxαx aP ¼ aN þ aS þ aW þ aE þ βP ΔxΔy, b ¼ Sm, P ΔxΔy

(90)

23

Radiative Transfer Equation and Solutions

961

Different differencing schemes can be obtained by defining different interpolation parameters, such as (1) step scheme (or first-order upwind scheme), αx = unitstep (μm) and αy = unitstep(ηm), and (2) diamond scheme (or central difference scheme), αx = αy = 1/2. The resulting linear systems in Eq. (89) can be solved element by element until the convergence, which can also be solved by sparse solvers. Since the source term contains radiative intensity of other directions, a global iteration is required to successively update the source terms for problems with scattering media and reflecting boundary conditions. The discretization presented above can be easily extended to 3D problems. The extension to unstructured mesh is presented in the next section.

3.3

Finite Volume Method

Raithby and Chui (Chui and Raithby 1992; Raithby and Chui 1990) firstly formulated FVM for solving radiative heat transfer problems. Chai and coworkers (Chai and Lee 1994; Chai et al. 1993) developed different variant implementation of FVM. A comprehensive review of the development of DOM and FVM was given recently by Coelho (2014). In the FVM method, both the angular domain and the spatial domain are discretized by using control volume integration. The angular discretization in FVM is thus different from the DOM. The angular domain discretization in FVM uses structured mesh as shown in Fig. 9, while the spatial domain can be structured or unstructured. Though different from DOM, the FVM can be formulated in a very similar fashion with DOM. As such the major difference between DOM and FVM lies in the angular discretization.

3.3.1 Angular Discretization Integrate the RTE (Eq. (12)) over a small control angle of Ωml centered at direction Ωml (the superscript m and l denotes the index of θ and φ discretization, respectively), and assuming the radiative intensity is constant in Ωml, it leads to

z

Physical coordinate system

θ

Control angle

Reference coordinate system

O

x

y

O

Fig. 9 Schematic of angular grid used in FVM discretization

ϕ

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J. M. Zhao and L. H. Liu

ð ΩdΩ  ∇I ml ðrÞ þ β Ωml I ml ðrÞ ¼ Ωml Sml ðrÞ

(91)

Ωml

where Sml(r) is given as Sml ðrÞ ¼ κ a I b ðrÞ þ  00  in which Φ Ωm l  Ωml ¼

Nφ X Nθ  00  00 κs X 0 0 I m l ðrÞΦ Ωm l  Ωml Ωm l 4π l0 ¼1 m0 ¼1 ð ð   0 0 Φ Ωm l  Ωml dΩ0 dΩ. ml

(92)

1 00 Ωm l Ω

Ωml Ωm0 l0

Dividing Ω to both sides of Eq. (91), an equation in the form of discreteordinate equation (Eq. (79)) is obtained, which is written as ml

ml

Ω  ∇I ml ðrÞ þ β I ml ðrÞ ¼ Sml ðrÞ where Ω

ml

(93)

is an averaged direction vector defined as Ω

ml

1 ¼ ml Ω

ð ΩdΩ Ω

(94)

ml

ml

Note that Ω can be calculated analytically (Murthy and Mathur 1998). Taking ml Ω as the equivalent discrete direction in DOM, the quantities, such as heat flux, incident radiation, etc., can be calculated the same way as in the DOM.

3.3.2 Spatial Discretization Since mathematical form of Eq. (93) is the same as the discrete-ordinate equation, the spatial FVM discretization procedure described in the previous section for DOM can be directly applied. Here only formulation on unstructured mesh is introduced, which can also be applied to the DOM for spatial discretization instability. Considering the unknowns are stored in the center of the control volume cell, integrating Eq. (93) over a spatial control volume and applying the Gaussian divergence theorem to the first term leads to Nf X

ml

ml ml Afi I ml fi Ω  nfi þ βP I P ΔV P ¼ SP ΔV P

(95)

i¼1

where the subscript fi denotes value at the i-th face of control volume centered at P, the subscript P denotes value at point P, n is the surface normal, A is area of the face, and ΔVP is the volume of the control volume centered at P. To complete the

23

Radiative Transfer Equation and Solutions

963

discretization, the intensity defined at surface should be interpolated to nodal values (volume center), which can be generally written as ml ml I ml fi ¼ αi I P þ ð1  αi ÞI Pi

(96a)

where Pi denotes the center of the neighboring cell of i-th face of the cell P. With this closure relation, the final FVM discretization can be written as aP I ml P ¼

Nf X

ml aPi I ml Pi þ b

(97)

i¼1

where the discrete coefficients are given as

aP ¼

Nf X ml Afi αi Ω  nfi þ βP ΔV P i¼1 ml

aPi ¼ Afi ðαi  1ÞΩ  nfi bml ¼ Sml P ΔV P

(98)

 ml  If the step scheme (or first-order upwind) is used, then αi ¼ unitstep Ω  nfi : Equation (97) for each control angle can be solved element by element and iterates until convergence.

3.4

Finite Element Method

Fiveland and Jesse (1994) were the first to apply the FEM to solve radiative heat transfer problems based on the differential form of RTE. Until recently, many variant implementations of FEM have been proposed (Liu 2004b; W. An et al. 2005; Zhao and Liu 2007a; Zhang et al. 2016). After angular discretization, as described in Sects. 3.2 and 3.3, the RTE becomes a set of partial differential equations, i.e., the discreteordinate equations, which can then be solved by common numerical method for solving partial differential equations. Similar to FVM, FEM is another versatile method that can be applied to solve a broad range of partial differential equations that appeared in scientific and engineering problems and hence is very appealing for multiphysics simulation. The feature of FEM is that it usually owns higher order of accuracy as compared to the FVM. In the FEM, the unknown radiative intensity is first approximated as a series of shape functions, which is then combined with the weighted residual approach to discretize the RTE; finally a sparse linear system is obtained and solved by general solvers. Here the FEM for solving the RTE is introduced.

964

J. M. Zhao and L. H. Liu

a

b 3

3

φi (x)

s

x

Txy

1

i–1

i

i+1

Trs

x

2

Δx

1

2

(0,0)

r

Fig. 10 (a) Schematic of nodal shape function of 1D linear elements in FEM. (b) Transform of linear elements defined on reference space to physical space

3.4.1 Function Approximation The solid angular space discretization is by common the discrete-ordinate approach, such as the SN sets or the FVM approach described in Sects. 3.2 and 3.3. The radiative intensity for each discrete direction Ωm can be approximated using the FEM shape functions ϕi at each solution nodes, namely, I~m ðxÞ ’

N sol X

I m, i ϕi ðxÞ

(99)

i¼1

where Im, i are the expansion coefficients, which are also the values of the radiative intensity of direction Ωm at node i (namely, Im,i = I(Ωm,xi)) due to the Kronecker delta property of the FEM shape functions. For example, the global shape function at node i for the 1D linear element can be written as 8 < ðx  xi1 Þ=Δx, ϕi ðxÞ ¼ ðxiþ1  xÞ=Δx, : 0,

xi1  x < xi xi  x < xiþ1 , otherwise

(100)

which is also graphically shown in Fig. 10a for better understanding. Generally, for multidimensional complex elements, the shape function on an element can be first defined in a reference space, and then transformed to the physical space, as shown in Fig. 10b, where a linear triangular element is taken as an example. For 2D triangular element, the three shape functions defined in reference space (r-t) can be written as Γ1 ðr, sÞ ¼ 1  r  s, Γ2 ðr, sÞ ¼ r, Γ3 ðr, sÞ ¼ s

(101)

which can be transformed to obtain the shape function in physical space (x-y) as ϕi ðxðr, sÞ, yðr, sÞÞ ¼ Γi ðr, sÞ, i ¼ 1, 2, 3

(102)

The coordinate system transformation from reference element Trs: {(0, 0), (1, 0), (0, 1)} to physical element Txy:fð^x 1 , ^y 1 Þ, ð^x 2 , ^y 2 Þ, ð^x 3 , ^y 3 Þg is defined as

23

Radiative Transfer Equation and Solutions

x¼

3 X

^x i Γi ðr, sÞ,

i¼1

965

y¼

3 X

^y i Γi ðr, sÞ

(103)

i¼1

where ^x i , ^y i , i = 1 , 2 , 3 denote the coordinates of the nodes that define the triangular element.

3.4.2 Weighted Residual Approach The discrete-ordinate equation (Eq. (79)) can be written as Ωm  ∇I m ðrÞ þ β I m ðrÞ ¼ Sm ðrÞ

(104)

where the source term S(r, Ω) is defined as Sm ð r Þ ¼ κ a I b ð r Þ þ

M κs X I m0 ðrÞ ΦðΩm0  Ωm Þ wm0 4π m0 ¼1

(105)

Using weighted residual approach, Eq. (104) is weighted by a set of weight functions Wj(r) and integrated over the solution domain, which leads to (Liu et al. 2008) 

     Ωm  ∇I m ðrÞ, W j ðrÞ þ β I m ðrÞ, W j ðrÞ ¼ Sm ðrÞ, W j ðrÞ ð where the inner product <  ,  > is defined as < f , g >¼ fgdV.

(106)

V

By substituting the approximated radiative intensity (Eq. (99)) into weighted residual approach equation (Eq. (106)) and choosing different set of weight functions, a discrete set of linear equations can be obtained, which can be written in matrix form as Km um ¼ hm (107)   where um ¼ um, i i¼1, Nsol ¼ ½I m ðri Þi¼1, Nsol , Km, and hm are conventionally called stiff matrix and load vector, respectively, which are different for different FEM discretization. The selection of the weighted function results in different FEM discretization schemes, such as for Galerkin scheme (Galerkin FEM), which chooses Wj = ϕj, and for least-squares scheme (LSFEM) to choose Wj = Ωm  ∇ϕj + βϕj. Note that the LSFEM formulation can also be derived based on functional minimization procedure. For the Galerkin FEM discretization, Km and hm are obtained as (Liu et al. 2008)   Km ¼ K m, ji j¼1, Nsol ;i¼1, Nsol ¼ < Ωm  ∇ϕi , ϕj > þ < βϕi , ϕj >   hm ¼ hm, j j¼1, Nsol ¼< Sm , ϕj >

(108a) (108b)

For the LSFEM discretization, Km and hm are obtained as (Zhao et al. 2012a)

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J. M. Zhao and L. H. Liu

  Km ¼ K m, ji j¼1, Nsol ; i¼1, Nsol ¼< Ωm  ∇ϕi , Ωm  ∇ϕj > þ < βϕi , βϕj >

(109a)

þ < βϕi , Ωm  ∇ϕj > þ < Ωm  ∇ϕi , βϕj >   hm ¼ hm, j j¼1, Nsol ¼< Sm , βϕj > þ < Sm , Ωm  ∇ϕj >

(109b)

It is noted that the stiff matrix produced by LSFEM is symmetric and positive definite, which is a very good numerical property. The accuracy in the imposing of this type of boundary condition is very important for the overall solution accuracy. One accurate method for imposing the Dirichlettype boundary condition is operator collocation approach. In this approach, the row of stiff matrix Km corresponding to the inflow boundary nodes is replaced with the discrete operator of the related boundary condition. Similar modification is also applied to the load vector hm. The modification algorithm is formulated as below (the modification is only conducted for nodes on the inflow boundary, namely, nw(rj)  Ωm < 0). K m, ji ¼ δji ,
 hm, j ¼ I 0 rj , Ωm

(110a) (110b)

where I0(rj, Ωm) stands for the radiative intensity at the boundary given by Eq. (80). The FEM has also been successfully applied to the second-order form of RTEs to avoid the stability problem caused by the convection-dominated property of the firstorder RTE, and this kind of FEM has been demonstrated to be numerically stable and accurate (Fiveland and Jessee 1994; Zhao and Liu 2007a; Zhang et al. 2016).

3.5

Solution Methods for RTE in Refractive Media

Many numerical methods have been developed for the solution of radiative transfer in gradient index media, which include the curved ray-tracing-based methods (Ben Abdallah and Le Dez 2000b; Ben Abdallah et al. 2001; Huang et al. 2002a, b; Liu 2004a; Wang et al. 2011) and the methods based on discretization of the GRTE. The ray tracing is usually cumbersome and time consuming in calculation. Lemonier and Le Dez (Lemonnier and Le Dez 2002) pioneered the discrete-ordinate method for solving the GRTE. Their work is for one-dimensional problem. Thereafter, Liu (2006) formulated the discrete-ordinate equation of GRTE for general multidimensional problems, which forms the basis for the solution of radiative transfer in gradient index media. It is like the role of discrete-ordinate equation of RTE in uniform index media. Based on the discrete-ordinate equation, the spatial

23

Radiative Transfer Equation and Solutions

967

discretization techniques, such as FVM and FEM, can be readily applied for solution. Besides the FEM and FVM, many other numerical methods have been developed to solve radiative heat transfer in gradient index media based on the discrete-ordinate equation of GRTE (Zhao and Liu 2007b; Sun and Li 2009; Asllanaj and Fumeron 2010; Zhang et al. 2015). In this section, only the solution techniques based on the discretization of the GRTE is introduced. Generally, the basic principle to solve the GRTE is the same as that for radiative transfer in uniform index media. As outlined in previous sections, the first is to discretize the angular space to transform the integral-differential equation into a set of partial differential equations, namely, the discrete-ordinate equations. The second step is to spatially discretize the discrete-ordinate equations, which can be done by common methods such as finite difference, FVM and FEM, etc. However, the GRTE differs from the RTE in uniform index media for containing two angular redistribution terms, which have derivatives with respect to angular variables. The discrete-ordinate equation of GRTE is the key for different solution methods. However, to determine the discrete-ordinate equations for GRTE is not so straightforward as that for the RTE (Lemonnier and Le Dez 2002; Liu 2006). In the following, the discrete-ordinate equations for GRTE are presented. The detailed derivation refers to Liu (2006) Liu and Tan (2006). Formally, the discreteordinate equations of GRTE can be written as   ∇n @ Ωm, n  ∇I ðr, Ωm, n Þ þ sin1 θ @θ fI ðr, ΩÞðξΩ  kÞg Ω¼Ωm, n  n h i ∇n m, n @ þ sin1 θ @φ þ κ ðI ðr, ΩÞ s1 Þ  ð þ κ ÞI ð r, Ω Þ a s Ω¼Ωm, n n N N φ     0 0 θ κs X X 0 0 0 0 n I r, Ωm , n Φ Ωm , n , Ωm, n wm ¼ κa Ib þ θ wφ 4π m0 ¼1n0 ¼1

(111)

where piecewise constant angular quadrature (PCA) is used to discrete the angular space, in which the total solid angle is divided uniformly in the polar θ and azimuthal φ directions, defined as θm ¼ ðm  1=2ÞΔθ, m ¼ 1,   , N θ

(112a)

φn ¼ ðn  1=2ÞΔφ, n ¼ 1,   , N φ

(112b)

where Δθ = π/Nθ and Δφ = 2π/Nφ are steps for the discretization of polar and azimuthal angles, respectively, and Nθ and Nφ are the corresponding number of divisions. For each discrete direction (m,n), the corresponding weight is m1=2 wm  cos θmþ1=2 , wnφ ¼ φnþ1=2  φn1=2 , θ ¼ cos θ

where θm + 1/2 = (θm + θm + 1)/2, φn + 1/2 = (φn + φn + 1)/2. The two angular redistribution terms can be formally discretized as

(113)

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J. M. Zhao and L. H. Liu



1 @ ∇n fI ðξΩ  kÞg  sin θ @θ n

Ω¼Ω

,

’

mþ1=2, n mþ1=2, n

χθ

I

m n

m1=2, n m1=2, n

 χθ wm θ

I

, (114a)



1 @ ∇n ðI ðr, ΩÞ s1 Þ  sin θ @φ n

Ω¼Ωm, n

’

, n1=2 I m, n1=2 χ φm, nþ1=2 I m, nþ1=2  χ m φ : wnφ (114b)

Details on determining the discrete coefficients of χ θ and χ φ were presented in Ref. Liu (2006). Substitute Eq. (114) into Eq. (64) and apply necessary closure relations (Liu 2006). The final discrete-ordinate equation of GRTE can be obtained and expressed in a form similar to the discrete-ordinate equation of RTE, i.e., m, n m, n Ωm, n  ∇I m, n þ β~ ðrÞI m, n ¼ S~ ðrÞ, (115a) m, n m, n where the modified extinction coefficient β~ ðrÞ and modified source term S~ ðrÞ

    1 1 m, n mþ1=2, n m1=2, n β~ ðrÞ ¼ m max χ θ , 0 þ m max χ θ ,0 wθ wθ     1 1 , n1=2 , 0 þ ðκ þ κ Þ þ n max χ φm, nþ1=2 , 0 þ n max χ m a s φ wφ wφ Nθ X φ κs X 0 0 0 0 0 n0 I m , n Φm , n ;m, n wm θ wφ 4π m0 ¼1n0 ¼1   1 mþ1=2, n þ m max χ θ , 0 I mþ1, n wθ   1 m1=2, n þ m max χ θ , 0 I m1, n wθ   1 þ n max χ φm, nþ1=2 , 0 I m, nþ1 wφ   1 , n1=2 , 0 I m, n1 þ n max χ m φ wφ N

m, n 2 S~ ðrÞ ¼ n κ a I b þ

mþ1=2, n

The recursion formula for χ θ mþ1=2, n χθ

(115a)



m1=2, n χθ 1=2, n

χθ

(115b)

and χ φm, nþ1=2 is given below.

 wm @ ðξΩÞ ∇n θ  ¼ n Ω¼Ωm, n sin θm @θ N þ1=2, n

¼ χθ θ

¼0

(116a) (116b)

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Radiative Transfer Equation and Solutions

 wnφ @s1 ∇n   ¼ sin θm @φ n Ω¼Ωm, n

1 ∇n χ φm, 1=2 ¼ χ φm, Nφ þ1=2 ¼ j  sin θm n

, nþ1=2 χm φ

χ φm, n1=2

969

(116c)

(116d)

Similar to the discrete-ordinate equations of RTE, Eq. (115) with boundary m, n m, n conditions is solved for each discrete direction. Since both β~ ðrÞ and S~ ðrÞ contain part of angular redistribution terms, it is different from the discrete-ordinate equation of RTE. The source term updating is always needed during the solution process. The spatial discretization techniques, such as FVM and FEM presented in the previous sections, can be readily applied to Eq. (115) for solution. Note that the discretization of the GRTE can also be conducted without relying on the discreteordinate equation. Recently, Zhang et al. (2012) developed a hybrid FEM/FVM technique to solve the GRTE, in which the angular domain is discretized using FEM and spatial domain is discretized using FVM, and the radiative intensity at all the directions is solved simultaneously at each spatial node. The idea of this approach follows the work of Coelho (2005) for solving the RTE.

4

Numerical Errors and Accuracy Improvement Strategies

All numerical methods suffer from numerical errors. The MC method suffers from statistic errors, while the DOM, FVM, and FEM suffer from space and angular discretization errors. Due to the significant importance in real applications, numerical errors for solving RTE have attracted the interest of many researchers (Chai et al. 1993; Ramankutty and Crosbie 1997; Coelho 2002b; Hunter and Guo 2015; Huang et al. 2011; Tagne Kamdem 2015). In this section, DOM is taken as an example method, and the numerical errors that appear in DOM and the related improvement strategies are discussed.

4.1

Origin of Numerical Errors in DOM

The “false scattering” and “ray effects” are terms to describe the characteristics of the numerical error observed in numerical results of DOM. Literally, “false scattering” means the effect of the numerical error behave like scattering process, which was usually considered to be equivalent to numerical diffusion (Chai et al. 1993). The “ray effects” stand for the unphysical bump that appeared in the numerical results, which was attributed to using of discrete number of directions to approximate the continuous angular variation of radiative intensity (Chai et al. 1993). In order to understand the origin of the error phenomenon, it is necessary to analyze the source of errors in DOM. By carefully checking the DOM discretization of RTE and the calculation of heat flux and incident radiation, three major sources of

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J. M. Zhao and L. H. Liu

errors can be identified: (1) error from differencing scheme, which is related to the discretization of a differential operator in spatial domain; (2) error from the discretization of scattering term, which is related to the discretization of an integral operator in angular domain; and (3) error from the calculation of heat flux or incident radiation, which is related to the discretization of another integral operator. Note that the third source is distinctly different from the second source, since it will appear even if the medium is non-scattering. What is the effect of these three sources of errors? What is the relation of these three sources of errors with “false scattering” and “ray effects,” which will be discussed in the following section?

4.2

Error from Differencing Scheme

A differencing scheme is required to discretize the differential operator in the discrete-ordinate equation of RTE (Eq. (79)). Take step scheme as an example, which is equivalent to the first-order upwind finite difference scheme. Assuming μm > 0, it can be discretized for the x-direction as μm

@I m I m ðxÞ  I m ðx  ΔxÞ ’ μm Δx @x

(117)

Using Taylor expansion, the right-hand side of the above discretization can be written as  2
 I m ðxÞ  I m ðx  ΔxÞ @I m 1 @ Im μm ¼ μm þ μm Δx þ O Δx2 Δx 2 @x @x2

(118)

As seen, an additional diffusion term appears (the second-order term), which has a diffusion coefficient of μmΔx/2. Namely, the dominant error for step scheme is a diffusion process. This numerical diffusion will smooth the radiative intensity distribution, making additional radiative heat flux transport from high-intensity region to low-intensity region, similar to the heat conduction process. It is of significant difference from the scattering process, such as it only smears radiative flux of one direction, while the scattering process usually transfers energy from one direction to another direction. As such, this error is preferable to be called numerical diffusion. Hunter and Guo (2015) derived the expression of numerical diffusion of several differencing schemes, including the step scheme, diamond scheme, and QUICK scheme. The numerical diffusion can be significantly reduced if a high-order scheme is applied. For a low-order scheme like the step scheme, the numerical diffusion will decrease with mesh refinement. Figure 11a shows the solved heat flux by DOM with step scheme at coarse grid (15  15) and fine grid (125  125), in which the data are extracted from the work by Coelho (2002b). The angular discretization is extremely fine; hence, the effect of angular discretization error can be neglected, and only the effect of numerical diffusion is observed. At coarse grid, big numerical diffusion appears, and the heat flux value is always greater than the exact value, which agrees with the analysis presented above.

23

a

Radiative Transfer Equation and Solutions

b

0.30

0.30

Exact STEP, 15×15 STEP, 125×125

0.22

τ L= 1

Exact CLAM, 15×15 CLAM, 125×125

0.26

q/σTw4

0.26

q/σTw4

971

0.22

ε w= 1

0.18

ω =1

0.14 0.5

0.6

0.18

0.7

0.8

x/L

0.9

1.0

0.14

0.5

0.6

0.7

x/L

0.8

0.9

1.0

Fig. 11 Incident heat flux along the bottom wall to illustrate numerical diffusion and ray effects. (a) Step scheme, Nθ = Nφ = 100 per octant, (b) CLAM scheme, S8 approximation is used. Data are exacted from Coelho (2002b). Insets show the shape of enclosure; the top wall is hot and others are cold

4.3

Scattering Term Discretization Error

The DOM approximates the in-scattering term as ð

I ðΩ0 , rÞΦðΩ0  ΩÞdΩ0 ’

M X

0

I m0 ðrP Þ Φm m wm0

(119)

m0 ¼1

4π

This approximation will inevitably introduce discretization errors. Since the scattering phase function and radiative intensity distribution are usually complicated, this angular quadrature may introduce significant errors for scattering media, e.g., the phase function with strong forward peak. The discretization error of this term is considered to alter the contribution of original scattering phase function. By this understanding, this error induces unphysical scattering, i.e., false scattering, which changes the coupling of radiative intensity among different directions unphysically. To improve the discretization accuracy of scattering phase function, several techniques have been proposed. The first is to modify the discrete phase function to ensure the exact energy conservation constraint, i.e., M X

0

Φm m wm0 ¼ 4π

(120)

m0 ¼1

The most common approach is to modify the phase function as follows (Liu et al. 2002): 0

e m m ¼ Φm0 m Φ

M 1 X 0 Φm m wm0 4π m0 ¼1

!1 (121)

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J. M. Zhao and L. H. Liu 0

e m m exactly satisfy the energy conservation relation. which will ensure Φ Hunter and Guo (2012a, 2012b) showed that the normalization given in Eq. (121) is not enough for strongly forward-scattering phase function, e.g., asymmetry factor greater than 0.9. In this case, another constraint on conservation of asymmetry factor g should also be satisfied for the modified scattering phase function to get reliable results, namely, M X

 0  0 Φm m wm0 cos Θm m ¼ 4πg

(122)

m0 ¼1

They then devised new schemes to normalize the discrete scattering phase function to ensure both the conservation of energy (Eq. (120)) and the conservation of asymmetry factor be satisfied (Hunter and Guo 2012a, 2014). Detailed normalization procedure refers to the pioneer work of Hunter and Guo (2012a). A recent work on comparison of the different normalization schemes for strongly forwardscattering phase function was given by Granate et al. (2016). The errors that arise from discretization of the scattering term was also called “angular false scattering” in Ref. Hunter and Guo (2015).

4.4

Error from Heat Flux Calculation

The phenomenon of ray effects is that the heat flux distribution contains unphysical bump pattern of errors, as shown in Fig. 11b. It mainly influences the solution accuracy of DOM when there are sharp gradients or discontinuities in the boundary conditions, temperature distribution, or radiative properties of the medium. Ray effects have been demonstrated to mainly rely on angular discretization (Lathrop 1968; Chai et al. 1993). Figure 11b shows the heat flux distribution along the bottom wall solved by CLAM scheme at a course (15  15) and fine (125  125) spatial grid; S8 approximation is used for angular quadrature. It is known that CLAM scheme is a second-order accurate and bounded non-oscillatory scheme. However, even for the fine grid that spatial discretization is considered to be sufficiently accurate, there are still strong unphysical bump patterns in the heat flux distribution. Hence the numerical error featured with the bump pattern is unrelated with the spatial discretization. Furthermore, this kind of bump patterns contained in heat flux distribution still exists for the media without scattering (Coelho 2002b). Hence it cannot be attributed to the error from the scattering term. It has a distinct origin. Here the bump pattern of numerical error is attributed to the inaccurate heat flux calculation. Figure 12a shows a typical configuration that has strong ray effects. In this case, only a confined load is located at the bottom, and the heat flux distribution along the top wall is to be calculated. Considering the media is non-scattering, the incident radiative intensity at the top wall can be traced back to the load. Point O denotes the center of the top wall and B is a point located with a distance from the center. The blue and red lines start from O, and B indicates the discrete-ordinate

23

Radiative Transfer Equation and Solutions

a Exact heat flux distribution

unphysical 'bumps' by ray effect

O A

973

b

ΔΩ

Discrete direcons

Fig. 12 Schematics to illustrate ray effects. (a) Boundary confined load, (b) inside (volumetric) confined load

directions, which are traced back to determine the intensities exactly. As can be seen, for point B, there are two directions intercepted with the load. As for the center point O, only one direction intercepts with the load. If angular quadrature is used to calculate the heat flux, it is obvious that heat flux at point B will be greater than that at the center point. This analysis agrees with the numerical results. Besides the confined boundary load, confined volumetric load (as shown in Fig. 12b) will also induce ray effects as studied by Coelho (2002b, 2004). The confinement of radiative intensity in a small solid angle is difficult to be accurately integrated, because only few discrete-ordinate directions will be located in the small solid angle to do integration. The ray effects can be mitigated by refining the angular discretization (Chai et al. 1993; Li et al. 2003); however, this approach requires considerable computational effort. It can also be effectively mitigated by the modified discreteordinate approach (Ramankutty and Crosbie 1997; Coelho 2002b; Coelho 2004), which treats the contributions from boundary load and volumetric load to heat flux separately by solving a different transfer equation. Recently, several new approaches were proposed to mitigate the ray effects in FVM and DOM. More effective way to mitigate the ray effects in DOM is still an important subject of research (Huang et al. 2011; Tagne Kamdem 2015).

5

Conclusions

In this chapter, the classical radiative transfer equation and several variant forms of radiative transfer equation, the different solution techniques for the radiative transfer equations, and the numerical errors on the solution of radiative transfer equation and the related improvement strategies are presented and discussed. The classical RTE implies the light propagates through straight line. To analyze radiative transfer in gradient index media where light propagates through curved lines, the GRTE should be applied. Under Cartesian coordinate the GRTE contains two terms of angular

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derivatives as compared to the RTE, which account for the effect of gradient refractive index. The classical RTE is in a form of a first-order integral partial differential equation. It can be considered as a special kind of convection-diffusion equation with convection-dominated property. This is also true for the GRTE. The convection-dominated property will induce numerical instability for numerical solution. The classical RTE can be transformed to second-order forms and avoids the stability problem. Numerical methods to solve radiative transfer can be classified into two groups, (1) methods based on stochastic simulation and (2) the deterministic methods, which are usually formulated based on the integral or differential form of RTE. The MCM is a typical method of the first group; it is versatile and reliable, but usually time consuming since a huge number of photons need to be traced and are inconvenient to be coupled with conduction and convection solvers; the latter are usually implemented using deterministic methods such as FVM and FEM. DOM is the typical method of the second group. The FVM can be considered as a special kind of DOM that the discrete-ordinate equations are obtained based on the FVM. FEM is usually more accurate than the FVM; furthermore, it is very versatile and promising for the simulation of multiphysics processes including radiative heat transfer. Three major sources of errors for numerical solution of RTE can be identified: (1) error from differencing scheme, which is related to the discretization of a differential operator; (2) error from the discretization of scattering term, which is related to the discretization of an angular integral operator; and (3) error from the calculation of heat flux or incident radiation, which is related to the discretization of another angular integral operator. The first will induce numerical diffusion, in which radiative energy diffuses to the same direction. The second will induce unphysically altered phase function and hence is the true “false scattering.” The third will induce unphysical bump pattern of errors in the flux distribution, also known as “ray effects,” which is attributed to inaccuracy of angular quadrature. But this is distinctly different from the second source, since it will appear even if the medium is non-scattering. Besides the errors mentioned above, it should be noted that the actual solution accuracy of radiative transfer problems also relies closely on the accuracy of measured material properties, such as absorption coefficient, scattering coefficient, and scattering phase function, which should be cared for the solution of real problems.

6

Cross-References

▶ A Prelude to the Fundamentals and Applications of Radiation Transfer ▶ Monte Carlo Methods for Radiative Transfer ▶ Radiative Plasma Heat Transfer ▶ Radiative Properties of Gases ▶ Radiative Properties of Particles ▶ Radiative Transfer in Combustion Systems

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Acknowledgements The authors thank the supports by National Nature Science Foundation of China (Nos. 51336002, 51421063). The support by the Fundamental Research Funds for the Central Universities (Grant No. HIT. NSRIF. 2013094, HIT.BRETIII.201415) are also greatly acknowledged.

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Lathrop KD (1968) Ray effects in discrete ordinates equations. Nucl Sci Eng 32:357–369 Lemonnier D, Le Dez V (2002) Discrete ordinates solution of radiative transfer across a slab with variable refractive index. J Quant Spectrosc Radiat Transf 73(2–5):195–204 Li H-S, Flamant G, Lu J-D (2003) Mitigation of ray effects in the discrete ordinates method. Numer Heat Transf B 43(5):445–466 Li B-W, Sun Y-S, Yu Y (2008) Iterative and direct Chebyshev collocation spectral methods for one-dimensional radiative heat transfer. Int J Heat Mass Transf 51(25–26):5887–5894 Liou KN (2002) An introduction to atmospheric radiation. Academic press Liu LH (2004a) Discrete curved ray-tracing method for radiative transfer in an absorbing-emitting semitransparent slab with variable spatial refractive index. J Quant Spectrosc Radiat Transf 83 (2):223–228 Liu LH (2004b) Finite element simulation of radiative heat transfer in absorbing and scattering media. J Thermophys Heat Transf 18(4):555–557 Liu LH (2006) Finite volume method for radiation heat transfer in graded index medium. J Thermophys Heat Transf 20(1):59–66 Liu LH, Tan HP (2006) Numerical simulation of radiative transfer in graded index media. Science Press, Beijing Liu LH, Tan JY (2007) Least-squares collocation meshless approach for radiative heat transfer in absorbing and scattering media. J Quant Spectrosc Radiat Transf 103(3):545–557 Liu LH, Ruan LM, Tan HP (2002) On the discrete ordinates method for radiative heat transfer in anisotropically scattering media. Int J Heat Mass Transf 45(15):3259–3262 Liu LH, Zhang L, Tan HP (2006) Radiative transfer equation for graded index medium in cylindrical and spherical coordinate systems. J Quant Spectrosc Radiat Transf 97(3):446–456 Liu LH, Zhao JM, Tan HP (2008) The finite element method and spectral element method for numerical simulation of radiative transfer equation. Science Press, Beijing Mahian O, Kianifar A, Kalogirou SA, Pop I, Wongwises S (2013) A review of the applications of nanofluids in solar energy. Int J Heat Mass Transf 57(2):582–594 Maruyama S (1993) Radiation heat transfer between arbitrary three-dimensional bodies with specular and diffuse surfaces. Numer Heat Transf A Appl 24(2):181–196 Mengüç MP, Iyer RK (1988) Modeling of radiative transfer using multiple spherical harmonics approximations. J Quant Spectrosc Radiat Transf 39(6):445–461 Mengüç MP, Viskanta R (1985) Radiative transfer in three-dimensional rectangular enclosures containing inhomogeneous, anisotropically scattering media. J Quant Spectrosc Radiat Transf 33(6):533–549 Modest MF (2013) Radiative heat transfer, 3rd edn. Academic Press, New York Modest MF, Haworth DC (2016) Radiative heat transfer in turbulent combustion systems: theory and applications. Springer International Publishing, Cham Murthy JY, Mathur SR (1998) Finite volume method for radiative heat transfer using unstructured meshes. J Thermophys Heat Transf 12(3):313–321 Pilon L, Berberoglu H, Kandilian R (2011) Radiation transfer in photobiological carbon dioxide fixation and fuel production by microalgae. J Quant Spectrosc Radiat Transf 112(17):2639–2660 Raithby GD, Chui EH (1990) A finite-volume method for predicting a radiant heat transfer in enclosures with participating media. J Heat Transf 112:415–423 Ramankutty MA, Crosbie AL (1997) Modified discrete ordinates solution of radiative transfer in two-dimensional rectangular enclosures. J Quant Spectrosc Radiat Transf 57(11):107–140 Sadat H (2006) On the use of a meshless method for solving radiative transfer with the discrete ordinates formulations. J Quant Spectrosc Radiat Transf 101(2):263–268 Siegel R, Howell JR (2002) Thermal radiation heat transfer, 4th edn. Taylor & Francis, New York Simmons FS (2000) Rocket exhaust plume phenomenology. Aerospace Corporation Song TH, Park CW 1992 Formulation and application of the second-order discrete ordinate method. In: Wang B-X (ed) Transport phenomena and science. Higher Education Press, Beijing, pp 833–841 Sun Y-S, Li B-W (2009) Chebyshev collocation spectral method for one-dimensional radiative heat transfer in graded index media. Int J Therm Sci 48:691–698

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Tagne Kamdem HT (2015) Ray effects elimination in discrete ordinates and finite volume methods. J Thermophys Heat Transf 29(2):306–318 Thurgood CP, Pollard A, Becker HA (1995) The TN quadrature set for the discrete ordinates method. J Heat Transf 117(4):1068–1070 Truelove JS (1988) Three-dimensional radiation in absorbing-emitting-scattering media using discrete-ordinate approximation. J Quant Spectrosc Radiat Transf 39(1):27–31 Viskanta R, Mengüç MP (1987) Radiation heat transfer in combustion systems. Prog Energy Combust Sci 13(2):97–160 Wang Z, Cheng Q, Wang G, Zhou H (2011) The DRESOR method for radiative heat transfer in a one-dimensional medium with variable refractive index. J Quant Spectrosc Radiat Transf 112(18):2835–2845 Wu C-Y, Hou M-F (2012) Solution of integral equations of intensity moments for radiative transfer in an anisotropically scattering medium with a linear refractive index. Int J Heat Mass Transf 55 (7–8):1863–1872 Xia XL, Huang Y, Tan HP (2002) Thermal emission and volumetric absorption of a graded index semitransparent medium layer. J Quant Spectrosc Radiat Transf 74(2):235–248 Zhang L, Zhao JM, Liu LH (2010) Finite element approach for radiative transfer in multi-layer graded index cylindrical medium with Fresnel surfaces. J Quant Spectrosc Radiat Transf 111(3):420–432 Zhang L, Zhao JM, Liu LH, Wang SY (2012) Hybrid finite volume/finite element method for radiative heat transfer in graded index media. J Quant Spectrosc Radiat Transf 113(14):1826–1835 Zhang Y, Yi H-L, Tan H-P (2015) Analysis of transient radiative transfer in two-dimensional scattering graded index medium with diffuse energy pulse irradiation. Int J Therm Sci 87:187–198 Zhang L, Zhao JM, Liu LH (2016) A new stabilized finite element formulation for solving radiative transfer equation. J Heat Transf 138(6):064502–064502 Zhao JM, Liu LH (2006) Least-squares spectral element method for radiative heat transfer in semitransparent media. Numer Heat Transf B 50(5):473–489 Zhao JM, Liu LH (2007a) Second order radiative transfer equation and its properties of numerical solution using finite element method. Numer Heat Transf B 51:391–409 Zhao JM, Liu LH (2007b) Solution of radiative heat transfer in graded index media by Least Square spectral element method. Int J Heat Mass Transf 50:2634–2642 Zhao JM, Tan JY, Liu LH (2012a) A deficiency problem of the least squares finite element method for solving radiative transfer in strongly inhomogeneous media. J Quant Spectrosc Radiat Transf 113(12):1488–1502 Zhao JM, Tan JY, Liu LH (2012b) On the derivation of vector radiative transfer equation for polarized radiative transport in graded index media. J Quant Spectrosc Radiat Transf 113(3):239–250 Zhao JM, Tan JY, Liu LH (2013) A second order radiative transfer equation and its solution by meshless method with application to strongly inhomogeneous media. J Comput Phys 232(1):431–455 Zhou H-C, Cheng Q (2004) The DRESOR method for the solution of the radiative transfer equation in gray plane-parallel media. In: Mengüç MP, Selçuk N (eds) Proceedings of the fourth international symposium on radiative transfer, Istanbul, pp 181–190 Zhou H-C, Lou C, Cheng Q, Jiang Z, He J, Huang B, Pei Z, Lu C (2005) Experimental investigations on visualization of three-dimensional temperature distributions in a large-scale pulverized-coal-fired boiler furnace. Proc Combust Inst 30(1):1699–1706 Zhu K-Y, Huang Y, Wang J (2011) Curved ray tracing method for one-dimensional radiative transfer in the linear-anisotropic scattering medium with graded index. J Quant Spectrosc Radiat Transf 112(3):377–383

Near-Field Thermal Radiation

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Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Fluctuational Electrodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Overview of Maxwell’s Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Thermal Stochastic Maxwell’s Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Thermal Emission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Propagating and Evanescent Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Near-Field Thermal Emission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Size Effect on the Emissivity of Thin Films . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Near-Field Radiative Heat Transfer Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Near-Field Radiative Heat Transfer Between Two Bulk Materials . . . . . . . . . . . . . . . . . 4.2 Near-Field Radiative Heat Transfer in Complex Geometries . . . . . . . . . . . . . . . . . . . . . . . 4.3 Overview of the Thermal Discrete Dipole Approximation (T-DDA) . . . . . . . . . . . . . . . 4.4 Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Cross-References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

980 982 982 985 988 988 989 994 997 1002 1003 1003 1008 1009 1015 1016 1016

Abstract

Near-field, or nanoscale, thermal radiation is a regime arising when at least one characteristic length of the problem, namely, the size of the bodies and/or their separation distance, is comparable to or smaller than the wavelength. The goal of this chapter is to provide the basic information necessary to understand and analyze near-field thermal radiation problems. The fluctuational electrodynamics framework, consisting of Maxwell’s equations augmented by fluctuating current M. Francoeur (*) Radiative Energy Transfer Laboratory, Department of Mechanical Engineering, University of Utah, Salt Lake City, UT, USA e-mail: [email protected] # Springer International Publishing AG, part of Springer Nature 2018 F. A. Kulacki (ed.), Handbook of Thermal Science and Engineering, https://doi.org/10.1007/978-3-319-26695-4_63

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sources representing thermal emission, is reviewed. Equations for the energy density and radiative heat flux expressed in terms of the temperature of a heat source are derived. The different electromagnetic modes emitted by a heat source, namely, propagating modes, evanescent modes generated by total internal reflection, and surface polaritons, are explained. Thermal emission in the near field of a heat source and far-field emission by a subwavelength heat source are discussed. Finally, radiative heat transfer exceeding the blackbody limit between two bulk materials is reviewed, and a general approach for predicting near-field radiative heat transfer in complex three-dimensional geometries, called the thermal discrete dipole approximation, is introduced. At the end of each section, references discussing advanced and specialized topics in near-field thermal radiation are listed.

1

Introduction

The classical theory of thermal radiation is based on the blackbody concept. A blackbody is defined as an ideal body absorbing and emitting the maximum amount of radiation irrespective of the direction and the wavelength (Planck 1991; Modest 2013; Howell et al. 2016). The blackbody concept is, however, based on a fundamental assumption listed in Chap. 1 of Planck’s book The Theory of Heat Radiation (Planck 1991): “Throughout the following discussion it will be assumed that the linear dimensions of all parts of space considered, as well as the radii of curvature of all surfaces under consideration, are large compared with the wavelengths of the rays considered.” When all characteristic lengths Lc, which include the size of the bodies and their separation distance, are larger than the wavelength λT, radiative transport can be treated as an incoherent process (“neglect the influence of diffraction” (Planck 1991)), and thermal emission can be conceptualized as a surface process. In this chapter, when referring to the wavelength as a characteristic length, the following definition is used: λT = 2898/T [μm], where the temperature T is in kelvins. This relation, called Wien’s displacement law (Modest 2013; Howell et al. 2016), provides the wavelength at which thermal emission is maximum at a given temperature. When the size of the bodies and/or their separation distance is comparable to or smaller than the wavelength, the wave characteristic of the energy carriers must be taken into account. Indeed, constructive and destructive wave interference may occur within subwavelength bodies and subwavelength separation gaps. In addition to these coherence effects, radiation heat transfer between bodies separated by subwavelength separation gaps may exceed by a few orders of magnitude the blackbody prediction due to tunneling of evanescent modes decaying exponentially within a distance of approximately a wavelength normal to the surface of a heat source. Finally, when the size of a heat source is comparable to or smaller than the wavelength, thermal emission may experience size effect in addition to wave interference. The size effect implies that thermal emission can no longer be conceptualized as a surface process and is thus a function of the size of the heat source.

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Coherence effects, emission and tunneling of evanescent modes, and volumetric thermal emission that are important in the near-field regime of thermal radiation (i.e., when Lc < λT ) are modeled via fluctuational electrodynamics (Rytov et al. 1989). e The fluctuational electrodynamics framework is based on Maxwell’s equations into which fluctuating currents representing thermal emission are added. The link between the fluctuating currents and the local temperature of a heat source is provided by the fluctuation-dissipation theorem, which is valid under the assumption of local thermodynamic equilibrium. Fluctuational electrodynamics is also consistent with the classical theory of thermal radiation (far-field regime) when all characteristic lengths are larger than the wavelength. Progress in fabrication of devices with subwavelength characteristic lengths has led to the conceptualization of many applications capitalizing on the near-field effects of thermal radiation such as coherent thermal emission (Greffet et al. 2002), daytime radiative cooling (Rephaeli et al. 2013; Raman et al. 2014), nearfield thermophotovoltaic power generation (DiMatteo et al. 2001; Whale and Cravalho 2002; Narayanaswamy and Chen 2003; Laroche et al. 2006; Park et al. 2008; Francoeur et al. 2011b; Bernardi et al. 2015), localized radiative cooling (Guha et al. 2012), heat flow regulation (Otey et al. 2010; Ben-Abdallah and Biehs 2015), and thermal imaging (DeWilde et al. 2006; Babuty et al. 2013; Jones and Raschke 2012; O’Callahan et al. 2014). One of the most remarkable effects in the near-field regime of thermal radiation is the possibility of enhancing radiative heat transfer beyond the far-field blackbody limit. Radiation heat transfer exceeding the blackbody limit has been experimentally confirmed in the scanning probesurface (Xu et al. 1994; Kittel et al. 2005, 2015), scanning probe-film (Worbes et al. 2013), microsphere-surface (Narayanaswamy et al. 2008; Shen et al. 2009; Rousseau et al. 2009; Shen et al. 2012), microsphere-film (Song et al. 2015b), microsphere-nanostructured surface (Shi et al. 2015), and surface-surface (Hargreaves 1969; Domoto et al. 1970; Hu et al. 2008; Ottens et al. 2011; Kralik et al. 2012; Feng et al. 2013; St-Gelais et al. 2014; Ijiro and Yamada 2015; Ito et al. 2015; Lim et al. 2015; St-Gelais et al. 2016; Song et al. 2016; Bernardi et al. 2016) configurations. The objective of this chapter is to provide the basic theoretical background necessary to understand and analyze near-field thermal radiation problems where at least one characteristic length is comparable to or smaller than the wavelength. At the end of each section, a list of references on advanced and specialized topics is provided. The rest of the chapter is structured as follows. Section 2 provides a brief review of Maxwell’s equations and an overview of the fluctuational electrodynamics framework. Section 3 is devoted to thermal emission. The different electromagnetic modes thermally generated are first described, and thermal emission in the near field of a heat source and far-field thermal emission by a subwavelength heat source is afterward discussed. Near-field radiative heat transfer calculations are finally presented in Sect. 4. This portion of the chapter details the case of two bulk materials separated by a subwavelength vacuum gap. In addition, a numerical method for solving near-field thermal radiation problems in complex, three-dimensional geometries, called the thermal discrete dipole approximation, is presented.

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Fluctuational Electrodynamics

This section provides an electrodynamics description of thermal radiation. First, an overview of the macroscopic Maxwell equations is presented. The inclusion of thermal emission into Maxwell’s equations via fluctuating currents (fluctuational electrodynamics) is afterward discussed. Equations for the energy density and radiative heat flux expressed in terms of the local temperature of a heat source are derived via the fluctuation-dissipation theorem.

2.1

Overview of Maxwell’s Equations

2.1.1

Maxwell’s Equations in the Time Domain and Constitutive Relations The Maxwell equations describe the interrelationship between electromagnetic fields, sources, and material properties. In the time domain, the differential form of Maxwell’s equations is given by (Chew 1995) ∇
Eðr, tÞ ¼  ∇
Hðr, tÞ ¼

@Bðr, tÞ @t

@Dðr, tÞ þ Je ðr, tÞ @t

(1) (2)

∇  Dðr, tÞ ¼ ρe ðr, tÞ

(3)

∇  Bðr, tÞ ¼ 0

(4)

where E and H are, respectively, the electric [V/m] and magnetic [A/m] field intensities, D and B are, respectively, the electric [C/m2] and magnetic [Wb/m2] flux densities, Je is the electric current density due to free electrons not bound to any atoms [A/m2], ρe is the electric charge density [C/m3], r is a position vector [m], while t is the time variable [s]. Equation 1 is Faraday’s law of induction that implies that a time-varying magnetic flux density generates an electric field with rotation. Ampère’s law, given by Eq. 2, states that an electric current density or a time-varying electric flux density generates a magnetic field with rotation. Equations 3 and 4 are Gauss’s law. Equation 3 shows that the divergence of the electric flux density at a given location r is proportional to the electric charge density at that location, while Eq. 4 implies that the divergence of the magnetic flux density is always zero due to the absence of magnetic charge density. The charge conservation, or continuity, equation is obtained by substituting Eq. 3 into the divergence of Eq. 2 @ρe ðr, tÞ þ ∇  Je ðr, tÞ ¼ 0 @t

(5)

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From now on, it will be assumed that there is no electric charge density. The electric flux density and magnetic flux density, also respectively referred to as the electric displacement and magnetic induction, are related to the electric and magnetic field intensities via the following constitutive relations Dðr, tÞ ¼ em Eðr, tÞ

(6)

Bðr, tÞ ¼ μm Hðr, tÞ

(7)

where em and μm are material properties called the electric permittivity [F/m] and magnetic permeability [N/A2]. The constitutive relations (6) and (7) are valid for homogeneous, isotropic, and linear media. The electric permittivity and magnetic permeability are related to the electric χ and magnetic χ m susceptibilities via the following relations em ¼ e0 ð1 þ χ Þ

(8)

μm ¼ μ0 ð1 þ χ m Þ

(9)

where e0 and μ0 are, respectively, the permittivity [8.854
1012 F/m] and permeability [4π
107 N/A2] of vacuum. The electric and magnetic susceptibilities are measures of the electric and magnetic polarization properties of a medium. The electric current density Je is related to the electric field intensity via Ohm’s law Je ðr, tÞ ¼ σ e Eðr, tÞ

(10)

where σ e is the electric conductivity [A/Vm].

2.1.2 Electromagnetic Energy Conservation In heat transfer analysis, the quantities of interest are not the electric and magnetic field intensities but the energy and power exchanged. By applying the conservation of energy principle to Maxwell’s equations, expressions for the radiative flux and the energy density in terms of electric and magnetic field intensities can be derived. Conservation of electromagnetic energy is obtained by subtracting the dot product of H with Eq. 1 to the dot product of E with Eq. 2 (Chen 2005; Zhang 2007) ∇  ½Eðr, tÞ
Hðr, tÞ ¼

    1     @ 1 em Eðr, tÞ  E r, t þ μm H r, t  H r, t @t 2 2 þ Eðr, tÞ  Je ðr, tÞ

(11)

where the vector ∇  (A
B) = B  (∇
A)  A  (∇
B) and the rela identity  tion u  du ¼ d 12 u2 have been used. In Eq. 11, ∇  [E
H] is the net energy flow into a differential control volume [W/m3], 12 em E  E is the electric energy density [J/m3], 12 μm H  H is the magnetic energy density [J/m3], and E  Je is the energy generated by Joule heating [W/m3]. The conservation of electromagnetic energy thus

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M. Francoeur

implies that the net energy flow into a differential control volume equals the rate of change of the stored electromagnetic energy plus the heat generated. Two quantities of interest in heat transfer analysis are contained within Eq. 11. The Poynting vector [W/m2], or radiative flux, provides the direction and rate of energy flow per unit surface area. From Eq. 11, the Poynting vector is defined Sðr, tÞ ¼ Eðr, tÞ
Hðr, tÞ

(12)

The energy density [J/m3] is the sum of the electric and magnetic energy densities 1 1 uðr, tÞ ¼ em Eðr, tÞ  Eðr, tÞ þ μm Hðr, tÞ  Hðr, tÞ 2 2

(13)

2.1.3 Maxwell’s Equations in the Frequency Domain Thermal emission is a broadband process, which means that multiple frequencies (or wavelengths) are excited at a given temperature due to random oscillations of charges triggered by thermal agitation. As such, it is more convenient to work with Maxwell’s equations in the frequency domain rather than in the time domain when treating thermal radiation problems. Maxwell’s equations in the frequency domain are obtained by assuming that the fields are time harmonic. A time-harmonic field A(r,t) can be written   Aðr, tÞ ¼ Re AðrÞeiωt

(14)

where Re refers to ffireal part, ω is the angular frequency [rad/s], and i is the complex pffiffiffiffiffiffi constant (i ¼ 1). The application of Eq. 14 to the Maxwell equations leads to ∇
EðrÞ ¼ iωBðrÞ

(15)

∇
HðrÞ ¼ iωDðrÞ þ Je ðrÞ

(16)

∇  DðrÞ ¼ 0

(17)

∇  B ðr Þ ¼ 0

(18)

Equations (15) to (18) are valid for a monochromatic field. A more general form of these expressions, where the fields are function   r andω, is obtained by Ð 1 of both applying the Fourier transform Aðr, tÞ ¼ 1 A r, ω eiωt dω to the Maxwell equations in the time domain. The resulting Maxwell’s curl equations (i.e., Faraday and Ampère’s laws) are given by ∇
Eðr, ωÞ ¼ iωμðωÞμ0 Hðr, ωÞ

(19)

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∇
Hðr, ωÞ ¼ iωeðωÞe0 Eðr, ωÞ

(20)

where Eqs. 6, 7, and 10 have been used. The dimensionless material properties e and μ are the relative electric permittivity and relative magnetic permeability, respectively. For nonmagnetic materials, μ takes a value of unity. Since the vast majority of naturally occurring materials are nonmagnetic in the thermal spectral band, it is assumed from now on that all media are nonmagnetic (μ = 1). The relative electric permittivity, often referred to as the dielectric function or dielectric constant, is a complex quantity defined e ð ωÞ ¼

em σe þi e0 ωe0

(21)

In this chapter, a complex number a is written as a = a0 + ia00. As such, the dielectric function e has a real e0 and an imaginary e00 component given, respectively, by em/e0 and σ e/ωe0. The refractive index n, with real and imaginary parts n0 and n00, is related to the dielectric function via the relationship e = n2.

2.2

Thermal Stochastic Maxwell’s Equations

2.2.1 Fluctuating Currents and Fluctuation-Dissipation Theorem The Maxwell equations as described in Sect. 2.1 are insufficient for near-field thermal radiation calculations as they do not account for the thermally generated electromagnetic field. Thermal emission is included into Maxwell’s equations via fluctuational electrodynamics (Rytov et al. 1989). This formalism provides the bridge between classical electrodynamics and heat transfer theories necessary for modeling near-field thermal radiation problems. Electromagnetic fields are generated by the out-of-phase oscillations of charges of opposite signs (dipoles). When a medium is at a temperature larger than 0 K, thermal agitation leads to a chaotic motion of charged particles which induces oscillating dipoles. These oscillating dipoles generate in turn a fluctuating electromagnetic field, called the thermal radiation field, as it originates from random thermal motion. From a macroscopic point of view, the thermal radiation field can be conceptualized as an electromagnetic field generated by macroscopic fluctuating currents. The fluctuational electrodynamics framework is based on this macroscopic description, where fluctuating currents are added into Maxwell’s equations to model the thermally generated electromagnetic field. For nonmagnetic media, as considered in this chapter, a fluctuating currents density Jfl [A/m2] due to electric dipole oscillations is added to the right-hand side of Ampère’s law (Eq. 20). The thermal stochastic Maxwell equations are thus given by: ∇
Eðr, ωÞ ¼ iωμ0 Hðr, ωÞ

(22)

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M. Francoeur

∇
Hðr, ωÞ ¼ iωeðωÞe0 Eðr, ωÞ þ J fl ðr, ωÞ

(23)

For magnetic media, a fluctuating current density due to magnetic dipole oscillations must be included in Faraday’s law (Eq. 19) (Rytov et al. 1989). The fluctuating current is a stochastic variable that is fully characterized by its first two moments. The first moment of the fluctuating current, which corresponds to its ensemble average, is zero. This implies that the mean thermally radiated electric and magnetic fields are also equal to zero. The second moment of the fluctuating current, or, equivalently, the ensemble average of the spatial correlation function of the fluctuating current, is given by the fluctuation-dissipation theorem (Rytov et al. 1989): D

 E 4ωe0 e00 ðωÞ J αfl ðr0 , ωÞJ βfl r00 , ω0 ¼ Θðω, T Þδðr0  r00 Þδðω  ω0 Þδαβ π

(24)

where hi denotes an ensemble average, the superscript * refers to complex conjugate, α and β are orthogonal components indicating the state of polarization of the fluctuating current, δ(r0  r00) and δ(ω  ω0) are Dirac functions, while δαβ is the Kronecker function. The term Θ(ω, T ) is the mean energy of an electromagnetic state at frequency ω and temperature T Θðω, T Þ ¼

ℏω expðℏω=kB T Þ  1

(25)

where kB is the Boltzmann constant [1.381
1023 J/K] and ℏ is the reduced Planck constant [1.055
1034 Js]. The fluctuation-dissipation theorem as given by Eq. 24 is limited to the following assumptions: the media are linear, isotropic, nonmagnetic, and defined by a dielectric function local in space. In addition, the fluctuation-dissipation is applicable when the media are in local thermodynamic equilibrium, where an equilibrium temperature T can be defined. The fluctuationdissipation theorem relates the fluctuating current generating the thermal electromagnetic field to the local temperature of a heat source. The fluctuation-dissipation theorem is thus the key for modeling thermal radiation problems via Maxwell’s equations. Fluctuational electrodynamics is a comprehensive framework that takes into account all near-field effects, namely, coherence, evanescent modes, and volumetric thermal emission. In addition, fluctuational electrodynamics is consistent with the formalism used in the classical theory of thermal radiation when all characteristic lengths, namely, the size of the bodies and their separation distance, are larger than the wavelength.

2.2.2 Radiative Heat Flux and Energy Density Volume integral expressions for the electric and magnetic fields in terms of the fluctuating current can be derived from Maxwell’s equations using the method of potentials (Peterson et al. 1998). The resulting electric and magnetic fields

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987

observed at location r due to fluctuating current sources located at r0 within a volume V are ð E Eðr, ωÞ ¼ iωμ0 G ðr, r0 , ωÞ  J fl ðr0 , ωÞd3 r0

(26)

V

ð

H

Hðr, ωÞ ¼ G ðr, r0 , ωÞ  J fl ðr0 , ωÞd3 r0

(27)

V Eð H Þ

where G is the electric (magnetic) dyadic Green’s function relating the electric (magnetic) field with a frequency ω observed at location r to a source located at r0 (Tai 1994). The electric and magnetic dyadic Green’s functions are related to each H

E

other via the relation G ¼ ∇
G . The monochromatic radiative heat flux [W/m2  (rad/s)] is calculated as the ensemble average of the Poynting vector (Eq. 12) (Chen 2005)    1  hSðr, ωÞi ¼ Re Eðr, ωÞ
H r, ω 2

(28)

where the ergodic hypothesis has been applied (i.e., time averaging is equivalent to ensemble averaging) (Mishchenko 2014).

Calculation of the radiative heat flux requires computations of terms Em H n that are expressed as a function of the fluctuating current J fl using Eqs. 26 and 27 ð ð D  

E fl  00 00 0 fl 0 0 Em ðr, ωÞH n r, ω ¼ iωμ0 d 3 r0 d 3 r00 GEmα ðr, r0 , ωÞGH ð r, r , ω Þ J ð r , ω ÞJ , ω r nβ α β V

V

(29) where m and n are orthogonal components indicating the state of polarization of the electric and magnetic fields (m 6¼ n). Indices mα and nβ imply that a summation is performed over all components. Substitution of the fluctuation-dissipation theorem (Eq. 24) into Eq. 29 provides an explicit expression relating the radiative heat flux to the local temperature T of a heat source

Em ðr, ωÞH n



r, ω



ð 4ik20 e00 ðωÞ 0 3 0 Θðω, T Þ GEmα ðr, r0 , ωÞGH ¼ nα ðr, r , ωÞd r π

(30)

V

The term k0 is the magnitude of the wavevector in vacuum [rad/m] calculated as pffiffiffiffiffiffiffiffiffi ω/c0, where c0 is the speed of light in vacuum given by 1= e0 μ0 = 2.998
108 m/s. Note that the total radiative heat flux in W/m2 is obtained by integrating Eq. 28 over the entire spectrum. The monochromatic energy density [J/m3  (rad/s)] is derived by taking the ensemble average of Eq. 13 and by assuming ergodicity (Mishchenko 2014)

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  1  

1 (31) huðr, ωÞi ¼ em Eðr, ωÞ  E r, ω þ μm Hðr, ωÞ  H r, ω 4 4

the same procedure as for the radiative heat flux, the terms Em Em and Following

Hm Hm necessary for computing the energy density are given by the following expressions after application of the fluctuation-dissipation theorem ð   4ω3 e00 ðωÞ 0 3 0 Θ ð ω, T Þ GEmα ðr, r0 , ωÞGE Em ðr, ωÞEm r, ω ¼ mα ðr, r , ωÞd r c40 πe0

(32)

V



ð   4ωe0 e00 ðωÞ 0 H 0 3 0 Θðω, T Þ GH H m ðr, ωÞH m r, ω ¼ (33) mα ðr, r , ωÞGmα ðr, r , ωÞd r π V

The only unknowns in the radiative heat flux and energy density expressions are the dyadic Green’s functions. Therefore, when using a volume integral representation of the electric and magnetic fields, solving a near-field thermal radiation problem essentially reduces to determining the appropriate dyadic Green’s functions for a specific geometry and boundary conditions.

2.3

Further Reading

• Recent discussions on the limit of applicability of fluctuational electrodynamics with respect to the separation gap (Kim et al. 2015; Chiloyan et al. 2015) • In-depth discussion of classical electrodynamics and Maxwell’s equations (Chew 1995; Griffiths 2014; Jackson 1998; Balanis 2012) • In-depth discussion of fluctuational electrodynamics, including magnetic materials (Rytov et al. 1989) • Discussion on the electrodynamics treatment of thermal radiation (Modest 2013; Howell et al. 2016; Chen 2005; Zhang 2007; Basu 2016; Joulain 2006; Joulain et al. 2005; Francoeur and Mengüç 2008) • Derivation of the fluctuation-dissipation theorem (Novotny and Hecht 2006; Landau and Lifshitz 1960; Narayanaswamy and Chen 2005a) • General description of dyadic Green’s functions in electromagnetics (Chew 1995; Tai 1994; Mishchenko 2014; Novotny and Hecht 2006)

3

Thermal Emission

This section focuses on thermal emission. Thermally generated propagating and evanescent modes are first discussed. The impact of evanescent modes on thermal emission in the near field of a heat source is afterward presented. Finally, a description of the size effect on far-field thermal emission by films of subwavelength thickness is provided.

24

3.1

Near-Field Thermal Radiation

989

Propagating and Evanescent Modes

3.1.1 Qualitative Description of Propagating and Evanescent Modes A heat source emits waves that are propagating away from its surface (propagating modes) and waves that are evanescently confined within a distance of approximately a wavelength normal to its surface (evanescent modes). These modes are schematically illustrated in Fig. 1, where it is assumed that the heat source, denoted as medium 1, is emitting in a vacuum, labeled as medium 0. In Fig. 1(a), a thermally generated wave is propagating within the heat source. At the interface 1–0 delimiting the heat source and the vacuum, the wave is incident at an angle θ1. Part of the wave is reflected back in the heat source at an angle θ1 (assuming that the interface 1–0 is perfectly smooth), while the remaining portion of Fig. 1 Electromagnetic modes emitted by a heat source: (a) propagating wave, (b) evanescent wave generated by total internal reflection (frustrated mode), and (c) surface polariton

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the wave is transmitted in the vacuum at an angle θ0. The transmission, or refraction, angle of the wave is determined from Snell’s law (Hecht 2002) n1 sin θ1 ¼ n0 sin θ0

(34)

where n1 and n0 are the refractive indices of the heat source and vacuum, respectively. Due to the propagative nature of the wave transmitted in the vacuum, heat is carried away from medium 1. Thermal emission of propagating waves a few wavelengths away from a heat source is correctly described by Planck’s distribution. In the near field (i.e., at a subwavelength distance), the wave characteristic of propagating modes cannot be ignored due to coherence effects (Chen 2005). Additionally, Planck’s theory assumes that thermal emission is a surface process, while fluctuational electrodynamics treats emission as a volumetric process. Indeed, Eqs. 30, 32, and 33 show that the radiative heat flux and the energy density at r are the results of fluctuating currents located at r0 within the volume V of the heat source. This implies that Planck’s theory cannot be applied for predicting thermal emission by features with characteristic dimensions comparable to or smaller than the wavelength λT. This topic is discussed in greater details in Sect. 3.3. Figure 1(b) depicts a thermally generated wave propagating in the heat source. The wave reaches the heat source-vacuum interface at an angle θ1 larger than the so-called critical angle for total internal reflection θcr (Novotny and Hecht 2006; Hecht 2002). Indeed, when n1 is larger than n0, there is a critical angle θcr beyond which no wave is transmitted in the vacuum. This critical angle can be calculated from Snell’s law (Eq. 34) by recognizing that the maximum angle the transmitted wave can take, θ0, is 90 . As such, the critical angle for total internal reflection is given by θcr = asin(n0/n1). Yet, despite the wave being totally reflected back in the heat source, solution of Maxwell’s equations at the interface 1–0 predicts the presence of an evanescent wave field decaying exponentially over a distance of about a wavelength in vacuum (Novotny and Hecht 2006). Evanescent waves do not play any role in far-field thermal emission as they do not propagate away from the heat source. However, when a dissipative body is brought within the evanescent wave field of the heat source, a net energy transfer occurs due to radiation tunneling (Zhang 2007). Evanescent modes can thus lead to thermal emission and radiative heat transfer exceeding the (far-field) blackbody limit, as these modes are not taken into account in the classical theory of thermal radiation. Figure 1(c) shows a schematic representation of a surface polariton, where the field is evanescent in both the heat source and the vacuum (Raether 1988; Maier 2007). Surface polaritons are different from evanescent waves generated by total internal reflection since they are induced by mechanical oscillations within the heat source. In metals and doped semiconductors, the out-of-phase longitudinal oscillations of free electrons relative to the positive ion cores (plasma oscillations) generate an electromagnetic field, and its evanescent component is a surface plasmonpolariton. Similarly, oscillations of transverse optical phonons in polar crystals generate an electromagnetic field, and its evanescent component is a surface phonon-polariton. When surface polaritons are excited at their resonant frequency,

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thermal emission in the near field of the heat source can become quasimonochromatic due to a large number of electromagnetic modes excited within a narrow spectral band (Joulain et al. 2005). As for evanescent waves generated by total internal reflection, surface polaritons do not contribute to heat transfer in the far field due to their confinements at an interface; they contribute to radiant energy exchange in the near field by tunneling.

3.1.2 Quantitative Description of Propagating and Evanescent Modes The distinction between propagating and evanescent modes is illustrated hereafter via a simple quantitative analysis. For that purpose, the interface shown in Fig. 1 is considered where it is assumed that the heat source is lossless (i.e., n001 ! 0) and that n1 > n0. Without loss of generality, a wave propagating in the x-z plane with the electric field oscillating along the y-axis is considered. Such a wave is said to be transverse electric (TE), since the electric field is oscillating in a plane transverse to the propagation plane. Conversely, if the oscillation of the magnetic field is transverse to the propagation plane, the wave is called transverse magnetic (TM). The electric fields associated with TE-polarized waves propagating in the heat source and the vacuum can be written E1 ¼ ^y E01 eiðk1 rωtÞ

(35)

E0 ¼ ^y E00 eiðk0 rωtÞ

(36)

where E01 and E00 are the electric field amplitudes in media 1 and 0, respectively, and k is the wavevector. The wavevector provides the propagation direction of a wave, and its magnitude k = |k| gives the periodicity of a wave in space [rad/m]. For a wave propagating in the x-z plane, the wavevector can be written as k ¼ kx ^x þ kz^z, where kx and kz are the x- and z-components of the wavevector, respectively, parallel and perpendicular to the interface 1–0. Here, the media are assumed to be infinite along the x-direction, such that the x-component of the wavevector is a pure real number and is invariant from one medium to another (Zhang 2007). The z-component of the wavevector, however, varies from one medium to another and is, in general, a complex number kz ¼ k0z þ ik00z. By inspecting Fig. 1, it is straightforward to express the parallel wavevector kx as a function of the incident angle in the heat source θ1 kx ¼ n1 k0 sin θ1

(37)

where k1 = n1k0 has been used. Equation 37 serves as a basis for delimiting modes emitted by the heat source that are propagating and evanescent in vacuum. Using the fact that the maximum value of sinθ1 leading to a propagating wave in vacuum is n0/n1 (condition for total internal reflection), it is concluded from Eq. 37 that propagating modes are described by parallel wavevectors kx < k0, since n0 = 1. Evanescent waves generated by total internal reflection are described by parallel

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wavevectors k0 < kx < n1k0, where the upper limit has been obtained using Eq. 37 by recognizing that the maximum incident angle θ1 is 90 . The generation of an evanescent wave by total internal reflection can be mathematically demonstrated via the electric field equation. The z-component of the wavevector in vacuum is calculated as follows kz0 ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k20  k2x

(38)

In the regime kx < k0, kz 0 is a pure real number (kz0 ¼ k0z0 ), such that the electric field in vacuum given by Eq. 36 is modified as E0 ¼ ^y E00 eiðkx xþkz 0 zωtÞ 0

(39)

Clearly, Eq. 39 describes a wave propagating in vacuum along both the x- and zdirections (i.e., there is periodicity along the x- and z-directions). In the regime k0 < kx < n1k0, kz0 is a pure imaginary number (kz0 = ik 00z0) according to Eq. 38 such that the electric field in vacuum becomes 00

E0 ¼ ^y E00 eiðkx xωtÞ ekz 0 z

(40)

Here, the wave is periodic, and thus propagating, along the x-direction only. The wave is exponentially decaying along the z-direction, normal to the interface 1–0. This implies that no energy is carried away from the surface of the heat source due to the evanescent wave, at least for the case shown in Fig. 1. This can be formally demonstrated by calculating the Poynting vector (i.e., radiative flux) due to the evanescent wave described by Eq. 40. For a TE-polarized wave, the x- and z-components of the Poynting vector in vacuum are  1  hSx0 i ¼ Re Ey0 H z 0 2

(41)

 1  hSz0 i ¼  Re Ey0 H x0 2

(42)

The magnetic field in vacuum is easily determined using Eq. 40 and Faraday’s law (Eq. 22)  E00 iðkx xωtÞ k00z0 z k00z0 ^ ^ H0 ¼ e e x þ kx z ωμ0 i

(43)

Inserting Eqs. 40 and 43 into Eq. 41, the x-component of the Poynting vector in vacuum is written hSx0 i ¼

kx 2 2k00z 0 z E e 2ωμ0 00

(44)

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Fig. 2 Schematic representation of radiation tunneling between a heat source and sink of same refractive index n1 separated by a vacuum gap. In every layer, the parallel wavevector kx is the same and satisfies k0 < kx < n1k0. This results in a wave that is evanescent in the vacuum gap and propagating in the heat source and sink

Equation 44 implies that energy is carried away along the x-direction parallel to the interface 1–0 due to the evanescent wave. Now, inserting Eqs. 40 and 43 into Eq. 42, the z-component of the Poynting vector in vacuum is given by  00  1 kz0 2 2k00z 0 z E e ¼0 hSz0 i ¼  Re i 2 ωμ0 00

(45)

Equation 45 thus confirms quantitatively that the energy carried away from the surface of a heat source due to an evanescent wave is zero for the case shown in Fig. 1. Energy transfer mediated by evanescent waves can occur through radiation tunneling (Zhang 2007). Consider the case where a heat sink of refractive index n1, and thus made of the same material as the heat source, is located within the evanescent wave field of the heat source, as shown in Fig. 2. In any layer, the evanescent wave is described by a parallel wavevector satisfying k0 < kx < n1k0. As discussed above, this corresponds to a wave propagating in the heat source and evanescent in the vacuum. Since the heat sink has the same refractive index n1 as the heat source, the wave is thus also propagating along the z-direction in the heat sink. When the wave is propagating along the z-direction, this means that the z-component of the Poynting vector in the heat sink is nonzero. This physically implies that energy is transferred from the heat source to the sink and that total internal reflection does not occur anymore in the heat source (frustrated total internal reflection). This mechanism through which energy is transferred from the heat source to the sink is called radiation tunneling. Note that evanescent waves generated by total internal reflection that are tunneled from one medium to another are also referred to as frustrated modes. Modes with parallel wavevector kx > n1k0 having evanescent fields decaying in both the heat source and vacuum can be thermally excited (see Fig. 1(c)). These

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Fig. 3 Schematic representation of a dispersion relation. The modes on the left-hand side of the vacuum light line are propagating in both the heat source and the vacuum. The modes between the vacuum and material light lines are propagating in the heat source and evanescent in the vacuum. The modes on the right-hand side of the material light line are evanescent in both the heat source and the vacuum

modes, called surface polaritons, are generated by mechanical oscillations within the heat source and are thus described by parallel wavevector exceeding the light line in vacuum (kx = k0) and the light line in the heat source (kx = n1k0). The different modes thermally generated are schematically shown in a dispersion relation (i.e., ω versus kx) in Fig. 3.

3.2

Near-Field Thermal Emission

In this section, thermal emission in the near field of a heat source (i.e., at a subwavelength distance) is analyzed. The geometry under consideration is shown in Fig. 4 (a polar coordinate system is used), where a bulk material (medium 1) modeled as a half-space is emitting in a vacuum (medium 0). The media are infinite in the ρ-direction and azimuthally symmetric (i.e., invariance along ϕ), such that only variations along the z-axis are considered. Thermal emission is calculated in vacuum at a distance z above the interface 1–0. From Sect. 3.1.2, it is clear that thermal emission in the near field of a heat source cannot be correctly characterized by the Poynting vector (i.e., radiative flux) along the z-direction, since the Poynting vector is a measure of the energy carried away from the surface. In other words, the Poynting vector along z is nonzero here only for propagating modes. As such, near-field thermal emission is quantified hereafter via

24

Near-Field Thermal Radiation

995

the energy density that takes into account all propagating and evanescent modes. The monochromatic energy density for the case shown in Fig. 4 is determined by using Eqs. 31, 32, and 33 and by substituting the appropriate dyadic Green’s functions for a single planar interface (Joulain et al. 2005; Sipe 1987) 1 huω ðzÞi ¼ 2 Θðω, T 1 Þ 8 k 4π ω 9 ð 3 < ð0 k dk h =   i 1



      k dk ρ ρ

2 þ 1  r TE 2 þ2 ρ ρ Im r TM þ Im r TE e2k00z 0 z 1  r TM
k20 01 01 01 01 : ; jkz0 j jk z 0 j 0

k0

(46) Note that the density of electromagnetic states [1/(m3  (rad/s))], or electromagnetic modes, is determined by dividing Eq. 46 by Θ(ω, T1) (Zhang 2007). In Eq. 46, TE r TM 01 and r 01 are the Fresnel reflection coefficients at the interface 0–1 in TM and TE polarizations r TM 01 ¼

e1 kz 0  e0 kz1 e1 kz0 þ e0 kz1

(47)

kz0  kz1 kz0 þ kz1

(48)

r TE 01 ¼

In Eq. 46, the integration is performed over the wavevector kρ parallel to the interface 1–0. As discussed in Sect. 3.1.2, the parallel wavevector is a pure real number and is invariant from one layer to another. Physically, the integration over the parallel wavevector can be interpreted as an integration over all modes, propagating and evanescent, contributing to thermal emission. For propagating waves, the kρ -integration can be converted into an integration over the propagation angle. The first term on the right-hand side of Eq. 46 represents the contribution from propagating

γ 2 waves where the kρ -integration is performed from 0 to k0. The terms 1  r 01 , where γ = TE or TM, represents the spectral absorptance of the heat source (1 – reflectivity of the surface) and therefore also represents the spectral Fig. 4 Thermal emission by a heat source (temperature T1) at a distance z in vacuum

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emittance of the heat source. The second term on the right-hand side of Eq. 46 is the contribution from evanescent modes that include frustrated modes (k0 < kρ < Re(n1) k0)γ and  surface modes (kρ > Re(n1)k0). By analogy with propagating modes, Im r 01 can be interpreted as the spectral emittance/absorptance for evanescent modes. 00 The exponential factor e2kz 0 z shows explicitly the evanescent nature of these modes; for a large distance z a few wavelengths away from the heat source (i.e., z  λT), 00 e2kz 0 z ! 0 such that the energy density due to evanescent modes becomes zero. Figure 5 shows the monochromatic energy density when the heat source is made of silicon carbide (SiC) and is maintained at a uniform temperature T1 of 300 K (Joulain et al. 2005). The energy density is calculated at distances z of 100 μm, 1 μm, and 100 nm above the interface 1–0 In the far field (100 μm), evanescent modes do not contribute to the energy density. Therefore, the thermal spectrum does not exceed the blackbody spectrum at 300 K. As the distance z approaches the interface 1–0, the energy density increases substantially due to the extraneous contribution from evanescent modes. At a distance of 100 nm, the energy density is quasi-monochromatic around a frequency of 1.786
1014 rad/s owing to the fact that SiC supports surface phonon-polaritons in the infrared. In this near-field regime where thermal emission is dominated by

Fig. 5 Monochromatic energy density calculated at distances z of 100 μm, 1 μm, and 100 nm above a SiC heat source maintained at a constant and uniform temperature of 300 K (Figure taken from Joulain et al. (2005))

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modes characterized by kρ >> Re(n1)k0, the energy density can be approximated as (Zhang 2007) h uω ð z Þ i

1 π 2 ωz3

Θðω, T 1 Þ

e001

je1 þ 1j2

(49)

Equation 49 shows that the energy density becomes extremely large when Re (e1) = 1. This corresponds to the resonance condition of surface polaritons at a single material-vacuum interface in TM polarization (for nonmagnetic materials, surface polaritons can only be excited in TM polarization) (Joulain et al. 2005; Raether 1988; Maier 2007). For SiC, Re(e1) = 1 occurs at a frequency of 1.786
1014 rad/s (Francoeur et al. 2010a) which corresponds to the peak observed in Fig. 5 at a distance of 100 nm. Eq. (49) also shows that the energy density in the near field varies as z3. An approximate cutoff for the kρ -integration for the evanescent contribution to the energy density in Eq. 46 can be determined from the physics of the problem. The penetration depth of evanescent modes in vacuum is estimated as δ0 |kz0|1 (Francoeur et al. 2010a). For large parallel wavevectors with kρ >> k0 dominating thermal emission in the near field, the z-component of the wavevector in vacuum can be approximated by kz 0 ikρ, such that kρ δ1 0 . Physically, it is possible to argue that the smallest evanescent mode contributing to thermal emission has a penetration depth in vacuum equal to the distance z where the energy density is calculated, such that the maximum contributing parallel wavevector is estimated as kρ z1. This result shows explicitly that the number of modes contributing to thermal emission, and thus the evanescent energy density, increases as z decreases.

3.3

Size Effect on the Emissivity of Thin Films

This section discusses how the size of the heat source affects its far-field thermal emission. Specifically, the size effect on the emissivity of thin films, which are nanostructures used in numerous engineering applications, such as solar cells, optical filters, and antireflection coatings, is analyzed. Emissivity is a surface radiative property defined as the ratio of the emissive power of a surface to that of a blackbody at the same condition. The concept of emissivity thus implies that thermal emission is a surface process. However, in reality, thermal radiation emission is a volumetric process that can often be approximated as a surface process. Waves leaving the surface of a heat source are the result of various phenomena such as emission, absorption, transmission, and reflection. These phenomena occur throughout the entire volume of the heat source. Yet, only a small portion of volume below the emitting surface generally contributes significantly to the emitted spectrum; here, this small portion is called the critical thickness. The emissivity of a film with a thickness larger than the critical thickness is referred to as the bulk emissivity. If the emitting medium is thinner than the critical thickness, the concept of emissivity, as defined in the classical theory of thermal radiation, is

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Fig. 6 Thermal emission in air by a film of thickness t1 and temperature T1 coated on a substrate

not valid anymore such that emissivity data reported in the literature are not be applicable to such thin layers. In this case, a thickness-dependent film emissivity can be defined as the emissive power of the layer to that of a blackbody at the same condition. Using fluctuational electrodynamics, where thermal emission is treated as a volumetric process, the thickness-dependent emissivity of films can be predicted. The geometry under consideration is shown in Fig. 6. A film, denoted as medium 1, of thickness t1 emitting in air (medium 2) is coated on a substrate (medium 0). The layered medium, with perfectly smooth and parallel interfaces, is infinite in the ρ-direction and azimuthally symmetric (i.e., invariance along ϕ). The emitting film is in local thermodynamic equilibrium and is maintained at a constant and uniform temperature T1 of 293 K. Using the fluctuational electrodynamics framework described in Sect. 2, the spectral, hemispherical emissive power of the film, which is equivalent to the emitted radiative heat flux, is derived by computing the z-component of the Poynting vector (Eqs. 28 and 30) Θðω, T 1 Þ qω ¼ 4π 2

kð0

kρ dkρ 0

X 

2 2  1  Rγ1  T γ1 γ¼TE, TM

(50)

where

γ 2 r γ21 þ r γ10 e2ikz1 t1 2

R ¼ 1

1 þ r γ r γ e2ikz1 t1 21 10

(51)

2 TE ikz1 t1

TE 2 Reðkz0 Þ tTE

21 t10 e

T ¼ 1

TE TE 2ik t Reðkz2 Þ 1 þ r 21 r 10 e z1 1

(52)



2 TM ikz1 t1

TM 2 Reðkz0 =e0 Þ e0 tTM

21 t10 e

T ¼ 1



TM TM 2ik t z1 1 Reðkz2 =e2 Þ e2 1 þ r 21 r 10 e

(53)

In Eqs. 52 and 53, tγij is the Fresnel transmission coefficient at the interface i-j in polarization state γ calculated as tTE ij ¼

2kzi kzi þ kzj

(54)

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tTM ij

2ej kzi ¼ ej kzi þ ei kzj

rffiffiffiffi ei ej

(55)

The terms Rγ1 and T γ1 are the field amplitude reflection and transmission coefficients

2for the

film

2 (Yeh 2005). The square of the magnitude of these coefficients, Rγ1 and T γ1 , represents the spectral, directional reflectivity and spectral, directional of layer 1, respectively. Accordingly, in Eq. 50, the term

γ 2 transmissivity

γ 2



1  R1  T 1 can be interpreted as the spectral, directional absorptivity of the film. From Eq. 51 through Eq. 53, it can be seen that for a bulk material, where t1 is large, Rγ1 and T γ1 reduce to r γ21 and 0, respectively. Therefore, the spectral, directional

2 absorptivity of a bulk material in polarization state γ is α0 γω ¼ 1  r γ21 .

3.3.1 Spectral, Directional Emissivity The spectral, directional emissivity of the film is the ratio of its actual spectral, directional emissive power to that of a blackbody at the same condition (Modest 2013) Iω e^0 ω ¼ I b, ω

(56)

where Iω and Ib,ω are the spectral intensities of the film and a blackbody, respectively. From Eq. 50, the spectral intensity Iω can be written I ω ¼ Θðω, T 1 Þ

ω2 8π 3 c20

X  γ¼TE, TM

2 2  1  Rγ1  T γ1

(57)

The spectral blackbody intensity is given by the Planck distribution I b, ω ¼ Θðω, T 1 Þ

ω2 4π 3 c20

(58)

Taking the ratio of Eqs. 57 and 58, the following expression for the spectral, directional emissivity is determined 1 e^0 ω ¼ 2

X 

2 2  1  Rγ1  T γ1 γ¼TE, TM

(59)

Equation 59 shows that for bodies in local thermodynamic equilibrium, the spectral, directional emissivity is equal to the spectral, directional absorptivity. In other words, Eq. 59 is Kirchhoff’s law that has been rigorously derived from fluctuational electrodynamics. For a bulk material, Eq. 59 reduces to 1 e^0 ω ¼ 2

X 

2  1  r γ21 γ¼TE, TM

(60)

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3.3.2 Spectral, Hemispherical Emissivity The spectral, hemispherical emissivity of the film is the ratio of its actual spectral, hemispherical emissive power to that of a blackbody at the same condition (Modest 2013): ^e ω ¼

qω qb , ω

(61)

where qω is calculated from Eq. 50 and qb,ω (= πIb,ω) is the spectral, hemispherical emissive power of a blackbody. The spectral, hemispherical emissivity is thus given by k2 ^e ω ¼ 0 2

kð0

kρ dkρ 0

X 

2 2  1  Rγ1  T γ1 γ¼TE, TM

(62)

3.3.3 Total, Hemispherical Emissivity The total, hemispherical emissivity of the film is defined as the ratio of its actual total, hemispherical emissive power to that of a blackbody at the same condition (Modest 2013) ^e ¼

q qb

(63)

The total, hemispherical emissive power of a blackbody is given by the StefanBoltzmann law (qb ¼ n22 σT 41, where n2 is the refractive index of medium 2), while the total emissive power of the film, q, is obtained by integrating Eq. 50 over all frequencies 1 ð

q¼

qω dω

(64)

0

3.3.4 Thickness-Dependent Emissivity Hereafter, the substrate is assumed to be a lossless medium with a frequencyindependent dielectric function of 1. The spectral, hemispherical emissivity of gold (Au) and SiC are shown in Fig. 7 for various film thicknesses; the results are compared against bulk predictions. For the case of Au, the emissivity decreases significantly when the thickness of the film increases until it converges to the emissivity of the bulk material. This counterintuitive behavior is due to the extraneous contribution of waves experiencing multiple reflections within the thin film, which are usually internally absorbed for metallic bulk materials (Edalatpour and Francoeur 2013; Narayanaswamy and Chen

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Fig. 7 Spectral, hemispherical emissivity for various film thickness t1: (a) Au film and (b) SiC film (Figure taken from Edalatpour and Francoeur (2013))

2004). The thickness above which the emissivity of Au does not exhibit any size effect (i.e., critical thickness tcr) is 81 nm. For SiC, the emissivity decreases when decreasing the thickness of the emitter due to a loss of source volume. Comparing the emissivity of a bulk of SiC and a bulk of Au, it is obvious that the reflectivity of dielectrics is not as large as that of metals. Additionally, dielectrics are internally poorly absorptive. Therefore, the contribution from waves experiencing multiple reflections within dielectrics exists in both thin and thick layers and is nearly

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Fig. 8 Total, hemispherical emissivity of various materials as a function of the film thickness t1 (Figure taken from Edalatpour and Francoeur (2013))

unaffected by the size of the heat source. As a consequence, the emissivity of dielectric films is a pure volumetric process. The dominant wavelength emitted λT by a body at 293 K is approximately 10 μm (corresponds to a frequency of 1.88
1014 rad/s). Since constructive and destructive wave interference is significant when the film thickness is approximately equal to the wavelength, some oscillations of the emissivity are observed when the thickness of the film corresponds to the dominant emitted wavelength of 10 μm. Here, the critical thickness tcr for SiC is estimated to be 6 cm. The total, hemispherical emissivity versus the film thickness of various materials, namely, SiC, cubic boron nitride (cBN), aluminum (Al), Au, silver (Ag), and copper (Cu), is shown in Fig. 8. By increasing the size of a metal film, its emissivity decreases rapidly such that it reaches the bulk value for a thickness of approximately 100 nm. Inversely, the emissivity of dielectric films increases with increasing the layer thickness, and the speed of convergence to the bulk emissivity is much slower than for metals. As a result, a film as thick as a few centimeters is required for the film emissivity to be 99% of the bulk emissivity. The emissivity of a metal bulk is generally much smaller than the bulk emissivity of dielectrics. However, for thicknesses below approximately 500 nm, the emissivity of metals can exceed the emissivity of dielectrics.

3.4

Further Reading

• Dyadic Green’s functions for a single planar interface: (Joulain et al. 2005; Sipe 1987) • Fundamentals of surface plasmon-polaritons: (Raether 1988; Maier 2007) • Further discussion on the impact of surface polaritons on near-field thermal emission: (Joulain et al. 2005)

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• Impact of surface polaritons in TE polarization on near-field thermal emission by effectively magnetic materials: (Petersen et al. 2013a, b) • Near-field thermal emission by thin films: (Biehs et al. 2007; Francoeur et al. 2010b) • Far-field thermal emission by structured materials (films, gratings, photonic crystals, metamaterials): (Greffet et al. 2002; Edalatpour and Francoeur 2013; Narayanaswamy and Chen 2004, 2005a; Lee and Zhang 2006, 2007; Lee et al. 2005, 2008; Chang et al. 2015; Wang et al. 2009, 2011; Blandre et al. 2016; Didari and Mengüç 2015a) • Experimental measurement of the spectral distribution of near-field thermal emission: (Babuty et al. 2013; Jones and Raschke 2012; O’Callahan et al. 2014)

4

Near-Field Radiative Heat Transfer Calculations

This section is devoted to near-field radiative heat transfer calculations. The case of two bulk materials, modeled as half-spaces, separated by a vacuum gap is first considered. An overview of near-field radiative heat transfer calculations in complex geometries is afterward provided. Finally, the thermal discrete dipole approximation, which is a numerical method for solving near-field radiative heat transfer problems in arbitrary three-dimensional geometries, is presented.

4.1

Near-Field Radiative Heat Transfer Between Two Bulk Materials

Near-field radiative heat transfer between two bulk materials of SiC, modeled as half-spaces, is discussed (Mulet et al. 2002). As shown in Fig. 9, the two half-spaces, labeled as media 1 and 2, are maintained at constant and uniform temperatures T1 = 300 K and T2 = 0 K. The vacuum gap of thickness d separating the half-spaces is denoted as medium 0. Only the variations of the flux along the z-axis are considered, since the geometry is azimuthally symmetric (i.e., invariance along ϕ) and infinite in the ρ-direction. Fig. 9 Near-field radiative heat transfer between two half-spaces of SiC separated by a vacuum gap of thickness d. The half-spaces are maintained at constant and uniform temperature T1 = 300 K and T2 = 0 K

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The monochromatic radiative heat flux along the z-direction is derived using Eqs. 28 and 30 and by substituting the dyadic Green’s functions for two planar interfaces (Joulain et al. 2005; Sipe 1987): qprop ω, 12 ¼

 X  1  Θðω, T 1 Þ  Θ ω, T 2 2 4π γ¼TE, TM 

γ 2 

2  kð0 1  r 01 1  r γ02
kρ dkρ

1  r γ r γ e2ik0 z0 d 2 01 02

(65)

0

qevan ω, 12 ¼

 X  1 Θðω, T 1 Þ  Θ ω, T 2 2 π γ¼TE, TM 1 ð




kρ dkρ e

2k00z0 d

    Im r γ01 Im r γ02

1  r γ r γ e2k00z0 d 2

(66)

01 02

k0

where the superscripts prop and evan refer to propagating and evanescent components of the radiative heat flux, respectively. The denominators in both Eqs. 65 and 66 account for multiple reflection within the vacuum gap of thickness d. Note that for the case shown in Fig. 9, Θ(ω, T2) = 0 for all frequencies since medium 2 is a heat sink at 0 K. The total heat flux q12 is obtained by integrating the sum of Eqs. 65 and 66 over all frequencies 1 ð

q12 ¼

h i evan dω qprop þ q ω, 12 ω, 12

(67)

0

As for the energy density, the radiative heat flux due to propagating modes involves a kρ -integration from 0 to k0. For the evanescent contribution to the flux, the integration over kρ is performed from k0 to infinity, such that frustrated modes and surface polaritons are taken into account. An approximate cutoff in the kρ -integration is determined using the penetration depth of evanescent modes in vacuum estimated as δ0 |kz0|1. For large parallel wavevectors with kρ >> k0 dominating radiative heat transfer in the near field, kz0 ikρ such that kρ δ1 0 . The smallest evanescent mode contributing to heat transfer has a penetration depth in vacuum equal to the separation gap d, such that the maximum contributing parallel wavevector is estimated as kρ d1. The radiative heat flux due to evanescent modes thus increases as d decreases, as expected. In the far-field limit (i.e., d >> λT), the radiative heat flux between the two halfspaces is independent of the gap thickness d (view factor of unity). This can be 00 mathematically demonstrated as follows. As d ! 1, e2kz0 d ! 0 such that the flux due to evanescent modes becomes zero. Additionally, radiation heat transfer in the far-field regime is incoherent, such that the denominator in Eq. 65 can be written as (Zhang 2007):

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1 1

!

1  r γ r γ e2ik0 z0 d 2 1  ργ01 ργ02 01 02

(68)

where ργij is the reflectivity at the interface i-j in polarization state γ, calculated from

2

the Fresnel coefficients as ργij ¼ r γij . The kρ -integration can also be expressed as an integration over the polar angle θ using the fact that kρ = k0 sin θ. By using the emissivity of medium j (^e γj ¼ 1  ργij for a bulk material), and the Planck blackbody intensity (I b, ω ðT Þ ¼ Θðω, T Þω2 =4π 3 c20), the monochromatic radiative heat flux in the far-field limit when d  λT is qFF ω, 12



¼ 2π I b, ω ðT 1 Þ  I b, ω ðT 2 Þ



π=2 ð

0

^e 1^e 2 cos θ sin θdθ 1  ργ01 ργ01

(69)

where it has been assumed that TE- and TM-polarized waves have equal contributions to the radiative heat flux. Equation 69 is the same expression as the one derived from a ray-tracing approach (Mulet et al. 2002). This demonstrates that the classical theory of thermal radiation based on Planck’s blackbody distribution is a limiting case in the fluctuational electrodynamics formalism. The monochromatic radiative heat flux is shown in Fig. 10 for gap thicknesses d of 1 μm, 100 nm, and 10 nm. These results are compared against the blackbody predictions and the flux obtained in the far-field regime. Note that the spectral variable in Fig. 10 is expressed in electron volt (eV), which is calculated from the angular frequency as E = ℏω/e, where e = 1.602
1019 J/eV. The results of Fig. 10 show an enhancement of the flux beyond the blackbody predictions due to tunneling of evanescent modes. In the extreme near field (gaps of 10 nm and 100 nm), the flux increases substantially at a frequency of approximately Fig. 10 Net monochromatic radiative heat flux between two half-spaces of SiC for separation gaps d of 1 μm, 100 nm, and 10 nm. The halfspaces of SiC are maintained at constant and uniform temperatures of 300 K and 0 K. Near-field radiative heat flux profiles are compared against far-field (d  λT) and blackbody predictions

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0.12 eV (ω = 1.786
1014 rad/s). This quasi-monochromatic enhancement is due to surface phonon-polaritons supported by SiC, as discussed in Sect. 3.2 (Mulet et al. 2002). For a 10 nm-thick vacuum gap, more than 97% of the flux is transferred around 0.12 eV due to a large number of electromagnetic modes excited within a narrow spectral band. The dependence of radiative heat transfer as a function of the gap thickness is analyzed in Fig. 11. Specifically, the total radiative heat transfer coefficient hr. [W/m2K] is reported for gaps from 1 nm to 100 nm. The near-field profile is compared against blackbody and far-field predictions. The radiative transfer coefficient is derived by assuming that medium 1 is at temperature T (300 K), while medium 2 is at T + δT, such that hr. is calculated as the net radiative heat flux between the two layers divided by δT as δT! 0 1 ð 1 @Θðω, T Þ X hr ¼ 2 dω π @T γ¼TE, TM 0 2 3 

2 

2  1 kð0  γ   γ  ð 1  r γ01 1  r γ02 Im r 01 Im r 02 7 00 6 þ kρ dkρ e2kz0 d 4 kρ dkρ 5

00 2 γ γ 2ik0 z0 d 2

4 1  r 01 r 02 e 1  r γ01 r γ02 e2kz0 d 0

k0

(70)

Fig. 11 Radiative heat transfer coefficient as a function of the vacuum gap thickness d (SiC, T = 300 K). The near-field curve shows that the radiative heat transfer coefficient follows a d2 power law, while both the far-field and blackbody curves do not vary as a function of the gap thickness

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Both the blackbody and far-field curves do not vary as a function of the vacuum gap thickness since the view factor between two half-spaces is unity. The radiative transfer coefficient calculated with Eq. 70 varies as d2 in the extreme near field. The d2 power law can also be derived by performing an asymptotic analysis of Eq. 70 in the limit that d! 0 (Mulet et al. 2002)

hr

1 e001 e002 @Θðω, T Þ 2 @T π 2 d je1 þ 1j2 je2 þ 1j2

(71)

As for the energy density, the radiative heat transfer coefficient is amplified at surface polariton resonance in TM polarization, arising when Re(e1) = 1 and Re (e2) = 1, of a single material-vacuum interface. The resonant enhancement of radiative heat transfer in the near field is maximized when media 1 and 2 are made of the same material. For structures with micro/nanosize dimensions, the d2 power observed here is not always true due to surface polariton coupling within the emitters/absorbers. The coexistence of d2, d3, and d4 regimes for thin films supporting surface phononpolaritons in the infrared such as SiC was discussed in Ben-Abdallah et al. (2009) and Francoeur et al. (2011a). The problem described in this section involving two half-spaces separated by a vacuum gap is of great importance in near-field radiative heat transfer calculations, as this geometry can also be used for estimating the heat transfer between curved objects via the proximity approximation. The proximity approximation is applicable when the radius of curvature R of the objects is much larger than their separation gap d (i.e., R >> d ). In addition, the objects must be optically thick. When these conditions are satisfied, the heat rate between curved objects can be approximated as a summation of local heat rates between two half-spaces with varying gap thicknesses. The proximity approximation has been successfully applied for calculating near-field radiative heat transfer between two spheres (Sasihitlu and Narayanaswamy 2011), between a sphere and a surface (Rousseau et al. 2009) and between a probe and a surface (Kim et al. 2015; Edalatpour and Francoeur 2016). Near-field radiative heat transfer exceeding the blackbody limit between large surfaces (5
5 mm2), made of intrinsic silicon, separated by a subwavelength vacuum gap has been experimentally demonstrated by Bernardi et al. (2016) and is shown in Fig. 12. For a 150 nm-thick vacuum gap, approximately 88% of the radiative heat flux between the heat source and sink is due to evanescent modes (frustrated modes). As such, the radiative heat flux exceeds the blackbody predictions, and a maximum enhancement over the blackbody limit by a factor of 8.4 is obtained at a temperature difference ΔT of 115.6 K. For a 3500 nm-thick gap, only 32% of the radiative heat flux is due to evanescent modes, which is not sufficient to exceed the blackbody limit.

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Fig. 12 Net total radiative heat flux between two 5
5 mm2 layers of intrinsic silicon for separation gaps d of 3500 nm and 150 nm as a function of the temperature difference ΔT = T1 – T2. The heat sink is maintained at a temperature T2 of 300 K. The symbols show the experimental radiative heat flux, while the lines are fluctuational electrodynamics predictions. Blackbody predictions are shown for comparison

4.2

Near-Field Radiative Heat Transfer in Complex Geometries

4.2.1 Analytical Methods The radiative heat flux Eqs. 65 and 66 have been obtained by using analytical expressions for the dyadic Green’s function for two planar interfaces. The methodology of deriving analytical expressions for the dyadic Green’s functions in a specific geometry has the advantage of providing exact results but is intractable when dealing with three-dimensional arbitrarily shaped objects. Over the past years, the method of dyadic Greens functions has been applied to various cases: two halfspaces (Francoeur and Mengüç 2008; Mulet et al. 2002; Polder and Van Hove 1971; Loomis and Maris 1994; Fu and Zhang 2006; Volokitin and Persson 2001), two films (Francoeur et al. 2010a; Ben-Abdallah et al. 2009; Francoeur et al. 2011a), two structured surfaces (Biehs et al. 2008), two nanoporous materials (Biehs et al. 2011), two gratings (Lussange et al. 2012; Yang and Wang 2016), one-dimensional layered media (Narayanaswamy and Chen 2005a; Francoeur et al. 2009; Zheng and Xuan 2011), cylindrical cavity (Hammonds 2006), two dipoles (Domingues et al. 2005; Chapuis et al. 2008, 2010), two large spheres (Narayanaswamy and Chen 2008; Krüger et al. 2012), dipole-surface (Mulet et al. 2001), dipole-structured surface (Biehs and Greffet 2010), sphere-surface (Krüger et al. 2012; Otey and Fan 2011), two long cylinders (Golyk et al. 2012), and N small objects (compared to the wavelength) modeled as electric point dipoles (Ben-Abdallah et al. 2011; Messina et al. 2013). In addition, a formalism based on a scattering matrix approach has been used to derive a general expression for the radiative heat flux between two arbitrarily shaped objects (Messina and Antezza 2011a, b). Yet, an explicit closed-form solution for the radiative heat flux has been provided only for the case of two slabs (Messina and Antezza 2011b).

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4.2.2 Numerical Methods Modeling of near-field radiative heat transfer between arbitrarily shaped objects can be accomplished via numerical methods. So far, the finite-difference frequency-domain method (Wen 2010), the finite-difference time-domain method (Rodriguez et al. 2011; Didari and Mengüç 2014, 2015a, b, 2016), the boundary element method (Rodriguez et al. 2012), the thermal discrete dipole approximation (Edalatpour and Francoeur 2016; Edalatpour and Francoeur 2014; Edalatpour et al. 2015, 2016), and a fluctuating volume-current method (Polimeridis et al. 2015) have been applied to numerical simulation of near-field thermal radiation problems. The finite-difference frequencydomain and finite-difference time-domain methods are based on discretizing the differential form of the thermal stochastic Maxwell equations. These approaches require discretization of the free space in addition to the objects. In the boundary element method, the surface integral form of the thermal stochastic Maxwell equations is discretized. The thermal discrete dipole approximation and the fluctuating volumecurrent method are based on discretizing the volume integral form of the thermal stochastic Maxwell equations. In Sect. 4.3, solution of near-field radiative heat transfer problems via the thermal discrete dipole approximation is presented.

4.3

Overview of the Thermal Discrete Dipole Approximation (T-DDA)

The thermal discrete dipole approximation (T-DDA) is a numerical method for nearfield radiative heat transfer calculations in three-dimensional, arbitrary geometries. The T-DDA is based on the discrete dipole approximation (DDA), a widely used method for predicting electromagnetic scattering and absorption by particles (Yurkin and Hoekstra 2007; Moghaddam et al. 2016). An overview of the T-DDA is provided hereafter. A general near-field radiative heat transfer formalism is established starting from the thermal stochastic Maxwell equations (Eqs. 22 and 23) and by using Fig. 13, where an arbitrary number of finite objects are embedded into a vacuum of volume V1. The objects occupy as a whole a finite interior volume V2. The entire threedimensional space is given by ℜ3 = V1 [ V2. Hereafter, it is assumed that the material of the interior region V2 is isotropic, linear, possibly inhomogeneous, nonmagnetic, and in local thermodynamic equilibrium. The electric field due to external sources, such as illumination by a laser or thermal emission from the surroundings, is modeled via an incident electric field Einc. The electric field everywhere in the exterior and interior regions satisfies the following vector wave equation derived from the thermal stochastic Maxwell equations (Edalatpour et al. 2016) ∇
∇
Eðr, ωÞ  k20 Eðr, ωÞ ¼ 0,

r  V1

(72)

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Fig. 13 Graphical representation of a general near-field radiative heat transfer problems involving arbitrarily shaped objects. The interior region of total volume V2 is composed of objects embedded into the exterior region of volume V1. Individual objects may have different, inhomogeneous dielectric functions and temperatures. Illumination of the objects by external sources is represented by the incident electric field vector Einc (Figure taken from Edalatpour et al. (2016))

∇
∇
Eðr, ωÞ  k22 Eðr, ωÞ ¼ iωμ0 J2fl ðr, ωÞ,

r  V2

(73)

where k2 is the magnitude of the wavevector in V2, which is, in the general case, a complex number, while the subscript 2 in J2fl emphasizes that the fluctuating current is in V2. Equations 72 and 73 can be combined into a single inhomogeneous equation that is applicable everywhere in ℜ3 ∇
∇
Eðr, ωÞ  k20 Eðr, ωÞ ¼ iωμ0 Jðr, ωÞ,

r  ℜ3

(74)

where the current J is an equivalent source function. In the interior region, its expression is given by Jðr, ωÞ ¼ J2fl ðr, ωÞ  iωe0 ½e2 ðrÞ  1Eðr, ωÞ,

r  V2

(75)

where the second term on the right-hand side of the equation is the source function for the scattered field. The equivalent current J vanishes in the exterior region due to the absence of scattering objects and since it is a vacuum (non-emitting). The solution of the inhomogeneous linear differential Eq. 74 is the sum of the solution of the homogeneous equation and a particular solution of the inhomogeneous equation. The homogeneous vector wave equation is given by ∇
∇
Einc ðr, ωÞ  k20 Einc ðr, ωÞ ¼ 0,

r  ℜ3

(76)

The solution of Eq. 76 provides the incident electric field that exists in ℜ3 in the absence of objects. The particular solution of Eq. 74 is the sum of the scattered and fluctuating fields generated by the equivalent current J. The sum of the scattered and fluctuating

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electric fields, determined using the free-space dyadic Green’s function G0 relating the electric field observed at r to a source located at r0 in V2, is ð E ðr, ωÞ þ E ðr, ωÞ ¼ iωμ0 sca

fl

G0 ðr, r0 , ωÞ  Jðr0 , ωÞd3 r0 ,

r  ℜ3

(77)

V2

where eik0 R G ðr, r , ωÞ ¼ 4πR 0

"

0

! i 1 þ I ðk0 RÞ2 k0 R 1

! # 3i ^ ^ 1 þ R R ð k 0 RÞ 2 k 0 R 3

(78) ^ ¼ ðr  r0 Þ=jr  r0 j, I is the unit dyadic In the above expressions, R = |r  r0|, R and denotes the outer product defined as the multiplication of the first vector by the conjugate transpose of the second vector. The volume integral equation for the (total) electric field at r is determined by adding the incident field to Eq. 77 ð Eðr, ωÞ ¼ iωμ0

G0 ðr, r0 , ωÞ  Jðr0 , ωÞd3 r0 þ Einc ðr, ωÞ,

r  ℜ3

(79)

V2

The magnetic field in ℜ3 can be obtained using Eq. 79 and Faraday’s law (Eq. 22). In the absence of objects, the (total) electric field is simply equal to the incident field. The T-DDA formulation is initiated by discretizing V2 into N cubical subvolumes. The size of the subvolumes must be smaller than all characteristic lengths of the problem, namely, the wavelength in V2 and vacuum as well as the object-object separation distance. In addition, the subvolume size must be small enough to represent accurately the object shape via a cubical lattice. When these conditions are fulfilled, the electric field, the dyadic Green’s functions, and the electromagnetic properties can be assumed uniform within a given subvolume. Under the approximation of uniform electric field, it is possible to conceptualize the subvolumes as electric point dipoles. The total dipole moment associated with a subvolume i of volume ΔVi is related to the equivalent current via i pi ¼ ω

ð

Jðr0 , ωÞd3 r0

(80)

ΔV i

The discretized volume integral equation for the electric field (79) can thus be written in terms of dipole moments as follows 1 k2 X 0 3 1 fl  pi  0 G p ¼ pi þ Einc i αi e0 j6¼i ij j e2, i þ 2 αCM i

(81)

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where the dyadic Green’s function Gij is evaluated between the center points of subvolumes i and j. The total dipole moment pi is the sum of two contributions, namely, an induced dipole moment pind ¼ ΔV i e0 e2, i  1 Ei and a thermally Ði fluctuating dipole moment pfli = ði=ωÞ ΔV i Jfl2 ðr0 , ωÞd3 r0 . The ensemble average of the spatial correlation function of the fluctuating dipole moments, derived from the fluctuation-dissipation theorem (Eq. 24), is D E 4e0 e00 ΔV i 2, i Θðω, T Þδðω  ω0 ÞI pfli ðωÞ pfli ðω0 Þ ¼ πω

(82)

and αi in Eq. 81 are the Clausius-Mossotti and radiative The terms αCM i polarizabilities αCM ¼ 3e0 ΔV i i αi ¼

e2 , i  1 e2 , i þ 2

αCM i  3 ik0 ai ð1  ik a Þ  1 1  αCM =2πe a ½ 0 0 i i e i 

(83) (84)

where ai is the radius of a sphere of volume ΔVi. The application of Eq. 81 to all N subvolumes in V2 results in a system of equations that can be written in a matrix form fdt

AP¼E

inc

þE

(85)

fdt

where A is the interaction matrix, E is a column vector containing the first term on inc the right-hand side of Eq. 81, E is the incident field column vector, while P is the column vector of unknown total dipole moments. The system of equations (85) is stochastic, and its direct solution provides the instantaneous total dipole moment in each subvolume. The monochromatic power dissipated [W/(rad/s)] in the absorbers is calculated



ω X    2 3 ind

ind k  Qabs, ω ¼ Im α1 i 0 tr pi pi 2 i  abs 3

(86)



ind where ergodicity has been assumed (Mishchenko 2014). In Eq. 86, tr pind is i pi the trace of the autocorrelation function of the induced dipole moment of subvolume i. This term can be calculated directly from the system of equations (85) such that there is no need to compute the instantaneous total dipole moment, which is a quantity that is not experimentally observable (Edalatpour et al. 2015). Note that the T-DDA formalism only involves numerical approximations. Therefore, the T-DDA can be considered as numerically exact since the results obtained from this method converge to the exact solution in the limit that N ! 1.

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Figure 14 shows the monochromatic thermal conductance [W/K  eV] between two spheres made of silica calculated with the T-DDA and the analytical solution based on deriving closed-form expressions for the dyadic Green’s functions (Narayanaswamy and Chen 2008). The monochromatic thermal conductance at a temperature T is Gω ðT Þ ¼ lim

δT!0

Qnet, ω δT

(87)

where hQnet,ωi is the net monochromatic heat rate between the spheres computed using Eq. 86. The diameters of the spheres are fixed at 0.5 μm, their separation distance is 0.5 μm, and the temperature is 400 K. Figure 14 shows that a good agreement between the T-DDA and the analytical solution is obtained. The T-DDA can also accommodate near-field radiative heat transfer calculations between arbitrarily shaped objects and a surface, modeled as a half-space (see Fig. 15) (Edalatpour and Francoeur 2016). When dealing with a surface, the volume integral equation for the total electric field (Eq. 79) is modified

Fig. 14 Spectral conductance between two 0.5 μm-diameter silica spheres separated by a 0.5 μmthick vacuum gap. The temperature T is fixed at 400 K, and each sphere is discretized into 552 subvolumes. The T-DDA predictions are compared against exact results obtained from the analytical solution for two spheres (Narayanaswamy and Chen 2008)

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Fig. 15 Graphical representation of a general near-field radiative heat transfer problem involving arbitrarily shaped objects and a surface. The interior region of total volume V2 is composed of objects embedded into a vacuum of volume V0. The surface is modeled as a half-space and has a volume V1. The incident electric field Einc accounts for illumination by external sources, while the surface field Esur is the electric field due to thermal emission by the surface (Figure taken from Edalatpour and Francoeur (2016))

Eðr, ωÞ ¼ iωμ0

 ð 0 R  G ðr, r0 , ωÞ þ G r, r0 , ω  Jðr0 , ωÞd3 r0

V2 inc

þE ðr, ωÞ þ E ðr, ωÞ, sur

(88)

r  V0 [ V2

R

where G is the reflection dyadic Green’s function representing the electric field generated at r due to radiation by the source J located at r0 after reflection by the surface, while Esur is the electric field thermally emitted by the surface. In this formalism, only the objects described by V2 are discretized; the surface interactions are treated analytically. After discretization of V2 into N cubical subvolumes, the volume integral equation for the electric is written in terms of dipole moments as follows: 1 k2 X 0 k2 X R 3 1 sur  CM pfli þ Einc pi  0 Gij  pj  0 Gij  pj ¼  i þ Ei αi e0 j6¼i e0 j e2, i þ 2 αi

(89)

The application of Eq. 89 to all subvolumes in V2 results in a system of equations that can be written in a matrix form:   fdt inc sur AþR P¼E þE þE sur

(90)

where E is the surface field column vector and R is the reflection-interaction matrix calculated from the reflection dyadic Green’s function. The net monochromatic heat

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Fig. 16 Net spectral heat rate between a sphere and a surface for separation gaps d of 100 μm and 100 nm obtained with the T-DDA and the exact solution (Otey and Fan 2011). The sphere is at a temperature T2 = 400 K, while the surface is at T1 = 300 K. The sphere and the surface are made of silica, and the sphere diameter is 1.6 μm. The number of subvolumes employed for discretizing the sphere varies between 11,536 and 33,552 depending on the frequency (Figure taken from Edalatpour and Francoeur (2016))

rate between the objects and the surface is calculated by using Eq. 86 (Edalatpour and Francoeur 2016). Figure 16 shows the net monochromatic heat rate between a sphere and a surface for separation gaps d of 100 μm and 100 nm obtained from the T-DDA with surface interaction. For comparison, the exact results obtained from the analytical solution for the sphere-surface configuration (Otey and Fan 2011) are plotted in Fig. 16. A thorough error and convergence analysis of the T-DDA has been provided in Ref. (Edalatpour et al. 2015). The T-DDA with surface interaction has been applied to analyze near-field radiative heat transfer between a complex-shaped probe and a surface (Edalatpour and Francoeur 2016).

4.4

Further Reading

• Additional information on the free-space dyadic Green’s function: (Novotny and Hecht 2006) • Dyadic Green’s functions for two planar interfaces: (Joulain et al. 2005; Sipe 1987) • Near-field radiative heat transfer between effectively magnetic materials: (Joulain et al. 2010; Francoeur et al. 2011a) • Alternative derivation of the near-field radiative heat flux between two halfspaces via the Landauer formalism: (Biehs et al. 2010)

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• Near-field radiative heat transfer between hyperbolic metamaterials: (Biehs et al. 2012, 2013) • Discussion on the maximum achievable near-field radiative heat transfer: (Pendry 1999; Basu and Zhang 2009; Ben-Abdallah and Joulain 2010; Miller et al. 2015) • Review article surveying different approaches for solving near-field radiative heat transfer problems: (Liu et al. 2015) • General review articles on near-field thermal radiation: (Joulain 2006; Joulain et al. 2005; Narayanaswamy and Chen 2005a; Edalatpour et al. 2016; Basu et al. 2009; Jones et al. 2013; Park and Zhang 2013; Song et al. 2015b).

5

Cross-References

▶ A Prelude to the Fundamentals and Applications of Radiation Transfer ▶ Design of Optical and Radiative Properties of Surfaces ▶ Thermal Transport in Micro- and Nanoscale Systems Acknowledgments The author acknowledges the financial support of the National Science Foundation (Grants No. CBET 1253577 and CBET 1605584) and the US Army Research Office (Grant No. W911NF-14-1-0210).

References Babuty A, Joulain K, Chapuis P-O, Greffet J-J, De Wilde Y (2013) Blackbody spectrum revisited in the near field. Phys Rev Lett 110:146103 Balanis CA (2012) Advanced engineering electromagnetics, 2nd edn. Wiley, New York Basu S (2016) Near-field radiative heat transfer at nanometer distances. Elsevier, Oxford Basu S, Zhang Z (2009) Maximum energy transfer in near-field thermal radiation at nanometer distances. J Appl Phys 105:093535 Basu S, Zhang ZM, CJ F (2009) Review of near-field thermal radiation and its application to energy conversion. Int J Energy Res 33:1203–1232 Ben-Abdallah P, Biehs S-A (2015) Contactless heat flux control with photonic devices. AIP Adv 5:053502 Ben-Abdallah P, Joulain K (2010) Fundamental limits for noncontact transfers between two bodies. Phys Rev B 82:121419 Ben-Abdallah P, Joulain K, Drevillon J, Domingues G (2009) Near-field heat transfer mediated by surface wave hybridization between two films. J Appl Phys 106:044306 Ben-Abdallah P, Biehs S-A, Joulain K (2011) Many-body radiative heat transfer theory. Phys Rev Lett 107:114301 Bernardi MP, Dupré O, Blandre E, Chapuis P-O, Vaillon R, Francoeur M (2015) Impacts of propagating, frustrated and surface modes on radiative, electrical and thermal losses in nanoscale-gap thermophotovoltaic power generators. Sci Rep 5:11626 Bernardi MP, Milovich D, Francoeur M (2016) Radiative heat transfer exceeding the blackbody limit between macroscale planar surfaces separated by a nanosize vacuum gap. Nat Commun 7:12900 Biehs S-A, Greffet J-J (2010) Near-field heat transfer between a nanoparticle and a rough surface. Phys Rev B 81:245414

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Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Modeling and Theoretical Fundamentals of Periodic Structures . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Numerical Methods for Modeling Optical Properties of Period Structures . . . . . . . . . 2.2 Anisotropic Rigorous Coupled-Wave Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Surface Plasmon Polaritons and Magnetic Polaritons in Nanostructures . . . . . . . . . . . 2.4 Polarization Dependence of Radiative Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Applications of Periodic Nano-/Microstructures and Metamaterials . . . . . . . . . . . . . . . . . . . . . . 4 Tailoring Thermal Radiation Using 2D Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Optical and Radiative Properties of Graphene and its Ribbons . . . . . . . . . . . . . . . . . . . . . 4.2 Graphene-Covered Metal Gratings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Hexagonal Boron Nitride-Covered Metal Gratings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Conclusion and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Cross-References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

Tailoring optical and radiative properties has attracted a great deal of attention in recent years due to its importance in advanced energy systems, nanophotonics, electro-optics, and nanomanufacturing. Micro-/nanostructured surfaces can interact with electromagnetic waves in a unique way by excitation of various optical resonances or polaritons that can modify the polarization-dependent directional and spectral radiative properties. Latest advances in graphene and other two-dimensional (2D) materials offer enormous potentials to revolutionize current microelectronic, optoelectronic, and energy harvesting systems. This chapter summarizes the recent advances in the design of optical and radiative properties B. Zhao · Z. M. Zhang (*) George W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA, USA e-mail: [email protected]; [email protected] # Springer International Publishing AG, part of Springer Nature 2018 F. A. Kulacki (ed.), Handbook of Thermal Science and Engineering, https://doi.org/10.1007/978-3-319-26695-4_58

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of micro-/nanostructures and 2D materials. The physical mechanisms that are behind the exotic behaviors are discussed, with an emphasis on various plasmonic and phononic polaritons. Anisotropic rigorous coupled-wave analysis is presented as a modeling technique that is suitable to simulate periodic multilayer structures involving anisotropic materials. The insights gained from this chapter may benefit the future development of energy harvesting systems, photodetectors, thermal management, local thermal management, and high-resolution thermal sensing.

1

Introduction

Radiation from a thermal source, such as an incandescent light bulb, is usually incoherent. However, in the late 1990s and at the beginning of the twenty-first century, researchers demonstrated that the thermal radiation from a periodically patterned grating surface with a period smaller than the typical wavelength of the thermal radiation can be coherent (to certain degree) in narrow wavelength bands as well as in a well-defined direction (Hesketh et al. 1986; Greffet et al. 2002). Since then, various subwavelength micro-/nanostructures have been proposed that can tailor optical and radiative properties both spectrally and directionally (Lee et al. 2005; Dahan et al. 2008; Cai and Shalaev 2009; Tan et al. 2016). This research area has gained increasing attention due to the widespread important applications (Zhang et al. 2003; Zhang 2007), such as photodetectors (Chang et al. 2010; Kulkarni et al. 2010; Li and Valentine 2014), band-pass filters (Ebbesen et al. 1998; Porto et al. 1999; Baida and Van Labeke 2002; Lee et al. 2008a), negative refraction (Pendry 2000; Shelby et al. 2001; Smith et al. 2004; Shalaev et al. 2005), anomalous reflection (Grady et al. 2013; Hou-Tong et al. 2016), clocking (Pendry et al. 2006; Schurig et al. 2006; Cai et al. 2007; Liu et al. 2009), surface-enhanced Raman spectroscopy (Stiles et al. 2008), medical therapy (Cai et al. 2008), thermophotovoltaics (TPV) emitter and absorber (Basu et al. 2007; Wang and Zhang 2012b; Watts et al. 2012; Boriskina et al. 2013), photovoltaic and solar thermal technology (Baxter et al. 2009; Atwater and Polman 2010; Cui et al. 2012; Fan 2014; Khodasevych et al. 2015), structural color printing (Cheng et al. 2015), as well as radiative cooling (Raman et al. 2014). Tailoring radiative properties generally relies on various resonance modes or surface waves in periodic micro-/nanostructures. Among them are gratings that support surface phonon polaritons (SPhPs) (Greffet et al. 2002) or surface plasmon polaritons (SPPs) (Raether 1988; Zayats et al. 2005; Williams et al. 2008), hyperbolic metamaterials that can empower hyperbolic modes and epsilon-near-zero modes (Krishnamoorthy et al. 2012; Poddubny et al. 2013; Othman et al. 2013), and metal/dielectric/metal structures and deep gratings that can enable magnetic polaritons (MPs) (Lee et al. 2008b; Zhao and Zhang 2014; Xuan and Zhang 2014). Meanwhile, various structures like photonic crystals (Luo et al. 2004), nanowire arrays (Zhang and Ye 2012), and nanoparticles (Didari and Menguc 2015) have also been extensively studied for their radiative properties. Figure 1 shows various metal/dielectric/metal structures that can support MPs from visible to

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Fig. 1 Schematic of metal/dielectric/metal emitters and absorbers with different top metal patterns over a ground plate separated by a dielectric spacer: (a) strip, (b) square, (c) cross, (d) circular disk, and (e) square ring. The geometry factors are illustrated in the figure (From Sakurai et al. 2014)

far-infrared region (Sakurai et al. 2014). The radiative properties of rough surfaces have been investigated and reviewed (Zhu et al. 2009). The optical and radiative properties of these period micro-/nanostructured surfaces can be dramatically different from the radiative properties of the counterpart rough or smooth surfaces due to the excitation of surface waves and polaritons. Emerging two-dimensional (2D) materials (Geim and Grigorieva 2013; Xia et al. 2014), including graphene (Novoselov et al. 2004; Lin and Connell 2012; Grigorenko et al. 2012; Koppens et al. 2014; Bonaccorso et al. 2015), hexagonal boron nitride (hBN) (Dai et al. 2014), transition metal dichalcogenides (Wang et al. 2015), and recently emerged black phosphors (Ling et al. 2015), offer exciting new possibilities to construct micro-/nanostructures with unprecedented optical and radiative properties. With a layer of carbon atoms arranged in a honeycomb lattice, graphene exhibits unique electronic, thermal, mechanical, and optical properties (Nair et al. 2008; Thongrattanasiri et al. 2012; Pop et al. 2012). It enables saturation absorption in the visible and near-infrared region, and low-loss, actively tunable surface plasmons in mid- and far-infrared region (Novoselov et al. 2004; Grigorenko et al. 2012; Nair et al. 2008), which have been demonstrated for their ability to control thermal radiation (Freitag et al. 2010). Hyperbolic 2D materials like hBN (Lin and Connell 2012; Dai et al. 2014, 2015b; Brar et al. 2014; Kumar et al. 2015) can support plentiful phononic resonance modes with unique radiative properties. Moreover, it is recently demonstrated that 2D materials can be used together with nano-/microstructures to assemble hybrid structures with unique tunable radiative properties by enabling plentiful coupling effects (Li et al. 2016). Optical and radiative properties essentially describe the way that light interacts with matter. When the characteristic length of the material or structure is much longer than the wavelength of light, the polarization and vector attributes of light may be neglected, and radiative properties of this scale can be solved using ray tracing or geometrical optics (Modest 2013; Howell et al. 2015). However, to

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calculate the radiative properties of subwavelength periodic nano-/microstructured surfaces, Maxwell’s equations have to be solved to fully consider the vector and polarization characters of electromagnetic waves (Zhang et al. 2003; Xuan 2014). Recent advances in 2D materials put new challenges on the numerical modeling of the optical and radiative properties due to their atomically thin thickness and natural anisotropy, especially when 2D materials are combined with periodic nanostructures. This chapter is intended to discuss the numerical modeling of periodic micro-/ nanostructures and different 2D materials, as well as the unique optical and radiative properties of typical subwavelength structures. An anisotropic rigorous coupledwave analysis (RCWA) algorithm is described that can be used to model multilayer periodic micro-/nanostructures involving anisotropic materials. Optical and radiative properties of different micro-/nanostructures, 2D materials, as well as various hybrid structures are reviewed with their potential applications. The physical mechanisms of the unique radiative properties are discussed. The arrangement of this chapter is listed in the following: Section 2 discusses the popular modeling techniques, with a focus on anisotropic RCWA. The basics are summarized with equations emphasizing the difference as compared to the original RCWA. The fundamentals of light diffraction by periodic structured surfaces are discussed to serve as the theoretical background. The optical and radiative properties micro-/nanostructures are reviewed, including the underlying mechanism of MPs in deep gratings, the radiative properties of SPPs in 2D nanostructures, and anisotropic metamaterials. Section 3 reviews the potential applications of tailored thermal radiative properties with an emphasis on the application of TPVs. Section 4 summarizes the recent research on using 2D materials with nanostructures to achieve unique radiative properties. Wavelength-selective perfect absorption in graphene and hBN-covered metal gratins are specifically discussed. The underlying mechanisms are discussed with an emphasis on the role of 2D materials in modifying the radiative properties of nano-/microstructures.

2

Modeling and Theoretical Fundamentals of Periodic Structures

2.1

Numerical Methods for Modeling Optical Properties of Period Structures

Maxwell’s equations describe how electromagnetic waves propagate in a certain medium and interact with objects in the medium. Together with the constitutive relations from which the permittivity and permeability of the medium can be obtained, radiative properties can be attained by studying how light interacts with various objects. Numerical methods are usually used to simulate the radiative properties of periodic structures since analytical solutions are seldom available due

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to the complex geometries (Xuan 2014). Computational electromagnetics models the interaction of electromagnetic fields with physical objects and the environment. It is widely used to calculate antenna performance, radar cross section, and electromagnetic wave propagation in a certain medium or structure. Various computational techniques have been developed. Choosing the right technique for solving a certain problem is important, because using an inappropriate one may make the computational time excessively long or even lead to incorrect results. For periodic structures, the most popular simulation tools are the finite-difference time-domain (FDTD) or Yee’s method (Kane 1966; Taflove and Hagness 2005), the finite-element method (FEM) (Davidson 2010), and the rigorous coupled-wave analysis (RCWA) method (Moharam and Gaylord 1981). FDTD is a time-domain method that uses wideband sources and computes a wideband response in one run, whereas RCWA is a frequency-domain method that has to calculate the system response for each frequency point. If only a narrowband response is of the interest, a very large number of time steps may be required for FDTD (Davidson 2010). Meanwhile, the properties of the material (namely, the permittivity and permeability) can be expediently expressed as a function of the frequency in the frequency domain, but it is more challenging in the time domain since a convolution is implied (Xuan 2014). FEM can be used in both time and frequency domain. All three methods are widely used. Commercial software such as Remcom XFDTD and Lumerical FDTD solutions, as well as a free-software package (Oskooi et al. 2010), are based on FDTD. COMSOL RF module and Ansoft HFSS are examples that use FEM, while RCWA programs are also accessible online free of charge (Liu and Fan 2012; Zhang 2014). FDTD and FEM methods discretize the structure or computation domain with mesh or grid and numerically solve the quantity of interest associated with each element or cell. For structures with small dimensions like 2D materials, accrete simulation using FEM and FDTD may become very challenging and timeconsuming since the dimension of the mesh needs to be very small in the out-ofplane direction, especially when structures with small characteristic lengths coexist with large structures, like the hybrid structures with 2D materials and micro-/ nanostructures. On the contrary, RCWA is a semi-analytical method that does not require a discretization of the structure. Instead, the electric or magnetic field in the structure is expressed as a Fourier series with the coefficients to be solved using boundary conditions. Since mesh is not needed, RCWA has advantages in modeling the structures that contain extremely small geometries.

2.2

Anisotropic Rigorous Coupled-Wave Analysis

RCWA has been widely used in modeling the radiative properties of periodic structures. Most available RCWA algorithms are for isotropic materials based on the widely used algorithms for one-dimensional (1D) grating proposed by Moharam and Gaylord (1981). Afterward, anisotropic RCWA has been studied by different researchers. Glytsis and Gaylord (1990) formulated RCWA for 1D anisotropic gratings, but the algorithm may have convergence issues for highly conducting

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Fig. 2 Illustration of the numerical model for general 2D periodic multilayer structures consisting of anisotropic materials (From Zhao and Zhang 2016)

metal gratings because of the way that Fourier factorizations were calculated. Later, Li (1996, 1998) reformulated the algorithm with the correct Fourier factorization rules. Continuous effort has been devoted and more general algorithms have been formulated to consider 2D arbitrary lattice configurations and permittivity tensor (Popov and Nevière 2001; Lin et al. 2002; Li 2003). In the rest of this section, an easy-to-implement algorithm to model a 2D multilayered periodic structure made of biaxial materials with a diagonal permittivity tensor is summarized. This algorithm is an extension of the public available RCWA code (Zhang 2014), and the core equations are summarized below and compared to the original RCWA, which has been presented in other works (Lee et al. 2008a; Zhang and Wang 2013; Chen and Tan 2010; Bräuer and Bryngdahl 1993). It also serves as a theoretical background of the later sections since the resonance mechanisms are easier to be understood in the frequency domain. The 2D anisotropic multilayered periodic structure is schematically shown in Fig. 2. The periodicity is characterized by Λx and Λy, which are the periods in the xand y-directions, respectively. Each layer in the structure can be either a grating or a continuous film by adjusting the lateral dimensions lx and ly. In the schematic, the first layer is a 2D grating and the rest are films. The medium where the wave is incident, the intermediate layers (total N layers), and the semi-infinite substrate can be categorized as Region I, II, and III, respectively, as indicated in the schematic. The incident medium with a dielectric function eI is usually vacuum or lossless dielectric and set to be isotropic. The incident wave with an electric field Einc is assumed to be linearly polarized. The plane of incidence indicated in transparent gray color is the plane determined by the incident wave vector kinc = (kx, inc, ky, inc,

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Design of Optical and Radiative Properties of Surfaces

1029

kz, inc) and the z-axis. A polar angle θ (the angle between kinc and the z-axis) and azimuthal angle ϕ (the angle between the x-axis and the plane of incidence) are used to depict the direction of kinc. Polarization angle ψ is defined as the angle between the electric field and the plane of incidence. In Region I, the electric field contains the incident and reflected fields and the expression is the same as the isotropic RCWA (Zhao et al. 2013):
 EI ¼ Einc exp ikx, inc x þ iky, inc y þ ikz, inc z   XX þ Ermn exp ikx, m x þ iky, n y  ikrz, mn z m

(1)

n

The time-harmonic term, exp(iωt), with ω being the angular frequency, is omitted hereafter. The second term on the right-hand side is the reflected wave. Ermn is the complex amplitude of the (m, n) order reflected wave, and its transverse wave vector components are determined by the Bloch-Floquet condition (Glytsis and Gaylord 1990): 2π Λx 2π ky, n ¼ ky, inc þ n Λy kx, m ¼ kx, inc þ m

(2)

where m and n denote the diffraction orders in the x- and y-directions, respectively, and they can take both positive and negative numbers. The z-components of the wave vector is

krz, mn

8 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi < k2I  k2x, m  k2y, n qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ : i k2 þ k2  k2 x, m y, n I

,

eI k20  k2x, m þ k2y, n

,

eI k20 < k2x, m þ k2y, n

(3)

where k0 = ω/c0 is the wave vector (magnitude) in vacuum with c0 being the speed of light in vacuum. In each layer of Region II, both the electromagnetic field and the dielectric function are expressed as Fourier series based on the periods in the x- and ydirections. The materials of each layer are assumed to be nonmagnetic, and thus the dielectric function of the layer can be described by a location-dependent permittivity tensor: 0

ej, x ej ðx, yÞ ¼ @ 0 0

0 ej, y 0

1 0 0 A ej, z

(4)

where j is the number of the layer in the structure ranging from 1 to N. Generally, e = e0 + ie00 is a complex number with e0 and e00 being its real and imaginary part, respectively. The electric and magnetic field can be expressed as

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EII ¼

XX


 χmn ðzÞexp ikx, m x þ iky, n y

m ffiffiffiffiffin r
 e0 X X HII ¼ i γmn ðzÞexp ikx, m x þ iky, n y μ0 m n

(5)

The unknowns χmn(z) and γmn(z) can be related by the following equations based on Maxwell’s equations, ∇  EII  iωμ0HII = 0 and ∇  HII þ iωe0 eEII ¼ 0 , where μ0 and e0 are the permeability and permittivity of vacuum, respectively: ! @χ y, mn ky, n X X kx, p γ y, pq  ky, q γ x, pq ¼ þ k0 γ x, mn @z k0 p q eord z, mp, nq
 @χ x, mn kx, m X X kx, p γ y, pq  ky, q γ x, pq ¼  k0 γ y, mn eord @z k0 p q z, mp, np  X X k0 χ x, pq @γ y, mn ky, n
¼ kx, m χ y, mn  ky, n χ x, mn  @z k0 einv x, m-p, n-q p q  X X k0 χ y, pq @γ x, mn kx, m
¼ kx, m χ y, mn  ky, n χ x, mn  @z k0 einv y, mp, nq p q

(6)

where p and q are two integers. The superscripts ord and inv indicate the coefficients of the Fourier series for e and its inverse, respectively, and the expressions are similar to Chen and Tan (2010). The inverse of the dielectric function is used for the sake of fast convergence of the algorithm according to the Fourier factorization rule (Li 1996). Note that different dielectric function components are used in Eq. 6 because of the anisotropy of the material. When all three components of the dielectric tensor are equal, Eq. 6 degenerates to the isotropic scenario presented in Chen and Tan (2010). The substrate (Region III) in general can be a biaxial medium. A transmitted plane wave (forward propagating wave) in this region with an in-plane wave vector t t (kx,m, ky,n) can have   two different kz, mn . If the electric field is Emn ¼ Etx, mn , Ety, mn , Etz, mn , then based on Maxwell’s equation (Yeh 1979), one obtains 1  2 0 t 1 eIII, x k20  k2y, n  ktz, mn kx, m ky, n kx, m ktz, mn C Ex, mn B C B   C t 2 CB B C@ Ey, mn A ¼ 0 B k x, m k y, n eIII, y k20  k2x, m  ktz, mn ky, n ktz, mn A Et @ z, mn kx, m ktz, mn ky, n ktz, mn eIII, z k20  k2x, m  k2y, n 0

(7) To have nontrivial plane-wave solutions, the determination of the matrix has to be zero and four solutions can be obtained. Two solutions of ktz, mn that correspond to the two forward propagating waves (Li 1998) are used to express the electric field in Region III as

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Design of Optical and Radiative Properties of Surfaces

EIII ¼

XX X m

n

σ¼1, 2

  Etσ, mn pσ, mn exp ikx, m x þ iky, n y þ iktσ , z, mn z

1031

(8)

in which 0  2    2 1 2 2 t 2 2 2 2 t  k e k  k  k e  k  k k k C B III, y 0 x, m σ , z, mn 0 III, z x, m y, n y, n σ , z, mn C B C B     2 C B pσ , mn ¼ N σ , mn B C kx, m ky, n ktσ , z, mn  kx, m ky, n eIII, z k20  k2x, m  k2y, n C B   C B  2 A @ 2 t t 2 2 t kx, m ky, n kσ, z, mn  kx, m kσ , z, mn eIII, y k0  kx, m  kσ , z, mn

(9) is the polarization vector for the electric field with diffraction order (m, n) and σ is the index for the two forward propagating waves. Note that Nσ,mn is a coefficient that normalizes Pσ,mn. The magnetic field for Regions I and III can be obtained from the electric field based on Maxwell’s equation. The complex coefficients in the field expressions are then solved through matrix manipulations by matching the tangential component of the electric and magnetic field between adjacent layers. Once the coefficients are solved, the spectral directional-hemispherical reflectance (R)`, spectral directionalhemispherical transmittance (T ), as well as the field distributions in each layer can be obtained. Note that if the substance is isotropic like the structure to be discussed in this work, the field expression in Eq. 8 will be the same as isotropic RCWA (Chen and Tan 2010). In this case, the only difference between the isotropic and anisotropic RCWA exists in Eq. 6. The spectral directional absorptance of a structure can be obtained from α = 1– R– T based on energy balance, and the spectral directional emissivity e is equal to the absorptance based on Kirchhoff’s law. For opaque structures, T = 0 and thus e = α = 1– R. To illustrate the local absorption profile inside the structure, the local power dissipation density in W/m3 can be calculated based on (Zhao and Zhang 2015b)    2

1 1 wðx, y, zÞ ¼ Re iωe0 eE  E ¼ e0 ω e00x jEx j2 þ e00y Ey  þ e00z jEz j2 2 2

(10)

in which E is the complex electric field obtained from RCWA. The absorptance of a certain volume or layer can be calculated by the ratio of the absorption inside the volume over the incident power (Zhao et al. 2014, 2015): ÐÐÐ α¼

ωðx, y, zÞdV

0:5c0 e0 jEinc j2 A cos θ

(11)

The denominator is the incident power on area A at a polar angle θ. Using Eq. 11, the absorptance inside a certain volume can be retrieved. The integration over the whole structure yields the same absorptance with 1-R-T.

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Surface Plasmon Polaritons and Magnetic Polaritons in Nanostructures

Generally speaking, the reason for the unique optical and radiative properties of the nano-/microstructures is the excitation of various polaritons, which are also of critical importance in near-field radiative heat transfer between two objects, as discussed in ▶ Chap. 24, “Near-Field Thermal Radiation.” Polaritons are quasiparticles resulting from strong coupling of electromagnetic waves with an electric or magnetic dipole-carrying excitation. One of such excitations is plasmons, which are quasiparticles associated with oscillations of plasma, i.e., a collection of charged particles inside the materials. The external electromagnetic wave can couple with a surface plasmon (SP), which is a collective oscillation of surface charges, to form a surface plasmon polariton (SPP) (Barnes et al. 2003; Zayats et al. 2005). SPPs have been intensively studied for applications in lithography (Srituravanich et al. 2004), chemical and bio-sensing (Homola 2008), extraordinary optical transmission (Barnes et al. 2003), and optical communication (Ozbay 2006). Upon the excitation of SPPs, the charges close to the surface are driven by the electric field and oscillate back and forth intensively. The field of an SPP is confined near the surface with the amplitude exponentially decaying away from the interface. Not only is a surface wave induced propagating along the interface with an amplitude exponentially decaying away from the interface (Raether 1988; Ghaemi et al. 1998; Barnes et al. 2003; Zayats et al. 2005; Marquier et al. 2007; Chen and Chen 2013), but the oscillation of charges also dissipates the electromagnetic energy into heat, creating strong absorption at the resonance frequency. SPPs are non-radiative surface waves since the required wave vector is larger than the free space wave vector, and they do not couple with propagating electromagnetic waves in vacuum (Zhang 2007). This can be seen from the magnitude of the wave vector of the SPs on a metal-dielectric interface that can be expressed as (Zhang 2007) jkSP j ¼ k0

rffiffiffiffiffiffiffiffiffiffiffiffiffiffi e1 e2 e1 þ e2

(12)

where k0 = ω/c0 is the magnitude of the wave vector in vacuum, and e1 and e2 here are the dielectric functions of the dielectric and metal, respectively. Note that Eq. 12 is for nonmagnetic materials and transverse magnetic (TM) waves. For transverse electric (TE) waves, the excitation of SPs requires a magnetic material with negative permeability. The wave vector of the electromagnetic waves must have a tangential component equal to kSP to excite SPPs. Since |kSP| is greater than k0, high index prisms can be used to increase the wave vector of the incident waves so that SPs can couple with incident light to excite SPPs. Another method is taking advantage of the diffracted light by periodic micro-/nanostructures, as will be explained in the following. Consider a 1D silver (Ag) grating on a semi-infinite Ag substrate shown in Fig. 3 (a). The grating has a periodicity in the x-direction described by Λ = 1.7 μm. The

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Design of Optical and Radiative Properties of Surfaces

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Fig. 3 (a) Schematic of the 1D grating with a period Λ, height or depth h, ridge width w, and trench width b. The equivalent LC circuit model is also shown with the capacitance C and inductance L. The wave vector kinc of the incident plane wave is in the x-z plane at an angle θ with respect to the z-axis. (b) Folded dispersion of SPPs for a Ag grating with Λ = 1.7 μm, h = 0.1 μm, and b = 0.595 μm. (c) Normal reflectance of TM waves for a Ag grating with the same geometries with (b)

grating height is h = 0.1 μm and the trench width is b = 0.595 μm. The incident wave is a plane wave with an oscillating magnetic field in the y-direction (i.e., TM wave). For this 1D grating, the Bloch-Floquet condition presented in Eq. 2 becomes kx , m = kx , inc + 2πm/Λ. Thus, the dispersion relation can be folded into the region for kx  π/Λ, and SPPs can be excited on a grating surface with propagating waves in air. Figure 3b shows the folded dispersion relation of SPPs for the given Ag grating. Unless mentioned specifically, the optical properties of Ag in this work are obtained using a Drude model (Zhang 2007; Modest 2013). The intersections of the folded dispersion with the vertical axis identify the location where SPPs can be excited for a normal incidence, as shown in Fig. 3c. Note that the plane of incidence is set as the x-z plane for normal incidence. Since the grating is 1D and the electric  field of TM waves is in the x-direction, θ = ψ = ϕ = 0 and ky = 0. The excitation of surface polaritons is responsible for the dips in the reflectance, whose frequency locations agree well with predictions of the dispersion curves. Figure 4 illustrates an instantaneous field distribution at ν = 5,727 cm1 corresponding to the first

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Fig. 4 Instantaneous field distribution at the excitation of SPP at ν = 5727 cm1 for an Ag grating with Λ = 1.7 μm, h = 0.1 μm, and b = 0.595 μm. The incident wave is at normal direction and TM-polarized. Color represents the relative magnetic field and the arrows show the electric field

reflectance dip in Fig. 3c. The colors represent direction and magnitude of the magnetic field and the arrows show the electric field. Note that this surface wave is a standing wave since the incident photon can couple with two surface plasmons at normal incidence: one corresponds to m = +1 with a wave vector ksp = 2π/Λ (right traveling) and the other corresponds to m = 1 with ksp = 2π/Λ (left traveling). The two surface waves form a standing wave with a wavelength that is nearly equal to the period of the grating. At oblique incidence, only one SPP can be excited and thus the surface wave is not a standing wave (Zhao et al. 2015). Note that e1 and e2 in Eq. 12 need to have different signs. For typical metal, since e2 has a negative real part whose magnitude is much larger than e1, the dispersion of SPP is very close to the light line, which is k0 = ω/c0. On the other hand, in some cases where e2 has a positive real part, another type of surface waves that has a similar dispersion called Wood’s anomaly (WA) (Nguyen-Huu et al. 2012) may be supported. Wood’s anomaly occurs when a diffraction order shows up at the grazing angle, and its dispersion can be expressed as |kk,mn| = k0, which is essentially the light line (Lee et al. 2008a). Both SPP and WA can be used to tailor the optical and radiative properties, as will be discussed later. Another type of polaritons that has been widely studied is magnetic polariton (MP). MPs refer to the strong coupling of the magnetic resonance inside a micro-/ nanostructure with the external electromagnetic waves (Wang and Zhang 2009, 2012a, b; Mattiucci et al. 2012; Pardo et al. 2011; Lee et al. 2008b; Wang et al. 2014). This type of resonance was initially discovered and experimentally demonstrated in split-ring resonator (Linden et al. 2004), metal-rod pairs (Podolskiy et al. 2002), and fishnet structures (Zhang et al. 2005) that are possessed of negative permeability in the microwave region. The radiative properties of MPs have also been experimentally studied as have been summarized in a recent review (Zhang and Wang 2013). Take the grating in Fig. 3a as an example, if the geometries change to Λ = 400 nm, h = 200 nm, and b = 5 nm, for TM waves, this deep grating can

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Design of Optical and Radiative Properties of Surfaces

1035

Fig. 5 Emittance for Ag deep gratings with Λ = 400 nm, h = 200 nm, and b = 5 nm: (a) Normal spectral emittance; (b) contour plot of the emittance in terms of the wavenumber and parallel component of the wave vector. The vertical line with kx = 0 represents normal direction and the diagonal represents grazing angle or light line (From Zhao and Zhang 2014)

support MP that is characterized by an emittance peak as high as 0.85 at the wavelength of 2.74 μm in the normal emittance spectrum shown in Fig. 5a. The emittance enhancement is remarkable since the emittance is less than 0.005 for a smooth Ag surface at this wavelength. Compared to SPPs, MPs show less angle dependence as shown in Fig. 5b. The excitation of this MP is caused by the oscillating magnetic field of the incident waves. Under the time-varying magnetic field parallel to the y-direction, an oscillating current is produced around the grooves in the x-z plane, generating a strong magnetic field according to Faraday’s law. Figures 6a, b show the instantaneous electromagnetic and current-density field when the MP occurs in the Ag grating. The magnetic field, represented by the color contour, is the square of the relative amplitude. Since the instantaneous electric and current-density field vectors oscillate with time, the direction of the arrows may reverse. The big arrows show the general direction of the vectors. The current-density vectors are obtained from J = σE where σ is the complex electrical conductivity of the material at the given location (Zhang 2007), and they form a closed loop around the trench. The strongest magnetic enhancement corresponding to the closed current loop is at the bottom of the trench, where the magnitude of magnetic field is more than 30 times that of the incident waves, showing a strong diamagnetic effect. Meanwhile, when the resonance happens, charges tend to accumulate at the upper corner of the grating, and this in turn creates a strong electric field around the trench opening. Based on the closed current loop, an equivalent LC circuit model (Wang and Zhang 2009, 2011, 2012b; Engheta 2007; Solymar and Shamonina 2009; Zhao and Zhang 2014) shown in Fig. 3a can be used to predict the magnetic resonance condition. The air in the trenches serves as a dielectric capacitor and the surrounding metallic material acts as an inductor. Since the walls on both sides of the groove are

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Fig. 6 (a) The electromagnetic field and (b) current-density distribution in the Ag grating with the same parameters as for Fig. 5 at λ = 2.74 μm. The color contour shows the relative magnitude of the y component of the magnetic field. The vectors show the direction and magnitude of the electric field in (a) and current density in (b) (From Zhao and Zhang 2014)

close to each other, mutual inductance Lm needs to be considered. It can be evaluated from the parallel-plate inductance formula and written as Lm = μ0hb/l, where μ0 is the permeability of vacuum and l is the length in the y-direction that can be set to unity for 1D gratings. The other contribution of the inductance comes from the kinetic energy of charges since charge current must accelerate to create the currents (Solymar and Shamonina 2009). Thus, kinetic inductance Lk is introduced and added to the mutual inductance to form the total inductance in the circuit. It can be obtained from the frequency-dependent complex impedance of the metal, Zk Rk  iωLk, where ω is the angular frequency. The impedance can be expressed as Zk = s/(σAeff), where s is the total length of the current path in the metal and Aeff is the effective cross-sectional area of the induced electric current. For the deep grating structure, s = 2h + b and Aeff = δl, where δ = λ/2πκ is the penetration depth of electric field, in which κ is the extinction coefficient, σ = iωee0 is the electrical conductivity. After some manipulations, the following expression can be obtained: Lk ¼ 

2h þ b e0
 e0 ω2 lδ e0 2 þ e00 2

The capacitance of the vacuum inside the trench can be approximated by

(13)

25

Design of Optical and Radiative Properties of Surfaces

C ¼ c 0 e0

hl b

1037

(14)

where c0 is a numerical factor between 0 and 1 accounting for the nonuniform charge distribution between the ridges of the grating (Wang and Zhang 2011; Zhou et al. 2006). Without using a full-wave simulation, c0 can be taken as an adjustable parameter that is about 0.5. Since resistance elements do not affect the resonance frequency, only the imaginary part of the total impedance of the LC circuit is considered and can be expressed by   1 Z tot ¼ iω Lk þ Lm  2 ω C

(15)

By setting Ztot = 0, one obtains the magnetic resonance wavelength as λR ¼ 2πc0

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðLk þ Lm ÞC

(16)

where c0 is the speed of light in vacuum, which is an implicit equation because Lk is frequency- or wavelength-dependent. Note that l does not show up in this equation and can be assumed unity in later discussions. If c0 = 0.5 is used in Eq. 16, the LC model yields a resonance at λR = 2.78 μm, which agrees well with the RCWA simulation (Zhao and Zhang 2014). For other structures like slit array and metal/ dielectric/metal structures (Wang and Zhang 2009, 2012b), the inductance and capacitance in the circuits can be obtained in a similar way.

2.4

Polarization Dependence of Radiative Properties

The shape of the periodic patterns in metamaterials can induce polarization dependence. For metamaterial absorbers designed for energy harvesting purpose, patterns with polarization-independent radiative properties may be preferred (Wang and Wang 2013). However, some shapes can induce polarization-dependent response and can be used to design metamaterials with polarization control ability. These structures have recently attracted a lot of attention for their potential applications in multiband and dual-band absorption (Sakurai et al. 2015), nonlinear optics, holography, chemical sensing, and anomalous refraction (Grady et al. 2013; Cui et al. 2014). Here, the focus is the anisotropic radiative properties of metamaterial structures. Consider a 2D anisotropic metal/dielectric/metal structure shown in Fig. 7a, b. The middle layer is an Al2O3 layer with a thickness d = 140 nm, sandwiched between the L-shape patterned 100-nm-thick (h) gold layer and a gold ground plane that is opaque. The top metallic pattern is at the center of the unit cell and repeats periodically with the same period Λx = Λy = 3.2 μm. The width w = 0.85 μm and the L-shape pattern have two arms with different lengths: lx = 1.275 μm and ly = 1.7 μm. The commercial Lumerical FDTD software is used to compute the radiative properties. At normal

1038

B. Zhao and Z. M. Zhang

Fig. 7 (a) Illustration of the L-shape metal/dielectric/metal structure, the plane of incidence, incident wave vector, electric field vector, polar angle θ, azimuthal angle ϕ, and polarization angle ψ. (b) The x-y plane view of the structure. (c) Normal reflectance contours obtained from FDTD simulations. (d) Simulated normal reflectance as a function of the polarization angle at λ = 4.9 μm and 7.0 μm (Reproduced reflectance from the simulated reflectance for ψ = 0 , ψ = 90 , and ψ = 45 using the three-polarization-angle method are shown with markers. From Zhao et al. 2016)

incidence, the reflectance shows highly polarization dependence as demonstrated in Fig. 7c, especially at the two resonances due to MPs at the wavelengths λ = 4.9 and 7.0 μm (Sakurai et al. 2015). Only at a certain polarization angle (i.e., the polarization of the eigenmodes) can the MPs be fully excited and cause minimum reflectance. It may be easy to identify this polarization angle for the symmetric structures. However, if the arms of the L shape are not equal, the dependence on polarization angle becomes more complicated, and three-polarization-angle method can be used. For a plane wave with arbitrary polarization incident from a vacuum to a medium, the reflectance is (Zhao et al. 2016)  2  2 Rψ ¼ r ss sin ψ þ r ps cos ψ  þ r sp sin ψ þ r pp cos ψ  ¼ RTE sin 2 ψ þ RTM cos 2 ψ þ RC sin ð2ψ Þ

(17)

Here, rss and rpp are the co-polarized reflection coefficients and rsp and rps are the cross-polarized reflection coefficients. The first and second subscripts describe the

25

Design of Optical and Radiative Properties of Surfaces

1039

polarization status of the reflected and incident waves, respectively. For isotropic medium, no cross polarization can occur; thus, rsp = rps = 0. RTE = |rsp|2 + |rss|2 2 and RTM = |rpp| + |rps|2 are the  reflectance for TE and TM waves, respectively, and  RC ¼ Re r ss r ps þ Re r pp r sp , which is generally nonzero but can be either positive or negative due to cross polarization. This means reflectance as a function of polarization angle can be determined once three unknowns are solved using three reflectance for different ψ. Eq. 17 can be recast as Rψ ¼ A sin ð2ψ þ χ Þ þ R (18) qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where A ¼ ðRTM  RTE Þ2 =4 þ RC 2 is the amplitude and R ¼ ðRTE þ RTM Þ=2 is the average reflectance for TE and TM waves. The phase χ  (π, π] is determined by sin χ = (RTM  RTE)/2A and cos χ = RC/A. Based on Eq. 18, the reflection maximum and minimum, which correspond to the two eigenmodes, occur at ψ 1 = π/4  χ/2 and ψ 2 = ψ 1 + π/2, respectively. Figure 7d shows the reflectance with markers as a function of polarization angle reproduced using the reflectance for TE, TM, and ψ = 45 . As the plot shows, the reproduced results agree with the fullwave simulation. ψ 1 = 18.6 and ψ 2 = 108.6 can thus be determined. Though the matrix method for anisotropic medium (Schubert 1996; Yariv and Yeh 2002) may be applied to find the polarizations for the eigenmodes, the three-polarization-angle method may be more convenient since the reflectance and transmittance can be identified directly.

3

Applications of Periodic Nano-/Microstructures and Metamaterials

Periodic nano-/microstructures and metamaterials have attractive characteristics for many applications such as bio- and chemical sensing, Raman spectroscopy, solar photovoltaics and TPVs, thermoelectric energy conversion, hot electron solar cells and photodetectors, and water splitting and undercooled boiling (Boriskina et al. 2013). This chapter discusses a specific application in TPV systems. Wavelengthselective emitters are of critical importance to improving the efficiency of TPV systems and solar TPV systems (Basu et al. 2007; Zhou et al. 2016; Bierman et al. 2016; Khodasevych et al. 2015). An ideal TPV emitter should not only be wavelength-selective, but also desired to be polarization-insensitive so that high emittance for both TE and TM waves can be achieved (Nguyen-Huu et al. 2012). Micro-/nanostructures of wide profile diversity can tailor thermal radiation by utilizing different physical mechanisms. Not only 1D gratings (Nguyen-Huu et al. 2012; Chen and Zhang 2007), V-groove gratings (Sergeant et al. 2010), and photonic crystals (Nagpal et al. 2008; Narayanaswamy and Chen 2004; Lee et al. 2007) but also various 2D nano-/microstructures (Heinzel et al. 2000; Pralle et al. 2002; Sai and Yugami 2004; Sai et al. 2003, 2005; Chen and Tan 2010; Song et al. 2016; Zhao and Fu 2016) and multilayer structures (Bouchon et al. 2012;

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Fig. 8 (a) Illustration of the numerical model for the 2D metal/dielectric/metal nanostructure. The parameters are Λx = Λy = 600 nm, lx = ly = 300 nm, and h = d = 60 nm. (b) Normal emittance spectra of the 2D structure and a 1D structure for both TE and TM waves, along with that of plain tungsten (From Zhao et al. 2013)

Wang et al. 2007; Cui et al. 2011; Hendrickson et al. 2012; Lévêque and Martin 2006; ; Liu et al. 2010, 2011b; Puscasu and Schaich 2008; Wang et al. 2012; Deng et al. 2015) have been investigated for their wavelength-selective properties. Here, a typical 2D metal/dielectric/metal periodic nanostructure is discussed for TPV applications (Zhao et al. 2013). Figure 8a is a schematic of the considered structure. The grating is made of rectangular tungsten (W) patches whose lateral dimensions are lx and ly with a height h and periods Λx and Λy. The periodic arrays of patches are on a thin dielectric film SiO2 of thickness d that is deposited on a tungsten substrate. Tungsten is chosen as the emitter material because it is corrosion resistant and can withstand high temperatures. In the present study, the geometric parameters are fixed as follows: Λx = Λy = 600 nm, lx = ly = 300 nm, and h = d = 60 nm. Figure 8b compares the normal emittance spectra for the 2D structure and a 1D counterpart of the proposed structure from Wang and Zhang (2012b) for both TE and TM waves. The emittance spectrum for plain tungsten is also shown. The spectrum for the 1D structure with TM waves and that for the 2D structures are very similar; both contain two major emission peaks (near 0.7 and 1.8 μm) that do not exist in the spectra for the TE wave or plain tungsten. The overall emittance at normal direction is the average of those for TE and TM waves. As an example, at λ = 1.7 μm, the normal overall emittance for the 2D structure is 0.85 and only 0.58 for the 1D structure. Therefore, the throughput and efficiency of the TPV system can be significantly improved with the 2D structure. The structure is designed for a TPV cell that has a bandgap at around 2 μm, such as In0.2Ga0.8Sb. For cells with a different bandgap, the emittance spectrum can be optimized by tuning the geometries of the structure. Τhe emittance peaks at λ = 1.83 μm and 0.7 μm are critical for the shape of the spectrum. The emittance peak around λ = 0.7 μm is because of Wood’s anomaly. It can be seen from Fig. 8b that only TM waves can cause a high Wood’s anomaly

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emittance peak for 1D gratings, but can be accomplished for both TE and TM waves for 2D gratings. Note that this peak is not due to SPP since the dielectric function of tungsten has a positive real part between 0.24 and 0.92 μm (Palik 1985). At longer wavelengths, however, the real part can go negative and SPPs can be excited. Figure 9a illustrates the contour plot of the emittance for the 2D structure at TM waves in terms of wavenumber and the x-component of the freespace wave vector. The bright oblique bands indicate the emittance enhancement due to SPPs or Wood’s anomaly, while the flat bright band around 5,400 cm1 is due to the excitation of MP as will be discussed later. The effect of the periodicity in the x- direction can be analyzed by considering n = 0. In this case, surface waves (can be SPPs or Wood’s anomaly) can exist once the following dispersion relation is satisfied:

Fig. 9 Emittance contour plots of the 2D grating/thinfilm structure from RCWA calculations along with the SPP dispersion relations: (a) TM waves; (b) TE waves (From Zhao et al. 2013)

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ksw

   2πm  ¼ kinc sin θ þ Λ 

(19)

x

This dispersion relation is solved with different m values (2, 1, 0, and +1) for surface waves excited between air and tungsten, and solutions of this equation for SPPs are marked as circles in Fig. 9a for demonstration purpose and the same for the following contour plots. Since SPPs are quite close the light line, the solutions are essentially the dispersion for Wood’s anomaly. These circles follow the bright emittance band well obtained from the 2D RCWA calculation. The periodicity in the y-direction (Λy) can affect the dispersion of the surface waves as well. Take m = 0 and n 6¼ 0 for simplicity. The incident wave will be diffracted into the y-direction, and the dispersion relation becomes

ksw

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   2πn 2 ¼ ðkinc sin θÞ2 þ Λy

(20)

The solutions of this equation for SPPs are marked on Fig. 9a with triangles and they agree well. Within the considered ranges of frequency and wave vector, only the surface waves associated with n = 1 show up when m = 0. This branch will fold with the grating period Λx in the x-direction and show up as m = 1, 2 orders. However, these higher orders also do not show up in the contour plot due to the intrinsic losses of tungsten. For TE incident waves, the periodicity in the y-direction plays a crucial role to excite surface waves and create a high emittance peak. Figure 9b shows the emittance contour for TE waves. For TE wave incidence with ϕ = 0 , the order n must not be zero since ky must have a nonzero real part. A bright resonance band can be observed whose resonance frequency increases with kx. This branch is identified as the surface wave with m = 0 and n = 1 and can be matched by the predictions from Eq. 20 as shown by the triangles. The emittance peak at λ = 1.83 μm is due to MP. When the MP is excited, the magnetic field is strongly enhanced in the dielectric layer inserted between the tungsten grating and tungsten substrate, as shown in Fig. 10. The instantaneous electric field and current density vectors are denoted by the arrows, while the magnetic field is represented by the color contour. The current density vector J = σE can be expressed as the sum of the conduction current and displacement current: Re(J) = Jcond + Jdisp = σ 0 Re(E)  σ 00 Im(E). For locations 1 and 2 (marked on Fig. 10), the displacement current is the dominant contribution and has the same direction as the electric field. For locations 3 and 4, the conduction and displacement current densities have opposite signs, but the magnitude of the displacement current density is more than twice greater than the conduction current density, make the full current density follow the direction of the displacement current density. Therefore, the displacement is of critical importance for the formation of the current loop. Since the instantaneous electric field vectors in Fig. 10 will oscillate with time, the direction of the arrows may reverse but should always be antiparallel.

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Fig. 10 The electromagnetic fields and current density distribution in the 2D structure for TM waves at normal incidence and λ = 1.83 μm. The fields are calculated at y = 0 in the x-z plane. The color shows the relative magnitude of the y component of the magnetic field. The vectors show the direction and magnitude of (a) the electric field and (b) current density (From Zhao et al. 2013)

4

Tailoring Thermal Radiation Using 2D Materials

Emerging 2D materials offer another exciting new element to construct metamaterials with unique thermal radiative properties (Grigorenko et al. 2012). Not only can 2D materials be patented to different shapes like ribbons and resonators to control optical and radiative properties, but they also can be used as an element to construct hybrid structures with other micro-/nanostructures to create exotic properties. Using a hybrid structure with a monolayer graphene covered on a metal grating, Zhao et al. (2014, 2015) and Zhao and Zhang (2015a) demonstrated strong absorption through excitation of SPPs and MPs. Fan’s group (Piper and Fan 2014; Zhu et al. 2016) utilized critical coupling in graphene/photonic crystal hybrid structures to achieve perfect absorption. Plasmonic optical nanoantennas with different shapes can also be hybridized with 2D materials to capture light efficiently (Echtermeyer et al. 2011; Yao et al. 2014). The hybrid structures may open a novel route to engineer radiative properties by enabling plentiful coupling effects between

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graphene plasmonic resonances with SPPs and MPs in traditional plasmonic metamaterials. This section reviews the recent work on optical and radiative properties of 2D materials and hybrid structures. Graphene and hBN will be discussed specifically.

4.1

Optical and Radiative Properties of Graphene and its Ribbons

With a layer of carbon atoms arranged in a honeycomb lattice, graphene exhibits unique electronic, thermal, mechanical, and optical properties (Nair et al. 2008; Thongrattanasiri et al. 2012; Pop et al. 2012). Since its discovery in 2004 (Novoselov et al. 2004), graphene has been extensively studied for potential applications in nanoelectronics, optoelectronics, plasmonics, transformation optics, and energy conversion (Vakil and Engheta 2011; Xia et al. 2009; Liu et al. 2011a; Yao et al. 2013; Echtermeyer et al. 2011; Miao et al. 2012; Messina and Ben-Abdallah 2013; Wu et al. 2009). Unlike in conventional solids where electrons are described by the Schrödinger equation, electrons in graphene are governed by the Dirac equation for 2D relativistic fermions (Mics et al. 2015). This enables a saturation absorption in the visible and near-infrared region and actively tunable surface plasmons in mid- and far-infrared region (Grigorenko et al. 2012; Nair et al. 2008) with relatively low loss compared to traditional plasmonic materials that have a lot of potential applications in electro-optics, optical communications, and energy conversion (Basov et al. 2014; Li et al. 2008; Fang et al. 2013a; Vakil and Engheta 2011; Xia et al. 2009; Liu et al. 2011a; Yao et al. 2013; Echtermeyer et al. 2011; Miao et al. 2012; Messina and Ben-Abdallah 2013; Wu et al. 2009). The radiative properties of graphene can be described by its sheet conductivity σ s, which consists of the contribution from intraband (Drude-like term) and interband transitions, i.e., σ s = σ D + σ I, respectively (Falkovsky 2008):   i 2e2 kB T μ ln 2 cosh σD ¼ ω þ i=τ πℏ2 2kB T

(21)

and "   # ð e2 ℏω 4ℏω 1 GðξÞ  Gðℏω=2Þ G dξ þi σI ¼ 2 π 0 4ℏ ðℏωÞ2  4ξ2

(22)

where G(ξ) = sinh(ξ/kBT )/[cosh(μ/kBT ) + cosh(ξ/kBT )]. Here, e is the electron charge, ℏ is the reduced Planck constant, and kB is the Boltzmann constant. Otherwise specified, the following parameters are used in all the calculations presented in this work: chemical potential μ = 0.3, relaxation time τ = 1013 s, and temperature T = 300 K. Since μ can be tuned by electrical gating or chemical doping, the optical properties of graphene can be actively tuned (Novoselov et al. 2004). Figure 11

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Fig. 11 Real and imaginary part of the sheet conductivity of graphene with different chemical potentials: (a) real part and (b) imaginary part

illustrates the effect of μ. As the plot shows, in the visible and near infrared, interband transitions dominate and graphene shows a wavelength-independent conductivity σ s = σ 0 = e2/(4ℏ), making graphene to have no plasmonic response but have wavelength-independent absorptivity of about 2.3% (Nair et al. 2008). In the mid- and far- infrared region, graphene can support highly confined SPs. For a graphene sheet surrounded by media with dielectric functions e1 and e2 on each side of graphene, respectively, the dispersion of the graphene surface plasmon satisfies e1 e2 iσ s qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼  ωe 2 2 2 2 0 kGSP  e1 k0 kGSP  e2 k0

(23)

where kGSP is the wave vector for the plasmon and e0 is the permittivity of vacuum. Figure 12a shows a schematic of graphene SPs.

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In the mid- and far-infrared region, the intraband transitions dominate, and the conductivity can be approximately expressed in a Drude-like model σ s = e2μτ/ (πℏ2  iπωτℏ2) (García de Abajo 2014). In this wavelength region, if graphene is surrounded by vacuum, i.e., e1 = e2 = 1, the dispersion can be simplified as (Vakil and Engheta 2011)

kGSP

ω ¼ c0

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4e0 2iωe0 e1  2  σ s μ0 σs

(24)

where μ0 is the vacuum permeability. Graphene resonators such as ribbons (Brar et al. 2013; Nikitin et al. 2012; Strait et al. 2013; Yan et al. 2013; Liu et al. 2014), disks (Yi et al. 2015; Koppens et al. 2011; Yan et al. 2012; Fang et al. 2013b) and cross shapes (Fallahi and PerruisseauCarrier 2012) have been studied for their coherent radiative properties. The plasmons in ribbons, for example, cannot propagate freely since they are reflected on the ribbon edges with a phase shift Δφ (Chen et al. 2013; Garcia-Pomar et al. 2013). The resonance can be described by a Fabry-Pérot model: Δφ þ ReðkGSP Þr ¼ mπ

(25)

where integer m denotes the resonance order, r is the width of the ribbon, and Re takes the real part of the complex quantity. By substituting Eq. 24, the resonance condition of suspended ribbons in vacuum can be expressed as (Du et al. 2014) sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e2 μðm  Δφ=π Þ ω¼ 2e0 ℏ2 r

(26)

At normal incidence, plasmons initiated from the two edges of the ribbon are out of phase, and thus only the plasmons associated with odd m’s can show up. The plasmons with even m’s can occur at oblique incidence since in-phase plasmons can be excited from the two edges (Du et al. 2014). It is recently found that Δφ is π/4 for free-standing ribbons in vacuum (Nikitin et al. 2014; Du et al. 2014). Figure 12(b) shows the absorptance (α) contour of a suspended graphene ribbon arrays at normal incidence obtained with RCWA. The dashed lines, from bottom to top, are the predictions from Eq. 26 with m = 1, 3, 5,pand ffiffi 7, respectively. The frequency of the resonance is inversely proportional to r . Note that the highest absorptance occurs for m = 1 at r  2 μm with a value near 0.35. For the other branches with higher orders, the absorption by graphene ribbons is even smaller.

4.2

Graphene-Covered Metal Gratings

As shown in Fig. 12b, though graphene can be patterned into different periodic resonators to boost the absorptance, high absorptance or emittance is difficult to

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Fig. 12 (a) Schematic of graphene surface plasmon (GSP). (b) The normal absorptance contour of a freestanding graphene ribbon array with Λ = 4 μm and μ = 0.3 eV, where the dashed lines are the predictions of Eq. 26 with Δφ = π/4 (From Zhao and Zhang 2015a)

achieve solely with graphene resonators due to its atomically thin thickness. One way to solve this problem is to combine 2D materials together with nano-/ microstructures. Here, a hybrid structure consists of graphene and grating is considered, as shown in Fig. 13a. The Ag grating geometries are Λ = 400 nm, b = 30 nm, and h = 200 nm. A monolayer graphene is covered on top of the grating. Figure 13b compares the absorptance spectra for the plain grating and graphene-covered grating. For TM waves at θ = 10 , there exist three distinct peaks at ν = 6,700 cm1 (1.49 μm), 18,350 cm1 (545 nm), and 20,930 cm1 (478 nm), which are associated, respectively, with the excitation of the fundamental MP (MP1), the second-order MP (MP2), and an SPP. Clearly, the graphene increases the peak absorptance significantly without shifting the peak locations. This is because of the unique properties of graphene at visible and near-infrared region. The effect of the graphene across the trench opening can be considered by modifying the LC circuit for grating, as shown in Fig. 13a. The additional impedance introduced by the graphene layer, ZG = b/σ s, is a real number since σ s is dominated by the interband contribution σI at the resonance wavelengths. This makes the graphene layer behave like a pure resistor across the trench. Since the impedance of graphene is real and very large, the resonance wavelength of MP is not affected. However, the graphene layer does add resistance to the circuit, making the resonance peaks broader (lower Q factor).

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Fig. 13 (a) Schematic of the graphene-covered 1D grating nanostructure for a plane TM wave incident at an angle of θ. The inset shows the corresponding LC circuit model for the hybrid structure. (b) Comparison of the absorptance of the graphenecovered and plain Ag grating with h = 200 nm, Λ = 400 nm, and b = 30 nm at incidence angle θ = 10 for TM waves (From Zhao et al. 2015)

The absorption enhancement is achieved by coupling MP resonance with the graphene. As Fig. 6 shows, at the excitation of MP, a strong electric field is created at the trench opening. Thus, graphene acts like a pure resistor surrounded by a strongly enhanced local electric field and thus dissipates significant power. The power dissipation profile at MP1 resonance for the plain and covered grating is depicted in Fig. 14. It is clear that the absorption in the Ag walls delineating the trench is not enhanced, but somewhat weakened by the added graphene layer. However, the local absorption in the graphene sheet across the opening of the trench is extremely strong with a maximum w on the order of 108W/m3, which is three orders of magnitude higher than the highest w in the grating region. In fact, one can use Eq. 11 to evaluate

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Fig. 14 Power dissipation profiles for the two structures when θ = 10 at MP1 resonance (ν = 6700 cm1): (a) The plain Ag grating and (b) the graphene-covered grating. The unit of w is 105 W/m3 and the scale bar limit is 6  105W/m3, beyond which w is shown in white color (From Zhao et al. 2015)

the absorptance in graphene, which is as high as 0.68 and nearly 30 times higher than the absorptance of a free-standing graphene sheet. Similar mechanism and phenomena hold for the MP2 and SPP resonance peaks, and the absorptance in graphene can be even higher at the SPP excitation (Zhao et al. 2014, 2015). Since graphene has a remarkably high carrier mobility up to 200,000 cm2 V1 s1 at room temperature for both electrons and holes (Novoselov et al. 2004), this enhanced absorption in graphene has potential applications in ultrafast optoelectronic devices such as transistors (Engel et al. 2012) and photodetectors (Xia et al. 2009; Mueller et al. 2010; Furchi et al. 2012; Fang et al. 2012). However, in the mid- and far-infrared region, graphene has plasmonic behavior, and the plasmonic resonance inside the graphene can strongly couple with the MP resonances inside metal gratings to form a new hybrid plasmonic system and create tunable coherent radiative properties. Consider a graphene ribbon-grating hybrid structure illustrated in Fig. 15a. The grating geometries are Λ = 4 μm, b = 300 nm, and h = 2 μm. The ribbon edges are touched with the Ag grating. For TM waves, the absorptance spectra of the plain and ribbon-covered structure are shown in Fig. 15b. For the plain grating, the MP resonance is at 1,041 cm1 with an absorptance of 0.35. After covering ribbons on the grating, the absorptance is boosted to 0.94, and the resonance shifts to 1,086 cm1, indicating a strong coupling between graphene and the grating that is quite different with previous case in the visible and nearinfrared region. The spectrum of the free-standing ribbon array without the grating is also given for comparison. In this case, the ribbon plasmon gives rise to a small absorption peak around 650 cm1. The resonances with and without graphene can be successfully modeled using the equivalent LC circuit shown in Fig. 13a. The graphene in this case serves as an inductor across the trench that lowers the total inductance in the circuit and shifts MP resonance to a higher frequency.

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Fig. 15 (a) Schematic of the hybrid grating-graphene ribbon plasmonic structure. (b) The absorptance spectrum for the plain Ag grating, freestanding ribbon array, and ribbon-covered grating (r = b). The geometries for the grating are given in the figure (From Zhao and Zhang 2015a)

To better explain the coupling effect, Fig. 16 compares the absorptance contours at normal incidence for four different configurations versus the trench width. The plain gratings support MP resonances as shown in Fig. 16a. The frequency of the MP increases with b and reaches an asymptotic value when b >0.25 μm because of the effect of kinetic inductance as has been explained previously. The dot markers are the predictions of the LC model. The absorptance contour stays the same if the graphene ribbon is covered only on the ridges of the grating, as shown in Fig. 16b. The graphene ribbons appear to have no effect on the resonance since they are in contact with the metal on one side. On the contrary, if the ribbons are suspended above the trench openings, as the case in the inset of Fig. 16c, the plasmonic resonances of the ribbons show up and the absorptance becomes more plentiful. pffiffiffi The bright bands that exhibit a dependence on 1= b are caused by the plasmons in

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Fig. 16 Absorptance contours for four different structures when Λ = 4 μm, h = 2 μm, and μ = 0.3 eV are fixed and b changes: (a) plain Ag grating; (b) graphene ribbons covering the Ag grating ridges; (c) graphene ribbons covering the trench opening (r = b); and (d) continuous graphene sheet covering the whole grating. The round markers are the predictions of Eq. 16 and the diamond markers are the predictions of Eq. 26 with Δφ =  π (From Zhao and Zhang 2015a)

ribbons. Note that the edges of the ribbons are always in touch with Ag. Although the shape of these bands is similar to the free-standing ribbon array resonances, the resonances in Fig. 16c cannot be predicted with Eq. 26 using the reflection phase shift π/4. In fact, since the edge is touched with Ag instead of vacuum, the tangential component of the electric field (Ez) at the edges needs to vanish at the boundary due to the high conductivity of Ag, resulting in a phase shift of -π for the plasmon waves. This can be further justified by the excellent agreement between the simulation and the prediction results of Eq. 26 with Δφ = π, which are shown as diamond markers. Note that the plasmons are all associated with odd m’s since the incidence

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Fig. 17 The absorptance spectra of the structure shown in Fig. 15 with different graphene chemical potentials (From Zhao and Zhang 2015a)

is normal. Covering the grating by a continuous graphene, as illustrated in Fig. 16d, yields the same absorptance contour as Fig. 16c. Again, the portion of the graphene on the ridges has little contribution on the resonance. Thus, in terms of the radiative properties, covering a continuous graphene is equivalent to suspending a periodic ribbon array over the grating trenches. Since the fabrication of the structure with continuous graphene on grating can be realized with the existing fabrication technique (Papasimakis et al. 2010), it provides a realistic way to experimentally achieve the coupling between MP and plasmons in ribbon array. The chemical potential of graphene can be used to actively modify the coupling picture by changing the dispersion of the graphene plasmons. For example, Fig. 17 shows the absorptance spectra for the structure shown in Fig. 15a with various μ. The case when μ = 0.3 eV corresponds to the solid spectrum in Fig. 15b. If μ changes to 0.28 eV or 0.32 eV, the ribbon plasmon shifts to a slightly lower or higher frequency, respectively, and the coupling strength decreases as the peak absorptance becomes lower. When μ is further decreases to μ = 0.2 eV, two bumps at 871 cm1 and 1,164 cm1 occur that are due to the plasmons associated with m = 1 and 3, respectively. However, neither of them can couple with the MP, whose absorptance drops down to about 0.4. Similarly, at μ = 0.5 eV, the plasmon in ribbons associated with m = 1 moves to 1,412 cm1 and totally decouples with the MP. Therefore, the chemical potential can be tuned to control plasmons in the ribbons to couple or decouple with the MP resonance in the grating. Besides the chemical potentials, geometry parameters like the ribbon width and the elevation can also be used to tune the radiative properties of the structure (Zhao and Zhang 2015a).

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4.3

Design of Optical and Radiative Properties of Surfaces

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Hexagonal Boron Nitride-Covered Metal Gratings

As a material that has a similar lattice structure with graphene, hexagonal boron nitride (hBN) has been used as an ideal substrate supporting high-quality graphene (Dean et al. 2010; Yan et al. 2016). While 2D plasmonic materials and semiconductors have been studied extensively because of their potential applications in microelectronic, optoelectronic, and photonic devices (Li et al. 2016), few studies have yet explored the potentials of using phononic 2D materials to achieve perfect absorption through coupling with nano-/microstructures, especially for materials like hBN that is a hyperbolic material. Considering that hBN can survive at 1,500  C in air (Liu et al. 2013), accomplishing strong absorption or emission bands in the infrared region with hBN films holds great significance for high-temperature energy harvesting applications (Khodasevych et al. 2015). Hyperbolic materials refer to uniaxial materials whose axial and tangential permittivities have opposite signs. The isofrequency surfaces obey a hyperbolic shape instead of a closed sphere for common isotropic materials. Subsequently, these materials can support propagating modes with very large tangential wave vectors and thus can have unique applications in sub-wavelength imaging (Caldwell et al. 2014; Dai et al. 2015a; Li et al. 2015) and heat transfer (Jacob et al. 2012; Nefedov et al. 2013; Biehs et al. 2012; Poddubny et al. 2013). The hyperbolic response of hBN is mainly caused by its optical phonon vibrations. The real part of the dielectric function of hBN is shown in Fig. 18a. The two mid-infrared Reststrahlen bands due to the optical phonon modes are evident. The in-plane phonon modes (ωTO,⊥ = 1,370 cm1 and ωLO,⊥ = 1,610 cm1) and out-of-plane phonon modes (ωTO,k = 780 cm1 and ωTO,k = 830 cm1) contribute to the in-plane (E lies in the x-y plane, denoted by ⊥) and out-of-plane (E parallel to the optical axis or the z-direction, denoted by ||) dielectric functions, respectively (Kumar et al. 2015): eξ ¼ e1, ξ 1 þ

ω2LO, ξ  ω2TO, ξ

ω2TO, ξ  iγ ξ ω  ω2

! (27)

where ξ = k, ⊥. The other parameters used are e1, jj = 2.95, γ jj = 4 cm1, e1, ⊥ = 4.87, and γ ⊥ = 5 cm1. Since the damping coefficients γ are rather small, the dielectric function becomes negative between the TO and LO phonon modes, making the in-plane and out-of-plane dielectric functions of hBN possess opposite signs in either Reststrahlen band. In the lower Reststrahlen band, e0jj ¼ e0z < 0 and e0⊥ ¼ e0x ¼ e0y > 0, hBN has type I hyperbolicity, while in the upper Reststrahlen band, e0x ¼ e0y < 0 and e0z > 0 , hBN holds type II hyperbolicity (Jacob 2014). Meanwhile, in the two regions with hyperbolicity, loss is negligibly small. The hyperbolic regions allow propagating waves with unbounded wave vectors as can be seen from the isofrequency surface for a uniaxial medium with the optical axis in the z-direction (Poddubny et al. 2013):

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Fig. 18 (a) Real part of the in-plane and out-of-plane dielectric function of hBN. The two hyperbolic regions are shaded and marked with the corresponding type of hyperbolicity. (b) Contour plot showing imaginary part of reflection coefficient of TM waves for a 30-nm-thick suspended hBN film in vacuum near the upper (type II) hyperbolic region. The predictions from Eq. 29 are overlaid as dashed curves (From Zhao and Zhang 2017)

k2x þ k2y k2z ω2 þ ¼ 2 ejj e⊥ c 0

(28)

where k = (kx, ky, kz) represents the allowed wave vector. If loss is neglected, in the frequency ranges, the hBN possesses hyperbolicity, Eq. 28 becomes a hyperboloid, and both kx and kz can theoretically be infinitely large. Note that Eq. 28 is for extraordinary waves or TM waves, which are the primary interest here. For TE waves, the isofrequency surface becomes a sphere described by jkj2 ¼ e⊥ ω2 =c20 ; thus, hBN behaves the same as an isotropic material with the ordinary dielectric function.

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For an hBN film, hyperbolic phonon polaritons (HPPs) are supported in the two Reststrahlen bands or hyperbolic regions. Their dispersion can be obtained from the reflection coefficient of TM waves for an hBN film of a thickness d suspended in vacuum. The dispersion of HPPs can be seen by the bright bands from the contour plots of the imaginary part of reflection coefficient of a suspended hBN film in kx-ω space (Kumar et al. 2015), shown in Fig. 18b for a 30-nm-thick hBN film in the frequency ranges near the upper hyperbolic region. For the lower upper hyperbolic region, similar dispersions exist, though not presented here. Multiple orders of HPPs exist in both hyperbolic regions, and more orders are allowed if the film thickness becomes larger. Alternatively, when kx >> k0, the dispersion can be approximately expressed as (Dai et al. 2015b) 1 k x ð ωÞ ¼ d

!# rffiffiffiffiffiffiffiffiffiffi" ejj 1  pπ 2 arctan pffiffiffiffiffiffiffiffiffiffiffiffiffi e⊥ ejj e⊥

(29)

where p is an integer indicating the number of orders of the HPPs and the dielectric functions are for hBN. The plus and minus signs are chosen, respectively, for the upper and lower hyperbolic bands based on the shape of the dispersion curves or the direction of the group velocity (Kumar et al. 2015). The prediction from Eq. 29 is overlaid on the contour plot as dashed lines in Fig. 18b with the corresponding p. The excellent agreement suggests that Eq. 29 can provide a convenient description of HPPs and will be referred to in later discussions. For both type I and type II regions, at very large |k|, the allowed wave vectors pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi approach the asymptotic lines described by kz ¼ e⊥ =ek kx (neglecting loss) for the structure considered here since ky = 0. In this case, the angle between z-axis and the Poynting vector or energy flux, the propagation angle of the polaritons, is orthogonal to the isofrequency curve that approximately equals to (Caldwell et al. 2014; Dai et al. 2015a; Li et al. 2015) rffiffiffiffiffiffiffiffiffiffi! e⊥ βðωÞ ¼ arctan  ejj

(30)

Equation 30 suggests that for a given frequency, the HPPs have a unique propagation direction. As will be shown, this property plays an important role in the absorption profile of HPPs. Figure 19a shows the geometric arrangement of the hBN/metal grating hybrid structure. Similar to previous cases, the grating is made of silver (Ag) and is periodic in the x-direction only with a period Λ, extending infinitely in the y-direction. Its trench width is b and height is h. The thickness of the covered hBN film is denoted as d. The substrate of the structure is also Ag and assumed to be semi-infinite. In reality, an Ag film whose thickness is much greater than the photon penetration depth can be deposited on another supporting substrate. Thus, the directional-spectral absorptance of the structure, α, can be calculated by α = 1R, where R is the directionalhemispherical reflectance of the whole structure that can be calculated using

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Fig. 19 (a) Schematic of the hBN/metal grating hybrid structure. (b) Absorptance spectra of plain Ag gratings (dashed line), hBN-covered Ag gratings (solid line), and a suspended 30-nm hBN film in vacuum. The geometries are Λ = 4 μm, b = 300 nm, d = 30 nm, and h = 1.76 μm (From Zhao and Zhang 2017)

anisotropic RCWA. Figure 19b shows the normal absorptance spectra of plain gratings (dashed lines) and hBN-covered Ag gratings (solid lines) for TM waves. The geometry parameters are set as Λ = 4 μm, b = 300 nm, h = 1.76 mm, and d = 30 nm. One absorptance peak can be identified on the plain Ag grating spectrum, and this peak is caused by the excitation of MP resonances as explained before (Zhao and Zhang 2014). In Ag gratings, a high absorptance peak due to MP relies on an efficient coupling between the surface waves on the trench walls that can be achieved only when the trench is very narrow (Zhao and Zhang 2014). Thus, for trench width b = 300 nm, the absorptance peaks for plain Ag gratings are far from unit in Fig. 19b. High absorptance is difficult to achieve with thin hBN films either, as demonstrated by the absorptance spectrum of a suspended 30-nm hBN film in the plot. Only a low absorptance peak at ωTO,⊥ is obtained, since no HPPs can be excited

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Fig. 20 Local power dissipation density profile at ν = 1436 cm1 corresponding to the high-absorptance resonance in Fig. 19b: (a) zoomed-in profile showing the details inside the hBN film and (b) the dissipation profile (enlarged by one order of magnitude) of the structure. The scale bar is in MW/m3 with an upper limit 1 MW/m3, beyond which w is displayed with white color. The propagation angle β is illustrated on (a). Note that the x and z scales are readjusted to show the profile clearly (From Zhao and Zhang 2017)

with propagating waves in air due to the requirement of large parallel wave vectors (kx). After covering the 30-nm hBN film on top of Ag gratings, very high absorptance can be obtained, although the frequency location is different from that of plain Ag grating. As shown in the plot, perfect absorption (α = 1) is achieved at ν = 1,436 cm1, which falls in the upper hyperbolic regions of hBN. This resonance is a hybrid hyperbolic phonon-plasmon polariton formed by strong coupling between plasmonic MP in the metal grating and HPPs in the hBN film. Figure 20 shows the local power dissipation profile at the excitation of the hybrid polariton. Figure 20a is a zoomed-in picture of the hBN film, while Fig. 20b displays the dissipation profile in the structure enlarged by one order of magnitude to show the dissipation in the grating. The unit of the contour scale is MW/m3, while for Fig. 20b, w is multiplied by a factor of 10 to show the dissipation near the surface of the Ag groove clearly. The dissipation in the grating is similar to Fig. 14a, indicating the excitation of MP. Compared to the grating, the absorption in the hBN film becomes much stronger. In fact, the maximum w in hBN is 2.4 MW/m3, which exceeds the maximum value of the scale bar of 1 MW/m3. The whole hBN film looks bright in Fig. 20b since w is multiplied by 10. If w is integrated in the hBN film according to Eq. 11, the absorptance of the hBN film is obtained as 0.91. Thus, most of the incoming power at the perfect and near-perfect absorption resonances is absorbed by the 30-nm hBN film instead of the grating.

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Not only is the dissipation much stronger in the hBN film, but it also shows a highly location-dependent feature as shown in Fig. 20a. The white dashed lines are the surface of the grating. The dissipation density follows a zigzag path in the hBN film that is symmetric about the center of the grating, indicating that the power is nonuniformly absorbed in the film. This unique dissipation profile is caused by the HPP waveguide modes (Caldwell et al. 2014; Dai et al. 2015a; Li et al. 2015). Due to the diffraction of the grating, electric fields with large wave vectors are generated, and they excite the hyperbolic polaritons in the hBN film. The polaritons predominantly initiate from the two upper corners of the grating because of the highly concentrated electric field therein that is about 50 times of the incident waves. The polariton rays propagate inside the film with a fixed angle with respect to the z-axis as described in Eq. 30 and experience a total internal reflection on the top surface of the hBN film (Dai et al. 2015a). Furthermore, p theffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi period of the zigzag pattern in the x-direction can be derived as 2d tan β ¼ 2d e⊥ =ejj . This period in real space agrees with the wave vectors of the polaritons. It should be noted that not one but multiple orders of HPPs are excited simultaneously at the resonance frequency due to grating diffraction. For a specific frequency, according to Eq. 29, the wave vectors of the multiplepHPPs ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiform an arithmetic progression with a common difference Δkx ¼ ðπ=dÞ ejj =e⊥. Thus, these HPPs produce an interference pattern with a spatial period 2π/Δkx (Dai et al. 2015a), which is the same as the aboveobtained period in real space, providing a complementary way to understand the periodic zigzag pattern. The HPPs dissipate their power as they propagate inside the film and finally vanish due to the loss of hBN. As mentioned previously, the intrinsic loss of hBN is actually very small in the hyperbolic regions, and this small loss is a critical advantage of hBN in subdiffractional focusing and imaging applications compared to hyperbolic metamaterials that are constructed with metallic nanowires or metal/dielectric multilayers (Caldwell et al. 2014; Dai et al. 2015a; Li et al. 2015). The low loss also permits long propagating length of HPPs. As Fig. 20a shows, the polaritons initiated from the grating corners experience more than five round trips before they totally disappear. Thus, the unique directional propagation of the polaritons allows multiple reflections and long light-matter interaction distances that make strong absorption in hBN possible. In the lower hyperbolic regions, similar phenomenon can occur and has been discussed in a recent publication (Zhao and Zhang 2016). Since the propagation angle of the HPPs depends on the frequency, geometry parameters can be used to manipulate the resonance condition and the location of the absorption in hBN films. These unique properties enabled by the hybrid hBN film and grating nano-/microstructures have potential applications in sub-wavelength imaging (Li et al. 2015) and surface-enhanced Raman spectroscopy (Stiles et al. 2008). Considering that gratings made of hightemperature materials like tungsten (Zhao et al. 2013) and SiC (Wang and Zhang 2011) also support MPs, the design presented here could be used to build stable perfect absorbers or spectral-selective emitters for high-temperature applications (Khodasevych et al. 2015).

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Conclusion and Outlook

Various numerical modeling methods are available for simulation of the optical and radiative properties of periodic nano-/microstructures. Thanks to the excitation of various polaritons such as MPs and SPPs, various micro-/nanostructures can be designed to control the optical and radiative properties for different practical application purposes. The emerging 2D materials provide even more plentiful resonances that can couple with the polaritons in the nano-/microstructures to yield exotic optical and radiative properties such as perfect absorption and tunable locationdependent absorption. Anisotropic RCWA is summarized as a promising and competitive method to model periodic micro-/nanostructures compared to FDTD and FEM. It can simulate the radiative properties and the field distribution in the frequency domain. Because of its meshless nature, it has advantages of modeling structures with small characteristic length like 2D materials. Metal gratings can be used to excite SPPs and MPs. The wave vector of SPs can be compensated by the diffraction effect and thus help the incident electromagnetic wave to couple with the SPs to form a SPP. Localized resonance, MPs, can be excited inside the trench of the grating, and LC model can be used to model the resonance frequency. Three-polarization-angle method is discussed as a method to deduct the polarization of the eigenmodes from the reflectance or transmittance of three arbitrary polarization angles. The method may benefit the design of optical and radiative properties with polarization dependence. A 2D metal/dielectric/metal structure can be used to achieve wavelength-selective emittance that can be used to improve the efficiency of TPV system. MPs, SPPs, and WAs are responsible for the selective emittance spectrum. Different with a 1D grating, the 2D structure is insensitive to the polarization angle. Graphene ribbons can be used to tune radiative properties. By using a graphene/ metal grating hybrid structure, strong absorption can be achieved in different frequency range. In the visible and near-infrared region, graphene couples with MPs and SPPs in grating and enhances the absorptance of the structure significantly without affecting the original resonance condition of MPs and SPPs. The majority of the power is dissipated by graphene and that can be favorable in ultrafast optoelectronic applications. In the mid- and far-infrared, graphene ribbon plasmons strongly couple with MPs in gratings to create a hybrid resonance with significantly high absorptance. The coupled resonance can be tuned by electrical gating or chemical doping of graphene. Perfect absorption is achieved in hBN/metal grating hybrid anisotropic structures due to phonon-plasmon polaritons. In the two hyperbolic regions of hBN, HPPs strongly couple with localized MPs, forming hybrid hyperbolic phonon-plasmon polaritons and achieving strong absorption. The majority of the power is dissipated inside the hBN film with a tunable location-dependent absorption profile. Micro-/nanostructures and 2D materials provide unique platforms on which various resonances can be created, and the hybridization of them may open a novel route to engineer optical and radiative properties by enabling plentiful

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coupling effects. The unique optical and radiative properties can be benefitted to a lot of exciting applications over a broadband frequency range from optical to infrared, such as solar thermal technique, photodetection, chemical sensing, local heating and cooling, thermal imaging, surface-enhanced Raman spectroscopy, medical therapy, and TPVs.

6

Cross-References

▶ A Prelude to the Fundamentals and Applications of Radiation Transfer ▶ Near-Field Thermal Radiation ▶ Thermal Transport in Micro- and Nanoscale Systems Acknowledgments The research was supported by the National Science Foundation (CBET1235975; CBET-1603761) and the US Department of Energy, Office of Science, Basic Energy Science (DE-FG02-06ER46343).

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Radiative Properties of Gases

26

Vladimir P. Solovjov, Brent W. Webb, and Frederic Andre

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Radiative Transfer in Gaseous Medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Variables, Constants, and Units in Gas Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Planck Blackbody Emissive Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Radiative Transfer Equation in Absorbing and Emitting Gaseous Medium . . . . . . . . 2 Gas Absorption Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Physical Nature of Gas Radiation Emission and Absorption . . . . . . . . . . . . . . . . . . . . . . 2.2 Molecular Vibrational-Rotational Energy Transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Rotations of Diatomic Molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Vibrations of Diatomic Molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Combined Vibration-Rotation Transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Line Positions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Line Intensities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8 Spectral Line Shape . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.9 Lorentz Profile of Collision Broadening (Pressure Broadening) . . . . . . . . . . . . . . . . . . 2.10 HITRAN Spectral Database http://www.cfa.harvard.edu/hitran/ . . . . . . . . . . . . . . . . . . 2.11 Line-by-Line Model (LBL) of Gas Absorption Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . 2.12 Uniform Narrow Band Model of Gas Absorption Spectrum . . . . . . . . . . . . . . . . . . . . . . 2.13 Spectral Absorption Coefficient and Absorption Cross Section . . . . . . . . . . . . . . . . . . . 2.14 The Line-by-Line Method of Solution of the Spectral RTE . . . . . . . . . . . . . . . . . . . . . . .

1071 1071 1072 1074 1075 1076 1076 1077 1079 1080 1081 1082 1083 1086 1087 1090 1096 1097 1098 1100

V. P. Solovjov (*) · B. W. Webb Mechanical Engineering Department, Brigham Young University, Provo, UT, USA e-mail: [email protected]; [email protected] F. Andre Centre de Thermique et d’Energétique de Lyon, INSA de Lyon, Villeurbanne, France e-mail: [email protected] # Springer International Publishing AG, part of Springer Nature 2018 F. A. Kulacki (ed.), Handbook of Thermal Science and Engineering, https://doi.org/10.1007/978-3-319-26695-4_59

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3 Narrow Band Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Principle of Statistical Narrow Band (SNB) Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 The SNB Model with Malkmus’ Distribution of Line Strengths for an Array of Lorentz Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 SNB Models in Nonuniform Gaseous Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Fictitious Gases and Mapping Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Additional Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Global Models of Gas Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Introduction to Global Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Gray Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 WSGG Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 The Spectral Group Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 WSGG RTE Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 SLW, ADF, and FSK Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7 Continuous Limit of the SLW Method in Uniform Media: the FSK Model . . . . . . . . 5 Conclusions and Perspectives of Gas Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1101 1102 1105 1107 1109 1110 1111 1111 1111 1112 1114 1115 1115 1124 1137 1138

Abstract

Radiation transfer in high-temperature gas systems is critical in many engineering applications. Understanding the fundamental physical phenomena associated with radiative transfer in these environments is thus critical to predicting the physical phenomena. This chapter seeks to present the fundamental physics of radiative transfer in high-temperature gases and review the viable methods for predicting the associated radiative transfer. The general physical statements of gas radiation are first formulated. It is shown that the principal properties of molecular gases needed for the radiative transfer equation are the gas spectral absorption cross section and the spectral absorption coefficient. Radiation constants and equations are explicitly written in terms of wavenumber for gas radiation. The fundamentals of the physical nature of gas radiation are presented to contextualize the spectral properties – what defines positions, strength, and shape of spectral lines at given temperature and pressure. The chapter provides the information needed to find and to read spectroscopic databases such as HITRAN and HITEMP and how to use the compiled data to assemble the gas absorption spectra for both the gas absorption cross section and the gas absorption coefficient. The principles of narrow band models and global models of gas radiation are formulated. The statistical narrow band model with Malkmus’ distribution function of line strength for an array of Lorentz lines is presented, and its application for modeling of radiation transfer in nonuniform media is explained. The wide range of global models of gas radiation starting from gray model and weightedsum-of-gray-gases model and their development into more advanced models such as SLW, ADF, and FSK is described. While more detailed attention is given to the SLW model, its relation to the FSK and ADF models is outlined. Finally, the application of global models for prediction of radiative transfer in nonuniform gaseous medium is presented.

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1

Introduction

1.1

Radiative Transfer in Gaseous Medium

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The ability to accurately model the thermal radiation heat transfer in gaseous media is important for characterizing the energy transfer in atmospheres of the Earth and other planets, astrophysics, and combustion systems such as furnaces, kilns, boilers, gas turbines, internal combustion engines, forest and building fires, atmospheric reentry of spaceships, etc. (Fig. 1). These problems of radiative transfer are distinguished by optical properties of molecular gaseous media, which is characterized by a complicated dependence on wavelength of the gas absorption coefficient. The gas absorption spectrum consists of millions of spectral lines whose behavior is strongly influenced by gas mole fraction and gas temperature and pressure. Understanding and modeling that dependence is critical to the ability to accurately predict the total radiation transfer in relevant systems. As will be explained in considerable detail in this chapter, because the spectral radiation character of gaseous media is described by potentially millions of absorption lines, modeling the gas radiation involves significant computation time and information storage. The modeling of radiation transfer in gaseous media requires the detailed spectral line description of gas absorption coefficient at a specified thermodynamic state. This information is compiled in spectral databases now widely available to the scientist or engineer. In general, modeling of radiation transfer in high-temperature gases requires the following: 1. Characterization of the spectral absorption coefficient with the help of appropriate physical statistical distribution functions. This permits the significant computational work foundational to the predictive effort to be done in advance.

Fig. 1 Gas-fired glass furnace

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2. Construction of a suitable spectral model for the gaseous medium. Models include, among others, line-by-line models, narrow band models (Elsasser and statistical narrow band), wide band models (box models and exponential wide models), and global models. 3. The efficient solution of the radiative transfer equation using as property input the spectral gas absorption coefficient (or spectral absorption cross section). The solution method may range in complexity and accuracy from benchmark solutions to approximate engineering solutions for application in comprehensive heat and mass transfer models.

1.2

Variables, Constants, and Units in Gas Radiation

To begin, this chapter will define the relevant variables, constants, and units which are relevant in the physical characterization of thermal radiation transfer in gases. In contrast to a mainstream description of radiation with the help of electromagnetic theory where the main spectral variables are wavelength λ and frequency v, the principle spectral variable most convenient to the study of gas radiation is the wavenumber η, measured in cm
1. Consequently, different constants and equations are used for characterization of gas radiation. Relationship between wavelength and wavenumber scales for gas radiation is shown in Fig. 2. The principal variables, constants, and units which appear in the modeling of radiation transfer in high-temperature gases are summarized here Penner (1959), Goody and Yung (1989): Spectral variables λ

1000 η¼ λ, ½μm

½μm

1  cm

wavelength wavenumber

Radiation constants h ¼ 6:62606896e
34

Planck constant, ħ ¼

k

Boltzmann constant

σ c0 n

½ J  s   J ¼ 1:38065404e
23 K   W ¼ 5:670400e
8 m2  K4 hmi ¼ 2:99792458e þ 8 s c0 ¼ c

h 2π

Stefan-Boltzmann constant speed of light in vacuum index of refraction; n ¼ 1 for gases

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visible

UV

IR Ebl (2000K)

l, mm 0.1

10

1

100

important spectral bands of combustion gases

Ebh (2000K)

2.7mm 1.9mm

4.3mm

15 μ m

gas absorption spectrum

h,cm-1

100000

10000

1000

100

Fig. 2 Relationship between wavelength and wavenumber scales for gas radiation

Radiation constants in terms of wavelength c1 ¼ 2πhc20  1024

¼ 3:741771e þ 8

  W  μm4 m2

hc0  106 k c3 ¼ 2897:8

¼ 14387:7516

½μm  K

c2 ¼

½μm  K

Radiation constants in terms of wavenumber 

C1 ¼

2πhc20

C2 ¼

hc0  102 k

C3

 10

8

¼ 3:741771e
8 ¼ 1:43877516 ¼ 1:961009

W 2
4 m  cm  K
1 cm  1 cm  K



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Planck Blackbody Emissive Power

The spectral intensity is the principal dependent variable in the prediction of radiative transfer. The emissive power of blackbody radiation is also an important parameter in the theory of radiation. These two variables are generally described as dependent on the frequency ν or the wavelength λ. However, in gas radiation, intensity of blackbody radiation is more conveniently given in terms of wavenumber η. The blackbody emissive power is not a universal distribution and cannot be obtained by a simple change of the independent spectral variable. The Planck spectral emissive power emitted by a blackbody at temperature T as a function of η is defined by the Planck blackbody distribution function (for unity index of refraction, n = 1): Ebη ðT Þ ¼ πI bη ðT Þ ¼

  2πhc η3 C η3 W  hc η 0  ¼ C2 η1 2
1 0 e T
1 m  cm n2 e nkT
1

(1)

with the total blackbody emissive power given by the Stefan-Boltzmann law: 1 ð

Eb ðT Þ ¼

Ebη ðT Þdη ¼ σT 4

(2)

0

The graph of Ebη(T) as a function of wavenumber η is illustrated in Fig. 3, revealing a continuous function of wavenumber with a maximum in blackbody emission. The maximum in the Planck spectral blackbody distribution occurs at wavenumber ηmax = C3T according to Wien’s law. Fractional blackbody emissive power (FBEP) The fractional blackbody emissive power represents the cumulative fraction of blackbody energy in the wavenumber rage from 0 to η:

Fig. 3 Planck spectral blackbody emissive power and fractional blackbody emissive power

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1 F0!η ðT Þ ¼ Eb ð T Þ

ðη Ebη ðT Þdη

(3)

0

Series expansion of the FBEP in terms of wavenumber η (Howell et al. 2011): F0!η ðT Þ ¼ 1


1 15 X e
V ðV ðV ðV þ 3Þ þ 6Þ þ 6Þ, π 4 m¼1 m4

where V ¼ mC2 η=T

(4)

Numerical calculation of the FBEP with help of truncated series provides very good accuracy with as few as 25 terms.

1.4

Radiative Transfer Equation in Absorbing and Emitting Gaseous Medium Ih (s,W)

s

W 0

s

Consider a gaseous medium assumed to be continuous and in local thermodynamic equilibrium, with ability to absorb and to emit radiation at any point in space. Propagation of radiation in absorbing and emitting molecular gaseous media along a path length s in a direction Ω is characterized by the spectral radiation intensity Iη(s, Ω), [W/(m2  sr  cm
1)] as a function of wavenumber η. The change in intensity along an arbitrary path s is governed by the spectral radiative transfer equation (RTE) Howell et al. (2016): @I η ðs, ΩÞ ¼
κη ðsÞI η ðs, ΩÞ þ κ η ðsÞI bη ðT ðsÞÞ @s

(5)

where Ibη(T(s)) = Ebη(T(s))/π is the spectral Planck blackbody intensity at the local gas temperature T(s). The local spectral absorption coefficient is defined as κη(s) = N(s)Y(s)Cη(ϕ(s)), [1/m], where Cη(ϕ(s)), [m2/mol] is the spectral gas absorption cross section. The symbol notation ϕ(s) = {T(s), Y(s), p(s)} is adopted here for definition of the local gas thermodynamic state at temperature T(s), total pressure P(s), and mole fraction Y(s). The term “spectral” denotes the dependence of radiation characteristics on wavenumber η according to the electromagnetic description of radiation. The physical meaning of the RTE is a description of the rate of change of radiation intensity along a path of propagation as influenced by local absorption and emission of radiation. While scattering also affects the radiation

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intensity along the path for purposes of simplicity, here scattering is neglected. The radiative intensity depends on the three coordinate directions, the two angular directions, and the wavenumber. The radiative transfer equation, subject to appropriate boundary conditions, may be solved for any spatial location in medium at any wavenumber η. The total intensity of radiation can then be found by integration over all wavenumbers 1 ð

I ðs, ΩÞ ¼

I η ðs, ΩÞdη

(6)

0

The total intensity is used to find the total radiative flux and the total divergence of the net radiative flux, which are involved in a general comprehensive energy transfer analysis. In general, emission and absorption are described by different coefficients. However, under the assumption of local thermal equilibrium, according to Kirchhoff’s law, both effects can be modeled with the help of the same spectral absorption coefficient κ η(s). More specific characterization of gas radiation can be revealed by statistical evaluation of the emission and absorption terms in the RTE, in which the reciprocal of the absorption coefficient 1/κη(s) has a physical sense of a mean path traveled by photon with a frequency ν = cη before its absorption by gas molecules. Solution of the RTE in gaseous media requires a detailed description of the highresolution absorption coefficient κη(s) at the corresponding local thermodynamic state ϕ(s). The contribution of scattering by molecular gases to radiative transfer is negligible. Scattering by non-agglomerated soot particles in combustion gases is also generally considered to be negligible.

2

Gas Absorption Spectra

2.1

Physical Nature of Gas Radiation Emission and Absorption

The effects of the physical processes of emission and absorption in the RTE for continuous media will now be treated as an interaction of radiation with a gaseous medium. Fundamentally, the mechanism is emission and absorption of photons by gas molecules comprising the medium (Fig. 4). According to the Bohr postulate, the energy of an emitted or absorbed photon hν corresponds to the following transitions Herzberg (1950): • From upper energy level E0 to lower energy level E00 of the molecule for emission • From lower energy level E00 to upper energy level E0 of the molecule for absorption E0
E00 ¼ hν

(7)

26

Radiative Properties of Gases

1077

photon is absorbed by a molecule

molecule emits a photon

hν

hν

upper energy level

E′

E′

lower energy level

E ′′

E ′′

Fig. 4 Transition in energy level for absorption and emission of radiation by gas molecules

In spectroscopy, this phenomenon is interpreted as interaction of light with molecular gases Harris and Bertolucci (1978). For modeling of the spectral absorption coefficient, we must know all possible energy levels important for thermal radiation in the gas. The transitions considered in gas radiation at moderate temperatures are so-called bound-bound transitions between non-dissociated molecular states when no electrons or ions are breaking away from the molecule. These transitions are vibrational and rotational molecular energy transitions. Classical mechanics describe the rotations and vibrations of molecules as a simple spring-mass model based on a free-body diagram for all possible vibrational and rotational degrees of freedom. The vibration and rotation degrees of freedom represent mechanisms for absorption/emission of energy. The possible degrees of freedom for CO, CO2, H2O, and CH4 molecules are illustrated in Fig. 5.

2.2

Molecular Vibrational-Rotational Energy Transitions

The equation E0
E00 = hν defines the frequency ν (or the wavenumber η = c/ν) of the corresponding energy transition in the molecule. The values of wavenumber at which emission or absorption of photons occurs define the molecular absorption spectrum. The first step in construction of the gas absorption spectrum consists of determining the frequencies of all significant energy transitions. Quantum mechanical treatment of the interaction of radiation with gaseous matter is based on the Schrödinger equation, which describes energy levels of molecules in contrast to classical electromagnetic treatment of radiation. The Schrödinger equation is the fundamental postulate of quantum mechanics; it describes the wave function corresponding to translation, vibration, and rotation of molecules (Howell et al. 2011):


ℏ @ 2 ψ ðr Þ þ V ðr Þψ ðr Þ ¼ Eψ ðr Þ 2m @r 2

(8)

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diatomic

CO

linear triatomic CO2

non-linear triatomic

H 2O

3 internal degrees of freedom (1 vibrational and 2 rotational)

6 internal degrees of freedom (4 vibrational and 2 rotational)

6 internal degrees of freedom (3 vibrational and 3 rotational)

spherical top CH4

Fig. 5 Vibrational and rotational degrees of freedom of principal combustion gas species

The equation above is the time-independent form of the Schrödinger equation describing the stationary states of molecules, where r is the intermolecular distance, V(r) is the potential function, and E is the total energy. The values of total energy Ej for which the Schrödinger equation has non-zero solutions ψ j(r) are called eigenvalues, and the corresponding functions ψ j(r) are called eigenfunctions. Sometimes two or more eigenfunctions correspond to the same eigenvalue. In this case, these eigenfunctions are called degenerate.

26

Radiative Properties of Gases

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Application of the Schrödinger equation to diatomic molecules, such as carbon monoxide, CO, illustrates the main characteristics of rotation and vibration of molecules.

2.3

Rotations of Diatomic Molecules

center mass

no rotation

Rotation of a diatomic molecule is modeled by a rigid rotor: a system of two atoms spaced by a fixed separation distance re rotated over their center of mass. Quantum mechanical analysis yields that the molecule can rotate only with discrete rotational energy levels Ej for which the Schrödinger equation has non-zero solutions:
 Ej ¼ FðJ Þ ¼ BJ ðJ þ 1Þ, cm
1

(9)

where J = 0, 1, 2, . . . are rotational quantum numbers and B is a constant. For the rotations related to the vibration state v, an additional index will be used F v ð J Þ ¼ Bv J ð J þ 1Þ

(10)

The selection rule for diatomic molecules, ΔJ = 1, allows rotational transitions to the neighboring energy level only: • Transitions from lower to upper level J0 J00 correspond to absorption and are called the R branch. • Transitions from upper to lower level J0 ! J00 correspond to emission and are called the P branch. The number of molecules NJ which are at the certain energy level J is called the rotational population. The rotational population determines the intensity of the rotational transition F(J ). The ratio of population at a level J to the population at the ground level J = 0 is represented by the formula (Levine 1975): BC2 NJ ¼ ð2J þ 1Þe
T JðJþ1Þ N0

(11)

1080

V. P. Solovjov et al.

Fig. 6 Ratio of the rotational level populations of CO fundamental band at different temperature

NJ 14 N0

T = 1500K

12 10

T = 1000K 8 6

T = 500K 4 2 0 0

10

20

30

40

50

As seen from this formula, the rotational population strongly depends on temperature – the higher the temperature, the higher the population of molecules at the higher energy levels. Figure 6 demonstrates the rotational population, illustrating that the maximum occurs at some intermediate values of J moving toward higher values with an increase of temperature. The patterns of intensities in the rotational molecular bands are explained by this dependence. Real molecules are not ideal rigid rotors, and centrifugal distortion of the molecules leads to deviations in the energy levels. These deviations are taken into account by the higher power distortion corrections with more constants depending on vibrational and electronic states (Bernath 2005).

2.4

Vibrations of Diatomic Molecules m1

m2

μ =

r re

r − re

m1 m2 m1 + m2

reduced mass

26

Radiative Properties of Gases

1081

Diatomic molecules have only one vibrational degree of freedom. Therefore, the form of the vibrational model is defined by the choice of the potential function V(r) in the one-dimensional Schrödinger equation


ℏ @ 2 ψ ðr Þ þ V ðr Þψ ðr Þ ¼ Eψ ðr Þ 2μ @r 2

(12)

The harmonic oscillator is the simplest model which treats molecular vibrations as a simple spring-mass system with the help of Hook’s law using the spring constant k and reduced mass μ. The potential function of harmonic oscillator is parabolic, given by V ðr Þ ¼ 2k ðr
r e Þ2. This yields the discrete vibrational energy values given by  sffiffiffi 1 h k , GðvÞ ¼ v þ 2 2π μ

v ¼ 0, 1, 2, . . .

(13)

The zero vibrational energy level (ground level) has some finite value (in contrast to a zero value of the zero rotational level). However, the harmonic oscillator is a poor model for diatomic molecules. The functional form of more advanced potential functions is developed by empirical methods. The potential function of Morse type,

βðr
re Þ 2 V ðr Þ ¼ D e 1
e , has an asymmetric form about a point re. The eigenvalues of the Schrödinger equation with the Morse potential define the vibrational energy levels  GðvÞ ¼

sffiffiffi sffiffiffi  1 h k 1 2 h k
xe v þ , v ¼ 0, 1, 2, . . . vþ 2 2π μ 2 2π μ

(14)

where the second term is a small anharmonic correction. The most general form of V(r) is a Dunham potential which represents the anharmonic potential function V(r) as a Taylor expansion about the equilibrium intermolecular distance re; and the energy levels are represented by an infinite series, the first two terms of which coincide with eigenvalues of the Morse potential. The transition from the ground state v = 0 to the first level v = 1 is called the fundamental transition. Further transitions from the ground state are as follows: v = 0 ! v = 2 is the first overtone, v = 0 ! v = 3 is the second overtone, etc. (see Fig. 7). Vibrational transitions from levels higher than the ground level are termed hot bands.

2.5

Combined Vibration-Rotation Transitions

The energy level of a molecule at the v vibrational and J rotational state is denoted by EυJ ¼ GðυÞ þ Fυ ðJ Þ

(15)

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V. P. Solovjov et al.

potential energy V (r)

Gu = vibrational level harmonic oscillator anharmonic potential (Morse type)

G3 G2

"hot bands" fundamental first second band overtone overtone

G1 G0 ground level

re

r intermolecular distance

quilibrium intermolecular distance

Fig. 7 The potential energy functions and the vibrational energy levels and transitions

The vibrational-rotational energy transitions of diatomic molecules, which take into account the interaction of vibration and rotation and the anharmonicity of the energy potential, are given by the Dunham equation in the form of a double power series

EυJ ¼

XX i

j

 Y ij

1 i υþ ½J ðJ þ 1Þj 2

(16)

The coefficients of expansion Yij are obtained by nonlinear least-squares fitting of experimental measurements of molecular transitions of different isotopes. Corresponding data are part of the well-established spectral databases HITRAN and HITEMP (Centeno et al. 2015; Rothman et al. 1992, 1998, 2010, 2013).

2.6

Line Positions

Gas absorption line spectral positions are directly related to the energy levels of the molecules through the fundamental postulate of quantum mechanics ΔE = hν. Line positions in the molecular spectrum depend on the difference between energies at all possible vibrational and rotational levels. Hence, the wavenumber η, [cm
1], of the transition v00 J00 ! v0 J0 is defined by ηðυ00 J 00 ! υ0 J 0 Þ ¼ Eυ00 J00
Eυ0 J 0 ¼ Gðυ0 Þ
Gðv00 Þ þ Fυ0 ðJ 0 Þ
Fυ00 ðJ 00 Þ ¼ η0 ðυ00 ! υ0 Þ þ Fυ0 ðJ 0 Þ
Fυ00 ðJ 00 Þ

(17) (18)

26

Radiative Properties of Gases

1083

3 10-18 R branch

2.5 10-18

P branch

2 10-18

1.5 10-18

1 10-18

5 10-19

2000

2050

2100

2150

2200

2250

Fig. 8 Fundamental band of the CO spectrum

where the wavenumber η0(v0 ! ν00) is called the vibrational band center without any corresponding line because the rotational transition 0 ! 0 is prohibited and the remaining terms define the detailed line positions for m = 1, 2, . . . (Fig. 8): R branch : ηðυ00 ! υ0 ,
mÞ ¼ η0 ðυ00 J 00 ! υ0 J 0 Þ þ Fυ0 ðmÞ
Fυ00 ðm
1Þ

(19)

P branch : ηðυ00 ! υ0 ,
mÞ ¼ η0 ðυ00 J 00 ! υ0 J 0 Þ þ Fυ0 ðm
1Þ
Fυ00 ðmÞ,

(20)

2.7

Line Intensities

The intensity of a spectral line (of any molecule, not just diatomic) at the wavelength ηul corresponding to transition u ! l is defined as the rate of change in energy of the molecule, which is proportional to the rate of change of number of molecules at the initial state. Three types of transitions between the energy levels are possible. The corresponding rate of change of the number of molecules Nl at the lower state is expressed with the Einstein coefficient (transition probability) for spontaneous emission Aul, the Einstein coefficient for induced emission Bul, and the Einstein coefficient for absorption Blu, as illustrated in the figure below.

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V. P. Solovjov et al.

Spontaneous emission @ Nl @t

Photons

Molecule before

Molecule after

Photons hν

¼ Aul N u Eu El

Induced emission @ Nl @t

¼ Bul N u uηul

2hν

hν Eu El

True absorption @ Nl @t

¼
Blu N l uηul

hν Eu El

The total rate of change of the number of molecules at the lower level for the combined effect is then determined from the relation @N l ¼ Aul N u þ ðBul N u
Blu N l Þuηul @t

(21)

A single-line intensity at standard reference temperature is given by      hηul N u N 0l hηul cm
1 Sul ðT ref Þ ¼ Bul N l 1
exp
, c N l N 0u kT ref molecule  cm
2

(22)

The Einstein coefficients can be found with the help of eigenfunctions corresponding to the lower and upper energy levels with the results organized in the dipole matrix ℜηη0 . Then line intensity of transition u ! l (this transition is denoted as transition from wavenumber η to η0 in Rothman et al. 1998) is  C2 E00η    I g exp
a η C2 ηηη0 8π 3 T ref 1
exp
Sηη0 ðT ref Þ ¼ ηηη0 ℜηη0 QðT Þ 3hc T ref   cm
1  10
36 , molecule  cm
2

(23)

where Ia is the natural terrestrial isotopic abundance (the fraction of occurrence of given isotope relative to all other isotopes of the same molecule which are present naturally in the Earth’s atmosphere).

26

Radiative Properties of Gases

1085

Intensities Sηη0 (Tref) are provided in the HITRAN database. The line intensity Sηη0 (T ) at an arbitrary temperature T is defined by scaling the intensity at the reference temperature Tref through the following expression (Rothmann et al. 1998): 

Sηη0 ðT Þ ¼ Sηη0 ðT ref Þ

QðT ref Þ exp
C2 E00 Qð T Þ



 C2 ηηη0  1
exp
  1 1 cm
1 T 
, C2 ηηη0 T T ref molecule  cm
2 1
exp
T ref

(24) where Sηη0 ðT ref Þ,

h

cm
1 moleculecm
2

i

is the line intensity at Tref (Tref = 296 K in HITRAN),

ηηη wavenumber of line center (weak dependence on p), C2 = hc0/k radiation constant, Q(Tref), Q(T ) partition function at Tref, and at T, 0 E00η lower energy h levelof transition i η ! η . 0

The term exp
C2 E00

1 T


T1ref

is related to the abovementioned ratio of the h  ih  i C2 ηηη0 C η 0 energy level populations, and the term 1
exp
T 1
exp
T2 refηη is due to effect of stimulated emission (Fig. 9).

Fig. 9 Part of the absorption spectrum of CO at 1,000 K

10

-17

10

-18

10

-19

10

-20

10

-21

10

-22

10

-23

1900

1950

2000

2050

2100

2150

2200

2250

2300

1086

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V. P. Solovjov et al.

Spectral Line Shape

m1

m2 μ=

m1 m2 m1 + m2

reduced mass

r re

r – re

The spectral line shape corresponding to a molecular transition is the dependence of intensity of a single vibration-rotation transition on frequency or wavenumber. If the vibrations of a molecule experience no damping during a transition, then the magnitude of vibrations is periodic with deflection pffiffiffiffiffiffiffiffi r
re  cos ω0t with a natural frequency of the spring-mass system ω0 ¼ k=μ. Transformation from the time domain to the frequency domain through a Fourier cosine transform indicates that the frequency dependence is concentrated at ω = ω0, which is the frequency of the vibrational transition. Deflection r
re  cos ω0t

Fc{r
re }  S0δ(ω
ω0) Fourier cosine transform

t

ω0

ω

In the case where damping is nonnegligible, the vibrations are proportional to the expression with r
re  e
γt cos ω0t, where ω0 is a quasi-frequency and γ describes the damping effect (natural and collision damping). A Fourier transform of exponentially decaying vibrations to the frequency domain features the classical Lorentz line profile shown below.

26

Radiative Properties of Gases

1087 Fc fr
r e g  S0 ðω
ωγ Þ2 þγ2 Fourier cosine transform

Deflection r
re  e
γt cos ω0t

0

t

ω0

2.9

ω

Lorentz Profile of Collision Broadening (Pressure Broadening) C

Lorentz profile of a single line

Si = area under curve

γi π (h −hi )2+ g i2

g i is a line halfwidth

Si

γi

ηi

η

The shape of spectral lines broadened due to the collision effect of randomly moving gas molecules is described by the Lorentz profile. The spectral absorption cross section corresponding to a transition η ! η0 with a wavenumber of the line center ηi = ηηη0 is described with a normalized shape function of the damped vibrations fL(η
ηi, T, p) , [1/cm
1] in terms of the wavenumber η instead of frequency ω:   Si ðp, T Þ γ i ðT, pÞ cm2 Cηi ðT, pÞ ¼ Si ðp, T Þf L ðη
ηi Þ ¼ , (25) π ðη
ηi Þ2 þ γ 2i ðT, pÞ molecule h i cm
1 integrated intensity (strength) of line i, where Si ðT Þ moleculecm
2 ηi[cm
1] line center (position) of line i, and γ i[cm
1] half-width at the half-maximum (HWHM) of line i.

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V. P. Solovjov et al.

Theoretically, the Lorentz profile extends to all wavenumbers on either side of the line. However, the Lorentz line profile is accurate only near the line center due to restrictions of the model of damped vibrations as modeled by a spring-mass system. Consequently, the shape of the wings of the Lorentz profile requires special attention. In any case, at some point, the line wings of collision broadening should be truncated to allow spectral windows – wavenumber ranges with no absorption. The pressure broadened line half-width γ(p,T) for a gas at total pressure p [atm], temperature T [K], partial pressure ps [atm], and mole fraction Y = ps/p diluted by air is calculated as γ L ðp, T Þ ¼ p

 nair   nself T ref T ref γ air ðpref , T ref Þ  ð1
Y Þ þ γ self ðpref , T ref Þ  Y T T (26)

where the half-width of lines broadened by the effect of collision of gas molecules with themselves (self-broadening) and by collision with nitrogen molecules (air-broadening) γ air( pref, Tref) , γ self( pref, Tref) , nair , nself may be found in literature (HITRAN and HITEMP). In the absence of other data, the coefficient of temperature dependence of the selfbroadened half-width has been assumed to be equal to that of the air-broadened halfwidth:  γ ðp, T Þ ¼

T ref T

nair  p  ½γ air ðpref , T ref Þ  ð1
Y Þ þ γ self ðpref , T ref Þ  Y 

(27)

Alternatively, the classical value of 0.5 has been used by default " #  0:5 T ref nair T ref γ ðp, T Þ ¼ p γ air ðpref , T ref Þ  ð1
Y Þ þ γ self ðpref , T ref Þ  Y T T (28) The kinetic theory of gases describes the dependence of Lorentz lines half-width on pressure and temperature as (Howell et al. 2011, p. 450) 2D2 p γ L ðp, T Þ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi c πMkT

(29)

Pressure (collision) broadening is the dominant cause of broadening of gas spectral lines in thermal radiation at terrestrial conditions. As seen from the previous equation, it becomes more important with an increase of total gas pressure and a decrease in temperature. The half-width at an arbitrary temperature and pressure can be calculated by relating it to the half-width at a predefined reference state at standard temperature T0 and pressure p0 as follows

26

Radiative Properties of Gases

1089

p γ L ðp, T Þ ¼ γ L ðp0 , T 0 Þ p0

rffiffiffiffiffi T0 T

(30)

which gives generally an accurate dependence on pressure and temperature.

2.9.1 Doppler Profile The frequency shift in the spectrum due to the Doppler effect caused by chaotically moving molecules is the other major source of spectral line broadening. The line shape associated with Doppler broadening is described by the equation (Howell et al. 2011) # rffiffiffiffiffiffiffi " ln2 ln2 Cηi ðT Þ ¼ Si ðp, T Þf D ðη
ηi Þ ¼ Si ðp, T Þ exp
ðη
ηi Þ 2 , γ D, i π γ D, i 1

(31)

where the Doppler half-width of a single line with a center at wavenumber ηi is η γ D, i ¼ i c

rffiffiffiffiffiffiffiffiffiffiffiffi kTln4 M

(32)

Note that the Doppler shape half-width increases with increasing temperature.

2.9.2 Voight Profile Because thermal radiation in gases can include situations with a range of total pressure from subatmospheric pressure (e.g., reentry problems, turbines) to elevated pressures (e.g., oxy-combustion) over a wide range of temperatures, both collision broadening and Doppler broadening might be important. Consequently, the convolution of Lorentz and Doppler profiles is more universal and can be more accurate. The combined spectrum is called the Voight profile and is defined by the following improper integral (Howell et al. 2011) Cηi ðT, pÞ ¼ Si ðp, T Þf V ðη
ηi Þ γ L, i ðp, T Þ ¼ Si ðp, T Þ pffiffiffiffiffi π3

1 ð

e
x dx (33)  xγ D, i ðp, T Þ 2 2 ðT, pÞ pffiffiffiffiffi η
η
þ γ i
1 L, i ln2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where the variable of integration is the parameter x ¼ v M=2kT. Integration cannot be performed in closed form and therefore, numerical integration is required. Numerical methods based on the Drayson or Humlicek algorithms have been used (Modest 2013). More recently, a faster and more accurate method is presented in Letchworth and Benner (2007); Birk (2013). Practical problems involving the prediction of gas radiation in gaseous media may feature broadening mechanisms dominated by either pressure effects or Doppler effects or may involve a combination of both broadening mechanisms. The suitable 2



1090 Fig. 10 Comparison of Lorentz and Doppler line profiles

V. P. Solovjov et al.

Doppler

Lorentz

choice of particular shape profile depends on the physical situation. Comparison of the relative shape of the profile demonstrates significant differences in the magnitude of lines. As seen in Fig. 10, the Lorentz line profile dominates over the Doppler profile in the wings of the line. Decay of the Lorentz line in the wings is slower – a characteristic of the adopted physical model of collision broadening caused by damping of vibrations of the idealized spring-mass system. In this case, the mathematical solution is in exponential form, which requires infinite time for return of the damped system to equilibrium. Fourier transform of this oscillating solution with an exponential solution yields a slowly decaying Lorentz profile over all wavenumbers, which can significantly affect other weaker lines in the absorption spectrum. Real spring-mass systems with damping reach equilibrium in finite time, and therefore, even a simple appropriate truncation of Lorentz lines can improve the accuracy of the absorption spectrum by allowing spectral windows – intervals of wavenumbers where no absorption occurs. The other reported method of improvement of accuracy of the Lorentz profile in the line wings is performed by correction of the line profile by the so-called χ-factor (Mlawer et al. 2012; Perrin and Hartmann 1989).

2.10 HITRAN Spectral Database http://www.cfa.harvard.edu/hitran/

26

Radiative Properties of Gases

1091

HITRAN is an acronym for high-resolution transmission molecular absorption database. HITRAN is a compilation of spectroscopic parameters that may be used to predict and simulate the transmission and emission of electromagnetic radiation in the atmosphere. The database is a long-running project initiated by the Air Force Cambridge Research Laboratories (AFCRL) in the late 1960s in response to the need for detailed knowledge of infrared properties of gases in the Earth’s atmosphere. Because this resource is the most widely used database for detailed spectroscopic radiation data for gases, considerable time will be spent here describing its use. The permanently updated versions of the HITRAN molecular spectroscopic database and associated compilation are available on an ftp site located at the Smithsonian Astrophysical Observatory in Cambridge, USA. It also includes the high-temperature extension to the spectral database, HITEMP. An interactive internet application called HITRANonline is available to registered users to access the current edition of HITRAN (Hill et al. 2016). The application HAPI provides additional features such as filtering, plotting, calculation of absorption and transmission, etc. (Kochanov et al. 2016). A video tutorial for the use of HITRANonline is available on YouTube. However, the calculation of accurate spectral data at an arbitrary thermodynamic state requires a customized approach. Each major release of HITRAN (nominally every 4 years) has been accompanied by a publication outlining the enhancements to the database. The most recent release was published as HITRAN 2012 (Rothman et al. 2013). The output file from HITRAN includes data in columns (100-character HITRAN line-transition format for editions before 2004 and 160 characters per record for editions starting with HITRAN-2014). The following table illustrates the content, format, and units used for spectroscopic parameters in the 100-character format: Column Mol Iso vij Sij Rij γ air γ self E00 nair δair iv0, iv00 q0 , q00 ierr

FORTRAN I2 I1 F12.6 E10.3 E10.3 F5.4 F5.4 F10.4 F4.2 F8.6 2I3 2A9 3I1

iref

3I2

Description and units Molecule number Isotopologue number (1 = most abundant) Wavenumber (center of the spectral line), cm
1 Line intensity @296 K, cm
1/molecule∙cm
2 Weighted transition moment-squared, Debye Air-broadened half-width (HWHM) @296 K, cm
1/atm Self-broadened half-width (HWHM) @296 K, cm
1/atm Lower state energy, cm
1 Coefficient of temperature dependence air-broadened half-width Air-broadened pressure shift of line transition @296 K, cm
1/atm Upper-state global quanta, lower-state global quanta indices Upper-state global quanta, lower-state global quanta Uncertainty indices for wavenumber, intensity, and air-broadened halfwidth Indices of references corresponding to wavenumber, intensity, and half-width

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V. P. Solovjov et al.

2.10.1

Compilation of the Gas Absorption Cross Section from Spectroscopic Data The lines from the spectral database can be ordered with respect to their wavenumber line centers: ηi
1 < ηi, [cm
1], where i is the number of the line in the list, with corresponding attribution of other parameters to this line. The resulting spectral absorption cross section at a given thermodynamic state is a superposition of all spectral lines Cη ðϕÞ ¼

X i



cm2 Si ðp, T Þf ðη
ηi , Y Þ, molecule

 (34)

where Si(T) integrated intensity (strength) of line i, ηi line center (position) of line i, f(η
ηi, Y ) = fη line shape (profile) of line i, and ϕ = {T, p, Y} symbolic notation of the given thermodynamic state. Theoretically, for calculation of the absorption cross section Cη(ϕ) at some wavenumber η, the contribution of all spectral lines should be included because direct application of any of the considered line profiles spreads the lines over all wavenumbers. However, inclusion of all lines in construction of the absorption spectrum is neither necessary nor wise because of the negligible contribution of numerous weak lines and because of the possible overprediction of the resulting line shape by contribution of inaccuracies in the shape of the line wings. Further, because of the enormous number of spectral lines in modern spectral databases, including all possible lines is computationally expensive, both for computer time and storage. Construction of the detailed gas absorption spectrum consists of following steps: 1. Reading the data from the spectral database at standard conditions. It should be noted that spectral data in HITRAN are presented for different isotopologues (isotopes) of the molecules with line intensities already factored by the natural abundance Ia. Therefore, for correct calculation of the spectrum, data for all isotopologues should be included. The majority of data is for the primary isotopes (whose abundance represents approximately 99%); the secondary isotopes include the strongest lines only. However, some isotopes can also play a significant role. Only selected isotopologues are used in some spectral compilations (Rivière and Soufiani 2012). The spectral data compiled in Denison and Webb (1994) and Solovjov and Webb (1998) included only the primary isotopes. 2. Selection of lines. Contemporary spectral databases include an enormous number of spectral lines (millions of lines for main gas combustion species). However, including all lines in the generation of the absorption spectrum is both unnecessary and computationally expensive. In practice, a cutoff criterion is generally

26

Radiative Properties of Gases

1093

imposed, and line strengths below the criterion are neglected in the generation of the detailed absorption spectrum. The appropriate cutoff criterion must be determined such that raising it does not appreciably affect the results. The appropriate cutoff criterion depends on the gas specie and temperature. A typical approach in combustion applications imposes a cutoff of 10
26
10
28cm
1/ (molecule  cm
2), thus dropping the lines with intensity below this value. Establishing a cutoff criterion can significantly reduce the number of lines used. For example, using the cutoff criterion described above reduced the number of lines for H2O from 60 million to 10 million without any noticeable influence on the accuracy. The number of lines included must increase with an increase of temperature due to the appearance of “hot lines,” corresponding to molecular transitions that originate from the non-ground state. Rivière and Soufinai (2012) used the criterion nabsSlu(T) > 10
10cm
2 in determining which spectral lines to retain, which is effectively equivalent to a cutoff criterion of approximately 10
27cm/molecule for H2O and CO2. In the case of CO, Rivière and Soufiani (2012) included all lines. In Pearson et al. (2014), a line intensity cutoff value of 10
26 cm
1/molecule/cm
2 was chosen for H2O at temperatures below 2,500 K and for CO at all temperatures, while a cutoff value of 10
27 cm
1/molecule/ cm
2 was chosen for H2O at temperatures of 2,500 K and above and for CO2 at all temperatures. Reducing the cutoff criterion below these values yielded no appreciate difference in the predictions. 3. Scaling of spectroscopic data at standard conditions to the given thermodynamic state ϕ = {T, p, Y}.

(a) Line center locations taken from the spectral database can be corrected to the given pressure using the relation ηi ðpÞ ¼ ηi, vac þ pδ

(35)

where the pressure shift δ can be found in the HITRAN database. It should be stated that the pressure-induced line shift δ usually is very small.

d

η ηi (p) ηi,vac

1094

V. P. Solovjov et al.

(b) Line intensity at a given state is calculated as  C2 ηηη0    1
exp
  QðT ref Þ 1 1 cm
1 T  , Sηη0 ðT Þ ¼ Sηη0 ðT ref Þ exp
C2 E00
C2 ηηη0 molecule  cm
2 QðT Þ T T ref 1
exp
T ref

(36) Calculation of the ratio of partition functions Q(Tref)/Q(T ) appearing in Eq. 36 at standard conditions and at a given nonstandard temperature can be performed using the additional data provided in HITRAN-2008 (Rothman et al. 2009). An alternative approach, recognized as a more accurate functional form of the partition function for H2O (Vidler and Tennyson 2000), differs from those calculated using HITRAN by at most 1.1% over the range of temperatures 300–3,000 K but at significantly higher computational cost. Vidler’s partition function adjustments are used by Rivière and Soufiani (2012). The data for partition function of the secondary isotopologues is not usually available, and therefore, the ratio of the partition functions is calculated with the help of data for the main isotopologues. The remainder of the parameters appearing in Eq. 36 can be taken from the HITRAN database. (c) Correction of the half-widths. The equation for scaling of the Lorentz line widths from reference conditions to a pressure and temperature of interest is  nair   nself T ref T ref γ L ðp, T Þ ¼ p γ air ðpref , T ref Þ  ð1
Y Þ þ γ self ðpref , T ref Þ  Y T T (37) where the half-width of lines broadened by the effect of collision gas molecules with itself (self-broadening) and by collision with nitrogen molecules (air-broadening) γ air( pref, Tref), γ self( pref, Tref), nair , nself is provided in the HITRAN database. The dependence on lower energy level of power coefficients of air-broadened half-widths nair provided by HITRAN were fit by polynomials for CO2 in Denison and Webb (1994) and for CO in Solovjov and Webb (1998). For those cases where data are not provided, the commonly used approximation is nair = nself. (d) Half-widths for mixtures of gases with broadening by different species included into the gas mixture can be calculated by equations presented in Rivière and Soufiani (2012). (e) Correction and truncation of the wings of spectral lines.

26

Radiative Properties of Gases

1095

γi

δηcut

η

ηi

To deal with the problem of inaccuracy in the wings of spectral lines, a universal truncation has been used for H2O and CO2 (Denison and Webb 1994), δηcut = 25 cm
1. A progressive truncation approach with correction of the intensity in the line wings was used by Rivière and Soufiani (2012): H2O

CO2 CO

Lines from HITEMP-2010 with parameters from for collisional broadening Function χ is used for correction of the line shapes at T = 300 K Progressive line wing cutoff: δηcut = 500cm
1 for T < 2100K δηcut = 200cm
1 for 2100K < T < 3300K δηcut = 100cm
1 for T > 3300K Lines from CDSD-4000 with δηcut = 50cm
1 for all cases χ correction function is used for temperature below 900 K Lines from HITEMP-2010 with δηcut = 500cm
1

Progressive cutoff of spectral lines was performed by Pearson et al. (2014) in different manner. In order to achieve more accurate results, the corrections to the Lorentz profile recommended for H2O and CO2 were used (the MT-CKD model; Mlawer et al. 2012; Perrin and Hartmann 1989). No such correction for CO exists at present. However, the MT-CKD model, which corrects the line shape for H2O, is only valid for atmospheric conditions. The correction to the line shape for CO2 is limited to a maximum temperature of 800 K and to a maximum pressure of 60 bars. In order to account for the inaccuracy of the Lorentz profile, the following procedure was used. The average number of line half-widths was calculated for which the Lorentzian profile yields data that is within an order of magnitude of the corrected line shape from the corresponding model at the highest temperature for which the corrected profile is valid. This spectral distance was then used as the line wing cutoff for all conditions. For H2O, this calculation was performed for T = 300 K and p = 1 atm, while for CO2, it was done at T = 800 K applicable to all total pressures investigated. A single-line wing cutoff was chosen for all mole fractions for each specie. Based on this

1096

V. P. Solovjov et al.

investigation, the line wing cutoff used was 2750 half-widths from line center for H2O and 600 half-widths for CO2. Because no correction factor data are available for CO, a line wing cutoff of 600 half-widths was also applied to CO.

2.11 Line-by-Line Model (LBL) of Gas Absorption Spectrum Integration of the RTE over the continuous spectral absorption coefficient κη for a radiating gas is generally considered computationally prohibitive for prediction of the radiative transfer (see a Single Line Model in Modest 2013).Therefore, the most accurate possible practical model is the line-by-line (LBL) model which is a discrete representation of the absorption spectrum with very fine details. In the LBL model, the spectrum is calculated at all line centers ηi and at several evenly distributed points between line centers. In Denison and Webb (1994) and Solovjov and Webb (1998), the interval between line centers was subdivided into four to eight subintervals. Therefore, the LBL spectrum is a discrete representation of the continuous spectrum by a set of pairs as defined and illustrated below:

η
η 

i ηi, j , Ci, j , ηi, j ¼ ηi þ j  iþ1 , Ci, j ¼ Cηi, j , m

(38)

i ¼ 0, 1, 2, . . . , j ¼ 0, 1, 2, . . . , m

where index j corresponds to partition of intervals between line centers. This double-indexed sequence can be renumbered into a single-ordered sequence: fηk , Ck g, ηk ¼ ηi, j , Ck ¼ Cηi, j , k ¼ i  m þ j, i ¼ 0, 1, 2, . . . , j ¼ 0, 1, 2, . . . , m

(39)

Ch Ch

Ch

i+1

i

Ch(Tg) Ch

i-1

Ci,j

hi-1

hi

hi, j

hi+1

h

26

Radiative Properties of Gases

1097

All line centers are included into discrete set of data {ηi,j,Ci,j}. Partition of the spectrum line behavior into the discrete wavenumbers ηj is generally nonuniform. Application of the LBL model to spectral integration is extremely computationally demanding. In general, this spectral model is used exclusively for development of benchmark solutions against which other approximate engineering models are compared.

2.12 Uniform Narrow Band Model of Gas Absorption Spectrum A simplified method of calculation is based on a narrow band representation of the actual high-resolution line spectrum (theoretically, with any degree of accuracy compared to the detailed line spectrum). In this approximate model, the detailed line spectrum is divided into narrow bands each of which includes a number of spectral lines. The narrow band model spectrum may be generated as follows: fηi , Ci g, ηi ¼ i  Δη, Ci ¼ Cηi , i ¼ 0, 1, 2, . . .

(40)

Ch (f)

hi = i · ∆ uniform subdivision

Ci-1

Ci

ηi−1

Δ

ηi

η

Formally, depending on the application, the narrow band representation can be treated as a box histogram spectrum as shown in the figure above, or it can be treated as a piecewise linear spectrum as shown in Figs. 11 and 12.

1098

V. P. Solovjov et al.

hi, cm−1 2150.000 2150.005 2150.010 2150.015 2150.020 2150.025 2150.030 2150.035 2150.040 2150.045 2150.050 2150.055 2150.060 2150.065 2150.070 2150.075 2150.080 2150.085 2150.090 …

Ci ,

cm2 molecule

1.4152e-21 1.4176e-21 1.4206e-21 1.4242e-21 1.4284e-21 1.4332e-21 1.4387e-21 1.4448e-21 1.4516e-21 1.4591e-21 1.4673e-21 1.4763e-21 1.4859e-21 1.4965e-21 1.5080e-21 1.5205e-21 1.5339e-21 1.5484e-21 1.5638e-21

Ch ,

10-18

cm2 molecule

10-19

10-20

10-21 2150

2155

2160

2165

2170

2175

…

Fig. 11 Absorption cross section of CO at 1,000 K and 1 atm (Pearson et al. 2014), tabulated with a uniform step Δη = 0.005 cm
1. The bands are extremely narrow, and the line peaks are accurately represented. C++ code for line-by-line calculation of the absorption cross section from HITEMP spectroscopic data following the development of this section is presented in (Pearson 2013)

The spectral resolution for a uniform step spectral representation used in different works usually is between 0.005 and 0.02 cm
1.

2.13 Spectral Absorption Coefficient and Absorption Cross Section The spectral absorption cross section constructed from high-resolution spectral data can be used for evaluation of the local absorption coefficient κ η(s) , [m
1], which appears in the radiative transfer equation (Eq. 5):
 κ η ðsÞ ¼ N ðsÞY ðsÞCη ðϕðsÞÞ, m
1

(41)

The absorption cross section at a thermodynamic state ϕ(s) = {T(s), Y(s), p(s)} was defined as Cη ðϕÞ ¼ Cη ðT, p, Y Þ ¼

X i



cm2 Si ðp, T Þf ðη
ηi , Y Þ molecule

 (42)

The following dimensions, units, and constants are involved in the definition of the components of the absorption coefficient at state ϕ(s) = {T, p, Y}, which is used in the SLW model of radiative transfer (see Sect. 4):

26

Radiative Properties of Gases

1099

10-15

Ch ,

cm2 molecule

10-16

10-17

10-18

10-19

10-20 2320

2322

2324

2326

2328

2330

Fig. 12 CO2 at 300 K, 0.01 atm (Pearson et al. 2014), Δη = 0.005 cm
1


 κη ðϕÞ, m
1 ¼ N ðT, pÞ  Y  Cη ðT, Y, pÞ       pg p mol molecule  ¼  NA 3 Ru T m mol p |fflffl{zfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} Y N ðp, T Þ  2   cm2 1 m  Cη ðT, Y, pÞ  molecule 10000 cm2 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} HITRAN absorption cross-section  pg ½Pa ð101325Þ½Pa    ¼ p½Pa Pa  m3 ð8:314Þ T ½K  |fflfflfflfflfflffl{zfflfflfflfflfflffl} mol  K Y |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} N ðT , pÞ of SLW model    2   molec cm2 1 m  ð6:022045e23Þ  Cη ðT, Y, pÞ  mol molec 10000 cm2 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} m2 Cη ðT , Y , pÞ½mol  spectral absorption cross-section used in SLW ðDenison and Webb 1994Þ

(43)

(44)

(45)

1100

V. P. Solovjov et al.

This modification of absorption cross section consists in moving Avogadro number NA from the molar density to the absorption cross section. Then absorption coefficient κη(ϕ) used in the SLW model is calculated as κ η ðϕÞ,

     2 1 mol m ð T, Y, p Þ ¼ N ðT, pÞ  Y  C η m m3 mol

(46)

where the SLW absorption cross section is obtained by scaling the HITRAN absorption cross section:  Cη ðT, Y, pÞ,

     m2 cm2 molecule  m2 ð6:022045e19Þ ¼ Cη ðT, Y, pÞ mol molecule mol  cm2 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} HITRAN

(47) and the molar density at total pressure p[atm] and at gas temperature T[K] is calculated as 

   mol K  mol p½atm N ðT, pÞ, ¼ 12187:27 3  m3 m  atm T ½K

(48)

The purpose of this scaling is to deal in the SLW method with more moderate magnitudes of absorption cross section (of orders 10
4
102) compared to very low magnitudes of HITRAN cross sections (of orders 10
27–10
16), which are less convenient for calculation.

2.14 The Line-by-Line Method of Solution of the Spectral RTE The line-by-line (LBL) method solves the spectral RTE for discrete wavenumber η in a finely resolved discretization of the spectrum: @I η ðs, ΩÞ ¼
κη ðsÞI η ðs, ΩÞ þ κ η ðsÞI bη ½T ðsÞ @s

(49)

After solution for all wavenumbers, the total intensity of radiation is obtained by the spectral integration over all wavenumbers. The narrow band approximation requires solution of the RTE at many (perhaps millions) points ηi in the numerical integration: 1 ð

I ðs, ΩÞ ¼

I η ðs, ΩÞdη  0

X i

I ηi ðs, ΩÞΔηi

(50)

26

3

Radiative Properties of Gases

1101

Narrow Band Models

Approximate models for the radiative properties of gases are not only required in heat transfer problems, but they also have applications in quantitative spectroscopy. However, in this case, models averaged over so-called narrow bands – spectral widths usually of the order of a few dozens of cm
1 – are required so as to comply with the spectral resolution of usual infrared spectrometers. The two most important areas of application of infrared spectroscopy in gases are the sounding of the atmosphere and the radiative characterization of flames – from small laboratory scales up to infrared signature of planes or rockets studies, which involves radiation paths in the gas that can attain several kilometers. In these applications, the purpose is most of the time to infer the scalar fields – temperature and species concentrations and total pressure – in the gaseous medium from its absorptivity, absorption spectroscopy, or radiative emission, emission spectroscopy. The main objective of narrow band models is to represent the radiative properties of the gas averaged over small spectral intervals, defined in such a way that the Planck function can be treated as a constant. This leads to narrow intervals that usually contain information about a few hundreds of high-resolution LBL data – around 2,500 if the narrow band has a width of 25 cm
1 and if the distance between two values in the LBL dataset is 10
2 cm
1; these values are typical for applications in high-temperature gases (Taine and Soufiani 1999). Obviously, narrow band models can be also used in prediction of heat transfer for problems of practical interest. They are especially useful when the problem under consideration involves non-gray walls – as, for instance, in radiative heat transfer in glass furnaces – or non-gray particles, such as soot, ash, etc. In these kinds of applications, the radiative transfer problem is first solved over each narrow band, and the total radiative intensity is calculated, in a second step, by summing the contributions of all the bands. Accordingly, narrow band approaches are more computationally efficient than line-by-line methods, but more computationally expensive than global models. They are also applicable in a wider range of possible configurations than full-spectrum techniques, due to their ability to account for any kind of boundary condition, gray or non-gray, as noted earlier. There are two main narrow band approaches – if we skip some recent developments such as the ‘-distribution method (André 2016a, b) – that still remains at the time of writing in a development stage: the correlated-k distribution method, usually referred to as the C-k model in high-temperature applications (Taine and Soufiani 1999; Soufiani and Taine 1999), or also CKD in the atmosphere and the statistical narrow band (SNB) modeling; these two approaches are however founded on the same ideas and assumptions. C-k/CKD techniques are based on the same concepts as full-spectrum k-distribution models. However, due to the fact that the radiative properties are considered at small spectral scales over which the Planck function is constant, they use simpler formulations. For instance, they do not require the definition of any reference thermophysical state to treat nonuniform media, nor do they require a distribution function that depends on some reference source

1102

V. P. Solovjov et al.

temperature. These approaches are otherwise rigorously the same as those described in the next chapter. They will not be developed further here, but the interested reader should refer to references Taine and Soufiani (1999) and Soufiani and Taine (1999) for more details on this particular technique. SNB approaches mostly differ from k-distribution methods by the fact that the random quantities considered in this model are not absorption coefficients but, rather, line strengths and centers. They mostly consist of a representation of an absorption spectrum as a random array of spectral lines with given prescribed distributions of line strengths. They require some specific techniques to be extended to nonuniform situations. SNB methods will be the main focus of this chapter. Some possible extensions of narrow band models to highly nonuniform media will be also discussed at the end of the present part.

3.1

Principle of Statistical Narrow Band (SNB) Models

The main idea behind narrow band models is that if one considers a blackbody source with spectral intensity Iη(0) located at the origin of a path of length L inside a uniform gas with spectral absorption coefficient κη and further, if one assumes that the Planck function is constant over a narrow spectral interval Δη, then one can write the band averaged radiative intensity I(L ) at the exit of the path as: ð ð 8     1 1 > I ð L Þ ¼ I ð 0 Þexp
xPκ L dη ¼ I ð 0 Þ exp
xPκ η L dη > η η > > Δη Δη < Δη Δη ð 1 > > I ð 0Þ ¼ I η ð0Þdη > > : Δη

(51)

Δη

ð In Eq. 51, the following notation has been used

1 Δη Δη

1 . . . dη ¼ η1
η0

ηð1

. . . dη, η0

Δη ¼ ½η0 , η1 : In that case, as soon as we are interested in characterizing the fraction of the incident blackbody radiation I(0) that is absorbed by a single spectral line averaged over the band Δη, and if the line profile is totally included inside the band, it does not matter to know precisely where the line is located in the spectrum, i.e., what is its line center. Indeed, in any case, the ratio I(L )/I(0), which is called the band averaged transmissivity of the single line, is given by: τΔη ðLÞ ¼

I ðLÞ 1 ¼1
I ð 0Þ Δη

þ1 ð


1




  1 W ðxPLSÞ 1
exp
xPSf η L dη ¼ 1
Δη

(52)

26

Radiative Properties of Gases

1103

where S is the line strength that depends principally on the radiative transition associated with the line and the temperature T of the gas, fη is the line profile that is a function of the thermophysical state of the gas, x is the molar fraction in absorbing species, and P is the total pressure in atm. In the case of Eq. 52, the spectral absorption coefficient, in cm
1atm
1, is κη = Sfη. Notice that the spectral integral in (52) extends from
1 to +1 because the line profile must be integrated from 0 to +1 on both sides of the line center. The quantity W that appears in Eq. 52 is called the equivalent black line width of the spectral line. It has the same unit as a spectral band width (cm
1) and is proportional to the total absorptivity of the single spectral line. The curve that depicts the dependence of the black line width with respect to the optical length xPL of the gas path is called the curve of growth (COG). The same name is used for arrays of absorption lines. If more than one spectral line appears in the spectrum, which is most often the case in practice, and in some situations this number can be very large, one assumes that the fraction of radiation absorbed by a given spectral line is statistically independent from the absorption by all the other lines. This assumption is reasonable as soon as line centers are considered to be independent from the others. Then, the transmissivity of the array can be approximated as the product of the transmissivities of each of the lines, each of which is represented by an index “i”; all spectral lines are assumed to share the same profile but are associated to statistically distinct line centers: Δη

N

τ ðLÞ ¼ ∏ i¼1

τΔη i ðLÞ ¼

  ! N X 1 1 ∏ 1
W ðxPLSi Þ ¼ exp ln 1
W ðxPLSi Þ Δη Δη i¼1 i¼1 N



(53) The idea is then to assume that the number of spectral lines N is large so that the occurrence of a particular value of line strength can be represented statistically by introducing P(S)dS as the fraction of spectral lines whose strengths is inside a small interval dS of values of line strengths S. This allows transforming the sum into an integral: 0 τΔη ðLÞ ¼ exp@N

þ1 ð

0

1   1 W ðxPLSÞ PðSÞdSA ln 1
Δη

(54)

Finally, using the fact that equivalent black line widths of single lines are small compared to the total width of the interval, we obtain, as lnð1
xÞ 
x: x!0

0 N τΔη ðLÞ ¼ exp@
Δη

þ1 ð

0

1

  W ðxPLÞ PðSÞW ðxPLSÞdSA ¼ exp
δ

(55)

1104

V. P. Solovjov et al.

where δ = Δη/N represents the mean distance between two spectral lines assumed to be uniformly distributed over the narrow band Δη and: þ1 ð

W ðxPLÞ ¼

PðSÞW ðxPLSÞdS

(56)

0

is the mean value of the equivalent black line width. Various distributions of line strengths were proposed in recent decades. Typical examples of gas spectral models are depicted in Figs. 13, 14, and 15. Among them, the one based on the so-called inverse S-tailed distribution proposed by Malkmus in Malkmus (1967) is undoubtedly the most widely encountered both in modeling radiation transfer in combustion applications and atmospheric sciences. A schematic

Fig. 13 Regular array of identically spaced Lorentz lines: Elsasser’s model

Fig. 14 Uniform statistical narrow band model. All Lorentz lines have the same intensity, but the distance between lines is not a constant

26

Radiative Properties of Gases

1105

Fig. 15 Random array of spectral lines, with distribution of line strengths P(S). For Malkmus’ distribution P(S), the transmissivity is given by Eq. 57

representation of the corresponding spectrum is plotted in Figs. 13, 14, and 15 together with some properties of the associated transmissivity model. This specific model will be focused in the following.

3.2

The SNB Model with Malkmus’ Distribution of Line Strengths for an Array of Lorentz Lines

The SNB model with Malkmus’ distribution of line strengths is probably the most widely used in modeling radiative transfer in combustion applications, for which line profiles can be realistically considered as Lorentzian. It provides the transmissivity of a uniform path of length L inside the gas as: 2

0sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 13 β 2πxPkL τΔη ðLÞ ¼ exp4
@ 1 þ
1A 5 π β

(57)

Accordingly, this model condenses all the information about the collection of spectral lines into two quantities: • k, in cm
1.atm
1, is the mean value of the absorption coefficient over the narrow band and represents the value of the transmissivity at the optically thin limit, when xPL ! 0. Furthermore, as spectral lines are assumed to have their profile completely included inside the band Δη, it is proportional to the mean value of line strengths. Consequently, this quantity only depends on temperature. • β = 2πγ L/δ, dimensionless, is called the overlapping parameter. It compares the mean distance between spectral lines δ, in cm
1, to their Lorentz HWHM γ L, in cm
1. Small values of this quantity correspond to spectra with almost nonoverlapping lines and can be encountered in low pressure situations, whereas high values are found when a strong overlap is observed, which is the case at high total pressures.

1106

V. P. Solovjov et al.

Notice that for high pressure situations, when β ! 1, the SNB-Malkmus approach simplifies to a gray gas model. This idea was used to develop some specific techniques for high pressure combustion chambers, such as the high pressure box model (HPBM) (Pierrot 1997). SNB-Malkmus model parameters can be estimated from spectroscopic databases directly from the definition, according to the SNB model theory. However, this method usually provides inaccurate results. Accordingly, it was proposed in Ludwig et al. (1973), to adjust these two quantities using a nonlinear least-squares fitting technique of the approximate model to curves-of-growth-evaluated LBL. Other methods, based on moments, can also be found in the literature (André and Vaillon 2007, 2012). Depending on the moments used to build the parameters, higher accuracy adjustments can be obtained “locally,” viz., for given sets of path lengths, by this approach. However, as the fitting technique accounts for a wider range of possible values of parameters xPL, this method was used to build most of the usual SNB-Malkmus model databases (Soufiani and Taine 1999). The SNB-Malkmus model has many strengths: (1) its parameters have clear physical meanings, as explained earlier; (2) it is usually accurate in uniform media, although errors of a few percent are not uncommon in this kind of approach; (3) it is computationally efficient, because compared to k-distribution methods, it does not require any kind of spectral integration. An example of narrow band transmissivity calculation using this model is shown in Fig. 16.

LBL Trasnmissivity (solid line) Relative difference Model / LBL (symbol)

1.0

0.8

0.6

0.4

0.2

LBL Trasnmissivity 1 - SNB with Malkmus' distribution P(S) / LBL

0.0 750

1000

1250 1500 1750 Wavenumber in cm–1

2000

2250

Fig. 16 Narrow band averaged transmissivity of a uniform column of 10% CO2–20% H2O and 70% N2 at 1 atmosphere and 300 K. The length of the gas path is 50 cm

26

Radiative Properties of Gases

1107

Furthermore, as was shown by Domoto (1974), one can provide an explicit formula for the corresponding k-distribution, equivalent to the ALBDF encountered in full-spectrum models as described in Sect. 4: ð   1 H k
κ η dη Δη 0Δη 2rffiffiffiffiffi0sffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffi13 2rffiffiffiffiffi0sffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffi131  1@ β xPk k 2β β @ xPk k A 5A @ A5 þ exp erfc4 erfc4 ¼
þ 2 2π k π 2π k xPk xPk

gðkÞ¼

(58) in which erfc represents the standard complementary error function. The derivative of Eq. 58 with respect to k is the inverse Laplace transform of the transmissivity (Domoto 1974) and is in this case inverse Gaussian: @gðkÞ 1 f ðk Þ ¼ ¼ @k 2πk

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi    2πβxPk β k xPk exp 2

k 2π k xPk

(59)

where k is expressed in cm
1. These results have led many early adopters of k-distribution methods (Lacis and Oinas 1991) to use the previous formulas in radiative heat transfer calculations instead of the “exact” versions. This is both due to its simplicity and analyticity but also to a lack of availability, at the time, of high-resolution LBL data to construct k-distribution model parameters over wide ranges of thermophysical states. This approach remains in use widely today and is usually referred to as the SNB-Ck method (Liu et al. 2001).

3.3

SNB Models in Nonuniform Gaseous Media

In many applications, gases are not uniform – homogeneous and isothermal – but are subject to gradients of temperature and species concentrations. Using SNB models to treat such problems requires extending the approach described in Sect. 3.2 to nonuniform paths. This point is the aim of the present part. SNB models are formulated in terms of transmissivity, and their extension to a nonuniform path consists primarily of the definition of methods to estimate pathaveraged or effective parameters and to use these quantities inside the same models as are used in uniform situations. There exist different techniques to define these effective parameters. The most common ones involve one (this mostly corresponds to a scaling approximation) or two path-averaged coefficients. The general form of these approximations can be found in Young (1975, 1977) together with a detailed derivation of all these techniques:

1108

8 > > > > >
> > 0 > > : 1 @W ð0,s Þ ¼ xðs0 ÞPðs0 Þkðs0 Þyðs0 Þ δ @s0

s0ð ¼s

s0 ¼0

3 1 @W ð0,s0 Þ 0 5 ds δ @s0

(60) where y is a function that both depends on local, viz., at location s’ along the nonuniform path, and path-averaged band model parameters between location 0, origin of the path, and s. Notice that this nonuniform formulation is mostly an extension of the formulation obtained at the spectral scale: Spectral formulation: @τη ð0, s0 Þ ¼
xðs0 ÞPðs0 Þκη ðs0 Þτη ð0, s0 Þ @s0

(61)

Band averaged formulation: @τΔη ð0, s0 Þ 1 @W ð0, s0 Þ Δη ¼
τ ð0, s0 Þ ¼
xðs0 ÞPðs0 Þkðs0 Þyðs0 ÞτΔη ð0, s0 Þ 0 @s δ @s0

(62)

Two of the most widely encountered methods to define the function y were proposed as the Curtis-Godson approximation (Godson 1953) and Lindquist-Simons (Lindquist and Simmons 1972). However, in practice, the Curtis-Godson approximation is undoubtedly the most widely used, due to its simplicity. This approach consists of treating nonuniformities by assuming that both line strengths and the HWHM are scaled. Each of these parameters possesses its own dependency function with respect to the local thermophysical states. More details can be found in Young (1975, 1977). The flowchart to apply the Curtis-Godson approximation is given in Fig. 17. In the case of mixtures of radiating species, the Curtis-Godson approximation is first applied to all the gases in the mixture considered separately, and the transmissivity of the mixture is then obtained as the product of the transmissivities of each of the nonuniform single molecule gas paths. This multiplication property arises from the statistical independence of spectral line centers from distinct molecular species. Many prior works were devoted to the assessment of the Curtis-Godson approximation, and other nonuniform methods, against LBL reference calculations. It was shown, for instance, to be among the most accurate for combustion applications in Soufiani et al. (1985). However, the accuracy of the Curtis-Godson approximation decreases quickly when large temperature and/or pressure gradients are encountered along the nonuniform path. Its use can nevertheless be justified in many small-scale problems. In infrared signature situations, more sophisticated techniques are required. They are described briefly in the next paragraph.

26

Radiative Properties of Gases

1109

Fig. 17 Flowchart for the application of the Curtis-Godson approximation

3.4

Fictitious Gases and Mapping Methods

All traditional models handle path nonuniformities by assuming that the behavior of gas spectra between distinct states is almost ideal, viz., scaled or correlated. In practice, this is rarely truly observed, especially when large temperature gradients are encountered along the nonuniform path. This is due to the appearance in the spectrum of the absorbing species of so-called hot lines – spectral lines associated with radiative transitions involving high energy levels of the molecule. These lines are almost absent of the spectrum at low temperature and appear slowly as the temperature increases. Scaling and correlation assumptions are not able to handle this physical phenomenon in a general frame, and some dedicated techniques were proposed to extend models to highly nonuniform media. Two of the most widely encountered models are the so-called “fictitious gases” and mapping methods. In the fictitious gases approach (Levi di Leon and Taine 1986; Rivière et al. 1992; Soufiani et al. 2002), the gas of a single molecule is treated as a mixture of fictitious gases whose spectra are built by compiling sets of lines of the molecule associated with given ranges of energy states of the transitions. Each fictitious gas is then modeled using the same methods as are used for all lines, and the properties of the

1110

V. P. Solovjov et al.

molecule are reconstructed by treating the mixture of fictitious gases as a mixture of distinct species, viz., by assuming that spectra associated with groups of spectral lines are statistically independent. The advantage of this technique is that it is relatively simple to apply. Mapping techniques (West et al. 1990; Zhang and Modest 2003; André et al. 2014) are founded on a discretization of the spectral axis into subintervals in such a way that over each of them, the assumption of scaling of gas spectra can be considered as true. This kind of approach was originally proposed to treat radiative transfer problems in the atmosphere (West et al. 1990) and was extended more recently to combustion applications. Mapping techniques are a little more complicated to apply than fictitious gases because they require defining criteria and methods to construct the subintervals. Once built, usual approaches are applied over each of these subintervals. Fictitious gases and mapping techniques usually provide a similar accuracy, but mapping techniques were shown (Zhang and Modest 2003) to be computationally more efficient than FG when applied in absorption coefficient form. When formulated in terms of transmissivities, both techniques are equivalent.

3.5

Additional Comments

Two of the most important advantages of SNB models, compared with k-distribution approaches, are that their parameters have clearly defined physical meaning and also that their computational efficiency is high as they do not require any summation – over wavenumbers or k-values. However, as they provide a model formulated in terms of transmissivity, they can only be applied together with the integral form of the RTE: Δη

Δη

Δη

ðs

I ðsÞ ¼ τ ð0, sÞI ð0Þ þ s0 ¼0

0  @τΔη s , s Δη  0  0 I b s ds @s0

(63)

Equation 63 can be discretized by representing the nonuniform path [0,s] as a sequence of uniform sub-paths. Each nonuniform transmissivity obtained by this method can be estimated using the Curtis-Godson approximation. Notice that Eq. 63 can be differentiated with respect to s to provide a differential form, but this formulation has not been used yet, to the best of the author’s knowledge, for radiative heat transfer calculations. The formulation of SNB models in terms of transmissivities thus restricts their use to RTE solvers based on ray tracing methods. Notice that in the case of nonblack walls, and when a deterministic RTE solver is used, some additional assumptions are further required to treat boundary conditions, which may lead to large errors when highly reflecting walls are involved (Pierrot 1997). Such limitations are circumvented when SNB models are used together with a Monte Carlo method, for which the optical path is traveled up to the absorption of radiative energy at some location inside the medium or at a wall.

26

Radiative Properties of Gases

1111

4

Global Models of Gas Radiation

4.1

Introduction to Global Models

In contrast to the LBL method, in which the objective is to find the spectral intensity of radiation Iη(s, Ω), the objective of global models is the determination of the total intensity of radiation 1 ð

I ¼ I ðs, ΩÞ ¼

I η ðs, ΩÞdη: 0

The difference between global models and line-by-line models may be summarized as follows. In the LBL model, the RTE is first solved for spectral intensity, after which the spectral intensity is integrated over all wavenumbers to find total quantities. In global models, the RTE is first integrated to reduce it to an equation for the total intensity (or the fractions of total intensity), after which the spectrally integrated RTE is solved for total intensity (or the fractions of total intensity which are then summed to get the total intensity).

4.2

Gray Model

The simplest global model is the gray gas model in which it is assumed that the gas absorption coefficient does not depend on wavenumber η and is represented by a single value κa (see Fig. 18). Spectral integration of the RTE over all wavenumbers in uniform media yields an equation for the total intensity I: @I ¼
κa I þ κ a I b @s

(64)

One must then specify the value of gray gas absorption coefficient κ a. The most popular assumption is the Planck mean absorption coefficient. The mean coefficient based on total emissivity of the equivalent path L may be specified as

κ

Fig. 18 Gray model of gas absorption coefficient

κη

κa η

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V. P. Solovjov et al.

e ¼ 1
e
κa L ,

1 1 κa ¼ ln L 1
e

The semi-gray model assumes different values for absorption and emission terms (Viskanta 2005): @I ¼
κa I þ κ P I b @s

(65)

The gray and semi-gray models do not usually produce satisfactory accuracy in the prediction of gas radiation.

4.3

WSGG Model

The weighted-sum-of-gray-gases (WSGG) method was introduced by Hottel (1954) for modeling of the gas total emissivity e. In this method, the total emissivity of the gas layer of thickness L at temperature T is approximated by the following expression:

e¼

1 I b ðT Þ

ð eη I bη ðT Þ 

n X

  aj  1
e
κj L

(66)

j¼1

where κj are so-called gray gases absorption coefficients, aj are the gray gases weights, and n is the number of gray gases in the model. The physical meaning of κj and aj may be described in terms of the high-resolution absorption spectrum as follows: aj can be viewed as the energy fraction of the blackbody spectral regions in which the X effective pressure absorption coefficient is given by κ j (see Fig. 19): aj ¼ E1b Eηi, j Δηi, j, where Δηi,j are the wavenumber intervals for which kj
Δ/2  i

k  kj + Δ/2. Perhaps because no spectral database at elevated temperature was available at the time of the development of this model, the parameters κj and aj were originally determined simply as coefficients which make the series expression fit experimental data. Correlations for κ j and aj have been developed by Smith et al. (1982). For an isothermal, homogeneous layer of gas, if the sum of gray gas weights aj , j = 1 , 2 , . . . , n is equal to unity, the total intensity will approach unity with an increase of the layer thickness L. However, for molecular gases, the total emissivity reaches a constant saturated value less than unity for relatively short paths. Therefore, a clear gas weight a0 is added to the weight balance, which formally corresponds to “windows” in the absorption spectrum (the absorption

26

Radiative Properties of Gases

1113

kh (T)

kj + D 2 D

kj kj – D 2

h Ebh (T)

h

Δhi,j

Fig. 19 Weighted-sum-of-gray-gases (WSGG) model of gas absorption coefficient

coefficient for which it is assumed κ 0 = 0). Then

n P

aj ¼ 1. As noted in Hottel and

j¼0

Sarofim (1967), for many purposes, a one-gray, one-clear gas approximation is adequate (this fact is a basis for construction of the SLW-1 model which absorption spectrum consists of a one-gray gas and of one-clear gas as in Solovjov et al. (2011, 2013)). The WSGG spectral model represents the absorption coefficient κ η by a histogram spectrum with a finite number of values of the absorption coefficient κ j , j = 1 , 2 , . . . , n. As a result, the spectral absorption coefficient is reordered with respect to their weights into an increasing stepwise function (see Fig. 20). The geometrical sense of reordering is as follows. The area under the reordered absorption coefficient is equal to the Planck mean absorption coefficient κP ¼

1 E b ðT Þ

1 ð

κ η Ebη ðT Þdη  0

n X j¼1

aj κ j

(67)

1114

V. P. Solovjov et al.

k

kh

kn

kj

kj

k0 = 0

η

1

0

a0

…

…

aj

an

Fig. 20 Reordered absorption coefficient and the WSGG histogram absorption spectrum

4.4

The Spectral Group Model

The spectral group model was developed from the WSGG model by Song and Viskanta (1986). In this approach, the wavenumber intervals Δηi,j of the WSGG model are used for spectral integration of the RTE yielding the RTE for gray gas intensities Ij: @I j ¼
κj I j þ wj κ j I b @s

(68)

where the gray gas intensity and the weighing factor of spectral group j are defined as Ij ¼

X ð i

Δ i, j

I η dη,

wj ðT Þ ¼

ð 1 X Ebη ðT Þdη Eb ð T Þ i Δi, j

(69)

If the medium is non-isothermal, the intervals Δηi,j may vary, and spectral integration of RTE yields additional terms due to Leibnitz’s rule of integration. As a first approximation, one can neglect these terms. To eliminate these additional terms, the spectral intervals Δηi,j are assumed to be kept fixed at any location. Then the gray gas absorption coefficients κi and corresponding weights wi must be modified. The dependence on temperature is expressed in terms of polynomials of temperature. The spectral group model was developed for calculation of the total intensity of radiation on a side of an open flame for which temperature and

26

Radiative Properties of Gases

1115

concentration distributions were prescribed. The intensity at s = L with a two-group model is expressed as

I ðLÞ ¼

L 2 ð X j¼1

wj κj e




ÐL

κ j ds0

s

ds

(70)

0

where κ i is the absorption coefficient and wi is the corresponding weight of group j, both of which are found by minimizing the deviation of the modeled emissivity n P aj  ð1
e
κj L Þ from the true (experimentally measured) emissivities for various j¼1

gas path lengths L. Special treatment was applied for the case of gas mixtures.

4.5

WSGG RTE Model

A method similar to the spectral group model was presented by Modest, 1991, referred to as the weighted-sum-of-gray-gases (WSGG) model. It was shown that the Hottel parameters aj and j that appear as the coefficients in the RTE spectrally integrated with respect to reordered absorption coefficient are completely equivalent to RTE of spectral group model: @I j ¼
κ j I j þ aj κj I b @s

(71)

This equation, subject to appropriate boundary conditions, can be solved for the gray gas intensities Ij with the help of any arbitrary RTE solver. Then the total intensity is found by a simple summation over all gray gases: I¼

n X

Ij

(72)

j¼0

This method was demonstrated applicable only for media bounded by black walls. The correlations for aj and κj developed by Smith et al. (1982) are usually used in this method. Recently, parameters aj and κj have been calculated based on the high-resolution spectral databases HITRAN and HITEMP (França et al. 2016).

4.6

SLW, ADF, and FSK Models

The spectral line weighted-sum-of-gray-gases (SLW) method was the first global method based on a global distribution function of absorption coefficient weighted by the Planck function. That distribution function, introduced by Denison and Webb

1116

V. P. Solovjov et al.

(1994), was termed the absorption line blackbody distribution function (ALBDF). In the SLW method, the RTE for gray gases in uniform medium is the same as in the spectral group model or WSGG RTE. The weights of the gray gases aj are calculated using the ALBDF. The SLW method was extended for gray (nonblack) boundary conditions. The method can be used with any number of gray gases, and with an increase in the number of gray gases, the SLW model approaches its continuous limit which can be formulated by two alternative equations (Solovjov et al. 2014). The ADF method (Pierrot et al. 1999; Rivière et al. 1996) differs from the SLW method principally in the way the gray gas weights aj are calculated. The FSK method (Modest 2013) starts with a continuous form of the spectrally integrated RTE and then discretization is used for integration to obtain the total intensity. The relationship between the SLW, FSK, and ADF models has been presented elsewhere (Solovjov and Webb 2011).

4.6.1 Absorption Line Blackbody Distribution Function (ALBDF) The absorption line blackbody distribution function (ALBDF) will now be described. Let one denotes the gas thermodynamic state as ϕg = {Tg, Y, p}, where Tg is the gas temperature, Y is the mole fraction, and p is a total pressure. The ALBDF as a function of cross section C defines the fraction of blackbody emissive power Eb ðT b Þ ¼ σT 4b emitted at the source temperature Tb that lies in the part of the spectrum where the absorption cross section Cη(Tg) at the gas thermodynamic state ϕg is below the prescribed value C Denison and Webb (1993a, b):   F C, ϕg , T b ¼

1 Eb ð T b Þ

ð Ebη ðT b Þdη

(73)

fη:Cη ðϕg Þ 0. 2. The arbitrary value of the absorption cross section Cj in the gray gas interval ~ j
1 < Cj < C ~ j can be chosen to represent the gray gas absorption coefficient C (in prior work as the geometric mean):

κ j ¼ NYCj ¼ NY

qffiffiffiffiffiffiffiffiffiffiffiffiffiffi ~ j
1 C ~ j , j ¼ 1, . . . , n gray gases C

κ 0 ¼ 0 clear gas ðno absorptionÞ

(76)

3. Gray gas wavenumber intervals are defined by intersection of the supplemental cross sections with Cη(Tg): n   o ~ j
1 < Cη ϕg < C ~j Δj ¼ η : C n   o ~ 0 ¼ Cmin Δ0 ¼ η : Cη ϕg  C

gray gas intervals clear gas intervals ðspectral windowsÞ

(77) The intervals Δj are used for spectral integration of monochromatic RTE. 4. Define the gray gasðintensity as spectral intensity integrated over gray gas  !  ! intervals I j s, Ω ¼ I η s, Ω dη. Δj

5. Spectral integration of the monochromatic radiative transfer (Eq. 4)  ! @  ! I η s, Ω ¼
κη I η s, Ω þ κη I bη ðT Þ @s

(78)

over the gray gas intervals Δj yields the gray gas RTEs and a clear gas RTE  ! @  ! I j s,Ω ¼
κ j I j s,Ω þ aj κ j I b ðT Þj ¼ 1, . . . , n @s @  ! I 0 s,Ω ¼ 0 @s

(79) (80)

where the gray gas and clear gas weights are calculated using the ALBDF:       ~ j
1 , ϕg , T b ¼ T , a0 ¼ F C ~ 0 , ϕg , T b ¼ T aj ¼ F C~ j , ϕg , T b ¼ T
F C (81)

26

Radiative Properties of Gases

1121

The weights for boundary conditions at the wall temperature Tw are:       aj ¼ F C~ j , ϕg , T b ¼ T w
F C~ j
1 , ϕg , T b ¼ T w , a0 ¼ F C~ 0 , ϕg , T b ¼ T w

(82)

6. After the gray gas RTEs are solved subject to boundary conditions, the total intensity is found by summation over all gray gases n  ! X  ! I s, Ω ¼ I j s,Ω

(83)

j¼0

4.6.3

The SLW Method for the Non-isothermal, Nonhomogeneous Medium: The Reference Approach

Spectral Integration of RTE in Nonuniform Media In nonuniform media, the spectrum of the gas absorption cross section Cη(ϕ) varies with the spatial location s if the thermodynamic state ϕ(s) depends on location. ~ j with the Consequently, the intersection of fixed supplemental cross sections C absorption cross sections ϕ1 = ϕ(s1) and ϕ2 = ϕ(s2) at different spatial locations s1 and s2 produces different gray gas wavenumber intervals Δj(s1) 6¼ Δj(s2). Therefore, in the spectral integration of the RTE, interchange of differentiation with respect to the spatial variable s and integration with respect to the wavenumber η over the gray gas intervals Δj(s) = {[ai,j(s), bi,j(s)]} which depend on location s, due to variation of the limits of integration will produce so-called Leibnitz terms: ð Δj ðsÞ

@ @ I η ðs, ΩÞdη ¼ @s @s

ð I η ðs, ΩÞdη Δj ðsÞ Leibnitz terms

zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{  X
 dbi, j ðsÞ
 dai, j ðsÞ
I η s, η ¼ ai, j ðsÞ, Ω þ I η s, η ¼ bi, j ðsÞ, Ω ds ds i This fact was previously observed in the spectral group model (see Sect. 4.4) by Song and Viskanta (1986). It is practically impossible to take into account these numerous Leibnitz terms, but their neglect can yield a significant error in prediction of radiation transfer in media with high nonuniformity. To eliminate Leibnitz terms from the spectrally integrated RTE, the gray gas intervals must be fixed for all spatial locations, yielding Δj(s) = Δj = const ð Δj

@ @ I η ðs, ΩÞdη ¼ @s @s

ð I η ðs, ΩÞdη þ Leibnitz terms Δj

1122

V. P. Solovjov et al.

The fixing of gray gas intervals can be done by choosing different supplemental ~ j ðsÞ at different locations in such a way that they define the same cross sections C spectral intervals Δj = const for all locations (see Fig. 24). In the SLW method, it is performed with the help of the reference approach under the assumption of correlated spectra. (Originally, this assumption was called as “ideal spectra behavior” in Denison and Webb 1994.)

Correlated Spectrum Assumption (“Ideal Spectra Behavior”) In the SLW method, one assumes that for the gas absorption spectrum at some chosen reference state Cη(ϕref) and at any arbitrary state Cη(ϕloc), the following relationship is true: for prescribed value of the reference cross section Cref, there exists Cloc such that the wavenumber intervals



η : Cref < Cη ðϕref Þ ¼ η : Cloc < Cη ðϕloc Þ

(84)

are the same. Because these intervals are equal, the values of the ALBDF at Cref and Cloc for the fixed blackbody source temperature Tb = Tref should also be equal:     F Cloc , ϕloc , T b ¼ T ref ¼ F Cref , ϕref , T b ¼ T ref

(85)

This implicit equation defines the local absorption cross sections, which generate the same spectral intervals by intersection with the absorption cross section at the local state (see Fig. 24).

The SLW Reference Approach The reference approach is developed for spectral integration of RTE in nonuniform medium. It keeps the spectral intervals of integration fixed at all spatial locations of the medium (see Fig. 24). Outline of the SLW Reference Approach in Nonuniform Media 1. Choose the reference state: ϕref = {Tref, Yref, pref} (usually the spaced averaged values). ~ ref , j ¼ 0, 1, 2, . . . , n. 2. Choose the set of the reference supplemental cross sections: C n jloc o 3. Find the set of the local supplemental cross sections C~ j by solving the implicit equations for j = 0,1,2,. . .,n:  ref   loc  ~ , T loc , T b ¼ T ref ~ , ϕref , T b ¼ T ref ¼ F C F C j j

(86)

1

)

aloc j

0 1

(

)

F C,Tg=Tloc,Tb=Tref

(

F C,Tg=Tref ,Tb=Tref

)

0

C

~ref C j–1

~ C jref

~loc C j–1

~ C jloc

C

Δj

Cη (Tloc )

)

Ebη (Tb = Tref )

Ebη (Tb = Tloc )

Cη (Tref

η

η

C ref j

C loc j

Radiative Properties of Gases

Fig. 24 The SLW reference approach based on correlated spectrum assumption

F

(

F C,Tg=Tref ,Tb=Tloc

C

26 1123

1124

V. P. Solovjov et al.

4. Use the local supplemental cross sections to calculate the local gray gas absorption coefficients: κ j ðsÞ ¼ N loc Y loc Cloc j ¼ N ðsÞY ðsÞ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ~ loc C~ loc C j j
1

(87)

5. Calculate the local weights of the gray gas absorption coefficients:  ref   ref  ~ , ϕref , T b ¼ T ðsÞ
F C ~ , ϕref , T b ¼ T ðsÞ ¼ F C aj ðsÞ ¼ aloc j j
1 j

(88)

The other way to calculate the local gray gas weighs is with the help of the found local supplemental cross sections (that is the main difference between the SLW and the ADF methods):  loc   loc  ~ ~ aj ðsÞ ¼ aloc j ¼ F C j , ϕloc , T b ¼ T ðsÞ
F C j
1 , ϕloc , T b ¼ T ðsÞ

(89)

6. RTE in non-isothermal, nonhomogeneous medium: @I j ðsÞ ¼
κ j ðsÞI j ðsÞ þ κj ðsÞaj ðsÞI b ½T ðsÞ @s

(90)

7. Boundary conditions (gray diffuse reflective and emitting boundaries at Tw):     ~ j , ϕref , T b ¼ T w
F C~ j
1 , ϕref , T b ¼ T w aj ð T w Þ ¼ F C

(91)

Recently, Andre et al. (2017) have developed the new co-monotonic global model, and as its particular case, the Rank Correlated SLW model (Solovjov et al. 2016), which do not require specification of the gas reference thermodynamic state.

4.7

Continuous Limit of the SLW Method in Uniform Media: the FSK Model

One may define the following forms of the gray gas intensity in the medium at ϕ = {T, Y, p} as

26

Radiative Properties of Gases

1125

  I j ðs, ΩÞ ¼ I C~ j , s, Ω  ΔCj

(92)

  I j ðs, ΩÞ ¼ I F~j , s, Ω  ΔFj

(93)

where the finite increments in absorption cross section and ALBDF are defined as ~ j
1 ~j
C ΔCj ¼ C

    ~ j , ϕ, T
F C~ j
1 , ϕ, T ΔFj ¼ F C

(94)

The gray gas RTEs may then be written as    @ ~ ~ j , s, Ω þ ΔFj NYCj I b ðT Þ I C j , s, Ω ¼
NYCj I C @s ΔCj       @ ~ I F j , s, Ω ¼
NYC Fj , ϕ, T I F~j , s, Ω þ NYC Fj , ϕ, T I b ðT Þ @s

(95)

(96)

The total intensity is calculated by the summation I ðs, ΩÞ ¼

n X   I C~ j , s, Ω  ΔCj

(97)

j¼0

I ðs, ΩÞ ¼

n X   I F~j , s, Ω  ΔFj

(98)

j¼0

In the limit when the number of gray gases approaches infinity, the finite increment RTEs become equations in terms of intensity I(C,s,Ω) as a continuous function of the variable C Solovjov et al. (2014) @ @FðC, ϕ, T Þ I ðC, s, ΩÞ ¼
NYCIðC, s, ΩÞ þ NYCIb ðT Þ @s @C

(99)

or as an equation for intensity I(F,s,Ω) as a function of the variable F @ I ðF, s, ΩÞ ¼
NYCðF, ϕ, T ÞI ðF, s, ΩÞ þ NYCðF, ϕ, T ÞI b ðT Þ @s

(100)

The second equation, if written in terms of absorption coefficient κ(F, ϕ, T ) = NYC(F, ϕ, T ), is known as the FSK model in uniform medium (Modest 2013): @ I ðF, s, ΩÞ ¼
κ ðF, ϕ, T ÞI ðF, s, ΩÞ þ κðF, ϕ, T ÞI b ðT Þ @s

(101)

1126

V. P. Solovjov et al.

Following integration, one obtains the total intensity 1 ð

I ðs, ΩÞ ¼

I ðC, s, ΩÞdC

(102)

I ðs, ΩÞ ¼ I ðF, s, ΩÞdF

(103)

0

ð1 0

Solution of the continuous RTE can be found analytically, and then integration can be performed using the integral quadratures.

4.7.1 SLW Modeling of Gas Mixtures Most practical problems involving gas radiation are in the media which consists of a mixture of different species. In this case, the effective absorption spectrum of the gas mixture, which includes spectral lines from each gas specie and absorption coefficient of gas mixture κη,mix, is a sum κ η, mix ¼ κη, 1 þ . . . þ κη, m þ . . . þ κ η, M where κ η,m are absorption coefficients of individual species m that are defined following Eq. 46 as κη, m ðϕÞ,

     2 1 mol m  C ð T, Y, p Þ ¼ N ðT, pÞ  Y m η , m 3 m m mol

with mole fraction Ym = pm/p, where pm is a partial pressure of specie m and p is a total pressure of gas mixture. In SLW modeling, the gas mixture can be treated as a single gas with absorption coefficient κ η,mix and absorption cross section Cη,mix, which are defined through the relationship   κ η, mix ¼ NCη, mix ¼ N Y 1 Cη, 1 þ . . . þ Y m Cη, m þ . . . þ Y M Cη, M In this case, the ALBDF function must either be calculated directly for the gas mixture absorption cross section Cη,mix, or it can be approximated with the help of ALBDFs of individual species (Fig. 25). The other approach developed in SLW modeling of gas mixtures (Denison and Webb 1995a, b, c; Solovjov and Webb 2000) is a repeated spectral integration of the RTE, which is based on inclusion of additional indices to the spectral parameters of RTE. These multiple indices denote the separate subdivision of the absorption cross sections of different species into gray gases. For simplicity, one may demonstrate this approach for the case of a binary gas mixture.

26

Radiative Properties of Gases

1127 10

10

Cmix,h =Y1C1,h +Y2C2,h

Y1C1,h 8

8

6

6

Y2C2,h

y

y 4

4

2

2

C

0

0 0

2

4

6

8

10 eta

12

14

16

18

0

2

4

6

8

10 eta

12

14

16

18

Fig. 25 Absorption cross section of binary gas mixture

Double Integration for Binary Gas Mixtures For a mixture of two gases, subdivision into gray gases for each component can be performed separately: n   o n   o ~ j
1 < Cη, 1 ϕg < C ~ j , j ¼ 1, 2, . . . , n and Δ1 ¼ η : Cη, 1 ϕg < C ~0 , Δ1j ¼ η : C 0 n   o n   o ~ i
1 < Cη, 2 ϕg < C ~ i , i ¼ 1, 2, . . . , n and Δ2 ¼ η : Cη, 2 ϕg < C ~0 , Δ2i ¼ η : C 0 where n is the number gray gases for each subdivision. Integration of spectral RTE over all intersections of gray gases Δ1j \ Δ2i yields the RTE for double-index intensities  !  ! @ I i, j s, Ω ¼
κ i, j I i, j s,Ω þ ai aj κi, j I b ðT Þ i ¼ 0, 1, . . . , n, j ¼ 0, 1, . . . , n @s (104) where the double-indexed gray gas absorption coefficient is κi, j = N(Y1Cj + Y2Ci) with gas mixture molar density N and mole fractions of gas species are ( p1 and p2 are partial pressure of gas species) N¼

p p p , Y1 ¼ 1 , Y2 ¼ 2 Ru T g p p

(105)

and the weights ai and aj are calculated with the help of the ALBDF of corresponding gas species by Eqs. 81–82. The total intensity can be found by summation with respect to both indices

1128

V. P. Solovjov et al. n X n  ! X I s, Ω ¼ i¼0 j¼0

ð

n X n  !  ! X I η s,Ω dη ¼ I ij s,Ω

Δ1j \Δ2j

(106)

i¼0 j¼0

This straightforward approach requires solution of (n + 1)2 gray gas equations for the RTE (including the clear gas). The method can be extended to gas mixtures with arbitrary number of species M. In this case, number of equations will be (n + 1)M. Mixture of Gases Treated as a Single Gas In a more efficient approach, the gas mixture may be treated as a single gas with a spectrum combined from the lines of individual species. Then the effective absorption coefficient of the gas mixture can be written as κmix, η ¼ κ 1, η þ κ 2, η þ    þ κm, η  p  ¼ Y 1 C1, η þ Y 2 C2, η þ    þ Y 2 CM, η Ru T g

(107)

As a result, the gas mixture is treated as a single gas with the effective absorption coefficient κmix, η ¼ NCmix, η

(108)

and with the mixture absorption cross section Cmix, η ¼ Y 1 C1, η þ Y 2 C2, η þ    þ Y m Cm, η

(109)

where N = p/RuTg is the gas mixture molar density, p is the gas mixture total pressure, pi are the gas species partial pressures, and Yi = pi/p the gas species mole fractions.   If the ALBDF of the gas mixture Fmix ðCÞ ¼ FCmix, η C, T g , T b , Y 1 , Y 2 with cross section Cmix,η is known, the traditional SLW model for a single gas can be applied. The approximation can be used to express the gas mixture ALBDF with the help of the ALBDF of individual species. If the absorption spectra of individual species are assumed to be uncorrelated (which is considered to be physically reasonable, Modest 2013), then this derivation yields the so-called multiplication approach (Solovjov and Webb 2000):   FY 1 C1, η þY 2 C2, η þ...þY M CM, η C, T g , T b , Y 1 , . . . , Y M    C C C ¼ F C1, η , T g , T b , Y 1 FC2, η , T g , T b , Y 2   FCm, η , Tg, Tb, Ym Y1 Y2 Ym (110) The other approach to treating the mixture as a single gas requires an additional restriction of constant ratio of the molar fractions of individual species. Then the

26

Radiative Properties of Gases

1129

ALBDF of the mixture is expressed in the form of convolution integral. For two species, such a convolution integral is written as (Solovjov and Webb 2000) 



F2ð¼1

FY 1 C1, η þY 2 C2, η C, T g , T b , Y 1 , Y 2 ¼

F C1, η F2 ¼0

  C
ð1
r ÞC0 dFC2, η ðC0 Þ r

(111)

The gray gas absorption coefficients of the gas mixture treated as a single gas require just a single partition and are calculated as κj ¼ N

qffiffiffiffiffiffiffiffiffiffiffiffiffiffi ~ j
1 C ~j C

(112)

with the gray gas weights of the gas mixture     ~ j , T g , T b , Y 1 , . . . , Y M
Fmix C ~ j
1 , T g , T b , Y 1 , . . . , Y M aj ¼ Fmix C

(113)

(Note that the mole fractions are not involved in the expressions.) Using the multiplication approach, the gray gas weights can be calculated as: ~ ~ Cj Cj a j ¼ F C1, η , T g , T b , Y 1   FCm, η , Tg, Tb, Ym Y1 Ym ~ ~ C j
1 C j
1
F C 1, η , T g , T b , Y 1   FCm, η , Tg, Tb, Ym Y1 Ym

(114)

If the gas mixture is treated as a single gas, the number of RTE equations in SLW modeling is n + 1 for any number of species in the gas mixture. The accuracy of the multiplication approach, in general, is not lower than the accuracy of repeated integration approach. The treatment of gas mixture as a single gas is very efficient for multicomponent gas mixtures which also involve non-gray particles. Non-scattering particles such as soot can be treated in the same manner as the gas species and can be incorporated into SLW modeling (Solovjov and Webb 2001, 2005). Direct calculation of the ALBDF for a mixture of H2O, CO2, CO, and soot for wide range of possible values of species mole fractions was performed in Wang et al. (2016), and data are presented in a form of a look-up table.

4.7.2 SLW Modeling of Boundary Conditions The most typical case of the spatial domain for combustion systems, which involve gas radiation, is when the gaseous medium is bounded by emitting and reflecting opaque walls. If the boundary surface is assumed to emit  and reflect diffusely, then ! the boundary condition for spectral radiation intensity I η rw , Ω at the boundary point indicated by the position vector r = rw is described by the equation (Howell et al. 2011):

1130

V. P. Solovjov et al. diffusively reflected incident

radiosity zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{ emitted ð  ! ! zfflfflffl}|fflfflffl{ zfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflffl{ πI η ðrw Þ ¼ πeðrw ÞI bη ðrw Þ þ ρðrw Þ I η rw , Ω n  Ω dΩ

(115)

Ω¼2π

!

n Ω >0

where e(rw) is the total wall emissivity and ρ(rw) = 1
e(rw) is the total wall reflectivity.

n

  Iη rw , Ω Ω

(

)

diffusively emitted radiation

surface of an opaque solid

rw 0

The SLW method allows modeling in the case of gray boundaries (in contrast to the spectral group model or WSGG RTE model, see Sects. 4.4 and 4.5 which can be formulated for the case of black walls only). Spectral integration of the boundary condition with gray wall properties e(rw) and ρ;(rw) over the same gray gas spectral intervals Δj defined by Eq. 77, which are used for spectral integration of RTE in Sect. 4.6.2 yields the gray gas boundary conditions: I j ðrw Þ ¼ aj ðrw Þeðrw ÞI b ðT ðrw ÞÞ þ

ρð r w Þ π

ð

 ! ! I j r,Ω n  Ω dΩ

(116)

Ω¼2π

!

n Ω >0

Because the wall temperature is Tw, the wall gray gas weights are calculated with the help of ALBDF with the blackbody source temperature at Tw:     ~ j , ϕg , T b ¼ T w
F C ~ j
1 , ϕg , T b ¼ T w aj ð r w Þ ¼ F C   ~ 0 , ϕg , T b ¼ T w a0 ð r w Þ ¼ F C

26

Radiative Properties of Gases

1131

4.7.3

Plane-Parallel Layer Between Two Gray Emitting and Diffusively Reflecting Walls Consider propagation of radiation in an absorbing and emitting uniform gaseous layer at temperature T between two parallel opaque emitting and reflective walls at temperature T0 and TL. Intensity of radiation Iη(s,Ω) along the path length s in a ! direction Ω ¼ ðφ, θÞ for axisymmetric (independent of azimuthal angle φ) planeparallel layer can be described in the following way. Define the directional cosine μ = cosθ as a positive variable μ >0 such that  μ¼

μ for 0 < θ < π=2
μ for π=2 < θ < π

(117)

and define forward and backward intensities I  η ðx, μÞ in the positive and negative directions:  I η ðx, θÞ ¼

Iþ η ðx, μÞ, I
η ðx, μÞ,

μ > 0, μ > 0,

for 0 < θ < π=2 for π=2 < θ < π

(118)

where x-axis is across the layer and dx = μds (Fig. 26). Then the gray gas RTEs for axisymmetric plane-parallel layer, 0< x < L and μ >0, can be written as

Iη− ( x,μ)

μ= 0 μ > 0 μ >0

Iη+ ( x,μ )

μ =1

μ =1

( )

I λ s,Ω Ω

μ =0 ds

θ

μ = cos θ x

0

Fig. 26 Plane-parallel layer

x

dx

L

1132

μ

V. P. Solovjov et al.

@I þ j ðx, μÞ ¼
κj I þ j ðx, μÞ þ κ j aj I b @x


μ

I b ¼ I b ðT Þ ¼ σT 4 =π

@I
j ðx, μÞ ¼
κ j I
j ðx, μÞ þ κ j aj I b @x

(119) (120)

Clear gas RTEs (Fig. 27): @I þ 0 ðx, μÞ ¼0 @x

(121)

@I
0 ðx, μÞ ¼0 @x

(122)

Boundary conditions for gray walls with total emissivities e0,eL are written as Iþ j ð0, μÞ

¼

ð0Þ e 0 aj I b ð T 0 Þ

ð1

þ 2ρ0 I
j ð0, μÞμdμ

(123)

0

I
j ðL, μÞ

¼

ðLÞ e L aj I b ð T L Þ

ð1

þ 2ρL I þ j ðL, μÞμdμ 0

x=0

ε 0 I bη (T 0)

wall emission

1

Iη+ (0, μ) = ε0 I bη (T0 ) + 2 ρ0 Iη− (0, μ ) μ d μ 0

μ =1

Iη− ( 0,μ )

Fig. 27 Boundary condition of plane-parallel layer at x = 0

diffuse reflection of incident radiation

(124)

26

Radiative Properties of Gases

1133

where the weights are calculated as: ð0Þ

aj

    ~ j
1 , ϕ, T b ¼ T 0 ~ j , ϕ, T b ¼ T 0
F C ¼F C   ð0Þ ~ 0 , ϕ, T b ¼ T 0 a0 ¼ F C

ð LÞ

aj

    ¼ F C~ j , ϕ, T b ¼ T L
F C~ j
1 , ϕ, T b ¼ T L   ð LÞ ~ 0 , ϕ, T b ¼ T L a0 ¼ F C

(125) (126) (127)

(128)

and ρ0 = 1
e0 and ρL = 1
eL are the total wall reflectivities. Analytical Solution in Terms of Integral Exponential Functions En(x) (Solovjov and Webb 2008) The gray gas directional wall fluxes j = 0,1,. . .,n (including clear gas, j = 0) are found to be: h i   h i   ð0Þ ðLÞ aj I b þ 2e0 aj I b ðT 0 Þ
aj I b ðT Þ E3 κ j L þ 4ρ0 eL aj I b ðT L Þ
aj Ib ðT Þ E23 κ j L   Fþ j ðLÞ ¼ π 1
4ρ0 ρL E23 κ j L

(129) h i   h i   ðLÞ ð0 Þ aj Ib þ 2eL aj Ib ðT L Þ
aj I b ðT Þ E3 κ j L þ 4ρL e0 aj I b ðT 0 Þ
aj I b ðT Þ E23 κ j L   F
j ð0Þ ¼ π 1
4ρL ρ0 E23 κ j L

(130) Directional gray gas intensities:   κ j 1 ð0Þ ð0Þ
ρ Iþ ð x, μ Þ ¼ e a I
a I þ F ð 0 Þ e
μ ðx
0Þ þ aj I b 0 j j b 0 j j b π   κ j 1 ð LÞ ð LÞ þ ρ ð x, μ Þ ¼ e a I
a I þ F ð L Þ e μ ðx
LÞ þ aj I b I
L j b L j j j b π

(131)

(132)

Directional gray gas radiative fluxes: h ρ0
i   ð0Þ Fþ F ð0Þ E3 κj x þ πaj I b ðT Þ ð x Þ ¼ 2π e a I ð T Þ
a I ð T Þ þ (133) 0 b 0 j b j j π j h  ρL þ i
ðLÞ Fj ðLÞ E3 κj ðL
xÞ þ πaj I b ðT Þ (134) F
ð x Þ ¼ 2π e a I ð T Þ
a I ð T Þ þ L b L j b j j π

1134

V. P. Solovjov et al.

Gray gas net radiative fluxes: h ρ0
i   ð0Þ
F ð 0Þ E3 κ j x ð x Þ
F ð x Þ ¼ 2π e a I ð T Þ
a I ð T Þ þ Fj ðxÞ ¼ Fþ 0 j b 0 j b j j π j h i
(135)  ρ ð LÞ
2π eL aj I b ðT L Þ
aj I b ðT Þ þ L Fþ j ðLÞ E3 κ j ðL
xÞ π Gray gas divergence of the net radiative fluxes j = 1,. . .,n (clear gas does not contribute): i   nh @ ρ ð0Þ Fj ðxÞ ¼ 2πκj e0 aj I b ðT 0 Þ
aj I b ðT Þ þ 0 F
j ð0Þ E2 κ j x @x π (136) h o ρL þ i
ð LÞ þ eL aj I b ðT L Þ
aj I b ðT Þ þ Fj ðLÞ E2 κj ðL
xÞ π

Qj ðxÞ ¼


The total quantities can be found by summation over all gray gases:

I  ðx, μÞ ¼

n X

I j ðx, μÞ

(137)

Fj ðxÞ

(138)

Qj ðxÞ

(139)

j¼0

Fð x Þ ¼

n X j¼0

QðxÞ ¼

n X j¼1

Example

Consider a plane-parallel layer of thickness L = 0.75 m filled with homogeneous isothermal water vapor at total temperature P = 1.0 atm, temperature T = 750 K, and mole fractionY H2 O ðxÞ ¼ 0:2. The boundaries are assumed to be gray with total emissivity e0 = eL = 0.7 at T0 = TL = 400K. The SLW model with a number of gray gases n = 25 is used for prediction of the total net radiative flux and total divergence of the net radiative flux. The ALBDF is calculated using correlations of Pearson et al. (2014). The gray gas absorption coefficients and corresponding weights are shown in Fig. 28a, b. Prediction of the total net radiative flux according to Eqs. 133, 134, 135, and 138 and prediction of the total divergence of net radiative flux according to Eqs. 136 and 139 are shown in Fig. 28b, c. Comparison with LBL solution based on the same spectral data which was used for evaluation of ALBDF demonstrates high accuracy of the SLW method.

26

a

Radiative Properties of Gases

1135

b

Kj, 1 m

aj

0.1

.1e3

aj

.1e2

0.08

1.

0.06

Kj

.1

0.04

.1e−1 0.02 .1e−2 0 0

c

0.2

0.4

0.6

0.8

1

0

0.8

1

-10

2

-15 Q(x), kW/m 3

2

0.6

-5

3

F(x), kW/m

0.4

d

4

1 0 -1

-20 -25 -30

-2

LBL

-3 -4

0.2

LBL

SLW

0

0.1

0.2

0.3

0.4

x, m

0.5

0.6

0.7

-35 -40

SLW

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

x, m

Fig. 28 (a) Gray gas absorption coefficients; (b) gray gas weights; (c) total net flux; (d) total divergence

4.7.4

Recent Advances (and Deficiencies) of Global Methods in Gas Radiation Modeling The closely related to each other methods such as WSGG, SLW, FSK, and ADF are recognized as well-established methods of global modeling in gas radiation. Since their introduction in the middle of 1990s, numerous improvements and enhancements of these methods have been developed, and they are still under development in pursuing of satisfaction of growing demands of comprehensive numerical modeling of heat mass transfer. The domain of modifications can be subdivided into three main categories: 1. Continuous update of the spectroscopic databases, radiation parameters, and distribution functions (ALBDF). The very first versions of the SLW method were based on room temperature spectroscopic data and their own extension of

1136

V. P. Solovjov et al.

gas absorption spectra to higher temperatures and generation of the so-called hot lines dealing with the tens of thousands of spectral lines. The recent model parameters are based on enormous progress in spectroscopy and deal with gas spectra which contain tens of millions of spectral lines in HITRAN and HITEMP spectroscopic databases. That leads to necessity of improvements of the mathematical technique dealing with enormous spectroscopic data and more sophisticated physical models of gas absorption spectra. The efforts are in improvement of accuracy, speed, amount of preprocessed data, and more efficient and accurate representation of the distribution functions in the form of mathematical correlations, of look-up tables, and of compilation of distribution functions for gas mixtures. The ultimate objective would be an easy and fast access to look-up table of the ALBDF data of LBL accuracy for any combination of gases and soot at arbitrary thermodynamic state of practical interest. The first steps in this direction are made by Modest and Haworth (2016) for ALBDF and Franca (2016) for WSGG parameters. In parallel of these developments, new engineering problems may involve molecular species at high temperature whose spectroscopic parameters are still not well known: it will be necessary to develop spectroscopic databases for these species and, probably, dedicated models. 2. Enhancement of effectiveness of the global models by construction of more compact models with reduced number of gray gases without loss of accuracy. These models are developed to reduce the number of gray gases in the SLW method when computational time for solution of radiation problem is critical, especially when radiation is only one part in the comprehensive modeling of heat and mass transfer. It can often be done with an acceptable loss of accuracy when it is better to have an approximate solution than to neglect radiation entirely or reduce it to a gray model. The SLW-1 model is a minimal SLW model, which consists of a single gray gas and a single clear gas, and is thus the most efficient version of gray gas models (Solovjov et al. 2011a, b). The justification of such a reduction of the number of gray gases was already mentioned in the early work of Hottel and Sarofim (1967). Despite the ultimate simplicity, the SLW-1 model can be very accurate if an optimized subdivision into gray gas and clear gas is performed Solovjov et al. (2013). The compact SLW model is a version of generalized SLW model whereby partition into gray gases is performed more efficiently with the help of the Gauss-Legendre integral quadratures by simultaneous partition of C and F variables and application of direct and inverse ALBDF (Solovjov et al. 2016). The number of gray gases can be reduced to three to five without noticeable loss of accuracy (Solovjov et al. 2016b). The main advantage of this approach is that it does not need iterative solution of implicit equations for local gray gas supplemental cross sections (Eq. 86 in the SLW reference approach). If an optimization technique is used, then excellent results can be obtained with two to three gray gases (Denison and Webb 1994). However, optimization, especially if sophisticated methods such as conjugate gradient methods are used, can be time-consuming, reducing the efficiency of the method.

26

Radiative Properties of Gases

1137

3. Modeling in nonuniform media with high temperature and concentration gradients is, probably, the most challenging problem in global methods. Simple neglect of Leibnitz terms as it was described in Sect. 4.6.3 can yield significantly inaccurate prediction. Significant progress is made recently by development of Rank Correlated SLW model which does not require specification of the gas reference thermodynamic state for its construction (Solovjov et al. 2017). This method still preserves SLW method as an efficient simple engineering approach. There is still a search for new approaches in gas radiation modeling. Multispectral framework (MSF) is a mapping technique based on functional data analysis. This method allows building spectral intervals over which gas spectra in distinct states can be reasonably considered as truly scaled (André et al. 2011, 2014). This framework leads to several variations of the SLW method, which try to improve correlation by artificial splitting of gas absorption spectrum into better correlated fictitious gases or groups. Among them are fictitious gases ADF (Rivière et al. 1996), multiscale FSK (MSFSK), multigroup FSK (MGFSK), and their combination MSMGFSK models by Modest et al. (2013). Disadvantage of these methods is in more complicated models, which significantly increase computational time making them not suitable for application in comprehensive heat and mass transfer modeling. However, development and improvement of existing methods based on assumption of correlated and scaled spectra is now reaching its limitations, since the real spectra are not fully either correlated or scaled. New approaches based on advanced statistical theories which take into account real partial correlation of gas absorption spectra become more promising for further progress in gas radiation modeling (André 2017).

5

Conclusions and Perspectives of Gas Radiation

Although many methods are already available for gas radiation modeling, it is still presently an active research field. Spectral databases are regularly updated and so approximate model databases also evolve. Further, additional species of engineering interest need data generated. With the recent and continuous advances in computational resources, more and more users are interested in approaches that allow treating non-gray boundaries or non-gray particles. The field on wide band approaches (not represented in details in the present review) is likely to know a considerable interest and specific developments in the future years (Modest 2013). Except in emission/absorption spectroscopy studies, gas radiation is not the only heat transfer mechanism involved in a given problem: chemical reactions, turbulent flows are usually coupled with radiation. There is some kind of “coupling” between these developments and the need for new methods of gas radiation (for instance, a very accurate chemical model requires highly accurate determination of local temperatures and thus of local radiative powers).

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References André F (2016a) The ‘-distribution method for modeling non-gray absorption in uniform and non-uniform gaseous media. J Quant Spectrosc Radiat Transf 179:19–32 André F (2016b) A polynomial chaos approach to narrow band modeling of radiative heat transfer in non-uniform gaseous media. J Quant Spectrosc Radiat Transf 175:17–29 André F (2017) Symmetry issues in the ‘-distribution method for modeling of non-gray absorption in uniform and non-uniform gaseous medium. J Quant Spectrosc Radiat Transf 190:78–87 André F, Vaillon R (2007) The k-moment method for modeling the blackbody weighted transmission function for narrow and wide band radiative properties of gases. J Quant Spectrosc Radiat Transf 108:1–16 Andre F, Vaillon R (2008) The spectral-line moment-based (SLMB) modeling of the wide band and global blackbody-weighted transmission function and cumulative distribution function of the absorption coefficient in uniform gaseous media. J Quant Spectrosc Radiat Transf 109:2401–2416 André F, Vaillon R (2012) Generalization of the k-moment method using the maximum entropy principle. Application to the NBKM and full spectrum SLMB gas radiation models. J Quant Spectrosc Radiat Transf 113:1508–1520 André F, Vaillon R, Galizzi C, Guo H, Gicquel O (2011) A multi-spectral reordering technique for the full spectrum SLMB modeling of radiative heat transfer in nonuniform media. J Quant Spectrosc Radiat Transf 112:394–411 André F, Hou L, Roger M, Vaillon R (2014) The multispectral gas radiation modeling: a new theoretical framework based on a multidimensional approach to k-distribution methods. J Quant Spectrosc Radiat Transf 147:178–195 Andre F, Solovjov VP, Lemonnier D, Webb BW (2017) Comonotonic global spectral models of gas radiation in non-uniform media based on arbitrary probability measures. Appl Math Model 50:741–754 Berk A (2013) Voigt equivalent widths. J Quant Spectrosc Radiat Transf 118:102–120 Bernath PF (2005) Spectra of atoms and molecules. Oxford University Press, New York Centeno FR, Brittes R, França FHR, Ofodike, Ezekoye OA (2015) Evaluation of gas radiation heat transfer in a 2D axisymmetric geometry using the line-by-line integration and WSGG models. J Quant Spectrosc Radiat Transf 156:1–11 Denison MK, Webb BW (1993a) A spectral line based weighted-sum-of-gray-gases model for arbitrary RTE solvers. ASME J Heat Transfer 115:1004–1012 Denison MK, Webb BW (1993b) An absorption-line blackbody distribution function for efficient calculation of total gas radiative transfer. J Quant Spectrosc Radiat Transf 50:499–510 Denison MK, Webb BW (1994) k-distributions and weighted-sum-of-gray-gases – a hybrid Model. ASME J Heat Transfer 2:19–24 Denison MK, Webb BW (1995a) The spectral line based weighted-sum-of-gray-gases model in non-isothermal non-homogeneous media. ASME J Heat Transfer 117:359–365 Denison MK, Webb BW (1995b) Development and application of an absorption-line blackbody distribution function for CO2. Int J Heat Mass Transfer 38:1813–1821 Denison MK, Webb BW (1995c) The spectral-line weighted-sum-of-gray-gases model for H2O/ CO2 mixtures. ASME J Heat Transfer 117:788–792 Domoto GA (1974) Frequency integration for radiative transfer problems involving homogeneous non-gray gases: the inverse transmission function. J Quant Spectrosc Radiat Transf 14:935–942 Godson WL (1953) The evaluation of infrared radiative fluxes due to atmospheric water vapour. Q J R Meteorol Soc 79:367–379 Goody RM, Yung YL (1989) Atmospheric radiation. Oxford University Press, New York Harris CH, Bertolucci MD (1978) Symmetry and spectroscopy. Oxford University Press, New York Herzberg G (1950) Molecular spectra and molecular structure. D. Van Nostrand, Princeton

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Hill C, Gordon IE, Kochanov RV, Barrett L, Wilzewski JS, Rothman LS (2016) HITANonline: an online interface and the flexible representation of spectroscopic data in the HITRAN database. J Quant Spectrosc Radiat Transf 177:4–14 http://albdf.byu.edu. Accessed 3 Aug 2016 http://www.cfa.harvard.edu/hitran/. HITRAN and HITEMP spectral databases website. Accessed 3 Aug 2016 Hottel HC (1954) Radiant heat transmission. In: McAdams WH (ed) Heat transmission. McGrawHill Book Company, New York Hottel HC, Sarofim AF (1967) Radiative transfer. McGraw-Hill, New York Howell JR, Siegel R, Mengüç MP (2011) Thermal radiation heat transfer, 5th edn. CRC Press, New York Howell JR, Mengüç MP, Siegel R (2016) Thermal radiation heat transfer, 6th edn. CRC Press, New York Kochanov RV, Gordon IE, Rothman LS, Wcislo P, Wilzewski JS (2016) HITRAN Application Programming Interface (HAPI): a comprehensive approach to working with spectroscopic data. J Quant Spectrosc Radiat Transf 177:15–30 Lacis A, Oinas V (1991) A description of the correlated k-distribution method for modeling nongray gaseous absorption, thermal emission, and multiple scattering in vertically inhomogeneous atmospheres. J Geophys Res 96:9027–9063 Letchworth KL, Benner DC (2007) Rapid and accurate calculation of the Voigt function. J Quant Spectrosc Radiat Transf 107:173–192 Levi Di Leon R, Taine J (1986) A fictive gas-method for accurate computations of low-resolution IR gas transmissivities: application to the 4.3 μm CO2 band. Rev Phys Appl 21:825–531 Levine IN (1975) Molecular spectroscopy. Wiley, New York Lindquist GH, Simmons FS (1972) A band model formulation for very nonuniform paths. J Quant Spectrosc Radiat Transf 12:807–820 Liu F, Smallwood GJ, Gülder ÖL (2001) Application of the statistical narrow-band correlated-k method to non-grey radiation in CO2-H2O mixtures: approximate treatment of overlapping bands. J Quant Spectrosc Radiat Transf 68:401–417 Ludwig CB, Malkmus W, Reardon JE, Thomson JAL (1973) Handbook of infrared radiation from combustion gases. Technical report NASA SP-3080, Washington, DC Malkmus W (1967) Random Lorentz band model with exponential-tailed S-1 line intensity distribution function. J Opt Soc Am 57:323–329 Mlawer EJ, Payne VH, Moncet J-L, Delamere JS, Alvarado MJ, Tobin DC (2012) Development and recent evaluation of the MT-CKD model of continuum absorption. Philos Trans R Soc A 370:2520–2556 Modest MF (2013) Radiative heat transfer, 3rd edn. Academic, ASME, New York Modest MF, Haworth DC (2016) Radiative heat transfer in turbulent combustion systems. Theory and applications. Springer, New York Modest MF, Zhang H (2000) The full-spectrum correlated-k distribution and its relationship to the weighted-sum-of-gray-gases method. In: Proceedings of 2000 IMECE HTD-366-1 Orlando, New York, pp 75–84 Pearson JT (2013) The development of updated and improved SLW model parameters and its application to comprehensive combustion prediction. PhD dissertation, Brigham Young University, Provo, Utah, USA Pearson JT, Webb BW, Solovjov VP, Ma J (2014) Efficient representation of the absorption line blackbody distribution function for H2O, CO2, and CO at variable temperature, mole fraction, and total pressure. J Quant Spectrosc Radiat Transf 138:82–96 Penner SS (1959) Quantitative molecular spectroscopy and gas emissivities. Addison-Wesley Publishing Company, London Perrin MY, Hartmann JM (1989) Temperature-dependent measurements and modeling of absorption by CO2-N2 mixtures in the far line-wings of the 4.3 μm CO2 band. J Quant Spectrosc Radiat Transf 42:311–317

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Pierrot L (1997) Développent, étude critique et validation de modèles de propriétés radiatives infrarouges de CO2 et H2O à hautes températures. Application au calcul des transferts dans les chambres aéronautiques et à la télédétection. PhD thesis, Ecole Centrale Paris Pierrot L, Rivière PH, Soufiani A, Taine J (1999) A fictitious-gas-based absorption distribution function global model for radiative transfer in hot gases. J Quant Spectrosc Radiat Transf 62:609–624 Rivière PH, Soufiani A (2012) Updated band model parameters for H2O, CO2, CH4 and CO radiation at high temperature. J Quant Spectrosc Radiat Transf 55:3349–3358 Rivière PH, Soufiani A, Taine J (1992) Correlated-k and fictitious gas methods for H2O near 2.7 μm. J Quant Spectrosc Radiat Transf 48:187–203 Rothman LS, Gamache RR, Tipping RH, Rinsland CP, Smith MAH, Chris Benner D, Malathy Devi V, Flaud JM, Camy-Peyret C, Perrin A, Goldman A, Massie ST, Brown LR (1992) The HITRAN molecular database: editions of 1991 and 1992a. J Quant Spectrosc Radiat Transf 48:469–507 Rothman LS, Rinsland CP, Goldman A, Massie ST, Edwards DP, Flaud JM, Perrin A, CamyPeyret C, Dana V, Mandin JY, Schroeder J, Mccann A, Gamache RR, Wattson RB, Yoshino K, Chance KV, Jucks KW, Brown LR, Nemtchinov V, Varanasi P (1998) The HITRAN molecular spectroscopic database and HAWKS (HITRAN atmospheric workstation): 1996 edition. J Quant Spectrosc Radiat Transf 60:665–710 Rothman LS, Gordon IE, Barbe A, ChrisBenner D, Bernath PF, Birk M, Boudon V, Brown LR, Campargue A, Champion J-P, Chance K, Coudert LH, Danaj V, Devi VM, Fally S, Flaud J-M, Gamache RR, Goldmanm A, Jacquemart D, Kleiner I, Lacome N, Lafferty WJ, Mandin J-Y, Massie ST, Mikhailenko SN, Miller CE, Moazzen-Ahmadi N, Naumenko OV, Nikitin AV, Orphal J, Perevalov VI, Perrin A, Predoi-Cross A, Rinsland CP, Rotger M, Simeckova M, Smith MAH, Sung K, Tashkun SA, Tennyson J, Toth RA, Vandaele AC, VanderAuwera J (2009) The HITRAN 2008 molecular spectroscopic database. J Quant Spectrosc Radiat Transf 110:533–572 Rothman LS, Gordon LE, Barber RJ, Dothe H, Gamache RR, Goldman A, Perevalov VI, Tashkun SA, Tennyson J (2010) HITEMP, the high-temperature molecular spectroscopic database. J Quant Spectrosc Radiat Transf 111:2139–2150 Rothman LS, Gordon IE, Babikov Y, Barbe A, ChrisBenner D, Bernath PF, Birk M, Bizzocchi L, Boudon V, Brown LR, Campargue A, Chance K, Cohen EA, Coudert LH, Devi VM, Drouin BJ, Fayt A, Flaud J-M, Gamache RR, Harrison JJ, Hartmann J-M, Hill C, Hodges JT, Jacquemart D, Jolly A, Lamouroux J, LeRoy RJ, Li G, Long DA, Lyulin OM, Mackie CJ, Massie ST, Mikhailenko S, Müller HSP, Naumenko OV, Nikitin AV, Orphal J, Perevalov V, Perrin A, Polovtseva ER, Richard C, Smith MAH, Starikova E, Sung K, Tashkun S, Tennyson J, Toon GC, Tyuterev VG, Wagner G (2013) The HITRAN2012 molecular spectroscopic database. J Quant Spectrosc Radiat Transf 130:4–50 Smith TF, Shen ZF, Friedman JN (1982) Evaluation of coefficients for the weighted sum of gray gases model. ASME J Heat Transfer 104:602–608 Solovjov VP, Webb BW (1998) Radiative transfer model parameters for carbon monoxide at high temperature. J Proc11th Int Heat Transfer Conf Kyongju, Korea 7:445–450 Solovjov VP, Webb BW (2000) SLW modeling of radiative transfer in multicomponent gas mixtures. J Quant Spectrosc Radiat Transf 65:655–672 Solovjov VP, Webb BW (2001) An efficient method for modeling of radiative transfer in multicomponent gas mixtures with soot particles. ASME J Heat Transfer 123:450–457 Solovjov VP, Webb BW (2005) The cumulative wavenumber method for modeling radiative transfer in gas mixtures with soot. J Quant Spectrosc Radiat Transf 93:273–287 Solovjov VP, Webb BW (2008) Multilayer modeling of radiative transfer by SLW and CW methods in non-isothermal gaseous media. J Quant Spectrosc Radiat Transf 109:245–257 Solovjov VP, Webb BW (2011) Global spectral methods in gas radiation: the exact limit of the SLW model and its relationship to the ADF and FSK methods. ASME J Heat Transfer 133:88–798

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Solovjov VP, Lemonnier D, Webb BW (2011a) The SLW-1 model for efficient prediction of radiative transfer in high temperature gases. J Quant Spectrosc Radiat Transf 112:1205–1212 Solovjov VP, Lemonnier D, Webb BW (2011b) SLW-1 modeling of radiative heat transfer in nonisothermal nonhomogeneous gas mixtures with soot. ASME J Heat Transfer 133:102701–3-9 Solovjov VP, Lemonnier D, Webb BW (2013) Efficient cumulative wavenumber method – the CW-1 Model of radiative transfer in the gaseous medium bounded by non-gray walls. J Quant Spectrosc Radiat Transf 63:2–9e Solovjov VP, Lemonnier D, Webb BW (2014) Extension of the exact SLW model to nonisothermal gaseous media. J Quant Spectrosc Radiat Transf 143:83–91 Solovjov VP, Andre F, Lemonnier D, Webb BW (2016a) The rank correlated SLW model of gas radiation in non uniform media. In: Proceedings of RAD-16, Begell House Solovjov VP, Andre F, Lemonnier D, Webb BW (2016b) The generalized SLW model. Eurotherm conference 105: computational thermal radiation in participating media V. J Phys Conf Ser 676:1–36. 012022 Solovjov VP, Andre F, Lemonnier D, Webb BW (2017a) The rank correlated SLW model of gas radiation in non uniform media. J Quant Spectrosc Radiat Transf. https://doi.org/10.1016/j. jqsrt.2017.01.034 Solovjov VP, Andre F, Lemonnier D, Webb BW (2017b) The scaled SLW model of gas radiation in non uniform media. In: Proceedings of CHT-17. ICHMT international symposium on advances in computational heat transfer. May 28–June 1, 2017, Napoli Song TH, Viskanta R (1986) Development of application of a spectral-group model to radiative heat transfer. ASME Paper No. 86-WA/HT-36 Soufiani A, Taine J (1999) High temperature gas radiative property parameters of statistical narrowband model for H2O, CO2 and CO and correlated-k (CK) model for H2O and CO2. Int J Heat Mass Transf 40:987–991 Soufiani A, Hartmann J-M, Taine J (1985) Validity of band-model calculations for CO2 and H2O applied to radiative properties and conductive-radiative transfer. J Quant Spectrosc Radiat Transf 33:243–257 Soufiani A, André F, Taine J (2002) A fictitious-gas based statistical narrow-band model for IR long-range sensing of H2O at high temperature. J Quant Spectrosc Radiat Transf 73:339–347 Taine J, Soufiani A (1999) Gas IR radiative properties: from spectroscopic data to approximate models. Adv Heat Transfer 33:295–414 Vidler M, Tennyson J (2000) Accurate partition function and thermodynamic data for water. J Chem Phys 113:9766–9771 Viskanta R (2005) Radiative transfer in combustion systems: fundamentals and application. Begell House, New York Wang C, Modest MF, He B (2016) Full-spectrum k-distribution look-up table for nonhomogeneous gas–soot mixtures. J Quant Spectrosc Radiat Transf 176:129–136 West R, Crisp D, Chen L (1990) Mapping transformations for broadband atmospheric radiation calculations. J Quant Spectrosc Radiat Transf 43:191–199 Young SJ (1975) Band model formulation for inhomogeneous optical paths. J Quant Spectrosc Radiat Transf 15:483–501 Young SJ (1977) Non isothermal band model theory. J Quant Spectrosc Radiat Transf 18:1–28 Zhang H, Modest MF (2003) Scalable multi-group full-spectrum correlated-k distributions for radiative transfer calculations. ASME J Heat Transfer 125:454–461

Radiative Properties of Particles

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Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Assumptions and Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Particle Size Versus Wavelength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Absorption and Scattering by an Ensemble of Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Shapes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Optical/Electromagnetic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Basic Formulations, Methodology, and Toolbox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Basic Formulations of Electromagnetic Scattering by Particles . . . . . . . . . . . . . . . . . . . . . 3.2 Radiative Properties of Particles: Derivation and Application . . . . . . . . . . . . . . . . . . . . . . 3.3 Methodology and Toolbox for Calculating the Radiative Properties of Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Predicting the Radiative Properties of Individual Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Arbitrarily Shaped Individual Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Regularly Shaped Individual Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Deriving the Radiative Properties of Particles from Experiments . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Analog Experiments on Model Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Laboratory Experiments on Real Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Cross-References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1144 1144 1144 1145 1146 1147 1148 1148 1150 1157 1158 1160 1166 1168 1169 1169 1170 1171

Abstract

This chapter aims at providing an overview on established theories, methods, and tools for dealing with the determination of the radiative properties of particles. The assumptions on the particulate media are provided to confine the treatise to R. Vaillon (*) CETHIL, UMR 5008, Univ Lyon, CNRS, INSA-Lyon, Université Claude Bernard Lyon 1, Villeurbanne, France Radiative Energy Transfer Laboratory, Department of Mechanical Engineering, University of Utah, Salt Lake City, UT, USA e-mail: [email protected]; [email protected] # Springer International Publishing AG, part of Springer Nature 2018 F. A. Kulacki (ed.), Handbook of Thermal Science and Engineering, https://doi.org/10.1007/978-3-319-26695-4_60

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configurations that are common and generic in contemporary thermal science and engineering problems. Basics of the physics of electromagnetic absorption and scattering by particles are summarized. The radiative properties of interest are derived and their use for thermal radiation transfer clarified. A general methodology and a toolbox for calculating the radiative properties of particles are introduced. The existing certified methods for predicting directly the properties of individual particles are briefly presented, separating the cases of particles with arbitrary shapes from those with regular shapes. Common approaches for deriving the radiative properties of particles from measurements are described.

1

Introduction

In thermal science and engineering, problems to be solved may involve particulate media and thus the need for determining their thermal properties. In this chapter, the focus is on the radiative properties of these media. In the following, a compendium of information required to calculate directly or to derive from experiments the radiative properties of particles are provided. Methods established long ago but also more recent ones are introduced briefly, with the leading aim to assist the reader in choosing the most appropriate path for solving his own specific problem.

2

Assumptions and Parameters

In many thermal science and engineering problems, media are composed of either natural or artificial discrete components of closed shape that will be generically called particles.

2.1

Particle Size Versus Wavelength

If the characteristic dimension of one of these particles (a, the radius R for a sphere) is much larger than the dominant wavelength of thermal radiation (Howell et al. 2016; Modest 2013), then the definition of a volumetric radiative property of that particle is not relevant. Instead the theories for radiation exchange between opaque solids or for radiative transfer in semitransparent media (involving the crossing of waves through interfaces that bound that large object) shall be used. In this chapter, particles whose size is smaller, comparable, or moderately larger than the dominant wavelength of radiation are considered. It is convenient to define a quantity called size parameter: x¼

ωa 2πa ¼ ¼ k0 a, c0 λ0

(1)

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where ω is the angular frequency of the electromagnetic (e.m.) wave under consideration, c0 is the speed of light in vacuum, and λ0 and k0 are the wavelength and the magnitude of the wave vector in vacuum, respectively. Size parameters considered in this treatise are mostly smaller than several dozens.

2.2

Absorption and Scattering by an Ensemble of Particles

In radiative heat transfer problems, the particulate media are generally composed of a large number of individual particles. Each particle in a volume may not have the same size, the same shape, and the same orientation in a fixed reference frame. Figure 1 illustrates that on its pathway, an incident e.m. wave (Einc) interacts with particles through two basic processes. Particle material constituents may convert the e.m. energy carried by the waves into another form of energy, through a process

a imaginary sphere

scattered e.m. radiation absorbed e.m. radiation

incident e.m. radiation

Einc

b

Einc

Fig. 1 Interaction of e.m. radiation with an ensemble of particles: absorption refers to conversion within the particles of a fraction of the incident energy. Scattering refers to the change of direction of the incident radiation in the solid angular space all around the particles. (a) In the most general case, it might be necessary to account for the illumination of particles by the incident field (whose amplitude may decay along its path) but also possibly by the field scattered by the neighboring particles. (b) The usual way for solving radiative heat transfer problems is to assign volumetric radiative properties to the particle ensemble, where each particle is assumed to be illuminated by the incident field only

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called absorption (this corresponds to radiation energy trapped in the imaginary sphere around the isolated particle depicted on Fig. 1a). Part of the incident e.m. radiation can be redirected in any direction away from the illuminated particle, through a process called scattering. For calculating radiative properties of the particle ensemble, the first stage is to be able to do it for a single particle. Then, in most engineering thermal radiation problems, it is quite desirable to avoid describing electromagnetic absorption and scattering events considering separately and exactly the characteristics (size, shape) and states (position and orientation as a function of time) of each individual particle. It is instead preferable to apply summation and averaging rules over distributions of these characteristics and states to get the time-averaged radiative properties representative of the particle ensemble. As illustrated in Fig. 1a, it would otherwise mean describing the interaction of the incident field as it meets progressively the particles and the fields scattered by each particle that may in turn be incident on the other neighboring particles. The wave nature of e.m. radiation would have to be taken into consideration, by making coherent addition of the e.m. fields coming from the different particles to evaluate the field scattered all around the particle ensemble. This configuration is usually called the dependent scattering regime. Instead, averaging and summing over the particles contained in a volume for which distributions of states and characteristics of the particles can be defined is a convenient way to make the modeling computationally affordable for radiative transfer. In this way, the radiative properties per unit volume are defined as if there was a unique scatterer representative over time of the population of particles, where each sample particle is assumed to be independently illuminated by the incident field (Fig. 1b). The summations over characteristics of single-scattering radiative properties assume that the intensities scattered by the each particle sample can be incoherently added. This configuration is referred to as the independent scattering regime. A popular criterion for applying the aforementioned incoherent summation rules in calculating the volumetric radiative properties of particulate media is based on the clearance-to-wavelength ratio c/λ0, where c is the average clearance between particles (Howell et al. 2016; Modest 2013). The limit between the dependent and independent scattering regimes is depicted on a map as a function of particle size parameter and particle volume fraction. This chart was proposed three decades ago in Tien and Drolen (1987) and never fully challenged (i.e., replaced) since then. It is indeed stated in Mishchenko et al. (2002) that it is quite difficult to provide general and definitive conditions for applying the summation rules associated with the independent scattering regime. In the following, it is assumed that the independent scattering regime holds.

2.3

Shapes

With the constant progress in electromagnetic scattering modeling and computation, the possible complex nature of the shape of particles is not a real concern anymore.

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However, it remains convenient to distinguish between regular shapes and irregular shapes, thus applying a classical and suitable categorization. Regular shapes mean spheres, coated spheres, cylinders, and spheroids. For these shapes, there exist analytic formulations of the radiative properties and robust numerical techniques. Irregular shapes mean any shape that does not fall in the previous category. In the latter case, analytic formulations are scarce, and thus numerical techniques solving for the complex electromagnetic field (amplitude and phase) are used, often requiring large computational resources. Before having recourse to models suited for arbitrarily shaped particles, in case orientation averaging is applied, it might be wise to check if an approximate modeling by a regularly shaped particle would be acceptable in a first step.

2.4

Optical/Electromagnetic Properties

The material constituting the particles determines the optical (electromagnetic) properties that rule the response of matter to excitation by electric and magnetic fields. For bulk (homogeneous) and isotropic materials, these properties are given by the complex refractive index m for optical properties and by the complex relative electric permittivity ϵ and magnetic permeability μ for electromagnetic properties. pffiffiffiffiffiffiffi Taking the temporal convention exp(iωt), where i ¼ 1 , ω is the angular frequency and t is time, and making the assumption that the material is nonmagnetic (μ = μ0), the relation between the complex refractive index m = mR + i mI and the complex relative permittivity ϵ = ϵ0 + iϵ00 is (Bohren and Huffman 1983): ϵ0 ¼ m2R  m2I

(2)

ϵ00 ¼ 2mR mI sp ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 0 00 0 ϵ þϵ þϵ mR ¼ 2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ϵ0 2 þ ϵ00 2  ϵ0 mI ¼ 2

(3) (4)

(5)

The dependence on frequency of these electromagnetic (ϵ) and optical (m) properties can be analytically formulated using dispersion relations such as Drude and Lorentz models (Bohren and Huffman 1983; Jackson 1998). Being response functions, the spectral distributions of the real and imaginary parts of the complex refractive index (permittivity) are not independent from each other: they are linked by the Kramers-Krönig relations (Jackson 1998). These are very convenient when one section of the spectrum is missing for either the real or the imaginary part. Optical properties have been experimentally determined over the years so that databases are available for the most common materials (dielectrics, metals, and

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semiconductors, mostly at room temperature). More details about them are given in the toolbox in Sect. 3.3. It is unfortunately not standard to find the variations with temperature of these properties. It may happen that the particle material is heterogeneous, i.e., with inclusions of a different material, defects, grains, etc. Unless the sub-particles need to be included in the modeling the same way the host particle is, which would make the problem more complex to solve, the common approach is to assign effective optical (electromagnetic) properties to the heterogeneous material, using averaging (homogenization) techniques. Maxwell-Garnett and Bruggeman are among the most known formulations for calculating effective permittivities (Bohren and Huffman 1983). In the following, the particles are assumed to be composed of homogeneous, isotropic, and nonmagnetic materials.

3

Basic Formulations, Methodology, and Toolbox

3.1

Basic Formulations of Electromagnetic Scattering by Particles

Before introducing the radiative properties of particles and their use in thermal radiation heat transfer, it is worth providing the very essentials about the physics of electromagnetic scattering by particles. In the most general case, the description of the interaction of an electromagnetic field with an isolated particle and further on with an ensemble of particles falls in the field of electromagnetics (Born and Wolf 1999; Jackson 1998). The macroscopic Maxwell’s equations and constitutive relations are the base of most developments for deriving radiative properties of particles (Mishchenko et al. 2006; Mishchenko 2014). This means dealing with complex electromagnetic fields, their magnitude and phase, in order to account for the wave nature of radiation (involving interference, diffraction, and in certain cases near-field expansions of the waves in the very vicinity of particles, but it is reminded that this treatise is restricted to the independent scattering regime). In the following, the basic ingredients commonly used for describing absorption and scattering by a single particle are briefly provided. Details are available in several monographs, e.g. Bohren and Huffman (1983), Mishchenko et al. (2006), and Mishchenko (2014).

3.1.1 The Amplitude Scattering Matrix An individual particle surrounded by a nonabsorbing medium is shown in Fig. 2. It is illuminated by a plane harmonic wave traveling in direction kinc = z. Radiation is scattered in the form of a spherical wave, potentially in any direction ksca (note that it is assumed that the surrounding medium is non-absorbing with n = 1, such that kinc = ksca = k0). The two components of the vectorial complex incident electric field (E inc) and scattered electric field (E sca) at point (P1) in the far field, at a

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e

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,inc

x

y

z e

,sca

Fig. 2 Scattering by an individual particle: frames and notations. Wave vectors, electric fields, and Poynting vectors used in the derivation of the radiation power absorbed and scattered by the particle are depicted for illustration

distance r from the particle such that ksca r  1, can be written (Bohren and Huffman 1983):   E inc ðtÞ ¼ Ek, inc ek, inc þ E⊥, inc e⊥, inc ¼ E0k, inc y þ E0⊥, inc x expðik0 z  iωtÞ ¼ Einc expðiωtÞ ¼ E0, inc expðik0 zÞexpðiωtÞ E sca ðtÞ

  ¼ E0k, sca ek, sca þ E0⊥, sca e⊥, sca

¼ Ek, sca ek, sca þ E⊥, sca e⊥, sca expðik0 r  iωtÞ ¼ Esca expðiωtÞ k0 r expðik0 r Þ expðiωtÞ ¼ E0, sca k0 r

(6)

(7)

where Ek and E⊥ are the complex polarization components which are collinear to unit vectors parallel and perpendicular to the plane formed by kinc and ksca, called the scattering plane. Einc and Esca are the time-independent complex incident and scattered fields. E0,inc and E0,sca are the time- and position-independent amplitude vectors for the plane incident and spherical scattered waves, respectively.

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The relation between the amplitude vectors E0,sca and E0,inc takes the form (adapted from Bohren and Huffman 1983; Mishchenko 2014): 

E0k, sca ðθ, ϕÞ E0⊥, sca ðθ, ϕÞ



 ¼

S2 S4

S3 S1



E0k, inc E0⊥, inc

 (8)

where the 2  2 matrix is called the amplitude scattering matrix. Its elements are function of the polar (θ) and azimuthal (ϕ) scattering angles and of the properties of the particle (shape and size, composition, orientation). Details on the steps resulting in that formulation can be found in Mishchenko (2014). The complex elements (Si) of the amplitude scattering matrix are the building blocks for fully describing scattering and absorption by a particle. Radiative properties of particles can be derived from them (see Sect. 3.2.2).

3.1.2 The Scattering Matrix Radiation power is the quantity of primary interest when deriving scattering properties for thermal radiation transfer problems. By knowing the scattered field anywhere far enough from the particle, the power and polarization state of scattered radiation can be quantified by the Stokes parameters (I,Q,U,V ) and the Stokes (column) vector I ¼ ðI, Q, U, V Þt . The Stokes vector of the far-field scattered radiation can be expressed as a function of the Stokes vector of the incident radiation by (Mishchenko 2014): 0

I sca

S11 1 B S21 ¼ 2 2B k0 r @ S31 S41

S12 S22 S32 S42

S13 S23 S33 S43

1 S14 S24 C CI S34 A inc S44

(9)

where the 4  4 matrix is called the scattering matrix or Stokes phase matrix. From the definition of the Stokes parameters, it results that all elements (Sij) of the scattering matrix are functions of the elements (Si) of the amplitude scattering matrix (Bohren and Huffman 1983; Mishchenko 2014). The expression for S11, related to the scattering phase function (see Sect. 3.2.2), is (Bohren and Huffman 1983): S11 ¼

3.2 3.2.1

 1 2 j S1 j þ j S2 j 2 þ j S3 j 2 þ j S4 j 2 2

(10)

Radiative Properties of Particles: Derivation and Application

Radiation Power Scattered and Absorbed by an Individual Particle To start with, in Fig. 2, an imaginary sphere of radius r containing an individual particle is depicted. Incident radiation is traveling in direction z with an electric field

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1151

Einc having polarization components along the x and y axes. Part of the incoming radiation is scattered in all directions around the particle (θ  [0,π] and ϕ  [0,2π]). At any point P2 on the sphere, the electromagnetic field can be modeled as the superposition of the incident field (without the particle) and the scattered field (resulting from the interaction of the incident field with the particle). That results in the total electric field Etot = Einc + Esca (similarly in the total magnetic field Htot = Hinc = + Hsca). Calculation of the net flux trough the surface of the whole sphere of the power carried by this total field provides the amount of power accumulated within the imaginary sphere, which turns out to be the radiation power absorbed by the particle (noted Wabs). The power carried by an e.m. wave is given by the time-averaged Poynting vector defined as (Mishchenko 2014): 1 S ¼ hS ðtÞi ¼ hE ðtÞ  ℋðtÞi ¼ ReðE  H Þ 2

(11)

where S ¼ 12 ½E  H  is the complex Poynting vector, H* is the complex conjugate of the magnetic field, and the notation 〈〉 stands for time averaging over a time much larger than the period of the waves (2π/ω). By summing the (net) flux of the timeaveraged Poynting vector of the total field all over the surface of the sphere (Surf), the power absorbed by the particle writes: Z W abs ¼ 

 1  Re Etot  Htot :ndA Surf 2

(12)

where n is the unit vector normal to the sphere and oriented outward. As accumulation within the imaginary sphere is expected, the minus sign ensures that the calculated quantity is positive. Considering that the total field is the sum of the incident and scattered fields, the time-averaged Poynting vector of the total field splits into three terms: 1 Stot ¼ Re½ðEinc þ Esca Þ  ðHinc þ Hsca Þ  ¼ Sinc þ Ssca þ Sinterf 2

(13)

 1  Sinc ¼ Re Einc  Hinc 2

(14)

 1  Ssca ¼ Re Esca  Hsca 2

(15)

 1  Sinterf Re Einc  Hsca þ Esca  Hinc 2

(16)

where:

The notation “interf” of the last term indicates that the cross terms in the development of Eq. 13 correspond to the interference between the incident and

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scattered waves. When developing Eq. 12 with the three terms, it is observed that the flux over the sphere of the time-averaged Poynting vector of the incident field (Sinc) is identically null: the flux at any entrance point on the sphere (P3-in) is compensated by the flux at the corresponding exit point (P3-out), since the medium surrounding the particle in nonabsorbing. Thus, only two terms remain in the development of the absorbed power: Z

Z

W abs ¼ 

Ssca :ndA  Surf

Sinterf :ndA

(17)

Surf

At this stage, it is worth noticing that the scattered radiation is exiting the sphere (e.g., at point P4 in Fig. 2), in such a way that the following positive quantity can be defined: Z W sca ¼

Ssca :ndA

(18)

Surf

as being the radiation power scattered by the particle. Thus, in Eq. 17, Wabs and Wsca can be gathered on the same side of the preceding equation to give: Z W abs þ W sca ¼ 

Sinterf :ndA ¼ W ext

(19)

Surf

The previous derivations resulting in the above equation have important meanings and implications. First, the sum of the radiation powers absorbed and scattered by the particle corresponds to the radiation power lost by the incident radiation after interaction with the particle, hence the notation Wext which refers to extinction. The radiation power scattered by the particle: Z W sca ¼

 1  Re Esca  Hsca :ndA 2 Surf

(20)

and the power taken out from the incident radiation: Z W ext ¼ 

 1  Re Einc  Hsca þ Esca  Hinc :ndA Surf 2

(21)

can be calculated if the incident field and the amplitude scattering matrix elements are known, as they provide the scattered field (Esca) everywhere in the far field as a function of the incident field (see Eq. 8). The magnetic fields are not additional unknowns since they are related to the electric fields: Hp ¼

1 kp  Ep ωμ0

(22)

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where “p” stands for either “inc” or “sca.” The absorbed radiation power is not evaluated directly but follows from Wabs = Wext  Wsca.

3.2.2 Radiative Properties: Definitions and Application Radiative properties of an individual particle In order to characterize the ability of the particle to absorb and scatter incident light independently from the amount of incoming radiation power, it is convenient to express the previously derived radiation extinction, scattering and absorbed powers as the product of a quantity Cp, where “p” stands for “ext,” or “sca” or “abs,” and of the incident power per unit area Iinc, as follows: W ext ¼ Cext I inc

(23)

W sca ¼ Csca I inc

(24)

W abs ¼ Cabs I inc

(25)

Cext, Csca, and Cabs are called the total extinction, scattering, and absorption cross-sections (however, these cross-sections are defined for a given radiation frequency: here total stands for the summation over all directions made to derive the extinction and scattered powers). Considering the units of W and Iinc (W and W m2, respectively), the cross-sections have the dimension of an area (m2). With the formulations of the radiation power scattered (in the far field) and lost after interaction with an individual particle (Eqs. 20 and 21), the derivation of the radiative properties is a matter of mathematical developments (see Bohren and Huffman 1983 for details). Only final expressions (reformulated from Bohren and Huffman 1983; Mishchenko 2014) are given here: Cext ¼

  4π  Im E ð θ ¼ 0 Þ:E 0 , sca 0, inc 2 k2 E0, inc

(26)

0

Csca ¼

1

Z

2 k2 E0, inc 0

2π Z π 0

0

E0, sca ðθ, ϕÞ 2 sin θdθdϕ

Cabs ¼ Cext  Csca

(27) (28)

It is worth noticing that: • The radiative properties can be calculated if the incident field and the scattered field at all angles (θ,ϕ), or alternatively the elements of the amplitude scattering matrix (from Eq. 8), are known. • The expression of the extinction cross-section indicates that it depends on the scattered field in the forward direction only, a relation known as the optical theorem.

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• For unpolarized incident radiation, the cross-sections should be calculated for the two polarization components of the incident field and their average taken. • The absorption cross-section is not calculated directly but derived from the extinction and scattering cross-sections. As for the angular distribution of scattered radiation, it is described by the scattering phase function: pðθ, ϕÞ ¼

2 4π E0, sca ðθ, ϕÞ

(29) k20 Csca R 2π R π which is a normalized quantity (such that ð4π Þ1 0 0 pðθ, ϕÞ sin θdθdϕ ¼ 1). The name “phase function” is common and somewhat confusing but is a shorter denomination for the probability density of the scattering polar and azimuthal angles (θ,ϕ). 2 In certain publications, the quantity E0, sca ðθ, ϕÞ =k20 is called differential scattering cross-section and expressed with the confusing notation dCsca/dΩ, where Ω stands for the solid angle (but it is not a derivative of the scattering function with the solid angle Ω). According to the definitions of the scattering matrices, the differential scattering cross-section is identically equal to the scattering matrix element S11 for unpolarized incident radiation, such that the phase function can be derived directly from the amplitude scattering matrix elements:

pðθ, ϕÞ ¼

4πS11 ¼ k20 Csca

  2π jS1 j2 þ jS2 j2 þ jS3 j2 þ jS4 j2 k20 Csca

(30)

where the dependence on (θ,ϕ) is implicit for the (amplitude) scattering matrix elements. The average cosine of the polar scattering angle θ, called asymmetry parameter, is given by the first moment of the scattering phase function (of the variable scattering polar angle): g¼ 1 g¼ 2

Z

π

1 4π

Z

2π Z π 0

pðθ, ϕÞ cos θ sin θdθdϕ

(31)

0

pðθÞ cos θ sin θ dθðazimuthal symmetryÞ

(32)

0

It is null if scattering is isotropic ( p(θ, ϕ) = 1) or if the phase function is symmetric about θ = π/2. A particle that scatters dominantly in the forward directions will have a positive asymmetry parameter (and conversely). Radiative properties for unpolarized radiative transfer Once the properties are calculated for an individual particle having given shape, size, and orientation in space, the radiative properties relevant to unpolarized radiative transfer can be

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derived for known distributions of shape realization, size, and orientation. Statistical averaging and summations can be applied following the recommendations given in (Mishchenko et al. 2006; Mishchenko 2014). For most of thermal radiation heat transfer problems of interest in this handbook, except when the size of bodies emitting radiation or the distance between bodies exchanging radiation are smaller than the thermal wavelength (see ▶ Chap. 24, “Near-Field Thermal Radiation”), the conditions for applying the radiative transfer equation (see ▶ Chap. 23, “Radiative Transfer Equation and Solutions”) hold. In that case, the averaging and summation rules specified in Sect. 2.2 to calculate volumetric radiative properties of particles are applied. For example, when a particle size distribution is considered, the scattering coefficient (σ s), extinction coefficient (β), and absorption coefficient (κ), the scattering phase function (Φ), and the asymmetry parameter (g) are derived from the scattering radiative properties calculated for an individual particle as: Z

1

σs ¼ Z

(33)

Cext ðaÞnðaÞda

(34)

Cabs ðaÞnðaÞda

(35)

1

β¼ Z

Csca ðaÞnðaÞda

0

0 1

κ¼ 0

Z 1 1 pða, θ, ϕÞCsca ðaÞnðaÞda σs 0 Z 1 1 g¼ gðaÞCsca ðaÞnðaÞda σs 0

Φðθ, ϕÞ ¼

(36) (37)

where n(a) is the number R 1 of particles per unit volume having a size comprised between a and a + da ( 0 nðaÞda is the total number of particles per unit volume). The above quantities derive from the summation (here over particle size) of radiative properties evaluated for individual particles (Csca, Cext, Cabs, p, g) and unpolarized incident e.m. radiation. Scattering regimes for radiative transfer The scattering, absorption, and extinction coefficients have the dimension of m1. Thus their inverse, lsca = 1/σ, labs = 1/κ and lext = 1/β are lengths. They are called scattering, absorption, and extinction mean free paths, respectively. For absorption, it corresponds to the distance where the radiation intensity has dropped by 1/e because of absorption only and is alternatively called attenuation length. For scattering, in the frame of the independent scattering regime, which models local particle ensembles by a representative scatterer, this is the average distance radiation travels between two effective scattering events (scattering is a collision event in the particulate (photon) transport modeling frame of the Boltzmann equation Chen 2005). The medium surrounding the particles might absorb as well (e.g., a nontransparent gaseous medium;

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see ▶ Chap. 26, “Radiative Properties of Gases”), but this configuration is not considered in the following. By assuming that radiation can travel at least partially through the participating medium of characteristic dimension L without being fully absorbed (labs  L), several scattering regimes relevant to radiative transfer can be defined: • lsca ~ L: on average, a single effective scattering event takes place on the radiation pathway through the medium. This is the single scattering regime for radiative heat transfer. • lsca  L: on average, several single scattering events are likely to take place on the radiation pathway through the medium. This is the multiple scattering regime for radiative heat transfer. In addition, since solving the radiative transfer equation with the in-scattering term requires additional computation time (because locally, radiation traveling in one direction depends on radiation coming from any direction because of scattering), it is useful to determine the relative weight of scattering to that of extinction. This is given by the scattering albedo: ωsca ¼

σs σs þ κ

(38)

When the scattering albedo tends toward zero, scattering plays a negligible role in radiation extinction, and thus only absorption properties are required. Approximate scattering phase functions The scattering phase function may exhibit large oscillatory behaviors as a function of the scattering angle (however, this potential feature is most likely to disappear for polydispersed and randomly oriented nonsymmetric particles). For radiative transfer direct (or inverse; see Sect. 5.2) calculations, it is often welcome to have recourse to an approximate model smoothing angular variations of the original phase function. The Dirac delta, HenyeyGreenstein, and truncated Legendre polynomial approximations are among the most used in thermal radiation heat transfer (Howell et al. 2016; Modest 2013). Radiative properties for polarized radiative transfer It is usually considered that polarization does not play a significant role for most problems in thermal science and engineering. When unpolarized volumetric thermal emission dominates, there is obviously no need to account for polarization modification of radiation throughout its path within a scattering medium. The situation is however different when dealing with polarized thermal emission from a surface that might be nonnegligible depending on material optical properties and surface roughness (Sandus 1965; Gurton and Dahmani 2005) and possibly very strong for periodic microstructures etched on materials supporting phonon-polaritons in the infrared (Greffet et al. 2002) and that is transmitted through a cold scattering medium. In that case, the radiative properties required in the vector (or polarized) radiative transfer equation (Mishchenko et al. 2006) needs to be calculated as a function of the radiative properties of individual particles. The averaging and summation rules of the independent scattering regime are thus applied to the elements of the scattering matrix (Sij) (Eq. 9).

27

3.3

Radiative Properties of Particles

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Methodology and Toolbox for Calculating the Radiative Properties of Particles

A methodology is proposed for dealing with the determination of the radiative properties of a particulate medium, using the diagram of Fig. 3 that depicts a series of suggested steps. They consist of: • Determining the characteristic length (L ) of the medium. • Determining the spectral interval for radiative transfer calculations [λ1, λ2]. If the minimum and maximum temperatures (Tmin,Tmax) within the medium are known, an estimate for covering properly the spectrum is given by [0.5λ0,max(Tmax), 5λ0,max(Tmin)], where λ0,max(T) stands for the wavelength in vacuum where the Planck function is maximum, given by Wien’s law (Howell et al. 2016; Modest 2013). • Collecting the characteristics of the particles: their shape and size (possibly distributions of size and shape characteristics), their orientation (random or fixed), and their composition (unique bulk material or with multicomponents). • Collecting in databases or calculating (with dispersion relations, effective medium mixing rules) the complex refractive index of the particles’ material. • Determining the size parameter interval from the wavelength and size distribution intervals.

Fig. 3 Suggested methodology for calculating the radiative properties of a particulate medium

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• Choosing a model that predicts the radiative properties. For an individual particle, the choice is based on the size parameter interval and shape of the particles (see Sect. 4). The radiative properties for particle ensembles derive from the application of averaging and summation rules (Sect. 3.2.2). A toolbox with the very essential resources for making the aforementioned calculations is proposed with: • A web portal with light scattering resources, called SCATTPORT (Wriedt 2016), that lists calculation methods and associated simulation programs for most of the models presented in Sect. 4, http://www.scattport.org/. • A short list of databases of the complex refractive index of bulk materials. The handbooks of Edward Palik are among the most popular resources (Palik 1998; Gervais and Palik 1991). Adachi’s work on semiconductor properties includes the optical constants (e.g., Adachi 2009). Some databases are available online, such as: • Heidelberg – Jena – St. Petersburg – Database of Optical Constants (HJPDOC) (http://www.mpia-hd.mpg.de/HJPDOC/) • Refractive index info (http://refractiveindex.info/) • SpringerMaterials – The Landolt-Börnstein Database (http://www.springer materials.com/) Guidelines for choosing a model that predicts the radiative properties of individual particles (Csca, Cext, Cabs, p, g) are given in Sect. 4, where a compendium of existing models are provided, by distinguishing between arbitrarily shaped and regularly shaped particles.

4

Predicting the Radiative Properties of Individual Particles

To predict the radiative properties of a particle ensemble, one has to calculate these properties for individual particles first. According to the developments summarized in Sect. 3, the averaging and summation rules of the so-called independent scattering regime allow calculating the radiative properties for populations of particles with known characteristic distributions. Thus, for a given incident e.m. planar wave with frequency (ω) and electric field (Einc), knowing all characteristic of the individual particle (shape, size, composition, orientation), the objective is to get the polarization components of scattered field (Esca) in the far-field region (see Fig. 4) or, similarly, the elements (Si) of the amplitude scattering matrix. The radiative properties (Csca, Cext, Cabs, p, g) derive from the knowledge of the scattered field all around the particle (see Eqs. 26, 27, 28, 29, 30, and 31). In the following, it will be assumed that the particle is made of a homogeneous, linear (the response of the material to an excitation is linear, i.e., the electric displacement is linearly proportional to the excitation electric field), isotropic, and nonmagnetic (μ = μ0) material (with ϵ = ϵ0 + iϵ00 or equivalently m = mR + imI) and is surrounded by a homogeneous, linear, isotropic, nonmagnetic (μe = μ0), and nonabsorbing medium (with ϵe ¼ ϵ0e or

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Radiative Properties of Particles

1159

=

+

=

Fig. 4 Absorption and scattering property determination for an individual particle: problem statement and notations

equivalently me = mR,e). The scattering problem is governed by the contrast in optical properties between the particle and its surroundings quantified by the ratio of the particle complex relative permittivity (refractive index)  to the surroundings real ~ ¼ m=mR, e ¼ mR =mR, e part of the relative permittivity (refractive index) ϵ~ ¼ ϵ0 =ϵ0e m þimI =mR, e Þ. By separating the volume inside the particle (V) to that outside (Ve), the distribution of permittivity (refractive index) ratios can be conveniently written: ~ ðrÞ ¼ 1, r  V e ϵ~ðrÞ ¼ m

(39)

~ ðrÞ ¼ mðrÞ=mR, e ¼ mR ðrÞ=mR, e þ imI ðrÞ=mR, e , r  V (40) ϵ~ðrÞ ¼ ϵ0 ðrÞ=ϵ0e or m The electric field E(r) and magnetic field H(r) at any point (r) inside and outside the particle are governed by (Mishchenko 2014): • The macroscopic Maxwell equations within the particle (volume V ) and outside the particle (volume Ve):

∇  EðrÞ ¼ iωμ0 HðrÞ ,rV ∇  HðrÞ ¼ iωϵðrÞe0 EðrÞ

∇  EðrÞ ¼ iωμ0 HðrÞ , r  Ve ∇  HðrÞ ¼ iωϵe e0 EðrÞ

(41) (42)

• The boundary conditions at the interface separating the particle (V ) from the surrounding medium (Ve) • And a radiation condition at infinity, • Constituting the standard electromagnetic problem for a plane-wave illumination.

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In the following, elements of general and specific methodologies to solve this problem and whenever possible the final result are provided. It makes sense to apply the usual distinction between regularly shaped and irregularly shaped particles. However, notwithstanding the historical development timeline, the general case of arbitrarily shaped particles is examined first, as the regularly shaped particles fall in the subcategory of special cases. For a large number of methods reported hereafter, there is a good chance to find corresponding computer codes at the SCATTPORT web portal (Wriedt 2016); hence, reference to it will not be repeated throughout the section.

4.1

Arbitrarily Shaped Individual Particles

4.1.1 Volume Integral Equations for the Total and Scattered Fields In the most general case of arbitrarily shaped particles, and within the assumptions described in preamble, vector wave (called also Helmholtz) equations can be derived from combining the Maxwell Eqs. 41 and 42 (Mishchenko 2014): ∇  ∇  EðrÞ  k2 ðrÞEðrÞ ¼ 0, r  V

(43)

(44) ∇  ∇  EðrÞ  k2e EðrÞ ¼ 0, r  V e pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi where kðrÞ ¼ ω μ0 ϵðrÞe0 and ke ¼ ω μ0 ϵe e0 are the wave vectors inside and outside the particle, respectively. A reformulation leads to a single inhomogeneous differential equation that governs the field everywhere (Mishchenko 2014): ∇  ∇  EðrÞ  k2e EðrÞ ¼ jðrÞ, r  V e [ V

(45)

with: jðrÞ ¼ k2e

 2   2  k ðrÞ ~ ðrÞ  1 EðrÞ  1 EðrÞ ¼ k2e m 2 ke

(46)

~ ðrÞ defined previously, the perturbation function j(r) Given the distribution of m is nonzero only within the particle volume (V ). The solution of this inhomogeneous differential equation is then the superposition: • Of the solution when the perturbation term is null (there is not any particle), which is naturally given by the incident electric field (Einc), • Of a particular solution when the perturbation is considered within the particle that should indeed coincide with the scattered field (Esca) and thus satisfy physical constraints (zero-valued infinitely far from the particle and energy conservation).

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Radiative Properties of Particles

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The scattered field can be expressed as the response at point r of the excitation at any point r0 within the particle volume (V ) (Mishchenko 2014): ð Esca ðrÞ ¼ Gðr, r0 Þjðr0 Þd 3 r0

(47)

V

where G(r,r0 ) is the free space dyadic Green’s function that is essentially the response in free space to a point source function (for the sake of simplicity, as its expression does not add to the understanding of the physics, it is not given here. It is available in a large number of references, e.g., Mishchenko 2014). The final expressions for the total and scattered fields are thus volume integral equations (VIE) (Mishchenko 2014): ð  2 0  ~ ðr Þ  1 Eðr0 Þd 3 r0 EðrÞ ¼ Einc ðrÞ þ k2e Gðr, r0 Þ m

(48)

V

ð  2 0  ~ ðr Þ  1 Eðr0 Þd3 r0 Esca ðrÞ ¼ k2e Gðr, r0 Þ m

(49)

V

With this specific formulation of the scattering problem, the determination of the radiative properties (essentially the extinction and scattering cross-sections via Eqs. 26 and 27) of an arbitrarily shaped individual particle of known geometry (given by the volume V it occupies in space) and known optical properties (given by ~) the ratio of the particle optical properties to those of the surrounding medium m subjected to a defined homogeneous plane wave (Einc) ends up in calculating the scattered field everywhere in the far field around the particle, which writes as an integral over the particle volume of the response to the excitation induced by the local internal electric field. The main challenges are the calculation of the integral for arbitrarily shaped particles and the determination of the unknown internal electric field. In the following, solution techniques that aim at addressing the scattering problem are introduced, referring to the VIE formulation whenever it is convenient. The methods that are not stating any prior simplifying assumption about the particle are presented first. Then the solutions that are built on approximations that make solving the problem easier are reviewed.

4.1.2 Methods Not Making Any Prior Assumption on the Particle Volume integral equation methods The Volume Integral Equation Methods are directly based on the VIE formulation of the scattering problem. The principle is to write a discretized form of the VIE for the total field (Eq. 48) by subdividing the particle volume (V ) into a number i = 1, N of (cubical) cells of volume ΔV and center ri as follows (Mishchenko 2014):

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Eðri Þ ¼ Einc ðri Þ þ k2e ΔV

N X   2      ~ rj  1 E rj , G ri , rj m

i ¼ 1, N

(50)

j¼1

Once the above system of linear equations is solved for the unknown internal fields at each discrete volume center ri, application of the VIE for the scattered field (Eq. 49) gives Esca ðrÞ ¼ k2e ΔV

N X   2      ~ rj  1 E rj , ðrÞ in the far field G r, rj m

(51)

j¼1

This approach is referred to as the method of moments (MoM). The numerical challenge is solving the system, in particular for particles with large size parameters, as achieving accuracy requires a minimum number of cells per wavelength. An equivalent method does not solve directly for the electric fields but instead deals with discrete volumes that are polarizable and as such are usually called dipoles (unless atomic units are considered, these are not real dipoles but an ensemble of dipoles that behaves as a single dipole). Each dipole at the volume center ri is assigned a polarizability (αi) (for the sake of simplicity, the medium is assumed to be isotropic, hence a scalar polarizability) and is subjected to an electric field which is the superposition of the incident electric field and the electric fields radiated by the rest of the dipoles. The dipole moment for a dipole can be expressed as Pi ¼ αi Einc, i þ

X

! Aij :Pj , i ¼ 1, N

(52)

j6¼i

where Aij.Pj is the field at dipole location ri caused by the dipole located at rj. Once an appropriate polarizability model is chosen for αi and the above system of linear equations is solved for the dipole moment Pi, the scattered field can be calculated at any point r in the far field by summing the fields generated by each dipole: Esca ðrÞ ¼

N eike r X Fðr, ri , Pi Þ ke r i¼1

(53)

where function F depends solely on the point position r, the dipole locations ri, and the previously calculated dipole moments Pi. This approach is referred to as the discrete dipole approximation (DDA). Details about the implementation, strengths, and limitations of DDA can be found in Kahnert (2003), Yurkin and Hoekstra (2007), and Kahnert (2016). T-matrix method This method is based on the development in series expansions of vector spherical wave functions of the incident and scattered fields (Mishchenko 2014):

27

Radiative Properties of Particles

Einc ðrÞ ¼

1163

1 X n h X

i amn Mð1Þ ðrÞ þ bmn Nð1Þ ðrÞ

(54)

n¼1 m¼n

Esca ðrÞ ¼

1 X n h X

pmn Mð3Þ ðrÞ þ qmn Nð3Þ ðrÞ

i (55)

n¼1 m¼n

where M and N are the vector spherical wave functions (exponents (1) and (3) stand for regular, finite at the origin, and radiating, ensuring that transverse and radial components decay properly as the distance r increases, respectively). The core of the methodology is the linear relationship between the coefficients pmn and qmn for the scattered wave and those (amn and bmn, which are known) for the incident field, given by a transition matrix (or T-matrix) (Mishchenko 2014): pmn ¼

qmn ¼

1 n0 X X  n0 ¼1 m0 ¼n0

12 T 11 mnm0 n0 am0 n0 þ T mnm0 n0 bmn

1 n0 X X  n0 ¼1 m0 ¼n0

22 T 21 mnm0 n0 am0 n0 þ T mnm0 n0 bmn





(56)

(57)

Once the T-matrix – which is independent of the incident field – is known, it can be used for various configurations of incidence and polarization of the incident field. One particular advantage is that analytical developments for averaging over orientation of the particle are possible (Mishchenko et al. 1996). A brief review of the various methods for computing the T-matrix is available in Kahnert (2016). The method being widely and increasingly used, a thematic reference database of peerreviewed T-matrix publications is regularly released (see the last to date Mishchenko et al. 2016). Other methods There are many more numerical solution techniques available for solving the standard scattering problem for arbitrarily shaped particles, such as the finite-difference time-domain (FDTD), finite-difference frequency-domain (FDFD), finite element (FEM), separation of variables (SVM), point-matching (PMM), nullfield method with discrete sources (NFM-DS) methods. An overview of most of them can be found in review articles and textbooks (Kahnert 2003, 2016; Mishchenko 2014).

4.1.3 Methods and Solutions Based on Prior Approximations The previously introduced methods do not require any a priori assumptions about the particle shape, dimensions, and optical properties. The a posteriori limitations of these methods derive from numerical issues that are nevertheless linked to the particle properties. In this section, approximations about the particle are made since they allow simplifying, sometimes tremendously, the solution methodology. The first simplification is to assume that the surrounding medium permittivity is that

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Fig. 5 (a) Rayleigh approximation where the particle volume is acting as a single dipole with the dipole moment P = αEinc. (b) Rayleigh-Gans approximation where the subvolumes of the particle act as Rayleigh scatterers. (c) Specific case of an aggregate of spheres, where each sphere is a Rayleigh scatterer

~ ¼ m and of vacuum (ϵe = 1 or equivalently mR,e = 1, which implies ϵ~ ¼ ϵ or m ke = k0). The Rayleigh approximation It applies to an arbitrarily shaped particle that has its largest characteristic dimension (a) much smaller than the wavelength in vacuum (x  1) and that is made of an optically not too dense and not too absorbing material (such that |m|x  1). Then the particle can be considered as a single electric dipole oscillating in phase with the incident field (Fig. 5a). In the case where the material is isotropic, the radiative properties are given by (Bohren and Huffman 1983; Van de Hulst 1981): S1 ð θ Þ ¼ 

ik30 α 4π

(58)

S2 ðθÞ ¼ S1 ðθÞ cos θ

(59)

8 Csca ¼ πk40 jαj2 3

(60)

Cabs ¼ k0 ImðαÞ

(61)

pð θ Þ ¼

 3 1 þ cos 2 θ 4

(62)

where α is the (scalar) polarizability of the small particle volume V, given by (Born and Wolf 1999): α ¼ 3V

ϵ1 m2  1 ¼ 3V 2 ϵþ2 m þ2

(63)

The Rayleigh-Gans approximation It results from assuming that the particle volume V can be subdivided into subvolumes that are all subjected to the same incident field and that behave as Rayleigh scatterers (Fig. 5b). This implies that the internal field is almost equal to the incident field everywhere in the particle. For this

27

Radiative Properties of Particles

1165

to be valid, the phase lag between any two points in the particle should be negligible, translating into the condition x|m  1|  1. The scattered field results from the interference of the contributions of all subvolumes and the amplitude scattering elements then write (Kerker 1969): S1 ðθ, ϕÞ ¼ 

ik30 m2  1 3 4π m2 þ 2

Z eiδ dV

(64)

V

S2 ðθ, ϕÞ ¼ S1 ðθ, ϕÞ cos θ

(65)

where δ is the phase (relative to a fixed reference) of the waves contributed by each subvolume at the point (θ,ϕ) in the far field. This configuration is sometimes called the Born approximation as it is equivalent to neglecting orders larger than one in the Born series expansion of the total (or scattered) field. In some references (e.g., Bohren and Huffman 1983), it is additionally made the approximation that the refractive index is not much different from one (|m  1|  1), which simplifies the above equations by replacing (m2 – 1)/(m2 + 2) by 2(m  1)/3. One remarkable result is that the absorption cross-section is the summation over the particle subvolumes of that of the Rayleigh approximation which gives Cabs = 2 k0V Im(m); i.e., absorption is proportional to the volume of the particle. One specific application of the Rayleigh-Gans approximation is for aggregates of spheres where each sphere contributes as a Rayleigh scatterer (Fig. 5c). For the aggregates whose geometry can be described by fractal scaling laws, this method is usually referred to as the Rayleigh-Debye-Gans for Fractal Aggregates (RDG-FA) approximation (Sorensen 2011). The geometrical optics approximation It applies to arbitrarily shaped particles that have characteristic dimensions (a) much larger than the wavelength in vacuum (x  1) and that are made of an optically not too soft material (such that (mR  1) x  1). In that case, the interaction of the incident electromagnetic wave with the particle can be modeled by the superposition of diffraction, reflection, and transmission by the particle undergone by localized rays (Van de Hulst 1981) (Fig. 6). Thus ray-tracing is used to determine the modifications in direction, amplitude, and phase of the field along the path of these rays. At the interface between the particle and its surroundings, Snell-Decartes’ and Fresnel’s laws provide the reflection and refraction angles for perfectly smooth surfaces, and the amplitude and phase of the reflected and transmitted waves, respectively (Born and Wolf 1999). The absorption Fig. 6 Elementary principles of ray-tracing for the geometrical optics approximation

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and scattering cross-sections result from a coherent sum of the contributions of an ensemble of rays that covers the geometric cross-section of the particle. The anomalous diffraction approximation It applies to arbitrarily shaped particles that are made of an optically soft and not too absorbing material (such that |m  1|  1) and have characteristic dimensions (a) much larger than the wavelength in vacuum (x  1) (Van de Hulst 1981). The principles of ray-tracing associated to geometrical optics hold, but as reflection is negligible owing to the small refractive index contrast, the incident rays that cross the particle are not deviated. The scattered field results from the superposition of transmission through the particle with large phase changes and diffraction. Cross-sections derive from the summation of the contributions from an ensemble of incident waves that covers the geometrical cross-section of the particle.

4.2

Regularly Shaped Individual Particles

Although historically the cases of regularly shaped particle were mostly investigated before most of the arbitrarily shaped particle cases, it makes sense that results for the former cases are essentially solutions of the latter cases for specific shapes.

4.2.1 The Lorenz-Mie Theory for Spheres The scattering problem for a homogeneous, linear, isotropic sphere was indeed addressed more than a century ago by Ludwig Lorenz (1890) and Gustav Mie (1908). The methodology, most commonly called Lorenz-Mie or Mie Theory (LMT), was used countless times in science and engineering problems involving electromagnetic absorption and scattering by spheres. The T-matrix method (Sect. 4.1.2) is somehow a generalization of the LMT, which expressed otherwise makes the LMT a sub-case of it. General standard case The Lorenz-Mie Theory applies to spheres of arbitrary size parameter (x, see Sect. 2.1) and complex refractive indices (m). The bases for the theory and their implications can be found in various monographs (e.g., Van de Hulst 1981; Bohren and Huffman 1983). Because of symmetries, only the elements S1 and S2 of the amplitude scattering matrix elements are non-null, meaning that scattering by such a sphere does not depolarize the incident radiation and are expressed as a function of the polar scattering angle θ, size parameter x, and complex refractive index m only, with infinite series expansions (Bohren and Huffman 1983): S1 ðx, m, θÞ ¼

1 X 2j þ 1 ðan ðx, mÞπ n ðθÞ þ bn ðx, mÞτn ðθÞÞ jðj þ 1Þ j¼1

(66)

S2 ðx, m, θÞ ¼

1 X 2j þ 1 ðan ðx, mÞτn ðθÞ þ bn ðx, mÞπ n ðθÞÞ jðj þ 1Þ j¼1

(67)

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where aj and bj, called the Lorenz-Mie coefficients, depend on the size parameter x and complex refractive index m via Ricatti-Bessel functions, and π j and τj are functions involving the Legendre polynomials. The extinction and scattering crosssections derive from these elements to give (Bohren and Huffman 1983): 1   2π X ð2j þ 1ÞRe aj ðx, mÞ þ bj ðx, mÞ 2 k0 j¼1

(68)

1  2 2  2π X ð2j þ 1Þ aj ðx, mÞ þ bj ðx, mÞ 2 k0 j¼1

(69)

Cext ðx, mÞ ¼

Csca ðx, mÞ ¼

According to Eq. 30, the phase function expression involves S1, S2, and Csca, which for Mie scattering can be written in the compact form (Howell et al. 2016; Modest 2013):

pðx, m, θÞ ¼

  2π jS1 j2 þ jS2 j2 k20 Csca ðx, mÞ

¼1þ

1 X

Aj ðx, mÞPj ð cos θÞ

(70)

j¼1

where Pj are the Legendre polynomials and the coefficients Aj are functions of the Mie coefficients aj and bj. The asymmetry parameter can be developed as a function of the Mie coefficients as well (Bohren and Huffman 1983). In practice, numerical calculations of the radiative properties for Mie scattering spheres rely on a choice of a truncation order (linked to the size parameter x) for the infinite series and on accurate algorithms for reckoning the mathematical functions involved. There are indeed special cases where approximate methods are better suited. First-order expansion of the Mie series (Rayleigh approximation) Omitting the terms in the series with order larger than one is acceptable for spheres with small-size parameters (x  1). If in addition the material is not too dense (|m|x  1), then the Mie coefficient b1 is negligible compared to a1; hence, in the Mie series expansions, a single term remains as follows (Bohren and Huffman 1983): S1 ðx, m, θÞ ¼ ix3

m2  1 m2 þ 2

S2 ðx, m, θÞ ¼ S1 ðx, m, θÞ cos θ 2 2 8 4 m  1 Csca ðx, mÞ ¼ πR x 2 3 m þ 2  2  m 1 2 Cabs ðx, mÞ ¼ πR 4x Im 2 m þ2

(71) (72) (73) (74)

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pðx, m, θÞ ¼

 3 1 þ cos 2 θ 4

(75)

These above results are fully consistent with the application of the Rayleigh approximation (Sect. 4.1.3) to the case of a sphere (where V = 4/3πR3 and k0 = x/R). For the polarization component perpendicular to the scattering plane, scattering is isotropic. The absorption cross-section is as expected proportional to the volume of the sphere. For materials having weak spectral variations of the complex index, the absorption  refractive  and scattering cross-sections vary as ω (1/λ0) and ω4 1=λ40 , respectively. Spheres with large-size parameters: geometrical optics approximation Calculations of the series expansion are converging with difficulty for large size parameters (the limit of the best algorithms is however reported to be currently x = 10.000 in Hergert and Wriedt (2012, Chap. 2)). In that case, the geometrical optics approximation that applies to particles with large size compared to the wavelength (see Sect. 4.1.3) is very well suited. One remarkable result is that the extinction cross-section is in that case found to be equal to twice the geometrical cross-section of the sphere (Cext = 2πR2) for sufficiently large-size parameters. This is explained by the effect of diffraction in addition to the geometrical obstruction to incident light. The resulting radiative properties for spheres can be also viewed as the asymptotic limit of the Lorenz-Mie formulas for large size parameters (Van de Hulst 1981; Ungut et al. 1981). Formulations of the radiative properties can be derived in the special cases of large opaque spheres (Modest 2013) or large transparent spheres (Bohren and Huffman 1983). Extensions There are plenty of extensions of the Mie theory, applying to a large number of cases beyond the standard configuration described above (Hergert and Wriedt 2012), such as multilayered concentric spheres, spheres illuminated by Gaussian beams using the generalized Mie theory (GMT) (Gouesbet and Gréhan 2011).

4.2.2 Infinitely Long Cylinders and Spheroids The solution for the case of infinitely long cylinders and spheroidal particles builds on the LMT for spheres. Expressions of the amplitude scattering matrix elements (all four) are developments in series (not reported here; see Kerker 1969; Bohren and Huffman 1983 for infinite cylinders, Asano and Yamamoto 1975 for spheroids). This case of infinite cylinders is of particular interest for calculating the radiative properties of fibrous materials (see a review in Baillis and Sacadura 2000). Spheroids are acceptable approximate shapes for some irregularly shaped particles.

5

Deriving the Radiative Properties of Particles from Experiments

Radiative properties of particles can be inferred from experiments instead of derived from direct calculations. It can be either for the assessment of theories and numerical methods or for the cases where a lack of prior knowledge and/or a quite complex nature of the particles make direct predictions impossible or very uncertain.

27

5.1

Radiative Properties of Particles

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Analog Experiments on Model Particles

In most applications, particles may be either too small or impractical to grab and maintain under an illumination for making controlled experiments. To overcome these difficulties, the scale invariance rule (SIR) in electromagnetic scattering is a convenient basis for allowing more handy experiments. For a scattering object surrounded by an infinite, homogeneous, linear, isotropic, and nonabsorbing medium, the SIR states that the scattering properties are only function of two ~ dimensionless variables: the size parameter x and the refractive index contrast m (refractive index of the object relative to that of the surrounding medium) (Mishchenko 2006). Thus, a single calculation or experiment for a given size parameter (x) can apply to an infinite number of particle size and wavelength pairs, provided the refractive index contrast is unchanged. The practical interest of the SIR for making controlled experiments in electromagnetic scattering is that one can shift the size of scatterers and wavelength to reach more favorable experimental conditions. The real particles are replaced with models – called analog targets – which must have in the new spectral range the exact same electromagnetic properties as in the original range, by using analog materials. The so-called microwave analog to light scattering experiments, pioneered by Greenberg et al. (1961) back to over half a century are essentially when the spectral shifting is from the visible or near-infrared light [0.4–2 μm] to microwaves [0.2–30 cm]. The amplitude and phase of the polarization components of the scattered electric field are derived from the subtraction of the measured incident field (configuration without the particle) to the measured total field when the particle is present. Thus the scattered field polarization components or amplitude scattering matrix elements derived from these experiments can be used to assess scattering theories and numerical methods. The methodology and some applications of these scaled experiments are reviewed in Mishchenko et al. (2000, Chap. 13), Gustafson (2009), and Vaillon and Geffrin (2014).

5.2

Laboratory Experiments on Real Particles

In thermal science and engineering, experiments aimed at inferring the radiative properties of particulate media are based on the following elements and principles (Fig. 7): • A source, spectrally extended on the wavelength interval of interest, or monochromatic (laser). Note that in emission measurements, the particulate medium is the source. • Transfer optics for splitting the radiation beam and collecting a reference power, for controlling the beam direction and polarization state. • A detection system for collecting the scattered light (including the reflected part) and transmitted light. It is comprised of analyzing optics (direction and

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Fig. 7 Elementary principles of transmission-scattering experiments on a particulate medium

polarization), a detector sensitive in the spectral range of interest. The collection can either be bidirectional by moving the detector around the particulate medium, or hemispherical. In these experiments, the intensities or fluxes that are measured can be used to build databases of properties for single scattering, or validate specific models for the radiative properties and radiative transfer, or infer the radiative properties (or parameters of a chosen model) using parameter estimation techniques (see, e.g., Baillis and Sacadura 2000; Aslan et al. 2003, 2006a, b; Mengüç 2003; Kozan et al. 2008; Muñoz and Hovenier 2011; Muñoz et al. 2012). The inference of parameters corresponds to an inverse problem that is ill-posed, particularly when the single scattering regime (Sect. 3.2.2) does not apply, as there may be different sets of parameters that match with the measurement data. In that case various techniques can be used to refine the parameter estimation (see ▶ Chap. 30, “Inverse Problems in Radiative Transfer”), such as restricting the number of parameters (e.g., by using a simplified model for the phase function, see Sect. 3.2.2).

6

Cross-References

▶ Inverse Problems in Radiative Transfer ▶ Near-Field Thermal Radiation ▶ Radiative Properties of Gases ▶ Radiative Transfer Equation and Solutions Acknowledgments The author was hosted by the Department of Mechanical Engineering at the University of Utah when this chapter was written. The financial support from the College of Engineering (W.W. Clyde Visiting Chair award) is acknowledged. The author is thankful to Olivier Dupré for having read carefully the manuscript and made helpful suggestions.

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References Adachi S (2009) Properties of semiconductor alloys: group-IV, III–Vand II–VI semiconductors. Wiley, Chichester, West Sussex, United Kingdom Asano S, Yamamoto G (1975) Light scattering by a spheroidal particle. Appl Opt 14(1):29–45 Aslan M, Yamada J, Mengüç MP (2003) Characterization of individual cotton fibers via lightscattering experiments. AIAA J Thermophys Heat Tran 17(4):442–449 Aslan M, Crofcheck C, Tao D, Mengüç MP (2006a) Evaluation of micro-bubble size and gas holdup in two-phase gas–liquid columns via scattered light measurements. J Quant Spectrosc Radiat Transf 101(3):527–539 Aslan M, Mengüç MP, Manickavasagam S, Saltiel G (2006b) Size and shape prediction of colloidal metal oxide MgBaFeO particles from light scattering measurements. J Nanopart Res 8(6):981–994 Baillis D, Sacadura JF (2000) Thermal radiation properties of dispersed media: theoretical prediction and experimental characterization. J Quant Spectrosc Radiat Transf 67(5):327–363 Bohren CF, Huffman DR (1983) Absorption and scattering of light by small particles. Wiley, New York Born M, Wolf E (1999) Principles of optics, 7th edn. England, Cambridge Chen G (2005) Nanoscale energy transport and conversion. Oxford University Press, New York Gervais F, Palik ED (1991) Handbook of optical constants of solids II. Academic, Boston Gouesbet G, Gréhan G (2011) The generalized Mie theories. Springer, Berlin Greenberg JM, Pedersen NE, Pedersen JC (1961) Microwave analog to the scattering of light by nonspherical particles. J Appl Phys 32(2):233–242 Greffet JJ, Carminati R, Joulain K, Mulet JP, Mainguy S, Chen Y (2002) Coherent emission of light by thermal sources. Nature 416:61–64 Gurton KP, Dahmani R (2005) Effect of surface roughness and complex indices of refraction on polarized thermal emission. Appl Opt 44(26):5361–1642 Gustafson BAS (2009) Scaled analogue experiments in electromagnetic scattering. Springer Light Scatt Rev 4:3–30 Hergert W, Wriedt T (2012) The Mie theory: basics and applications. Springer series in optical sciences. Springer, Berlin Howell JR, Mengüç MP, Siegel R (2016) Thermal radiation heat transfer, 6th edn. CRC Press, Boca Raton Jackson JD (1998) Classical electrodynamics. Wiley, New York Kahnert FM (2003) Numerical methods in electromagnetic scattering theory. J Quant Spectrosc Radiat Transf 79:775–824 Kahnert FM (2016) Numerical solutions of the macroscopic Maxwell equations for scattering by non-spherical particles: a tutorial review. J Quant Spectrosc Radiat Transf 178:22–37 Kerker M (1969) The scattering of light and other electromagnetic radiation. Academic, New York Kozan M, Thangala J, Bogale R, Mengüç MP, Sunkara MK (2008) In-situ characterization of dispersion stability of wo3 nanoparticles and nanowires. J Nanopart Res 10(4):599–612 Mengüç MP (2003) Characterization of fine particles via elliptically-polarized light scattering. Purdue Heat Transfer Celebration, West Lafayette Mishchenko MI (2006) Scale invariance rule in electromagnetic scattering. J Quant Spectrosc Radiat Transf 101(3):411–415 Mishchenko MI (2014) Electromagnetic scattering by particles and particle groups: an introduction. Cambridge University Press, Cambridge, United Kingdom Mishchenko MI, Travis LD, Mackowski DW (1996) T-matrix computations of light scattering by nonspherical particles: a review. J Quant Spectrosc Radiat Transf 55(5):535–575 Mishchenko MI, Hovenier JW, Travis LD (2000) Light scattering by non–spherical particles. Theory, measurements and applications. Academic, New York Mishchenko MI, Travis LD, Lacis AA (2002) Scattering, absorption, and emission of light by small particles. Cambridge University Press, Cambridge, UK

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Mishchenko MI, Travis LD, Lacis AA (2006) Multiple scattering of light by particles: radiative transfer and coherent backscattering. Cambridge University Press, Cambridge Mishchenko MI, Zakharova NT, Khlebtsov NG, Videen G, Wriedt T (2016) Comprehensive thematic t-matrix reference database: a 2014–2015 update. J Quant Spectrosc Radiat Transf 178:276–283 Modest MF (2013) Radiative heat transfer, 3rd edn. Academic, New York Muñoz O, Hovenier JW (2011) Laboratory measurements of single light scattering by ensembles of randomly oriented small irregular particles in air. A review. J Quant Spectrosc Radiat Transf 112(11):1646–1657 Muñoz O, Moreno F, Guirado D, Dabrovska DD, Volten H, Hovenier JW (2012) The Amsterdam Granada light scattering database. J Quant Spectrosc Radiat Transf 113:565–574 Palik ED (1998) Handbook of optical constants of solids. Academic, San Diego Sandus O (1965) A review of emission polarization. Appl Opt 4(12):1634–1642 Sorensen CM (2011) Light scattering by fractal aggregates: a review. Aerosol Sci Technol 35:648–687 Tien CL, Drolen BL (1987) Thermal radiation in particulate media with dependent and independent scattering. Annu Rev Heat Tran 1(1):1–32 Ungut A, Gréhan G, Gouesbet G (1981) Comparisons between geometrical optics and lorenz-mie theory. Appl Opt 20(17):2911–2918 Vaillon R, Geffrin JM (2014) Recent advances in microwave analog to light scattering experiments. J Quant Spectrosc Radiat Transf 146:100–105 Van de Hulst HC (1981) Light scattering by small particles. Wiley, New York, USA Wriedt T (2016) Scattport. URL http://www.scattport.org/. Accessed 28 Oct 2016 Yurkin MA, Hoekstra AG (2007) The discrete dipole approximation: an overview and recent developments. J Quant Spectrosc Radiat Transf 106(1):558–589

Radiative Transfer in Combustion Systems

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Pedro J. Coelho

Contents 1 2 3 4

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Laminar Flames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Turbulent Flames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Laboratory Combustion Chambers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Gas-Fired Combustion Chambers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Liquid-Fired and Solid-Fired Combustion Chambers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Industrial Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Gas Turbine Combustors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Industrial Furnaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Utility Boilers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Other Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Cross-References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

Thermal radiation is an important heat transfer mode in many combustion systems. This article addresses experimental and computational works on radiative transfer in these systems. Attention is restricted to works where thermal radiation was accurately measured or simulated. The effects of radiative transfer in laminar flames and a few works where these effects were investigated are discussed first. Then, turbulent free flames are addressed. The importance of non-gray models for reliable prediction of thermal radiation, especially in

P. J. Coelho (*) LAETA, IDMEC, Department of Mechanical Engineering, Instituto Superior Técnico, Universidade de Lisboa, Lisbon, Portugal e-mail: [email protected] # Springer International Publishing AG, part of Springer Nature 2018 F. A. Kulacki (ed.), Handbook of Thermal Science and Engineering, https://doi.org/10.1007/978-3-319-26695-4_61

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nonluminous gaseous flames, is emphasized. The interaction between turbulence and radiation is also highlighted. Confined turbulent flames in laboratory combustion chambers are addressed next, beginning with gaseous flames and then discussing radiative transfer in liquid spray and coal flames. Finally, industrial applications are considered, namely, gas turbine combustors, industrial furnaces, and utility boilers, and a brief reference to other applications is made. A few general conclusions are summarized in the last section.

1

Introduction

Thermal radiation is the dominant heat transfer mode in many combustion systems, particularly in the case of large furnaces and boilers (Hottel and Sarofim 1967). Its role may be of secondary importance in small nonluminous flames, or in small combustors, but it generally influences the temperature of the medium. Hence, even if radiative heat fluxes are small, it is still important to account for radiation in the numerical simulation of combustion systems when an accurate estimation of the temperature field is required, e.g., in the prediction of pollutant emissions, which are strongly dependent on the temperature. Accordingly, thermal radiation needs to be taken into account in the vast majority of combustion problems. A relatively old but comprehensive review of thermal radiation in combustion systems was published by Viskanta and Mengüç (1987). More recently, Viskanta (2005) and Modest and Haworth (2016) published books on this subject, and Coelho et al. (2015) presented guidelines for the selection of a radiation model and for the evaluation of the radiative properties of the medium in the numerical simulation of reactive flows. Thermal radiation appears as a source/sink term in the energy conservation equation and is expressed by the divergence of the radiative heat flux vector. The energy conservation equation may be written as follows, taking the specific enthalpy, h, which includes the sensible and chemical enthalpy, as the dependent variable (Poinsot and Veynante 2005):
 @ ðρ hÞ @ ρ uj h @p @p @ui @jq, j @qR, j þ þ uj ¼ þ τij   @t @t @xj @xj @xj @xj @xj

(1)

where xj is the coordinate in jth direction; t the time; uj the velocity component in jth direction; ρ the density; p the pressure; t the time; τij the viscous tensor; jq,j the jth component of the heat flux vector, which includes heat conduction and the heat flux due to diffusion of species with different enthalpies; and qR,j the jth component of the radiative heat flux vector. The divergence of the radiative heat flux vector is evaluated as follows (Howell et al. 2011; Modest 2013): @qR, j ¼ @xj

ð1 0

 κλ 4π I bλ 



ð 4π

I λ dΩ dλ

(2)

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where κλ is the spectral absorption coefficient, Iλ the spectral radiation intensity, and Ibλ the blackbody spectral radiation intensity. The outer integration is carried out over all wavelengths, λ, and the inner one over all directions, Ω being the solid angle. Thermal radiation influences the enthalpy field and thereby the temperature and density fields. The density field then influences the flow field, which also appears in Eq. 1. Therefore, the governing equations for mass, momentum, mass fractions of chemical species, and energy are coupled and need to be solved simultaneously along with radiation. In the case of liquid or solid fuels, additional equations are needed to describe the transport of mass, momentum, and energy of the dispersed phase. If the flow is turbulent, and apart from a few geometrically and physically simple academic problems, which may be solved using direct numerical simulation (DNS), the governing equations are either averaged in time or filtered. Time averaging yields the so-called Reynolds-averaged or Favre-averaged Navier-Stokes equations. The acronym RANS is used below to denote the simulations based on these equations, even though Favre averaging is usually employed in the case of reactive flows. The filtered governing equations are obtained in the case of large eddy simulations (LES). These equations include additional unknowns and need closure assumptions, associated with the turbulence and combustion models. It is apparent from the above explanation that the numerical simulation of reactive flows, particularly in the case of turbulent flows, is a complex and computationally demanding task, whose difficulty is enhanced due to thermal radiation. As far as thermal radiation is concerned, the challenge is to evaluate the radiative source term of the energy equation with an accuracy similar to that of the other terms of the governing equations and with a minimal increase in computational time. This requires a wise selection of the radiation model (solution method of the radiative transfer equation) and of the model for the calculation of the radiative properties of the medium (radiative properties of gases, radiative properties of particles). Guidelines on this issue may be found elsewhere (Coelho et al. 2015) and will not be addressed here. The main focus of this article is on the application of accurate radiation models to the simulation of combustion systems. Rather than a thorough review on the subject, which may be found elsewhere (Viskanta 2005; Modest and Haworth 2016), only works in which thermal radiation was accurately measured or simulated are addressed to illustrate its role in combustion systems. Laminar and turbulent free flames are described first, and applications to combustion chambers and industrial combustion systems are presented thereafter.

2

Laminar Flames

The influence of radiative transfer in turbulent flames is usually more important than in laminar flames, but the latter cannot be ignored. Therefore, the main effects found in laminar flames are described first. A few experimental and theoretical studies on this subject have been reported, along with numerical simulation of one- and

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two-dimensional premixed, non-premixed, and partially premixed flames. In general, these simulations are carried out using detailed chemistry and accurate thermal and transport property models. The computational requirements of laminar flame simulations are typically much lower than in the case of turbulent flames, and complexities arising from the turbulence-chemistry interaction are absent. Therefore, accurate methods for the solution of the radiative transfer equation (RTE) and for the calculation of radiative properties may be used without excessive computational cost. Studies of premixed laminar flames have shown that heat losses due to radiation influence the laminar flame speed, the flammability limits, and the critical stretch rate at which extinction occurs in premixed flames (Ju et al. 2000; Jayachandran et al. 2014). Other studies of laminar diffusion flames revealed that flame extinction due to radiation occurs at low stretch rates (see, e.g., Wang et al. 2007). These effects are due to the decrease of the flame temperature due to radiation, which then influences all the temperature-dependent phenomena. An optically thin approximation (OTA) may be sufficient to obtain satisfactory predictions in the case of small, weakly radiating flames, but otherwise absorption needs to be taken into account. An investigation of the effect of the radiation model in a two-dimensional axisymmetric laminar diffusion flame of methane with coflow of air was reported by Liu et al. (2004a). The OTA and the discrete ordinates method (DOM) were used along with several implementations of the statistical narrowband correlated-k (SNBCK) model. Soot was also considered. Predictions of temperature and soot concentration were in good agreement with experimental data. The effect of soot absorption was marginal in the studied flame, and so the OTA, which neglects absorption of emitted radiation, under predicts the centerline flame temperature by only 17 K in comparison with the solution of the RTE using the DOM, which accounts for absorption. The effects of gas and soot radiation on soot formation in counter flow ethylene diffusion flames were numerically investigated in another study (Liu et al. 2004b). Reasonable predictions of soot volume fraction were obtained. The influence of radiation on the temperature and on the soot volume fraction is illustrated in Fig. 1. The flame temperature and soot volume fraction were more affected by radiation from the gas than from soot. Moreover, radiation from soot was less important than normally found in coflow flames per unit soot volume fraction. A more recent investigation of the role of radiation on soot production is reported in Demarco et al. (2013) who simulated 24 two-dimensional axisymmetric laminar diffusion flames of several hydrocarbons burning in air. Normal flames (fuel injected through a central tube with a coflow of air) and inverse flames (same configuration, but with air in the central tube and fuel coflow) at atmospheric and microgravity conditions were considered. The finite volume method (FVM) and the OTA, gray (based on the Planck-mean absorption coefficient), full-spectrum correlated-k (FSCK), and SNBCK models were used to calculate thermal radiation. Predictions were in good agreement with available experimental data. Both soot and gas radiation were important, except for normal flames at atmospheric pressure, in which soot radiation was negligible. The OTA was found to be acceptable when

Fig. 1 Predicted temperature and soot volume fraction profiles along the stagnation streamline of a counter flow ethylene laminar diffusion flame. XO,o denotes the oxygen molar fraction in the oxidizer (Reprinted from J Quant Spectrosc Radiat Transf, 84, Liu et al., Effects of gas and soot radiation on soot formation in counterflow ethylene diffusion flames, 501–511, Copyright (2004b), with permission from Elsevier.)

T, K

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2500 with radiation 2250 without radiation 2000 optically thin 1750 1500 1250 1000 750 500 Fuel 250 –0.4 –0.2 0.0 0.2

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0.0 -0.10 -0.05 0.00 0.05 0.10 0.15 0.20 0.25 0.30 x, cm

the optical thickness of the medium, based on the radius of the flame and on the Planck-mean absorption coefficient, did not exceed 0.05. These conditions occurred for all inverse flames and most normal flames at atmospheric pressure. When radiation absorption is significant, the OTA and the gray model are not satisfactory, while the FSCK and SNBCK perform well. It was also found that radiation from soot could be modeled as gray, in contrast with gas radiation. Guo and Smallwood (2007) studied the interaction between soot and NO in a laminar axisymmetric diffusion flame of ethylene with coflow of air. Radiative transfer was calculated using the methods employed by Liu et al. (2004a, b). They concluded that soot formation significantly suppresses the formation of NO due to the thermal effect induced by radiation and to a chemical effect associated to consumption of acetylene in soot formation. The latter process reduces the concentration of the CH radical that is involved in the formation of NO via the prompt mechanism, which was the dominant NO formation mechanism for the studied conditions. One-dimensional counter flow laminar methane/air diffusion flames at high pressure were investigated by Zhu and Gore (2005). They performed calculations for adiabatic and optically thin conditions and compared them with an analytical solution of the RTE with the radiative properties evaluated using the weighted-sumof-gray-gases (WSGG) model. Figure 2 shows the influence of radiation on the predicted peak temperature for various pressures. When the pressure rises from 1 to

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Fig. 2 Influence of radiation on the predicted peak temperature for various pressures in a one-dimensional counter flow laminar methane/air diffusion flames (Reprinted from Combust Flame, 141, Zhu and Gore, Radiation effects on combustion and pollutant emissions of high-pressure opposed flow methane/air diffusion flames, 118–130, Copyright (2005), with permission from Elseiver.)

Adiabatic Emission/Absorption Emission

2400

u =10 cm/s 2300

2200

2100

2000 0

10

20 30 Pressure, p (atm)

40

40 atm, the peak temperature increases by 270 K for adiabatic conditions, but only by 220 K if both emission and absorption are considered. The effects of radiation increase with pressure, as expected. If radiation is ignored, the temperature is overestimated by 90 K at 40 atm, in contrast with 40 K at 1 atm. If radiative emission is considered but absorption is neglected, radiation is underestimated by 110 K at 40 atm and by 100 K at 1 atm. It was also concluded that absorption is more significant at low stretching rates and that neglecting radiation significantly overestimates soot and NO emissions.

3

Turbulent Flames

The numerical simulation of turbulent flames is far more complex and computationally demanding than that of laminar flames. Even in the case of two-dimensional axisymmetric nonluminous turbulent diffusion gaseous flames, turbulence represents a major challenge. Direct numerical simulation is not feasible, except in relatively simple geometrical and physical problems. Therefore, RANS or LES are usually employed, as pointed out in the introduction, and this leads to the appearance of new unknowns in the governing equations. Closure of these equations requires a turbulence model, a combustion model, and to account for turbulence-combustion and turbulence-radiation interaction (TRI). Turbulence-combustion interaction arises from the strongly nonlinear dependence of the reaction rates of the species on the temperature, and turbulence-radiation interaction is mainly due to the nonlinear dependence of radiative emission on the temperature. Hence, the mean value of the reaction rate of a chemical species is rather different from the reaction rate of that species calculated using the mean temperature and the mean mass fractions of the species. Similarly, the mean radiative emission is different from the radiative emission calculated using the mean values of temperature and mass fractions.

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As far as radiation is concerned, many authors used a simple OTA, and most calculations ignored TRI or took only the temperature self-correlation into account (see Coelho 2007, for a comprehensive review of TRI in reactive flows). The temperature self-correlation is defined as the ratio of the mean value of the fourth power of temperature to the mean temperature raised to the fourth power. Only a few works considered TRI in more detail, most of them relying on the optically thin fluctuation approximation (OTFA). The OTFA, which should not be confused with the OTA, states that the mean radiative absorption in the RTE may be expressed as the product of the mean absorption coefficient by the mean radiation intensity, i.e., the correlation between the absorption coefficient and the radiation intensity is neglected. This is justified by the expected weak correlation between those two quantities, since the absorption coefficient depends only on the local temperature and chemical composition, while the radiation intensity depends on the temperature and chemical composition along the optical path. While the simulation of turbulent flames requires the simultaneous solution of the governing equations for mass, momentum, energy, and radiation, in addition to the equations associated with the turbulence and combustion models, useful insight into the role of radiation and TRI may be achieved by performing decoupled simulations. In this case, the flame structure is experimentally determined or computed using a measured fraction of radiative heat loss. Thermal radiation is then evaluated using the temperature and chemical composition as input data. These decoupled simulations are addressed first, before considering coupled ones. Faeth and co-workers were among the firsts to perform a comprehensive experimental and computational investigation of the structure of turbulent free jet diffusion flames and of the influence of radiation on the flame structure and heat fluxes (Faeth et al. 1989). The simulation of the flame structure was carried out using the measured fraction of radiative heat loss as input and decoupled from the radiative transfer calculations, which were carried out in a post-processing stage. The integral form of the RTE was solved along lines of sight for a statistically significant number of realizations of instantaneous temperature and species concentration profiles, generated using a stochastic method, thus allowing TRI to be accounted for. The radiative properties of the medium were calculated using the statistical narrowband (SNB) model. In all studied flames, turbulence contributed to increase the mean spectral radiation intensity. The effect of turbulent fluctuations on the mean spectral radiation intensity was about 10% for carbon monoxide/air flames, 10–30% for methane/air flames, of the order of 100% for hydrogen/air flames, ranged between 40% and 100% in acetylene/air flames, and between 50% and 300% in ethylene/air flames. While these ranges may be influenced by the simplifying assumptions used in the employed stochastic method, they provide a good indication of the impact of TRI. In the last 25 years, a joint effort has been undertaken by several investigators over the world to provide highly accurate experimental data for velocity, temperature, and species concentration in turbulent diffusion flames, acquired by advanced optical diagnostic techniques. These data are aimed at improving the understanding of the physics of turbulent flames and at providing an accurate database for

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validation of mathematical models. The reports of the International Workshop on Measurement and Computation of Turbulent Nonpremixed Flames give detailed information on this work. These data were extended by Gore and co-workers (Zheng et al. 2003a, b), who measured spectral radiation intensities for several of these flames using a fast infrared array spectrometer. They also performed predictions of the spectral radiation intensity by integrating the RTE along lines of sight and using the SNB model, an improved stochastic method, and a tomography-like technique to account for TRI. Figure 3 shows an example of their results for the well-known flame D (Barlow and Frank 1998). The plotted data correspond to the mean and root mean square (rms) of the spectral radiation intensity measured at three different planes normal to the flame axis (x/d = 30, 45 and 60, where d is the burner diameter and x the axial distance to the burner), at the end of lines of sight at different distances from the flame axis (r denotes the distance from the point under consideration to the flame axis). A very satisfactory agreement between predictions and measurements was obtained. The results of Gore and co-workers include predictions of the probability density function (PDF), power spectral density, and autocorrelation coefficient of the spectral radiation intensities. The predictions are typically within 10% of

Flame D, d = 7.2mm, Re = 22,400 x/d 30 45 60

Il (W/m2 sr µm)

6000

Mea. Cal.

4000

2000

l = 4.27 µm

0 0.9

0.6

2

I'l / Il

Fig. 3 Predicted and measured spectral mean and rms of the radiation intensity at a wavelength λ = 4.27 mm for Sandia’s flame D (Reprinted from J Heat Transf, 125, Spectral radiation properties of partially premixed turbulent flames, 1065–1073, Copyright (2003b), with permission from ASME)

0.3

0.0 0.00

0.02

0.04

0.06 r/x

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the experimental data, which is within the experimental uncertainty, including the uncertainty in the data used in the SNB model. Coelho (2004) investigated the accuracy of the OTFA for the well-known flame D (Barlow and Frank 1998) using experimental data of temperature and species concentrations as input for the radiative calculations. These were performed by solving the integral form of the RTE along lines of sight and using the SNB and the correlated-k (CK) models to evaluate the radiative properties of the medium. Although the measured fraction of radiative heat loss in flame D is small (5.1%), and the temperatures may be relatively well predicted using the OTA, the computed radiative heat loss depends significantly on how TRI is modeled. The accuracy of the OTFA was assessed by means of comparison with a reference solution obtained by taking the average values of the solution of the integral form of the RTE along lines of sight for a large number of realizations. A stochastic method was employed to generate the instantaneous temperature and species concentration fields. It was found that the results predicted by OTFA are in good agreement with the reference solution, with the largest differences being about 4%. A comparison between the predicted and measured nondimensional radiant power along the axial direction is shown in Fig. 4. The results obtained by neglecting TRI, denoted by number 1, and those that neglect the fluctuations of the absorption coefficient of the medium (number 2) clearly under predict the radiant power calculated using TRI (number 3) and the stochastic method (number 4), the latter being relatively close to the experimental data. As far as coupled turbulent reactive flows/radiative transfer simulations are concerned, the conserved scalar/prescribed probability density function (PDF) formulation and the joint composition PDF transport model are often employed. Flame D was simulated by Coelho et al. (2003) who used the first approach, while Li and

Fig. 4 Predicted and measured normalized radiative power along the axial direction of Sandia’s flame D (circles, experimental data; triangles, ray tracing/ CK; solid lines, ray tracing/ SNB) (Reprinted from Combust Flame, 136, Coelho, Detailed numerical simulation of radiative transfer in a non-luminous turbulent jet diffusion flame, 481–492, Copyright (2004), with permission from Elsevier.)

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Modest (2003) relied on the second one. The former calculations were performed using the Reynolds stress model, the laminar flamelet model, the DOM, and the spectral line-based weighted-sum-of-gray-gases (SLW) models for radiation, along with the OTFA. The predicted fraction of radiative loss was 5.3%, which closely matches the experimental data. The latter calculations were carried out using the k-e model, the P1 approximation, and the FSCK method for radiation, along with the OTFA. The calculated fraction of radiative loss (4.6%) was also in good agreement with the experiments. Several other authors modeled flame D in the framework of RANS, using different approaches, and a comparison between their predictions of the fraction of radiative loss is reported in Coelho (2012). It was found that the OTA significantly overestimates the fraction of radiative heat loss, regardless of the turbulence and combustion models employed. Moreover, if the medium is treated as gray and the Planck-mean absorption coefficient is employed in the solution of the RTE using the DOM, the predicted fraction of radiative heat loss is still too high, demonstrating that the non-gray nature of gas emission needs to be taken into account to satisfactorily estimate the fraction of radiative loss. In addition, all the calculations carried out using global models (WSGG, SLW, FSCK) or band models, which account for the non-gray gaseous emission, perform better than the gray gas assumption, no matter whether the OTA is assumed or the RTE is solved. When the non-gray gas radiation is taken into account, the predicted mean temperatures are higher than in the case of calculations that assume a gray medium or use the OTA, but lower than in the case of adiabatic calculations. The mean temperatures decrease due to TRI in comparison with those calculated without considering TRI. The NO formation via the thermal mechanism is highly sensitive to the temperature field, and therefore little differences in the temperature field arising from different methods employed to simulate radiation may yield significantly different NO concentrations (Habibi et al. 2007). Even though the heat loss due to radiation is small for flame D, the predicted fraction of radiative loss was 10.5% using the simple OTA and the Planck-mean absorption coefficient (Frank et al. 2000) and 3.7% using a Monte Carlo method, the line-by-line (LBL) method, and full TRI (Wang and Modest 2008). Despite this, the temperature differences between these two cases are relatively small and do not exceed 100 K. Li and Modest (2003) performed several simulations for fictitious flames similar to flame D, but with a fuel nozzle diameter that was doubled and quadrupled, while maintaining the Reynolds number fixed. This leads to an optical thickness of the medium that is two and four times greater than the original one, thereby increasing the role of radiation. The temperature drop due to radiation is higher for the largest flames, as expected, and TRI contributes to about one third of the peak temperature drop. The Monte Carlo method employed by Wang and Modest (2008) relaxes the OTFA and accounts for both emission and absorption TRI. However, absorption TRI was found to be marginal, for both flame D and the artificially scaled ones, except in the optically thick spectral regions at the flame sheet, and the OTFA is acceptable for calculation of overall quantities, e.g., the net radiative heat loss.

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Thermal radiation is stronger in luminous flames due to the effect of soot radiation. This may overshadow the effect of gas radiation, depending on the concentrations of soot and emitting gaseous species, and TRI effects may be more important than in nonluminous flames. Gore et al. (1992) were among the firsts to perform a coupled flame structure/radiation simulation using an accurate treatment for radiation. They employed the time-averaged integral form of the RTE using the SNB model and the OTFA. They reported a good agreement with experimental data and showed that uncoupled calculations based on the assumption of a uniform fraction of radiative heat loss strongly under predict the temperature and the radiation intensity. More recently, Tessé et al. (2004), Wang et al. (2005), and Mehta et al. (2010a, b) reported simulations of turbulent sooting flames, all of them based on the Monte Carlo method for radiation and accounting for both emission and absorption TRI. Tessé et al. (2004) simulated an ethylene-air jet flame and found that both the net and emitted radiative power are higher with TRI than without TRI, as also observed in nonluminous flames. The total fraction of radiative loss is 30%, which significantly exceeds the values found in nonluminous flames. Wang et al. (2005) studied an oxygen-enriched turbulent jet propane flame. The influence of non-gray gas radiation was found to be important, even in the case of strong soot radiation, and needs to be accounted for to correctly predict the radiative heat fluxes, soot and NOx formation upstream, and within the flame zone. The non-gray gas radiation is more important downstream than upstream and affects soot burnout. The temperature and emission of NOx decrease due to radiation from soot, especially at the flame tip. Mehta et al. (2010a, b) studied six flames that differ in the Reynolds number, fuel (ethylene and 90% methane-10% ethylene), and oxygen concentration in the oxidizer. Soot contributed by up to 70% of the emitted radiation, and more than 90% of that fraction left the domain. However, only 40–60% of the emitted gas radiation left the domain, the remainder being absorbed. The results further showed that absorption TRI is negligible for all studied flames, in contrast with emission TRI, which is important for all of them. Radiative emission increases by 30–60%, while the net radiative heat loss increases by 45–90%, due to TRI. These values exceed those typically found in nonluminous flames. In a recent study, Pal et al. (2015) compared the accuracy and computational requirements of several radiation models in the prediction of flame D, and two similar flames scaled by a factor of 4, one of them including soot. They used the Monte Carlo method, FVM, P1 and P3 approximations as RTE solvers, as well as the OTA, and the LBL, FSCK, and gray assumption for radiative gas properties. Accurate results were obtained for flame D, no matter the radiation model employed, as long as non-gray gas properties are considered, in agreement with past work. Even NO predictions are little sensitive to the radiation model, provided that non-gray gas properties and TRI are taken into account. As far as the other two flames are concerned, and taking the Monte Carlo/LBL results as reference, the OTA yields poor results, while the FVM and P3 models along with FSCK perform better than the P1 model. Radiation without TRI causes a decrease of the peak flame temperature by about 150 K for the luminous flame, 100 K for the nonluminous scaled flame, and

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30–45 K for flame D. TRI causes an additional decrease of the peak flame temperature by about 150 K, 100 K, and 25 K, respectively. The large temperature drops observed for optically thicker flames have a major impact in NO concentration. In statistically stationary simulations, the Monte Carlo/LBL calculations were surprisingly found to be less expensive than the FVM ones, no matter the model used to compute the radiative properties of the gases. All turbulent flame simulations described above were carried out in the framework of RANS. A few investigations of radiation and TRI were performed using DNS and LES, but they are restricted to academic configurations, as reviewed in Coelho (2012) and Modest and Haworth (2016). An exception is the work of Gupta et al. (2013), who simulated four flames: flame D, the fourfold scaled flame D, and these two flames with artificial soot. They found that TRI contributes to a significant part of radiative emission, even for flame D, and that the contribution increases for optically thicker flames. An interesting finding was that subgrid-scaled fluctuations of emission TRI were larger than resolved fluctuations, in contrast with former expectations based on uncoupled combustion/radiation simulations (Coelho 2009). Absorption TRI was negligible, except for high optical thickness. Moreover, the contribution of subgrid-scale fluctuations for absorption TRI was always negligible.

4

Laboratory Combustion Chambers

4.1

Gas-Fired Combustion Chambers

Most simulations of turbulent reactive flows in laboratory combustors and industrial applications that account for radiative heat transfer rely on a gray medium assumption and ignore TRI, or consider only the temperature self-correlation, despite the important role that non-gray effects and TRI may have, as described above. Even when WSGG or a band model is employed, most authors solve the RTE using a gray absorption coefficient, which is determined from Beer’s law (see, e.g., Modest 2013) using the total emissivity of the model evaluated from the WSGG or from the band model. This procedure corresponds to a gray implementation of the WSGG model. However, Beer’s law only holds on a spectral basis. If the medium were gray, Beer’s law could be applied, but this is never the case in combustion systems, due to the spectral nature of gaseous radiation. Accordingly, there is no theoretical basis to calculate the gray absorption coefficient using this method. Moreover, application of Beer’s law requires the specification of the mean beam length, which is usually estimated from the overall size of the domain or from the size of the control volume under consideration. However, the WSGG, as well as other global or band models, may be implemented as a non-gray model by solving the RTE for all the gray gases and for the clear gas. In this case, Beer’s law is no longer required, and there is no need to estimate a mean beam length. Obviously, the non-gray implementation is more computationally expensive. A comparison between the results obtained using the gray and non-gray implementations of the WSGG for an idealized three-dimensional rectangular

Radiative Transfer in Combustion Systems

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1400 z =0.4 m 1350 T, K

Fig. 5 Predicted gas temperature profiles at different locations of a threedimensional rectangular furnace. The three top lines at every location represent non-gray implementations of the WSGG model, which differ in the coefficients of the model and in the boundary conditions, while the three bottom ones correspond to gray implementations (Reprinted from Int J Heat Mass Transf, 41, Liu et al., A comparative study of radiative heat transfer modelling in gas-fired furnaces using the simple grey gas and the weighted-sum-of-grey-gases models, 3357–3371, Copyright (1998), with permission from Elsevier.)

1300 1250 1200 1150 1400 z =2 m 1350

T, K

28

1300 1250 1200 1150 1300 z =3.6 m

T, K

1250 1200 1150 1100 0.0

0.5

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1.5

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enclosure was reported by Liu et al. (1998). The medium is homogeneous and a uniform radiative heat source was prescribed. Figure 5 compares the gas temperatures profiles at three different locations. It is clear that there are significant differences between the gray and non-gray implementations. The errors introduced by the gray implementation may be quite large, as demonstrated by several authors for benchmark problems (e.g., Goutiere et al. 2000; Coelho 2002; Demarco et al. 2011). However, these errors are attenuated in sooting media, becoming smaller with the increase of soot concentration (Bressloff 1999), since the variation of the spectral absorption coefficient of soot with the wavelength is much smoother than that of gaseous species. Song and Viskanta (1987) were the first authors to use a non-gray model and to account for TRI, along with the OTFA, in a coupled CFD/radiative transfer simulation of a combustion chamber. Adams and Smith (1995) and Hartick et al. (1996)

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also accounted for TRI, relying on the OTFA, but the media were treated as gray. Experimental data for validation purposes was limited in all these publications. None of them found a significant influence of TRI on the temperature, but Song and Viskanta (1987) speculated that TRI would become significant if the size of the flame were large in comparison with the volume of the furnace, and Hartick et al. (1996) found a significant influence on NO formation and emission. One of the earliest detailed experimental in-flame data sets that includes velocity, temperature, species concentration, as well as wall heat flux measurements, along with well characterized boundary conditions, was obtained in the BERL furnace (Sayre et al. 1994a). Sayre et al. (1994b) solved the integral form of the RTE along lines of sight using the exponential wideband (EWB) model. The measured temperature and species concentration were used as input data for the radiative transfer calculations. The predicted total radiation intensity was in good agreement with the experimental data. Lallemant et al. (1996) report additional predictions based on a gray gas assumption, several versions of the WSGG model, the SLW model, and the CK method. The gray gas model performs poorly. In contrast, the predictions of the non-gray WSGG model are in reasonable agreement with the experimental data, but there are noticeable differences between different versions. The CK method yields results close to those of the EWB model, and in satisfactory agreement with the data, while the SLW model did not perform so well, particularly when band overlapping was taken into account. However, these results should be regarded with some caution. First, they did not account for TRI. Second, the coefficients of the various models were derived using different experimental spectroscopic data, and therefore the results do not necessarily reflect the accuracy of the models, but are influenced by the accuracy of those data. If this were not the case, the SLW model with band overlapping should yield accurate results. The BERL furnace has been simulated by various researchers using coupled CFD/radiation solvers, but most of them treated the medium as gray and ignored TRI. However, Krishnamoorthy (2013) compared gray and non-gray calculations using the DOM and the WSGG model and concluded that the incident radiative flux profiles were clearly different for the two cases, particularly in the regions of significant temperature gradients. However, the axial temperature profiles were similar, with maximum differences of 50 K away from the burner. The temperature self-correlation had a marginal effect on the radiative heat fluxes incident on the wall. The first coupled CFD/radiative transfer simulations based on the PDF transport model for combustion modeling that accounted for non-gray radiation and TRI in a combustion chamber were presented by Mazumder and Modest (1999). A methane flame burning in air was stabilized by a bluff body placed in a channel. Both the velocity and the composition were included in the joint PDF. The P1 approximation, the EWB model, and the OTFA for TRI were considered in the radiative transfer calculations. The increase of radiative emission due to TRI caused a decrease of about 100 K in the flame temperatures and an increase of 40% of the incident heat fluxes on the wall. The joint velocity-composition PDF transport method is computationally expensive and difficult to apply to general 2D and 3D problems, and subsequent works by

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Modest and co-workers relied on the k-e model for the calculation of the turbulent flow field and on the composition PDF transport method for combustion modeling, considering only the temperature and the mass fraction of the species as random variables. The finite volume method was used to solve the fluid flow equations and the Monte Carlo to so solve the transport PDF equation. These models were applied by Li and Modest (2002) to simulate a methane-air flame in an axisymmetric combustor. Radiation was modeled using again the P1 approximation, but the FSCK method was employed to calculate the radiative properties of the medium, along with the TRI and the OTFA. The role of different TRI correlations was investigated by performing additional uncoupled CFD/radiation calculations and revealed that the absorption coefficient-Planck function correlation is the most important one. The need to reduce CO2 emissions has motivated the scientific community to investigate CO2 capture and sequestration techniques, and oxy-fuel combustion. In this context, a comprehensive set of experiments was carried out in a threedimensional rectangular combustion chamber burning natural gas in an atmosphere of oxygen (Lallemant et al. 2000). This combustion chamber was the subject of several CFD simulations, but radiation was generally modeled assuming that the medium is gray and ignoring TRI. An exception is the work of Zhao et al. (2013) who used the composition PDF transport model for combustion and the Monte Carlo method for radiation. The radiative properties of the medium were evaluated using LBL databases. A comparison between results obtained using full TRI, emission TRI only (OTFA), and no TRI was reported. Although strong local effects of TRI were found, the overall influence on the predicted mean temperature and species molar fractions was negligible. This was explained by the small fluctuations of temperature and species concentrations outside of the turbulent flame. The methods used by Zhao et al. (2013) for the calculation of the thermal radiation are the most accurate ones, but they are computationally expensive and become unaffordable for 3D simulations and industrial applications, as well as for LES. Centeno et al. (2015) assessed the accuracy of the much more economical DOM and WSGG model by means of comparison with LBL calculations for a two-dimensional axisymmetric combustion chamber. Two different versions of the WSGG model were used, one based on a constant pw/pc ratio, pw being the partial pressure of H2O and pc the partial pressure of CO2, and the other one allowing for arbitrary ratios, using coefficients based on the most recent spectroscopic databases. The influence of TRI was not considered in this work. The temperature and species concentration fields were prescribed using the solution of a global simulation. Figure 6 shows the radiative heat source evaluated using the LBL method, which is taken as the reference solution, and the relative error of the two different versions of the WSGG model. It can be seen that the results of both WSGG versions are in rather satisfactory agreement with the reference LBL solution, with maximum errors of about 8% and average errors of about 2%, with a slightly better performance of the model that allows for arbitrary pw/pc ratios. The performance of the WSGG is rather satisfactory, given the strong non-uniformity of temperature and species concentration fields in the studied problem, and significantly better than that reported by other

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0.25

a

a

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Fig. 6 Predicted radiative heat source (a) and relative error of the WSGG models with fixed (b) and arbitrary (c) pw/pc ratios (Reprinted from J Quant Spectrosc Radiat Transf, 156, Centeno et al., Evaluation of gas radiation heat transfer in a 2D axisymmetric geometry using the line-by-line integration and WSGG models, 1–11, Copyright (2015), with permission from Elsevier.)

authors, but it is important to realize that the spectroscopic database used for the LBL and WSGG models was the same, in contrast to many earlier works available in the literature. Moreover, the WSGG with fixed and varying pw/pc ratio were about 17,000 and 3400 times faster than LBL model, respectively. All the previously mentioned works on laboratory combustion chambers were carried out using RANS. Poitou et al. (2012) performed LES of a premixed propane/ air flame stabilized by a triangular flame holder and confined in a three-dimensional rectangular chamber. They used the DOM and the FSCK model for radiation and accounted for TRI using the OTFA. The temperature levels of the burnt gas decreased by 100–150 K, while those of the unburnt gas increased by a similar amount due to radiation. All TRI correlations were found to be important and must be taken into account. The total radiative heat loss increased by 7.4% due to TRI, but maximum local changes of the radiative source term were about 20%.

4.2

Liquid-Fired and Solid-Fired Combustion Chambers

The combustion of liquid and solid fuels has also been the subject of intensive research. However, most works treat radiation in a relatively crude way. Tsuji et al. (2003) reported detailed measurements carried out in furnace No. 1 of the International Flame Research Foundation using different fuels: natural gas, light fuel oil, heavy fuel oil, and coal. The measured radiative heat fluxes along the axial direction were approximately uniform for all fuels, being higher (350–390 kW/m2) for the coal and heavy fuel oil, due to radiation from soot and coal particles, than for natural

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gas and light fuel oil. Predictions based on the solution of the integral form of the RTE using the EWB model were also reported for a radial profile. The computed radiation intensity was much lower than that measured if only gas radiation was accounted for. Assuming that the measured particulate concentration is soot, good predictions were obtained by adjusting the constant that appears in the spectral soot absorption coefficient derived from the Rayleigh theory. The first spray simulations that accounted for non-gray gas radiation were presented by Choi and Baek (1996) for a cylindrical combustion chamber. They used the DOM and the WSGG model, but did not account for radiation from the droplets. This limitation was later relaxed by Baek et al. (2002), who accounted for gray radiation from droplets and soot and used geometric optics to determine the radiative properties of the droplets. The influence of radiation from droplets and soot on species concentration, including NO, was marginal for the studied conditions. In contrast, an experimental study of kerosene spray flames at pressures up to 13 bars revealed that radiation from soot provides the main contribution to the total flame radiation, while gas radiation has a minor influence (Fischer and Moss 1998). Recently, Roy et al. (2016) presented high-pressure spray simulations for a laboratory-scale diesel engine. They compared four different methods for radiation, namely, the Monte Carlo/LBL, P1/FSCK, P1/gray gas, and OTA. The radiative properties of droplets were evaluated using geometric optics. The temperature distribution was affected by radiation, but the evolution of the liquid spray was not. Spectral effects are expected to be less important in the case of solid fuels, due to the continuum emission spectrum of solid particles. Therefore, earlier studies of coal-fired combustors treat the medium as gray (Mengüç and Viskanta 1987; Varma and Mengüç 1989). The non-gray WSGG model was firstly used by Yu et al. (2001) in the modeling of a pulverized coal-fired combustor. Geometric optics was used to evaluate the radiative properties of the particles, which were assumed to scatter isotropically. Calculations were performed using the DOM. Predictions of the incident heat fluxes on the wall, temperature and species concentrations obtained using the non-gray gas model were in good agreement with experimental data, while gray gas calculations were not so satisfactory. Marakis et al. (2000) assumed gray gas radiation and prescribed the absorption coefficient of the gases, but used Mie theory to determine the radiative properties of coal, char, and fly-ash particles and the Rayleigh theory for soot, in the simulation of an axisymmetric combustion chamber. The complex refractive index was taken from experimental data, accounting for the spectral variation. The temperature field was prescribed, i.e., only radiative calculations were made using the Monte Carlo and the P1 methods. The concentration of the particles was prescribed, as well as the particle size distribution. The P1 approximation was found to be accurate enough for the studied conditions, due to the high optical thickness of the medium. Calculations with strong forward scattering, isotropic scattering, and without scattering were performed. The first case is the most realistic one, but it is computationally more expensive. Predictions without scattering were better than those with isotropic scattering, except in the case of high mass concentration and small particles, in which both simplifications performed poorly.

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Oxy-fuel coal combustion has been the subject of significant research during the last decade. Andersson et al. (2008) reported an experimental study of the combustion of lignite in a cylindrical combustor in an atmosphere of air or a mixture of oxygen and recycled flue gas. Measurements of the total radiation intensity were compared with predictions of the gaseous radiation based on the solution of the integral form of the RTE along with the SNB model for several optical paths. A comparison between air-fuel and oxy-fuel combustion revealed that particle radiation is a significant fraction of the total emitted radiation, which does not change much with the composition of the oxidizer. In oxy-fuel combustion, the overlap between gas and particle radiation reduces the influence of gas radiation in the CO2-rich atmosphere. The total radiation intensities were also similar in the two cases, provided that the temperatures were also similar. A more detailed computational study of the same problem was reported by Johansson et al. (2013), who included radiation from char, ash, and soot in the model. Mie theory was used to calculate the absorption and scattering coefficients of char and ash particles, and isotropic radiation was assumed. The Rayleigh theory was used for soot. They concluded that the complex refractive index and the load of the ash particles have a large influence on the radiative heat fluxes. Moreover, these fluxes are significantly higher for non-scattering than for isotropically scattering media. Additional measurements and computations are presented in Bäckström et al. (2014). An experimental investigation of co-firing coal and biomass under oxy-fuel combustion conditions is presented in Smart et al. (2010). The measurements were carried out in a scaled version of an air-staged burner of the International Flame Research Foundation. The peak radiative heat fluxes decreased with the increase of the recycle ratio. The peaks were lower with biomass co-firing in comparison with the parent coals. Edge et al. (2011) presented RANS and LES of air and oxy-fuel pulverized coal combustion for two different combustion chambers. They used the DOM, the gray WSGG implementation, or the FSCK model for gaseous radiation and geometric optics for particle radiation. The incident radiative fluxes were found to be significantly different in RANS and LES. The differences were attributed to the ability of LES to account for intermittency. A subsequent study based on similar models simulated oxy-coal combustion in a pilot-scale furnace (Clements et al. 2015). The predicted temperature and radiative fluxes changed only marginally with the gas radiation model for RANS calculations. However, a better agreement with experimental data was achieved using LES for both air-fired and oxy-fuel coal combustion. Recently, Wu et al. (2016) modeled a pulverized coal-fired laboratory-scale combustion chamber using the Monte Carlo method. Coupled simulations were performed considering gray gas radiation from gases and particles and the OTA. Then, a snapshot of that simulation was used to analyze the importance of detailed gas property models. The gray gas assumption under predicted reabsorption of radiation leading to differences up to 50% relatively to LBL calculations of the radiative source term. The flame lift-off height was found to be sensitive to radiative heat transfer.

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Radiation in a circulating fluidized bed test rig was studied by Ates et al. (2016) using the method of lines solution of the DOM, the gray gas assumption, and the SLW model for gas radiation and Mie theory/geometric optics for the particles. Differences between the SLW and the gray gas predictions were marginal, so that the gray gas approximation was considered satisfactory. Calculations performed without scattering are more satisfactory than those with isotropic scattering. The former are closer to the reference forward anisotropic solution than the latter, in agreement with the conclusions of Marakis et al. (2000). When scattering is neglected, the incident heat fluxes are almost unchanged, while the radiative source term is slightly overpredicted.

5

Industrial Applications

5.1

Gas Turbine Combustors

The vast majority of gas turbine combustor simulations neglected radiative heat transfer, which is acceptable for aeronautical applications, due to the relatively small size of the combustion chamber and the small impact of thermal radiation on the temperature field. Still, the influence that radiative heat transfer may have on the temperature of the liner motivated a few investigations in the late 1980s and early 1990s, based on the gray gas approximation. In addition, experimental and numerical studies of a propane/air diffusion flame in a small gas turbine combustor model were carried out by Krebs et al. (1994, 1996). Measurements of the temperature and CO2 concentration were performed for several radial profiles and used as input data for radiative transfer calculations. The integral form of the RTE was solved using the SNB model and accounting for TRI with the OTFA. The importance of the correlations that appear in the time-averaged integral form of the RTE was evaluated. It was shown that turbulent fluctuations increase the transmissivity of the medium. The OTFA was accurate if the optical thickness of the medium based on the turbulent integral length scale does not exceed 0.3. More recently, Amaya (2010) performed LES of a reverse flow annular combustor of a helicopter. The DOM was used along with several non-gray gas property models. First, uncoupled calculations were performed for radiation using temperature and species concentration fields taken from an instantaneous snapshot of LES. Reference results were computed using the SNBCK model. The WSGG, based on coefficients derived from a different spectroscopic database, did not yield accurate results in some zones, but is more accurate than the gray gas model. The FSCK model provided accurate results and is much faster than the SNBCK model. Second, coupled fluid flow/radiation simulations were performed. The inclusion of radiation redistributes energy in the domain, but does not significantly change the energy content of the medium, and the mean temperature at the exit plane decreased only 0.4% due to radiation. Coupled simulations are required to obtain accurate predictions of the incident radiative heat fluxes on the wall.

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Industrial Furnaces

In contrast to gas turbine combustors, thermal radiation plays a major role in industrial furnaces. Despite this, the gray gas assumption has been used in most cases. Early exceptions are the works of Song and Viskanta (1988), Hoogendoorn (1996), and Song et al. (1997). Song and Viskanta (1988) simulated an industrial natural gas-fired furnace using the methods described in Song and Viskanta (1987) for radiative transfer. Hoogendoorn (1996) reported the simulation of an industrial gas-fired glass furnace using the discrete transfer method and the SNB model. Radiative transfer calculations at four different locations were performed by neglecting TRI (method 1) and compared with two other methods, one that accounts only for the temperature self-correlation (method 2) and a stochastic model (method 3). The stochastic model, which is expected to be the most accurate one, yields a heat flux 27% higher than method 1 and 23% higher than method 2 at a location close to the flow inlet. The differences become much smaller at larger distances from the inlet. Song et al. (1997) simulated a side-port-fired regenerative glass melting furnace. The spectral radiation intensity was calculated at several locations using the SNB model for gas properties. More recently, Lee et al. (2007) simulated a municipal waste incinerator and compared adiabatic results with those calculated with gray and non-gray gas radiation, the latter being based on the WSGG model, but neglected particle radiation. Radiation reduced the temperature, particularly in the primary combustion zone, the temperature gradient, and the NO emission. The temperature is further reduced when non-gray gas radiation effects are included. Stefanidis et al. (2008) simulated a steam cracking furnace using the DOM and the EWB model for radiation. However, they used Beer’s law to determine the absorption coefficient of the medium in every band, and this assumption of gray band radiation may significantly reduce the accuracy of the results. A walking beam reheating furnace was studied by Han et al. (2009) using the FVM and the non-gray implementation of the WSGG. Klason et al. (2008) studied a fixed bed biomass laboratory furnace and a largescale grate-fired biomass furnace. They used several different models for radiation: the P1/gray gas approximation, FVM/gray gas, and FVM/SLW. The Mie theory was used to determine the radiative properties of char and fly ash. The P1/gray gas models predicted a temperature field in the furnace similar to the other models. However, the predictions of the radiative heat transfer to the fuel bed were not so good, and the FVM/gray gas and FVM/SLW should be used. Particle scattering had a minor effect on the temperature field, but affected the heat transfer rate to the fuel bed by 15%. Bäckström et al. (2015) reported an experimental and numerical investigation of a scaled pilot of a rotary kiln furnace used in the production of iron ore pellets. Eleven different fuels were used including natural gas, heavy fuel oil, several coals, and coal/ biomass mixtures. Radiation was modeled using the methods formerly employed by Johansson et al. (2013). The radiation intensity was calculated along radial profiles using the measurements of gas and particle temperatures, species concentration, and projected particle surface area. The calculations showed that radiation from the particles was dominant, the contribution from gas radiation being small.

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An industrial refinery furnace was simulated by Pedot et al. (2016) using the DOM and the SNBCK models for radiation. Thermal radiation was found to be the dominant heat transfer mode, controlling the temperature of the fuel pipes, despite the coupling between radiation and convection in the gaseous medium and conduction in the pipes. Predictions of the incident radiative heat fluxes were in good agreement with the available measurements.

5.3

Utility Boilers

Thermal radiation is also the dominant mode of heat transfer in utility boilers. In the case of natural gas-fired boilers, gas radiation should be modeled as non-gray. This is illustrated in the work of Borjini et al. (2007), who performed uncoupled radiative transfer calculations in an idealized utility boiler using the FVM and a narrowband-based WSGG model. The temperature and the species concentration fields, including soot concentration, were prescribed. In the case of fuel oil or coalfired boilers, continuum radiation from the droplets and particles, respectively, may dominate and in such a case a gray assumption may be acceptable for gas radiation. Only a few authors considered non-gray gas radiation in the simulation of radiative transfer in utility boilers. Among them, Ahluwalia and Im (1994) investigated the influence of the ash composition, ash content, and coal preparation on the radiative heat fluxes and compared their predictions with measurements in a coalfired boiler. They used a hybrid method that selects the DOM, P1 approximation or modified differential approximation to solve the RTE, depending on the value of the absorption coefficient of the medium, and spectral data of H2O, CO2, CO, char, ash and soot to define the radiative properties. The agreement between the predictions and the measurements was reasonable. Ströhle (2003) modeled a front wall pulverized coal-fired utility boiler using the DOM, two different models for gas radiation (gray WSGG model and wideband CK model), and geometric optics for particles. Although the radiative source terms calculated by the two different gas property models exhibit differences up to 100% in regions where absorption is dominant, about 0.2 m downstream of the burners, the predicted temperature differences are only about 40 K, since radiative heat transfer is small in that region. The gray gas model predicts wall heat fluxes along the height of the boiler that are 12% higher near the second level of burners, where the highest temperatures occur, leading to a significant temperature decrease at the bottom of the boiler and a decrease of 40 K at the exit, in comparison with the non-gray model. Overall, radiation from the particles attenuates the errors introduced by the gray gas assumption, as expected. Oxy-coal combustion in a full-scale boiler was studied by Nakod et al. (2013) who compared gray and non-gray WSGG formulations for gas radiation. Radiation from particles was assumed as gray, and the absorption and scattering efficiencies of coal, char, and ash particles were prescribed. Large volume regions were observed where gas radiation dominates. Differences of about 10% were found between gray

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500000 450000 400000 350000 300000 250000 200000 150000 100000 50000 0

Perry Gray

Perry:5GG

Chalmers:5GG

Fig. 7 Predicted incident heat fluxes (W/m2) on the walls of a full-scale boiler. The results on the left were evaluated using a gray WSGG model and the others using two different non-gray versions (Reprinted from Appl Therm Eng, 54, Nakod et al., A comparative evaluation of gray and non-gray radiation modelling strategies in oxy-coal combustion simulations, 422–432, Copyright (2013), with permission from Elsevier.)

and non-gray predictions of the incident heat fluxes on the walls, as shown in Fig. 7 for oxy-firing with dry flue gas recycle, and differences of 50 K were predicted for the average outlet gas temperatures.

5.4

Other Applications

Other interesting studies of radiative transfer in industrial combustion systems are available in the literature. Radiative transfer in diesel engines has received little attention, although the subject is not new (Mengüç et al. 1985), and the few works that addressed this topic are surveyed in Modest and Haworth (2016). This reference discusses also radiative transfer in high-speed propulsion systems. Radiation in porous burners is briefly addressed in a review of porous media combustion (Mujeebu et al. 2010). Radiative transfer in fires is discussed in Sundén and Faghri (2008).

6

Concluding Remarks

Accurate methods are available to solve the RTE and to calculate the radiative properties of gases and particles, for both laminar and turbulent reactive flows. Although accurate methods may be used in the simulation of laminar flames, a simple OTA is often sufficient if soot is not present and a very accurate prediction of the temperature field is not required. However, if soot is present or the prediction of highly sensitive temperature phenomena, such as the formation of NO, is required, a

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more precise treatment of radiative transfer should be used, taking non-gray gas radiation effects into account. In the case of turbulent reactive flows, the challenge is to select an RTE solver and radiative property models for gases and particles that provide an accuracy similar to that of the other physical models, e.g., turbulence and combustion models, with minimum computational cost. This is not an easy task, and although a few guidelines have been provided, there are no general rules. The OTA is seldom adequate. In the case of gaseous fuels, a non-gray model for the gas radiative properties is generally required for accurate predictions, unless soot is present and soot radiation dominates over gas radiation. TRI effects play a role, and it has been shown for free flames and laboratory combustors that they contribute to decrease the temperatures. However, these effects have generally been ignored in industrial applications. In the case of liquid and solid fuels, the importance of non-gray gas radiation effects depends on how important gas radiation is, and this depends on the problem under consideration. It is clear, however, that non-gray gas models are required whenever gas radiation is important, and the gray gas approximation based on Beer’s law should not be used.

7

Cross-References

▶ Radiative Properties of Gases ▶ Radiative Properties of Particles ▶ Radiative Transfer Equation and Solutions Acknowledgments This work was supported by the Portuguese Science and Technology Foundation (FCT), through IDMEC, under LAETA, project UID/EMS/50022/2013.

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Ströhle J (2003) Spectral modelling of radiative heat transfer in industrial furnaces. PhD thesis, University of Stuttgart Sundén B, Faghri M (eds) (2008) Transport phenomena in fires. WIT Press, Southampton Tessé L, Dupoirieux F, Taine J (2004) Monte Carlo modelling of radiative transfer in a turbulent sooty flame. Int J Heat Mass Transf 47:555–572 Tsuji H, Gupta AK, Hasegawa T, Katsuki M, Kishimoto K, Morita M (2003) High temperature air combustion: from energy conservation to pollution reduction. CRC Press, Boca Raton Varma KR, Mengüç MP (1989) Effects of particulate concentrations on temperature and heat flux distributions in a pulverized-coal fired furnace. Int J Energy Res 13:555–572 Viskanta R (2005) Radiative transfer in combustion systems: fundamentals & applications. Begell House, New York Viskanta R, Mengüç MP (1987) Radiation heat transfer in combustion systems. Prog Energy Combust Sci 13:97–160 Wang A, Modest MF (2008) Monte Carlo simulation of radiative heat transfer and turbulence interactions in methane/air jet flames. J Quant Spectrosc Radiat Transf 109:269–279 Wang L, Haworth DC, Turns SR, Modest MF (2005) Interactions among soot, thermal radiation, and NOx emissions in oxygen-enriched turbulent nonpremixed flames: a computational fluid dynamics modeling study. Combust Flame 141:170–179 Wang HY, Chen WH, Law CK (2007) Extinction of counterflow diffusion flames with radiative heat loss and nonunity Lewis numbers. Combust Flame 148:100–116 Wu B, Roy SP, Modest MF, Zhao X (2016) Monte Carlo modeling of radiative transfer in a pulverized coal jet flame. In: Proceedings of the 8th international symposium radiative transfer, Cappadocia Yu MJ, Baek SW, Kang SJ (2001) Modeling of pulverized coal combustion with non-gray gas radiation effects. Combust Sci Technol 166:151–174 Zhao XY, Haworth DC, Ren T, Modest MF (2013) A transport probability density function/photon Monte Carlo method for high-temperature oxy-natural gas combustion with spectral gas and wall radiation. Combust Theory Model 17:354–381 Zheng Y, Barlow RS, Gore JP (2003a) Measurements and calculations of spectral radiation intensities for turbulent non-premixed and partially premixed flames. J Heat Transf 125:678–686 Zheng Y, Barlow RS, Gore JP (2003b) Spectral radiation properties of partially premixed turbulent flames. J Heat Transf 125:1065–1073 Zhu XL, Gore JP (2005) Radiation effects on combustion and pollutant emissions of high-pressure opposed flow methane/air diffusion flames. Combust Flame 141:118–130

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Monte Carlo Methods for Radiative Transfer Hakan Ertürk and John R. Howell

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Energy Equation for Radiative Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Statistical Representation of Physical Events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Wavelength of Emission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Direction of Emission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Location of Emission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Absorption and Scattering by Participating Medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Absorption or Reflection by a Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Pseudorandom Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Statistical Uncertainty of Monte Carlo Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 General Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Preprocessing: Defining Geometry, Properties, and Boundary Conditions . . . . . . . . 6.2 Monte Carlo Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Post-Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Ray Tracing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Collision-Based Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Path Length-Based Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Reverse Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Implementing Spectral Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Probability-Based Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Bandwise Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Full-Spectrum K-Distribution Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Performance Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 General Guidelines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Selecting Different Approaches or Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1202 1204 1206 1206 1208 1209 1210 1214 1216 1216 1217 1217 1219 1220 1221 1221 1227 1230 1231 1232 1233 1234 1235 1235 1236

H. Ertürk (*) Department of Mechanical Engineering, Boğaziçi University, Istanbul, Turkey e-mail: [email protected] J. R. Howell Department of Mechanical Engineering, The University of Texas at Austin, Austin, TX, USA e-mail: [email protected] # Springer International Publishing AG, part of Springer Nature 2018 F. A. Kulacki (ed.), Handbook of Thermal Science and Engineering, https://doi.org/10.1007/978-3-319-26695-4_57

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9.3 Smoothing Exchange Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4 Parallel Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Cross-References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1237 1238 1240 1241 1241

Abstract

The solution of the radiative transfer equation is challenging, especially in the presence of a participating medium, wavelength- and direction-dependent properties, or a complex geometry. The Monte Carlo method that relies on statistical sampling of photon bundles using pseudorandom numbers and probability distributions which are derived based on physical laws is a powerful and robust approach to solving the radiative transfer equation. While the method is computationally demanding even for simple surface exchange problems, introducing complex phenomena does not significantly increase formulation complexity or the required computational power. Therefore, the method has become one of the most widely adopted solution techniques with the increasing computation capacity in the last decades. This chapter introduces the method, presenting general guidelines to adopt it for solution of radiative transfer problems, discussing how to introduce further phenomena such as wavelength- or direction-dependent properties, and improving computational performance. The method is known for its flexibility and can be applied in many different ways. Different strategies are discussed, considering advantages or disadvantages of each for different problems.

1

Introduction

The Monte Carlo method is a generalized technique to statistically simulate a system’s behavior in terms of a Markov chain. A Markov chain represents a sequence of events, of which the probability of each succeeding event is independent of the prior events. The method was developed in the 1940s by Stanislaw Ulam at the Los Alamos National Laboratory, who developed the method for modeling neutron transport (Metropolis 1987). At the time they were investigating the shielding of nuclear radiation and were trying to identify the distance that neutrons would travel through different materials. While the problem had not been solved deterministically, Ulam came up with the idea of using random numerical experiments. His colleague John von Neumann implemented the idea in a program to be executed by the ENIAC computer. As the method was developed as a part of nuclear weapon development, its development was classified, and they come up with the code name “Monte Carlo” referring to the Monte Carlo Casino, where Ulam’s uncle gambled (Metropolis 1987). The method is considered as a computationally demanding approach as it relies on statistical sampling. However, with the improvements in computer

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technology in the late twentieth century, the method gained more popularity. As a result, the method has been applied to different problems in a wide variety in fields of science and engineering such as genetics (Nielsen and Wakeley 2001), ecology and environmental sciences (Goodrum et al. 1996), marketing (Sherman et al. 1999), demographics (Renshaw and Gibson 1998), finance (Boyle et al. 1997, Fournié et al. 1999), and mechanics (Papadrakakis and Lagaros 2002). In the area of thermal sciences, it has been applied to simulation of thermal radiation problems (Howell 1998), heat diffusion problems (Haji-Sheikh and Sparrow 1967), rarefied gas flows (Bird 1994, Bird 1998), thermal transport by phonons (Mazumder and Majumdar 2001) or electrons (Fischetti and Laux 1993), for microscale heat conduction, and for modeling electron beam processing (Wong and Mengüç 2004). The method was first used for solution of thermal radiation problems by Howell and Perlmutter (1964). Since then, it has become one of the most widely accepted and fundamental solution techniques for thermal radiation problems (▶ Chap. 23, “Radiative Transfer Equation and Solutions”). The method’s accuracy is verified or validated through many studies such as by Ertürk et al. (1997), and it is often used to produce benchmark solutions such as in Hsu and Farmer (1997). There is a broad literature on the use and applications of the Monte Carlo method in thermal radiation problems. Some of the highlights are presented in Howell (1998, 1968), Farmer and Howell (1998), Walters and Buckius (1994), Wong and Mengüç (2008), and Yang et al. (1995). The application of the method for modeling thermal radiation is based on simulating a finite number of photon bundles. Therefore, the method is often referred as the “photon Monte Carlo method.” The photon bundles are comprised of photons of the same frequency or wavelength, and they carry a finite amount of radiative energy. Physical events such as emission, reflection, absorption, and scattering that happen in the “life” of a photon bundle are all decided using the probability density functions derived from physical laws and are sampled using random numbers. Once a large enough number of samples is used, the method can produce solutions within the statistical accuracy limits no matter how complex the problem, which makes it the most versatile solution technique for radiation problems. On the other hand, this also brings the main drawback for the Monte Carlo method, which has always been the required large computational expense. The usual tendency is that the method requires a considerable amount of computation even for a simple surface exchange problem. However, adding further complexities does not require a significant amount of additional computation. Therefore, the method is advantageous over the others for problems that include many complex phenomena, such as wavelength- and direction-dependent properties, and complex geometries with shading or blockage effects. The method gained even more popularity in recent years with the increase in computational power and the introduction of parallel processing using large-scale grids, multi-core processors, multi-processors, and graphics processors.

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Energy Equation for Radiative Transport

A generic formulation of a radiative transport problem can be carried out using an exchange factor formulation for a wide variety of problems with non-gray, nondiffuse surfaces that enclose absorbing, emitting, and anisotropically scattering medium. For a general enclosure problem, as shown in Fig. 1, where a participating medium is enclosed within radiating walls, the spectral radiative heat flux for differential surface, dAj, can be defined by the following radiative energy balance equation. N X

 
 qr, λ rj dAj ¼ eλ rj Eb, λ T rj dAj  k¼1

ð  V

ð Ak

eλ ðrk ÞEb, λ ½T ðrk Þ dΓλ, dAk dAj dAk

4κλ ðrm Þ  Eb, λ ½T ðrm Þ dΓλ, dV m dAj dV m (1)

where dΓλ, dAk dAj , the spectral, differential exchange factor between the differential elements dAk and dAj, represents the net ratio of radiative heat emitted from dAk at a wavelength λ to the part of it that is absorbed by dAj by all means including all intermediate reflections or scattering. Similarly, dΓλ, dV m dAj is the spectral, differential exchange factor between the differential volume dVm and area dAj. This represents the ratio of radiative heat emitted from dVm at a wavelength λ to the part of it that is absorbed by dAj by all means. Here, the rj represents the position of dAj, and N is the total number of surfaces in the enclosure. The similar formulation for the divergence of spectral radiative heat flux for a volume element dVm can be defined as N ð X ∇qr, λ ðrm ÞdV m ¼ 4κ λ ðrm ÞEb, λ ½T ðrm Þ dV m  eλ ðrk ÞEb, λ ½T ðrk Þ dΓλ, dAk dV m dAk k¼1 Ak Ð  V 4κλ ðrn ÞEb, λ ½T ðrn Þ dΓλ, dV n dV m dV n

(2) Fig. 1 A typical radiation exchange problem with a participating medium [described by Eq. (5)] enclosed within radiating walls [described by Eq. (6)]

Surface k dA k

Participating Medium

Surface N rk dV m Surface 1 rm

dV n rn rj

Surface 2

dA j

Surface j

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The solution of Eqs. (1) and (2) is almost always carried out numerically. For that, the system is divided into subsurfaces and volumes. For a numerical solution, the discretized versions of these equations should be considered and solved. For a discretized system that is comprised of a total of Ns sub-areas and Ng sub-volumes, these equations can be rewritten as Ns X
 q r , i , j Aj ¼ e i , j Eb , i T j Aj  ei, k Eb, i ðT k Þ Γi, kj Ak k¼1



Ng X

4κi, m Eb, i ðT m Þ Γi, mj V m

(3)

m¼1

∇qr, i, m V m ¼ 4κ i, m Eb, i ðT m Þ V m 

Ns X

ei, k Eb, i ðT k ÞΓi, km Ak

k¼1



Ng X

4κi, n Eb, i ðT n Þ Γi, nm V n

(4)

n¼1

where the subscript i denotes the wavelength band Δλ around λi and Γi,kj is the spectral exchange factor from surface element k to surface element j for the band λi. The total quantities for radiative heat flux and its divergence can then be estimated by numerical integration through the spectrum. For a gray system, these equations can be further simplified as Ng Ns X X
 q r , j Aj ¼ e j E b T j A j  ek Eb ðT k Þ Γkj Ak  4κm Eb ðT m Þ Γmj V m

∇qr, m V m ¼ 4κ m Eb ðT m Þ V m 

(5)

m¼1

k¼1 Ns X

ek Eb ðT k ÞΓkm Ak 

Ng X

4κn Eb ðT n Þ Γnm V n (6)

n¼1

k¼1

where Γkj is the exchange factor from surface element k to surface element j and Γmj is the exchange factor from volume element m to surface element j. Here, the temperature, and heat flux or divergence of heat flux, within a sub-element is considered to be uniform. Each sub-element is denoted by a subscript. There are certain rules the exchange factors must follow, as for geometric configuration factors. These rules are the summation rule and reciprocity, and they are in a way a manifestation of the laws of thermodynamics. The summation rule that is valid for an enclosure, such as the one shown in Fig. 1, represents the conservation of emitted radiative energy and is Ns X k¼1

Γjk þ

Ng X n¼1

Γjn ¼ 1

(7)

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In words, it states that all the energy emitted from a surface/volume element must be absorbed either by the other medium or surface elements within the enclosure or the emitting sub-element itself. The reciprocity relation valid between the surfaces is ej Γjk Aj ¼ ek Γkj Ak

(8)

and it warranties that the net radiant heat exchange between two surfaces at identical temperatures would be zero. Unlike the case with configuration factors, the reciprocity of exchange factors does not only involve the geometry but also properties as exchange factors define the radiant exchange. For a system with known exchange factors, a radiative energy balance equation based on Eq. (5) or (6) can be written for each sub-element. A boundary condition, either a temperature or flux/divergence of flux, is prescribed for every sub-element in a regular analysis problem leading to a linear system of equations with an equal number of equations and unknowns. Solution of the resulting system is straightforward using a linear solver. The exchange factors for a surface exchange problem with diffuse-gray surfaces can be simply defined based on geometric view factors (configuration factors). However, for a system with participating medium and non-diffuse walls, the calculation of the exchange factors is not straight forward. The Monte Carlo method can be considered as the most generic approach for estimating the exchange factors even for systems with participating media, complex geometries, and directional and wavelength-dependent properties.

3

Statistical Representation of Physical Events

The general idea behind the Monte Carlo method is to model a complex system by representing individual physical events statistically through random sampling. Radiative heat transfer can be considered as energy transport by means of photons, and the samples considered for the application of the method to model radiative transfer are photon bundles as explained earlier. Therefore, the physical behavior of these photons or photon bundles must be represented accurately. A bundle is emitted at a particular wavelength or frequency into a particular direction, and it travels a certain distance within a participating medium before it is absorbed or scattered. Moreover, if it intercepts a surface before that distance, it is either reflected or absorbed. Therefore, a probabilistic representation of all these events must be defined and considered to statistically model the behavior.

3.1

Wavelength of Emission

Consider radiative emission from a surface that is at a temperature T. The surface emits, eλEb,λ(T )dλ, at wavelength λ, with eλ being the spectral hemispherical

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emissivity and Eb,λ(Τ), the blackbody spectral emissive power described by Planck’s distribution for temperature T. The probability of this surface to emit at a specific wavelength λ is eλ Eb, λ ðT Þdλ eλ Eb, λ ðT Þdλ Pλ ðλÞdλ ¼ Ð 1 0 ¼ e E ð T Þdλ eσT 4 λ b, λ 0

(9)

where e is the hemispherical total emissivity of the surface, σ = 5.67108 W/m2K4 is the Stefan-Boltzmann constant, and the function Pλ(λ) is the probability density function (PDF) that represents the ratio of radiation emitted at the particular wavelength λ to the total emitted radiation. As it represents a probability, the PDF always has a value smaller than unity, and moreover the sum of all the probabilities, the area under the PDF curve, should be equal to unity. In order to statistically simulate the wavelength of emission in a domain defined from λmin to λmax, two random numbers varying between 0 and 1, R1 and R2, are required. Using these random numbers, two wavelength values, λ1 and λ2, should be estimated, one from the relation Pλ(λ1) = R1 and the other from λ2 = (λmax  λmin) R2 + λmin. If λ2 is smaller than λ1, the wavelength λ2 is accepted as the wavelength of emission. Otherwise, the two random numbers are rejected, and another two are generated to estimate λ1 and λ2 until the condition is satisfied. The process of simulating physical events using a PDF and two random numbers as explained is an expensive one as depending on the PDF, many generated random numbers may be rejected until the appropriate pair is found. An alternative approach necessitates the definition of a cumulative distribution function (CDF) as Rλ ð λ Þ ¼

ðλ 0

Pλ ðλ0 Þdλ0 ¼

1 eσT 4

ðλ 0

eλ Eb, λ ðT Þdλ0

(10)

which denotes the probability of selecting a wavelength in the range 0 to λ. The value of the CDF monotonically increases from 0 to 1 as λ increases from λmin to λmax or 0 to 1, for the particular case. Therefore, a relation can be developed for the CDF, Rλ(λ), that can be inverted for λ(Rλ), where Rλ represents a random number varying between 0 and 1. The relation λ(Rλ) is referred as a random number relation (RNR). If N random numbers are generated, by using the RNR, N wavelength values can be calculated. If NΔλ represents the number in each Δλ increment, then N Δλ =N ΔRλ ¼ Δλ Δλ

(11)

When a small Δλ or a large enough N is used, the quantity ΔRλ/Δλ approaches dRλ/dλ, which is simply the PDF or Pλ (λ). Therefore, if a large enough number of samples is used, the event can be simulated using pseudorandom numbers satisfying the PDF that describes the event. The similar PDF can be defined for emission of radiation from a volume as

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4κ λ Eb, λ ðT ÞdλdV κλ Eb, λ ðT Þdλ ¼ Pλ ðλÞdλ ¼ Ð 1 0 4κ E ð T Þdλ dV κp σT 4 λ b, λ 0

(12)

where κλ and κ p are the spectral absorption coefficient and Planck mean absorption coefficient of the emitting medium, respectively. Then the corresponding CDF becomes ðλ

1 Rλ ðλÞ ¼ Pλ ðλ Þdλ ¼ κ p σT 4 0 0

0

ðλ 0

κλ Eb, λ ðT Þdλ0

(13)

It might not be possible to derive an analytical expression for the RNR, λ(Rλ). In such a case, one might consider fitting a curve to represent the CDF or Rλ(λ) that can be inverted to obtain an approximate relation for RNR.

3.2

Direction of Emission

A similar approach can be applied for physical events that must be described by multiple independent variables. Consider the PDF for direction of emission from a surface that represents the probability of emission through a solid angle described by a polar angle θ and an azimuthal angle φ eλ ðθ, φÞ I b, λ cos θ sin θ dθ dφ dA Pθ φ ðθ, φÞdθ dφ ¼ Ð 2π Ð π=2 eλ ðθ0 , φ0 Þ I b, λ cos θ0 sin θ0 dθ0 dφ0 dA 0 0 ¼

(14)

eλ ðθ, φÞ I b, λ cos θ sin θ dθ dφ dA eλ Eb, λ dA

where eλ(θ,φ) is the spectral directional emissivity of the surface. For a diffuse surface, the PDF can be separated into two PDFs, one for the polar angle, Pθ (θ), another for the azimuthal angle, Pφ (φ), as Pθ φ (θ,φ) = Pθ (θ) Pφ (φ) where Pθ ðθÞdθ ¼ 2 cos θ sin θ dθ

(15)

and Pφ ðφÞdφ ¼

dφ 2π

(16)

Then the CDFs for the polar and azimuthal angle become Rθ ð θ Þ ¼

ðθ 0

2 cos θ0 sin θ0 dθ0 ¼ sin 2 θ

(17)

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and Rφ ð φÞ ¼

ðφ 0

dφ0 φ ¼ 2π 2π

(18)

both of which can be inverted easily to give an analytical expression for the corresponding RNR. For the case where the emission is considered from a participating medium, the PDF must be modified to consider the change in the polar angle limits so that the CDF becomes Rθ ð θ Þ ¼

3.3

1 2

ðθ

1 2 sin θ0 dθ0 ¼ ð1  cos θÞ 2

0

(19)

Location of Emission

There might be cases where the PDF considered is defined in terms of two independent variables, but it is not possible to expand it as a multiplication of two separate PDFs. To illustrate this, consider the PDF that defines the location of emission from a rectangular surface defined over a Cartesian x–y plane with xmin  x  xmax, ymin  y  ymax. Consider a diffuse-gray surface with a blackbody emissive power distribution of Eb[T(x,y)]. The PDF can be defined as e Eb ½T ðx, yÞ Px y ðx, yÞ ¼ Ð xmax Ð ymax e Eb ½T ðx0 , y0 Þ dy0 dx0 xmin y

(20)

min

and then the CDFs are Ðx Rx ð x Þ ¼

Ð ymax

min Ð xxmax xmin

Ð yymin max ymin

e Eb ½T ðx0 , y0 Þ dy0 dx0 e Eb ½T ðx0 , y0 Þ dy0 dx0

(21)

which can be inverted for a RNR, x(Rx), and Ðy y

e Eb fT ½xðRx Þ, y0 gdy0

ymin

e Eb fT ½xðRx Þ, y0 gdy0

Ry ðRx , yÞ ¼ Ð ymin max

(22)

which can be inverted for a RNR, y(Rx, Ry), respectively. The x coordinate of the point of emission is only dependent on a random number, while the y coordinate depends on both random numbers. The Eqs. (21) and (22) are often termed the marginal and conditional distributions of the PDF, respectively.

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Absorption and Scattering by Participating Medium

Consider radiative transfer through absorbing-emitting medium as shown in Fig. 2. The radiation intensity is attenuated along the path S according to the Beer-Lambert law. dI λ ¼ κ λ dS Iλ

(23)

where κλ is the spectral absorption coefficient describing the medium’s absorption per unit length. The fraction that is absorbed by the medium through a distance S can be defined as the absorptance.  ðS  I λ ð 0Þ  I λ ð SÞ 0 0 ¼ 1  exp  κλ ðS ÞdS αλ ðSÞ ¼ I λ ð 0Þ 0

(24)

The spectral absorptance goes from 0 and approaches 1 exponentially as the path length S increases, and it represents the cumulative attenuation rate. Therefore, it can be considered as the CDF for absorption distance, so that Ra(Sa) = αλ(Sa). For a system with uniform absorption coefficient throughout the medium (κλ = constant), the RNR for absorption path length is Sa ¼ 

1 ln Ra κλ

(25)

Therefore, once the photon bundle travels a path of Sa, it must be absorbed by the medium. For the cases where the absorption coefficient is not uniform, rather than Eq. (25), the absorption path length, Sa, is defined by an implicit equation ð Sa 0

κλ dS ¼ ln Ra

(26)

Fig. 2 Radiative transfer through participating medium

sˆ

dS I l(S) S 0 I l(0)

b l=k l+ss, l

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For a discretized system, where absorption coefficient is assumed to be uniform within each sub-volume, the integral term can be replaced by a summation so that the photon bundle travels if it satisfies the inequality defined as ð Sa 0

κ λ dS ffi

X

κλ, m ΔSm < ln Ra

(27)

m

where κ λ,m represents the absorption coefficient within sub-volume m and ΔSm represents the path the photon bundle travels within sub-volume m. In both cases discussed, the absorption path length, Sa, represents the total path that the photon bundle travels before it is absorbed by the medium, and it might be comprised of numerous segments where reflection or scattering events might redirect the path until the photon bundle travels Sa. For an absorbing-emitting and scattering medium, the distance the photon bundle should travel before it is scattered or scattering path length can be estimated similarly to the absorption path length. For a system with uniform scattering coefficient (σ s,λ = constant), the RNR for scattering path length is Ss ¼ 

1 ln Rs σ s, λ

(28)

and for the case where the scattering coefficient varies through the medium, the scattering path length is given by the implicit equation ð Ss 0

σ s, λ dS ¼ ln Rs

(29)

For a discretized system, where the scattering coefficient is assumed to be uniform within each sub-volume, the integral term can be replaced by a summation, and the scattering path length is estimated from the inequality ð Ss 0

σ s, λ dS ffi

X

σ s, λ, m ΔSm < ln Rs

(30)

m

Once the photon bundle is scattered, the scattering angles must be determined. The scattering phase function, Φλ(θs,φs), describes the distribution of energy into different directions due to scattering relative to the isotropic scattering case. Therefore, it can also be considered as the probability of photon bundle scattering into a direction described by the scattering angles θs and φs. The scattering angles are defined with respect to the original direction vector, ^s . Therefore, the PDF can be represented in terms of scattering phase function as Pθ φ ðθs , φs Þdθs dφs ¼ Φλ ðθs , φs Þ sin θs dθs dφs

(31)

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The corresponding CDFs that can be inverted for the respective RNRs are Ð φs Ð π

R φ ðφs Þ ¼ Ð02π Ð0π 0

0

Φλ ðθ, φÞ sin θdθdφ Φλ ðθ, φÞ sin θdθdφ

(32)

and Ð θs Φλ ðθ, φÞ sin θdθdφ R θ ðθs Þ ¼ Ð0π 0 Φλ ðθ, φÞ sin θdθdφ

(33)

Scattering is a phenomenon that is most frequently observed in particulate media. Two frequently used models to represent scattering in particulate media are Rayleigh scattering and Mie scattering (▶ Chap. 27, “Radiative Properties of Particles”). While Rayleigh scattering is valid when the size of the scatterers is smaller than the wavelength of radiation, Mie theory is applied for the cases where scatterer size is of the same order as the wavelength of radiation. Let us exemplify the derivation of RNRs for the scattering angle for the phase functions representing Rayleigh scattering and Mie scattering. The phase function representing Rayleigh scattering is ΦR ðθ Þ ¼

 3
1 þ cos 2 θ 4

(34)

Mie scattering theory relies on analytical solution of Maxwell’s equations for a spherical scatterer. The exact phase function based on Mie scattering theory is represented in terms of a linear combination of orthogonal Legendre polynomials. The use of the exact representation is often too cumbersome even for numerical solution. A good approximation to the exact phase function for Mie scattering is proposed by Henyey and Greenstein (1941). The Henyey-Greenstein phase function is ΦHG ðθÞ ¼

1  g2

(35)

ð1 þ g2  2g cos θÞ3=2

where g is the asymmetry factor that can be estimated by integrating the exact Mie scattering phase function. The asymmetry factor ranges from 1 to 1, and it is 0 for limiting case of isotropic scattering. While its value is positive for forward scattering, negative values represent backscattering. In order to achieve a better approximation, it is often used in a modified form, or several Henyey-Greenstein phase functions are used in combination with different asymmetry parameters. The CDF for the polar angle of scattering is derived once Eq. (33) is applied for Rayleigh scattering phase function presented in Eq. (34): Rθ , R ð θ Þ ¼ 

3 4

 cos θ þ

cos 3 θ 4  3 3

 (36)

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Similarly, the following expression results from using Eq. (33) for the HenyeyGreenstein phase function presented in Eq. (35). " # 1  g2 1 1 Rθ, HG ðθÞ ¼   2g ð1 þ g2  2g cos θÞ1=2 1  g

(37)

For both phase functions, the CDF for the azimuthal angle of scattering is identical to the expression presented by Eq. (18). An alternative approach relies on attenuation of radiative intensity rather than treating absorption and scattering separately, as discussed above. Based on this approach, the CDF of extinction distance or the distance photon bundles should travel before either absorption or scattering occurs is  ð Se  0 0 RS, e ðSe Þ ¼ 1  exp  βλ ðS ÞdS 0

(38)

For an idealized system with uniform extinction coefficient, βλ = κλ + σ s,λ = constant, an explicit RNR can be derived as before: Se ¼ 

1 ln Re βλ

(39)

A similar approach to those presented by Eqs. (26) and (29) can be applied for nonuniform medium properties as ð Se βλ dS ¼ ln Re (40) 0

and for a discretized system, the integral term can be replaced by a summation so that the extinction path length is defined by the inequality ð Se X βλ dS ffi βλ, m ΔSm < ln Re (41) 0

m

Here, once the photon bundle travels as far as the extinction path length, it is either absorbed or scattered. The single scattering albedo defines how strongly scattering the medium is, and it is ωλ ¼

σ s, λ κ λ þ σ s, λ

(42)

Therefore, single scattering albedo that defines the probability of a scattering event must be used to decide on the event. Another random number is generated, Rω, and it is compared to single scattering albedo. If the random number is greater than the single scattering albedo (Rω > ωλ), the photon bundle is absorbed; otherwise it is scattered. If it is scattered, the scattering angle must be identified as explained before.

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H. Ertürk and J. R. Howell

Absorption or Reflection by a Surface

If the photon bundle is incident upon an opaque surface, it is either reflected or absorbed. For a diffuse surface, where the surface roughness can be considered significant with respect to the wavelength of incident radiation, reflectivity can be used for identifying whether the photon bundle is reflected or absorbed. The reflectivity of the surface can be considered as a property describing the probability of reflection, and it can be used for identifying if the photon bundle is reflected. For that, a random number, Rρ, is generated and is compared with the reflectivity of the surface. If the generated random number is greater than the reflectivity (Rρ > ρλ), the photon bundle is absorbed by the surface. If the random number is less than or equal to the reflectivity (Rρ  ρλ), it is reflected, in which case the direction of reflection must be identified. For a diffuse surface, the PDFs and CDFs defining the reflection angles are identical to the emission angles. Therefore, the same RNRs for emission polar and azimuthal angles that are defined by Eqs. (17) and (18) can be used, determining reflection polar, θr, and azimuthal angles, φr, respectively. If the surfaces have directionally dependent properties (▶ Chap. 25, “Design of Optical and Radiative Properties of Surfaces”), it is more appropriate to use reflectivity in determining the reflection or absorption of the photon bundle than absorptivity, as directional dependent behavior is more completely represented in terms of reflectivity. The photon bundle’s angle of incidence with respect to the surface (θi,φi) must be identified first in order to decide if the incident photon bundle is absorbed or reflected. Estimation of the angle of incidence is explained in detail in the upcoming section on ray tracing. Once the angle of incidence is known, a random number can be generated and compared against the directional reflectivity. If the random number is smaller or equal to directional reflectivity (Rρ ρλ(θi,φi)), the photon bundle is reflected. Otherwise, if the random number is larger than the directional reflectivity (Rρ >ρλ(θi,φi)), the bundle is absorbed by the surface. It must be noted that the directional reflectivity referred here, ρλ(θi,φi), is reflectivity that is known as the directional to hemispherical reflectivity (Howell et al. 2016). If the photon bundle is reflected by the surface, the reflection direction must be identified. Considering an incident intensity of Ii,λ(θi,φi), the reflected intensity in a direction (θr,φr) is I r, λ ðθi , φi , θr , φr Þ ¼ ρλ ðθi , φi , θr , φr Þ I i, λ ðθi , φi Þ cos θi sin θi dθi dφi

(43)

where ρλ(θi, φi, θr, φr) is the spectral, bidirectional reflectivity of the surface between directions (θi,φi) and (θr,φr). The probability of the surface reflecting into direction (θr,φr) is Pθr φr ðθr , φr Þdθr dφr ¼ ρλ ðθi , φi , θr , φr Þ I i, λ ðθi , φi Þ cos θi sin θi dθi dφi cos θr sin θr dθr dφr Ð 2π Ð π=2 I i, λ ðθi , φi Þ cos θi sin θi dθi dφi 0 0 ρλ ðθi , φi , θr , φr Þ cos θr sin θr dθr dφr (44)

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which can be simplified to ρ ðθi , φi , θr , φr Þ cos θr sin θr dθr dφr Pθr φr ðθr , φr Þdθr dφr ¼ Ð 2π Ð π=2λ ρλ ðθi , φi , θr , φr Þ cos θr sin θr dθr dφr 0 0

(45)

Then the corresponding CDF for azimuthal angle of reflection becomes Ð φr Ð π=2

0 R φr ðφr Þ ¼ Ð 2π0 Ð π=2 0

0

ρλ ðθi , φi , θr , φÞ cos θr sin θr dθr dφ

ρλ ðθi , φi , θr , φr Þ cos θr sin θr dθr dφr

(46)

If the dependence of bidirectional reflectivity based on azimuthal angle is weak or does not exist, as in the case of an optically smooth, perfect dielectric surface, the reflectivity depends only on the polar angle of incidence. In such a case, Eq. (46) would yield an RNR that is identical to the one presented in Eq. (18). The CDF for polar angle of reflection is Ð θr

R θr ðθr Þ ¼ Ð π=20 0

ρλ ðθi , φi , θ, φr Þ cos θ sin θ dθ

ρλ ðθi , φi , θr , φr Þ cos θr sin θr dθr

(47)

in which case, the azimuthal angle of reflection is given by Eq. (46). When optically smooth or polished surfaces are encountered in a system, it is very common to treat these surfaces as specularly reflecting or mirrorlike reflecting surfaces. For determining direction of reflected photon bundle, rules of geometric optics are used rather than relying on a statistical model. For such a surface, the polar angle of reflection can be defined simply as θr ¼ θi

(48)

and the azimuthal angle of reflection becomes φr ¼ φ i þ π

(49)

Considering that diffuse reflection and specular reflection are two idealistic limiting cases (although some rough surfaces are backscattering), they are often used in combination to represent a more realistic behavior. In such a case the reflectivity is assumed to have diffuse and a specular component and can be defined as ρ ¼ ρd þ ρs

(50)

where subscripts d and s denote the diffuse and specular components, respectively. If the reflectivity of a surface is modeled relying in this approach, and a photon bundle is reflecting from the surface, a random number, Rs/d, can be used to decide whether the reflection behavior would be diffuse or specular. If the generated random number

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H. Ertürk and J. R. Howell

is smaller than or equal to ρd/(ρd + ρs), (Rs/d  ρd/[ρd + ρs]), the new direction of the photon bundle after reflection can be computed for a diffuse reflection. If it is the opposite case (Rs/d > ρd/[ρd + ρs]), the reflection is considered to be a specular reflection.

4

Pseudorandom Numbers

Consider what is meant by a random number. A random number can be defined as a number chosen from a large set of numbers of equal intervals without a sequence. The simplest way to “generate” random numbers might be writing down a series of numbers of equal intervals on small pieces of paper, putting all the papers in a bag, and, after mixing them up, picking a number at a time. Applying this method of generating random numbers is certainly impractical while carrying out a computational simulation. The standard way to generate a random number that is to be used during a computation is through the use of a piece of code that is developed to perform the task. It might sound quite unacceptable to expect a computer, arguably the most precise and deterministic machine produced by mankind, to generate a true random number; therefore, random numbers generated as such are usually referred as pseudorandom numbers. These algorithms are capable of creating a finite sequence of pseudorandom numbers for a given initial seed. Therefore, for simulations necessitating large quantities of pseudorandom numbers, one should be careful about the limitations of the so-called “random number generators.” One other important aspect is the distribution of the random numbers. The most common random number generators are those that would generate random numbers with a uniform or normal distribution. While simulating a physical event using a Monte Carlo method, a random number generator that generates random numbers with a uniform distribution must be used. The topic of generating pseudorandom numbers is a broad area, and it is beyond the interests of this study. Further information about the topic is available at literature such as Press et al. (1992), Hammersley and Handscomb (1964), and Taussky and Todd (1956).

5

Statistical Uncertainty of Monte Carlo Simulations

One of the most powerful aspects of Monte Carlo method is that it is a statistical simulation technique and the error in results can be estimated in terms of statistical uncertainty, whereas defining such a confidence level is not possible for most other methods. The most practical way of estimating the error in the value is to subdivide the calculation of the mean of the results into a group of sub-means. The central limit theorem, which states that the statistical distribution of the sub-means should be distributed in a Gaussian distribution about the overall mean, is then applicable. For a Gaussian distribution, the measure of the fluctuations in the means is called the variance. Therefore, instead of making a single Monte Carlo simulation that uses

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Monte Carlo Methods for Radiative Transfer

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N sample photon histories, M simulations, using N/M samples each, should be carried out to calculate a property, P. These simulations must utilize pseudorandom numbers that are generated by using a different seed so that they do not simply repeat each one. If P is distributed based on the Gaussian around the overall mean, P, an estimate of the error can be made in terms of variance. The variance, γ 2, is 2 !2 3 M
M M X X X  1 1 1 2 4 γ2 ¼ Pi  P ¼ Pi 2  Pi 5 M  1 i¼1 M  1 i¼1 M i¼1

(51)

The variance, the square of the standard deviation, is an estimate of the mean square deviation of the sample mean P from the true mean assuming the true mean could have been calculated using an infinite number of samples. From the statistical theory, the probability of the sample mean lying within γ of the true mean is 68%, 2γ of the true mean is 95%, and 3γ of the true mean is 99.7% for a Gaussian distribution. From Eq. (51), it can be observed that to reduce the standard deviation by half, the number of simulations, M, and at the same time the number of samples, N, should be quadrupled. Usually this will result in quadrupling the CPU time.

6

General Outline

Monte Carlo method can be used for many different problems. Out of these, enclosure problems, where radiative heat transfer of a closed system comprised of surfaces and medium, are one of the most frequently encountered ones. A general radiative heat transfer solver based on Monte Carlo method for enclosure problems comprises three main steps: 1. Preprocessing 2. Monte Carlo simulation 3. Post-processing Each step is explained in detail in the following subsections.

6.1

Preprocessing: Defining Geometry, Properties, and Boundary Conditions

A first natural step toward a Monte Carlo simulation is the preprocessing step, where the geometry, the properties of surfaces, the medium, and their boundary conditions are defined. While simple geometries can easily be modeled in the form of basic shapes, for modeling more complex geometries, computer-aided design (CAD) tools can be used for modeling more complex geometries. A commercial CAD tool is used by Farmer and Howell (1994a) where they described how a finite element-based

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geometric model can be used for Monte Carlo method in detail. Another alternative is using parametric definitions for representing surfaces such as using nonuniform rational B-splines (NURBS) as explained by Daun and Hollands (2001). While fundamental information is briefly introduced here for the completeness of the discussion, interested readers should refer to standard CAD literature and text, such as Rogers and Adams (1989), for more general and detailed description of defining surfaces parametrically. For a Cartesian coordinate system, a surface can be defined as z = f(x,y), and the corresponding vector representation of a point over a surface can be presented as ^ r ¼ x^i þ y ^j þ f ðx, yÞ k

(52)

A more general representation relies on a parameter set (u,v) that can be more useful leading to a vector representation of ^ r ¼ xðu, vÞ^i þ yðu, vÞ^j þ zðu, vÞk

(53)

For such a surface, at a given point defined by the parameter set (u,v), the local tangential unit vectors are ^t 1 ¼ @r=@u j@r=@uj

(54)

^t 2 ¼ @r=@v j@r=@vj

(55)

and

Then the unit normal vector is defined using the cross product of the two tangential vectors as ^t  ^t 2  ^¼ 1 n ^t 1  ^t 2 

(56)

Once the surfaces and enclosed volume are defined along with their unit normal and tangential vectors, both the surfaces and the enclosed volume must be divided into subareas and volumes. It is common to consider the temperature, flux, or flux divergence to be uniform within a sub-element, as is the case in the formulations presented by Eqs. (3)–(6). Therefore, the accuracy of such a discretized solution relies on the number of sub-elements used in the simulation. The radiative transport equation only has first derivative and integral terms, rather than higher-order derivatives that exist in the other modes of heat transfer. Hence, to be able to approximate the higher order derivatives more accurately, the number of sub-elements used has often more significance in multimode problems than a pure radiation problem. However, the number of sub-elements used in the simulation is still very important

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for the solution of radiative heat transfer by Monte Carlo method, and checking the solution for the grid independence is a must. Following the definition of the geometry, the properties of the surfaces and the medium must be defined. While data bases based on experimental data can be used for surface properties (▶ Chap. 25, “Design of Optical and Radiative Properties of Surfaces”) and for particulate media (▶ Chap. 27, “Radiative Properties of Particles”), the properties can be estimated relying on theory as explained in the relevant text in the absence of these (Howell et al. 2016). For gas radiation, different models varying from line by line data (Wang and Modest 2007), narrow band models (Cherkaoui et al. 1996), wide band models (Farmer and Howell 1994a), weightedsum-of-gray-gases (WSGG) models (Maurente et al. 2007) or contemporary models such as full spectrum k-distribution method (FSK) (Wang et al. 2007) can be used (▶ Chap. 26, “Radiative Properties of Gases”). As the application of gas spectral models is a cumbersome task, more information about the use of some these methods are presented in the upcoming sections. The number of photon bundles that must be emitted from each sub-element must also be determined during the preprocessing step. If the radiative emission for each sub-element is known, the number of samples that must be emitted from each sub-element can be determined based on the total number of photon samples simulated and the ratio of emitted radiative energy per sub-element to that of complete system. If this information, radiative emission of each element, is not available before the simulation, a reference temperature can be used for the prediction. Then the number of photons that must be used for each surface would be distributed based on emissivity-area product for surface elements and absorption coefficient-volume product for the volume elements. Based on this approach, the behavior of larger elements or elements with higher emissivity that are expected to emit more energy would be simulated with a larger number of samples leading to a statistically better representation. Once the geometry, boundary conditions, properties of the materials, and the number of photon bundles that are to be emitted from each sub-element are identified, the Monte Carlo simulation can be carried out.

6.2

Monte Carlo Simulation

As explained earlier, one great advantage of the Monte Carlo method is its flexibility. Therefore, the method can be applied in many different ways depending on the problem and the boundary conditions. However, a generic approach would target estimating the exchange factors between all the sub-elements of the considered system to be used with the formulations presented by Eqs. (3)–(6). In such an approach, photon samples are emitted from each sub-element, and they are tracked until they are absorbed by a sub-surface or a sub-volume. The so-called “ray tracing” algorithms are used for tracking the behavior of photon bundles relying on random numbers and probability distributions of the relevant events from their emission to their absorption. Therefore, the ray tracing algorithm is the most critical part of a

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Fig. 3 The pseudo-code for the Monte Carlo simulation

Monte Carlo Simulation Photon emission from surfaces is=1,Ns ip =1,Ne(is) Ray Tracing Algorithm end end Photon emission from volumes ig=1,Ng ip=1,Ne(Ns+ig) Ray Tracing Algorithm end end end Monte Carlo Simulation

radiative heat transfer model based on the Monte Carlo method. More detail about ray tracing algorithms is presented in the upcoming sections. A counter is utilized through the ray tracing algorithm to keep record of where photon bundles are absorbed. At the end of the simulation, when tracking of all the photon bundles emitted from all sub-elements is finalized, the exchange factors can be estimated. Considering that Ne( j) represents the number of photon bundles emitted from a sub-element j, the pseudo-code for Monte Carlo simulation of a system with Ns sub-surfaces and Ng sub-volumes is presented in Fig. 3. The efficiency of the ray tracing algorithm used is critical as it may be used millions of times throughout the simulation to achieve good statistics. To assure convergence within a specified variance, Eq. (51) can be computed after each Monte Carlo simulation comprised of N sample bundles. This on-the-fly computation avoids the necessity of guessing the number of samples that will provide useful results. When the variance γ 2 is within the required value, the ray tracing simulations can be terminated.

6.3

Post-Processing

Following the Monte Carlo simulation, the exchange factors are estimated, and the unknown radiative heat flux or temperature distributions for surfaces and unknown divergence of radiation heat flux or temperature distributions for the medium can be identified during the post-processing. For that, a system of equations comprised of

29

Monte Carlo Methods for Radiative Transfer

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radiative energy balances over each subsurface and sub-volume must be established and solved. Following the solution of the set of equations, the radiative energy balance for the system must be controlled to ensure that the solution satisfies the first law of thermodynamics.

7

Ray Tracing

The Monte Carlo method relies on statistical sampling of photon bundles for the solution of radiative transfer problems. Therefore, it is required to simulate the behavior of each bundle from its emission until it is absorbed using pseudorandom numbers and probability distributions for each particular event. As the statistical accuracy of the Monte Carlo method is significantly dependent upon the number of samples used, the efficiency of each of these simulations is critical. Although using a basic and simple ray tracing algorithm may suffice, the use of sophisticated ray tracing algorithms might be preferable to increase efficiency in many cases. Ray tracing is utilized in different areas, and one of its main application areas is computer graphics. There exists significant ray tracing know-how developed by the computer graphics community. While the objective of this section is to introduce the basics of ray tracing, for further information the interested readers should refer to relevant text in the field such as Glassner (1989), where ways of increasing efficiency are discussed. A typical ray tracing algorithm is comprised of steps that determine the point and direction of emission, identification of the surface the photon bundle will be intercepting, reflection or absorption by surfaces, and scattering or absorption by the intervening participating medium. If the photon bundle is reflected or scattered, the new direction must be identified. The ray tracing ceases when the energy contained in the bundle is transferred to the medium or surface. As mentioned earlier, the Monte Carlo method is very flexible, and it can be applied in many different ways. Moreover, some would be advantageous based on the problem and the boundary conditions considered. While some special cases are considered in upcoming sections, in this particular section, a general outline of ray tracing will be provided based on the classification originally presented by Farmer and Howell (1998). Their classification considered the collision-based algorithm and the path length-based algorithm, forward and reverse algorithms, which is also adapted here. While the basic versions of these algorithms are introduced here by step-by-step instructions, they can be implemented in many different ways. These are not included here, and an interested reader should refer to Farmer and Howell (1998) for a more detailed discussion of different options of these algorithms.

7.1

Collision-Based Algorithm

The basic idea behind the collision-based algorithm is that the photon bundles are considered as a whole throughout the ray tracing, and they are not split into smaller pieces, and their energy is not distributed. Therefore, when the bundle undergoes a

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reflection or absorbed by medium, the whole energy contained is assumed to experience that event. The flowchart of one variation of a collision-based algorithm is presented in Fig. 4, and a brief outline can be summarized as follows. The collision-based algorithm starts with the emission of the bundle. If the emission is from a surface for a three-dimensional system, the point of emission is determined by using two pseudorandom numbers using Eqs. (21) and (22). For a volume element, a third pseudorandom number must be used with an additional CDF that can be derived similar to Eqs. (21) and (22). Once the point of emission, re, is identified, the polar and azimuthal emission angles are estimated through Eqs. (17) and (18) by using two pseudorandom numbers for a diffuse emitter. For emission from a volume element, Eq. (19) must be used instead of Eq. (17) for determining ^ , and unit polar angle of emission. Considering that the unit normal vector, n tangential vectors, ^t 1 and ^t 2 , are known for the surface at the point of emission,

New emission from ie Emission point (re) Eqs. (21, 22) Emission angles: (θ, j) Eqs. (17-19) sˆ = cos θ nˆ + sin θ (sinj tˆ1 + cos j tˆ 2 ) −1

Absorption pathlength: Sa = −k l ln Ra −1 Scattering pathlength: S s = − s s,l ln Rs

Directed surface and path to interception (ds)

Hit the surface? (Sa>ds and Ss>ds) Scattering θs, js rs = re + d s sˆ sˆ = sˆ s re=rs Sa=Sa-Ss

Yes

rr = re + d ssˆ θi, ji

No Yes

Scattered? (Sa > Ss) No

ra = re + d s sˆ

Does it reflect? R r≤r l(θ i, j i) No

Yes

Reflection θr, jr sˆ = sˆ r re=rr Sa=Sa-ds Ss=Ss-ds

Absorbing sub-element (ia) Na(ie,ia)= Na(ie,ia)+1

Fig. 4 Flow chart for collision-based ray tracing algorithm using absorption and scattering path lengths

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Monte Carlo Methods for Radiative Transfer

1223

the unit vector describing the direction of the photon bundle or the directional vector, ^s , is
 ^s ¼ cos θ n ^ þ sin θ cos φ ^t 1 þ sin φ ^t 2

(57)

Once the point and direction of emission are known, the surface the photon bundle is directed toward must be identified. A simple approach relies on estimating the photon bundle’s path to its point of interception with surfaces. The governing equation for the path of a photon bundle emitted at point re, travelling a distance d in a direction ^s can be simply presented as r ¼ re þ ^s d

(58)

The point of interception of the photon bundle with a surface can be estimated by equating the vector equation of the photon bundle, Eq. (58), with the parametric equation of the surface, Eq. (53). ^ ¼ ^r e þ ^s ds xðus , vs Þ^i þ yðus , vs Þ^j þ zðus , vs Þk

(59)

Equation (59) is a vector equation, yielding three scalar equations with three unknowns, which are the parameter pair representing point of intersection (us,vs) and the path photon travels before intercepting the surface (ds). The unknowns can be found from the solution of the system. Equation (59) must be solved for all surfaces ^ < 0, and the intercepting point must be checked so that in the system that satisfy^s  n it lies within the bounds of the surface (umin  us  umax and vmin  vs  vmax). From the surfaces satisfying these two criteria, the surface with the shortest ds can be identified as the intercepting surface. Once the directed surface and the path length to interception are identified, using two pseudorandom numbers, the absorption and scattering path lengths, Sa and Ss, are estimated by using Eqs. (25)–(27), (28)–(30), respectively. The next event will be decided based on the shortest path length, comparing all three path lengths (ds, Sa, and Ss). The photon bundle is scattered if Ss is the shortest (Ss < ds and Ss < Sa), and the point where the photon bundle is scattered can be estimated as rs ¼ re þ ^s Ss

(60)

The scattering polar and azimuthal angles (θs and φs) are identified from Eqs. (32) and (33) by using two pseudorandom numbers. Then, the scattered direction of the photon is identified based on the direction of the photon ^s and two unit vectors (^e 1 and ^e 2 ) that are orthogonal to each other and to the original direction vector ^s. ^s s ¼ cos θs ^s þ sin θs ð cos φs ^e 1 þ sin φs ^e 2 Þ

(61)

For a given ^s, the first of these vectors can be identified utilizing an arbitrary unit vector such as ^i by

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H. Ertürk and J. R. Howell

^s  ^i ^e 1 ¼   ^s  ^i 

(62)

^s  ^e 1 j^s  ^e 1 j

(63)

and the second one by ^e 2 ¼

Once the scattering point and new direction vector is identified, the ray tracing algorithm returns to the step that identifies the directed surface and the path length for intercepting that surface. While a new scattering path should be estimated using a new pseudorandom number, the absorption path length must be updated by subtracting the travelled path from the original absorption path (Sa = Sa  Ss). The photon bundle hits the surface if ds < Sa and ds < Ss. The position of interception is defined by rr ¼ re þ ^s ds

(64)

For a diffusely reflecting and opaque surface, a pseudorandom number is generated and compared with the reflectivity of the surface to identify whether the photon is reflected. If the surface reflectivity is smaller than or equal to the random number (ρλ  Rρ), the photon is reflected. Otherwise, the photon bundle is absorbed. For a diffuse reflector, the reflection angles are estimated from Eqs. (15) and (16), and the direction of the reflected photon bundle is estimated by Eq. (57). If the surface is not diffuse, the angle of incidence must be determined. Considering the direction vector of incident photon (^s i ¼ ^s) and the unit normal and tangential vectors of the surface ^ i ,^t 1, i ,^t 2, i ), the polar and azimuthal angle of incidence can be found from the (n following vector equation
 ^s i ¼  cos θi n ^ i  sin θi cos φi ^t 1, i þ sin φi ^t 2, i

(65)

The three unknown direction cosines (cos θi, sin θi sin φi, sin θi cos φi) can be found through the solution of three scalar equations. If the surface is specularly reflecting and properties are not direction dependent, there is no need to calculate the angles (θi, φi) as the direction of the reflected photon can be identified by the direction cosines only. Considering that in such a case, θr = θi and φr = φi + π, the reflected bundle’s direction would be
 ^s r ¼ cos θr n ^ i þ sin θr cos φr ^t 1, i þ sin φr ^t 2, i
 ^ i þ sin θi  cos φi ^t 1, i þ sin φi ^t 2, i ¼ cos θi n

(66)

The calculation of angles should be avoided unless it is absolutely necessary, as execution of inverse trigonometric functions is expensive, and this operation when repeated many times will increase the processing time. However, if the surface have

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directional reflectivity, ρλ(θi, φi), then the incidence angles (θi, φi) must be calculated. It is determined if the photon is reflected or not comparing a pseudorandom number with the directional reflectivity. If the random number is smaller than or equal to directional reflectivity (Rρ  ρλ(θi,φi)), the photon bundle is reflected; otherwise it is absorbed for opaque surfaces. The reflection angles must then be calculated by using Eqs. (46) and (47). Equation (57) is used for predicting the direction of the reflected photon bundle. The absorption and scattering path lengths must be updated by subtracting the already travelled path (Sa = Sa  ds, Ss = Ss  ds), and the ray tracing for the reflected photon bundle continues, with the new origin of the photon being rr. If the photon is absorbed by the surface, the photon will be absorbed by the subsurface, ia, containing point rr. Considering that the emitting sub-element’s index is ie, it is recorded that a photon bundle emitted by sub-element ie is absorbed by subsurface ia via a counter. N a ðie , ia Þ ¼ N a ðie , ia Þ þ 1

(67)

If the absorption path length is the shortest (Sa < ds and Sa < Ss), it is assumed that the photon bundle is absorbed by the medium. The point where photon bundle is absorbed will be identified by ra ¼ re þ ^s Sa

(68)

and the index of absorbing sub-volume containing point ra can be identified as ia. The counter presented in Eq. (67) is used to record that a photon bundle emitted by sub-element ie is absorbed by sub-volume ia. Once all the photon bundles or samples are simulated from their emission to their absorption, the exchange factors can be calculated as Γjk ¼

N a ðj, kÞ N e ðjÞ

(69)

The explained algorithm can also be modified by using an extinction path length, Se, rather than absorption and scattering path lengths. In such a case, if the extinction path length, Se, is smaller than ds, a pseudorandom number is compared to scattering albedo to decide on whether photon bundle is absorbed or scattered. The rest of the modified algorithm, whose flowchart is presented in Fig. 5, is very similar to the original one shown in Fig. 4. One other frequently adopted version of the collision-based algorithm considers accounting for energy parcels that are transferred via photon bundles carrying them, rather than relying on exchange factors and solution of system of equations during post-processing. This version is more advantageous to use when the surface temperatures are known, and the radiative heat flux distribution over the surfaces and variation of divergence of radiative heat flux within the medium are sought.

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New emission from ie Emission point (re) Eqs. (21, 22) Emission angles: (θ, j) Eqs. (17-19) sˆ = cos θ nˆ + sin θ (sin j tˆ1 + cos j tˆ 2 ) Extinction pathlength: Se = −bl−1 ln Re Directed surface and path to interception (ds)

Hit the surface? (Se >ds)

Scattering θs, js rs = re + d s sˆ sˆ = sˆ s re=rs

rr = re + d ssˆ θi, ji

Yes

No Yes

Scattered? (Rω ≤ ωl)

Does it reflect? Rr ≤ rl(θi,ji)

No

Yes

Reflection θr, jr sˆ = sˆ r re=rr Se=Se-ds

No Absorbing sub-element (ia) Na(ie,ia)= Na(ie,ia)+1

ra = re + d s sˆ

Fig. 5 Flow chart for collision-based ray tracing algorithm using absorption and scattering path lengths

Another case where this approach can be practical is when the surface temperatures are either defined or surfaces are reradiating and the medium is in radiative equilibrium. In such a case, once a photon bundle is absorbed in a volume element that is in radiative equilibrium or a surface that is reradiating, the absorbed energy of the photon bundle is recorded, and a photon bundle of the same energy is emitted from the position where the previous bundle is absorbed. No extra photons are emitted from these reradiating subsurfaces or sub-volumes in radiative equilibrium. Converting the emitted (or absorbed since they are identical) energy to emissive power, the temperature of the sub-element can be estimated without the need of solving a system of equations. For a sub-element emitting Qe, with a Planck mean absorption coefficient κ p, and a volume of V, the medium temperature is

Qe Tg ¼ 4κp σV

1=4

(70)

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Path Length-Based Algorithm

It is assumed that the photons forming the photon bundles act together from their emission to absorption in the collision-based algorithm and the photon bundles are considered to have a fixed energy throughout the ray tracing. This is not the case in the path length-based algorithm, where the energy of the photon bundles is distributed to the encountered surfaces and volume elements as they travel through the system. In a way, the photons forming the bundles act separately, and some are absorbed by the medium or the encountered surfaces, while the rest travel through the medium or reflect. While many features of the collision-based algorithm and path length-based algorithm are similar, the execution of the algorithms for a system with homogeneous properties differs significantly. The basic features of a path length-based ray tracing algorithm is presented in Fig. 6. Similar to the collision-based algorithm, the path length-based algorithm starts with emission of a photon bundle. The energy rate of photon bundle, ep, which represents the ratio of energy of the bundle at any time to its emission energy, is set to 1 at emission. The point of emission can be determined using Eqs. (21) and (22), using pseudorandom numbers. The emission angles are calculated using Eqs. (17) and (18) for diffusely emitting surfaces and by using Eqs. (18) and (19) for participating media. The direction of emission can be defined based on emission angles, unit normal, and tangential vectors by Eq. (57). The scattering path length can be identified using Eq. (28). Next, it is required to estimate the path length to the boundary of the volume element, where the photon bundle is traveling. For a photon bundle travelling in sub-volume j, if this path, dv,j, is larger than the scattering path length, Ss, there will be no scattering within the volume element, and the photon travels to the boundary of the medium element. As the photon bundle travels a path of dv,j, within volume element j, part of its energy is absorbed by the medium based on the Beer-Lambert law. Assuming that ep is the rate of energy of the photon bundle as it enters the volume j, the energy transferred to the medium would be [1  exp(κλ dv,j)] times the energy of the photon bundle as it enters the volume j. If the photon bundle is emitted by the sub-element ie, the portion of energy absorbed by the volume element j can be recorded as 
 N a ðie , jÞ ¼ N a ðie , jÞ þ ep 1  exp κ λ, j dv, j

(71)

The energy rate of the photon bundle as it arrives at the boundary of the volume element j is updated to account for the energy transferred to the medium along path dv,j as
 ep ¼ ep exp κλ, j d v, j

(72)

The updated energy rate of photon bundle, ep, must be checked at this point. If it is less than a prespecified value, ξ, which is usually defined on the order of 103–105, the remaining energy is added to the volume element j:

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New emission from ie ep=1 Emission point (re) Eqs. (21, 22) Emission angles: (θ, j) Eqs. (17-19) sˆ = cos θ nˆ + sin θ (sin jtˆ1 + cos jˆt2)

s −1 Scattering pathlength: S s = −ss,l ln Rs

Determine sub volume j Path to sub-volume boundary, dv,j

Scattered in volume? (Ss ξ

Yes

No Fig. 6 Flow chart for path length-based ray tracing algorithm

N a ðie , jÞ ¼ N a ðie , jÞ þ ep

(73)

so that energy conservation is achieved at the end of the simulation. Since the energy of the photon bundle is depleted, the ray tracing for the particular photon bundle is terminated, and simulation of the next photon bundle is initiated.

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If the updated energy rate of the photon bundle is higher than ξ, ray tracing for the photon bundle continues. The scattering path length is updated, considering the travelled distance by S s ¼ S s  d v, j

(74)

The boundary of the volume element j can be either a surface of the enclosure or the interface between volume element j and the neighboring volume element, k. If it is a surface of the enclosure, the surface incidence angles (θi, φi) must be predicted by Eq. (65), as explained in the previous section so that the amount of energy transferred from the photon bundle to the surface can be defined in terms of the directional reflectivity of the surface. The point of reflection must be identified by Eq. (64), replacing ds term by dv,j, and the sub-element where the photon bundle is incident (ir) must be identified based on the point of reflection. Considering that the surface is opaque, the energy of the photon bundle transferred to the reflecting subarea can be recorded as N a ðie , ir Þ ¼ N a ðie , ir Þ þ ep ½1  ρλ ðθi , φi Þ

(75)

and the energy rate of the photon bundle must be updated to account for the energy absorbed by the surface: e p ¼ e p ρλ ð θ i , φ i Þ

(76)

For a diffuse surface, there is obviously no need to calculate the incidence angles, in which case hemispherical the reflectivity, ρλ, should be used in Eqs. (75) and (76), instead of the directional reflectivity. Following absorption and updating of the energy rate of the photon bundle, it is checked with respect to specified threshold, ξ. If it is smaller than the threshold value, the remaining energy is added to subarea ir via Eq. (73), replacing j with ir. The ray tracing for the photon bundle terminates, and simulation of the next photon bundle proceeds. If ep is larger than ξ, then reflection angles and direction of reflected bundle are determined considering that the photon is reflected at a position rr. Eqs. (17), (18), and (57) must be used for diffuse surfaces; Eqs. (46), (47), and (57) must be used for directional surfaces; and Eq. (66) must be used for specularly reflecting surfaces. The ray tracing continues as explained earlier with identifying the path to the volume element boundary the photon is directed toward. If the boundary of the volume element j is an interface between volume elements j and k, then the ray tracing continues with updating of the position of the photon bundle along the same direction. If the path of the photon bundle to the sub-volume’s boundary, dv,j, is larger than the scattering path length, Ss, then the photon bundle scatters after travelling Ss within the sub-volume j. The energy absorbed by the sub-volume due to the Beer-Lambert law is recorded by Eq. (71), replacing dv,j term by Ss. Similarly, the energy rate of the photon bundle is updated by Eq. (72), replacing dv,j term by Ss. If the updated ep is smaller than threshold value ξ, the remaining energy of the bundle is added to the volume element by Eq. (73), and the ray tracing for the

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particular bundle terminates, succeeded by simulation of the next bundle. If the updated ep is larger than ξ, then the ray tracing continues by updating the position of the photon bundle using Eq. (60), determining the scattering angle using Eqs. (32) and (33), and the direction of the scattered photon bundle by using Eqs. (61)–(63). A new scattering path length is calculated by Eq. (28), and the ray tracing continues with determination of the path to the sub-volume’s boundary. Ray tracing of the photon bundles continues until the energy rate of the photons degrades beyond the specified threshold value. The path length-based ray tracing algorithm can be implemented in different ways such as using a randomly estimated extinction path length and scattering albedo. This approach would require the use of two pseudorandom numbers and would be more computationally expensive than the presented algorithm. The implementation of a path length-based algorithm for surface exchange problems should be altered significantly from its implementation for participating medium that is explained in detail. The computational model does not have to include the medium and the volume elements representing the medium. In such a case, once the direction of the photon is defined, the algorithm must identify the surface where the photon bundle is directed and the path to the point of interception, similar to the collision-based algorithm. Therefore, the photon bundle’s energy reduces as it reflects from the surfaces, and the ray tracing continues as its energy depletes beyond a predefined threshold value. This approach complicates energy bookkeeping but may provide improved statistics since some energy is deposited within every surface and volume element encountered during the tracing of each bundle. Farmer and Howell (1998) found that improvement in computer time to reach a given specified variance between the collision and path length methods depended on the particular problem.

7.3

Reverse Algorithm

The algorithms considered so far rely on tracing the photon bundles from their emission to absorption based on different approaches. Therefore, they are often referred as forward ray tracing algorithms, and the methods relying on them are referred to as the forward Monte Carlo method, where the solution for a complete system is considered. However, there are problems where a relatively small portion of a system is considered, as in the case of a sensor problem. In these problems, the forward Monte Carlo method becomes very inefficient. The so-called reverse Monte Carlo method, where the photon bundles are traced from their absorption back to their emission points, is developed to improve computational efficiency for these cases. This approach is based on the reciprocity principle outlined by Case (1957), and it enables investigation of a limited part of a system without the need of considering the whole system. However, the analysis can also be extended to cover the complete system. The application of the reverse algorithm was first introduced to participating medium problems by Walters and Buckius (1992), where a path length-based ray tracing approach is considered.

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Sj ′ Il(S=0)=Il,k(rw, Ωw)

0

dS ′

Il(S)=Il,k(rj, Ωj) dS sˆ j

bl= kl+ ss,l

Sj

Fig. 7 The ray tracing with reverse algorithm in an absorbing and scattering medium

The application of a reverse Monte Carlo method for a surface problem is straightforward. For a participating medium problem such as the one shown in Fig. 7, the reverse Monte Carlo method considers the intensity at a given point rj as a mean of intensities based on sample intensities as Np

 1 X I λ r j ,  Ωj ¼ I λ, k r j ,  Ω j N p k¼1

(77)

Here the individual sample intensities, Iλ,k(rj,Ωj), are estimated based on probabilistically traced rays in reverse that can be defined by
 I λ, k rj ,  Ωj ¼

 Ð S eλ ðrw ÞI λ, b ½T ðrw Þexp  0 k κ λ ðr0 ÞdS0  Ð o ÐS n S þ 0 k κλ ðr0 ÞI λ, b ½T ðr0 Þexp  S0k κλ ðr00 ÞdS00 dS0

(78)

assuming that the ray originates from a surface at point rw, as shown in Fig. 7. The integrals in Eq. (78) can be replaced for a discretized system by summations relying on the path lengths determined by ray tracing in the reverse direction up to a surface position rw, where the photon is emitted. Considering ray tracing of Np photon bundle samples originating from point of interest rj, the local intensity considering each sample can be estimated based on the estimated paths and the discretized version of Eq. (78). Through averaging of different sample photon’s paths by Eq. (77), for a system with known temperatures, the radiative flux or its divergence can be identified at the point of interest, rj.

8

Implementing Spectral Properties

Radiative heat transfer problems with participating media, particularly molecular gases (▶ Chap. 26, “Radiative Properties of Gases”), exhibit strong spectral behavior. The spectral properties of surfaces (▶ Chap. 25, “Design of Optical and Radiative Properties of Surfaces”), particles (▶ Chap. 27, “Radiative Properties of Particles”), and gases (▶ Chap. 26, “Radiative Properties of Gases”) are discussed

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extensively in the relevant chapters. Therefore, rather than covering each one of these spectral models, a general framework will be provided to explain how to implement spectral properties within the Monte Carlo method. The spectral properties can be implemented by three approaches. The first approach relies on probability-based sampling of the emission bands so that the spectral integral to estimate total quantities is handled implicitly by the Monte Carlo method. In the second approach, the simulations for each wavelength band are carried out separately, and the spectral integral is executed following the simulation. The third approach is the more recently developed full-spectrum k-distribution method (FSK) and its closely related spectral line weighted-sum-of-gray-gases method (▶ Chap. 26, “Radiative Properties of Gases”) that have similarities to the first approach but present a practical way of executing the spectral integral. While a general overview is explained here, the details of the implementation of gas spectral properties in Monte Carlo method rely strongly on the spectral model used, and it is difficult to unify the approaches. Therefore, more details of implementation using different models can be found in related literature such as Farmer and Howell (1994a) for wide band model, Cherkaoui et al. (1996) for narrow band model, Maurente et al. (2007) for the spectral line-based weighted sum of gray gases, and Wang and Modest (2007) and Wang et al. (2007) for full-spectrum k-distribution and line-by-line methods.

8.1

Probability-Based Modeling

Probability-based modeling of spectral properties relies on random sampling for determining the wavelength of each emitted photon bundle. The sampling is carried out based on the PDFs represented by Eqs. (9) and (12). A pseudorandom number is used with the RNR obtained by inverting the CDFs presented by Eqs. (10) and (13). These CDFs do not usually yield an invertible analytical expression and in the absence of a RNR sampling through using the PDF will be computationally inefficient as mentioned earlier. A more efficient way of sampling would be to use a function fitted to the corresponding RNR. In such an approach, the corresponding PDFs and CDFs must be estimated during preprocessing, where the RNR can be described in the form of an analytical function. The spectral properties for most materials are either in the form of numerical data or described in terms of complex and piecewise functional forms. Moreover, they have strong dependence on wavelength, with sharp spikes at certain wavelengths. Therefore, it is usually not possible to represent the spectral properties and the corresponding PDF in terms of a function fit. However, the CDF is a relatively smooth function, monotonically increasing from 0 to 1, and a functional fit representation of the RNR based on a numerically estimated CDF is more straightforward. This approach can be easily applied to surfaces or a medium comprised of participating gas and particles. However, for systems with unknown temperature,

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pressure, and concentration field, the amount of data that must be generated during the preprocessing step and stored during the Monte Carlo simulation can be significant especially if a line-by-line model is used. In such a case, memory requirement can be reduced by using a database of spectral absorption coefficients at certain wavelengths, temperatures, and pressures that can be interpolated to represent the spectral absorption coefficient at different conditions. A similar approach is also applicable to narrow band and wide band models. There are several advantages to probability-based modeling of spectral properties. As the probability distributions of emission are considered, more photon bundle samples are used for the more dominant or important wavelength bands, improving the overall solution accuracy by minimizing the statistical variance for these dominant bands. This will improve the computational efficiency, as simulation of an excessive number of sample photons is prevented for wavelengths bands that are relatively less important. Besides, the computational efficiency of the Monte Carlo method is not strongly dependent on the spectral model employed as the CDF and RNR are established in the preprocessing step. Therefore, even a line-by-line model can be utilized with no significant penalty of excessive computational time. If the system considered has a known temperature field in the medium and at the boundaries, or the medium is in radiative equilibrium, the Monte Carlo method can be implemented assigning energy values to the photon bundle samples, and the radiative heat fluxes at the boundaries and its divergence in the medium can be calculated directly. However, if the temperature field is not known, use of the exchange factor formulation would be adequate. In such a case, the solution would be iterative as probabilistic spectral sampling relies on temperature field within the system. Enough photon samples should be used in such a case, so that the statistical variance in the spectral exchange factors is within adequate limits.

8.2

Bandwise Modeling

Another approach is simulating the bands separately for estimating the spectral exchange factors and solving the resulting set of equations before integrating for the total heat flux and its divergence. While the application of this approach is straightforward and its implementation is very similar to that of most other radiative transfer equation solvers, it has some limitations. As the approach necessitates a solution for each band, it is not feasible when there are too many bands contributing to the radiative transfer, as in the case of an implementation using line-by-line data or narrow band models. A hybrid of bandwise and probabilistic modeling can be implemented for wide band models, where each band can be considered by a separate simulation and the spectral absorption coefficient variation within a band can be considered by probabilistic modeling. However, the bandwise modeling approach is best suited for the weighted-sum-of-gray-gases model, where on the order of tens of bands are considered at most.

1234

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H. Ertürk and J. R. Howell

Full-Spectrum K-Distribution Method

The more recently developed k-distribution method considers reordering based on occurrence of the same absorption coefficient many times at different wavelengths. The solution of the radiative transfer equation will be identical for a homogeneous medium for all these wavelengths, where the absorption coefficient is identical. This is considered in the k-distribution method, and when the entire spectrum is considered, the method is termed the full-spectrum k-distribution (FSK) method. The spectral integral must be carried out with small wavelength increments to capture change in spectral absorption coefficient with wavelength accurately. Therefore, representing the behavior of spectral absorption coefficient in terms of a relatively smooth and monotonically increasing CDF would be beneficial. For a homogeneous gas, the CDF for the full-spectrum k-distribution can be defined as gð κ Þ ¼

π σT 4

ð1 λ¼0 κ λ κ

I b, λ dλ

(79)

Therefore, integrating over g-space is more convenient and requires fewer data points, and the method considers the solution of the radiative transfer equation in g-space, rather than the wavelength, wavenumber, or frequency space. The radiative transfer equation in an absorbing-emitting medium can be presented in g-space as
 dI g ¼ κg Ib  Ig dS

(80)

where the total intensity can be estimated by integrating over g as I¼

ð1 0

I g dg

(81)

Considering the formulation in g-space rather than wavelength space, the cumulative distribution function used to determine the wavelength must be modified for the Monte Carlo implementation. The resulting cumulative distribution can be expressed as Ðg Rg ¼ Ð01 0

κ g I b ðT Þdg0 κg I b ðT Þdg

(82)

Carrying out the sampling based on Eq. (82) becomes more efficient as the spectral behavior can be reproduced with fewer sample photon bundles.

29

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Performance Considerations

Solution of radiative transfer problems considering relatively more complex phenomena is possible with the Monte Carlo method without compromising the accuracy of the solution. Moreover, the implementing of these complex physical phenomena, such as spectral and directional properties, anisotropic scattering, and complex geometries, does not require substantial additional effort. However, the method is known to be computationally demanding even for simple surface exchange problems. While the computational demand for a more complex problem does not significantly increase, it is still considered to be challenging to use the method as a radiative solver for a multimode, multi-physics problem. Therefore, achieving computational efficiency as much as possible is very important while implementing the method.

9.1

General Guidelines

The efficiency of the probabilistic representations of the physical events has significant importance on computational performance as often millions of photon bundle samples are simulated to achieve results with reasonably small statistical variance. Therefore, computational efficiency of these simple but repeating operations must be considered while implementing these probabilistic representations. A good example is estimation of absorption, scattering, or extinction path lengths that must be calculated for every photon for problems with participating medium. For a homogeneous medium, these calculations are based on Eqs. (25), (28), and (39), and at least one division operation is executed for these equations to estimate the probabilistic path lengths. Considering that the division operation is relatively more computationally expensive than multiplication, the value of the reciprocal of the absorption, scattering, or extinction coefficient can be estimated once, and these quantities can be multiplied to estimate the path length during ray tracing to improve computational efficiency. Another such example is trigonometric or inverse trigonometric functions that are also known to be relatively more computationally expensive. These operations should be avoided unless their use is absolutely necessary. The random number relations for polar angle usually result in an expression for cosine or sine of the polar angle. Considering that the direction unit vector requires sine and cosine of the polar angle, it is not required to estimate the polar angle unless it is needed for determining the directional properties. Moreover, conversions between sine and cosine can be handled without using inverse trigonometric functions. However, the azimuthal angle is predicted through the random number relations, and the use of at least one trigonometric function is inevitable, while the conversion can be handled similarly.

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Selecting Different Approaches or Algorithms

As mentioned earlier, the Monte Carlo method can be applied in many different ways. While an alternative application might be advantageous in terms of computational efficiency and ease of implementation for a particular problem, it might be the opposite case for another problem. In this section different implementations presented earlier are evaluated for different types of more frequently encountered problems along with other means of performance improvement. Two basic ray tracing algorithms, collision-based and path length-based algorithms, have been presented in the earlier sections. The exponential decay of intensity in an absorbing and scattering medium can be reproduced by the collision-based ray tracing algorithm by simulating the behavior of photon bundle samples emitted in a particular direction. Photon bundles will be absorbed or scattered based on the randomly estimated path lengths based on the corresponding probability distributions. Therefore, when substantial sample photon bundle numbers are used, the decay can be reproduced with acceptable statistical variance. On the other hand, the decay due to absorption is imposed implicitly in the path lengthbased ray tracing algorithm, and the similar representation would require using fewer number of samples to achieve acceptable statistical variance. Therefore, there is an opportunity to improve the performance using the path length-based method. For a system with nonhomogeneous properties, where both algorithms have to estimate the path lengths through each volume element, using the path length algorithm would help in achieving solutions of acceptable statistical variance with simulation of fewer photon bundles. However, for a system with homogenous medium properties, the formulation effort and number of operations required for a single photo bundle simulation of a collision-based algorithm can be less than it is for path length algorithm. While it would be hard to comment on the comparative efficiency of algorithms in such a case, it is recommended to consider further aspects such as the formulation effort required or long-term development plans, if any. The Monte Carlo method becomes inefficient when a small portion of a large system is of interest as it is required to solve for the radiative transfer in the complete system using a forward Monte Carlo method. However, significant efficiency improvement can be achieved through using the reverse Monte Carlo method, limiting the analysis to the area of interest. It was shown that one order of magnitude lower statistical uncertainty can be achieved using one order of magnitude less computation time by using reverse Monte Carlo method in a surface problem where signal readings of light pipe radiation thermometers for a wafer annealing system is considered (Ertürk and Howell 2010). For a two-dimensional sensor problem with an absorbing, scattering medium subject to collimated radiation, similar statistical uncertainty is achieved using four orders of magnitude fewer photon samples (Modest 2003). One other pitfall of the Monte Carlo methods is in treating problems with optically thick media. While the Monte Carlo method is considered to be a powerful

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tool for modeling ballistic transport, radiative transport exhibits diffusion-like behavior in optically thick media, where the mean free path of photons become much shorter than the geometric length scales. When the Monte Carlo method is used for modeling a system with optically thick media, the path lengths of the photon bundles become very short, and too many photon bundle samples are required to estimate the distribution over a region. Therefore, it is not advisable to use the straightforward Monte Carlo method for modeling an optically thick system exhibiting diffusive behavior. Some techniques for ameliorating the problem of short mean free paths were developed for neutron diffusion problems, where shielding of neutrons requires materials with large neutron cross-sections, resulting in very short mean free paths. For such cases, biasing methods were developed that effectively increase the mean free path of particles while proportionally decreasing their energy. This allows better statistics for absorption or scattering data and is lumped under the class of “biasing” methods (Hammersley and Handscomb 1964). For thermal radiation, the transport can be represented by a diffusive equation according to the Rosseland diffusion approximation, and the diffusion method relying on this approximation can be used for the solution of radiative transfer of such systems. While this approximation is quite accurate within the medium, it fails to accurately represent the regions close to boundaries. Therefore, for systems with optically thick media, a hybrid method suggested by Farmer and Howell (1994b) can be utilized. In this approach, the system is divided into two regions: a region that is comprised of the surfaces and relatively thin layer of medium around them and the remaining medium. In the first region, the radiative transport can be modeled by Monte Carlo method. The method does not suffer from computational inefficiency caused by the presence of optically thick medium if the thickness of the medium modeled is selected carefully so that it does not represent an optically thick system. The remaining region, which is optically thick, can be represented by the diffusion method based on the Rosseland approximation.

9.3

Smoothing Exchange Factors

While the Monte Carlo method implicitly forces the summation rule of exchange factors as a manifestation of conservation of energy, if a forward Monte Carlo algorithm is used, then precisely satisfying reciprocity might necessitate the use of a very large number of photon bundle samples for the simulation. The statistical noise in the exchange factors estimated by utilizing a relatively smaller number of samples can be smoothed by considering both summation and reciprocity rules. This idea was first proposed by Larsen and Howell (1986), who applied leastsquares-based smoothing for direct exchange areas. A similar approach was later developed by Daun et al. (2005) using maximum likelihood approximation, showing that the accuracy of the method can be increased without increasing computational expense.

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Parallel Processing

Monte Carlo methods have long been acknowledged for their suitability for parallel processing using multiprocessors or computer clusters. The parallel processing concept was originally applicable by the use of super computers that are comprised of multiple-interconnected processors, operating in parallel. However, with the introduction of processors relying on multi-core architecture, parallel processing became possible not only by using workstations with multi-core multiprocessors but also using ordinary desktop or mobile computers. Moreover, through the use of graphic processing units (GPU) that have hundreds of cores, the opportunities for increasing the efficiency of parallel implementations of Monte Carlo methods have been further increased. The metric of evaluation for increase in efficiency of parallel processing is often considered as speedup that defines the ratio of processing time required to simulate a given system using a single processor to the processing time required to simulate the same system using parallel processing considering the same level of statistical uncertainty. While the maximum speedup of a parallel processing scheme is identical to the number of processing cores used, this limit can only be approached. The parallel-processed simulation comprises of distributed and undistributed parts. The increase in efficiency will be the result of reduction of the execution time of the distributed parts. Besides, there is processing time required for exchanging information among processors while distributing and gathering the information. Therefore, the processing time required for the distributed parts must be significantly larger than the time required for undistributed parts to maximize the speedup by parallel processing. Moreover, the time required for exchanging information among processors must be minimized. Parallel implementation of the Monte Carlo method is discussed in detail by Farmer and Howell (1998), and they listed several alternative approaches. Based on their classification, parallelization can be carried out by distributing the simulations, spectral bands, photon bundle samples, the spatial domain, or a hybrid of these to the available processing nodes. As explained earlier, one great advantage of using the Monte Carlo method is its capability of estimation of the uncertainty along with the prediction of system behavior. The statistical uncertainty of a Monte Carlo solution can be estimated in terms of a variance by executing a number of simulations with fewer samples rather than a single simulation using the same total number of samples. The results can be averaged, and the variance can be estimated by using Eq. (51). The first parallelization strategy considers distributing these individual simulations to different processors to achieve reduction in computation time. Enough samples must be used in each of these simulations to represent the system behavior adequately, and this limits the number of such simulations to an order of tens of simulations. If the system used to execute these simulations has hundreds of processors, as in the case of a GPU or supercomputer, then the effectiveness of this approach would be limited compared to the capability of the system. However, this approach would be quite reasonable

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when using systems with tens of processors, as in the case of mainstream workstations, with its minimal communication requirement between processors and ease of implementation. It was discussed earlier that one way of modeling the spectral behavior is simulating the spectral bands separately and estimating the exchange factors in each spectral band. The spectral equations can then be solved simultaneously, calculating the total quantities in a post-processing step. Distributing the spectral bands can be considered as a parallelization strategy if such a spectral modeling approach is adopted. The effectiveness limit of this strategy depends on not only the system used for executing the simulations but also the spectral model adopted. For example, the effectiveness limit of this approach for a system with tens of processing cores can be reached if the weighted sum of gray-gases-based model is used, when the number of gases matches the number of processors. Although simulating each line separately will not be practical if a line-by-line model is used, if such an approach is adopted using a supercomputer with hundreds of processors, significant improvement in efficiency can be observed. One drawback about this strategy is in regard to the strong dependence of simulation times on radiative properties of the medium and surfaces. If the system is optically thick, or strongly scattering or reflecting at some wavelengths, the simulation times for these will be significantly larger than it is for the others. Then, the unbalance in processing times for simulations executed in parallel will limit the effectiveness of parallelization using this scheme. This can be prevented by balancing the number of optically thin and thick bands allocated to different processing cores, so that the total processing time for each core is similar. Considering that the behavior of individual photon bundles is independent from each other, a more flexible and scalable parallelization strategy is distributing the simulated photon bundle samples. In this approach, the simulation of photon bundles is distributed to the processing cores so that all the processing cores of the system can be utilized equally and the speedup can be maximized. Once the simulation of the bundles by each processing core is completed, the communication of the behavior of individual photon bundles is necessary as the results of simulation must consider all the photon bundles. However, excessive communication would limit the speedup. While using separate counters for each processing core will help minimize the communication, this would lead to a significant memory requirement. Therefore, an effective communication strategy must be adopted, balancing the requirement for memory and maximizing the speedup. Another very popular parallelization strategy is distributing the spatial domain to the processing cores. This strategy is widely adopted for parallelization implementations of Eulerian computational fluid dynamics (CFD) and computational heat transfer (CHT) methods. However, for Lagrangian methods that rely on tracking of photon bundles or particles, adopting this strategy with a similar implementation with Eulerian CFD/CHT methods would necessitate communication of information between processing cores. In a Monte Carlo model, photons emitted or absorbed by different sub-elements are traced for forward or reverse algorithms,

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respectively. Therefore, a more reasonable spatial domain-based strategy would be distributing the emitting or absorbing sub-elements or regions to different processing cores and carrying out ray tracing for the photon bundles emitted or absorbed by the corresponding sub-element by using the assigned processing core. The load over each processing core must be balanced by assigning the sub-elements so that the total number of photon bundle samples each core is processing is similar to maximize the speedup. Finally, a hybrid of these approaches can be used to minimize communication among processors, balance the workload among processors and minimize the idle time on processors, and maximize the speedup. A hybrid approach can be preferable for cases where there is a mismatch between the number of processing cores and the desired number of parallel distributions based on the preferred strategy. An example is using weighted-sum-of-gray-gases model considering ten gray gases using a system with 100 processing nodes. Then ten simulations can be distributed for each of the ten gases considered, whose results are averaged, and a variance can be estimated accordingly during post-processing.

10

Conclusions

The Monte Carlo methods can simulate physical systems in many different fields of science and engineering. Photon bundles are used as samples for solution of radiative transfer problems, and their behavior is statistically simulated relying on pseudorandom numbers and probability distributions based on physical laws. The method has been used for solution of radiative transfer since the space programs in the 1960s, and it is gaining more popularity due to increasing computational power. Being a statistical approach, the solution uncertainty for Monte Carlo methods can be quantified, and it depends on the number of samples used for the simulation. Therefore, achieving a reasonable variance might necessitate a considerable computation time even for a relatively simple surface exchange problem. However, the method can handle many different complex physical phenomena, such as a participating medium, spectral or directional properties, and anisotropic scattering, with ease. Introducing these will not significantly increase the computational expense or formulation effort. The method is very flexible, as it can be applied in many different ways; however, depending on the problem, one approach might be more advantageous than another. Therefore, the nature of the problem must be considered while deciding on the solution strategy. Moreover, the method is highly parallelizable, and with the use of graphics processors; multi-core, multi-processor systems; or high-performance computing grids, significant improvements on solution performance can be obtained. The flexibility and reliability of the method and its capability of handling a wide variety of phenomena have led to an increase in the popularity of the method. Considering the increasing computational capacity in the last decades, the method is expected to be the ultimate solution approach for radiative heat transfer.

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Cross-References

▶ Design of Optical and Radiative Properties of Surfaces ▶ Radiative Properties of Gases ▶ Radiative Properties of Particles ▶ Radiative Transfer Equation and Solutions

References Bird GA (1994) Molecular gas dynamics and the direct simulation of gas flows. Clarendon, Oxford Bird GA (1998) Recent advances and current challenges for DSMC. Comput Math Appl 35(1):1–14 Boyle P, Broadie M, Glasserman P (1997) Monte Carlo methods for security pricing. J Econ Dyn Control 21:1267–1321 Case KM (1957) Transfer problems and the reciprocity principle. Rev Mod Phys 29(4):651 Cherkaoui M, Dufresne JL, Fournier R, Grandpeix JY, Lahellec A (1996) Monte Carlo simulation of radiation in gases with a narrow-band model and a net-exchange formulation. J Heat Transf 118(2):401–407 Daun KJ, Hollands KGT (2001) Infinitesimal area radiative area analysis using parametric surface representation through NURBS. J Heat Transf 123(2):249–256 Daun KJ, Morton DP, Howell JR (2005) Smoothing Monte Carlo exchange factors through constrained maximum likelihood estimation. J Heat Transf 127(10):1124–1128 Ertürk H, Howell JR (2010) Efficient signal transport model for remote thermometry in full-scale thermal processing systems. IEEE Trans Semicond Manuf 23(1):132–140 Ertürk H, Arınç F, Selçuk N (1997) Accuracy of Monte Carlo method re-examined on a box-shaped furnace problem. In Mengüç MP (ed) Proceedings of second international symposium on radiative heat transfer, Kusadasi, Turkey, Jul 21–25, 1997. Begell House, New York, pp. 85–95 Farmer JT, Howell JR (1994a) Monte Carlo prediction of radiative heat transfer in inhomogeneous, anisotropic, nongray media. J Thermophys Heat Transf 8(1):133–139 Farmer JT, Howell JR (1994b) Hybrid Monte Carlo/diffusion methods for enhanced solution of radiative transfer in optically thick nongray media. ASME Publications-HTD 276:203–203 Farmer JT, Howell JR (1998) Comparison of Monte Carlo strategies for radiative transfer in participating media. Adv Heat Tran 31:333–429 Fischetti MV, Laux SE (1993) Monte Carlo study of electron transport in silicon inversion layers. Phys Rev B 48:2244 Fournié E, Lasry JM, Lebuchoux J, Lions PL, Touzi N (1999) Applications of Malliavin calculus to Monte Carlo methods in finance. Finance Stochast 3(4):391–412 Glassner AS (ed) (1989) An introduction to ray tracing. Academic Press, San Diego, CA, USA Goodrum PE, Diamond GL, Hassett JM, Johnson DL (1996) Monte Carlo modeling of childhood lead exposure: development of a probabilistic methodology for use with the USEPA IEUBK model for lead in children. Hum Ecol Risk Assess 2:681–708 Haji-Sheikh A, Sparrow EM (1967) The solution of heat conduction problems by probability methods. J Heat Transf 89(2):121–130 Hammersley JM, Handscomb DC (1964) Monte Carlo methods. Springer, The Netherlands Henyey LG, Greenstein JL (1941) Diffuse radiation in the galaxy. Astrophys J 93:70–83 Howell JR (1968) Applications of Monte Carlo to heat transfer problems. Adv Heat Tran 5:1–54 Howell JR (1998) The Monte Carlo method in radiative heat transfer. J Heat Transf 120(3):547–560 Howell JR, Perlmutter M (1964) Monte Carlo solution of thermal transfer through radiant media between gray walls. J Heat Transf 86(1):116–122 Howell JR, Mengüç MP, Siegel R (2016) Thermal radiation heat transfer, 6th edn. CRC Press, Boca Raton

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Hsu PF, Farmer JT (1997) Benchmark solutions of radiative heat transfer within nonhomogeneous participating media using the Monte Carlo and YIX method. J Heat Transf 119(1):185–188 Larsen ME, Howell JR (1986) Least-squares smoothing of direct-exchange areas in zonal analysis. J Heat Transf 108(1):239–242 Maurente A, Vielmo HA, França FHR (2007) A Monte Carlo implementation to solve radiation heat transfer in non-uniform media with spectrally dependent properties. J Quant Spectrosc Radiat Transf 108(2):295–307 Mazumder S, Majumdar A (2001) Monte Carlo study of phonon transport in solid thin films including dispersion and polarization. J Heat Transf 123:749–759 Metropolis N (1987) The beginning of the Monte Carlo method. Los Alamos Sci 15(584):125–130 Modest MF (2003) Backward Monte Carlo simulations in radiative heat transfer. J Heat Transf 125(1):57–62 Nielsen R, Wakeley J (2001) Distinguishing migration from isolation: a Markov chain Monte Carlo approach. Genetics 158(2):885–896 Papadrakakis M, Lagaros ND (2002) Reliability-based structural optimization using neural networks and Monte Carlo simulation. Comput Methods Appl Mech Eng 191:3491–3507 Press WH, Teukolsky SA, Vetterling WT, Flannery BP (1992) Numerical Recipes. Cambridge University Press, Cambridge Renshaw E, Gibson GJ (1998) Can Markov chain Monte Carlo be usefully applied to stochastic processes with hidden birth times? Inverse Prob 14:1581–1606 Rogers DF, Adams JA (1989) Mathematical elements for computer graphics. McGraw-Hill, New York Sherman RP, Ho YYK, Dalal SR (1999) Conditions for convergence of Monte Carlo EM sequences with an application to product diffusion modeling. Econ J 2(2):248–267 Taussky O, Todd J (1956) Generation and testing of pseudo-random numbers. In: Symposium on Monte Carlo methods, vol 11. Wiley, New York Walters DV, Buckius RO (1992) Rigorous development for radiation heat transfer in nonhomogeneous absorbing, emitting and scattering media. Int J Heat Mass Transf 35(12):3323–3333 Walters DV, Buckius RO (1994) Monte Carlo methods for radiative heat transfer in scattering media. Annu Rev Heat Transf 5(5) Wang A, Modest MF (2007) Spectral Monte Carlo models for nongray radiation analyses in inhomogeneous participating media. Int J Heat Mass Transf 50(19):3877–3889 Wang L, Yang J, Modest MF, Haworth DC (2007) Application of the full-spectrum k-distribution method to photon Monte Carlo solvers. J Quant Spectrosc Radiat Transf 104(2):297–304 Wong BT, Mengüç MP (2004) Monte Carlo methods in radiative transfer and electron-beam processing. J Quant Spectrosc Radiat Transf 84(4):437–450 Wong BT, Mengüç MP (2008) Thermal transport for applications in micro/Nanomachining. Springer, New York Yang WJ, Taniguchi H, Kazuhiko K, Wen-Jei Yang HT (1995) Radiative heat transfer by the Monte Carlo method. Adv Heat Tran 27

Inverse Problems in Radiative Transfer

30

Kyle J. Daun

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Overview of Inverse Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 What is Inverse Analysis? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Types of Inverse Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Solution Methods for Inverse Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Linear Regularization Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Nonlinear Programming Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Metaheuristic Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Bayesian Inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Does Least-Squares Minimization Constitute Inverse Analysis? . . . . . . . . . . . . . . . . . . . 4 Radiant Enclosure Design Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Linear Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Nonlinear Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Case Study: Inverse Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Parameter Estimation Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Linear Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Nonlinear Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Case Study: Parameter Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Conclusions and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Cross-References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

Inverse problems are ubiquitous in all areas of radiative heat transfer. They can broadly be categorized as inverse design problems, with the goal of inferring a design configuration that satisfies an engineering requirement, and parameter K. J. Daun (*) Department of Mechanical and Mechatronics Engineering, University of Waterloo, Waterloo, ON, Canada e-mail: [email protected] # Springer International Publishing AG, part of Springer Nature 2018 F. A. Kulacki (ed.), Handbook of Thermal Science and Engineering, https://doi.org/10.1007/978-3-319-26695-4_64

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estimation problems, in which an unknown parameter or set of parameters is inferred from measurement data. Both problem types are mathematically ill-posed, due to the fact that the available information is either barely adequate or inadequate to identify a unique or stable solution. This chapter reviews the mathematical properties of inverse problems, along with inverse analysis schemes that have been used to solve inverse problems that arise in radiative transfer. This is followed by a summary of inverse design and parameter estimation problems reported in the literature, along with detailed case studies for an inverse boundary condition design problem and a parametric estimation problem focused on inferring the soot aggregate size distribution from light scattering measurements.

1

Introduction

Inverse problems can be defined as problems of inference, in which the goal is to infer some unknown quantity from specified information. They can be subcategorized as parameter estimation problems, in which the objective is to infer a parameter or parameter distribution from indirect measurement data, or inverse design problems, in which the goal is to infer a design configuration that satisfies an imposed design objective. Inverse problems are distinct from generic inference problems because they are mathematically ill-posed, which makes them challenging to solve. In the context of thermal sciences, early work focused mainly on the inverse heat conduction problem (Beck 1968), in which the objective is to infer a surface heat flux, or some other remote quantity, from the time-response of subsurface thermocouples. Likewise, the earliest examples of inverse analysis applied to thermal radiation were parameter estimation problems, and focused on inferring the property or distribution of a property within a participating medium from intensity measurements made at the periphery; the first studies were carried out by Özişik’s group in the late 1980s (Ho and Özşik 1988), although flame tomography (Santoro et al. 1981) and X-ray based medical tomography (Cormack 1973) considerably predate this work. The radiant enclosure design problem, in which the goal is usually to infer an enclosure design that produces a desired heat flux and temperature distribution on a particular surface (called the “design surface”), was conceived in mid- to late 1990s (Fedorov et al. 1998; Jones 1999; Harutunian et al. 1995), and further developed by Howell’s group at the University of Texas at Austin in the early 2000s. Subsequent interest in inverse problems of both types has exploded: two major textbooks on thermal radiation have devoted chapters on inverse analysis (Howell et al. 2016; Modest 2013), and many articles have been published on the topic. An important caveat, however, is that a majority of these papers are limited to “numerical experiments,” while only a small fraction feature physical experimentation or promote schemes that have otherwise been implemented in the physical world. While it may still be argued that these problems are interesting and worthy of study from a theoretical standpoint, many of these papers exclude a thorough discussion of the mathematical properties underlying inverse problems.

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This chapter presents a pedagogical review of inverse analysis and how it applies to radiative transfer. It begins with general overview of inverse problems, mathematical ill-posedness, and solution techniques that have been used to solve inverse problems involving thermal radiation. Subsequent sections focus on the radiant enclosure design problem and the parameter estimation problem, with detailed examples to highlight the mathematical properties of both problem classes.

2

Overview of Inverse Analysis

2.1

What is Inverse Analysis?

The common objective of all inverse problems is to infer some unknown property or attribute from specified, albeit indirect, information. While it may be tempting to simply classify inverse problems as “problems of inference,” by this definition one could argue that virtually every problem is an inverse problem, since the goal of most problems in science and engineering is to infer something from some specified information. Taking things to a logical extreme: if one looks out the window and see blue sky and sunshine, one could infer that it is not raining. Does this constitute an inverse problem? Clearly not. A key property of inverse problems is that they are mathematically ill-posed. This term originates from Hadamard (1923), who formally defined well-posed problems as those that: have a solution, which is unique, and stable to small perturbations to the input data. Problems that violate the: (i) existence, (ii) uniqueness, and (iii) stability criteria are thus mathematically ill-posed. In all cases, ill-posedness arises from an information deficit. Parameter estimation problems are often ill-posed when the information contained in the measured data is barely adequate to specify a unique solution, making it highly sensitive to measurement nose (violating the stability criterion), or is altogether insufficient to specify a unique solution, in which case multiple scenarios exist that could explain the data (violating the uniqueness criterion). As an example of parameter inference, Bohren and Huffman (1983) present an example of a knight hunting dragons in a forest. The knight could easily anticipate the footprints made by different species of dragon, if this information is known ahead of time. Inferring the species of dragon from the footprints may be more difficult, however, particularly if different species of dragons leave similar footprints (Fig. 1). The problem is even harder if the footprints are smeared in the mud, which violates the stability criterion since a small change to the footprint may suggest a different species of dragon. The problem becomes impossible to solve if the footprint is indistinguishable for two different species of dragon. In this example, predicting the footprint left by a dragon is the forward, well-posed problem, while inferring the dragon species from the footprint is the inverse, ill-posed problem. Note that, if each candidate species of dragon left a distinct footprint, the problem would not be mathematically ill-posed, and therefore would not be classified as an inverse problem.

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Fig. 1 A knight hunting dragons in a forest can be conceived as an inverse problem. Predicting the footprint left by a species of dragon is the well-posed forward problem, but inferring the species from a footprint is mathematically ill-posed if the footprints left by different species are similar. The difficulty of the problem is exacerbated if the footprints are smeared in mud, which is conceptually similar to measurement noise

Inverse design problems often have multiple solutions that produce the desired outcome, violating the uniqueness criterion, and, quite often, have no feasible soliton whatsoever, which violates the existence criterion. As a trivial example, shopping for orange juice could be cast as a design problem, in which the objective is to purchase 1 L of orange juice for the lowest possible cost. The problem is straightforward unless the grocery store stocks two competing brands of orange juice at the same cost, which violates the uniqueness criterion. On the other hand, the store may be sold out of orange juice, which violates the existence criterion. The ill-posedness of both examples is caused by an information deficit, and, without exception, inverse analysis techniques address this ill-posedness by introducing more information into the analysis. In the case of parameter estimation problems, the additional information takes the form of some prior knowledge of the unknown parameters (e.g., the species of dragon likely to inhabit the forest), while in design problems, the designer may specify additional desired attributes that tilts the outcome toward one of multiple candidate solutions (e.g., one brand of orange juice tastes better). In a more technical context, consider again the inverse heat conduction and computed tomography examples mentioned at the beginning of this chapter. Both are governed by integral equations of the first kind (IFKs), ð gðsÞ ¼ f ðtÞkðs, tÞdt

(1)

where g(s) is the measurement data, f(t) is the unknown parameter, and k(s,t) is the kernel function. Predicting g(s) from a specified f(t) by integration is the well-posed forward problem. Inferring f(t) from g(s) requires deconvolution or “unfolding” of

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the integral equation (Wing and Zhart 1991), which is the ill-posed inverse problem. The ill-posedness arises from the “smoothing” or “blending” action of the kernel. In the forward problem, variations in f(t) are “blended out” by the integration into much smaller variations in g(s). While the exact solution for f(t) can, in principle, be recovered by deconvolution, if g(s) is perturbed in some way (e.g., with measurement noise, or by discretizing the problem), these perturbations tend to overwhelm the small variations in f(t) that correspond to the “true” problem physics, and are amplified into large artifacts by the deconvolution process.

2.2

Types of Inverse Problems

Understanding the type and fundamental properties of the inverse problem is an essential first step, since this determines the appropriate solution methodology. Inverse problems that can be expressed as an IFK like Eq. 1 are “linear,” since f and g are linearly-related. Most often these problems are solved in matrix form, Ax = b, where x and b are discrete approximations of f and g, respectively, and the elements of A correspond to k in some way. In the simplest representation ð bi ¼ f ðsi Þ ¼ gðtÞkðsi , tÞdt 

n X

n   X Aij g tj ¼ Aij xj

j¼1

(2)

j¼1

which is written for each element of b to form an (n  n) matrix equation. (For now assume that the number of unknowns in x equals the number of knowns in b, so A is a square matrix.) Writing the integral equation in discrete form facilitates solution and also explicates the underlying ill-posedness of the problem via the properties of A. Specifically, the smoothing properties of k makes A ill-conditioned, which can be observed through a singular value decomposition, A = UΣVT, where U and V are orthonormal matrices whose column vectors form a basis for the data space and solution space, respectively, and Σ is a diagonal matrix containing the singular values, arranged in descending order, i.e., Σ = diag[σ1, σ2, σ3, . . ., σn], σ1  σ2  . . .  σn. Discretizing a linear IFK usually produces an A matrix having singular values that decay continuously over several orders of magnitude. An example singular value spectrum is shown in Fig. 2. In this scenario, an “exact” solution can be recovered if the “exact” data is known and the linear system is consistent, i.e., x

exact

n uT bexact X j ¼ vj σj j¼1

(3)

The summation builds xexact in a way analogous to a Fourier series; the first summation terms correspond to the lowest frequency components, while the “sharpest” features are resolved with the latter terms. Hansen (1999) further

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K. J. Daun 100

−5

10

σi

Fig. 2 A singular value spectrum from a discretized integral equation of the first kind. The singular values decay continuously over several orders of magnitude, indicating the ill-posedness of the problem

10−10

10−15

10−20 0

5

10

15

20 i

25

30

35

40

elucidates the relationship between Fourier series and SVD. Even though the singular values become very small as j increases, the magnitudes of the terms in the numerator, the “Fourier coefficients,” |ujTbexact|, decay faster for a smoothlyvarying x, and consequently the summation converges. This is called the “discrete Picard criterion.” If the data is perturbed with an error vector, b = bexact + δb, however, x¼

n uT ðbexact þ δbÞ X j j¼1

σj

vj ¼

|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} xexact

n uT bexact X j j¼1

σj

vj þ

|fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl} xexact

n uT δb X j j¼1

σj

vj

(4)

|fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl} δx

In this case, the discrete Picard criterion is violated because the singular values decay faster than the |ujTδb| terms, which makes x large in magnitude (due to ||δx||) and oscillatory. Equivalently, using the properties of vector and matrix norms, it can be shown that   kδbk kδxk kδbk  CondðAÞ kAkA1  ¼ kxk kxk kxk

(5)

where Cond(A) = σ min/σ max = σ 1/σ n is the condition number, which is large for discrete ill-posed problems. (|||| denotes the Euclidean norm.) An interesting scenario arises when there are more unknowns than equations, i.e., if x  ℜn and b  ℜm, A  ℜmn and m < n. In this case, A is rank-deficient and the singular value decomposition produces U  ℜmn, V  ℜnn, and Σ  ℜmn. Equivalently, one can conceive of a matrix problem having n equations and n unknowns but only m < n independent equations, but in this case, U  ℜnn and

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S  ℜnn, and the last n-m column vectors of U and the last n-m singular values are zero, or “null.” Equation 3 becomes

x¼

m uT b n X X j vj þ Cj vj σj j¼1 j¼mþ1

|fflfflfflfflfflffl{zfflfflfflfflfflffl} xr

(6)

|fflfflfflfflfflffl{zfflfflfflfflfflffl} xn

where {Cj, j = m + 1. . .n} is a set of arbitrary constants. The solution has two components: xr and xn, which belong to the range and nullspace of A, respectively. The first component is also the x having the smallest Euclidean norm that solves Ax = b, while the second component is formed by a linear combination of the last m-n column vectors of V. This scenario violates Hadamard’s uniqueness criteria, since there exists an infinite set of candidate solutions for x that satisfy Ax = b. In this scenario, A is rank-deficient and has a nontrivial null-space. In contrast to the ill-conditioned but full-rank case, in which the matrix equation provides barely enough information to specify a unique (albeit unstable) solution, in the rankdeficient case, the matrix equation provides no information about a key component of the physical solution, xn. A convenient geometric interpretation for these scenarios comes from plotting the residual norm squared, ||Ax  b||2, for a two-dimensional case; the contours are ellipsoids centered on the exact solution, xexact = A1bexact, or more generally xexact = A#bexact, where A# is the pseudoinverse of A. The principal axes of the ellipse are aligned with the column vectors of V and have lengths inversely proportional to the singular values, as shown for a well-conditioned case in Fig. 3a. In an ill-conditioned case, shown in Fig. 3b, the ellipse axis corresponding to the smallest singular value is very long, and the square of the residual norm has a very long, shallow topography surrounding xexact. In this scenario, any candidate solution x along the “valley” can be substituted into the measurement equation to obtain a small residual. In the context of parameter estimation, this residual may be smaller than measurement noise, while in the case of inverse design, these points may represent multiple candidate solutions. Conversely, if one were to contaminate b with small perturbations, the resulting solutions are widely-distributed within the elliptical contours of the residual norm squared, highlighting the violation of Hadamard’s stability criterion. (In fact, these ellipses correspond to confidence intervals via the chi-squared statistic.) Finally, the A matrix is rank-deficient if its rows are identical, as shown in Fig. 3c. In contrast to Fig. 3a, b, ||Ax  b||2 has a trough shape, and is invariant in the v2 direction. (The contours can be conceived as ellipses with one infinite principal axis that corresponds to σ 2 = 0.) As defined above, the xr solution component is obtained from the first summation term, in this case xr = u1Tb/σ 1v1. Adding xn = Cv2 to xr defines a locus of solutions aligned with the “bottom” of the valley. In a topographic sense, the “floor” of this valley is

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0.25

5

a

0.2 0.15

3 V1/σ1

0.1

2 1

0

X2

X2

0.05

−0.05

0 −1

V2 /σ2

−0.1

−2

−0.15

−3

−0.2

−4

−0.25

b

4

0.3

0.4

0.5

0.6

−5

0.7

−4

−2

X1

0

2

4

X1 1.2

c

1 0.8

V2

0.6

X2

0.4 0.2

Xr

0

u1Tb v σ1 1

−0.2 −0.4 −0.6 −0.8

−0.5

0

0.5 X1

1

Fig. 3 Contour plots of ||Ax  b||22 for: (a) a well-conditioned problem; (b) an ill-conditioned problem; and (c) a rank-deficient problem. Contours are ellipses with principal axes aligned in the directions of the V column vectors and lengths proportional to the inverse of the corresponding singular values. In the well-posed and ill-conditioned cases, xexact is marked with a cross, surrounded by trial solutions generated by perturbing bexact with a randomly generated vector δb having the same expected value. (Note the difference in axis scale.) The solution to the rankdeficient problem is the dotted line defined by the sum of xr = (uiTb/σ1)v1 and xn = Cv2

perfectly flat, highlighting the violation of Hadamard’s uniqueness criterion. While these results apply in two dimensions, they can be conceptually extended to ndimensional hyperspace, in which case the “exact” solution is surrounded by ndimensional hyperellipsoids. These figures also highlight a key property of linear inverse problems: with the exception of the rank-deficient case, the square of the residual norm is convex, meaning that it has one global minimum. This also is often, but not always, the case for nonlinear inverse problems. Nonlinear problems arise when the unknown variables cannot be expressed as a linear function of the known variables; for example, nonlinear IFKs have the form

30

Inverse Problems in Radiative Transfer 6

1251 6

a

5.5

b

5

5 4.5

4

X2

X2

4 3.5

3

3 2

2.5 2

1

1.5 1

1

1.2 1.4 1.6 1.8

2

2.2 2.4 2.6 2.8

3

0 −4

−3

−2

−1

0

1

2

3

4

X1

X1

Fig. 4 (a) A nonlinear inverse function that violates Hadamard’s stability criterion, but is convex over the function domain. (Note that the contours of ||A(x)  b||2 are not ellipses.) In some scenarios, it is possible to have a nonconvex residual norm, like the one shown in (b). In this function, the two local minima have the same value, and violate Hadamard’s uniqueness criterion

ð gðsÞ ¼ f ðtÞk½s, t, f ðtÞdt

(7)

and may be discretized as b = a(x). Figure 4a shows that, in contrast to the linear case, the contours are not ellipses. Nevertheless, the function is convex over the plotted region, indicating that Hadamard’s uniqueness criterion is satisfied, but the stability criterion is violated. On the other hand, Fig. 4b shows a scenario in which the residual norm is nonconvex; instead of a global minimum, there are multiple local minima. This may violate Hadamard’s uniqueness criterion, if multiple minima exist that represent alternative viable solutions to the governing equations. The convexity/ nonconvexity of ||a(x)  b||2 strongly influences the solution technique that should be used, as discussed in the next section.

3

Solution Methods for Inverse Problems

Just as there are many types of inverse problems, an equally large and diverse suite of analysis tools have been developed for solving them. All inverse problems derive their ill-posedness from an information deficit, and likewise all inverse analysis techniques address this ill-posedness by introducing additional information about the expected or desired solution attributes into the solution procedure. This information is called “prior information” since it is known prior to the inference procedure. Since the information contained in the ill-posed governing equations is barely adequate or inadequate to specify a unique solution, the information added during the solution scheme strongly influences the recovered solution. Accordingly, it is crucial for the analyst to be fully aware of how the solution schemes introduce prior information into the analysis.

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It is also important to use “the right tool for the job,” which, in this context, means selecting an algorithm that is compatible with the nature of the ill-posedness, size of the problem, and number and type of variables. At a minimum, the following questions should be considered: • What is the nature of the ill-posedness? Is ||Ax  b||2 or ||A(x)  b||2 convex or nonconvex? (Linear inverse problems are always convex, as are many nonlinear inverse problems.) • What additional information can be added to mitigate the ill-posedness? • Is uncertainty quantification important? (It usually is important, but often ignored.) • How many variables does the problem have? Are these variables continuous, discrete, or a mixture of both types? • How costly are the model equations to solve, and what is the computational budget for solving the problem? Can the problem be solved deterministically, or stochastically using a Monte Carlo method? By skipping this essential step, the analyst risks selecting an inappropriate analytical tool, leading to excessive computational time and questionable results. Inverse solution methods can be broadly categorized as: linear regularization schemes, nonlinear programming methods, and metaheuristics. Bayesian methods for solving stochastic inverse problems are also considered.

3.1

Linear Regularization Techniques

These are the oldest class of techniques for solving linear inverse problems; most of these approaches exploit the spectral structure of the coefficient matrix. Three techniques are briefly summarized: truncated singular value decomposition (TSVD); Tikhonov regularization; and conjugate gradient regularization. A more detailed description is available elsewhere in the literature, e.g., Hansen (1999). The singular value decomposition described in the previous section presents an obvious regularization approach: if the first summation terms in Eq. 3 fill out the “main part” of the solution, while the ill-posedness is associated with the small singular values and the high-frequency solution components represented by the last summation terms, why not ignore these last terms? In truncated singular value decomposition (TSVD), the last p summation terms are excluded from the reconstruction, so the regularized solution is given by xp ¼

np T X uj b j¼1

σj

vj

(8)

Prior information is added in an implicit way. Progressively truncating the higherorder summation terms promotes a solution that becomes less oscillatory (i.e.,

30

Inverse Problems in Radiative Transfer

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neighboring elements in x do not vary significantly) and small in magnitude. The amount of prior information introduced into the analysis is controlled by p, which is called a regularization parameter. Since the summation terms relate to progressively higher frequency components, TSVD can also be interpreted as a bandpass filter. Tikhonov regularization (Tikhonov and Arsenin 1977) is based on minimizing a quadratic function FðxÞ ¼ kAx  bk2 þ λ2 kLxk2

(9)

where L is a smoothing matrix and λ is the regularization parameter. Minimizing the first term promotes a small residual of the measurement equations, while minimizing the second term, the “regularizing” or “penalty” term, promotes a desired solution attribute that depends on the choice of L. In standard, or zeroth-order Tikhonov, L is chosen as the identity matrix, and minimizing the second term promotes a solution that has a small norm. In first-order Tikhonov regularization, L is a discrete approximation of the derivative operator, 2

1 1 ⋱ 

1 60 L¼6 4⋮ 0

0 1 ⋱ 0

 ⋱ ⋱ 1

3 0 ⋮ 7 7  ℜðn1Þn 0 5 1

(10)

so ||Lx||2 is minimized by a spatially-smooth solution. Because the function is quadratic, the minimum can be found by setting ∇F(x) = 0 and solving the normal equations 

 AT A þ λ2 LT L x ¼ AT b

(11)

although this should not be done in practice because Cond(ATA) = Cond(A)2. Instead, the problem can be rewritten as a linear least-squares minimization problem (   2 )  A b    x xλ ¼ arg minx fFðxÞgarg minx  λL 0 

(12)

and solved through singular-value decomposition. (The notation arg minx[F(x)] means “the value of x that minimizes F(x).”) There are a multitude of ways to conceptualize how Tikhonov regularization incorporates prior information into the analysis. The least-squares function in Eq. 12 explicitly shows that the regularizing term introduces an additional set of equations, Lx = 0, into the analysis, the influence of which relative to Ax = b is determined by λ. In terms of topography, adding ||Lx||2 to ||Ax  b||2 “steepens the valley” that surrounds xexact, which reduces the magnitude of δx. Making λ too large pushes xλ away from xexact and toward the minimizer of ||Lx||. Finally, in the case of zerothorder Tikhonov, it can be shown that

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xλ ¼

n X j¼1

f j ðλÞ

uTj b vj , σj

f j ðλÞ ¼

σ 2j σ 2j þ λ2

(13)

where {fj(λ), j = 1. . .n} are the filter factors. If λ > σ j, fj = 0, highlights the relationship between Tikhonov regularization and TSVD. Similar relations can be derived for other choices of L (Hansen 1999). Conjugate gradient (CG) regularization also works by minimizing a residual norm, F(x) = ||Ax  b||2, which is equivalent to solving the normal equations AT Ax ¼ AT b

(14)

While Tikhonov regularization does this in one step, conjugate gradient regularization carries out the minimization iteratively. Starting from the null vector x0 = {0}, each successive iteration is updated according to xkþ1 ¼ xk þ αk pk

(15)

where αk is the step size and pk is the search direction. There are many possible schemes for choosing αk and pk; the conjugate gradient scheme generates a sequence of “noninterfering” search directions that are mutually-conjugate with respect to ATA, i.e., they satisfy  T pk AT A pk ¼ 0,

i 6¼ j

(16)

The step size is then found by setting @f(xk + αkpk)/ @αk = 0, which can be done analytically for a quadratic function. In the case of a well-conditioned matrix, and using exact arithmetic, exactly n steps are required to arrive at x = A1b because the search directions are noninterfering; in other words, progress made in one search direction is not “undone” in a subsequent search direction. If x0 = 0, it can also be shown that solution norm, ||xk||, increases monotonically with each iteration, while the residual norm ||Axk  b|| drops monotonically with increasing k. For ill-conditioned matrices and a perturbed b = bexact + δb, however, the CG iteration schemes will converge to x = xexact + δx, where δx is large. Instead, as a regularization scheme CG exploits semiconvergence; during the first few iterations, xk approaches xexact, but with subsequent iterations, xk hones onto xexact + δx, so iterations are terminated before ||Ax  b|| is minimized and the iteration number, k, is the regularization parameter. This property is due to the fact that the search directions are generated in a sequence that roughly matches the order of column vectors in V for an SVD of ATA, so each xk is conceptually similar to the TSVD solution that would be obtained by excluding the last n-k summation terms in Eq. 3. Like TSVD, x becomes progressively larger

30

Inverse Problems in Radiative Transfer

1255

and more oscillatory with each successive iteration, so in CG, the prior information is introduced implicitly by truncating the number of regularization steps. CG regularization is particularly well-suited to large-scale linear problems, since it involves repeated vector products and avoids matrix decomposition or inversion. In all of the above linear regularization schemes, the analyst must determine the amount of prior information with which to supplement Ax = b via the regularization parameter. This must be done with care: sufficient regularization must be used to overcome the ill-posedness of the model equations, but using too much regularization overwhelms the often meagre information contained in these equations. To this end a number of parameter-choice methods have been developed to elucidate the tradeoff between a solution that minimizes ||Ax  b|| and one that complies with prior knowledge. These include: the L-curve curvature method (Hansen and O’Leary 1993); Morozov’s discrepancy principle (Morozov 1968); and generalized crossvalidation (Golub et al. 1979).

3.2

Nonlinear Programming Methods

The linear regularization techniques presented in the previous chapter can only be applied directly to linear problems. In many scenarios, however, the knowns and unknowns are related in a nonlinear way, and the objective is often to solve a nonlinear least-squares problem of the form   m  2 1 1X 2 x ¼ arg minx ½FðxÞ ¼ arg minx kf ðxÞk ¼ aj ð xÞ  b j 2 2 j¼1

(17)

where f(x) = a(x)  b is the residual vector. In problems where F(x) is convex, the solution can be found iteratively through nonlinear programming. This procedure follows Eq. 15, starting from some initial guess x0. At each iteration, the search direction pk is chosen based on the local curvature of F(xk), and a step size is then selected, either through univariate minimization (a “line search”) or by a nonconvergent series (e.g., αk = α0k1.) These algorithms are classified as nonlinear programming (NLP) methods (Bertsekas 1999; Gill et al. 1986). The most obvious choice for pk is the direction of steepest descent, i.e., k p = ∇F(xk), but this approach often requires a large number of iterations because progress in successive search directions partially “cancels out” progress made in previous iterations, as shown in Fig. 5a. A better choice considers both the first- and second-order curvature of f(xk). A locally-quadratic model of F(x) can be constructed through a Taylor-series expansion,       1 TT   T F xk þ pk  F xk þ pk ∇F xk þ pk ∇2 F xk pk 2

(18)

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K. J. Daun

Fig. 5 Minimization of a convex nonlinear function by nonlinear programming (NLP): (a) steepest descent; (b) Newton’s method; (c) conjugate gradient minimization. Steepest descent requires the most iterations because progress made in one search direction is cancelled out due to “back-tracking.” Newton’s method uses the fewest steps but requires the second-order sensitivities. The search directions in conjugate gradient are mutually-conjugate and noninterfering. In the case of an ndimensional linear problem, Newton’s method and CG would require one step and n steps, respectively

where ∇2F(xk) is the Hessian matrix, 2

@ 2 F=@x21 6   @ 2 F=@x2 @x1 ∇2 F xk ¼ 6 4⋮ @ 2 F=@xn @x1

@ 2 F=@x1 @x2 @ 2 F=@x22 @ F=@xn @x2 2

  ⋱ 

3 @ 2 F=@x1 @xn @ 2 F=@x2 @xn 7 7 5 ⋮ 2 2 @ F=@xn

(19)

Taking the gradient of Eq. 19, neglecting derivatives higher than second-order, and setting ∇F(xk + pk) = 0 results in

30

Inverse Problems in Radiative Transfer

1257

    ∇2 F xk pk ¼ ∇F xk

(20)

This direction is called “Newton’s direction,” and the corresponding solution scheme is called Newton’s method. (The search direction is self-scaling, so αk = 1.) This approach generally requires the fewest steps to reach a stationary point of F(x), as shown in Fig. 5b, and only one step if a(x) = Ax and F(x) is quadratic. A major drawback of Newton’s method is the computational expense needed to calculate the second-order sensitivities. While this can sometimes be done analytically, more often they must be approximated by finite differences, and if F(x) is costly to evaluate, the expense of calculating the Hessian at every iteration negates the advantage of needing fewer iterations. Accordingly, two techniques have been developed that only use first-order sensitivities to approximate the Hessian: the quasi-Newton method (sometimes called the variable metric method), and the Gauss–Newton method. In the quasi-Newton method, the gradient information is “built up” over successive iterations to develop an approximation for the Hessian, ∇2F(xk)  Bk, and the search direction is then found by solving Bkpk = ∇F(xk). The algorithm is initiated from B0 = I so p0 = ∇F(xk), i.e., the steepest descent direction, but in most cases, Bk quickly converges to the Hessian in only a few iterations. In contrast to Newton’s method, a separate scheme (e.g., nonconvergent series, line-search) must also be used to select αk at each iteration. The Gauss–Newton method exploits the special structure of the least-sum-ofsquares function, Eq. 17. It can be shown that ∇FðxÞ ¼ JðxÞT f ðxÞ

(21)

where J(x) is the Jacobian matrix, 2

@f 1 =x1 6 @f 2 =x1 Jð xÞ ¼ 6 4 ⋮ @f m =x1

@f 1 =x2 @f 2 =x2 ⋮ @f m =x2

3 @f 1 =xn @f 2 =xn 7 7 ⋮ 5    @f m =xn  

(22)

The second-order sensitivities are given by ∇2 FðxÞ ¼ JðxÞT JðxÞ þ

m X

f j ðxÞ∇2 f j ðxÞ  JðxÞT JðxÞ

(23)

j¼1

since the second-order terms vanish close to x*. In this case, Newton’s direction is approximated by solving  T    T   J xk J xk pk ¼ J xk f xk

(24)

which is the Gauss–Newton method. A comparison of Eqs. 24 and 14 shows that these are the normal equations found by setting the gradient of ||J(x)p  f(x)||2 equal to zero, and in the linear case, a(x) = Ax and J(x) = A. In the case of ill-posed

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K. J. Daun

problems, J(x)TJ(x) is usually ill-conditioned or singular, especially near the minimum of F(x). Alternatively, in the first few steps, Eq. 24 may return a poor choice for pk since F(x) may not resemble a quadratic function far away from its minimum. In these scenarios, calculation of the search direction can be stabilized using zerothorder Tikhonov regularization h  T   i  T   J xk J xk þ λI pk ¼ J xk f xk

(25)

where λ is usually chosen heuristically. This is the Levenberg–Marquardt method. It is important to note that, in contrast to Tikhonov regularization, the regularization parameter does not affect the curvature of F(x) by adding information to the problem, and therefore does not actually regularize the problem (Aster et al. 2013). Finally, the linear conjugate gradient algorithm described above can also be applied to nonlinear problems in which the search directions are generated by replacing f = Ax  b in the linear problem with ∇F(x). Because F(x) is not generally quadratic, more than n steps are required to minimize F(x), but typically the number of steps is O (n) (Fig. 5c). As with the linear case, key advantages of nonlinear conjugate gradient minimization is that it only requires calculation of vector products, so it is well-suited for large-scale problems.

3.3

Metaheuristic Methods

Nonlinear programming is well-suited for methods in which ||a(x)  b||2 is convex (i.e., one global minimum, cf. Fig. 4a) and the variables are nondiscrete. Some nonlinear inverse problems involve nonconvex functions, like Fig. 4b, or have discrete variables. A suite of techniques called “metaheuristic methods” have been developed for solving these problems. In contrast to nonlinear programming techniques, which are derived from rigorous mathematical analysis often based on a quadratic model of the objective function, and have a deterministic outcome, metaheuristic algorithms are heuristically-derived and include a stochastic component to prevent the algorithm from becoming stuck in a local minimum. Many of these algorithms are inspired by physical or biological processes. This section briefly summarizes four types of metaheuristics that have been used to solve inverse problems in radiation heat transfer: simulated annealing, genetic algorithms, Taboo search, and particle swarm optimization. Simulated annealing (SA) (Kirkpatrick et al. 1983) is motivated by the rearrangement of atoms and molecules during phase change (e.g., the annealing of metals). At each iteration, a candidate update is generated, xk+1,c, often by sampling from an n-dimensional normal distribution centered on xk. The candidate point is accepted with a probability determined by the Metropolis criterion (Metropolis et al. 1953)

30

Inverse Problems in Radiative Transfer



P x

kþ1, c



1259

   ΔF xkþ1, c , xk / exp  Tk

(26)

where ΔF = F(xk)  F(xk+1,c) is the improvement realized by the new point and Tk is an annealing temperature. The acceptance of xk+1 is determined by comparing P(xk+1,c) with a random number drawn from a uniform distribution between 0 and 1. Progress of the SA algorithm mirrors the rearrangement of atoms during the annealing of metals: the random thermal motion of atoms allows them to move into higher energy states at high temperatures, but this becomes less probable at lower temperatures. Thus, if the metal is quenched quickly, the atoms become locked in high-energy configurations (dislocations), while quenching the metal slowly allows the atoms to form a low-energy crystal lattice. The algorithm performance depends on the annealing temperature and how it changes with iteration: dropping the temperature slowly increases the likelihood of identifying a good local minimum, but the temperature must decay quickly enough for computational expediency. An example SA optimization path is shown in Fig. 6. This algorithm highlights some key differences between NLP and metaheuristics: (i) In contrast to NLP algorithms, which always generate a downhill direction, according to Eq. 26, an uphill direction may be accepted with a probability that increases with Tk; (ii) while NLP algorithms generate xk+1 deterministically, in SA both the candidate update xk+1,c and its acceptance are generated randomly; and (iii) while the convergence properties of NLP algorithms are well characterized, the performance of the SA algorithm depends strongly on the choice

b

3

3

2

2

1

1 X2

X2

a

0

0

−1

−1

−2

−2

−3 −3

−2

−1

0 X1

1

2

3

−3 −3

−2

−1

0

1

2

3

X1

Fig. 6 Minimization of a nonconvex function by simulated annealing. In contrast to NLP methods, which are deterministic, metaheuristic algorithms contain a random element and can occasionally accept an uphill direction. (a) shows that “fast annealing” becomes trapped in a shallow local minimum, while in (b), the temperature is dropped more slowly, which prevents the algorithm from getting trapped in the local minimum

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of heuristics, i.e., the annealing schedule and those that control how the candidate updates are generated. These properties broadly apply to all metaheuristics. Genetic algorithms (Mitchell 1996) are inspired by evolution in the natural world, i.e., “survival of the fittest.” The first step is to encode the variables onto a chromosome: in the case of continuous variables, this is often done using a binary representation, so each bit corresponds to a “gene” on the chromosome and a variable is represented by multiple genes. The algorithm then proceeds as follows: 1. An initial population of candidate solutions is generated, usually by random sampling, and the corresponding functions F(x) are evaluated. The size of the population depends on the size of the solution space. 2. A mating pool consisting of the best candidates in the population is selected and candidates are organized into mating pairs at random. New chromosomes are generated through a crossover operation of genes, which represents the exchange of genetic material during reproduction. 3. An optional “mutation step” can take place, in which a small subset of genes on the chromosomes are randomly perturbed. These steps continue until a computational budget is exceeded. Many variations of the above steps have been developed that can improve solution quality and computational efficiency, depending on the attributes of the inverse problem. The underlying strategy of the tabu search algorithm (Glover 1986) is to “remember” previous search paths, and to avoid returning to recently-visited subdomains (i.e., around a previously-visited local minimum). The algorithm performs a sequence of local or “neighborhood” downhill searches, where neighborhood defines a subset of solution space in which the candidate solutions share the same core attributes. A candidate list of potential solutions within the neighborhood is generated, and the best of these candidates is chosen as the solution. Since this solution may be worse than the incumbent solution, it is possible to make “uphill” steps, which allows an algorithm to escape from a local minimum. A memory structure is used to keep track of previous solutions, which are excluded from the current search for a set period of iterations. (These directions become “tabu,” which gives the algorithm its name.) In particle swarm optimization (PSO) (Kennedy and Eberhart 1995), a population of candidate solutions, each of which is called a “particle,” is dispersed over the search space. The particles move through the search space with a trajectory influenced by the best position visited by the particle, but also the best positions visited by other members of the population. The communication of information between the individual particles results in a “swarm behavior.” Information between the particles is communicated through a social network structure, and the interconnectedness of the structure determines how quickly the particles swarm, and the quality of the final solution. A high degree of interconnectedness causes rapid swarming, but may also lead to an incomplete search of the problem space and a shallow local minimum. On the other hand, a less interconnected structure can provide a more thorough search and a higher quality solution, but is more time-consuming.

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Inverse Problems in Radiative Transfer

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As a final note, it is important to debunk a commonly-held belief that metaheuristics algorithms return a global minimum. They are often categorized as “global search algorithms” because, unlike NLP algorithms, they contain mechanisms to avoid becoming trapped in local minima. Nevertheless, there is no way to guarantee that the solution returned by a metaheuristics algorithm is the true global minimum; in general, the only scenario in which the recovered minimum can be established as the global minimum is if the function can be shown to be convex.

3.4

Bayesian Inference

The previous sections showed how linear regularization, NLP, and metaheuristics algorithms can be used to recover a solution x that best satisfies some specified b, generally by minimizing ||a(x)  b||2. Because inverse problems are ill-posed, however, many candidate solutions exist that solve the model equations within the tolerances prescribed by measurement noise or design flexibility. In this context, least-squares minimization is deficient since it only provides a single answer to the inverse problem, and little to no information about the existence of other candidate solutions. An awareness of multiple candidate solutions is important in inverse design problems, since these represent feasible design alternatives, and parameter estimation problems, since the existence of multiple solutions implies uncertainty in the recovered variables. Bayesian inference (von Toussaint 2011; Kaipio and Somersalo 2005) directly addresses this need by conceiving x and b (and often ancillary model parameters) as random variables that obey probability densities related by Bayes’ equation pðxjbÞ ¼

pðbjxÞppr ðxÞ pðbÞ

(27)

In Eq. 27, p(x|b) is the posterior density of x conditional on b, p(b|x) is the likelihood of b occurring for a hypothetical x, ppr(x) is the probability of x based on prior information, and p(b) is the evidence, which scales the right-hand side of Eq. 27 in order to satisfy the Law of Total Probabilities. (The terms “prior” and “posterior” denote “before” and “after” the instant at which the information contained in b is incorporated into the estimate of x.) The distribution width of p(b|x) reflects the uncertainty in the measurement data in a parameter estimation problem, or the “looseness” of the design tolerance in an inverse design problem, while that of p(x|b) corresponds to the uncertainty in the inferred parameters or the range of acceptable design parameters. The remainder of this discussion focuses on parameter estimation, with the understanding that it can be extended to inverse design. In the context of parameter estimation, Bayes’ equation reflects the fact that the measurement data is not deterministic, but is better described by a distribution centered on a mean value. If measurement errors are caused by a sequence of unrelated random events, the measurement data often obeys a normal distribution,

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and the probability of observing a single measurement bj conditional on a hypothetical set of measurement parameters is (  2 )   b  a ð x Þ 1 j j p bj x ¼ qffiffiffiffiffiffiffiffiffiffi exp  2σ 2j 2πσ 2j

(28)

where σ j is the distribution width. (Often bj is the average of a set of measurements, in which case σ j is the standard deviation of the mean.) If the measurement noise is uncorrelated with the noise contaminating the other measurements and their distribution widths are equal, i.e. the noise is homoskedastic, then the joint probability of observing a set of measurements in b is given by   pðbjxÞ ¼ ∏ p bj x ¼ m

j¼1

(

kb  aðxÞk2 exp  m=2 2σ 2 ð2πσ 2 Þ

)

1

(29)

In the absence of prior information, Eq. 27 shows p(x|b) / p(b|x) and the most likely x conditional on the data in b is found by maximizing Eq. 29, which, subject to the assumption of independent and identically-distributed measurement noise, also minimizes ||b  a(x)||2. This is the maximum likelihood estimate, xMLE = arg maxx[p(b|x)]. The ill-posedness of inverse problems gives ||b  a(x)||2 a flat, shallow topography, which corresponds to a wide p(x|b). For 2D problems, this can be visualized by plotting the contours of the joint probability density p(x|b), which appear similar to those in Figs. 3 and 4. It is often convenient to derive univariate probability densities for each xj by marginalizing out the remaining n1 variables from the joint probability   p xj b ¼

ð ð

ð ð ...

x1 x2

xj1 xjþ1

ð . . . pðxjbÞdx1 dx2 . . . dxj1 dxjþ1 . . . dxn

(30)

xn

These univariate distributions can then plotted or summarized by a set of credibility intervals that contain a specified probability, e.g., xj,90%  [a, b]. Using these techniques, the impact of the ill-posedness on the recovered variables can be seen explicitly, as shown in Fig. 7. Quantifying and visualizing the ill-posedness of an inverse problem in the context of probability is an important advantage offered by Bayesian inference over deterministic inverse techniques. Arguably, the main advantage of the Bayesian methodology, however, is that it presents a mathematically robust and explicit way to incorporate information into the inference procedure and directly addresses the information deficit underlying the ill-posedness of the governing equations. This is done through the prior, ppr(x), which conditions the likelihood and “steepens” the contours of the posterior density, p(x|b) / p(b|x)ppr(x). When prior information is

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0.4

1.5 1 px1

px1

0.3 0.2

0.5

0.1 0

0

5

X2

X2

5

0

−5 −5

0 X1

5

0

0.05

0.1 px2

0.15

0.2

0

−5 −5

0 X1

5 0

0.2

0.4 px2

0.6

0.8

Fig. 7 Joint probability densities corresponding to an ill-conditioned measurement equation Ax = b, along with marginalized probability densities and 90% credibility intervals. The plot on the left assumes no prior information, hence log[p(x|b)] = log[p(b|x)]. The plot on the right adds a smoothness prior, ppr(x) = exp.(λ2||Lx||2), which “steepens” the contours of log[p(x|b)] = log[p(b| x)ppr(x)] and narrows the credibility intervals. Note that the prior also shifts xMAP away from xMLE

considered, the most probable value for x is the maximum a posteriori estimate, xMAP = arg maxx[p(b|x)ppr(x)], but again, a key advantage of Bayesian inference is that the joint posterior probability density p(x|b) or its marginalized univariate densities, p(xj|b), provide a direct indication of the uncertainty in x, which is essential in inverse analysis. Bayesian priors are sometimes derived from the results of previous experiments, or they can be defined heuristically to promote a known solution attribute. If the elements of x are known to be spatially-smooth, for example, then ! 1 ppr ðxÞ ¼ ppr, smooth ðxÞ / exp  2 kLxk 2σ pr

(31)

where L is a 2D version of Eq. 10, i.e., a discrete approximation of the ∇ operator, and σ pr2 is a heuristically-defined variance that quantifies the analyst’s “belief” in the prior information. If the measurement model is linear, i.e., Ax = b, and the elements of b are corrupted with normally-distributed error having a variance of σ 2, it can be shown that   xMAP ¼ arg maxx pðbjxÞppr ðxÞ ¼ arg minx ln pðbjxÞppr ðxÞ    2  A b    ¼ arg minx  x σ=σ pr L 0 

(32)

which is the same as Tikhonov regularization, Eq. 12, where λ = σ/σ pr expresses the reliability of the prior information relative to the measurement data.

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Does Least-Squares Minimization Constitute Inverse Analysis?

Many of the examples of inverse analysis of radiative systems recover x by minimizing ||a(x)  b||2 using an NLP or a metaheuristic algorithm like the ones described in Sects. 3.2 and 3.3. Does this actually constitute inverse analysis? Recall that inverse analysis problems are distinct from ordinary inference problems because they are mathematically ill-posed, and consequently the art of inverse analysis lies in augmenting the information contained in the model equations with “prior” information. If a problem can be solved by minimizing the residual without adding information, then it is not ill-posed and does not constitute an inverse problem. The shallow topographies typical of “true” nonlinear inverse problems, cf. Fig. 4, present numerical challenges associated with ill-conditioned or singular Hessians. In this context, it has been argued that nonlinear conjugate gradient and Levenberg–Marquardt algorithms constitute inverse techniques, since they have features for dealing with an ill-conditioned Hessian or JTJ in the vicinity of x* = arg minx||Ax  b||2: in the case of CG, semiconvergence is exploited by terminating iterations before a local minimum is reached, while in Levenberg–Marquardt, the regularization parameter λ stabilizes the calculation of pk. In these cases, one may tenuously argue that prior information is introduced, albeit in a very subtle way, since these schemes indirectly promote a small solution norm. Rarely, however, do studies establish that these techniques function as inverse analysis algorithms. If CG iterations can be carried out to convergence and arrive at a sensible value of x, then the underlying problem is not ill-posed; simply using CG to minimize a least-squares function does not, in itself, constitute inverse analysis. Likewise, in some cases, Gauss–Newton iterations do not diverge in the last few iterations because of ill-posedness, but rather diverge immediately because the quadratic model that forms the basis for the search direction is invalid far away from x*. In this scenario, LM does not act like an inverse analysis technique. Often, analysts turn to metaheuristics to avoid the numerical issues associated with an ill-conditioned Hessian or JTJ, since these algorithms are “derivative-free.” Like CG and LM iteration, metaheuristics do not directly introduce any additional information to the problem, and do not actually avoid the ill-posedness of the inverse problem. Consequently, the ill-posedness will manifest as a large set of candidate solutions identified by the metaheuristic algorithm. Metaheuristics can be useful in a design context, since these solutions represent feasible design alternatives. They are less useful for parameter estimation, however, since the set of recovered solutions cannot be used to infer statistical properties of x. This question is further clarified in the context of Bayesian inference. The ill-posedness of the inverse problem is realized by a broad likelihood density in p(b|x), due to the properties of the model equation, a(x) = b. The prior information introduced through ppr(x) alters the topography of the posterior probability density, p(x|b), and narrows the credibility intervals. On the other hand, using a CG/LM/metaheuristic to find xMLE by maximizing p(b|x) does nothing to address the ill-posedness of the problem.

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In summary, when solving inverse problems, the analyst must consider the origin of the ill-posedness, and what additional information can be imposed to mitigate the ill-posedness.

4

Radiant Enclosure Design Problems

As noted in the introduction, inverse problems in radiative transfer can broadly be categorized as either inverse design or parameter estimation problems. This section begins by considering inverse design problems, specifically those involving radiant enclosures; these are appealing because they have a simple physical interpretation of the mathematical ill-posedness, as well as important practical applications. In conventional, or “forward” radiant enclosure analysis, the enclosure geometry is prescribed and one boundary condition, either temperature or heat flux, is imposed on each surface. The goal is then to recover the complementary set of unknown boundary conditions. Such a scenario is shown schematically in Fig. 8a. Under these circumstances, the problem can be written as a matrix equation Ax = b, where

Fig. 8 Types of radiant enclosure design problems: (a) the forward problem, in which one boundary condition is specified over each surface and the objective is to infer the remaining surface; (b) the inverse boundary condition design problem, in which both conditions are specified over the design surface; (c) the inverse source term problem, in which the goal is to infer the emissive power or heat source distribution in a participating medium; and (d) the geometry design problem. In each case, the goal is to produce the desired conditions over the design surface. In the implicit formulations, one boundary condition is specified over the design surface, while the remaining boundary condition is used to specify an objective function

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b contains the known boundary conditions, the elements of A are derived from view factors, and x contains the unknown boundary conditions, or an intermediate vector of radiosities that can be postprocessed for the unknown boundary conditions. Provided that one boundary condition is specified over each surface, A is always full-rank (Baranoski et al. 2001) and consequently has a unique and stable solution. This can be conceived as the well-posed forward problem. In industrial settings, however, the objective is usually to find the enclosure configuration that achieves a specified heat flux and temperature distribution over a particular surface, e.g., a product being heat-treated. This surface is called the design surface. In problems involving transient heating, this scenario occurs when both the temperature and the heating rate are specified over a product. The goal of the inverse design problem is to satisfy the design surface conditions by altering the heat flux or temperature distribution over the other enclosure surfaces, changing the enclosure geometry, or arranging the location of heat sources within a participating medium. In these cases, shown schematically in Fig. 8b–d, the governing equations are mathematically ill-posed, and one of the inverse analysis techniques described in Sect. 3 must be employed to solve them.

4.1

Linear Problems

The simplest variation of the radiant enclosure design problems is the boundary condition design problem, shown in Fig. 8b. In the case of an enclosure of graydiffuse surfaces enveloping a transparent medium, the radiosity distribution over the enclosure surfaces is governed, in part, by a Fredholm integral equation of the firstkind having a form ðb

J ðuÞ ¼ gðuÞ J ðu0 Þkðu, u0 Þdu0

(33)

a

where J(u) is the radiosity distribution, g(u) depends on the emissivity of the surface, and k(u,u0 ) is the view factor between two infinitesimal elements at u and u0 divided by du0 . This inverse design problem can then be solved in one of two ways (Daun and Howell 2005): In the direct formulation, both the temperature and the heat flux are specified over the design surface, and an ill-conditioned matrix equation Ax = b is formed by discretizing the Fredholm IFK. The unknown boundary condition is then solved using one of the linear regularization schemes in Sect. 3.1. Harutunian et al. (1995) solved the inverse boundary condition design problem in its direct form, using a modified version of TSVD to regularize the problem (Hansen et al. 1992). Candidate designs were visualized by systematically changing the regularization parameter, p. Subsequent work by França et al. (2003), Ertürk et al. (2002b), Leduc

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et al. (2004), among others, used TSVD, CG, and Tikhonov regularization to solve the inverse boundary condition design problem in direct form. (In these examples, CG is used as a linear regularization technique as described in Sect. 3.1, as opposed to an NLP method in Sect. 3.2.) In each of these cases, the regularization parameter is varied systematically to obtain a set of design alternatives. Ertürk et al. (2008) used CG regularization to determine the heater settings in an axisymmetric rapid thermal processing chamber; the feasibility of this solution was then established on an experimental test-rig. A major drawback of this approach is that it does not allow for bound constraints, however, which limits the usefulness of these results. Under certain circumstances, the inverse source term problem, shown in Fig. 8c, can also be written as a linear inverse problem, Ax = b, where x typically contains the emissive power of volumetric elements within a participating medium and b specifies the designed boundary conditions over the design surface. Kudo et al. (1996) and França et al. (1999) used the direct formulation to estimate the temperature and heat source distribution within a participating medium required to produce the desired distribution over a design surface. Exchange factors between the volume and surface elements were calculated using Monte Carlo, and the inverse problem was solved using TSVD. In the indirect formulation, the inverse design problem is solved as an optimization problem. Conceptually, all optimization problems consist of two distinct steps: First, the engineering design problem is transformed into minimization problem by defining an objective function, F(x), which quantifies the “goodness” of a particular design, along with a vector of design constraints c(x)  0 that define the problem domain. In the context of inverse boundary condition design problems, one boundary condition is specified over each enclosure surface, including one of the design surface conditions (say, the temperature, TDS). The remaining design surface boundary condition (say, the heat flux, qDS) is used to define a least-squares objective function 2 1 1  FðxÞ ¼ kf ðxÞk2 ¼ qDS ðxÞ  qtarget DS 2 2

(34)

The goal, then, is to solve x ¼ arg minx FðxÞ such that cðxÞ  0

(35)

where x* specifies the optimal design configuration, e.g., the distribution of the unknown boundary condition over the remaining surfaces. The second step is to solve the constrained minimization problem using NLP (Sect. 3.2) or with a metaheuristics algorithm (Sect. 3.3), which involves repeated solution of the forward problem. The choice of minimization algorithm depends on: (i) the number and type of variables, (ii) the number and type of constraints, (iii) the topography of F(x) (e.g., smooth and convex), and (iv) the computational effort required to evaluate F(x).

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While the indirect formulation does not require solving the ill-conditioned matrix equation directly, this does not, in itself, avoid the ill-posedness of the underlying problem. While the ill-posedness manifests through the ill-conditioned A matrix in the direct formulation, and is addressed through regularization, in the case of the indirect formulation, it is reflected by the shallow topography of F(x), and is often reduced by limiting the degrees of freedom involved in the optimization problem. In many cases, this must be done for ease of deployment, e.g., use a limited number of isothermal panel heaters as opposed to one heater surface over which the temperature varies continuously. By specifying constraints, the designer introduces information into the inverse problem, which addresses the information deficit that causes the ill-posedness. In many cases, however, the problem remains sufficiently ill-posed such that a multitude of solutions may exist that minimize or “almost” minimize F(x). Fedorov et al. (1998) solved for the optimal panel heater temperatures needed to produce a desired temperature and heat flux distribution over a design surface moving on a conveyor through an industrial oven. The problem was solved in steady state, using an advection term to account for the change in sensible energy as the load moves through the furnace. The least-squares objective function was minimized using the Levenberg–Marquardt method with bound constraints on the heater temperatures. Daun et al. (2003a, 2005) solved an inverse boundary condition design problem using both the direct (TSVD, Tikhonov) and indirect (Newton’s method with a nonnegativity constraint) formulations, and highlighted the relationship between these two approaches. In this class of problem, the gradient vector and Hessian matrix can be readily calculated through direct differentiation of the governing equations. Specifically, if x contains the “design parameters” (e.g., the heat flux distribution over the heater surfaces), it is possible to compute qDS = A1x because A is well-conditioned. The first- and second-order sensitivities needed to calculate the gradient and Hessian can then be found by direct differentiation, qDS0 = A1x0 , etc. The solutions obtained through optimization were generally considered superior since it was possible to apply bound constraints. Metaheuristic schemes have also been applied to solve the inverse boundary condition design problem, including simulated annealing (Porter et al. 2006), tabu search (Porter et al. 2006), genetic algorithms (Safavinejad et al. 2009; Chopade et al. 2012), and particle swarm optimization (Lee and Kim 2015). A key advantage of metaheuristics is that they allow visualization of different candidate solutions that all satisfy conditions on the design surface. A drawback, however, is that they are more computationally-intensive compared to NLP, because metaheuristics do not use information about the local objective function curvature to generate a new design configuration. Algorithm performance also depends on the choice of heuristic parameters, which is mainly based on the analyst’s experience and trial and error. Most importantly, many of these studies fail to consider that the inverse boundary condition estimation problem is linear, and, in the absence of bound constraints, F(x) is convex. Consequently, it may be questionable whether metaheuristics is really the “right tool for the job” for many instances of the inverse boundary condition design problem.

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Inverse Problems in Radiative Transfer

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Nonlinear Problems

In many scenarios, it is impossible to reduce the problem into linear form, Ax = b, e.g., radiant enclosure problems involving transient or multimode heat transfer. In some cases, it is possible to linearize the equations and solve a sequence of linear inverse problems Ax = b(x) in the direct formulation. França et al. (2001) used this approach to optimize the heater settings in an inverse boundary condition design problem for an enclosure containing a conducting and participating medium. The heater settings were found by repeatedly solving Axk = b(xk1) using TSVD. Likewise, Ertürk et al. (2002a) used a similar scheme to solve for the transient heater settings of a batch furnace needed to uniformly heat a design surface according to a specified temperature history. At each time step, a linear inverse problem was defined by specifying the design surface temperature as well as the heat flux required to increase the temperature at the prescribed rate. The heater settings at each time step were then found through CG regularization. More recently, Mossi et al. (2008) used iterative TSVD regularization to solve the inverse boundary condition design problem for a radiant enclosure containing a turbulent, nonparticipating medium. Daun et al. (2006b) carried out a comparative study in which the goal was also to determine the transient heater settings in a batch furnace. In addition to CG regularization, this study also considered Tikhonov and TSVD regularization, as well as simulated annealing in the indirect formulation, i.e., find the optimal heater settings at each time step by minimizing Eq. 34 subject to a nonnegativity constraint, where qDStarget is the net radiant heat flux needed to increase the sensible energy of the design surface. These schemes were compared to an “all at once” minimization of an objective function defined as the residual between the desired temperature history and the one realized using particular heater settings. In order to obtain temporallysmooth heater inputs, these were parameterized as B-splines with the design parameters as control points, and minimization was carried out using the quasi-Newton method. This reduced-order parameterization constitutes prior information (i.e., the heater settings should change smoothly with time) which reduces the ill-posedness of the inverse problem. The literature is replete with studies that use the indirect formulation to solve nonlinear radiant enclosure design problems. Hosseini Sarvari et al. (2003), for example, solved an inverse boundary condition problem for an enclosure containing a conducting and radiating medium. In this work, the heat flux distribution over the heater surface was found by minimizing Eq. 34 using the conjugate gradient method, with sensitivities obtained through direct differentiation of the radiative transfer equation. Kim and Baek (2007) solved a similar boundary condition design problem using the Levenberg–Marquardt method, while Kowsary and Pooladvand (2007) compared the CG and quasi-Newton methods for solving this problem. As discussed in Sect. 3.2, it is important to distinguish between CG as a linear regularization scheme and as an NLP scheme for well-posed optimization problems; in the latter case, the CG iterations converge to x* = arg minx[F(x)]. It is also important to note that since NLP algorithms do not influence the topography of F(x), the choice of solution scheme should not influence the solution.

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While the NLP algorithms take different paths, they all arrive at the same destination, x*, as shown in Fig. 4. Unfortunately, many authors, reviewers, and journal editors fail to appreciate this important point, and consequently a large number of studies have been published that are simply permutations of different NLP algorithms and RTE solution schemes. (Due to the shallow topography of F(x), the solution may be strongly sensitive to the stopping criterion, typically ||∇F(x*)|| < ε, or other numerical artifacts that may be introduced in the solution procedure, but these are distinct from the NLP method.) In contrast, Kowsary and Pooladvand (2007) made a hybrid objective function by adding a first-order Tikhonov functional, equivalent to λ2||Lx||2, to the objective function defined in Eq. 34 to reduce the oscillations in the recovered heat flux distributions over the heater surfaces. In this case, information is being added (an oscillatory solution is undesired) through the Tikhonov functional, which alters the topography of F(x). The solution x* = arg minx F(x) should not depend on the minimization algorithm, since the minimization algorithm does not introduce any information into the analysis. This observation can also be extended to metaheuristics schemes used to solve enclosure design problems (e.g., Kim and Baek (2004) and Pourshaghaghy et al. (2006)), but with a caveat. Conceptually and ideally, the metaheuristics algorithm should identify a single global minimizer of Eq. 35. An advantage of metaheuristics is that they also often identify a set of alternate solutions that are nearly optimal, however, which is useful for nonconvex objective functions. Enclosure geometric optimization problems, shown schematically in Fig. 8d, are also nonlinear. In the case of diffuse-walled enclosures, the radiosity problem reduces to a nonlinear IFK, ðb

J ðuÞ ¼ gðuÞ J ðu0 Þkðu, u0 , xÞdu0

(36)

a

where x specifies the enclosure geometry. Because the problem is nonlinear, it is always solved in the indirect formulation. Daun et al. (2003c), Hosseini Sarvari (2007), and Farahmand et al. (2012) parameterized the enclosure geometry using B-splines, and optimized the location of the control points that produced a uniform irradiation of the design surface. Daun et al. (2003c) used Newton’s method with objective function sensitivities calculated from an analytical differentiation of the kernel, k(u, u0 , x), while Hosseini Sarvari (2007) and Farahmand et al. (2012) both used metaheuristics algorithms, which are derivative-free. In these examples, the inverse problem is regularized by the choice of a low-order parameterization, which steepens the topography of F(x) around x*. In contrast, Tan and Liu (2009) used a meshless method to represent the geometry involving a large number of degrees of freedom. They then augmented a least-squares objective function, Eq. 34, with a zeroth-order Tikhonov-type functional that promoted a small solution norm, and minimized the composite objective function using CG. While the above examples focus on diffuse-walled enclosures, practical engineering problems often feature partially- or fully-specular surfaces, since the

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geometry of these surfaces have a much stronger influence on the radiation within the enclosure compared to diffuse surfaces. While geometric optics treatments (Winston 1991) admit analytical solutions in a limited number of cases, e.g., various conic sections, more generally the enclosure problem must be solved using a Monte Carlo (MC) type ray-tracing algorithm. Monte Carlo techniques are extremely flexible: they can solve complicated problems, such as enclosures that contain obstructions as well as spectrally- and directionally-dependent surface properties, with relative ease compared to deterministic schemes. In the context of optimization, there are two complicating factors: (i) analytical solutions to the objective function sensitivities are not available; (ii) the MC-derived objective function is a statistical estimate of the deterministic (but unknown) objective function, i.e., FðxÞ ¼ F~ðxÞ þer, where er is a random error. Consequently, Daun et al. (2003b) employed a stochastic programming technique called the Kiefer–Wolfowitz algorithm (Kiefer and Wolfowitz 1952), a type of steepest descent that balances the truncation error in the forward-difference estimate of the gradient, which decreases with the finite difference interval, with the stochastic error caused by er, which increases with interval size. For computational expediency, the sample variance in F~ðxÞ is initially large, and gradually drops as xk approaches x* by increasing the number of samples (photon bundles) according to a power law. Daun et al. (2003b) demonstrated this scheme on the 2D enclosure problem shown in Fig. 9, as well as a 2D imaging furnace containing a faceted reflector surface. Subsequent work by Marston et al. (2012) used a quasi-Monte Carlo scheme to reduce the variance in the objective function estimate, thereby requiring fewer bundles and greatly accelerating convergence time. Rukolaine (2015) presents a semianalytical approach for optimizing the geometry of enclosures containing specular and diffuse-specular surfaces. Sensitivities are calculated using adjoint analysis, and optimization is carried out with conjugate gradient minimization. Nanotechnology represents an emerging new frontier for the inverse geometry design problem. Hajimirza et al. (2012), for example, optimized the geometry of

Fig. 9 Geometric optimization of a radiant enclosure, Fig. 8d, using Monte Carlo ray-tracing and Kiefer–Wolfowitz optimization (Adapted from Daun et al. (2003b))

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periodic nanostructured amorphous silicon solar cells textured with rectangular metallic nano-patterns. This was done by finding the geometric parameters (a-Si layer thickness, metal nano-pattern layout) that maximized the solar absorption enhancement factor, calculated through a finite difference time domain (FDTD) solution of Maxwell’s equations. They compared the performance of nonlinear programming (quasi-Newton) and metaheuristics (simulated annealing, tabu search). Under certain conditions, the objective function is nonconvex, and consequently the metahuristics schemes provide a superior solution compared to the quasi-Newton algorithm, which becomes trapped in a local minimum.

4.3

Case Study: Inverse Design

To demonstrate the explicit and implicit solution methodologies, consider the inverse boundary condition design problem shown in Fig. 10, inspired by the rapid thermal processing chamber testbed examined by Ertürk et al. (2008). The objective is to identify the heater setting configuration that produces the desired temperature and heat flux over the design surface. (The problem can be scaled so that all quantities are dimensionless.) All enclosure surfaces are black, and all surfaces other than the design surface and heater surfaces are cold. With these simplifications, the problem reduces to identifying the emissive power distribution over the heater surface, Eb(r2), that produces the required (uniform) irradiation over each location of the design surface, G1 = 1. These quantities are related by the Fredholm IFK R ð2

G1 ¼

  Eb r 2 kðr 1 , r 2 Þdr 2

(37)

0

Fig. 10 Example inverse boundary condition design problem, inspired by a rapid thermal processing chamber

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The kernel is defined by

k ðr 1 , r 2 Þ ¼

dFdr1dr2 dr 2

#   "   2 2r 2 L 2 L 2 r2 þ þ1 r1 r1 r 21 r 1 ¼ ( ) 2  2  2  2 3=2 r2 r2 L þ r1 þ 1  4 r1 r1

(38)

where dFdr1dr2 is the view factor between an infinitesimal ring element on the design surface and an infinitesimal ring element on the heater surface. In the explicit methodology, the IFK is transformed into a matrix equation Ax = b by discretizing the domains of r1 and r2, and writing

bi ¼ G 1 

n X j¼1



Eb r 2j



r2j þΔr ð 2 =2

r2j Δr 2 =2

kðr i , r Þdr ¼

n X

xj Aij

(39)

j¼1

for every discrete r1 value. In this example, it is assumed that both surfaces are discretized into 50 elements. As one would expect, A is ill-conditioned, and Fig. 11 shows that the singular values of A decay continuously over several orders of magnitude. Mathematically, the ill-conditioning is due to the smoothing or blending action of k(r1, r2) during integration, as discussed in Sect. 2.2. Physically, this smoothing is due to the fact that any given location over the design surface is irradiated by the entire heater surface, so emission from one ring element is blended with emission from all the other elements. Consequently, the irradiation at a given location on the design surface is insensitive to small variations in the radiosity over the heater surface.

Fig. 11 The singular values of the A matrix decay continuously over several orders of magnitude, due to the smoothing action of the kernel in Eq. 38. Labeled singular values correspond to the solutions shown in Fig. 12

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Since the irradiation of the design surface is a linear function of the radiosity/ emissive power distribution over the heater surface, Eb(r2) can be solved using any of the linear regularization techniques described in Sect. 3.1. Figure 12 shows the results obtained from truncated singular value decomposition, with various values of regularization parameter p. (The corresponding singular values are indicated in Fig. 11, and the solutions are constructed using the summation terms up to and including k = n  p.) Solutions obtained using very few summation terms are spatially-smooth, as shown in Fig. 12a, but do not produce the desired irradiation over the design surface. The design surface irradiation approaches the desired condition as regularization is decreased and more summation terms are retained, but the solution becomes increasingly oscillatory, and would be difficult to deploy on an actual furnace. It should be noted that TSVD and other regularization techniques only produce a subset of the candidate design solutions that satisfy Ax = b within a reasonable tolerance. The domain of candidate solutions is contained within a hyperellipsoid centered on xexact analogous to Fig. 3a for the 2D case.

Fig. 12 Emissive power distribution over the heater surface, and corresponding design surface irradiation, found using truncated singular value decomposition. As the regularization parameter p increases, the emissive power distribution becomes smoother, but the residual between the desired and realized design surface irradiation grows. (Singular values corresponding to each solution are shown in Fig. 11)

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In the implicit formulation, a candidate x is specified that represents Eb(r2) over the heater surface, and the resulting irradiation over the design surface is used to define an objective function 2 1 FðxÞ ¼ Gtarget  G1 ðxÞ2 1 2

(40)

In the simplest scheme, each element of x corresponds to the emissive power of an annular ring on the heater surface, as was the case in the explicit formulation, so G1(x) = Ax and minimizing F(x) is the same as solving the linear least-squares minimization problem x* = arg minx(||Ax  b||22). As discussed in Sect. 3.2; however, this objective function is problematic to minimize because JTJ = ATA is ill-conditioned. While one could adopt Levenberg–Marquardt technique to carry out the minimization, a major disadvantage of this approach is that there is no clear link between regularizing the search direction in Eq. 25 and promoting desired solution attributes. A better approach is to add a regularizing function to Eq. 40, following Kowsary and Pooladvand (2007). Alternatively, one could reduce the degrees of freedom by discretizing the heater surface into four isothermal rings, so that

bi ¼ Gi ¼

4 X j¼1

ð

r2j, max

Aij xj ,

Aij ¼ Fdij ¼

kðr i , r Þdr

(41)

r 2j, min

where [r2j,min, r2j,max] are the minimum and maximum radii of the jth heating element, and the (m  4) matrix equation is overdetermined. This reduces the ill-posedness of this inverse design problem considerably, and also makes the solution easier to implement from a design perspective. The unconstrained minimization problem can then be solved by any NLP algorithm, or equivalently through the normal equations, x* = (ATA)1ATb. Figure 13a shows the optimal heater settings, while the corresponding design surface irradiation is plotted in Fig. 13c. While the overdetermined problem has a unique solution, the problem remains ill-posed since Cond(∇2F(x*)) = Cond(ATA) = O (104); the last two singular values are very small, and consequently, a family of candidate solutions could be constructed by carrying out an SVD on ATA and using the last two columns vectors of V as an orthonormal basis, following Eq. 6. A key advantage of the implicit formulation is that it is possible to impose constraints on the design variables, which ensure that the optimal solution can be implemented in an industrial setting. Accordingly, Eq. 40 is minimized with nonnegativity constraints on the design parameters, using a trust-region reflective algorithm (Moré and Sorensen 1983). The optimal heater settings are plotted in Fig. 13b, while Fig. 13c shows that the irradiation over the design surface is only slightly further away from the target irradiation compared to the unconstrained case. It is important to keep in mind, however, that if one were to implement this solution in an industrial setting, the design surface irradiation would be

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Fig. 13 Solution of the inverse boundary condition design problem through (a) unconstrained and (b) constrained minimization of the objective function defined by Eq. 40. The corresponding design surface irradiation is shown in (c) for both cases. The irradiation realized through the unconstrained optimization is slightly closer to the target irradiation compared to the constrained optimization

different from the target distribution due to the assumptions made to facilitate the enclosure analysis (i.e., diffuse-gray surfaces). In reality, the simplifying assumptions needed to facilitate analysis coupled with the inherent uncertainty in the radiative properties of surfaces and participating media will degrade the quality of the optimized solution (Erturk et al. 2008; Amiri et al. 2013).

5

Parameter Estimation Problems

While the objective in inverse radiant enclosure design problems is to infer the enclosure configuration that produces the desired conditions over the design surface, in parameter estimation problems the goal is to infer some property or attribute from indirect experimental measurements. Like inverse design problems, parameter estimation problems can be categorized as either being linear or nonlinear.

5.1

Linear Problems

Arguably, the most prominent linear parameter estimation problem in thermal radiation is tomography with negligible scattering. Under these circumstances, the radiative transfer equation simplifies to

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2L 3 ð I L, λ ¼ I 0, λ exp4 κλ ðuÞdu5,

(42)

0

where I0,λ and IL,λ are the incident and exit intensities, and κλ(u) is the spectral absorption coefficient at a location u along the optical path, which scales with the local species concentration of interest. Equation 42 can be rewritten as 

ðL



bi ¼ ln I 0, λ =I L, λ ¼ κ λ ðuÞdu ¼ 0

n X

Aij xj

(43)

j¼1

The objective of axisymmetric deconvolution is to infer the radial distribution of κλ(r), from projected measurements, b( y), made through the tomography domain. A classic example is the inference of soot volume fraction concentration within an axisymmetric laminar diffusion flame through line-of-sight-attenuation measurements made with a laser or some other collimated light source. It can be shown that κ λ(r) and b( y) are related through a Volterra integral equation of the first kind called Abel’s integral equation, ðR bð y Þ ¼ 2 y

κλ ðr Þr pffiffiffiffiffiffiffiffiffiffiffiffiffiffi dr r 2  y2

(44)

Unlike most IFKs, Abel’s equation has an analytical solution, 1 κ λ ðr Þ ¼ π

ðR y

db=dy pffiffiffiffiffiffiffiffiffiffiffiffiffiffi dy y2  r 2

(45)

but it cannot be used to analyze experimental data because db/dy is not available, and finite difference approximations of the derivative amplify the measurement noise in b( y) to an unacceptable level. Instead, a more common route is to discretize the problem domain into annular elements, as shown in Fig. 14a, and approximate the integral in Eq. 44 as a summation assuming that κλ(r) is uniform over each annular element. Writing this equation for each projection produces a lower-triangular matrix equation, Ax = b, which is ill-conditioned due to the smoothing action of the kernel. Consequently, solving for x by back substitution, a process called “onion peeling,” amplifies small amounts of measurement noise in b = bexact + δb into large perturbations in the recovered field variable, x = xexact + δx. The condition number of A increases with the spatial resolution of the measurement data, which exacerbates the error amplification. One approach is to use Abel three-point inversion (Dasch 1992), which assumes that the projected data is locally cubic at each measurement point. Hall and Bonczyk (1990) inferred both the soot volume fraction

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Fig. 14 1D (axisymmetric) tomography problem, in which the goal is to infer a field variable from intensity measurements made through the tomography field. (a) The governing Volterra IFK is discretized by dividing the tomography field into annular elements. (b) The resulting lowertriangular matrix equation, Ax = b, is ill-conditioned, so small perturbations in the measurement data are amplified into large variations in the recovered field variable. (x, b = solid lines, xexact, bexact = dashed lines)

and soot temperature from a luminous laminar diffusion flame with negligible scattering by solving a sequence of linear equations AIxI = bI, and AII(xI)xII = bII. In the first measurement, bI is obtained by measuring the intensity with and without back illumination, AI is the onion peeling matrix, and xI represents the soot volume fraction. A second matrix equation is derived that relates the emission measurement to the local emissive power of each annulus, and the elements of AII are defined using the soot volume fraction of xI. Both deconvolutions are carried out using a Fourier-transform based algorithm that suppresses the high-frequency solution components associated with measurement noise; the fact that these high-frequency components should be absent from the solution constitutes prior information. Mengüç and Dutta (1994) also used a Fourier-transform based algorithm to reconstruct the radial distribution of the extinction coefficient in an axisymmetric flame; while the majority of CST experiments presume negligible scattering, this technique involves inverting an integral equation that relates the scattering coefficient to the angular distribution of scattered light measured at the periphery of the domain. Again, the prior knowledge is that the scattering coefficient distribution is spatially-smooth. First-order Tikhonov regularization is effective at stabilizing the deconvolution (Daun et al. 2006a; Åkesson and Daun 2008); the measurement information contained in Ax = b is augmented with prior information that the radial distribution of the field variable, κ λ(r), is smooth due to spatial diffusion. Tikhonov regularization has been widely adopted by other combustion researchers, and is particularly

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effective in scenarios in which the projected data has high spatial-refinement (Kashif et al. 2012). The above technique has been extended to tomography problems in two- and three-dimensions. Most often, the tomography field is discretized into pixels or voxels, and the unknown concentration is assumed to be uniform within each voxel. If the axial and angular spacing of projection measurements is uniform and dense, the 2D- or 3D-tomography problem has similar mathematical properties as the 1D axisymmetric problem. For example, Floyd and Kempf (2011) obtained a 3D reconstruction of CH* concentration within a matrix burner based on 2D chemiluminescence images using the algebraic reconstruction technique (ART), an iterative linear regularization scheme that stabilizes the inverse problem by suppressing or truncating high-frequency solution components. More often, however, the number of voxels, n, far exceeds the number of optical paths, m, due to the cost and complexity of the apparatus or the limited optical access afforded by the tomography field. For example, Wright et al. (2010) reconstructed the fuel concentration within a 2D cross section of an internal combustion engine cylinder: the cross section was subdivided into 1844 pixels, and was transected by 32 fiber-based diode lasers. In the rank-deficient scenario, the species concentration is governed by a matrix equation Ax = b, where A  ℜ(mn), and Rank(A)  m due to the rank nullity theorem. In this case, the solution will have a nontrivial nullspace component as described in Eq. 6. Figure 15 shows an example problem in which the nullspace component is inferred from the exact solution, xn = x  xr; in practice, however, constructing xn from the orthogonal basis formed by the last n-m column vectors of V depends wholly on specifying specific prior information. More thorough discussions of the mathematical aspects of rank-deficient chemical species tomography are provided by McCann et al. (2015) and Daun et al. (2016).

Fig. 15 The solution to a rank-deficient chemical species tomography problem has components belonging to the rank and nullspace of A. The rank component can be found from the measurement equations, but the nullspace component relies entirely on prior information specified by the analyst. In this case, the set of {Cj} is inferred from knowledge of xexact, but any set of {Cj} would satisfy Ax = b

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Nonlinear Problems

More often the equations that relate the measurements and the unknown parameters are nonlinear, which is usually the case when extinction, emission, and/or in-scattering terms are important in the radiative transfer equation. Under these conditions, the unknown parameters, x, are recovered by minimizing a least-squares function 2 1 FðxÞ ¼ bmeas  bmod ðxÞ 2

(46)

where bmeas and bmod are vectors of the measured and modeled data, respectively. Ill-posedness manifests in the shallow topography of the objective function surrounding x*, which means that a large number of solutions exist that can explain the measurement data in the context of measurement error, cf. Figs. 3a and 4a. Equivalently, the shallow curvature sensitizes the location of x* to small amount of measurement noise that contaminates bmeas. Many studies have been reported in the literature in which the objective is to infer the properties of a participating medium from intensity measurements made at the periphery of the medium, or temperature measurements made at locations within the medium, via the radiative transfer equation. A subset of these studies is described below, to give the reader an overview of the research carried out in this area. The majority of these studies consider simulated experiments in which x is specified, the model equations are used to generate measurement data which is then contaminated with noise, b = bmod(x) + δb, (forward problem), and a least-squares minimization is carried out to recover the parameters (inverse problem). A smaller number of studies use these simulated experiments to prototype true experiments; this is an important distinction, since very few studies consider the propagation of model error during the inversion procedure, which can be considerable in view of the standard simplifying assumptions made when solving the radiative transfer equations as well as the uncertainty in ancillary model parameters that are not the focus of the inference. The objective of many experiments is to infer the extinction coefficient, scattering albedo, and scattering phase function parameters of a participating media based on exit intensities. McCormick (1992) and Özşik and Orlande (2000) summarize the early work in this area. Ho and Özşik (1988) recovered the optical thickness and scattering albedo within an enclosure from measured exit intensities at various angles; this was done by minimizing Eq. 46 using the Levenberg–Marquardt algorithm, cf. Sect. 3.2. They also calculated confidence intervals; if the data in b are contaminated with independent and identically-distributed measurement noise having a variance of σ2, so that Cov(b) = σ2I, it can be shown that Cov(x) = σ2[J(x*)TJ(x*)]1 (Aster et al. 2013), cf. Fig. 3a. Silva Neto and Özşik (1995) extended this treatment to recover phase function parameters, assuming a modified Henyey–Greenstein phase function. As mentioned in Sect. 2.5, LM minimization presents an expedient way to find x*, but it does not affect J(x*)TJ(x*) and thus does not alter the covariance estimate. This highlights the fact that LM is not a true inverse method, since it does not supplement the information in the measurement equations with additional information.

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Hendricks and Howell (1996) inferred the absorption, scattering, and phase function parameters reticulated porous ceramics based on spectral hemispherical transmittance and reflectance measurements; least-squares minimization was carried out using the Levenberg–Marquardt method. They compared the suitability of two candidate dual-parameter phase functions based on the relative magnitudes of their residual norms. Randrianalisoa et al. (2006) recovered the extinction coefficient, the scattering albedo, and the scattering phase function of fused quartz containing closed cells from bidirectional transmissivities measured over a range of exit directions. Leastsquares minimization was carried out using Gauss–Newton method, cf. Sect. 3.2. Numerical difficulties caused by the ill-conditioning of JTJ were surmounted by computing an initial guess for the extinction measurement from transmittances measured in the normal direction. This reduced the number of degrees of freedoms in the multiparameter least-squares minimization, and thus improved the conditioning of JTJ. Deiveegan et al. (2006) compared LM, Bayesian inference, genetic algorithms, and an artificial neural network (ANN) to recover the absorption and scattering coefficients of participating medium contained between infinite parallel plate, as well as the surface emissivities of the bounding plates based on temperatures and heat fluxes measured at the boundary. Their results highlight a major advantage of the Bayesian approach, in that it quantifies the uncertainties in the recovered parameters, which is rarely mentioned in other studies. They also found the metaheuristics (GA and ANN) schemes to be more robust and less sensitive to convergence problems associated with the difficult function topography. Lee et al. (2008) compared the performance of particle swarm optimization and repulsive particle swarm optimization on a similar problem. Ren et al. (2015) retrieved temperature and concentration of a homogeneous gas layer at high temperature from spectral transmittance measurements made with a Fourier transform infrared (FTIR) spectrometer. The modeled data in Eq. 46, bmod(x), was generated using a spectral line database and then convolved by the FTIR instrument response data; the temperature and concentration were then retrieved through LM minimization. The recovered parameters were within 5% of independently-estimated values in most cases, although different temperatures and concentrations were recovered using different spectral databases even though these solutions reproduced the measurement data with a small residual. This highlights the importance of considering the impact of model error (i.e., some of the hightemperature spectral line databases are thought to be incorrect), which may be small in magnitude but have a significant impact on the recovered parameters due to the ill-conditioning of the underlying problem. In a subsequent study, Ren and Modest (2016) used LM regularization to determine the average concentration and temperature profile across a planar layer of CO2 using spectral emittance measurements. In order to infer these parameters, the least-squares function had to be augmented with a first-order Tikhonov regularization functional, which promotes a spatially-smooth temperature profile. This is an example of how supplementing the measurement data with additional prior information alters the topography of the

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objective function (i.e., reduces ||[J(x*)TJ(x*)]1||) and consequently reduces the covariance in the recovered parameters. In the context of nanoscale radiative heat transfer, Charnigo et al. (2012) presented a technique for inferring the size and agglomeration of nanoparticles deposited on a metallic film. The approach is based on generating polarations with the air/metal interface by evanescent waves, which are then scattered by the nanoparticles. Bayesian inference is used to derive marginalized probability densities and credibility intervals. They demonstrated this technique using numericallysimulated scattering data generated with the T-matrix method. Diffuse optical tomography is a particularly important subset of parameter estimation problems. In these experiments, visible or near infrared light sources (usually diode lasers) emit collimated light into a highly scattering medium, and the objective is to infer the spatial distribution of the scattering and absorption coefficients based on the response of sensors located around the periphery of the tomography domain. A classic example is the detection of cancerous tumors, which have absorption and scattering coefficients different from healthy tissue. Recent work in this area has been summarized by Haisch (2012) and Charette et al. (2008). This type of tomography is distinct from the linear tomography problems discussed in Sect. 5.1 due to the predominance of scattering, which obfuscates the relationship between the transmitters and receivers. This is an example of “soft field” tomography, as opposed to “hard field” tomography in which there is an unambiguous relationship between pairs of transmitters and receivers. The scattering between the transmitter and receiver acts as a blending function in the same way as the kernel in a linear IFK; consequently, the nonlinear measurement equations are ill-conditioned and require regularization. Nonlinear conjugate gradient minimization, cf. Sect. 3.2, is particularly suitable, since it is computationally-efficient and exploits the semiconvergence property of ill-posed problems. The first few steps fill in the low-frequency solution components, while iterations are terminated before x* is reached. This limits the maximum obtainable spatial resolution (i.e., the “sharpness” of the reconstruction), but it also suppresses measurement noise. Diffuse optical tomography is sometimes carried out in the frequency domain, using pulsed light sources. The model equation is derived from the time-dependent radiative transfer equation, which accounts for the finite rate of photon propagation through the media. The additional information contained in the time-resolved measurements reduces the ill-posedness of the underlying inverse problem.

5.3

Case Study: Parameter Estimation

To better understand the issues involved in inverse parameter estimation, consider the problem of inferring the morphology of aerosolized soot aggregates from multiangle elastic light scattering data. Soot consists of nanospheres called primary particles, which assemble into fractal-like aggregates through primary particle collisions within the aerosol. A sample transmission electron micrograph of a soot

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aggregate is shown in Fig. 16a. The aggregate structure is often described by the fractal relationship  D N p ¼ kf 2Rg =dp f

(47)

where Rg is the radius of gyration (loosely, the “size”) of the soot aggregate, Np is the number of primary particles per aggregate, dp is the primary particle diameter, and kf and Df are fractal parameters. One way to characterize the size distribution of aerosolized soot aggregates is though multiangle elastic light scattering (MAELS), in which a collimated light source (usually a laser) is shone through a soot-laden aerosol, and the scattered light is measured at various scattering angles, often using a goniometer as shown in Fig. 16b. It can be shown that the scattering phase function of soot aggregates is a unique function of Rg, Df, and kf (Sorensen 2001). In the hypothetical case where an aerosol consists of a single size class of soot aggregates, these parameters can be inferred unambiguously from angular scattering data. More often, however, a sootladen aerosol contains aggregates that obey a size distribution p(Rg), or equivalently p(Np), and the observed light scattering at angle θ are given by 1 ð

bð θ Þ ¼ C

    k θ, Rg p Rg dRg

(48)

0

where C is a coefficient that depends on the particle volume fraction, optical collection efficiency, excitation beam intensity, among other parameters, and the kernel k(θ, Rg) is derived from light-scattering theory. The kernel is also a function of other “nuisance” parameters like kf, Df, and dp, which are often presumed known and/or may not be the focus of the inference procedure. The smoothing action of the kernel makes inverting Eq. 48 extremely ill-posed, because the scattering cross section of soot aggregates increases geometrically with respect to Rg. Consequently, the scattering contributions from many smaller aggregates are overwhelmed by those

Fig. 16 (a) A TEM micrograph shows a soot aggregate, consisting of Np nanospheres called primary particles arranged in a fractal-like pattern. (b) A schematic of a multiangle elastic light scattering experiment used to infer the size distribution of aggregates in a soot-laden aerosol

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from a few large aggregates, and conversely, the observed angular scattering data is insensitive to variations of p(Rg) for small Rg. If p(Rg) is the unknown and C and k(θ,Rg) are considered known, Eq. 48 is a linear Fredholm IFK. Writing this equation for m multiple discrete measurement angles, and expressing the continuous function p(θ) as a discrete linear parameterization, n   X   p Rg  x j β j Rg

(49)

j¼1

transforms Eq. 48 into a matrix equation, Ax = b, where A is ill-conditioned. A common choice is to discretize Rg into n uniformly-spaced segments between 0 and some maximum Rg and then use basis functions βj(Rg) that are delta functions equal to unity over a range [Rg,j,min, Rg,j,max] and are otherwise zero. This is equivalent to representing p(Rg) using strips, as shown in Fig. 17. When written this way, x can be recovered using classical linear regularization schemes such as first-order Tikhonov, which presume a degree of smoothness among the elements of x (Burr et al. 2011; di Stasio et al. 2006). There are two important drawbacks with this approach: First, linear regularization by itself only provides a single candidate distribution, and ignores the existence of other candidate distributions, which may be very different from each other. Second, because the even with regularization, the inverse problem remains extremely ill-posed (Burr et al. 2011). One way to reduce the ill-posedness of the problem is to use a lower-order parametrization of p(Rg). For example, the competing agglomeration and fracturing processes in diffusion-limited aerosol transport often lead to a self-preserving size distribution (Sorensen 2001), which can be modeled as a lognormal distribution

Fig. 17 The Fredholm IFK deconvolution problem can be rewritten in a matrix equation by parameterizing p(Rg) using strips. The resulting A matrix is ill-conditioned, and has singular values that resemble those in Fig. 2

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"     2 # log Rg  log μg 1 p Rg , x ¼ pffiffiffiffiffi exp  ,  2 Rg 2π logσ g 2 logσ g 



1285

 T x ¼ μg , σ g

(50)

where the distribution parameters μg and σ g are inferred in the inverse problem. This constitutes incorporation of prior information, i.e., that the size distribution should be lognormal, in order to mitigate the ill-posedness of the inverse problem. This parameterization transforms the linear IFK, Eq. 48, into a nonlinear IFK having the general form of Eq. 7 since b(θ) is a nonlinear function of the unknowns, μg and σ g. Huber et al. (2016) used how Bayesian analysis to estimate posterior probability densities for μg and σ g from highly-resolved angular scattering data. They first derived the likelihood function using a detailed analysis of the measurement noise, " # 1 ½b  aðxÞT Γ1 ½ b  a ð x Þ  b pðbjxÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi exp  2 ð2π Þm=2 DetðΓb Þ

(51)

where Γb is the measurement noise covariance matrix and a(x) is found by carrying out the integration in Eq. 48 for a lognormal p(Rg, x) with x = [μg, σ g]T. Measurement noise consisted of photonic shot noise and electronic noise, which affect each measurement angle independently, and process noise due to variations in laser fluence and particle concentrations between measurements, which contributes to a nontrivial covariance between the measurement angles. An additional unknown calibration error is masked with 2% white noise, with the understanding that this provides a conservative estimate of the credibility bounds. An “artificial” 2D problem is generated by treating the nuisance parameters (C, Df, kf, and dp in Eqs. 47 and 48) as deterministic, and using uninformative priors for μg and σ g, i.e., ppr(μg) = ppr(σ g) = 1, so p(x|b) = p(b|x) and xMAP = xMLE. Figure 18 shows a contour plot of the probability density of p(x|b), marginalized probability densities for x1 = μg and x2 = σ g, and 90% credibility intervals. Marginalization is done in two ways: a Markov-Chain Monte Carlo sampling procedure, which uses Metropolis–Hastings criterion to generate samples that become ergodic to p(x|b); and an approximating normal distribution is constructed from the Jacobian, x ~ Ν(xMAP, Γx), Γx = J(xMAP)Γb1 J(xMAP). It is straightforward to extend the analysis to consider uncertainty in the nuisance parameters, which provides a more realistic reflection of the uncertainty in the inferred quantities of interest. Increasing the number of degrees of freedom also increases the ill-posedness of the problem, so it is often necessary to incorporate additional information through a Bayesian prior. The prior should be defined in a way that incorporates all available “testable” prior information (roughly speaking, information that can be empirically confirmed) while avoiding subjective information that could unduly bias the inferred parameters. The Principle of Maximum Entropy (Jaynes 1957; von Toussaint 2011) can be used to maximize the information entropy of the prior distribution subject to constraints related to the testable information. If point estimates and corresponding uncertainties are available, the

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Fig. 18 Using a lognormal parameterization for p(Rg) reduces the ill-posedness of the inference problem and makes it problem nonlinear. Contours correspond to posterior probability densities p(x|b) = p(b|x) for ppr = 1. Marginalization is carried out using Markov-Chain Monte Carlo and with a Gaussian approximation. Shaded regions of the marginalized probability densities denote 90% credibility intervals (Huber et al. 2016)

information entropy is maximized by a normal distribution centered on the estimate with a standard deviation equal to the uncertainty. Gaussian priors are also convenient, since they can readily be combined with the likelihood function following the example in Sect. 3.4. Figure 19 shows an example in which uncertainties in C and Df are considered by adding them to the unknown vector x = [μg, σ g, Df, C]T. The posterior probability densities obtained using an uninformative prior are wide, due to the ill-posedness of the inference problem. The maximum likelihood estimate deviates from the exact solution, denoted by vertical red lines, due to random noise contaminating the measurements. Specifying a Gaussian prior for Df centered at 1.7 with a standard deviation of 0.05 dramatically narrows the posterior probability density. The prior and marginalized posteriors for Df are almost indistinguishable, highlighting that the estimate for Df relies almost entirely on the prior information. While the likelihood function derived by Huber et al. (2016) considers measurement noise and an unknown calibration error, their study excludes uncertainty associated with the scattering model used to derive the kernel. Most researchers derive k(θ, Rg) from Rayleigh–Debye–Gans fractal aggregate (RDF-FA) theory, which is computationally-efficient and analytically-tractable, but can differ considerably from scattering predictions derived from more accurate but costly numerical techniques. (An error of 5–15% is typical, depending on the aggregate size, fractal structure, and scattering angle, e.g., Berg and Sorensen (2013).) Model error must be included in order to accurately assess the uncertainty in the estimated quantities of interest. Huber et al. (2016) highlight this by comparing the estimated size parameters found from experimental light scattering data using kernels derived from three variants of RDG-FA theory. The size distribution parameters recovered from

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Fig. 19 Uncertainties in the nuisance parameters, C and Df, can be incorporated by adding them to the unknown vector x, but results in a broad posterior probability density (gray curves). Specifying a Gaussian prior on Df (green curve) narrows the distributions (blue curves). The similarity between the posterior density and prior for Df shows that the measurement equations contain very little information about this parameter relative to the prior (Huber et al. 2016)

experimental data differ considerably, even though simulated datasets generated by substituting the same size distribution into the three measurement equations are nearly indistinguishable.

6

Conclusions and Outlook

Inverse problems are ubiquitous in radiative transfer: the goal of inverse design is to determine the configuration of a thermal system that produces a desired engineering outcome, while in parameter estimation quantities of interest are inferred from indirect experimental measurements. Both types of problems are mathematically ill-posed due to an information deficit. In the case of inverse design problems, multiple solutions may exist that satisfy the design specifications, or the design problem may not have a solution at all. Parameter estimation problems are often ill-posed because multiple candidate parameter sets exist that, when substituted into the measurement equations, produce simulated data that is nearly indistinguishable from the measurements, particularly in the presence of noise. A diverse suite of techniques have been applied to inverse problems arising in every subarea of radiative transfer. Each of these techniques addresses the

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information deficient that makes the inverse problem ill-posed, by injecting additional prior information into the analysis. There are many ways to do this: using a lower-order parameterization (e.g., reducing the number of heaters in a radiant enclosure design problem); linear regularization schemes that promote smooth solutions by suppressing error amplification by small singular values; and Bayesian priors, which explicate the role of prior information in the inference procedure. The inverse analysis technique should be chosen based on the mathematical properties of the ill-posed problem (linear, nonlinear, convex, etc.), the prior information available, the number and type of variables, and the cost of evaluating the forward problem. While many schemes produce a single solution, it is often important to consider the existence of other candidate solutions, since these represent alternative design options, or reflect uncertainty in parameter estimation problems. If the inverse problem is linear, the solution space can be explored by varying the regularization parameter or examining the orthonormal basis formed from a singular value decomposition of the coefficient matrix. A similar procedure can be done using the Jacobian matrix for nonlinear inverse problems, while most metaheuristic techniques inherently generate a family of candidate solutions. In the context of Bayesian inference, the existence of multiple solutions is reflected in the posterior probability density, and univariate marginalized densities and credibility intervals. A review of the current literature shows inverse analysis to be a very dynamic area within radiative transfer. The emergence of nanoscale radiation heat transfer, in particular, has resulted in inverse design problems involving engineered nanostructures for energy conversion and diagnostics, as well as parameter estimation problems focused on characterizing the attributes of aerosolized nanoparticles based on absorption, emission, and scattering measurements. Inverse problems arising in more traditional areas of radiative transfer remain important, e.g., for understanding climate change, developing cleaner, more efficient energy conversion technologies, and new manufacturing processes. Advancements in optics and electronics, e.g., hyperspectral imaging and new light sources, also lay a foundation for novel diagnostics at the macro-, micro-, and nanoscales to help further scientific understanding and solve important engineering problems. New inverse analysis algorithms are being developed to realize the full potential of this technology. These algorithms exploit improvements in computational power as well as improved solution techniques for the radiative transfer equation, which allow for multiple objective function evaluations needed to carry out design optimization and the high-order integrations required to calculate the evidence and marginalize probability densities for Bayesian inference. Researchers in thermal radiation also stand to benefit from emerging techniques in inverse analysis. In the case of parameter estimation, for example, new design-of-experiment techniques are being developed that maximize the information content of the measurement data, thereby reducing the underlying ill-posedness of the underlying inference problem. Likewise, new statistical tools help guide the development of physical models based on experimental data, in order to prevent over-tuning in the presence of measurement noise and model parameter uncertainty. In summary, inverse analysis will remain an important area of thermal radiation research for years to come.

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Cross-References

▶ Monte Carlo Methods for Radiative Transfer ▶ Near-Field Thermal Radiation ▶ Design of Optical and Radiative Properties of Surfaces ▶ Radiative Properties of Gases ▶ Radiative Properties of Particles ▶ Radiative Transfer Equation and Solutions ▶ Radiative Transfer in Combustion Systems

References Åkesson EO, Daun KJ (2008) Parameter selection methods for axisymmetric flame tomography through Tikhonov regularization. Appl Opt 47:407–416 Amiri H, Mansouri SH, Coelho PJ (2013) Inverse optimal design of radiant enclosures with participating media: a parametric study. Heat Transf Eng 34:288–302 Aster CR, Borchers B, Thurber CH (2013) Parameter estimation and inverse problems, 2nd edn. Academic Press, San Diego Baranoski G, Bramley R, Rokne JG (2001) Examining the spectrum of radiative transfer systems. Int Comm Heat Mass Transf 28:519–525 Beck JV (1968) Surface heat flux determination using an integral method. Nucl Eng Des 7:170–178 Berg MJ, Sorensen CM (2013) Internal fields of soot fractal aggregates. J Opt Soc Am A 30:1947–1955 Bertsekas DP (1999) Nonlinear programming, 2nd edn. Athena Scientific, Belmont Bohren CF, Huffman DR (1983) Absorption and scattering of light by small particles. Wiley, New York Burr DW, Daun KJ, Link O, Thomson KA (2011) Determination of the soot aggregate size distribution from elastic light scattering through Bayesian inference. J Quant Spectrosc Radiat Transf 112:1099–1107 Charette A, Boulanger J, Kim HK (2008) An overview on recent radiation transport algorithm development for optical tomography imaging. J Quant Spectrosc Radiat Transf 109:2743–2766 Charnigo R et al (2012) Credible intervals for nanoparticle characteristics. J Quant Spectrosc Radiat Transf 113:182–193 Chopade RP, Mishra SC, Mahanta P, Maruyama S (2012) Estimation of power of heaters in a radiant furnace for uniform thermal conditions on 3-D irregular shaped objects. Int J Heat Mass Transf 55:4340–4351 Cormack AM (1973) Reconstructions of densities from their projections with applications in radiological physics. Phys Med Bio 18:195–207 Dasch CJ (1992) One-dimensional tomography: a comparison of Abel, onion-peeling, and filtered backprojection methods. Appl Opt 31:1146–1152 Daun KJ, Howell JR (2005) Inverse design methods for radiative transfer systems. J Quant Spectrosc Radiat Transf 93:43–60 Daun KJ, Howell JR, Morton DP (2003a) Design of radiant enclosures using inverse and non-linear programming techniques. Inverse Prob Eng 11:541–560 Daun KJ, Howell JR, Morton DP (2003b) Geometric optimization of radiant enclosures containing specular surfaces. J. Heat Transf. 125:845–851 Daun KJ, Howell JR, Morton DP (2003c) Geometric optimization of radiant enclosures through nonlinear programming. Numer Heat Transf, Part B 43:203–219 Daun KJ, Thomson KA, Liu F, Smallwood GJ (2006a) Deconvolution of axisymmetric flame properties using Tikhonov regularization. Appl Opt 45:4638–4646

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Daun KJ et al (2006b) Comparison of methods for inverse Design of Radiant Enclosures. ASME J Heat Transf 128:269–282 Daun KJ, Grauer SJ, Hadwin PJ (2016) Chemical species tomography of turbulent flows: discrete ill-posed and rank deficient problems and the use of prior information. J Quant Spectrosc Radiat Transf 172:58–74 Deiveegan M, Balaj C, Venkateshan SP (2006) Comparison of various methods for simultaneous retrieval of surface emissivities and gas properties in gray participating media. ASME J Heat Transf 128:829–837 Ertürk H, Ezekoye OA, Howell JR (2002a) Comparison of three regularized solution techniques in a three-dimensional inverse radiation problem. J Quant Spectrosc Radiat Transf 73:307–316 Ertürk H, Ezekoye OA, Howell JR (2002b) The application of an inverse formulation in the design of boundary conditions for transient radiating enclosures. ASME J Heat Transf 124:1095–1102 Ertürk H, Gamba MEOA, Howell JR (2008) Validation of inverse boundary condition design in a thermometry test bed. J Quant Spectrosc Radiat Transf 109:317–326 Farahmand A, Payan S, Hosseini Sarvari S (2012) Geometric optimization of radiative enclosures using PSO algorithm. Int J Therm Sci 60:61–69 Fedorov AG, Lee KH, Viskanta R (1998) Inverse optimal design of the radiant heating in materials processing and manufacturing. J Mater Eng Perform 7:719–726 Floyd J, Kempf AM (2011) Computed tomography of Chemiluminescence (CTC): high resolution and instantaneous 3-D measurements of a matrix burner. Proc Combust Inst 11:751–758 França F, Ezekoye O & Howell J 1999 Inverse determination of heat source distribution in radiative systems with participating media. In: 33rd national heat transfer conference NHTC’99, Albuquerque, NM, USA, 15–17 August 1999 França FHR, Ezekoye OA, Howell JR (2001) Inverse boundary design combining radiation and convection heat transfer. ASME J Heat Transf 123:884–891 França FHR, Howell JR, Ezekoye OA, Morales JC (2003) Inverse design of thermal systems with dominant radiative transfer. Adv Heat Transf 36:1–110 Gill PE, Murray W, Wright MH (1986) Practical optimization. Academic Press, San Diego Glover F (1986) Future paths for integer programming and links to artificial intelligence. Comput Oper Res 5:533–549 Golub GH, Heath MT, Wahba G (1979) Ceneralized cross-validation as a method for choosing a good ridge parameter. Technometrics 21:215–223 Hadamard J (1923) Lectures on Cauchy's problem in linear differential equations. Yale University Press, New Haven Haisch C (2012) Optical tomography. Annu Rev Anal Chem 5:57–77 Hajimirza S, El Hitti G, Heltzel A, Howell J (2012) Using inverse analysis to find optimum nanoscale radiative surface patterns to enhance solar cell performance. Int J Therm Sci 62:93–102 Hall RJ, Bonczyk PA (1990) Sooting flame thermometry using emission/absorption tomography. Appl Opt 29:4590–4598 Hansen PC (1999) Rank-deficient and discrete ill-posed problems: numerical aspects of linear inversion. SIAM, Philadelphia Hansen PC, O'Leary DP (1993) The use of the L-curve in the regularization of discrete ill-posed problems. J Sci Comput 14:1487–1503 Hansen PC, Sekii S, Shibahashi H (1992) The modified truncated SVD method for regularization in general form. SIAM J Sci Stat Comput 13:1142–1150 Harutunian V, Morales JC, Howell JR (1995) Radiation exchange within an enclosure of diffusegray surfaces: the inverse problem. In: National heat transfer conference, Portland OR, 1995 Hendricks TJ, Howell JR (1996) Absorption/scattering coefficients and scattering phase functions in reticulated porous ceramics. ASME J Heat Transf 118:79–87 Ho C-H, Özşik MN (1988) Inverse radiation problems in inhomogeneous media. J Quant Spectrosc Radiat Transf 40:533–560 Hosseini Sarvari SM (2007) Optimal geometry design of radiative enclosures using the genetic algorithm. Numer Heat Transf, Part A 52:127–143

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Hosseini Sarvari S, Howell J, Mansouri S (2003) Inverse boundary design conduction-radiation problem in irregular two-dimensional domains. Numer Heat Transf, Part B 44:209–224 Howell JR, Mengüç MP, Siegel R (2016) Thermal radiation heat transfer, 6th edn. CRC Press, Boca Raton Huber FJT, Will S, Daun KJ (2016) Sizing aerosolized fractal nanoparticle aggregates through Bayesian analysis of wide-angle light scattering (WALS) data. J Quant Spectrosc Radiat Transf 184:27–39 Jaynes ET (1957) Information theory and statistical mechanics. Phys Rev 106:620–630 Jones MR (1999) Inverse analysis of radiative transfer systems. ASME J Heat Transf 121:481–484 Kaipio J, Somersalo E (2005) Statistical and computational inverse problems. Springer, Berlin Kashif M et al (2012) Soot volume fraction fields in unsteady axis-symmetric flames by continuous laser extinction technique. Opt Express 20:28742–28751 Kennedy J, Eberhart R (1995) Particle swarm optimization. Perth WA (ed) IEEE international conference on neural networks, pp. 1942–1948 Kiefer J, Wolfowitz J (1952) Stochastic estimation of the maximum of a regression function. Ann Math Stat 23:462–466 Kim KW, Baek SW (2004) Inverse surface radiation analysis in an axisymmetric cylinderical enclosure using a hybrid genetic algorithm. Numer Heat Transf, Part A 46:367–381 Kim KW, Baek SW (2007) Inverse radiation–conduction design problem in a participating concentric cylindrical medium. Int J Heat Mass Transf 50:2828–2837 Kirkpatrick S, Gelatt CD Jr, Vecchi MP (1983) Optimization by simulated annealing. Science 220:671–680 Kowsary F, Pooladvand KPA (2007) Regularized variable metric method versus the conjugate gradient method in solution of radiative boundary design problem. J Quant Spectrosc Radiat Transf 108:277–294 Kudo K et al (1996) Solution of the inverse radiative load problem using the singular value decomposition technique. JSME Int J, Ser B 39:808–814 Leduc G, Monchoux F, Thellier F (2004) Inverse radiative design in human thermal environment. Int J Heat Mass Transf 47:3291–3300 Lee KH, Kim KW (2015) Performance comparison of particle swarm optimization and genetic algorithm for inverse surface radiation problem. Int J Heat Mass Transf 88:330–337 Lee KH, Baek SW, Kim KW (2008) Inverse radiation analysis using repulsive particle swarm optimization algorithm. Int J Heat Mass Transf 51:2772–2783 Marston AJ, Daun KJ, Collins MR (2012) Geometric optimization of radiant enclosures containing specularly-reflecting surfaces through quasi-Monte Carlo simulation. Numer Heat Transf, Part A 59:81–97 McCann H, Wright P, Daun K (2015) Chemical species tomography. In: Industrial Tomography: Systems and applications. Woodhead Publishing, Sawston, pp 135–174 McCormick NJ (1992) Inverse radiative transfer problems: a review. Nucl Sci Eng 112:185–198 Mengüç PM, Dutta P (1994) Scattering tomography and its application to sooting diffusion flames. ASME J Heat Transf 144:144–151 Metropolis N et al (1953) Equations of state calculations by fast computing machines. J Chem Phys 21:1087–1092 Mitchell M (1996) An introduction to genetic algorithms. MIT Press, Cambridge Modest MF (2013) Radiative heat transfer, 3rd edn. Academic Press, San Diego Moré J, Sorensen D (1983) Computing a trust region step. SIAM J Sci Stat Comput 3:553–572 Morozov VA (1968) On the discrepancy principle for solving operator equations by the method of regularization. Zhurnal Vychislitel'noi Matematiki i Matematicheskoi Fiziki 8:295–309 Mossi AC, Vielmo HA, França FHR, Howell JR (2008) Inverse design involving combined radiative and turbulent convective heat transfer. Int. J. Heat Mass Trans. 51:3217–3226 Özşik MN, Orlande HRB (2000) Inverse heat transfer: fundamentals and applications. CRC Press, Boca Raton

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Porter JM, Larsen ME, Wesley BJ, Howell JR (2006) Metaheuristic optimization of a discrete Array of radiant heaters. ASME J Heat Transf 128:1031–1040 Pourshaghaghy A, Pooladvand K, Kowsary F, Karimi-Zand K (2006) An inverse radiation boundary design problem for an enclosure filled with an emitting, absorbing, and scattering media. Int Commun Heat Mass Transf 33:381–390 Randrianalisoa J, Baillis D, Pilon L (2006) Improved inverse method for radiative characteristics of closed-cell absorbing porous media. AIAA J Thermo Heat Transf 20:871–883 Ren T, Modest MF (2016) Temperature profile inversion from carbon-dioxide spectral intensities through Tikhonov regularization. AIAA J Thermo Heat Transf 30:211–218 Ren T, Modest MF, Fateev A, Clausen S (2015) An inverse radiation model for optical determination of temperature and species concentration: development and validation. J Quant Spectrosc Radiat Transf 151:198–209 Rukolaine SA (2015) Shape optimization of radiant enclosures with specular-diffuse surfaces by means of a random search and gradient minimization. J Quant Spectrosc Radiat Transf 151:174–191 Safavinejad A, Mansouri SH, Sakurai A, Maruyama S (2009) Optimal number and location of heaters in 2-D radiant enclosures composed of specular and diffuse surfaces using micro-genetic algorithm. Appl Therm Eng 29:1075–1085 Santoro RJ, Semerjian HJ, Emmerman P, Goulard R (1981) Optical tomography for flow field diagnostics. Int J Heat Mass Transf 24:1139–1150 Silva Neto AJ, Özşik MN (1995) An inverse problem of simultaneous estimation of radiation phase function, albedo and optical thickness. J Quant Spectrosc Radiat Transf 53:397–409 Sorensen C (2001) Light scattering from fractal aggregates: a review. Aerosol Sci Technol 35:648–687 di Stasio S et al (2006) Synchrotron SAXS 〈in situ〉identification of three different size modes for soot nanoparticles in a diffusion flame. Carbon 44:1267–1279 Tan JY, Liu LH (2009) Inverse geometry design of radiating enclosure filled with participating media using a meshless method. Numer Heat Transf, Part A 56:132–152 Tikhonov AN, Arsenin VY (1977) Solution of ill-posed problems. Winston and Sons, Washington, DC von Toussaint U (2011) Bayesian inference in physics. Rev Mod Phys 83:943–999 Wing GM, Zhart JD (1991) A primer on integral equations of the first kind: the problem of deconvolution and unfolding. SIAM, Philadelphia Winston R (1991) Nonimaging optics. Sci Am 264:76–81 Wright P et al (2010) High speed chemical species tomography in a multi-cylinder automotive engine. Chem Eng J 158:2–10

Part V Heat Transfer Equipment

Introduction and Classification of Heat Transfer Equipment

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Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Classification Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Types of Heat Exchangers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Recuperators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Regenerators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Direct Type Exchangers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Electrical Resistance Heaters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Heat Transfer Media and Applications Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Boilers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Condensers and Evaporators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Shell-and-Tube Heat Exchangers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Coolers, Refrigerators, and Chillers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Radiators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Furnaces and Ovens . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7 Dryers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8 Heat Pumps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Cross-References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

Efficiency trends in process industries, power generation, manufacturing, and transportation are associated nowadays with significant increase in temperature and pressure of the working media resulting in higher heat fluxes and fluids Y. Chudnovsky (*) Gas Technology Institute, Des Plaines, IL, USA e-mail: [email protected] D. P. Sekulic Department of Mechanical Engineering, University of Kentucky, Lexington, KY, USA e-mail: [email protected] # Springer International Publishing AG, part of Springer Nature 2018 F. A. Kulacki (ed.), Handbook of Thermal Science and Engineering, https://doi.org/10.1007/978-3-319-26695-4_18

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specific enthalpies, as well as increasingly demanding design requirements. Those changes encourage development of new types of heat transfer equipment capable to meet the more severe conditions of the existing operations, as well as to satisfy multiple engineering applications in heating and cooling where traditional heat exchangers cannot be used cost-effectively. So the heat transfer equipment can be considered as a spectrum of special design systems that allow transferring the thermal energy from one heat carrier to another, wherein the heat transfer media may be either liquid or gaseous.

1

Introduction

The core component characterizing any heat transfer system is a heat exchanger. A heat exchanger is a system component enabling transfer of heat from one subsystem/ matter to another with or without their direct contact as per any specific application requirements. Arguably, the most known heat exchanger is a car radiator that transfers heat from the car engine to the ambient air indirectly by circulating solution of water and ethylene glycol (antifreeze). This heat transfer process helps preventing an engine from overheating. Possibly, the most known heat transfer system is a boiler (steam generator or water heater) that transfers heat produced by fuel combustion burner or electrical heater to the liquid (in most applications – water) flow. In this case, the role of heat exchanger is played typically by tubular bundle or other heat transfer hardware shape that is heated by combustion products or electricity.

2

Classification Approach

The selection of heat exchange equipment is an extremely important task driven by a need to provide required features to the technological or manufacturing process. Prior to selecting the heat transfer equipment a thorough evaluation of the thermal application would be necessary to identify the key parameters of the thermal process such as heat exchanging media, thermophysical properties, flow rates, temperature and pressure of the streams, etc. Different types and constructive aspects of the heat transfer equipment will provide the specific end user with required thermal performance and operating efficiency. Heat exchangers can be classified according to various criteria. Figure 1 presents the classification and nomenclature of heat exchangers that was suggested by R. Shah (1981). For example, depending on the dominant heat transfer contact, the heat exchangers can be divided into two groups: direct and indirect, see Fig. 1. Furthermore, the criteria may include the number of fluids exchanging heat, the heat transfer surface compactness, hardware construction, flow arrangement, and heat transfer mechanism. The detailed elaboration of the various classifications and associated heat exchanger types are given in Shah and Sekulic (2003). All heat exchangers have design features depending on the application such as flows

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Classification according to transfer process Direct contact type

Indirect contact type Direct transfer type Single-phase

Storage type

Fluidized bed

Immiscible fluids

Gas-liquid

Liquid-vapor

Multiphase

Classification according to number of fluids Two-fluid

Three-fluid N-fluid (N > 3)

Classification according to surface compactness Gas-to-fluid Compact (β ≥ 700m2/m3)

Liquid-to-liquid and phase-change

Noncompact Compact (β < 700m2/m3) (β ≥ 400m2/m3)

Noncompact (β < 400m2/m3)

Classification according to construction Plate-type

Tubular PHE

Extended surface

Regenerative

Spiral Plate coil Printed circuit

Gasketed Welded Brazed

Rotary Fixed-matrix Rotating hoods Plate-fun Tube-fin Double-pipe Shell-and-tube Spiral tube Pipe coils Heat-pipe Ordinary wall separating wall Crossflow Parallelflow to tubes to tubes

Classification according to flow arrangements Single-pass

Multipass

Counterflow Parallelflow Crossflow Split-flow Divided-flow Extended surface

Shell-and-tube

CrossCross- Compound Parallel counterflow Split-flow Divided-flow counterflow parallelflow flow m-shell passes n-tube passes

Plate Fluid 1 m passes Fluid 2 n passes

Classification according to heat transfer mechanisms Single-phase convection Single-phase convection Two-phase convection Combined convection on one side, two-phase and radiative heat transfer on both sides on both sides convection on other side

Fig. 1 Classification of heat exchangers (Shah 1981)

arrangement, layout and configuration of the heat exchange interface, temperature difference, construction materials, etc. There is a number of heat exchanger design handbooks and manufacturers’ references available for the engineers and researchers, so the authors would recommend the readers to look at some of the widely used books such as Kakac et al.

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(2012) or Hewitt Ed. (2008) for a greater level of details relating to a wide spectrum of the heat exchanger types and selection criteria.

3

Types of Heat Exchangers

For the purpose of this brief review, let us provide just a basic consideration of the most typical heat exchanger types. Based on the transfer mechanisms, heat exchangers may be classified by three basic heat transfer mechanisms: conduction, convection, and thermal radiation. In practice, these phenomena are not isolated, are in some combination, and coexist at the same time during the heat transfer process. So, as per practical characterization of the heat exchangers they may be combined into the specific classification groups such as into specific classification groups such as recuperative, regenerative, mixing, and electrical heating.

3.1

Recuperators

In recuperative heat exchangers, the transfer of heat is accomplished through single layer or multilayer wall-dividing heat exchanging media and operating under steady state or unsteady thermal conditions. The continuously operating heat exchangers that work with steady conditions at the heat exchanger inlet and outlet ports could be considered as the exchangers with a steady-state thermal condition. Unless there is a perturbation of such thermal conditions, the steady state persists. The transfer of heat from one medium to another in a recuperative heat exchanger occurs with a simultaneous forced motion of media (fluids) without (or with) phase change of one (or both) heat transfer agent(s). Special types of heat exchangers may be combined into separate subgroups, such as spray cooling heat exchangers and recuperative systems with the dispersed flows. In the first subgroup, the transfer of heat through the separating wall is accompanied by the processes of heat and mass transfer on the external surface that is spayed by liquid (e.g., water). The second subgroup is characterized by doping of the heat transfer flow with a dispersed solid medium of small volume concentration that affects the heat transfer conditions and contributes to the enhancement of heat transfer. Majority of continuously acting recuperative heat exchangers can be classified as heat exchangers with steady-state thermal conditions. They may be divided as per their design specifics, for example their design specifics, for example coiled, modular, shell-and-tube, surface enhanced (ribbed, finned, studded), plane and finned, plate and framed, cellular, and 3D printed.

3.2

Regenerators

In regenerative heat exchangers, the heat transfer surface is also used to transfer heat from one medium to another. However, this surface, specifically packing or bed,

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serves as the intermediate heat storage medium where heat from the hot fluid is intermittently stored before it is transferred to the cold fluid. To accomplish this, the hot fluid is brought into contact with the heat storage medium followed by the displacement of the hot fluid with the cold one absorbing the heat. Processes of heating or cooling in the regenerators are not steady state, but synchronized recurrent, transient thermal processes. Usually the regenerators are employed for the industrial furnaces, MHD generators, steam production, and petrochemical applications. In the periodically acting heat exchangers, during certain time period the processes of heating, evaporation, cooling, and condensing can be achieved consecutively. In the heating or cooling segments of operation, the variation of the heated substance’s temperature occurs over time. The heating (cooling) medium, as a rule, is supplied continuously with practically unchanged inlet parameters but substantially time-varying temperatures at the outlets, especially in case of liquid and gaseous heat transfer agents. Thus, such type of heat exchanger should be considered as having unsteady thermal conditions.

3.3

Direct Type Exchangers

For mixing (or direct type) heat exchangers there is no special heat transfer surface separating fluids, i.e., it is not required for heat exchange between working fluids. Heat exchange in this case occurs on the free interface between the heat transfer fluids, and it is accompanied by a mass exchange, a change in enthalpy of mixture or each fluid flow, and/or by a change in the moisture content of gaseous medium. Mixing heat exchangers can be hollow and supplied with the packing. The surface of specific packing serves only for organizing and arrangement of the liquid film and it is not acting as a heat transfer surface. Depending on the application and designation, gas-liquid mixing exchangers are called scrubbers, cooling towers, spraying chambers, and mixing water heaters. Scrubbers serve for cooling, dying or humidifying, gaseous flow clean up form dust and other admixtures. The spraying chambers provide cooling, dehumidifying and humidifying of the air flow for air conditioning systems. Cooling towers reject heat from the circulating water loops at the thermal power generation stations, oil refineries, petrochemical plants and commercial building heating, ventilation, and air conditioning (HVAC) systems. The mixing steam- and water-to-water exchangers provide water heating for the hot water supply, the condensation of exhaust steam and so on.

3.4

Electrical Resistance Heaters

In heat exchangers with electrical resistance heating the electrical energy is used as the heating source. The heat transfer conditions differ from the heat transfer conditions in the above mentioned heat exchangers. Electrical energy is converted into thermal energy in an electric resistor, in the electric or plasma arcs of a direct or an

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indirect heating, or in the systems of induction and dielectric heating. The electrical resistant heaters and induction heaters are the most popular types on the industrial heating marketplace.

4

Heat Transfer Media and Applications Spectrum

Each group of heat exchangers, except the ones with the electrical heating, can be classified based on the heat transfer medium: steam-to-liquid, liquid-to-liquid, gasto-liquid, gas to gas, steam-to gas, and with dispersed media. Heat exchange processes have a highest importance in power machinery, petrochemical, glass, papermaking, metallurgical, food processing, transportation, and many other industries, so the most useful classification is on the basis of the intended applications. That is why the heat exchangers have design features highly depending on the application, i.e., the working media flow direction, the coolant temperature gradient, the working fluids and heat exchanger hardware materials, the heat exchange surface configuration, etc. A large diversity of practical applications for heat transfer equipment as shown in Fig. 2 makes a wide spectrum of heat exchanger designs and configurations. Heat exchangers intended for industrial applications/requirements depend on specific conditions of application, hence are very diverse. The main requirements are: (i) to provide the highest heat transfer coefficient with the smallest possible hydraulic resistance (pressure drop); (ii) to feature compactness and low material usage; (iii) to possess reliability and integrity; (iv) to have low fouling ability; (v) to satisfy unification of units and parts; (vi) to feature manufacturability and potential for advanced and digital manufacturing, accommodating of the wide range of operating temperatures, pressures, etc. When creating a new and more efficient heat exchangers one tends to: reduce the unit costs of materials, to eliminate/reduce the involvement of labor, to minimize manufacturing cost and reduce expended energy at work, all compared with the same performance of commercially available heat exchangers. The unit costs for heat exchangers may be estimated as the total costs divided by the heat transferred at the given conditions. The intensity of the heat transfer process or specific heat transfer rate density of a heat exchanger is the quantity of heat transferred per unit of time, through a unit of heat exchange surface at a given heat transfer duty. The intensity of the heat transfer process is characterized by the overall heat transfer coefficient and is affected by a form of heat transfer surfaces; equivalent diameter and channels arrangement, providing optimal fluid media movement and velocity; the average temperature difference; and the presence of turbulators in channels; fins, studs, etc. The multiple heat transfer enhancement techniques are employed to improve the heat transfer performance. These techniques, e.g., may be associated with changes in hydrodynamic parameters of the fluid flow near the heat exchange surface, such as vibrations of the surface creating flow pulsations, a blowing of a gas stream or a working fluid sucked through the porous wall, an imposition of electrical and/or magnetic fields on the flow, the heat exchange surface pollution prevention by strong turbulence in the flow, etc.

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Fig. 2 Applications of heat transfer equipment (Kakac et al. 2012)

Thus, the specific requirements and strong regulations for certain residential, commercial, and industrial applications have led, over the decades, to numerous designs and various types that are unique to particular markets. Below are summarized the descriptions and practical illustrations of key applications for heat transfer equipment and associated heat exchangers.

4.1

Boilers

Boilers are the heat transfer devices used to generate hot water or steam by using the electricity or gaseous, solid, and liquid fuel. Many applications employ boilers as

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water heaters or steam generators. Various sizes of boilers are ranging from residential to commercial and/or industrial scale. There are boilers that use the renewable energy such as solar, geothermal, etc., or hot exhaust streams from industrial or power generation installations such as furnaces or gas turbines. Figure 3 illustrates three typical configurations. Boilers are very complex systems that may include (i) water treatment and working fluid makeup subsystems (to prevent the surface fouling and plugins), (ii) flue gas treatments (to mitigate and comply with the local environmental regulations); furthermore, they may include additional recuperators, superheaters, and so forth. The details on different boilers design, specifications, critical operating conditions, and requirements could be found in corresponding Original Equipment Manufacturer (OEM) manuals and in a wide spectrum of the publicly available handbooks and technical publications such as online Boiler Book by Cleaver Brooks (2011). Special place in the boilers family is occupied by the utility boilers that generate high pressure and high temperature steam for the Rankine Cycle power plants of a typical capacity in the range of 350–500 MWe. The most popular type of the utility boilers is a radiant one that combines a radiation-type furnace with the steam superheating as illustrated in Fig. 4. Boilers may be supplied by a wide spectrum of fuels, from high-rank coals to low-rank coal wastes, and from oil and natural gas to the so-called opportunity fuels such as petroleum coke, biomass, and oil shale. Figure 5 illustrates a flexible circulating fluidized bed (CFB) boiler for the power generation featuring capacities of up to 660 MWe. Over the past two decades, CFB technology has demonstrated its ability to efficiently utilize a wide variety of fuels while still meeting stringent stack emission limits. The most complete collection of the specific details and information regarding utility boilers and steam generation could be found in Steam by Tome (2015).

4.2

Condensers and Evaporators

Condensers and evaporators are essential part of any industry. In many heat transfer systems, a condenser serves as a unit to condense a fluid from its gaseous to its liquid state. This is accomplished by cooling the fluid below the condensing temperature at the process pressure. In this process, the enthalpy of phase change (often expressed in technical lingo as the latent heat) is given up by the condensing medium and is transferred to the condenser coolant. Condenser designs are typically designs of heat exchangers and come in many sizes and variations ranging from small to very large industrial-scale units employed in plant processes. A simple example of the condenser at a substantially smaller scale is the condensing unit of the residential HVAC systems. The condensers are also widely employed across the diversity of cooling and refrigeration applications, chemical processes such as distillation, as well as in applications intended to improve the energy efficiency of gas-fired installations such as boilers, in thermal power plants (steam condensers), etc. Use of cooling water or

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Fig. 3 Examples of typical commercial and industrial boilers (a) Hurst performance series low pressure firebox boiler (Hurst Boiler – www.hurstboiler.com), (b) Scotch marine firetube wetback boiler (Hurst Boiler – www.hurstboiler.com), (c) CleaverBrooks FLX watertube boiler system (CleaverBrooks – http://www.cleaverbrooks.com/)

surrounding air as the cooling medium is common for condensing applications. However, there are some evaporative condensers on the market that improve the heat rejection process by using the cooling effect of evaporation. The steam or vapor to be condensed is circulated through a condensing coil, surface of which is continually wetted on the outside by recirculating water. Ambient air is pulled over the coil, causing a portion of the recirculating water to evaporate. The evaporation significantly enhances a rate of heat transfer from the coil and improves the condensation inside it. Figure 6 illustrates the typical layouts of the evaporative cooling condensers. Evaporators are heat exchangers that enable a phase transition from the liquid state to vaporous or gaseous state of the colder fluid at the expense of thermal energy transferred from the hotter fluid. When the process of a phase change takes place at

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Attemperator

Steam Drum

Primary Reheater

Secondary Superheater Platen Superheater

Final Reheater

Coal Silos

Primary Superheater

Economizer SCR HydroJet Boiler Cleaning System

NOX Ports

Furnace

AireJet Burners Coal Feeders

Air Heater

B&W TM Roll Wheel Pulverizers Dry Bottom Ash System

Forced Draft Fan

Primary Air Fans

Forced Draft Fan

Fig. 4 Typical Babcock&Wilcox Carolina-type radiant boiler (Babcock&Wilcox – http://www. babcock.com/products/radiant-boiler)

Fig. 5 Flexible CFB boiler overview (GE Power – https://www.gepower.com/steam/products/ boilers/circulating-fluidised-bed.html)

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Introduction and Classification of Heat Transfer Equipment

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the liquid interface, it is called evaporation. In case of a phase change taking place within the volume of the bulk liquid – with the vapor bubble formation – it is called boiling. Both a homogeneous liquid and a mixture of liquid components could be handled by evaporators. Evaporators usually carry different names, depending on the nature of the processes and related applications such as, but not limited to, • • • • • • • • • • • •

Falling or raising film evaporators Forced circulation evaporators Plate or compact evaporators Thermal and mechanical vapor recompression Flash evaporators Short or long tube verticals Multiple effect evaporators Thin film evaporators Vacuum evaporators Thermally accelerated falling film evaporators Low temperature evaporators Other

Evaporators are used in a wide spectrum of applications such as for manufacturing pharmaceuticals, food processing, pulp and paper, petrochemicals, wastewater treatment, and various materials processing. Evaporation is a process used to concentrate a solution of a nonvolatile solute and a volatile solvent to produce a concentrated solution or to extract a particular component from it. Thus, the evaporator design is essential for providing parameters that are required by the process or the dedicated application. Evaporators may have different designs including thinfilms, plate-and-frames, jacketed tanks, and others; however, a tubular design dominates in the field nowadays. Figure 7 illustrates several typical evaporators available on the market.

4.3

Shell-and-Tube Heat Exchangers

Shell-and-tube heat exchangers are the most popular type and dominating group of heat exchangers. There are many technical publications, handbooks, and operating manuals publicly available in both print and online. ® One of the most demanded shell-and-tube type exchanger is HELIXCHANGER that uses the industry proven, enhanced heat transfer technology to provide solutions for common issues such as shell-side fouling, high pressure drop, and vibration ® described in Master et al. (2003). In the HELIXCHANGER heat exchanger, quadrant shaped baffle plates are placed at an angle to the tube axis in a sequential arrangement to create a helical flow pattern as shown in Fig. 8 below. Whether a need is to extend the operating period between cleaning campaigns of crude preheat exchangers, to reduce the capital cost in a petrochemical plant or to improve energy efficiency of feed/effluent towers, the end-user traditionally

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Y. Chudnovsky and D. P. Sekulic

Fig. 6 Evaporative condensers: (a) and (b) – Eco ATWB closed circuit cooler by Evapco; (c) – Cube DTC condenser by SPX; (d) – Recold MC condenser by SPX; (SPX – http://www.spx.com and Evapco – www.evapco.com)

selects an advanced shell-and-tube heat exchanger technology as the most costeffective solution.

4.4

Coolers, Refrigerators, and Chillers

Coolers, refrigerators, and chillers belong to a group of systems that provide cooling and freezing services to residential, commercial, and industrial needs. A great variety of applications may be listed. For example, process cooling intended to reject certain amount of heat from a chemical process, dough chilling prior to a baking process, cryogenic biomedical treatments, and so forth. There are also many other processes and services that may require cooling, refrigerating, and chilling such as space

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Fig. 7 Evaporative coils and condensers: (a) – HVAC condensing coils by Diversified Heat Transfer; (b) – SST-B Series evaporator by Evapco; (c) – Falling-film evaporating system by B&P Engineering (Diversified Heat Transfer, Evapco and B&P Engineering)

cooling during the summer time, food refrigerating to extend their storage life, beverages chilling, ice making, etc. Most of those systems employ the heat exchangers as a core of their designs. For a conventional refrigeration cycle, the system uses at least two heat exchangers – evaporator and condenser. However, there are many sophisticated and extremely productive refrigeration solutions for variety of applications that can be found in the open domain or requested from the OEMs. Figure 9 illustrates a couple of commercially available refrigeration systems.

4.5

Radiators

Radiators belong to a type of heat exchangers serving as components of engineering systems designed to use circulating air to cool the fluid inside the tubes or channels of the exchanger. Using air (under forced or natural convection) as a cooling medium significantly expands the application range of radiators. Usually radiators are simple in design that often makes their maintenance quite easy. Radiators are widely employed in automobiles for removing thermal energy in the form of heat from an engine coolant, in residential space heating, and in electronic cooling systems as heat sinks. Light and/or nonexpensive materials are frequently used for radiators.

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Y. Chudnovsky and D. P. Sekulic

®

Fig. 8 HELIXCHANGER internal and external views (https://www.cbi.com/What-We-Do/Tech nology/Heat-Transfer-Equipment/HELIXCHANGER-Heat-Exchanger)

Traditionally, automobile radiators were made using copper and brass heat transfer surface cores. But cost-effective and lower weight aluminum alloy cores, using plastic header tanks, are replacing them. The major reason for this change is that aluminum radiators have smaller mass for the same load, hence are more cost effective than the copper-brass ones. Residential space heating radiators are usually made of aluminum, cast iron, and in some applications bimetallic. Figure 10 illustrates several typical radiators.

4.6

Furnaces and Ovens

Furnaces and ovens are the type of heat transfer equipment intended for heating of a product directly or indirectly by an electricity source (induction heating, electrothermal fluidized bed, etc.) or using a fossil fuel (gaseous or liquid) combustion products. The heat is supplied into the furnace or oven cavity by an electrically

31

Introduction and Classification of Heat Transfer Equipment

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Condenser

a

Liquid Separator Economizer

Evaporator

Quantum TMLX Compressor

b

POWER PANEL

HIGH-TEMP CUTOUT (HT1)

PURGE TANK

HI-PRESSURE CUTOUT SWITCH (HP1)

RUPTURE GENERATOR OUTLET DISK BOX

CONDENSER GENERATOR

EVAPORATOR

SOLUTION RETURN FROM GENERATOR

AUTO DECRYSTALLIZATION PIPE (ADC)

OPTIVIEW CONTROL PANEL

REFRIGERANT PUMP

REFRIGERANT LEVEL SWITCH (1F)

REFRIGERANT PUMP CUTOUT SWITCH (3F)

REFRIGERANT OUTLET BOX

GAS SEPARATOR

SOLUTION PUMP

PURGE PUMP

OIL TRAP

SOLUTION SIGHT GLASS

ABSORBER

Fig. 9 Commercial refrigeration systems: (a) – packaged PowerPac Ammonia Chiller by Frick; (b) – single stage YIA Absorption Chiller by York (Frick and York, both Johnston Controls family – http://www.johnsoncontrols.com/buildings/our-brands/frick)

heated element or by fuel-fired burners. Residential and commercial furnaces provide for space heating, while residential and commercial ovens are traditionally used for heat processing (cooking) purposes and food service applications. The industrial furnaces are widely used for metal heat treatment (under vacuum and with protective atmosphere), glass melting, steelmaking, chemical calcining, and construction materials production. Industrial ovens are employed in food processing (particularly in

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Y. Chudnovsky and D. P. Sekulic

Fig. 10 Radiators: (a) – automobile radiators (fin-tube and plate-fin types); (b) – CPU cooling radiator; (c) – residential space heating radiators (cast iron); (d) – residential space heating radiators (aluminum or bimetallic) (open web-domain)

wholesale baking applications), composite materials processing, etc. To improve efficiency of the fuel fired furnaces and ovens, the waste heat recovery techniques are usually employed by using the heat exchangers of recuperative or regenerative type for the purpose of combustion air preheating or ancillary heating services that are available at the site. A wide spectrum of cost-effective heating solutions with or without waste heat recovery systems can be found in popular handbooks of the major combustion equipment manufacturers such as North American Combustion Handbook (1997) or the John Zink Hamworthy Combustion Handbook (2013). Figure 11 illustrates commercial and industrial furnaces (a, b) and a wholesale bread baking ovens (c,d).

4.7

Dryers

Dryers constitute a heat transfer equipment that is intended to remove certain amount of liquid (usually water) from a product being dried in a process. The majority of residential and commercial clothing dryers feature a rotating, perforated drum, and a fan or an air blower that supplies the hot air heated by electricity or a gas-fired burner. Those dryers may be considered as a type of direct heat exchangers that facilitate water removal by evaporation from wet products (clothes or fabrics) by

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Introduction and Classification of Heat Transfer Equipment

1311

Fig. 11 Furnaces and ovens: (a) – commercial heat treating furnaces; (b) – interior of industrial gas-fired re-heat furnace; (c) – overview of the gas-fired wholesale tunnel bread baking oven; (d) – pita bread baking oven interior – (open web-domain)

blowing through the hot air. The industrial drying systems’ design and configuration strictly depends on the drying application and product to be dried. The dryers could be designed as rotary drums or kilns, as batch type or tunnels, etc. For example, the paper drying employs rotary cans that are heated by the condensing steam while the paper web is moving outside of the can. The food drying applications usually utilize the tunnel-conveyer-type of dryers. The product moves along the tunnel, traveling while position on the perforated conveyer. The hot air is supplied along the dryer in zone-by-zone manner, allowing for the drying process temperature control. The paint drying occurs in the chambers that are protected from the process hazards. Figure 12 illustrates a few drying applications.

4.8

Heat Pumps

Heat pumps are attractive due to high efficiency of their operation. For example, for climates with moderate heating and cooling needs, heat pumps offer an energyefficient alternative to furnaces and air conditioners. Similar to a refrigerator, heat pumps use electricity to transfer heat from a colder space to a warmer space. During the heating season, heat pumps transfer thermal energy in form of heat from cool outdoors into a warm house. During a cooling season, heat pumps move heat from a

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Y. Chudnovsky and D. P. Sekulic

d

Gearwheel

Transmission Devices

Riding Wheel

Riding Roller

Raising Plate

Dryer Shell

Fig. 12 Drying applications: (a) – commercial laundry dryers; (b) – steam heated belt tunnel dryer; (c) – drying section of the papermaking line; (d) – rotary dryer (kiln) – (open web-domain)

cool house into the warm outdoors. Because they transfer heat rather than generate thermal energy, heat pumps can provide an equivalent space conditioning at as little as one quarter of the cost of operating conventional heating or cooling appliances. There are three types of heat pumps on the marketplace: air-to-air, water source, and geothermal. Hence, heat pumps transfer heat from air, water, or ground outside, say, a home, and direct it for use inside. High-efficiency heat pumps also dehumidify moist air better than the standard central-air conditioner systems, resulting in less energy usage and more cooling comfort in summer months. A relatively new type of a heat pump for residential systems is the absorption heat pump, also called a gas-fired heat pump. Absorption heat pumps use transferred heat as their energy source and can be driven with a wide variety of heat sources. As opposed to the residential and commercial applications, industrial heat pumps could serve as an active heat transfer and waste heat recovery equipment. As such, they allow the temperature of a waste heat stream to be increased to a higher, more useful temperature, therefore improve the energy efficiency of the entire technological process. The work required to drive a heat pump depends on how much the temperature of the waste heat is increased; in contrast, a steam turbine produces increasing amounts of work as the pressure range over which it operates increases. More specific details on heat pumps and absorption chillers can be found in Herold, Radermacher, and Klein (1996). Figure 13 below illustrates the commercially available heat pumps. The subchapters below discuss the specific details of the heat exchanger fundamentals, design and operations with respect to different state and properties of working media, as well as process efficiency and overall sustainability.

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Introduction and Classification of Heat Transfer Equipment

1313

Fig. 13 Heat pumps: (a) – residential heat pump for heating and air conditioning by American Standard; (b) – industrial heat pump Unitop by Friotherm (http://www.friotherm.com/en/products/ unitop/)

5

Cross-References

▶ Applications of Flow-Induced Vibration in Porous Media ▶ Boiling and Two-Phase Flow in Narrow Channels ▶ Boiling on Enhanced Surfaces ▶ Compact Heat Exchangers ▶ Design of Optical and Radiative Properties of Surfaces ▶ Design of Thermal Systems ▶ Electrohydrodynamically Augmented Internal Forced Convection ▶ Energy Efficiency and Advanced Heat Recovery Technologies ▶ Enhancement of Convective Heat Transfer ▶ Evaporative Heat Exchangers ▶ Film and Dropwise Condensation ▶ Flow Boiling in Tubes ▶ Full-Coverage Effusion Cooling in External Forced Convection: Sparse and Dense Hole Arrays ▶ Fundamental Equations for Two-Phase Flow in Tubes ▶ Heat Exchanger Fundamentals: Analysis and Theory of Design ▶ Heat Exchangers Fouling, Cleaning, and Maintenance ▶ Heat Pipes and Thermosyphons ▶ Heat Transfer in Rotating Flows ▶ Heat Transfer Media and Their Properties ▶ Introduction and Classification of Heat Transfer Equipment ▶ Process Intensification ▶ Radiative Transfer in Combustion Systems ▶ Single-Phase Heat Exchangers ▶ Turbulence Effects on Convective Heat Transfer ▶ Two-Phase Heat Exchangers

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References Baukal CE (2013) The John Zink Hamworthy combustion handbook, 2nd edn , three-volume set. CRC Press/Taylor & Francois Group, Boca Raton, 1184 p Cleaver Brooks (2011) Boiler book (online version – http://www.cleaver-brooks.com/ReferenceCenter/Resource-Library/Boiler-Book/Index.aspx), Thomasville, Georgia, 1140p Herold KE, Radermacher R, Klein SA (1996) Absorption Chillers and Heat Pumps. CRC Press/ Taylor & Francois Group, Boca Raton, 356p Hewitt GF (ed) (2008) Heat exchanger design handbook, five-volume set. Begell House Inc, Danbury http://www.babcock.com/products/radiant-boiler. Accessed 27 May 2017 http://www.cbi.com/What-We-Do/Technology/Heat-Transfer-Equipment/HELIXCHANGER% c2%ae-Heat-Exchanger. Accessed 27 May 2017 http://www.friotherm.com/en/products/unitop/. Accessed 27 May 2017 https://www.gepower.com/steam/products/boilers/circulating-fluidised-bed.html. Accessed 27 May 2017 http://www.johnsoncontrols.com/buildings/our-brands/frick. Accessed 27 May 2017 Kakaç S, Liu H, Pramuanjaroenkij A (2012) Heat exchangers: selection, rating and thermal design, 3rd edn. CRC Press/Taylor & Francois Group, Boca Raton, 631 p Master BI, Chunangady KS, Pushpanathanz V (2003) Fouling mitigation using helixchanger heat exchangers. In: Watkinson P, Muller-Steinhagen H, Malayer MR (eds) Proceedings of ECI conference on heat exchanger fouling and cleaning: fundamentals and applications. Curran Associates, Red Hook, pp. 1–7 North American Manufacturing Company (1997) North American combustion handbook, 3rd edn, two-volume set, North American Manufacturing Company, Ohio, 982 Shah, R. K. (1981), Classification of heat exchangers, in Heat exchangers: thermal-hydraulic fundamentals and design by S. Kakac¸ A. E. Bergles, and F. Mayinger, eds., Hemisphere Publishing, Washington, DC, pp.9–46 Shah RK, Sekulic DP (2003) Fundamentals of heat exchanger design. Wiley, 976 pp Tome GL (ed) (2015) Steam: its generation and use, 42nd edn. The Babcock & Wilcox Company, Charlotte, 1200 p www.cleaverbrooks.com. Accessed 27 May 2017 www.evapco.com. Accessed 27 May 2017 www.hurstboiler.com. Accessed 27 May 2017 www.spx.com. Accessed 27 May 2017

Heat Exchanger Fundamentals: Analysis and Theory of Design

32

Ahmad Fakheri

Contents 1 Introduction: General Heat Exchanger Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Log Mean Temperature Difference Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 The Effectiveness: NTU Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 The Heat Exchanger Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 The Sizing Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 The Rating Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Heat Exchangers Networks (HENs) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Heat Exchangers in Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Multi-pass Heat Exchangers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Compact Heat Exchangers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Pressure Drop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Microchannel Heat Exchangers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Cross-References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1316 1320 1325 1328 1332 1332 1335 1336 1338 1342 1345 1350 1351 1352

Abstract

This chapter provides an overview of how different heat exchanger types, problems, and networks are analyzed. Heat exchangers are categorized by shape, flow arrangement, area to volume ratio, and channel size. The problem type depends on what information is available and what is sought. The heat exchanger networks can be arranged in numerous ways. For analyzing heat exchangers, three

A. Fakheri (*) Department of Mechanical Engineering, Bradley University, Peoria, IL, USA e-mail: [email protected] # Springer International Publishing AG, part of Springer Nature 2018 F. A. Kulacki (ed.), Handbook of Thermal Science and Engineering, https://doi.org/10.1007/978-3-319-26695-4_19

1315

1316

A. Fakheri

methods, LMTD correction factor, ε-NTU, and heat exchanger efficiency, are reviewed in this chapter. In their basic form, all three methods are based on the first law of thermodynamics, using average fluid properties and transversally averaged temperature, considering only axial temperature change. Since all three methods use the same equations and are based on the same assumptions, they differ only in the way that the results are presented. Historically, the choice of the method used depended on the type of problem under consideration; however using computers and equations solvers, each of the methods can be used to solve any heat exchanger problem, as long as the expressions for LMTD correction factors, ε-NTU relations, or the heat exchanger efficiency expressions are available. The chapter also provides an overview of the compact heat exchangers as well as microchannel heat exchangers.

1

Introduction: General Heat Exchanger Analysis

The general heat exchanger analysis involves examination of the average behavior of fluids. Perhaps the simplest heat exchanger is a double-pipe heat exchanger where two concentric tubes carry the two, the hot and cold fluids. The flow arrangement in a double-pipe heat exchanger can be parallel flow, where both fluids enter the heat exchanger at the same end and move in the same direction or counterflow, where the hot and cold fluids enter the heat exchanger at opposite ends and flow in opposite directions. The general heat transfer analysis for heat exchangers is demonstrated first by considering both counterflow and parallel flow heat exchangers shown in Figs. 1 and 2. The derivations are very similar and are presented in parallel with the differences indicated. The hot fluid enters at T1 and exits at T2, and the cold fluid enters at t1 and leaves at t2. The temperature distribution for both types are shown in Figs. 3 and 4. Consider a differential control volume of length Δx at some distance x from the cold fluid inlet which provides area dA for the transfer of heat. At this location, the hot fluid temperature is T, and the cold fluid is at temperature t. Assuming the hot fluid is Fig. 1 (a) Counterflow heat exchanger. (b) Parallel flow heat exchanger

a

T2

T

T1

t1

t

t2

b T1

T

t1

t

T t2

32

Heat Exchanger Fundamentals: Analysis and Theory of Design

1317

a T1

Hot

t2 cold

T2

t1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

b T1 Hot

T2 t2

cold

t1 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Fig. 2 (a) Temperature variation of the hot and cold fluids in a counterflow heat exchanger. (b) Temperature variation of the hot and cold fluids in a parallel flow heat exchanger

Fig. 3 Shell and tube heat exchanger, 1–2 TEMA E

flowing in the outer pipe, the heat transferred from the hot fluid to the cold fluid (δq) is first transferred by convection in the hot fluid to the outer surface of the inner pipe, then transferred through the pipe wall by conduction and then transferred to the cold fluid by convection, and is

1318

A. Fakheri

1 0.2

0.9

0.1

0.4 0.8

0.8 R=2.5

F 0.7

2 1.5

1.0

0.4

0.5 P

0.6 0.5 0.4

0

0.1

0.2

0.3

0.6

0.7

0.8

0.9

1

Fig. 4 F correction factor for 1 shell 2 M tube passes

δq ¼ UdAðT  tÞ

(1)

Integrating this equation along the heat transfer area and assuming the overall heat transfer coefficient U to be constant, a frequently made assumption, then Eq. (1) becomes q ¼ UAðT  tÞ

(2)

where ðT  tÞ ¼

1 A

ð ðT  tÞdA

(3)

A

is the driving temperature difference in the heat exchanger and is known as the effective or average temperature difference in the heat exchanger. As will be seen, the different methods for analysis of heat exchangers differ essentially on how this temperature difference is calculated. In the analysis below, we assume the heat loss to the ambient is zero, and therefore the heat transfer from the hot to cold fluid, δq, is positive and equal to the amount of heat that the hot fluid loses or the amount of heat that the cold fluid gains. Note that by definition

32

Heat Exchanger Fundamentals: Analysis and Theory of Design

1319

dT ¼ T ðx þ ΔxÞ  T ðxÞ

(4)

dt ¼ tðx þ ΔxÞ  tðxÞ

(5)

and the heat capacity of a fluid is denoted _ p C ¼ mc

(6)

Applying the first law to the control volumes shown in Figs. 1 and 2, the analysis for counterflow and parallel flow heat exchangers is shown side by side in the table below, since they are very similar. Counter Flow (dT > 0 & dt > 0)

Parallel Flow (dT < 0 & dt > 0)

δq ¼ Ch dT

δq ¼ Ch dT

(7)

δq ¼ Cc dt

δq ¼ Cc dt

(8)

Solving for the temperature differentials of the hot and cold fluid from the above two equations, δq Ch

dT ¼

dt ¼

dT ¼ 

δq Cc

dt ¼

δq Ch

δq Cc

(9) (10)

Subtracting Eq. (10) from Eq. (9) 

1 1  dðT  tÞ ¼ δq Ch Cc



  1 1 dðT  tÞ ¼ δq   Ch Cc

(11)

Substituting for δq from Eq. (1) into Eq. (11)  d ðT  tÞ ¼ UdAðT  tÞ

1 1  Ch Cc



  1 1 dðT  tÞ ¼ UdAðT  tÞ   Ch Cc

(12)

Integrating Eqs. (7), (8), (11), and (12) from the inlet (x = 0) to the outlet, (x = L) q ¼ C h ðT L  T 0 Þ

q ¼ Ch ðT L  T 0 Þ

q ¼ Cc ðtL  t0 Þ q ¼ Cc ðtL  t0 Þ  ðT  tÞL  ðT  tÞ0 ¼ q

1 1  Ch Cc



(13) (14)

  1 1 ðT  tÞL  ðT  tÞ0 ¼ q   (15) Ch Cc

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A. Fakheri

  ðT  tÞL 1 1 ln ¼ UA  Ch Cc ðT  t Þ0

  ðT  tÞL 1 1 ¼ UA   ln Ch Cc ðT  t Þ0

(16)

Equations (13), (14), (15), and (16) will be rearranged in three different ways to arrive at the three primary methods for analyzing heat exchangers.

2

Log Mean Temperature Difference Approach

The log mean temperature difference (LMTD) is the earliest method for analyzing heat exchangers. It provides a convenient method for heat exchanger selection (sizing problems) in which the temperatures are known or can be determined and the size of the heat exchanger is required. Dividing Eq. (15) by (16) and solving for the rate of heat transfer q ¼ UA

ðΔT ÞL  ðΔT Þ0 ¼ UA LMTD ðΔT ÞL ln ðΔT Þ0

(17)

where (ΔT )L is the temperature difference between the hot and cold fluids at one end and (ΔT )0 is the temperature difference at the opposite end of the heat exchanger. Comparing this equation with Eq. (2) shows that for counterflow and parallel flow heat exchangers, the effective or mean temperature difference is the LMTD. For a counterflow heat exchanger, shown in Fig. 3, the LMTD is LMTD ¼

ðT 2  t1 Þ  ðT 1  t2 Þ ðT 2  t 1 Þ ln ðT 1  t 2 Þ

(18)

and for a parallel flow heat exchanger, shown in Fig. 4, the LMTD is LMTD ¼

ðT 1  t1 Þ  ðT 2  t2 Þ ðT 1  t 1 Þ ln ðT 2  t 2 Þ

(19)

Heat transfer from a fluid flowing in a pipe with a constant surface temperature (isothermal pipe) is also found from the same equation. The appearance of the LMTD as the driving temperature difference for three different cases and its simplicity led to its adoption for other types of heat exchangers by the introduction of a correction factor, F. It was assumed that the heat transfer for all heat exchangers is q ¼ U A F LMTD

(20)

32

Heat Exchanger Fundamentals: Analysis and Theory of Design

1321

where the LMTD used is that for a counterflow heat exchanger having the same inlet and exit fluid temperatures as the heat exchanger being considered. Comparing this with Eq. (2) ðT  tÞ ¼ F LMTD

(21)

indicates that the relevant driving temperature potential in the heat exchanger is less than that for the counterflow case and is expressed as the product of F and the LMTD. Underwood (1934) and Bowman et al. (1940) compiled the available data and presented a series of equations and charts for the determination of F for a variety of heat exchangers by choosing to express F in terms of two nondimensional variables: R¼

T1  T2 t2  t1

(22)

P¼

t2  t1 T 1  t1

(23)

and

Note that R and P can also be defined as R¼

t2  t1 T1  T2

(24)

P¼

T1  T2 T 1  t1

(25)

and

Although Underwood (1934) and Bowman et al. (1940) do not provide a rationale for the selection of R and P, the two variables were later interpreted as the capacity ratio and a measure of heat exchanger effectiveness, respectively (Taborek 1990). For a single-shell double tube pass heat exchanger, also known as 1–2 TEMA E, shown in Fig. 3, Underwood (1934) provided the following expression for the correction factor: pffiffiffiffiffiffiffiffiffiffiffiffiffiffi R2 þ 1 F¼ R1

ln

1P 1  PR

  pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2  1  R þ R2 þ 1 P ln   pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2  1  R  R2 þ 1 P

(26)

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A. Fakheri

1

0.9

F

0.8

0.7

0.6

0.5

0

0.1

0.2

0.3

0.4

0.5 W

0.6

0.7

0.8

0.9

1

Fig. 5 F correction factor for N shell 2 M tube passes (M  N)

This expression is also plotted in Fig. 5. Underwood (1934) also showed that the validity of Eq. (26) is independent of the flow direction in the first tube pass, i.e., being parallel or counter to the general flow direction of the fluid on the shell side. Bowman et al. (1940) analyzed the four-tube pass arrangement and concluded that the correction factor was not significantly different than that given for the 1–2 case, and therefore the same equation is used for any number of even tube passes. Note that for a shell and tube heat exchanger, the value of F is independent of whether the hot or cold fluids are flowing in the tube or shell. The decision as to which fluid should be on the tube or shell side depends on a number of factors. In general, if the fluid is corrosive, or at very high pressure, or has significant fouling potential, it should be on the tube side (Wolverine Tube Heat Transfer Data Book). Example 1

Water at the rate of 3 kg/s is to be heated from 35  C to 90  C by 7 kg/s of oil cp = 1500 J/kg K supplied at 150  C using shell and tube heat exchanger having the overall heat transfer coefficient of 120 W/m2 K. Determine the total heat transfer area. The heat transfer is given by q ¼ Cc ðt2  t1 Þ ¼ 12558:0ð90  35Þ ¼ 690690:0 q ¼ Cc ðt1  t1 Þ ¼ Ch ðT 1  T 2 Þ

32

Heat Exchanger Fundamentals: Analysis and Theory of Design

T2 ¼ T1 

1323

Cc ðt2  t1 Þ ¼ 84:2 Ch

LMTD from Eq. (18): LMTD ¼

ðT 2  t1 Þ  ðT 1  t2 Þ ð84:2  35:0Þ  ð150:0  90:0Þ ¼ 54:43 ¼ ðT 2  t1 Þ ð84:2  35:0Þ ln ln ð150:0  90:0Þ ðT 1  t2 Þ

Find P and R from (Eqs. 22 and 23): T 1  T 2 150:0  84:2 ¼ 1:20 ¼ 90:0  35:0 t2  t1 t2  t1 90:0  35:0 P¼ ¼ 0:48 . ¼ T 1  t1 150:0  35:0 1P pffiffiffiffiffiffiffiffi ln R2 þ1   1  PR F from Eq. (26) F ¼ R1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 0:74 2  1  R þ R2 þ 1 P ln   pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2  1  R  R2 þ 1 P and UA from q = UA F LMTD. R¼

and

The solution is shown in the Table 1. As can be seen in Fig. 7 for each capacity ratio R, there is a maximum value of P at which F approaches zero. Therefore if a large temperature change for a fluid is needed, it may not always be possible to accomplish the objective in one heat exchanger. In situations like this, heat exchangers are connected in series, resulting, for example, in N shell 2 N tube TEMA E, heat exchangers. For N identical multishell and tube heat exchangers connected in series, with each shell having 2 M tubes (M an integer N), the LMTD correction factor for the series can be combined into a single general expression (Fakheri 2003a): FN, 2NM ¼

SlnW 1 þ W  S þ SW ln 1 þ W þ S  SW

(27)

where  W¼

1  PR 1P

1=N

(28)

and pffiffiffiffiffiffiffiffiffiffiffiffiffiffi R2 þ 1 S¼ R1

(29)

1324 Table 1 Example 1 solutions

A. Fakheri

t1 t2 T1 T2 t T mw cw mo co Ch Cc Cr Q P R LMTD F UA U A

35.0 90.0 150.0 84.2 62.5 117.1 3.0 4186.0 7.0 1500.0 10,500.0 12,558.0 0.8 690,690.00 0.48 1.20 54.43 0.74 17,051.63 120.00 142.10



C C  C  C  C  C Kg/s J/kg K Kg/s J/kg K J/Ks J/Ks – W – – K – W/K W/m2 K W/m2 K 

and P and R are based on the inlet and exit temperatures of the overall system min½ðT 1  T 2 Þ, ðt2  t1 Þ T 1  t1

(30)

max½ðT 1  T 2 Þ, ðt2  t1 Þ 1 ¼ min½ðT 1  T 2 Þ, ðt2  t1 Þ Cr

(31)

P¼ and R¼

For a balanced flow heat exchanger, R = 1. N  NP N  NP þ P pffiffiffi 1  W 1 F¼ 2 W 1 W þ pffiffiffi 1W 2 ln W 1  pffiffiffi 1W 2

W¼

(32)

32

Heat Exchanger Fundamentals: Analysis and Theory of Design

1325

This expression combines different expressions given for multi-shell and tube heat exchangers by Bowman et al. (1940) into a single general equation and is plotted in Fig. 5. For a cross flow heat exchanger with one mixed and one unmixed fluid 1P ln  1  PR  F¼ 1 ð1  RÞln 1 þ lnð1  PRÞ R

(33)

and for a cross flow heat exchanger with both fluids mixed, an implicit formula for the calculation of F is 1P  1P ðiþjÞ 1 1 ln 1PR ði þ jÞ! 1  PR X X ð1ÞðiþjÞ ðFÞðiþjÞ ðRÞj F¼ i!ði þ 1Þ!j!ðj þ 1Þ! R1 PR  P i¼0 j¼0 ln

(34)

which must be solved iteratively for a given values of P and R. The LMTD method is convenient to use in heat exchanger analysis when the inlet and the outlet temperatures of the hot and cold fluids are either known or can be determined from an energy balance. Once the mass flow rates and the overall heat transfer coefficient are available, the heat transfer surface area of the heat exchanger can be determined. Charts are available in many heat transfer texts from which LMTD correction factors for a variety of heat exchangers can be calculated. As can be seen from the results for shell and tube heat exchangers, the LMTD correction factor F is a strong function of P and R, which reduces the accuracy of reading the charts, particularly in the steep part of the curves (F < 0.8). The LMTD correction factor charts and formulas can be found in a number of sources including Hewitt (1990). A more accurate and convenient approach is using heat exchanger software or online tools (Chemical Engineering Calculations) that allows quick calculation of F for different shell and tube and cross flow heat exchangers.

3

The Effectiveness: NTU Method

The effectiveness-NTU method (NTU – number of transfer units) is used for cases where fluid mass flow rates, the inlet temperatures, and the type and size of the heat exchanger are specified, and the amount of heat transfer and exit temperatures are to be determined. This is called a rating problem and since the exit temperatures are unknown, the use of LMTD approach would require an iterative approach. To avoid this procedure, Kays and London (1984) proposed the effectiveness-NTU method.

1326

A. Fakheri

For any heat exchanger, absent energy loss to the surroundings: Ch ðT 1  T 2 Þ ¼ Cc ðt2  t1 Þ

(35)

This equation also shows that the fluid with higher capacity will experience the smaller temperature change. The second law dictates that the maximum temperature change that a fluid may experience cannot exceed the difference between the inlet temperatures of hot and cold fluids, otherwise, either the cold fluid will exit the heat exchanger at a temperature higher than the inlet temperature of the hot fluid or the hot fluid will exit the heat exchanger at a temperature lower than the inlet temperature of the cold fluid, both of which violate the second law of thermodynamics. If (T1  t1) is experienced by a fluid in a heat exchanger, it has to be the fluid with smaller capacity; otherwise the fluid with lower capacity will experience even larger temperature change, which will then violate the second law. Therefore, the maximum amount of heat that can be possibly transferred in any heat exchanger is qmax ¼ Cmin ðT 1  t1 Þ

(36)

Note this maximum rate of heat transfer. The actual heat transfer rate in a heat exchanger will be less than this value; therefore the heat exchanger effectiveness can be defined as e¼

q qmax

¼

q : Cmin ðT 1  t1 Þ

(37)

Substituting for heat transfer in terms of heat capacity and temperature difference the effectiveness can be written e¼

ðΔT Þfluid with min C ðT 1  t 1 Þ

¼

ðΔT Þfluidwith fluid max C Cr ðT 1  t1 Þ

(38)

where Cr ¼

Cmin Cmax

(39)

is the capacity ratio. From Eq. (38) effectiveness can be viewed as the temperature change experienced by the fluid having the smaller capacity, which will experience the larger temperature change (ΔTmax), divided by the maximum temperature difference, or it is the ratio of the actual maximum temperature change and the theoretically possible maximum temperature change. It is also equal to the temperature change experienced by the fluid having the larger capacity (ΔTmin), divided by the maximum temperature difference times the capacity ratio. Therefore, if the effectiveness is known, the temperature changes, and thus the exit temperatures of both fluids can be quickly calculated.

32

Heat Exchanger Fundamentals: Analysis and Theory of Design

1327

Equation (37) can be rearranged into q ¼ UA

e ðT 1  t1 Þ NTU

(40)

Again comparing this with Eq. (2), we recognize that in this approach the mean effective temperature difference in the heat exchanger (relevant driving temperature potential) is expressed ðT  tÞ ¼

e ðT 1  t1 Þ NTU

(41)

which is the product of effectiveness divided by NTU times the maximum temperature difference, where the number of transfer units (NTU) is defined as NTU ¼

UA Cmin

(42)

The effectiveness of a heat exchanger depends on the geometry of the heat exchanger as well as the flow arrangement. Therefore, different types of heat exchangers have different effectiveness relations. Below, we derive the expression for the effectiveness of the counterflow heat exchanger. Rearranging Eq. (16) h i UA C1 C1c T 1  t2 h (43) ¼e T 2  t1 Adding and subtracting t1 to the numerator and T1 to the denominator and dividing numerator and denominator by T1t1 t2  t1 h i UA C1 C1c T 1  t1 h ¼e T1  T2 1 T 1  t1 1

(44)

Substituting for temperature differences from Eqs. (13) and (14) in terms of heat transfer Cmin q h i 1 1 UA  Cc Cmin ðT 1  t1 Þ Ch Cc ¼e Cmin q 1 Ch Cmin ðT 1  t1 Þ 1

(45)

Regardless of whether Cc = Cmin or Ch = Cmin, it is easy to show that Eq. (45) simplifies to e¼

1  eNTU½1Cr  1  Cr eNTU½1Cr 

(46)

1328

A. Fakheri

Table 2 Heat exchanger effectiveness expressions Parallel-flow

ð1þCr Þ e ¼ 1exp½NTU ð1þCr Þ

Counterflow Shell and tube

1exp½NTU ð1Cr Þ e ¼ 1C r exp½NTU ð1Cr Þ

   1=2 1þexp NTUð1þC2r Þ1=2 1

e ¼ 2 1 þ Cr þ 1 þ C2r 1=2 1exp NTU ð1þC2r Þ

Balanced counterflow

NTU e ¼ 1þNTU

Single stream Cr = 0 Mixed

e = 1  eNTU h i e ¼ 1  exp  C1r ½1  exp½Cr NTU 

Unmixed

e ¼ C1r ½1  exp½Cr ½1  expðNTU Þ

Both mixed

NTU Cr NTU þ 1exp ½Cr NTU   1 h

i e ¼ 1  exp 1Cr NTU :22 exp Cr NTU :78  1 e ¼

Both unmixed

NTU 1exp½NTU 

Using a similar approach, but generally significantly more involved, the relation between effectiveness and NTU for many other heat exchangers has been developed, and some of the more widely used expressions are given in Table 2. For all heat exchangers, effectiveness ranges from 0 to 1. It increases rapidly with NTU for small values (up to about NTU =1.5) but rather slowly for NTU > 3. Example 2

In Example 1, determine the exit temperature of water, if the oil flow rate drops by 20%. In this case, water at the rate of 3 kg/s enters the shell and tube heat exchanger at 35  C and is heated by oil at a rate of 5.6 kg/s (cp = 1500 J/kg K) supplied at 150  C. The overall heat transfer coefficient is 120 W/m2 K, and the heat exchanger area as calculated in Example 1 is 142.13 m2. The Example 2 solution is given in Table 3.

4

The Heat Exchanger Efficiency

A third method for analyzing heat exchangers is using the concept of heat exchanger efficiency (Fakheri 2002, 2003b, c, 2006, 2007, 2008, 2010). The heat exchanger efficiency is defined as the ratio of the actual rate of heat transfer in the heat exchanger, q, and the optimal rate of heat transfer, qopt, η¼

q q   ¼ qopt U A Tt

(47)

The optimum (maximum) rate of the heat transfer is the product of UA of the heat exchanger under consideration and the arithmetic mean temperature difference (AMTD) in the heat exchanger:

32

Heat Exchanger Fundamentals: Analysis and Theory of Design

1329

Table 3 Example 2 solutions From Example 1 t1 T1 mw cw mo initially co U A If the oil flow rate drops by 20% mo Ch Cc Cr NTU e qmax q t2 T2 Verify the area using LMTD P R LMTD F UA A

35 150 3.0 4186.0 7 1500 120 142.13



C C Kg/s J/kg K Kg/s J/kg K W/m2 K m2 

5.60 8400.00 12,558.00 0.67 2.03 0.64 966,000.00 623,014.26 84.61 75.83

Kg/s J/Ks J/Ks – – – W W  C  C

0.43 1.50 52.15 0.70 17,055.10 142.13

– –  C – W/K m2

AMTD ¼ T  t ¼

T 1 þ T 2 t1 þ t2  2 2

(48)

which is the difference between the average temperatures of hot and cold fluids. Note that the optimum rate of heat transfer is different than the maximum rate of heat transfer used to define heat exchanger effectiveness. The optimum heat transfer rate takes place in a balanced counterflow heat exchanger (Fakheri 2003b). The rate of heat transfer in any heat exchanger for the same UA and AMTD is always less than the optimum value of the heat transfer rate (η  1) [5]. Again comparing this with Eq. (2) shows that the driving temperature potential in a heat exchanger can be expressed as the product of heat exchanger efficiency and mean temperature difference:   ðT  t Þ ¼ η T  t

(49)

1330

A. Fakheri

Therefore, the heat transfer in a heat exchanger can be calculated from   q ¼ ηNTUCmin T  t

(50)

Starting with Eq. (16) for a counterflow heat exchanger UA T 1  t2 ¼e T 2  t1

h

i

1 1 C h C c

(51)

Rearranging this equation by subtracting denominators from the numerators and adding the numerators to the denominators results in h i UA

1 C

C1

c h ðT 1  t 2 Þ  ðT 2  t 1 Þ e 1 h i ¼ ðT 1  t 2 Þ þ ðT 2  t 1 Þ UA C1 C1c h e þ1

(52)

Substituting for temperature differences in the numerator from Eqs. (13) and (14) and in the denominator from Eq. (48) and rearranging the right-hand side   q q NTU Cmin Cmin NTU Cmin Cmin     C Cc Ch Cc 2 2 e Ch Cc e  h  (53) ¼ Cmin Cmin 2 Tt NTU Cmin Cmin NTU Ch  Cc Ch  Cc 2 2 e þe Regardless of which fluid has the minimum capacity, Eq. (53) simplifies to 

 NTU ð1  Cr Þ   2 UA T  t NTU ð1  Cr Þ 2

tanh q¼

(54)

Comparing this with Eq. (47), we conclude that for a counterflow heat exchanger, the efficiency is 

 NTU tanh ð1  C r Þ 2 η¼ NTU ð1  Cr Þ 2

(55)

It is interesting that the expression for the efficiency of a counterflow heat exchangers has the same form as the efficiency of a constant area insulated tip fin. Similar analysis for parallel flow, shell and tube, and single-stream heat exchangers results in equations similar to Eq. (53). Therefore, the efficiency of all heat

32

Heat Exchanger Fundamentals: Analysis and Theory of Design

1331

exchangers only depends on one nondimensional parameter and can be expressed by the general expression: η¼

tanhðFaÞ Fa

(56)

where the fin analogy number, Fa, is a nondimensional group that characterizes the performance of different heat exchangers and is given in Table 4 for the four heat exchanger types. The general form of the fin analogy number can therefore be written as 1

ð1 þ mCr n Þn Fa ¼ NTU 2

(57)

As shown later, the expressions for heat exchanger efficiencies can be obtained from the available expressions for heat exchangers effectiveness. Fakheri (2006) used regression analysis, to arrive at approximate values for m and n for different cross flow heat exchangers. Note that the m and n values given in Table 5 for the cross flow heat exchangers are not necessarily the absolute minimum errors, and minor adjustments were made to provide uniformity in the results. The first four results are exact, and thus the maximum error is zero. The maximum error reported for the cross flow heat exchangers is for the range 0 < NTU < 3.0 and 0  Cr  1 and typically occurred at large NTU values that correspond to large Fa values or low heat exchanger efficiency.

Table 4 Fin analogy number of various heat exchangers Counter Fa ¼

Parallel

rÞ NTU ð1C 2

Fa ¼

rÞ NTU ð1þC 2

Single stream Fa ¼

NTU 2

Single shell pffiffiffiffiffiffiffiffiffiffi NTU 1þCr 2 Fa ¼ 2

Table 5 Efficiency expressions for various heat exchangers  1 η ¼

Tanh NTU NTU

ð1þmCr n Þn 2

1 ð1þmCr n Þn 2

Counterflow Parallel Single stream (Cr = 0) Single shell and tube Cross flow, Cmax unmixed, Cmin mixed Cross flow, Cmax mixed, Cmin unmixed Cross flow, both mixed Cross flow, both unmixed

m -1 1 1 1. 1.2 1.35 1.2 0.1

n 1 1 1 2 4.4 4.02 2 0.37

Max error (%) 0 0 0 0 θ0 > θr. This figure qualitatively shows the hysteresis in transition boiling. If the transition boiling curve is obtained by starting from the film boiling part of the curve and reducing the wall superheat, the boiling curve will follow the line d-e-f-g. The heat flux–wall superheat curve deviates from the pure film boiling curve before it reaches the minimum heat flux point, and with further reduction in wall superheat, the gradient of the heat flux–wall superheat curve becomes negative. The part of the boiling curve between points e and MFB, i.e., where the curve deviates from the film boiling line, but the heat flux is larger than the minimum heat flux and the wall superheat is larger than the wall superheat at the

42

Transition and Film Boiling

1703

Fig. 5 Hysteresis in the transition boiling regime for a liquid–solid pair with significantly different contact angle hysteresis

minimum film boiling point, has been named film–transition boiling (Ramilison and Lienhard 1987). During film boiling, as will be explained later, a vapor film separates the heated surface from liquid. Hydrodynamic waves form on the liquid–vapor interphase and play an important role in the phenomenology of film boiling. While in the film boiling zone (i.e., for points to the right of point e in Fig. 5) there is virtually no liquid–surface direct contact, in the film–transition boiling region, some liquid–surface contact takes place near the nodes of the waves that form at the vapor–liquid interphase in film boiling. As the wall superheat is reduced and the transition boiling part of the boiling curve is reached, the area patches covered by liquid increase in extent and number, and film boiling occurs on the part of the surface covered by vapor, while nucleate boiling takes place on areas covered by liquid. The expansion of the surface patches that are covered by liquid evidently involves the advancing contact angle. Thus, when the transition boiling region is approached from the film boiling side, one may assume that the contact angle in effect is close to θa. If the transition boiling zone of the boiling curve is approached from the nucleate boiling side by increasing the surface temperature (recall that the transition boiling part of the curve can only be obtained in steady-state experiments if the surface temperature is controlled), the boiling curve will follow the a-b-c line. The peak heat flux (q00max, 1 in Fig. 5) and the surface temperature at which the peak heat flux occurs surpass the peak heat flux (q00max, 2 in Fig. 5) and the peak heat flux temperature, respectively, which are measured when the transition boiling curve is obtained by starting from the film boiling regime and reducing the wall superheat. The phenomenology that leads to this apparent surpassing of CHF can be understood by remembering that at and beyond the peak heat flux, the dry patches that occur on the heated surface tend to expand at the expense of the liquid that is in contact with the wall. The dry patches support an essentially film boiling regime, while the areas covered by liquid support fully developed nucleate boiling representative of the

1704

S. M. Ghiaasiaan

Zone III regime in Fig. 2. The process is thus likely to involve a receding contact angle, and when θr < θa, the situation is similar to boiling on a surface with better wettability. Better wettability, equivalent to smaller contact angle, helps the spreading of the liquid film which is followed by vigorous bubble nucleation and the blowing away of the liquid. The fully developed nucleate boiling (i.e., the boiling regime in Zone III of Fig. 2) is thus extended beyond the CHF point at least for some portion of the heated surface. The heat transfer regime on the transition boiling curve near the peak heat flux point is a combination of film boiling and fully developed nucleate boiling and is often called the nucleate–transition boiling (Ramilison and Lienhard 1987). Ramilison and Lienhard noted that no hysteresis in transition boiling occurred in experiments with acetone boiling on a Teflon-coated surface, where near-perfect wetting took place. The existence of the aforementioned hysteresis in transition boiling under steadystate conditions has been disputed by some researchers, however. Experimental studies by Auracher and coworkers (Blum et al. 1996; Hohl et al. 2001; Auracher and Marquardt 2002) and Ohtaki and Koizumi (2006), for example, have shown no hysteresis when the experiments are performed in true steady state, and the heated surface is maintained clean. Figures 6 and 7 show pool boiling curves for FC-72

Fig. 6 Steady-state boiling curve of FC-72 on a copperheated disk (P = 0.13MPa) (Auracher and Buchholz 2005)

Wall heat flux (W / cm2)

25 20

Increasing temperature

15

Decreasing temperature

10 5 0 0

80

160 Wall heat flux (W/cm2)

Fig. 7 Steady-state boiling curve of water on a copperheated disk (P = 0.99bars) (Auracher and Marquardt 2002)

20 40 60 Wall superheat (°C)

decreasing increasing

140 120 100 80 60 40 20 0 0

20

40 60 80 Wall superheat (°C)

100

120

42

Transition and Film Boiling

1705

boiling on a copper-heated block electroplated with a 20-micron-thick nickel layer and distilled water boiling on a copper-heated disk electroplated with a 50-micronthick nickel layer, respectively. The wall temperature was controlled by comparing the measured temperature close to the boiling surface with a set point value and using the difference in a controller for adjusting the power of the electric heating system. In this way, carefully controlled steady-state or transient experiments could be performed. The experiments depicted in Figs. 6 and 7 were thus performed under true steady-state conditions, with clean heated surfaces. The FC-72 and nickel pair represent a well-wetting condition with a small contact angle, while water and nickel represent a partially wetting pair (θa  79∘ , θr  34∘, Faghri and Zhang 2006). No hysteresis can be observed in these figures and other similar tests. However, in transient experiments hysteresis can be observed in most parts of the boiling curve. Figures 8 and 9 show clearly the important effect of transient experimentation on the boiling curve. Auracher and Marquardt (2002) argue that boiling curves measured under steady-state conditions with a proper control and a clean heater surface do not exhibit a hysteresis in the transition boiling region. The hysteresis resulting from transient heating and cooling in transition boiling had been noted earlier by Bui and Dhir (1985a), as shown in Fig. 10.

100 2 K/s 10 K/s 20 K/s 40 K/s steady-state

90 80

4 K/s 15 K/s 30 K/s 50 K/s

Wall heat flux (W / cm2)

70 60 50 40 30 20 10 0 0

10

20

30 40 Wall superheat (°C)

50

60

70

Fig. 8 Boiling curves for FC-72 during transient heating experiments (P = 0.13MPa) (Hohl et al. 2001)

1706

S. M. Ghiaasiaan 25 steady-state

Wall heat flux (W / cm2)

2 K/s

20

4 K/s no control

15

10

5

0 0

10

20 30 40 Wall superheat (°C)

50

60

Fig. 9 Boiling curves for FC-72 during transient heating experiments (P = 0.13MPa) (Hohl et al. 2001)

100

Wall heat flux (W / cm2)

Fig. 10 Boiling curves in steady-state and transient experiments for water boiling on a vertical copper surface (Bui and Dhir 1985a; Dhir 1991)

Steady state Transient cooling Steady state Transient cooling

Surfaces are clean and no oxide

10 Smooth

E-600 rough

1

10

100

200

Wall superheat (°C)

Further evidence questioning the occurrence of hysteresis in steady-state conditions has been provided by Ohtaki and Koizumi (2006), who performed experiments with water, using the top surface of a large and thick copper block (15  15 mm2 area and 60 mm thick) to ensure that due to the large thermal capacity of the block, the temperature of the block remained in a time-smoothed steady state. They did not observe hysteresis in the transition boiling regime.

42

2.3

Transition and Film Boiling

1707

Parametric Effects

Some important parametric effects on the pool boiling curve, in particular with respect to the transition and film boiling regimes, are now discussed. The discussion will address CHF and nucleate boiling as well, as these regimes are directly relevant to the transition boiling regime and its range of occurrence. The effect of surface wettability with respect to the transition boiling as well as CHF and MFB points has already been discussed. In nucleate boiling, better wettability (smaller contact angle) leads to smaller bubble departure diameter but higher bubble departure frequency. Moreover, increased surface wettability shifts the nucleate boiling line toward the right. Thus, with increased surface wettability, decreasing nucleate boiling heat transfer coefficients (for the same Tw  Tsat) are obtained. Increased surface wettability also increases the maximum heat flux as well as the minimum film boiling temperature (Roy Chowdhury and Winterton 1985; Liaw and Dhir 1986). Figure 11 depicts the results of quenching experiments of Vakarelski et al. (2012). In these experiments 20-mm-diameter stainless steel

a

600

Sphere temperature(°C)

Film Boiling Leidenfrost Point 400 Nucleate 200

HL

SHL

0 0

b Sphere temperature(°C)

Fig. 11 Sphere temperature versus cooling time for 20-mm steel spheres held in water at 22  C in the experiments of Vakarelski et al. (2012). (a) Superhydrophilic sphere (SHL) (contact angle 160 )

20

40 Cooling time (s)

60

80

400 Film Boiling 300 Leidenfrost Point 200 Nucleate 100

HB

Cassie

SHB

0 0

20

40 Cooling time (s)

60

80

1708

S. M. Ghiaasiaan

spheres, with their surfaces modified in order to obtain various levels of hydrophobicity, were quenched in water. The surface conditions included smooth hydrophilic produced by cleaning with organic reagents, smooth hydrophobic produced by silanization with trichloro(1H,1H,2H,2H-perfluorooctyl)silane, textured superhydrophobic produced by treatment with a commercial superhydrophobic coating agent, and textured superhydrophilic produced by plasma cleaning of the surfaces previously treated with the superhydrophobic agent. For the superhydrophilic (SHL) sphere, the maximum temperature was not high enough to initiate film boiling. For hydrophilic (HL) (θ0  30∘) and hydrophobic (HB) (θ0  100∘) spheres, the film boiling mode on the surface ended with the collapse of the vapor layer and an explosive release of bubbles (Fig. 11). However, for the superhydrophobic (SHB) sphere (θ0  160∘), no vapor-layer collapse was observed. Film boiling remained stable throughout the boiling process. Nucleate boiling, CHF, and transition boiling regimes were completely bypassed, and film boiling remained stable until the surface was sufficiently cooled so that liquid–surface contact was established without bubble nucleation. For the superhydrophobic (SHB) surface case, the film boiling was replaced with the Cassie-state liquid–surface contact. (The Cassie or Cassie–Baxter state refers to conditions where liquid resides on a rough surface without completely penetrating the surface grooves and has been named after Cassie and Baxter (1944) who modeled the contact angle on composite surfaces and surfaces with microgroves. A useful discussion can be found in Gopalan and Kandlikar (2014)). Similar conclusions were arrived at more recently by Fan et al. (2016) based on quenching experiments with subcooled water. The hydrophobicity of surfaces can be manipulated by the deposition of nanoparticle films (nanocoating) (Kim et al. 2002; Hwang and Kaviany 2006; Forrest et al. 2010). By applying a layer-by-layer assembly method using polymer/SiO2 nanoparticle layers deposited on nickel wires and stainless steel plates, Forrest et al. (2010) could obtain static contact angles with water in the range of 3–141 . The critical heat flux and nucleate boiling heat flux (the latter for given wall superheats) both monotonically improved with better wettability and up to 100% improvement in CHF and better than 100% improvement in the nucleate boiling heat flux were obtained. Increased surface roughness tends to move the nucleate and transition lines to the left, implying improvement in the nucleate boiling heat transfer characteristics. Figure 10 depicts the data of Bui and Dhir (1985a). In the aforementioned experiments of Ohtaki and Koizumi (2006), surface roughness effect in transition boiling was small, however. Surface roughness also affects the minimum film boiling, as will be discussed later. The effect of surface roughness on heat transfer in stable film boiling is small and is primarily due to increased surface area and the change in surface radiative emissivity. Surface contamination (deposition and oxidation) and improved surface wettability both move the nucleate boiling part of the boiling curve to the right, i.e., a higher wall superheat will be needed for any given heat flux. Liquid pool subcooling improves heat transfer in all boiling regimes, as shown qualitatively in Fig. 12, except for the fully developed nucleate boiling region where its effect is small due to the overwhelming contribution of bubble generation (i.e., phase change) to heat transfer in comparison with convection. Nucleate boiling is the preferred

42

Transition and Film Boiling

1709

Fig. 12 The effect of liquid pool subcooling on the boiling curve

lng w ″

Subcooled Saturated

In (Tw –Tsat)

mode of heat transfer for many thermal cooling systems, since it can sustain large heat fluxes with low heated surface temperatures. Furthermore, in the transition boiling, a fraction of the surface is subject to a heat transfer regime similar to fully developed nucleate boiling. In the fully developed nucleate boiling zone (the slugs and jet zones), the heat transfer coefficient is insensitive to surface orientation. In the partial nucleate boiling zone, however, heat transfer is affected by orientation.

2.4

Film Boiling

Hydrodynamic models with minor adjustments have done well in predicting the pool film boiling heat transfer in many situations. Film boiling models and correlations for some important heated surface configurations are now reviewed.

2.4.1 Film Boiling on Vertical Flat Surfaces Film boiling on a vertical and flat surface is among the simplest film boiling configurations and can be solved analytically, provided that the vapor film remains laminar, coherent, and smooth-surfaced. These assumptions are not always realistic, and their effects will be discussed later. However, as will be shown, the solutions for the idealized vapor film can be modified in order to compensate for some of these unrealistic assumptions. Coherent and Laminar Film with Smooth Vapor–Liquid Interface Figure 13a depicts the configuration of the vapor film, which flows upward on the vertical surface, and its thickness grows due to evaporation as it rises. The following assumptions are made: (a) the process is steady state; (b) the liquid and vapor are incompressible; (c) the vapor film is laminar; (d) the liquid is infinitely large and except for the flow that is caused by thermal and momentum interaction with the

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Fig. 13 Idealize laminar film boiling on a vertical heated surface: (a) film configuration, (b) an infinitesimally thin slice, (c) temperature profiles, and (d) velocity boundary conditions at liquid–vapor interface

u

a

ν

b

δ+dδ rLuLdd

δ dz

rLnLdx

rvnvdx g

z

rvuvdd

δ

y

d

c Linear profile

Dynamic interphase

u Stagnant interphase

Tw

Tsat

δ

y

δ

y

rising vapor film, the liquid is stagnant; (e) when the liquid is subcooled, natural convection takes place in the liquid as a result of liquid temperature variation caused by thermal interaction with the vapor film, and Boussinesq approximation can be applied for the liquid; (f) the flow field is two dimensional; (g) radiation heat transfer effects are negligible in the liquid; and (h) the thermophysical properties are constant. The mass, momentum, and energy conservation equations for the vapor phase will then be @uv @vv þ ¼0 @z @y uv

(1)

@uv @uv @ 2 uv gð ρL  ρv Þ þ vv ¼ νv 2 þ ρv z @y @y

(2)

@T v @T v @2Tv þ vv ¼ αv @z @y @y2

(3)

uv

The mass, momentum, and energy conservation equations for the liquid phase will then be @uL @vL þ ¼0 @z @y

(4)

42

Transition and Film Boiling

uL

1711


 @uL @uL @ 2 uL þ vL ¼ νv 2 þ gβL T L  T L, 1 @z @y @y

(5)

@T L @T L @2TL þ vL ¼ αL @z @y @y2

(6)

uL

The boundary conditions for these equations are At y = 0 uv ¼ v v ¼ 0 T v ¼ T w

(7)

uL ¼ vL ¼ 0, T L ¼ T L, 1

(8)

uv ¼ uL , T v ¼ T L ¼ T sat ðPÞ

(9)

    dδ dδ ρg v v  uv ¼ ρf v L  uL dz dz

(10)

At y ! 1

At y = δ

@uL @uv ¼ μg @y @y   dδ @T L ¼ ρg vv  uv hfg  kf dz @y μf

kg

@T v þ q00rad @y

(11)

(12)

where q00rad is the radiative heat flux from the wall to the liquid–vapor interphase. Note that Eq. 10 can be derived by performing a mass balance on the control volume depicted in Fig. 13b. The above equations can be simplified if the liquid is saturated and radiation effect is neglected. In this case, TL = Tsat(P) everywhere, Eq. 6 will be redundant, and Eq. 5 will simplify to uL

@uL @uL @ 2 uL þ vL ¼v 2 @z @y @y

(13)

Also, neglecting the effect of superheating of the vapor, Eq. 12 can be replaced with 2 δ 3   ð @T v d4 ρ uv ðyÞdy5 kg ¼ hfg dz v @y y¼δ 0

(14)

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Koh (1962) derived a similarity solution for the above-simplified set of equations and showed that the behavior of the vapor film as well as the heat transfer process depends on the vapor Prandtl number, Prv, and the forthcoming two dimensionless parameters: CPv ðT w  T sat Þ hfg Prv ½ðρv μv Þ=ðρL μL Þ

1= 2

(15) (16)

Parametric calculations by Koh (1962) showed that interfacial shear is important 1= except for vanishingly small values of ½ðρv μv Þ=ðρL μL Þ 2 . The parametric calculations also indicated that the temperature profile across the film is approximately linear only when the film is very thin and becomes increasingly nonlinear as the film thickness increases. The vapor film momentum equation can be significantly simplified by bearing in mind that the vapor film that forms on a vertical wall tends to rise because of buoyancy, much like the boundary layer that forms on vertical surfaces during free convection, and in most cases, the inertia of the vapor is insignificant. The left side of Eq. 2 will then vanish, and Eq. 2 reduces to μv

d 2 uv þ gðρL  ρv Þ ¼ 0: dy2

(17)

For a contiguous laminar film with a smooth surface rising in stagnant liquid in steady state, the boundary conditions for this equation are uv ¼ 0 at y ¼ 0

(18)

uv ¼ 0 at y ¼ δ for the stagnant interphaseði:e:, noslipÞ,

(19)

duv ¼ 0 at y ¼ δ for the dynamic interphaseði:e:, zero shear stressÞ dy

(20)

Either of the assumed boundary conditions at y = δ decouples the momentum equation of the vapor film from the liquid (see Fig. 13d). The vapor film can then be modeled by using the integral technique, described in standard heat transfer textbooks (see, e.g., Ghiaasiaan (2011)). Bromley (1950) performed such analysis. The solution of Eq. 17 with these boundary conditions gives the velocity profile in the vapor film: uv ð y Þ ¼

gΔρ  C1 δy  y2 2μv

(21)

42

Transition and Film Boiling

1713

where C1 = 1 for the stagnant interphase, and C1 = 2 for the dynamic interphase. If it is assumed that the temperature profile across the vapor film is linear, Eq. 14 gives 2 δ 3 ð d4 T w  T sat ρv uv ðyÞdy5 ¼ kv hfg dz δ

(22)

0

One can now combine Eqs. 21 and 22, to derive a differential equation for δ. The solution of the latter differential equation leads to  δ¼

8 kv ðT w  T sat Þμv z 3ðC1 =2  1=3Þ ρv hfg gΔρ

1=4 (23)

Knowing δ, and assuming a linear temperature distribution across the vapor film, one can calculate the local film boiling heat transfer coefficient from hFB = kv/δ. The average heat transfer coefficient for a vertical surface of length L can then be found ÐL from hFB ¼ L1 hdz, and this integration yields (Bromley 1950) 0

 hFB ¼ C

ρv hfg gΔρk3v ðT w  T sat Þμv L

1=4 (24)

where C = 0.663 for the stagnant interphase and C = 0.943 for the dynamic interphase. The stagnant interphase is evidently appropriate when the liquid viscous resistance is much larger than the vapor viscous resistance. This condition holds 1= when ½ðρv μv Þ=ðρL μL Þ 2 10, (b) the entire thermal resistance of the vapor film occurred in the vapor film’s viscous sublayer, and (c) the temperature profile in the viscous sublayer was linear. Hsu and Westwater compared their model with data representing five different fluids boiling on the outside of vertical tubes with 5–16 cm heights and noted that the data could be predicted within 32%. The most serious shortcoming of Eq. 24, however, is that it does not account for the intermittency of the vapor film. Experimental observations show that the vapor film on long, heated surfaces does not remain smooth and coherent. Interfacial waves develop, and the vapor film becomes intermittent before the film grows sufficiently Fig. 14 Film boiling on a long, vertical surface

a

b

Turbulent Film

S Vapor Film

Liquid

Laminar Film

Vapor bluge

Contiguous Film

Film with Intermittent Bulges

42

Transition and Film Boiling

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thick to turn turbulent. Based on experimental observations, Bailey (1971) suggested that the vapor film supports a spatially intermittent structure. At the bottom of each spatial interval, the vapor film is initiated and grows, until it becomes unstable and eventually is disrupted by the time it reaches the top of the interval. Following its disruption, a fresh film is initiated in the next interval. The vapor film remains laminar in the aforementioned intervals. Based on the argument that the intermittency results from hydrodynamic instability, Leonard et al. (1978) proposed qffiffiffiffiffiffi that for σ vertical surfaces, L in Eq. 24 should be replaced with S  λcr ¼ 2π gΔρ , with λcr representing the critical (neutral) wavelength according to the two-dimensional Rayleigh–Taylor instability (for a discussion of linear instability analysis of interfacial waves that are of interest in boiling, see Ghiaasiaan (2017)). With this substitution, the modified Bromley correlation is obtained, which agrees with inverted annular film boiling data in vertical tubes (Hsu and Graham 1986). Bui and Dhir (1985a) studied subcooled and saturated film boiling of atmospheric water on a vertical surface. Their visual observations showed an intermittent, but considerably more complicated, vapor film behavior. In low subcooling experiments, three-dimensional waves with large and small amplitudes developed on the vapor–liquid interface (Fig. 14b). The amplitude of the large waves was of the order of a few centimeters and grew with distance from the heated surface leading edge. The peaks of the three-dimensional waves evolved into bulges that resembled bubbles which were attached to the surface, and their height was one or two orders of magnitude larger than the thickness of the surrounding vapor film. The bulges acted as vapor sinks for the vapor flowing in the film and grew in size as they moved upward due to buoyancy. The local heat transfer coefficient was highly transient as a result of intermittent exposure to vapor film and vapor bulges. The vapor path was thus interrupted rendering the distance between two adjacent bulges to be the effective average vapor path length, in agreement with the aforementioned vapor film intermittency argument. Waves with small and large amplitudes, and intermittency with respect to film hydrodynamics as well as heat transfer, were also noted in experiments dealing with subcooled film boiling on vertical surfaces (Vijaykumar and Dhir 1992a, b). The local liquid-side heat flux varied along the film and the bulge and had its maximum at the wave peaks (i.e., on the bulges). Based on these observations, Bui and Dhir (1985b) developed a mechanistic model that separately accounts for heat transfer between the surface and the liquid phase through the vapor film and the bulges and assumes that the bulges are separated by the wavelength of three-dimensional waves that form on the vapor–liquid interface. This wavelength was obtained to be λ3D 

pffiffiffi 2λd

(27)

where λd is the two-dimensional interfacial fastest-growing wavelengths. Bui and Dhir calculated λd using a linear two-dimensional instability analysis (Bui 1984). A more recent predictive method for film boiling on flat and long vertical surfaces has been proposed by Nishio and Ohtake (1993). For saturated film boiling, and

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using the integral method analysis of Nishio and Ohtake (1992) for film boiling on a vertical surface in subcooled liquid, they showed that when ρg =ρf TMFB (or q00w > q00MFB ), it is assumed that essentially no macroscopic physical contact between liquid takes place. However, as mentioned earlier, experimental observations have shown that some liquid–surface contact occurs at least in the portion of the boiling curve designated as film–transition boiling in Fig. 5. A phenomenon closely related to MFB is the Leidenfrost process, first reported in 1756 by the German scientist J.G. Leidenfrost. It refers to the dancing motion of a liquid droplet on a very hot surface, which takes place because of the formation of a vapor cushion that is typically of the order of 10 μm thick (Biance et al. 2003) between the droplet and the hot surface. If the surface temperature is gradually reduced, eventually Leidenfrost point (LP) is reached, when the surface temperature reaches the Leidenfrost temperature, at which point the heat transfer rate between the droplet and the surface is at its minimum. With further lowering of the temperature of the surface, more extensive contact between the droplet and surface takes place, and stable film boiling is terminated. A practical experimental method for determining the LP is by measuring the lifetime of droplets of a particular liquid with identical initial conditions as a function of the heated surface temperature. Figure 17, as an example, shows the evaporation curve for saturated water droplets with initial diameters of 3.5 mm dropped on hot sapphire and stainless steel surfaces (Nagi and Nishio 1996). Figure 18 shows qualitatively the boiling regimes and their dependence on wall superheat that occur during the Leidenfrost process. Each evaporation curve represents the variation of droplet lifetime as a function of wall superheat, and the point representing the maximum lifetime represents the LP. For the points on an evaporation curve to the left of the LP, there is partial contact between the liquid and surface which leads to faster evaporation and shorter droplet lifetime. For the points to the right of the LP, on the other hand, there is stable film boiling, and there is essentially no physical contact between the liquid and surface. For wall superheats larger than the wall superheat

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Fig. 17 Evaporation curve for water droplets with 3.5 mm initial diameter (after Nagi and Nishio 1996)

120

Evaporation time, s

100 80 60 SUS304

Sapphire 40 20 0

TRmax

0

50

100

150

200

Ttls

250

Wall superheat, °C

at the LP point, however, with increasing the surface temperature, the heat transfer rate increases, and therefore the droplet lifetime becomes shorter. The main difference between LP obtained with small liquid droplets and MFB in pool boiling is that in experiments with sessile droplets, the area underneath the droplet where a vapor film forms is typically too small to allow for the development of interfacial waves, whereas in pool boiling experiments, the surface area is often large enough to allow for the development of liquid–vapor interfacial waves. However, this discrepancy is important if the vapor film hydrodynamic instability, to be discussed shortly, is the mechanism leading to MFB. Furthermore, experiments have shown that the LP point is not sensitive to initial droplet size (Gottfried et al. 1966; Patel and Bell 1966; Nishio 1983). In terms of the physical processes that lead to MFB, a number of different mechanisms have been proposed and used for model development. The proposed models can generally be divided into two categories: 1. Heat flux-controlled models. Hydrodynamically, controlled vapor film instability models are examples of this group of models (Zuber 1959; Berenson 1961; Lienhard and Wong 1964). According to these models, during stable film boiling, the vapor film–liquid interface supports Taylor waves that grow and lead to the formation and release of bubbles. If the vapor generation rate is reduced to a level that the periodic formation and release of bubbles are no longer sustainable, then the vapor film will collapse at some points. The lowest vapor generation rate (equivalently, the lowest heat flux) that can sustain a coherent and stable vapor film will represent the MFB conditions. The hydrodynamic models are widely used but suffer from a number of shortcomings. Hydrodynamic models that assume the occurrence of a coherent vapor film do not

42

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Fig. 18 Sessile droplet evaporation curve and the associated boiling regimes (after Bernardin and Mudawar 1999)

explicitly account for the effect of surface conditions and properties. Experiment has shown that surface wettability and cleanliness have a significant effect on MFB (Chowdury and Winterton 1985, Bui and Dhir 1985a; Bernardin and Mudawar 1999). These models also apply when the surface size is large enough to support interfacial waves and depend on surface configuration. Some experimental data suggest that surface configuration has little effect on MFB, however (Nishio et al. 1987). 2. Temperature-controlled models. In these models, MFB is assumed to occur when the surface temperature, and consequently the temperature of the liquid layer

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interacting with it, exceeds some threshold. Recall that the liquid in contact with a hot surface becomes superheated during nucleate boiling. The local superheat that is needed for heterogeneous bubble nucleation is typically small. The MFB point, however, typically represents significantly higher wall superheats in comparison with heterogeneous nucleation in pool boiling. (Cryogenic fluids are an exception, however, and MFB happens for them at relatively low wall superheats.) These assumed threshold temperatures include the liquid spinodal temperature or the temperature that can cause homogeneous vapor nucleation. Mechanisms that have been used as the bases of models for MFB also include the hypothesis that MFB occurs when the temperature of the liquid in contact with surface reaches a level that causes perfect spreading (Olek and Zvirin 1988). Based on the observation that sessile droplets after impacting a hot surface separate from the surface while they are at their maximum spreading, and using an expression for the dependence of contact angle on temperature suggested by Adamson and Ling (1964), Olek et al. suggested that the temperature representing zero contact angle can be used as an upper limit for MFB. Segev and Bankoff (1980) also have proposed a mechanism for MFB based on the adsorption characteristics of the surface. In this hypothesis, a non-evaporating precursor (adsorbed) liquid film must spread on the surface in advance of a thicker liquid film that wets the surface and evaporates. The precursor film thickness decreases as the surface temperature increases, and the MFB occurs at a threshold temperature at which the precursor film thickness goes through a sharp decline. The above mechanisms have not found wide acceptance, however. The effect of the deposition of nanoparticle films of surfaces and the consequent improvement in CHF and nucleate boiling heat flux were already mentioned in Sect. 2.3. Deposited nanoparticles appear to increase the minimum film boiling temperature and minimum heat flux. Kim et al. (2010) performed quenching experiments with metallic spheres and cylindrical rodlets in 0.1% volume aqueous nanofluids (water containing 0.1% volume nanoparticles) under atmospheric pressure conditions. The rodlets were vertically plunged in the coolant liquid pool. The deposition of nanoparticles increased MFB (quenching) temperature and heat flux, as well as the quench front speed. Figure 19 depicts their quenching results with a 4.8-mm-diameter rodlet cooled in subcooled alumina nanofluid. The depicted data show the experimental measurements with the rodlet in pure water and in four repetitions of the experiment, this time with the nanofluid. As noted, in the first pass (the first test with nanofluid), the temperature profile is identical to pure water. In the subsequent passes, however, the temperature profiles show that CHF and MFB heat flux are both enhanced as a result of the deposition of nanoparticles on the surface. The improvement diminishes after the third pass, however, apparently due to the diminishing rate of particle deposition. Experimental data also suggest that the effect of liquid velocity on MFB conditions is insignificant (Sakurai 1984; Nishio 1987).

42

Transition and Film Boiling

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Fig. 19 Center temperature histories during quenching of 4.8-mm-diameter metallic rodlets quenched by vertically plunging in pure water and aqueous 0.1% volume alumina nanofluid (Kim et al. 2010). The water subcooling was 20  C

2.6.2 Predictive Methods for MFB Correlations for MFB that represent temperature-controlled MFB phenomenology include a reduced-state Leidenfrost temperature correlation by Baumeister and Simon (1973):

T MFB

8 2 ! 39 4=3 1=3 = 4 27 < 10 ð ρ =A Þ w w 5 ¼ T cr 1  exp40:52 ; 32 : σ

(64)

where Aw and ρw are the atomic number and density (in grams per cubic centimeter) of the heated surface, TMFB and Tcr are the MFB and the critical temperature of the fluid in absolute scale, and σ is the liquid–vapor surface tension (in dynes per centimeter). The correlation evidently depends on the fluid-solid pair properties and has been tested against some refrigerant, cryogenic, and high surface tension fluids such as mercury. This correlation also agreed quite well with the experimental data of Bernardin and Mudawar (1999) representing Leidenfrost phenomenon of acetone, benzene, F-72, and water on aluminum surfaces with polished, particleblasted, and rough-sanded surfaces. Kalinin et al. (1969) have proposed the following correlation which accounts for the effect of the solid surface thermophysical properties, as well as the subcooling of the liquid, and has been found to agree well with cryogenic fluid experiments for pool as well as forced flow boiling (Pron’ko and Bulanova 1978):   ðρCkÞf 0:25 ðρCkÞf T MFB  T sat ¼ 0:165 þ 2:5 þ T cr  T L ðρCkÞw ðρCkÞw

(65)

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The correlation is suitable for well-wetting surface–liquid pairs, and its range of validity, when applied to internal flow, is (Groeneveld and Snoek 1986) 103
0 was, hpL

  q00pL 5:9  107 ð1  xb ÞρL hfg ΔT s kL 1  eλle =rb ¼ ¼ xb T s σ Tw  Ts

(24)

where Eqs. (20), (22), and (23) are used to calculate rb, le, and λ, respectively. The subscript “pL” represents the refrigerant and “plain” lubricant mixture in order to differentiate it from refrigerant/nanolubricants, which are discussed in Section 8.2.

7.2

Convective Boiling of Refrigerant/Lubricant Mixtures

Convective boiling of a volatile fluid, like a refrigerant, is affected by the addition of lubricant primarily because lubricant significantly increases the viscosity of the mixture and changes the equilibrium vapor pressure. These effects strengthen as the local lubricant mass fraction (wl) increases as the flow quality (xq) increases, wl ¼

wb 1  xq

(25)

Here, wb is the bulk lubricant mass fraction for the all-liquid condition at zero quality. Thome (1995) outlines a methodology for predicting the saturated temperature– pressure relationship for a refrigerant/lubricant mixture, 1 ¼ Tr

lnðPr Þ  b þ c

A2 xq A1

(26)

Thome (1995) provides c and b as fourth-degree polynomials in the local lubricant mass fraction in the refrigerant liquid (wl), c ¼ 2300:2½K þ 182:5½Kwl  724:2½Kw2l þ 3868:0½Kw3l  5268:9½Kw4l b ¼ 15:146  0:722wl þ 2:391w2l  13:779w3l þ 17:066w4l (27)

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M. A. Kedzierski

All of the coefficients of the c (units of K) and b polynomials with the exception of the constant terms were taken from Thome (1995). The constant terms of the polynomials were adjusted by Sawant et al. (2007) to the lubricant-free R410A expression given in Eq. (26) when wl = 0. Choi et al. (2001) found that the two-phase pressure drop of refrigerant/lubricant mixtures could be predicted acceptably well with standard correlations using the local refrigerant/lubricant mixture viscosity in the calculation of the Reynolds number and pure-refrigerant properties elsewhere. Considering this, Sawant et al. (2007) showed, and Bigi and Cremaschi (2016) verified, that refrigerant/lubricant convective boiling in a microfin tube could be predicted to with 20% by using the Hamilton et al. (2008) correlation (Eq. (16)) with the refrigerant/lubricant mixture liquid viscosity and liquid density evaluated at the local lubricant mass fraction to calculate Re and Pr for use in Eq. (16). Use of the lubricant properties in the correlation caused between a 0.3% and a 0.8% reduction in the Nusselt number as compared to results using the lubricant-free R410A properties alone. Being that Sawant et al. (2007) showed that for qualities less than 50%, the lubricant changed the saturation temperature of R410A by less than 0.022 K; neglecting the effect on pressure may yield acceptable results. It is expected that a fair prediction for convective refrigerant/lubricant boiling in smooth tubes may be achieved by using the local mixture viscosity in the calculation of Re and Pr in the Zou et al. (2010) model, i.e., Eqs. (11)–(15). For smooth tubes, Chaddock and Mathur (1980) propose that refrigerant/ lubricant flow boiling when nucleate boiling is fully suppressed be correlated in the form,  n 1 h ¼ hc C (28) χ tt where C and n are paired correlating constants for a particular refrigerant/lubricant mixture and lubricant concentration. The Lockhart–Martinelli parameter (χ tt) varies with quality, xq, consists of liquid and vapor fluid properties, and is given in the Nomenclature. For the χ tt in Eq. (28), the vapor density and the vapor dynamic viscosity are evaluated as pure refrigerant vapor. In addition, the liquid density, the liquid dynamic viscosity, and the quality are evaluated as mixed refrigerant/lubricant properties at the lubricant mass fraction. When evaluating Eq. (12) for use in Eq. (28), only the dynamic viscosity is evaluated as a refrigerant/lubricant mixture. All other properties in Eq. (12) are taken as pure liquid-refrigerant values. Table 1 provides values for C and n, as taken from Chaddock and Mathur (1980), for various

Table 1 Chaddock and Mathur (1980) refrigerant/lubricant flow boiling constants for Eq. (28) Mass % oil in R22 0 1.0 2.9 5.7

C 3.90 4.72 4.36 4.97

n 0.62 0.59 0.60 0.59

% of data fitted to within 35% 93 95 88 89

44

Mixture Boiling

1815

percent mass fractions of a naphthenic oil, with a nominal kinematic viscosity of 21.58  106 m2 s1 at 100  C, in R22. In addition to the above discussed lubricant effects, Thome (1995) discusses the influence of the specific heat of the lubricant on the enthalpy change of the refrigerant/lubricant mixture and provides a methodology for its calculation. The effect of lubricant-specific heat on refrigerant/lubricant flow boiling becomes more significant for larger lubricant concentrations and larger thermodynamic qualities. The lubricant transports heat that is not used to evaporate refrigerant. Consequently, the heat capacity of the lubricant is a source of loss of available energy for phase change, which leads to a degradation in heat transfer.

8

Nanofluid Pool Boiling

8.1

Water-Based Nanofluid Boiling

The boiling data with water-based nanofluids is inconsistent. Some measurements show enhancement with respect to pure water and others show boiling degradations (Barber et al. 2011). The reason for this is that due to the relatively low viscosity of water, it is difficult to have a stable nanofluid that does not exhibit settling. Settling of nanoparticles onto the boiling service has the potential to fill active cavity sites on the surface, thus reducing the boiling performance. Nanofluid stability can be improved with a surfactant, but the surfactant may contribute to boiling enhancement by reducing the surface tension of water. Consequently, successful, generalized modeling of water-based nanofluids has not been accomplished, which makes reliable prediction of water-based nanofluids difficult.

8.2

Prediction of Refrigerant/Nanolubricant Pool Boiling

Because of the heat transfer degradation caused by the addition of lubricant to refrigerant, enhancement techniques for refrigerant/lubricant boiling are of high interest. A cost-effective way of enhancing the boiling performance of refrigerant/ lubricant mixtures is to add nanoparticles to the lubricant. For example, Kedzierski and Gong (2009) have shown that copper-oxide nanoparticles can increase the refrigerant/lubricant pool boiling heat flux by as much as 245%, and Kedzierski (2011, 2012) has shown that aluminum-oxide nanoparticles can yield similar enhancements for refrigerant/lubricant pool boiling. The existence of the lubricant excess layer provides an opportunity for heat transfer enhancement by providing an environment where the nanoparticles can be in stable Brownian motion. Nanofluid boiling with less viscous base fluids, like water, typically results in the cavities of the boiling surface being filled with trapped nanoparticles resulting in a heat transfer degradation. The lubricant excess layer prevents nanoparticle settling and allows the nanoparticles to be suspended above the surface.

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Equation 29 gives the pool boiling heat flux for a refrigerant/nanolubricant mixture (q00np ) normalized by the pool boiling heat flux for a refrigerant/lubricant mixture (q00PL ) of the same lubricant without nanoparticles (Kedzierski 2012), q00np q00PL

 N  1:45  109 ½s  m1  Anps  σνL ρv xb G ¼1þ  3=2   00 Dnp qn ρL ρnp  ρL gð1  xb Þ2

(29)

where q00n is equal to q00PL normalized by 1 Wm2, i.e., q00n ¼ q00PL =1Wm2 in order to make it nondimensional. Properties included in Eq. (29) are the neat refrigerant surface tension (σ), the neat refrigerant vapor density (ρv), the neat lubricant liquid density (ρL), the neat lubricant liquid kinematic viscosity (νL), and the nanoparticle density (ρnp). The bulk mass fraction of the lubricant in the refrigerant is xb. The pool boiling model above is based on the assumption that the enhancement is due to surface work on bubbles as caused by momentum transfer between growing bubbles and nanoparticles suspended within the lubricant excess layer. Both q00np and q00PL are based on the projected area of the boiling surface. The parameter remaining to be described is the nanoparticle surface density (Nnp/As)G, which is surface geometry dependent as the subscript “G” indicates. The Nnp/As is obtained by calculating the entire active surface area of the evaporator (As) and dividing it into the total number of nanoparticles (Nnp) charged to the evaporator, which can be obtained by assuming a spherical particle diameter of an average size. Figure 6 shows the nanoparticle surface density for three different surface geometries. For the smooth surface, (Nnp/As)G is equal to Nnp/As. However, enhanced surfaces are not expected to have uniform bubble distribution over the entire surface due to gradients in temperature and heat flux. Consequently, geometrydependent nanoparticle surface density expressions are required for enhanced surfaces. For example, for a rectangular-finned surface with 826 fins per meter (fpm)

Refrigerant\lubricant above nanolubricant excess layer

Lubricant excess layer with nanoparticles

smooth surface

Fin

Lubricant excess layer with nanoparticles Fin

Lubricant excess layer with nanoparticles

reentrant cavity rectangular fin

Fig. 6 Illustration of nanoparticle surface density for three surfaces

44

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with overall fin-height and fin-tip-width of 0.76 mm and 0.36 mm, respectively, the expression for (Nnp/As)G is (Kedzierski 2012),   1:47   2 N np  8 00 2:53 N np 20  1  10 ¼ 4:15  10 qn m As  G As

(30)

The leading constant includes the surface geometry effects for a rectangularfinned surface. For Eq. (30), the units for both (Nnp/As)G and Nnp/As are nanoparticles per square meter. The nanoparticle surface density for a reentrant cavity surface reflects a greater effectiveness of the surface as compared to the rectangular-finned surface (Kedzierski 2015),   1:47   2 N np  8 00 2:53 N np 20  1  10 ¼ 4:15  10 q m þ 0:00017q00n n As  G As

(31)

The additional term is a function of the normalized heat flux to account for the increased interaction between bubble and nanoparticles with increase in heat flux as more of the surface becomes active with bubble nucleation. For Eq. (31), the units for both (Nnp/As)G and Nnp/As are nanoparticles per square meter.

9

Pool Boiling with Additives

9.1

Additives for Boiling Water

Wasekar and Manglik (1999) review nucleate boiling enhancements with additives that work to reduce the surface tension of water. They postulate that boiling enhancement can occur with the addition of additives due to the reduction in the surface tension, while boiling degradation can occur due to the reduction in the bubble contact angle. Because both the surface tension and the contact angle decrease with the increase in the additive concentration, the result can be a negligible boiling enhancement. Wen and Wang (2002) were able to predict the effect of the additive that they studied with the Mikic–Rohsenow (1969) model by replacing the active site density expression with one that reflected its increase as due to surface tension reduction. The site density correlation (na) of Wen and Wang (2002) was,  m 1 na ¼ C ð1  cos ΘÞ rc

(32)

where rc is the critical site radius for bubble nucleation, Θ is the bubble contact angle, and C and m are fitting constants that are specific to the additive. For example,

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Wen and Wang (2002) found the constants C and m to be 5  105 and 6, respectively, for one commercial surfactant.

9.2

Additives for Boiling Refrigerant

Kedzierski (2007) has shown that lubricant-based refrigerant additives can either improve refrigerant/lubricant pool boiling by as much as 95% or have essentially no effect. In this study, lubricant excess surface density measurements along with surface tension measurements and other surface chemistry analysis were done to support the opinion that the additive can form a monolayer between the wall and the lubricant/additive excess layer if the lubricant and additive are sufficiently dissimilar chemically. The general conclusion was that for an enhancement mechanism to occur, the additive must be more viscous than the existing chiller lubricant and reside at the immediate wall in a monolayer. It was also hypothesized that a monolayer will not form if the additive and the refrigerant oil are too chemically similar, e.g., both naphthenic based or both synthetic based. For this case, represented as system 1 in Fig. 7, the additive will have little influence on the refrigerant/lubricant pool boiling because it remains well mixed in the excess layer with the lubricant and is typically only 2% by mass fraction of the lubricant charge. For system 2 to exist, it must evolve from system 1. The evolution can occur spontaneously only if the change from system 1 to system 2 results in a reduction of system surface energy (Rosen 1978). The requirement for system 2 to exist can be expressed in terms of surface energies by applying the analysis of spreading coefficients given by Rosen (1978), aγ m2 b þ aγ Am2 þ aγ wA < aγ wm1 þ aγ m1 b

(33)

Here, a is the surface area and γ m2 b is the interfacial free (surface) energy per unit area at the lubricant/additive mixture 2 – bulk refrigerant/lubricant/additive mixture interface. Similarly, γ Am2 , γ wA, γ wm1 , and γ m1 b are the surface energies of the additive –

∼bulk refrigerant/lubricant/additive liquid∼ lubricant/additive excess layer

ag m,b

ag wm

1

System 1: Additive and lubricant mixed in excess layer

∼bulk refrigerant/lubricant/additive liquid∼ lubricant/additive

ag Am

2

additive monolayer

ag m b 2

ag wA

System 2: Additive and lubricant mixed in excess layer with additional additive monolayer at surface

Fig. 7 Two possible surface energy systems for the refrigerant/lubricant/additive mixture

44

Mixture Boiling

1819

lubricant/additive mixture 2, the wall – additive, the wall – lubricant/additive mixture 1, and the lubricant/additive mixture 1 – bulk refrigerant/lubricant/additive interfaces, respectively. Subscripts 1 and 2 on the lubricant–additive mixture represent slightly different compositions of the two excess layers to account for some loss of additive to the monolayer in system 2. By assuming that the additive monolayer does not significantly deplete the lubricant/additive excess layer of additive, γ m1 b and γ m2 b are approximately equal for the two systems. Many of the additive and the lubricant/additive mixture fluid properties are similar because they are essentially both lubricants. By neglecting the surface energy between the additive and the lubricant/additive mixture and using Young’s equation (Adamson and Gast 1997) to quantify the surface energies associated with the additive and the lubricant, Eq. (33) simplifies to, γ Av > γ Lv (34) Equation 34 shows that the requirement for a pure additive monolayer to exist at the surface is that the liquid–vapor surface tension of the additive is greater than that of the existing lubricant. In summary, for a boiling enhancement to be possible with a lubricant-based additive, the following three criteria must be satisfied: (1) Eq. (34) is true; (2) the additive and lubricant are chemically dissimilar; and (3) the viscosity of the additive must be larger than the lubricant viscosity. The first two requirements make it possible for the additive to be immediately on the boiling surface. The third requirement provides the boiling enhancement by increasing the lubricant viscosity at the wall (Kedzierski 2001).

10

Cross-References

▶ Boiling and Two-Phase Flow in Narrow Channels ▶ Boiling in Reagent and Polymeric Solutions ▶ Boiling on Enhanced Surfaces ▶ Flow Boiling in Tubes ▶ Fundamental Equations for Two-Phase Flow in Tubes ▶ Nucleate Pool Boiling

References Adamson AW, Gast AP (1997) Physical chemistry of surfaces, 6th edn. Interscience Publishers, New York, p. 11 Barber J, Brutin D, Tadris L (2011) A review on boiling heat transfer enhancement with nanofluids. Nanoscale Res Lett 6:280 Bigi AAM, Cremaschi L (2016) A comparison between recent experimental results and existing correlations for microfin tubes for refrigerant and nanolubricants mixtures two phase flow boiling. 16th international refrigeration and air conditioning conference, Purdue, 11–14 July, paper 2340 Bigi AMA, Wong T, Deokar P, Cremaschi L (2015) Experimental investigation on heat transfer and thermophysical properties of mixtures of Al2O3 nanolubricants and refrigerant R410A. 2015

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ASHRAE Transactions, ASHRAE conference paper no. 15714, ASHRAE Winter Conference, Chicago, IL, 24–28 Jan Brown JS (2013) Fourth ASHRAE/NIST refrigerants Conference: “moving towards sustainability”. HVAC&R Research 19(2):101–102 Chaddock JB, Mathur AP (1980) Heat transfer to oil-refrigerant mixtures evaporating in tubes. 2nd multiphase flow and heat transfer symposium, Clean Energy Research Institute, University of Miami, April 1979; published in Multiphase transport: fundamentals, reactor safety, applications, Hemisphere, Washington, pp. 861–884 Chen JC (1966) Correlation for boiling heat transfer to saturated fluids in convective flow. I&EC Process Design and Development 5(3):322–329 Choi JY, Kedzierski MA, Domanski PA (2001) Generalized pressure drop correlation for evaporation and condensation in smooth and micro-fin tubes, IIF-IIR Commission B1, Paderborn, pp. B4.9–B4.16 Clift R, Grace JR, Weber ME (1979) Bubbles, drops, and particles. Academic Press, NY, p. 20 Cooper MG (1984) Saturation nucleate pool boiling- a simple correlation. Department of Engineering Science, Oxford University, England 86:785–793 Cremaschi L, Hwang Y, Radermacher R (2005) Experimental investigation of oil retention in air conditioning systems. Int J Refrig 28(7):1018–1028 Cremaschi L, Wong T, Bigi AAM (2014) Thermodynamic and heat transfer proprieties of Al2O3 nanolubricants. 15th international refrigeration and air conditioning conference at Purdue, paper no 2463, 14–17 July, West Lafayette, IN. Available online at: http://docs.lib.purdue.edu/cgi/ viewcontent.cgi?article=2499&context=iracc Cremaschi L, Bigi AAM, Wong T, Deokar P (2015) Thermodynamic properties of Al2O3 nanolubricants: part 1, effects on the two phase pressure drop. Sci Technol Built Environ, 21: 607–620, https://doi.org/10.1080/23744731.2015.1023165 (online), ISSN: 2374-4731 print / 2374-474X online Cremaschi L, Molinaroli L, Andres C (2016) Experimental analysis and modeling of lubricant effects in microchannel evaporators working with low global warming potential refrigerants. Sci Technol Built Environ 22: 1–14, https://doi.org/10.1080/23744731.2016.118865, ISSN: 23744731 print / 2374-474X online Deokar P, Cremaschi L, Wong T, Criscuolo G (2016) Effect of nanoparticles aspect ratio on the two phase flow boiling heat transfer coefficient and pressure drop of refrigerant and nanolubricants mixtures in a 9.5 mm micro-fin tube. Proceedings of the 16th international refrigeration and air conditioning conference at Purdue University, West Lafayette, IN, 11–14 July, paper no. 2098, pp. 1–10 Hamilton RL, Crosser OK (1962) Thermal conductivity of heterogeneous two-component systems. Ind Eng Chem Fundamen 1(3):187–191 Hamilton LJ, Kedzierski MA, Kaul MP (2008) Horizontal convective boiling of pure and mixed refrigerants within a micro-fin tube. J Heat Transf 15(3):211–226 Incropera FP, DeWitt DP (2002) Fundamentals of heat and mass transfer, 5th edn. John Wiley & Sons, New York Jung DS, McLinden M, Radermacher R (1989) Measurement techniques for horizontal flow boiling heat transfer with pure and mixed refrigerants. Exp Heat Transf 2:237–255 Kedzierski MA (2001) The effect of lubricant concentration, miscibility and viscosity on R134a pool boiling. Int J Refrig 24(4):348–366 Kedzierski MA (2002) Use of fluorescence to measure the lubricant excess surface density during pool boiling. Int J Refrig 25:1110–1122 Kedzierski MA (2003a) A semi-theoretical model for predicting R123/lubricant mixture pool boiling heat transfer. Int J Refrig 26:337–348 Kedzierski MA (2003b) Improved thermal boundary layer parameter for semi-theoretical refrigerant/lubricant pool boiling model. Proceedings of international congress of refrigeration, ICR0504, Washington, DC Kedzierski MA (2007) Effect of refrigerant oil additive on R134a and R123 boiling heat transfer performance. Int J Refrig 30:144–154

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Kedzierski MA (2011) Effect of Al2O3 Nanolubricant on R134a pool boiling heat transfer. Int J Refrig 34(2):498–508 Kedzierski MA (2012) R134a/AlO Nanolubricant mixture pool boiling on a rectangular finned surface. ASME J Heat Transf 134:121501 Kedzierski MA (2013) Viscosity and density of aluminum oxide nanolubricant. Int J Refrig 36(4):1333–1340 Kedzierski MA (2015) Effect of concentration on R134a/Al2O3 Nanolubricant mixture boiling on a reentrant cavity surface. Int J Refrig 49:36–38 Kedzierski MA, Goncalves JM (1999) Horizontal convective condensation of alternative refrigerants within a micro-fin tube. J Heat Transf 6(2–4):161–178 Kedzierski MA, Gong M (2009) Effect of CuO Nanolubricant on R134a pool boiling heat transfer. Int J Refrig 25:1110–1122 Kedzierski MA, Kang DY (2016) Horizontal convective boiling of R448A, R449A, and R452B within a micro-fin tube. Sci TechnolBuilt Environ 22(8):1090–1103. https://doi.org/10.1080/ 23744731.2016.1186460 Kedzierski MA, Kim JH, Didion DA (1992) In: Kim JH, Nelson RA, Hashemi A (eds) Causes of the apparent heat transfer degradation for refrigerant mixtures, two-phase flow and heat transfer, vol 197. HTD, ASME, New York, pp. 149–158 Kedzierski MA, Brignoli R, Quine K, Brown JS (2016) Viscosity, density, and thermal conductivity of aluminum oxide and zinc oxide Nanolubricants. Int J Refrig 74:3–11. https://doi.org/10.1016/ j.ijrefrig.2016.10.003 Laesecke A (2002) Private communications. NIST, Boulder Lemmon EW, Huber ML, McLinden MO (2013) NIST Standard Reference Database 23, Version 9.1. Private Communications with McLinden, National Institute of Standards and Technology, Boulder Maxwell JC (1954) A treatise on electricity and magnetism, vol 1, 3rd edn. Dover, New York, p. 440 Mikic BB, Rohsenow WM (1969) A new correlation of pool boiling data including the effect of heating surface characteristics. J Heat Transf 83:245–250 Radermacher R, Cremaschi L, Schwentker RA (2006) Modeling of oil retention in the suction line and evaporator of air conditioning systems. HVAC & R Research Journal 12(1):35–56 Reid RC, Prausnitz JM, Sherwood TK (1977) The properties of gases and liquids, 3rd edn. McGraw-Hill, New York, p. 460 Rosen MJ (1978) Surfactants and interfacial phenomena. John Wiley & Sons, New York, p. 57 Sawant NN, Kedzierski MA, Brown JS (2007) Effect of lubricant on R410A horizontal flow boiling. NISTIR 7456. U.S. Department of Commerce, Washington, D.C. Schluender EU (1983) Heat transfer in nucleate boiling of mixtures. Int Chem Eng 23(4):589–599 Shock RAW (1982) Boiling in multicomponent fluids. In: Multiphase Science and Technology, vol 1. Hemisphere Publishing Corporation, New York, pp. 281–386 Thome JR (1989) Prediction of the mixture effect on boiling in vertical thermosiphon reboilers. Heat Transf Eng 12(2):29–38 Thome JR (1990) Enhanced boiling heat transfer. Hemisphere Publishing Corporation, Washington, D.C. Thome JR (1995) Comprehensive thermodynamic approach to modeling refrigerant-lubricating oil mixtures. Int J HVAC&R Res 1(2):110–126 Thome JR (1999) Flow boiling inside microfin tubes: recent results and design methods. In: Kakac S et al (eds) Heat transfer enhancement of heat exchangers, Series E: applied sciences, vol 355. Kluwer Academic Publishers, Dordrecht, pp. 467–486 Tipler PL (1978) Modern physics. Worth Pub, New York, p. 320 Wasekar VM, Manglik RM (1999) A review of enhanced heat transfer in nucleate pool boiling of aqueous surfactant and polymeric solutions. J Enhanced Heat Transf 6:135–150 Wen DS, Wang BX (2002) Effects of surface wettability on nucleate pool boiling heat transfer for surfactant solutions. Int J Heat Mass Transf 45:1739–1747 Zou X, Gong MQ, Chen GF, Sun ZH, Zhang Y, Wu JF (2010) Experimental study on saturated flow boiling heat transfer of R170/R290 mixtures in a horizontal tube. Int J Refrig 33(2):371–380

45

Boiling in Reagent and Polymeric Solutions Raj M. Manglik

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Nucleate Boiling in Water . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Nucleate Boiling in Reagent and Polymeric Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Molecular Dynamics of Water-Soluble Additives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Dynamic Gas-Liquid Interfacial Tension Relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Liquid-Solid Surface Wetting and Electrokinetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Nucleate Boiling Heat Transfer in Aqueous Reagent Solutions . . . . . . . . . . . . . . . . . . . . 3.5 Nucleate Boiling Heat Transfer in Aqueous Polymeric Solutions . . . . . . . . . . . . . . . . . . 4 Concluding Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1824 1826 1828 1828 1831 1836 1839 1842 1845 1845

Abstract

Multiphase interfaces and the convective mass, momentum, and heat transport across their boundaries fundamentally govern boiling heat transfer. This interfacial activity is further altered and is quite complex in reagent and polymeric aqueous solutions. At the gas-liquid-solid boundaries, wetting at the liquid-solid interface is modified by physisorption of solute, and the liquid-vapor interfacial tension displays a time-dependent behavior due to solute adsorption. At the microscale, the transient transport of surface-active additives (reagents or polymers) in the liquid modulates dynamically the solid-liquid-vapor interfaces during nucleation and subsequent vapor bubble growth dynamics. This essentially characterizes macroscale heat transport and the ebullient signature of boiling. The

R. M. Manglik (*) Thermal-Fluids and Thermal Processing Laboratory, Department of Mechanical and Materials Engineering, University of Cincinnati, Cincinnati, OH, USA e-mail: [email protected] # Springer International Publishing AG, part of Springer Nature 2018 F. A. Kulacki (ed.), Handbook of Thermal Science and Engineering, https://doi.org/10.1007/978-3-319-26695-4_45

1823

1824

R. M. Manglik

molecular characteristics of reagents thus affect and control this process; in fact, surface wetting and surface tension can be decoupled. An overview of the current state of the art of boiling with reagents and polymers is presented in this chapter. Particular emphasis is given to the role of microscale interfacial changes on boiling heat transfer, which are triggered at the molecular scale by the adsorption-desorption of the soluble additive at the liquid-vapor interface and its electrokinetics at the liquid-solid interface. Nomenclature

C C* Csf Ccmc CMC g h hsurf n q00 q00w Ra Rq Rp T Tsat ΔT θ θa σ τ τg τw ζ

1

concentration of reagent or polymeric additives [wppm] or [mol/cc] critical or overlap concentration of polymer [wppm] or [mol/cc] surface-fluid constant in Rohsenow (1952) correlation [
] critical micelle concentration [wppm] or [mol/cc] critical micelle concentration [wppm] or [mol/cc] gravitational acceleration [N] heat transfer coefficient [W/m2 K] heat transfer coefficient of surfactant aqueous solution [W/m2 K] exponent constant in Rohsenow (1952) correlation [
] surface heat flux [W/m2] wall heat flux [W/m2] average roughness [μm] root-mean-square or rms roughness [μm] peak-to-mean roughness [μm] temperature [K] or [ C] saturation temperature [K] or [ C] temperature difference; wall superheat; [K] or [ C] liquid-solid contact angle [ ] advancing contact angle [ ] gas-liquid interfacial tension [N/m] surface age or time scale [s] bubble growth time [s] bubble incubation or waiting period [s] streaming zeta potential [mV]

Introduction

Phase-change heat transfer and nucleate boiling have increasingly found many new applications because very large heat transfer rates can be sustained in the liquid-vapor phase-change process with relatively small driving temperature differences. A wide range of engineering systems that span many different spatial scales (from very large scale to microscale) employ this mode of heat transfer. They include, among others, energy conversion, refrigeration and air-conditioning, chemical thermal processing, heat treatment and manufacturing, microelectronic cooling, and numerous emerging miniaturized devices (MEMS, micro heat pipes, lab-on-chips, etc.). Over the past two

45

Boiling in Reagent and Polymeric Solutions

LIQUID POOL g

vapor bubble

Marangoni convection [Liquid-gas interfacial tension]

1825

Bubble translation and coalescence / breakup [Gas-liquid interfacial tension & viscous interaction] Bubble size and departure frequency [Gas-liquid interfacial tension]

Nucleation & micro-layer evaporation [Liquid-solid interfacial tension or wetting] HEATED SURFACE

q” > 0

Cavity activation [Liquidsolid interfacial tension or wetting]

Fig. 1 Schematic delineation of the role of different interfacial properties at the various gas-liquid and liquid-solid interfaces that come into play during nucleation and bubble growth dynamics in boiling at a rough surface (Manglik and Jog 2009)

centuries and more, much of the research effort has been directed toward unraveling of many fundamental issues associated with the different applications (Bergles 1981; Nishikawa 1987; Manglik 2006; Manglik and Jog 2009). Moreover, the current imperatives of conservation and sustainable consumption as well as production of different forms of energy not only provide new research incentives but also add new dimensions to this work (Manglik 2006, 2011). In multiphase systems (co-existing gas, liquid, or solid states), many different interfaces can exist, depending upon which state is finely dispersed in another (Jaycock and Parfitt 1981; Myers 1999; Hunter 2001). Note that the contact region between two homogeneous phases over which intensive properties change from those of one phase to that of the other essentially defines an interface. The stable boundary demarcating this region tends to alter the interface area by virtue of its interfacial free energy (Jaycock and Parfitt 1981; Myers 1999). The dynamic energy or interfacial force balances, particularly between gas-liquid and solid-liquid interfaces, characterize the vapor bubble activity or ebullience in heterogeneous boiling and flow evaporation. The primary mechanisms that govern interfacial dynamics at the gas-liquid-solid interfaces during bubble nucleation and growth in surface boiling are schematically depicted in Fig. 1. Even though the basic phase-change process in nucleate boiling is quite complex and involves many different variables (Bergles 1988; Carey 2008; Rohsenow 1952), a set of primary functional determinants can be characterized by the transient microscale transport mechanisms at (a) liquid-solid interface, or surface wetting, and (b) liquid-vapor interface, or the gas-liquid interfacial tension (commonly referred to as “surface tension”). For instance, the surface wetting at the

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R. M. Manglik

liquid-solid interface influences micro-cavity activation, nucleation of embryonic vapor bubbles, thin-film spreading, and micro-layer evaporation (Dhir 2001; Han and Griffith 1965; Hsu 1962). Similarly, the time-variant and evolving liquid-vapor interface is affected by the surface tension behavior that determines post-nucleation bubble dynamics (bubble size, shape, frequency, coalescence, breakup, and translation) and where Marangoni flow or micro-convection may also occur (Kenning 1999; Shoji 2004; Wasekar and Manglik 2003). In the post-departure bubble dynamics, besides the interfacial tension, the shear force at the liquid-vapor interface or viscous interaction also plays a role (Sadhal et al. 1997). Characterization of these mechanisms along with the modeling and scaling of the underlying physics, in order to develop more accurate predictive correlations or models for the heat and mass transport during the boiling process, has been the focus of many investigations over the past several decades (Jakob 1936; Manglik and Kraus 1996; McAdams et al. 1949; Shoji 2004; Subramani et al. 2008). The ability to passively control surface wetting at the liquid-solid interface along with the dynamic interfacial tension at the liquid-vapor interface is indeed attractively offered by the molecular-scale adsorption dynamics and electrokinetics of surface-active additives in the boiling system. The latter include aqueous solutions of a variety of reagents, such as soluble surfactants and polymers (Wasekar and Manglik 2000; Zhang and Manglik 2003, 2005b). In pure liquids (say distilled water), the interfacial or surface tension and wetting behaviors are linked, and a low-surface tension liquid has high wetting and vice versa. By adding reagents in water, for example, these two interfacial properties can be passively decoupled to the extent that low-surface tension and yet low wetting can manifest (Fuerstenau 2002; Somasundaran 2002; Rosen 2004; Manglik 2011). This behavior, however, is dependent upon the molecular structure of the reagent and its electrokinetics in solution. Such a control strategy has transformational consequences for a variety of thermal processing applications where nucleate boiling is involved. These range from miniature or microscale thermal management devices to large-scale terrestrial and space-based heat exchangers.

2

Nucleate Boiling in Water

As a useful prelude to delineating phase-change in aqueous reagent or polymeric solutions, it is necessary to dwell briefly on nucleate boiling in pure water. The pioneering work of Nukiyama (1934) and Jakob and Fritz (1931) may be recalled as a starting period in this enterprise. While Nukiyama presented the complete boiling curve, Jakob and Fritz reported one of the earliest discussions of interfacial effects in their study of bubble dynamics and role of heater surface roughness and nucleation sites on boiling history. It is especially noteworthy that the latter continues to be a critical issue in the attempts to develop refined models for predicting heterogeneous bubble nucleation and boiling. The interplay of surface cavities (or surface roughness, necessary for providing nucleation sites) and liquid wetting and its evolved understanding have led to the development of a variety of novel enhanced boiling

45

Boiling in Reagent and Polymeric Solutions

Wall Heat Flux q" [kW/m2]

100

Vachon et al. (1968) n =1

1827

H2O-Cu (polished) Csf = 0.0128 H2O-Cu (lapped) Csf = 0.0147 H2O-Cu (scored) Csf = 0.015

H2O-Brass Csf = 0.015 n = 0.81 Pioro (1999)

Rohsenow (1952) Borishanskii (1969) Cooper (1984) Cornwell & Houston (1994) Run 1 Run 2 Run 3 Run 4

Distilled water Tsat = 100˚C p = 1 atm Rp = 0.244 μm Rq = 0.260 μm Ra = 0.225 μm

10 8 Wall Superheat ∆T [K]

10

Fig. 2 Nucleate boiling data for water at saturated temperature (atmospheric pressure) from a cylindrical horizontal heater (Athavale and Manglik 2009) and predictions from different correlations

surfaces (Bergles 1998; Manglik 2003, 2006; Manglik and Bergles, 2004; Webb and Kim 2005). The prediction of heat transfer in the nucleate pool boiling regime from heated surfaces, where the heaters used in most engineering devices inherently have (and should have) surface roughness, provides the baseline for the discussion on boiling with reagents and polymers. The nucleate boiling curve, as predicted from the more frequently used correlations, is graphed in Fig. 2 along with a set of saturated (atmospheric pressure) pool boiling data for water (Athavale and Manglik 2009). Results are for an electrically heated “rough” cylinder (average roughness Ra  0.225 μm, rms roughness Rq = 0.260 μm, peak-to-mean surface roughness Rp = 0.244 μm). That there is considerable scatter in the prediction envelope is evident from the plots. The data are seen to lie in the middle of the Rohsenow (1952) and Cooper (1984a,b) correlation results, where the former overpredicts and the latter underpredicts the data. While all correlations account for surface roughness effects, the ability to clearly identify and quantify them remains a challenge (Manglik and Jog 2009). This is perhaps best exemplified by the Rohsenow correlation and the selection of the adjustable

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R. M. Manglik

constants (n and Csf), which have since been much revised from the original listings (Vachon et al. 1968; Pioro 1999; Pioro et al. 2004a,b) to account for many different liquid-surface properties. While nucleate boiling heat transfer for saturated water generally scales as q00 / ΔT 3, effective characterization and quantification of heater surface roughness effects on the transport process remains a challenge. Heater surface roughness provides the much needed active and stable cavities for boiling nucleation. The cavity sizes, or more importantly their distribution and topological makeup after accounting for the liquid-surface wetting, provide the quantification of the nucleation site density on a heated surface, which in turn has been proposed as a predictive input (Rohsenow 1952; Marto et al. 1968; Mikiç and Rohsenow 1969; Singh et al. 1976). However, as pointed out in a recent review (Manglik and Jog 2009), while surface roughness does alter boiling behavior, in itself it is only symptomatic and difficult to scale and quantify predictively. Wetting, or the absence thereof, alters the surface geometry for each liquid-heater system, as the roughness may be composed of shallow, conical, reentrant, and a combination set of cavities; some of these get snuffed out under highly wetting conditions. A solution to this heater surface characteristic has been offered in the enhanced heat transfer arena (Bergles 1997; Manglik 2003; Webb and Kim 2005; Bergles and Manglik 2013), where premanufactured structured or enhanced boiling surfaces have been developed. For general heater roughness, the added surface wetting in reagent or polymeric solutions poses additional quantification uncertainties.

3

Nucleate Boiling in Reagent and Polymeric Solutions

The primary determinants of the general boiling problem can essentially be classified under three broad categories: heater, fluid, and heater-fluid interface (Nelson et al. 1996). For nucleate boiling in aqueous reagent solutions, the potential mechanisms that may be involved are depicted as a complex conjugate problem in Fig. 3. Besides the effects of heater geometry, its surface characteristics and wall heat flux level, bulk concentration of additive, surfactant chemistry (ionic nature and molecular weight), dynamic surface tension, surface wetting and nucleation cavity distribution, Marangoni convection, surfactant adsorption and desorption, and foaming are considered to have a significant influence (Zhang and Manglik 2004a, 2005a). Also, the bubble dynamics (inception and gestation ! growth ! departure) is found to be considerably altered with reduced departure diameters, increased frequencies, and decreased coalescence. Scaling and correlating the heat transfer with suitable descriptive parameters for these effects outlined in Fig. 3, however, remains elusive because of the evident complex nature of the problem.

3.1

Molecular Dynamics of Water-Soluble Additives

A variety of different surface-active agents (surfactants, polymers, electrolytes, soluble nanoparticles, etc.) can be added in small concentrations to alter the

45

Boiling in Reagent and Polymeric Solutions

Nucleate Pool Boiling Bubble growth rate Departure frequency Bubble departure size Nucleation -Active nucleation site density -Bubble nucleation rate and bubble size

1829

Dynamic Surface Tension (adsorption/desorption at the liquidvapor interface, bulk chemistry)

PhysicoChemical Properties •Ionic Nature

Surfactant Additive

•Molecular structure •Ethoxylation

Wettability or Contact Angle (physisorption at the solid-liquid interface, surface chemistry)

•Molecular Weight •Micelle

Microlayer thickness Heat flux removal rate

Other factors

•Heater size and its characteristics

Bubble collapse and merge

•Heat flux level (superheat)

Boiling hysteresis

•Far surface features and foaming

•Fluid properties

Fig. 3 The different conceptual mechanisms that are involved in scaling, modeling, and correlating nucleate boiling of aqueous solutions of reagents and polymers (Zhang and Manglik 2005a)

interfacial properties of liquids (Manglik 2006; Gatne et al. 2009; Wasekar and Manglik 2000; Rosen 2004). Among these, surfactants are particularly effective additives as they have a unique long-chain molecular structure composed of a hydrophilic head and a hydrophobic tail, as depicted in Fig. 4a. Surfactants are generally categorized as anionics, cationics, nonionics, and zwitterionics (Holmberg et al. 2003), depending upon their hydrophilic molecular part, which tends to be ionizable, polar, and polarizable. They all have a natural tendency to adsorb at interfaces when added in water, at both liquid-vapor and liquid-solid interfaces in a boiling system. Moreover, at elevated concentrations in the solvent, their solution tends to form an association colloid, where the reagent molecules form aggregates or micelles. The concentration at which micelles form is referred to as the critical micelle concentration or CMC. In nucleate boiling of such solutions, the reagent molecular dynamics significantly alters the interfacial properties. This occurs at both liquid-vapor and liquid-solid interfaces in the vicinity of the nucleating, then growing and translating bubble, as depicted schematically in Fig. 1. The molecular mobility of the reagent or polymeric additive in solution primarily determines the altered interfacial properties, and this in turn is concentration dependent and has a temporal behavior. The dynamic changes that can occur at the two primary interfaces during the phase-change process and the accompanied bubbling activity during boiling are functionally illustrated in Fig. 4b, c. The depicted processes can be summarized as follows: (1) Surfactant adsorption-desorption at the vapor-liquid interface alters the interfacial or surface tension, which decreases continually with increasing concentrations till the critical micelle concentration or CMC is attained. This interfacial

1830 Fig. 4 Molecular makeup and interfacial mobility of reagents and surfactants in aqueous solutions: (a) reagent molecule and its monomer and micelle structure, (b) physisorption of reagent molecules at the solid-liquid interface, and (c) adsorption of reagent molecular at an evolving liquid-vapor interface

R. M. Manglik Hydrophilic head

Hydrophobic tail

C

(a)

monomer

Ions

Hemimicelle

Reversed Hemimicelle

micelle Micelle (bilayer)

Hydrophilic

Hydrophobic

C

(b)

C

(c) tension relaxation is caused by both a diffusion-rate-dependent process and a surface-age- or time-dependent process. They are typically affected by the type of surfactant, its diffusion-adsorption or molecular kinetics, micellar dynamics, level of ethoxylation, and bulk concentration in solution (Holmberg et al. 2003; Manglik et al. 2001; Hunter 2001). (2) The physisorption of surfactant molecules at the solid-liquid interface changes the surface wetting behavior. With increasing concentration, in many aqueous surfactant solutions, the resulting adsorption behavior and surface state can be categorized into four different regimes (Fuerstenau 2002): (i) low-concentration adsorption as individual ions; (ii) sharp increase in the adsorption density due to self-association of adsorbed surfactant ions and the formation of hemimicelles; (iii) adsorption as reverse hemimicelles, with their polar heads oriented both toward the surface and liquid, to render the surface increasingly hydrophilic; and (iv) as the CMC is approached, the adsorption becomes independent of the bulk concentration, and the surfactant molecules form a bilayer on the surface to make it strongly hydrophilic. Furthermore, as observed in some recent experimental studies (Zhang and Manglik 2005a, 2004a; Wasekar and Manglik 2000) and also pointed out by

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Boiling in Reagent and Polymeric Solutions

1831

Hoffmann and Rehage (1986), dilute solutions of both ionic and nonionic surfactants usually behave as Newtonian liquids, and their viscosity is generally close to that of the solvent. In the case of polymeric additives, however, the rheology tends to be non-Newtonian, and the solvent viscosity not only increases substantially but is also shear rate dependent. Most polymers are large molecules, macromolecules, or agglomerates of smaller chemical units called monomers and are broadly classified as biological or nonbiological macromolecules. The higher viscosity of their aqueous solution tends to increase with concentration as well as the molecular weight of the polymer, and often their fluid motion is characterized by a shear-rate-dependent rheology (Carreau et al. 1997; Bird et al. 1987). Furthermore, perhaps with the exception of some polymeric surfactants, such as hydroxyethyl cellulose (HEC) (Athavale et al. 2012) and polyethylene oxide (PEO), most aqueous polymer solutions do not show any significant change in surface tension σ (Zhang and Manglik 2005b; Manglik et al. 2001; Hu et al. 1991). The viscosity of the polymer solution, however, can considerably influence the measurement of surface tension, particularly in relatively higher viscosity solutions and at higher bubble frequency where interfacial viscosity compensation is needed (Zhang and Manglik 2005b; Manglik et al. 2001; Fainerman et al. 1993). Surface tension relaxation or reduction in gas-liquid interfacial tension in reagentladen aqueous solutions is brought about by the molecular adsorption of the additives at the interface (Zhang and Manglik 2005a; Rosen 2004; Holmberg et al. 2003; Miller et al. 1994). The time scales of this process vary from order of a fraction of seconds to minutes, depending upon the reagent or polymer chemistry (as described by molecular weight, ionic character, molecular structure, etc.) and the additive concentration in solution (Rosen 2004). This transient molecular adsorption behavior also causes temporal variations in surface tension at the gas-liquid interface and wetting at the liquid-solid interface. The typical time scales for the boiling bubble dynamics in water are of the order of 10–100 ms (Zhang and Manglik 2005a, 2004b), and as a result, a rather complex interfacial behavior manifests in ebullient phase-change heat transfer in reagent solutions.

3.2

Dynamic Gas-Liquid Interfacial Tension Relaxation

In an equilibrium solution of a surfactant, when the bulk is disturbed either by surface expansion or contraction, as in the case of nucleate boiling, reagent molecules naturally adsorb at the newly formed and expanding gas(vapor)-liquid interface; some desorption may also occur, depending upon their ionic and molecular structure. A functional description of this transient two-step process (Defay and Petré 1971) is as follows: (i) molecular exchange between the surface layer and subsurface layer or sub-layer by adsorption-desorption and (ii) molecular exchange between the sub-layer and bulk solution by diffusion. The subsurface is effectively a “boundary layer” that separates the bulk fluid domain, where only diffusion occurs, from the domain near the interface where only adsorption-desorption occurs. This

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Fig. 5 Effect of reagent concentration and temperature on the change in equilibrium interfacial tension σ in aqueous solutions of three different surfactants (Razafindralambo et al. 1995; Zhang and Manglik 2005a)

90 SDS CTAB Triton X-305

80

T = 23◦C

σ[mN/m]

70

60 T = 80◦C 50

40 Razafindralambo et al. (1995), CTAB

30 100

101

102

103

104

C [wppm]

process is dictated by the molecular mobility of the reagent and it reduces interfacial tension, and its time-dependent nature manifests in a dynamic surface tension behavior at an evolving vapor-liquid interface of a bubble; to attain equilibrium conditions, a finite time is required for complete interfacial relaxation. As evident from the plots in Fig. 5, surface or interfacial tension σ of aqueous solutions of different surfactants decreases with increasing concentrations till CMC, when the reagent molecules agglomerate to form micelles; with C  CMC, σ remains unchanged. The adsorption behavior of all reagents changes significantly around their respective CMC, and the micellization is a property of each solute. Different shapes, sizes, and ionic orientation of micelles can be formed, depending upon the type of reagent, its packing, concentration, temperature, presence of other ions, and water-soluble organic compounds in the solution (Rosen 2004; Holmberg et al. 2003; Zhang and Manglik 2004b; Myers 1999). The adsorption isotherms of Fig. 5 further reveal that σ is also temperature dependent. The σ – C variation data at bulk temperatures of 23  C and 80  C for the three surfactants, namely, anionic SDS, cationic CTAB, and nonionic Triton X-305 (Zhang and Manglik 2005a, 2004b; Razafindralambo et al. 1995), clearly indicate this. While σ is generally lower at the higher temperature, in both cases, σ also decreases with increasing C and asymptotically attains a minimum constant beyond CMC. The reagent adsorption at the newly created vapor-liquid interface of a nucleated and growing bubble is furthermore temporally modulated and has a distinct

45

Boiling in Reagent and Polymeric Solutions

Fig. 6 The temporal (or interfacial surface-age dependent) variation of the gas-liquid interfacial tension or the dynamic surface tension of aqueous solutions of three different reagents at three different concentrations (Zhang and Manglik 2005a)

1833

80 SDS

CTAB

Triton X-305

C/CCMC= 0.5 C/CCMC= 1

70

C/CCMC= 2.0

σ [m N /m ]

60

50

40 Typical bubble frequencies in nucleate boiling of water

30 10–2

10–1 100 Surface Age [s]

101

characteristic time scale. For each reagent, a finite time period is required in order to attain an equilibrium condition between the adsorbed concentration at the interface and the bulk concentration. This gives rise to the dynamic surface tension (DST) behavior or a σ – τ variation at a given bulk concentration where eventually the equilibrium interfacial tension relaxation is attained after a long time period (Rosen 2004; Manglik et al. 2001). The data graphed in Fig. 6 clearly illustrates this timedependent interfacial tension relaxation behavior, which is representative of that shown in Fig. 4c. Molecular mobility of the reagent dictates this process, as a lower molecular mass reagent diffuses faster than its higher molecular mass counterpart. These characteristics, therefore, suggest that the temporal σ relaxation, or the DST, is perhaps the more critical determinant, and not the equilibrium or static surface tension value, of the altered nucleate boiling in aqueous reagent or polymeric solutions. The time scales for equilibrium (or total relaxation) at a new interface are typically of the order of 50–300 ms, whereas those for bubble formation and departure in nucleate boiling ~10–100 ms (Wasekar and Manglik 2003; Zhang and Manglik 2004a,b). Because of the σ – τ characteristics of the data in Fig. 6, when a bubble starts to form at a nucleating site, the initial gas-liquid interfacial tension is equal or close to that of the solvent. It then reduces continually with τ as reagent molecules migrate to and adsorb at the interface; desorption may also occur at the evolving interface till an equilibrium condition is reached and surface tension

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R. M. Manglik

tw

tg

Fig. 7 Schematic representation of the molecular transport of a reagent in solution at an evolving liquid-vapor bubble interface during nucleate boiling (not to scale)

Water (s = 72.3)

SDS Solution (s = 37.5) [mN/m]

DMF (s = 37.1)

Fig. 8 The effects of dynamic surface tension on the pre-departure shape and size of bubbles evolving in an air-liquid experiment (Manglik et al. 2006b; Manglik 2011)

becomes constant. This time-dependent adsorption-desorption around a nucleated and growing bubble is conceptually depicted in Fig. 7. The dynamic surface tension effects on bubble formation and departure are further demonstrated and clarified by the sample results of single-bubble experiments (Manglik 2011) presented in Fig. 8. In this figure, a larger bubble is seen to develop in aqueous SDS (sodium dodecyl sulfate) solution as compared to that in a pure liquid (DMF or N, N-dimenthylformamide), even though both liquids have the same bulk equilibrium σ value (~37 mN/m). Varying surface tension relaxation is also found in aqueous solutions of some types of polymer additives (Athavale et al. 2012; Zhang and Manglik 2005b). An illustrative set of data for the polymer HEC QP-300 is presented in Fig. 9. The variation in equilibrium surface tension, or the liquid-vapor interfacial tension, with different bulk concentrations of the polymer is graphed. The surface tension is seen to continually decrease from that of the solvent (water) to a minimum asymptotic value. The concentration that demarcates the lower inflection point in the σ – C isotherm (or the point of transition to the minimum surface tension asymptote) generally coincides with the critical polymer concentration (CPC), which is also

45

Boiling in Reagent and Polymeric Solutions

1835

Fig. 9 Equilibrium gas-liquid interfacial tension (surface tension) and its variation with polymer concentration in aqueous solution of HEC QP-300 (Data: open symbols from Zhang and Manglik (2005b); closed symbols from Athavale et al. (2012))

referred to as the overlap concentration C* (Zhang and Manglik 2005b; De Gennes 1979; Broseta et al. 1986; Cosgrove and Griffiths 1994). At CPC or C*, polymer agglomeration or coil entanglements begin to form in the semi-dilute regime of the solution. This interfacial tension relaxation is a diffusion-rate-dependent behavior, which is generally governed by the bulk concentration and diffusion-adsorption kinetics of the polymer-solvent systems (Miller and Neogi 1985). Besides the adsorption isotherm data of Fig. 9, the CPC or critical overlap concentration C* can also be determined from the intrinsic viscosity (Athavale et al. 2012). The inverse function of the latter approximately represents the concentration within the polymer, or its overlap concentration in a solvent, exceeding which molecules will touch and interpenetrate to form a semi-dilute solution (De Gennes 1979). Furthermore, the time-dependent liquid-vapor interfacial tension relaxation behavior is seen in the σ – τ plots of Fig. 10. The change in surface tension σ with surface age τ, or the time period of a newly formed bubble from inception to departure, is graphed (aqueous solutions of HEC QP-300; 1.0  10
9  C  4.0  10
9 mol/cc). It is seen that a surface age of τ > 1.0 s is needed for complete interface relaxation, i.e., to attain an equilibrium between the surface and bulk concentrations. For τ < 5 ms, the interfacial tension essentially corresponds to that of the solvent (water), whereas the interim period of 5 ms < τ < 1.0 s is characterized by sharp gradients in σ. In Fig. 10, the data are also mapped by the adsorption isotherm fit using the method given by

1836

R. M. Manglik 75

HEC QP-300, T = 23°C

σ [x 10-3 N/m]

73

1.0 x 10-9 mol/cc 2.5 x 10-9 mol/cc 4.0 x 10-9 mol/cc

σw = 72.4 x 10-3 N/m

71

69

67

65 10-3

στ→ ∞ = 72.4 x 10-3 N/m

10-2

10-1

100

τ [s] Fig. 10 Time-dependent change in the surface (gas-liquid interfacial) tension or the dynamic surface tension signature for different concentration of the polymer HEC QP-300 in aqueous solutions (Athavale et al. 2012)

Hua and Rosen (1988), so as to chart the complete dynamic surface tension behavior. Also, there is little difference between the data for 2.5  10
9 and 4.0  10
9 mol/cc solutions, suggesting micelle agglomeration or coil entanglements of the polymer at these concentrations.

3.3

Liquid-Solid Surface Wetting and Electrokinetics

As depicted in Fig. 4b, the molecular physisorption at the solid-liquid interface or electrokinetics and micellar dynamics of the reagent in solution alters the surface wettability. It also varies with concentration and is typically measured by the liquidsolid interface contact angle. A typical set of contact angle data for three different surfactants are graphed in Fig. 11. It is seen that nonionic surfactants (Triton X-305), because of their lack of charge, undergo a different adsorption process than that for ionic surfactants (anionic SDS and cationic CTAB). Generally with ionic reagents, the contact angle in all cases reaches a lower plateau around CMC where bilayers begin to form on the surface. For nonionic surfactants, however, this minimum constant value is attained with much lower concentrations than the CMC in solution. It has been suggested (Levitz 2002) that because of the rather weak direct interactions of the polar chain in nonionic reagents, it is possible for them to build and rebuild adsorption layers below CMC. The low contact angle trough at

45

Boiling in Reagent and Polymeric Solutions

Fig. 11 Variation of contact angle θ with concentration in different surfactant solutions (Zhang and Manglik 2005a)

1837

90

Contact angle [deg]

80

70 CMC

60

CMC

50

40

SDS (Anionic) CTAB (Cationic) Triton X-305 (Nonionic)

CMC

30

101

102

103

104

Concentration [wppm]

concentrations C < CMC can also be attributed to the absence of any electrical repulsion that could oppose molecular aggregation, which is unlike that associated with ionic surfactants (Miller and Neogi 1985). Moreover, the continuous decrease in contact angle in solutions of nonionic Triton X-305, with relatively low concentrations prior to reaching a constant value, has been attributed to their polar head size. The presence of 30 ethylene oxide (EO) groups in its molecular chain yields a much larger overall size of the polar head. It generally increases with the number of EO groups, which in turn control the hydrophilic-hydrophobic balance of the surfactant molecule (Holmberg et al. 2003). Altered liquid-solid interface wetting is also manifest in aqueous polymeric solutions, and this is evident from Fig. 12. The change in wetting at a metallic substrate (stainless steel) in HEC QP-300 solutions is characterized by the contact angle θ and its variation with concentration C. Surface wetting is seen to increase (decrease in θ) with increasing C till a lower constant-value asymptote is attained. The minimum contact angle plateau is further seen to be attained when C > C* (the overlap concentration), where molecular agglomeration of the polymer begins to form in solution. This is representative of the typical physisorption behavior of surface-active solutes at liquid-solid interfaces, where wetting is influenced by the kinetics of interfacial molecular adsorption (Kwok and Neumann 1999; Lee et al. 2008). Another example of increased wetting behavior is seen in different grades and anilide derivatives of polyethylene carboxylic acid (PEO or PE-CO2-OH) in aqueous solution. Figure 13 presents the change in the advancing contact angle θa (or cosθa, which is proportional to the interfacial free energy as governed by Young’s equation) with the pH of the solution (Wilson and Whitesides 1988). Selective PEO grades and derivatives are seen to promote large hydrophobicity (θ > 90 ) compared to others that contrastingly make the aqueous solutions more hydrophilic (θ < 90 ).

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R. M. Manglik

Fig. 12 Contact angle θ variations with concentration of the polymer HEC QP-300 in aqueous solutions (Athavale et al. 2012)

HEC QP-300, T = 23oC

θw = 77o

77

Surface: Stainless Steel

θ [degrees]

76

75 θC > cpc = 74o

74

73 0.1

1.0 C [x 10-9 mol/cc]

Fig. 13 Variation of advancing contact angle cosθa with pH in different waterpolymer (anilide derivatives of PE-CO2-OH) solutions (Wilson and Whitesides 1988)

O

PE.OS(CH3)3

PE

–0.5

NH

120 PE.H

O

O

0.0

PE

PE

NH

H O

cosθa 2

NH

90 CO2H

CO2H

θa

0.5

60 PE.CO2H O PE

CO2H

NH

30 0

1.0 2

4

6

8 pH

10

12

14

45

Boiling in Reagent and Polymeric Solutions

1839

Most interestingly the data for anilide derivatives (ortho, meta, and para) of PE-(ONH)-CO2H display very dramatic changes in cosθa, where variations from large hydrophobicity to hydrophilicity with varying pH of their water solutions can be observed. Electrokinetic effects manifest when one phase is caused to move tangentially past another phase. In a colloidal system of finely dispersed reagent in solution, many different interfaces can exist, depending upon which state (gas, liquid, or solid) is finely dispersed in another (Hunter 2001), and the interaction between two phases generally develops a potential difference between them. With the presence of ions, or excess electrons, or ionogenic groups in one or both phases, there is a tendency for the electric charges to distribute themselves in a particular direction at the interface (Hunter 2001, 1981; Evans and Wennerström 1999). An electrokinetic boundary layer, or an electric double layer (EDL), develops when a solid surface containing immobilized electrical charges comes in contact with an aqueous solution of mobile ions. For solid-liquid interfacial interactions, the consequent forces can be typically characterized in terms of the electrostatic potential that can fundamentally be quantified by the average potential in the surface of shear or what is often referred to as the streaming zeta potential ζ (Hunter 1981). Thus, the electrokinetics and physisorption of ionic surfactants at the solid-liquid interface, which alters the surface wettability behavior considerably, can be correlated by the ionic exchange in the EDL that is directly reflected in the change in ζ. In correlating surface wettability, or variations in liquid-solid contact angle, with the change in zeta potential ζ (which is an electrokinetic control parameter for the stability of hydrophobic colloids), distinct regions of change in adsorption and corresponding wetting variations are seen that are associated with the aggregation mode of adsorbed ions at the solid-water interface. The data presented in Fig. 14 provide illustrative examples, where the variation in zeta potential and contact angle with concentration in aqueous SDS (anionic) and CTAB (cationic) solutions are graphed. SDS solutions are seen to display a stronger adsorption than CTAB, which is reflected in the magnitude of zeta potential and the larger changes in contact angle. After the point of zeta potential reversal (PZR), also referred to as the isoelectric point (IEP) (Fuerstenau 2002; Hunter 1981), the slope of the ζ curve becomes negative for the anionic SDS and positive for the cationic CTAB because of the opposite charges they carry. This suggests that some adsorption may take place in reverse orientation to form reverse hemimicelles (Fuerstenau 2002), to render the surface increasingly hydrophilic. A bilayer is formed near CMC, and the contact angle tends to be constant as the surface becomes highly hydrophilic.

3.4

Nucleate Boiling Heat Transfer in Aqueous Reagent Solutions

The altered pool boiling behavior over a cylindrical heater, where both enhanced heat transfer and suppressed boiling are seen, is represented by the variation in the relative heat transfer coefficient h in Fig. 15. Data for three different surfactant solutions in water (anionic SDS, cationic CTAB, and nonionic Triton X-305) are

1840

R. M. Manglik 101 90

103

102

80

-30

70

60

-10 50

PZR

0

Zeta Potential [mv]

Contact angle [deg]

-20

40

30

20 101

Contact angle, CTAB Contact angle, SDS Zeta potential (Vanjara and Dixit, 1996), CTAB Zeta potential (Sakagami et al. 2002), SDS 102

103

10

20 104

Concentration [wppm]

Fig. 14 Correspondence between streaming zeta potential and liquid-solid contact angle in typical cationic CTAB and anionic SDS surfactant solutions (Manglik 2011)

graphed (Wasekar and Manglik 2000, 2002; Zhang and Manglik 2004a, 2005a). The results in each case have been normalized with those for deionized distilled water with same boiling conditions. The heat transfer coefficient is seen to increase with concentration (0 < C  CMC or Ccmc), and an optimum enhancement is obtained in solutions at or near CMC (C/Ccmc) = 1. But with C > CMC, the enhancement decreases and the heat transfer even deteriorates below that in pure water, particularly at low heat fluxes. Furthermore, early ONB (onset of nucleate boiling or bubble incipience) with C  CMC, but delayed incipience when C > CMC, has been reported (Zhang and Manglik 2004b). A thermal hysteresis was also observed in the latter case, which is characteristic of high wettability (Bar-Cohen 1992). In fact, a much larger temperature overshoot was observed in SDS solutions with very high concentrations (CCMC) as compared to that in Triton X-305 and CTAB solutions (in that order) with corresponding concentrations. This in essence follows the surface wettability trends seen with the contact angle variations observed in Fig. 11 with these reagents.

45

Boiling in Reagent and Polymeric Solutions

Fig. 15 Nucleate boiling heat transfer results for aqueous solutions of three different reagents (anionic SDS, cationic CTAB, and nonionic Triton X-305) (Manglik 2011)

1841

0.9 0.8

C/Cc.m.c.= 0.5

C/Cc.m.c.= 1

C/Cc.m.c.= 2

SDS CTAB Triton X-305

0.7

(hsurf - hwater) / hwater

0.6

Tsat =100°C

0.5 0.4 0.3 0.2 0.1 0.0 -0.1 -0.2

101

102

q" w

(kW/m2)

The primary boiling control, as stated earlier and identified in Fig. 1, is directly related to nucleation site activation and density (influenced by wetting) and bubble dynamics (growth through departure, influenced by interfacial tension). The altered bubbling behavior in the nucleate boiling regime with typical aqueous reagent solutions is photographically presented in Fig. 16. Compared to pure water, boiling in SDS and CTAB solutions is more vigorous with activation of smaller-sized, more regularly shaped bubbles that nucleate in a cluster of sites. They have a significantly higher departure frequency, with virtually no coalescence of either the neighboring or sliding bubbles when C < CMC. However, when C  CMC, foaming patches begin to occur, the area of liquid-only coverage on the heater surface increases, and slightly larger bubbles are formed. This is indicative of a change in surface wetting, and it is more evident in the SDS solutions. Boiling with Triton X-305, on the other hand, shows much smaller-sized bubbles in pre-CMC solutions, and considerably fewer and larger-sized bubbles are formed with increasing concentrations. This behavior contrasts from not only that of water but also of SDS and CTAB and is predicative of the continuous increase in wetting (or decrease in contact angle θ) with increasing concentration of this surfactant. The nucleation and bubbling behavior in aqueous solutions of reagents cannot be explained simply by the dynamic surface tension relaxation effects alone. If this were

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Fig. 16 Bubbling behavior during saturated nucleate pool boiling of distilled water and aqueous solutions of anionic SDS, cationic CTAB, and nonionic Triton X-305 (Zhang and Manglik 2005a)

Water

SDS

22.2 mm

CTAB

Triton X-305

C/CCMC = 0.5

C/CCMC = 1

C/CCMC = 2 20 kW/m2

so, then the smallest-sized bubbles would be seen in C  CMC solutions, where σ reaches the lowest possible value. Instead, because of the adsorption of surfactant molecules and their different orientations in the adsorption layer, the heater surface has higher wetting with increasing concentration and larger nucleation cavities get flooded. Fewer bubbles are thus nucleated, and they tend to have relatively larger departure diameters. Such bubble nucleation and growth departure behavior, and its interrelationship with reagent-induced wetting as well as dynamic interfacial tension changes, thus fundamentally scale and correspond with the reagent molecular dynamics.

3.5

Nucleate Boiling Heat Transfer in Aqueous Polymeric Solutions

In the case of boiling in polymeric solutions, the general phase-change and associated bubbling behavior tends to be more complex, and it is altered by not only changes in surface tension and heater surface wettability but the solution rheology as well. While wetting of the heater surface controls nucleation and the site density thereof, the post-nucleated bubble dynamics is affected by both the gas-liquid interfacial tension relaxation and the shear-dependent viscosity variations of the polymeric solution (Zhang and Manglik 2005a,b; Manglik and Jog 2009;

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Boiling in Reagent and Polymeric Solutions

1843

Fig. 17 Nucleate boiling heat transfer performance of aqueous HEC QP-300 solutions, relative to that of distilled, deionized water, and its variation with polymer concentration and wall heat flux (Zhang and Manglik 2005b; Athavale et al. 2012)

Athavale et al. 2012). For example, aqueous solutions of hydroxyethyl cellulose HEC QP-300, except in very low concentrations, generally display a viscous pseudoplastic behavior. At low shear rates the solution rheology tends to be Newtonian, but with a significantly higher viscosity than the solvent. With increasing shear rates, however, the shear-thinning non-Newtonian viscosity manifests (Athavale et al. 2012). This has complex implications for bubble dynamics, respectively, in the partial and fully developed nucleate boiling regimes, where the bubbling frequencies and extent of vapor generation tend to have different shear rates. The effects of heat flux q00w and molar concentration C on the nucleate boiling heat transfer performance in typical aqueous polymer (HEC QP-300) solutions are graphed in Fig. 17. The data are those reported by Athavale et al. (2012), and the variation in the heat transfer coefficient, quantified by the dimensionless ratio [(h
hw/hw)] with the heater wall heat flux q00w , is graphed. In general, the boiling heat transfer is enhanced in weak solutions with perhaps the highest boiling performance is attained with critical or overlap concentration (CPC or C*) of the polymer (Zhang and Manglik 2005b; Athavale et al. 2012). However, an anomalous boiling behavior is seen in solutions with higher concentration. There is consistent and virtually constant enhancement of about 20% in the 1.0  10
9 mol/cc (~ C*) HEC concentration aqueous solution (Fig. 17). The more curiously unexpected set of results are those for solutions with C = 2.5  10
9 and 4.0  10
9 mol/cc. In these two example cases, there is a reduction (or degradation with respect to pure water) in heat

1844

R. M. Manglik

transfer when q00w < 30 kW/m2, but much larger enhancement than that with C* when q00w > 100 kW/m2. The higher of the two concentrations produces about 6% decrease in heat transfer when compared with water for 7 < q00w < 30 kW/m2. The boiling performance seen in Fig. 17 is a direct consequence of the higher polymeric solution viscosity, which is ~2–3 times that of the solvent water (Athavale et al. 2012). At relatively low shear rates (typically the case with bubbling in the early or low heat flux partial boiling regime), the high viscous drag effects at the liquid-vapor interface of ebullient transport would be significant. With increasing heat flux and hence larger vapor generation, however, the interfacial shear rate increases and thus the viscosity of the shear-thinning solutions also decreases. In fact, at very high shear rates the higher concentration solution even becomes less viscous than the lower C ones (Athavale et al. 2012). The retarding viscous forces at the bubble-liquid pool interface consequently become less significant, and the low-surface tension-driven enhancement, characterized by smaller and higher-frequency bubble generation, is reestablished with peak heat transfer performance attained when q00w > 100 kW/m2. On the other hand, in the case of aqueous solutions of a different polymer, Carbopol 934, and as reported by Zhang and Manglik (2005b), there is no enhancement in heat transfer. Contrarily, there is significant deterioration in the boiling performance. It has been observed that the ebullience or bubbling activity in Carbopol 934 solutions is very different from that in water as well as HEC solutions. There is considerable bubble suppression, along with dispersed vapor explosions in some regions of the heater surface. This kind of bubbling activity was also observed (Bang et al. 1997) in boiling of dilute polyethylene oxide (PEO) solutions. Moreover, there is delayed incipience, or ONB, and the sparsely formed bubbles have a slower departure frequency. This is essentially due to the increased viscosity of Carbopol 934 solutions, which lends to a higher drag resistance for the nucleating and departing bubbles to overcome. The bubbling process in HEC QP-300 solutions is contrastingly more vigorous, with smaller-sized and more regularly shaped bubbles that have a reduced tendency for coalescence when C < C*. There is an early inception of bubbles with a faster covering of the heating surface and a higher bubble departure frequency, which is essentially the outcome of reduced surface tension at the liquid-vapor interface. Also, molecular adsorption on the heating surface may contribute to the formation of new sites (Levitskiy et al. 1996), which in turn would explain the increase in number of bubbles as there was no change in the measured surface wettability (Manglik et al. 2003). In general, the reduction in dynamic surface tension (which decreases the required superheat for the onset of boiling) and the macromolecular adsorption on the heating surface (which could contribute to the formation of new nucleation sites and increased bubble frequency) are perhaps two main factors for the enhancement of boiling heat transfer in low-concentration (C < C*), surface-active HEC QP-300 solutions. The decreases in the nucleate boiling heat transfer coefficients in Carbopol 934 solutions, as well as those of HEC QP-300 with concentrations C > C*, on the other hand, are possibly associated with the substantial increase in the liquid viscosity that tends to suppress micro-convection in the bubble boundary layer as well as retard the growth of vapor bubbles.

45

4

Boiling in Reagent and Polymeric Solutions

1845

Concluding Observations

Interfacial properties at the liquid-vapor (dynamic and equilibrium surface tensions) and solid-liquid (surface wetting or contact angle) interfaces characterize the adsorption behaviors of reagent and polymeric additives in their aqueous solutions. Surfactants tend to lower the surface tension of water considerably, and their adsorption-desorption process at the liquid-vapor interface is time-dependent. This manifests in a dynamic surface tension behavior, which eventually reduces to an equilibrium value after a long time span. Also, the reagent physisorption process and electrokinetics tend to follow a characteristic adsorption isotherm at the solid-liquid interface, and the consequent changes in surface wetting or contact angle correlate well with the adsorption characterization or zeta potential. These phenomena substantially alter the saturated nucleate pool boiling performance of aqueous solutions of reagents. Enhanced heat transfer is generally obtained in solutions with C  CMC, but there is a decrease in heat transfer, and even deterioration below that of pure water, when C > CMC. Besides the heat flux (or wall superheat) levels, the surfactant interfacial phenomena at both the liquid-vapor and solid-liquid interfaces govern the boiling bubble dynamics. In aqueous polymer solutions, the boiling behavior is altered essentially due to the changes in rheological and interfacial characteristics. They are generally more viscous than water; with a distinct shear rate-dependent non-Newtonian shearthinning rheology, they also tend to exhibit a surface-active nature. This is particularly the case with HEC QP-300 in water solutions, which show relaxation of both dynamic and equilibrium surface tension. As a consequence, with C < the critical polymer or overlap concentration C*, there is considerable heat transfer enhancement. In C > C* solutions, however, there is a significant decrease in the heat transfer. This is possibly due to the retardation of vapor bubble growth and suppression of micro-convection in the boundary layer because of the high viscosity of highconcentration solutions. The substantially viscous Carbopol 934 solutions, with little gas-liquid interfacial relaxation, however tend to exhibit poor or deteriorated nucleate boiling heat transfer. Acknowledgments The facilities and other support from the Thermal-Fluids & Thermal Processing Laboratory, University of Cincinnati; the US Department of Energy, ARPA-E; as well as the collegial and collaborative discussions with Professor Milind A. Jog, University of Cincinnati, are gratefully acknowledged.

References Athavale A, Manglik RM (2009) Experiments in pool boiling from a horizontal cylindrical heater in water and polymeric aqueous solutions, Report No. TFTPL-09-PR. University of Cincinnati, Cincinnati Athavale AD, Manglik RM, Jog MA (2012) An experimental investigation of nucleate pool boiling in aqueous solutions of a polymer. AIChE J 58(3):668–677 Bang KH, Kim MH, Jeun GD (1997) Boiling characteristics of dilute polymer solutions and implications for the suppression of vapor explosions. Nucl Eng Des 177(1–3):255–264

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Bar-Cohen A (1992) Hysteresis phenomena at the onset of nucleate boiling. In: Pool and external flow boiling. ASME, New York, pp 1–14 Bergles AE (1981) two-phase flow and heat transfer, 1756–1981. Heat Transfer Eng 2(3–4):101–114 Bergles AE (1988) Fundamentals of boiling and evaporation. In: Kakaç S, Bergles AE, Fernandes EO (eds) two-phase flow heat exchangers: thermal-hydraulic fundamentals and design. Kluwer, Dordrecht, pp 159–200 Bergles AE (1997) Enhancement of pool boiling. Int J Refrig 20(8):545–551 Bergles AE (1998) Techniques to enhance heat transfer. In: Rohsenow WM, Hartnett JP, Cho YI (eds) Handbook of heat transfer, 3rd edn. McGraw-Hill, New York, p 11 Bergles AE, Manglik RM (2013) Current progress and new developments in enhanced heat and mass transfer. J Enhanc Heat Transfer 20(1):1–15 Bird RB, Armstrong RC, Hassager O (1987) Dynamics of polymeric liquids: Vol. 1 – fluid mechanics, vol 1. Wiley, New York Broseta D, Leibler L, Lapp A, Strazielle C (1986) Universal properties of semi-dilute polymer solutions: a comparison between experiments and theory. Europhys Lett 2(9):733–737 Carey VP (2008) Liquid-vapor phase-change phenomena, 2nd edn. Taylor & Francis, New York Carreau PJ, De Kee DCR, Chhabra RP (1997) Rheology of polymeric systems: principles and applications. Hanser Gardner Publications, New York Cooper MG (1984a) Heat flow rates in saturated nucleate pool boiling – a wide-ranging examination using reduced properties. In: Hartnett JP, Irvines TF Jr (eds) Advances in heat transfer, vol 16. Academic Press, New York, pp 157–239 Cooper MG (1984b) Saturated nucleate pool boiling: a simple correlation. In: First U.K. national conference on heat transfer. Institute of Chemical Engineers, University of Leeds, pp 785–793 Cosgrove T, Griffiths PC (1994) The critical overlap concentration measured by pulsed field gradient nuclear magnetic resonance technique. Polymer 35(3):509–513 De Gennes P (1979) Scaling concepts in polymer physics. Cornell University Press, Ithaca Defay R, Petré G (1971) Dynamic surface tension. In: Matijevic E (ed) Surface and colloid science. Wiley, New York, pp 27–81 Dhir VK (2001) Numerical simulation of pool-boiling heat transfer. AIChE J 47(4):813–834 Evans DF, Wennerström H (1999) The colloidal domain – where physics, chemistry, and biology meet, 2nd edn. Wiley-VCH, New York Fainerman VB, Makievski AV, Miller R (1993) The measurement of dynamic surface tensions of highly viscous liquids by the maximum bubble pressure method. Colloids Surf A Physicochem Eng Asp 75(1):229–235 Fuerstenau DW (2002) Equilibrium and nonequilibrium phenomena associated with the adsorption of ionic surfactants at solid-water interfaces. J Colloid Interface Sci 256(1):79–90 Gatne KP, Jog MA, Manglik RM (2009) Surfactant-induced modification of low weber number droplet impact dynamics. Langmuir 25(14):8122–8130 Han CY, Griffith P (1965) The mechanism of heat transfer in nucleate pool boiling-part I: bubble initiation, growth and departure. Int J Heat Mass Transf 8(6):887–904 Hoffmann H, Rehage H (1986) Rheology of surfactant solutions. In: Zana R (ed) Surfactant solutions – new methods of investigation, vol 22. Marcel Dekker, New York, pp 209–239 Holmberg K, Jönsson B, Kronberg B, Lindman B (2003) Surfactants and polymers in aqueous solution, 2nd edn. Wiley, New York Hsu YY (1962) On the size of range of active nucleation cavities on a heating surface. J Heat Transfer 84:207 Hu RYZ, Wang ATA, Hartnett JP (1991) Surface tension measurement of aqueous polymer solutions. Exp Thermal Fluid Sci 4(6):723–729 Hua XY, Rosen MJ (1988) Dynamic surface tension of aqueous surfactant solutions. I basic parameters. J Colloid Interface Sci 124(2):652–659 Hunter RJ (1981) Zeta potential in colloid science-principles and applications. Academic Press, New York Hunter RJ (2001) Foundations of colloid science, 2nd edn. Oxford University Press, Oxford

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Jakob M (1936) Heat transfer in evaporation and condensation – I. Mech Eng 58:643–660 Jakob M, Fritz W (1931) Versuche über den Verdampfungsvorgang. Forschung auf dem Gebiete des Ingenieurwesens 2:435–447 Jaycock MJ, Parfitt GD (1981) Chemistry of interfaces. Ellis Horwood, Chichester Kenning DBR (1999) What do we really know about nucleate boiling. In: IMechE Trans, 6th UK national heat transfer conference, Edinburgh, pp 143–167 Kwok DY, Neumann AW (1999) Contact angle measurement and contact angle interpretation. Adv Colloid Interf Sci 81(3):167–249 Lee KS, Ivanova N, Starov VM, Hilal N, Dutschk V (2008) Kinetics of wetting and spreading by aqueous surfactant solutions. Adv Colloid Interf Sci 144(1–2):54–65 Levitskiy SP, Khusid BM, Shul'man ZP (1996) Growth of vapor bubbles in boiling polymer solutions-II. Nucleate boiling heat transfer. Int J Heat Mass Transf 39(3):639–644 Levitz PE (2002) Adsorption of non ionic surfactants at the solid/water interface. Colloids Surf A Physicochem Eng Asp 205(1–2):31–38 Manglik RM (2003) Heat transfer enhancement. In: Bejan A, Kraus AD (eds) Heat transfer handbook. Wiley, Hoboken, p 14 Manglik RM (2006) On the advancements in boiling, two-phase flow heat transfer, and interfacial phenomena. J Heat Transf 128(12):1237–1242 Manglik RM (2011) Molecular-to-macro-scale control of interfacial behavior in Ebullient phase change in aqueous solutions of reagents. Int J Transp Phenom 12(3–4):229–243 Manglik RM, Bergles AE (2004) Enhanced heat and mass transfer in the new millennium: a review of the 2001 literature. J Enhanc Heat Transfer 11(2):87–118 Manglik RM, Jog MA (2009) Molecular-to-large-scale heat transfer with multiphase interfaces: current status and new directions. J Heat Transf 131(12):121001–121011 Manglik RM, Kraus AD (1996) Process, enhanced, and multiphase heat transfer. Begell House, New York Manglik RM, Wasekar VM, Zhang J (2001) Dynamic and equilibrium surface tension of Aqueous surfactant and polymeric solutions. Exp Thermal Fluid Sci 25(1–2):55–64 Manglik RM, Bahl M, Vishnubhatla S, Zhang J (2003) Interfacial and rheological characterization of aqueous surfactant and polymer solutions. University of Cincinnati, Cincinnati Manglik RM, Jog MA, Subramani A, Gatne K (2006) Mili-scale visualization of bubble growthtranslation and droplet impact dynamics. J Heat Transf 128(8):736 Marto PJ, Moulson JA, Maynard MD (1968) Nucleate pool boiling of nitrogen with different surface conditions. J Heat Transf 90(4):437–444 McAdams WH, Kennel WE, Mindon CS, Carl R, Picornel PM, Dew JE (1949) Heat transfer to water with surface boiling. Ind Eng Chem 41:1945–1953 Mikiç BB, Rohsenow WM (1969) A new correlation of pool-boiling data including effect of heating surface characteristics. J Heat Transf 9(2):245–250 Miller CA, Neogi P (1985) Interfacial phenomena: equilibrium and dynamic effects. Marcel Dekker, New York Miller R, Joos P, Fainerman VB (1994) Dynamic surface and interfacial tensions of surfactant and polymer solutions. Adv Colloid Interf Sci 49(1):249–302 Myers D (1999) Surfaces, interfaces, and colloids. 2nd edn. Wiley-VCH, New York Nelson RA, Kenning DBR, Shoji M (1996) Nonlinear dynamics in boiling phenomena. Jpn Heat Transfer Soc J 35(136):22–34 Nishikawa K (1987) Historical developments in the research of boiling heat transfer. JSME Int J 30 (264):897–905 Nukiyama S (1934) The maximum and minimum values of the heat Q transmitted from metal to boiling water under atmospheric pressure. J Jpn Soc Mech Eng 37:367–374. (Translation in Int J Heat Mass Transfer 369:1419–1433, 1966) Pioro IL (1999) Experimental evaluation of constants for the Rohsenow pool boiling correlation. Int J Heat Mass Transf 42(11):2003–2013

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Pioro IL, Rohsenow W, Doerffer SS (2004a) Nucleate pool-boiling heat transfer. I: review of parametric effects of boiling surface. Int J Heat Mass Transf 47(23):5033–5044 Pioro IL, Rohsenow W, Doerffer SS (2004b) Nucleate pool-boiling heat transfer. II: assessment of prediction methods. Int J Heat Mass Transf 47(23):5045–5057 Razafindralambo H, Blecker C, Delhaye S, Paquot M (1995) Application of the quasi-static mode of the drop volume technique to the determination of fundamental surfactant properties. J Colloid Interface Sci 174(2):373–377 Rohsenow WM (1952) A method of correlating heat transfer data for surface boiling of liquids. Trans ASME 74(3):969–976 Rosen MJ (2004) Surfactants and interfacial phenomena, 3rd edn. Wiley-Interscience, Hoboken Sadhal SS, Ayyaswamy PS, Chung JN (1997) Transport phenomena with drops and bubbles. Springer, New York Sakagami K, Yoshimura T, Esumi K (2002) Simultaneous adsorption of poly(1-vinylpyrrolidoneco-acrylic acid) and sodium dodecyl sulphate at alumina/water interface. Langmuir 18:6049–6053 Shoji M (2004) Studies of boiling chaos: a review. Int J Heat Mass Transf 47(6–7):1105–1128 Singh A, Mikiç BB, Rohsenow WM (1976) Active sites in boiling. J Heat Transfer 98(3):401–406 Somasundaran P (2002) Simple colloids in simple environments explored in the past, complex nanoids in dynamic systems to be conquered next: some enigmas, challenges, and strategies. J Colloid Interface Sci 256(1):3–15 Subramani A, Jog MA, Manglik RM (2008) Air-water ebullience systems: visualizing single bubble to wave instability signatures. J Heat Transfer 129(8):080905–080901 Vachon RI, Nix GH, Tanger GE (1968) Evaluation of constants for the Rohsenow pool-boiling correlation. J Heat Transf 90(2):239–247 Vanjara AK, Dixit SG (1996) Adsorption of alkyltrimethylammonium bromide and alkylpyridinium chloride surfactant series on polytetrafluoroethylene powder. J Colloid Interface Sci 177:359–363 Wasekar VM, Manglik RM (2000) Pool boiling heat transfer in aqueous solutions of an anionic surfactant. J Heat Transf 122(4):708–715 Wasekar VM, Manglik RM (2002) The influence of additive molecular weight and ionic nature on the pool boiling performance of aqueous surfactant solutions. Int J Heat Mass Transf 45 (3):483–493 Wasekar VM, Manglik RM (2003) Short-time-transient surfactant dynamics and marangoni convection around boiling nuclei. J Heat Transf 125(5):858–866 Webb RL, Kim N-H (2005) Principles of enhanced heat transfer, 2nd edn. Taylor & Francis, Boca Raton Wilson MD, Whitesides GM (1988) The Anthranilate Amide of "Polyethylene Carboxylic Acid" shows an exceptionally large change with pH in its wettability by water. J Am Chem Soc 110 (26):8718–8719 Zhang J, Manglik RM (2003) Visualization of ebullient dynamics in aqueous surfactant solutions. J Heat Transfer 125(4):547 Zhang J, Manglik RM (2004a) Effect of ethoxylation and molecular weight of cationic surfactants on nucleate boiling in aqueous solutions. J Heat Transf 126(1):34–42 Zhang J, Manglik RM (2004b) Experimental and computational study of nucleate pool boiling heat transfer in aqueous surfactant and polymer solutions. University of Cincinnati, Cincinnati Zhang J, Manglik RM (2005a) Additive adsorption and interfacial characteristics of nucleate pool boiling in aqueous surfactant solutions. J Heat Transfer 127(7):684–691 Zhang J, Manglik RM (2005b) Nucleate pool boiling of aqueous polymer solutions on a cylindrical heater. J Non-Newtonian Fluid Mech 125(2–3):185–196

Fundamental Equations for Two-Phase Flow in Tubes

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Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Basic Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Two-Phase Flow Patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Vertical Tube . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Horizontal Tube . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Two-Phase Flow Pattern Maps and Transition Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Two-Phase Flow Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Volume Averaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Multidimensional Two-Phase Flow Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Interfacial Area Transport Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 One-Dimensional Two-Fluid Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Homogeneous Flow Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Void Fraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Void Fraction Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Homogeneous Flow Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 One-Dimensional Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Drift Flux Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Empirical Correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Models for Stratified Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Two-Phase Pressure Drop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Gravitational Pressure Drop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Frictional Pressure Drop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Acceleration Pressure Drop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Cross-References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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M. Kawaji (*) City College of New York, New York, NY, USA University of Toronto, Toronto, ON, Canada e-mail: [email protected] # Springer International Publishing AG, part of Springer Nature 2018 F. A. Kulacki (ed.), Handbook of Thermal Science and Engineering, https://doi.org/10.1007/978-3-319-26695-4_46

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Abstract

Two-phase flow of gas and liquid is often encountered in the design and operation of heat exchangers, oil/gas transport lines, chemical and bioreactors, and mass transfer equipment. The two-phase pressure drop governs the pumping requirement in forced-circulation systems, while the pressure drop dictates the circulation rate and, hence, various system parameters in natural-circulation systems. All three components of pressure drop (gravitational, frictional, and accelerational) are dependent on void fraction or quality, so the design of energy systems and their performance are highly dependent on accurate predictions of both the two-phase pressure drop and void fraction. In this chapter, basic parameters are defined first, followed by descriptions of two-phase flow patterns, flow pattern maps and transition criteria, the conservation equations used in two-phase flow analyses, and the correlations and models available for predicting void fraction and pressure drop in simple flow channel geometries such as circular and noncircular tubes. In particular, advanced two-phase flow models including multidimensional two-fluid models and the constitutive relations for interfacial transfer terms are presented. Examples of two-dimensional and one-dimensional two-fluid models applied to predict radial void fraction distributions in bubbly flow and interfacial wave characteristics in inverted annular flow, respectively, are also described. Keywords

Two-phase flow · Void fraction · Flow pattern · Pressure drop · Gas-liquid flow · Conservation equations · Two-fluid model · Constitutive relations

1

Introduction

Two-phase flow of gas and liquid is often encountered in the design and operation of heat exchangers, oil/gas transport lines, chemical and bioreactors, and mass transfer equipment. In forced-circulation systems, the pressure drop governs the pumping requirement, and in natural-circulation systems, the pressure drop dictates the circulation rate and, hence, the other system parameters. In steady two-phase flow, all three components of pressure drop (gravitational, frictional, and accelerational) are dependent on void fraction or quality, so the design of energy systems and their trouble-free operation are highly dependent on accurate predictions of both the overall pressure drop and void fraction. In power plants, highpressure steam and water flow as a mixture through piping systems and vessels of a wide range of sizes and orientations. It is necessary to predict the void fraction and pressure drop in the heat transport loops of nuclear reactors for design, operation, and safety. The predictive models and correlations that have been developed so far are mostly based on experimental data obtained in small-scale experiments under ambient conditions. Thus, there is still some uncertainty in the applicability of these models and correlations to two-phase flows in large-scale systems at high-pressure/ high-temperature conditions.

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In this chapter, basic parameters are defined first, followed by descriptions of two-phase flow patterns, flow pattern maps and transition criteria, the conservation equations used in two-phase flow analyses, and the correlations and models available for predicting void fraction and pressure drop in simple flow channel geometries such as circular and noncircular tubes.

1.1

Basic Parameters

1.1.1 Void Fraction Void fraction usually represents a volume fraction of the gaseous phase in a flow channel; however, it can also be defined in several different ways, either instantaneous or time-average void fraction and local, area, or volume-average void fraction. The local, instantaneous void fraction, αi,loc, is either zero or unity, as it represents the presence or absence of the gas phase at a point in space. On the other hand, if the two-phase flow inside a channel of cross-sectional area, A, could be instantaneously frozen and a picture of the fluid in the channel volume or over the cross section taken, then the area or volume occupied by the gas phase is the instantaneous area-average or volume-average void fraction. Such instantaneous void fraction can then be time averaged to obtain the time and area- or volumeaverage void fraction: Ð Instantaneous, area-average, αi, A ¼

A αi, loc dA

Ð Instantaneous, volume-average, αi, V ¼

V αi, loc dA

Ð Time-averaged, local void fraction, α ¼

(1)

A V τ αi, loc dt

Ð Time and area averaged void fraction, αA ¼

τ

τ αi, A dt

Ð Time and volume averaged void fraction, αV ¼

τ

τ αi, V dt

τ

(2) (3) (4) (5)

Another averaging method is ensemble averaging where an ergodic principle is applied. This principle states that repeating an experiment N times is equivalent to performing N experiments at one time. So, let us consider N statistically identical experiments, and measurement of the instantaneous, local void fraction, αi,loc, N times. If a gas is detected NG times, then the local, ensemble-average void fraction is given by αi = NG/N. In this chapter, void fraction will be represented by the same symbol, α or αG, regardless of different definitions given above. The liquid volume fraction is also given by αL= 1–α or 1–αG.

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The local time-average void fraction can vary strongly across the channel cross section in horizontal tubes and under some conditions in bubbly flow in vertical tubes as discussed later in Sect. 4.1. The void fraction distributions in vertical bubbly flow can show wall-peaking or core-peaking behavior depending on the average void fraction, how the gas is injected into the tube, surface tension of the fluid system, and bubble size, even under the same flow conditions. For practical applications, although knowledge of the time and area- or volume-average void fraction is usually sufficient, details such as the bubble size and phase distributions are important for a better understanding of the two-phase flow structure and validation of multidimensional two-phase flow models.

1.1.2 Flow Rates and Velocities If Q_ G and Q_ L are the volumetric flow rates of the gas and liquid phases, respectively, and A is the cross-sectional area of the flow channel, then the superficial velocities of gas and liquid, jG and jL, are given by jG ¼

Q_ G A

(6)

jL ¼

Q_ L A

(7)

The average velocities of gas and liquid flowing in the channel are different from the superficial velocities and can be expressed in terms of void fraction and superficial velocities as follows: jG α

(8)

jL ð1  αÞ

(9)

uG ¼ uL ¼

Using the above velocities, various other parameters can be defined. The slip ratio (or velocity ratio), S, is the ratio of the average gas and liquid velocities, S = uG/uL, and is related to the void fraction as follows: S¼

jG =α j ð1  αÞ ¼ G jL =ð1  αÞ jL α

(10)

_ and _ is given by the product of the volumetric flow rate, Q, The mass flow rate, m, density, ρ, as follows: m_ G ¼ ρG Q_ G ¼ ρG jG A

(11)

m_ L ¼ ρL Q_ L ¼ ρL jL A

(12)

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Fundamental Equations for Two-Phase Flow in Tubes

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The mass flux or mass velocity is defined as the mass flow rate per unit crosssectional area, GG ¼

m_ G ¼ ρG jG A

(13)

GL ¼

m_ L ¼ ρL jL A

(14)

1.1.3 Flow Quality Flow quality, x, is defined as the gas or vapor mass flow fraction, x¼

GG m_ G ¼ m_ G þ m_ L GG þ GL

(15)

and ranges in value from zero (single-phase liquid flow) to unity (single-phase gas flow). In diabatic two-phase flows, in which gas and liquid are both saturated and a change of phase occurs due to heat addition or removal, thermodynamic or equilibrium quality can be defined,  xeq ¼

hTP  hf hfg

 (16)

where hTP is the enthalpy of the two-phase mixture, hf is the enthalpy of saturated liquid, and hfg is the latent heat of vaporization (or the difference between the enthalpies of saturated vapor and liquid). The thermodynamic quality can take negative values for subcooled conditions and exceed unity for superheated vapor.

2

Two-Phase Flow Patterns

Two-phase flow patterns (or sometimes called two-phase flow regimes) were extensively investigated in transparent, vertical, and horizontal tubes mainly by visual observation including high-speed photography. Based on these observations, each flow pattern has been given a distinct name descriptive of the unique characteristics of the flow. To lessen the subjectivity of flow pattern identification by visual observation, alternate and less subjective flow pattern determination methods have also been used such as the measurement of void fraction fluctuations and analysis of their frequency spectra or probability density functions (Jones and Zuber 1975; Noghrehkar et al. 1999) and the amplitude or frequency of pressure drop fluctuations (Hubbard and Dukler 1966; Weisman et al. 1979), among others. For example, a probability density function of instantaneous void fraction data for bubbly flow would show a single peak at low void fraction, typically below 30%, and for annular flow a peak at avoid fraction over 80%. In slug flow where gas slugs of high void

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Bubbly flow

Plug flow (Taylor bubble)

Slug flow

Froth flow

(or churn-turbulent)

Annular flow

Annular-mistflow

Fig. 1 Sketches of typical two-phase flow patterns observed in a vertical tube (Reproduced from Rouhani and Sohal 1983, with permission from Elsevier)

fraction and liquid slugs of low void fraction alternately flow past the measurement point, the instantaneous void fraction data would show periodic variations resulting in the probability density function showing a bimodal distribution with two peaks, one at high (≳80%) and the other at low (≲30%) void fraction values (Noghrehkar et al. 1999). A recent review of flow patterns in both adiabatic two-phase flow and boiling and condensing flows can be found in Cheng et al. (2008).

2.1

Vertical Tube

Typical flow patterns in a vertical tube are named bubble (or bubbly) flow, slug flow, churn flow, and annular flow as shown in Fig. 1. If the gas slugs have a smooth bullet shape, they are called Taylor bubbles, and the flow pattern is named plug flow to distinguish it from the slug flow pattern. The churn flow pattern may also be called froth or churn-turbulent flow. It is characterized by a highly irregular geometry of the gas-liquid interface. Annular flow is characterized by a smooth or wavy liquid film flowing on the channel wall and a gas flowing through the core. If there are many droplets entrained in the gas core, the flow pattern is called annularmist flow.

2.2

Horizontal Tube

In a horizontal tube, gravitational force acts normal to the flow direction affecting the flow patterns strongly. Typical flow patterns observed include stratified flow, wavy flow, slug flow, and annular flow as shown in Fig. 2. The stratified flow pattern may be called stratified-smooth or stratified-wavy flow depending on the smooth or wavy profile of the gas-liquid interface. Plug and slug flows feature the gaseous phase flowing close to the upper part of the tube wall and liquid phase flowing along the bottom tube wall.

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Fundamental Equations for Two-Phase Flow in Tubes

Fig. 2 Two-phase flow patterns observed in a horizontal tube (Reproduced from Rouhani and Sohal 1983, with permission from Elsevier)

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Stratified smooth flow

Stratified wavy flow

Plug flow

Slug flow

Annular flow

Dispersed bubble flow

2.3

Two-Phase Flow Pattern Maps and Transition Criteria

Based on the two-phase flow patterns visually observed at different gas and liquid flow rates, many flow pattern maps have been proposed for vertical and horizontal two-phase flows. The flow pattern maps can be specific to a particular fluid system such as air and water or generalized which can be applied to different fluid systems. The simplest coordinate system employed to construct specific flow pattern maps uses superficial gas and liquid velocities, jG vs . jL(or UGS vs . ULS). Generalized flow pattern maps also use other coordinatesystems such as mass fluxes by Baker (1954) and superficial momentum fluxes ρG j2G vs:ρL j2L by Hewitt and Roberts (1969), among others (Collier 1972; Collier and Thome 1994). It is also noted that although the boundaries between adjacent flow patterns are often indicated by solid lines, the flow pattern transition should be considered to occur somewhat gradually as indicated by shaded lines in some flow pattern maps such as Mandhane et al.’s (1974).

2.3.1 Vertical Tube A flow pattern map for a vertical tube was proposed by Hewitt and Roberts (1969) employing gas and liquid momentum fluxes, ρG j2G and ρL j2L , based on superficial velocities. Their map was developed based on air-water flow experiments at atmospheric pressure and steam-water two-phase flow experiments at high pressure in small diameter pipes. Taitel et al. (1980) proposed a different map as shown in Fig. 3 using the superficial liquid and gas velocities. Their flow pattern map was based on adiabatic air-water flow experiments performed in a vertical, 5 cm IDtube at near atmospheric

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FINELY DISPERSED BUBBLE (II)

C

B

ULS (m/s)

Fig. 3 Taitel et al.’s (1980) vertical flow pattern map based on adiabatic air-water flow experiments (Reproduced with permission from John Wiley & Sons, Inc.)

M. Kawaji

E D

A

1.0

D D

BUBBLE (I)

D ANNULAR (V) SLUG OR CHURN (IV) E

0.1 SLUG (III) A

0.01

100

l E/D = 50

0.1

500 200

1.0 10.0 UGS (m/s)

100

Table 1 Flow pattern transition criteria proposed by Taitel et al. (1980) Flow pattern transition Existence of bubble flow

Criterion h 2 2 i1=4 ρ gD 4:36  ðρ Lρ Þσ L

Bubble to slug flow Bubble to dispersed bubble flow

G

ð17Þ h

G Þσ U LS ¼ 3:0 U GS  1:15 gðρL ρρ 2 L   ıE Mffi pUffiffiffiffi þ 0:22 ð19Þ D ¼ 40:6

i1=4

ð18Þ

gD

Slug to churn flow Churn to annular flow

 U LS þ U GS ¼ 4:0 pffiffiffiffi UGS ρG ½σgðρL ρG Þ1=4

¼ 3:1

D0:429 ðσ =ρL Þ υ0:072 L

0:089

h

gðρL ρG Þ ρL

i0:446

ð20Þ

ð21Þ

pressure and 25  C. To predict the flow patterns for a wide range of two-phase flows, Taitel et al. (1980) proposed analytical expressions for flow pattern transitions by considering the physical mechanisms responsible for each transition as summarized in Table 1. Their transition criteria shown in Fig. 3 by solid lines are in good agreement with the boundaries between the experimentally observed flow patterns. Because the transition criteria proposed by Taitel et al. (1980) involve physical properties of fluids, tube diameter, and superficial gas and liquid velocities, they can be applied to two-phase flow systems involving different gases and liquids other than air and water. Mishima and Ishii (1984) also performed a similar analysis and proposed flow pattern transition criteria different from Taitel et al.’s (1980) as shown in Table 2.

46

Fundamental Equations for Two-Phase Flow in Tubes

1857

Table 2 Flow pattern transition criteria proposed by Mishima and Ishii (1984) for a vertical tube Bubbly to slug flow Slug to churn flow

α = 0.3 (22) 2

30:75 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðΔρgD=ρL Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi5 α  1  0:8134 ð23Þ 1=18 jþ0:75 ðΔρgD=ρL ÞðΔρgD3 =ρf ν2L Þ qffiffiffiffiffiffiffiffiffi jG ¼ ΔρgD ρ ðα  0:11Þ withαsatisfying Eq:23 ð24Þ ðCo 1Þjþ0:35

Churn to annular flow

G

10

Superficial Gas Velocity (m/s)

Mishima and Ishii (1984) Dukler and Taitel (1977)

1 ANNULAR

SLUG BUBBLY

0.1

D : 2.54 cm CHURN D : 30.48 cm

0.01 0.01

0.1

1

10

100

Superficial Liquid Velocity (m/s) Fig. 4 Comparison of flow pattern transition boundaries predicted by Mishima and Ishii (1984) and Dukler and Taitel (1977) (Reproduced with permission from Elsevier)

Their transition predictions agreed quite well with the atmospheric air-water flow pattern data from Govier and Aziz (1972), Hewitt and Roberts (1969), Oshinowo and Charles (1974), and Wallis (1969), among others. The churn-to-annular flow transition was especially well predicted. When plotted, Mishima and Ishii’s (1984) transition criteria showed similarities and differences in comparison with those of Dukler and Taitel (1977) as shown in Fig. 4 for air-water two-phase flow in 2.54 cm and 30.48 cm vertical pipes at 1 atmosphere. In contrast with adiabatic two-phase flows, the flow quality and void fraction vary continuously in boiling and condensing flows in a vertical tube. As a result, the flow pattern also changes axially as illustrated in Fig. 5 for boiling flow. Flow pattern maps for boiling flows have been suggested in the past using the coordinates such as the mass flux and steam weight fraction by Bergles et al. (1968). Although many flow pattern maps have been developed based on adiabatic two-phase flow data, they have often been used to predict the vapor-liquid two-phase flow patterns in boiling flows as well.

1858

M. Kawaji

Fig. 5 Flow pattern evolution in boiling flow in a heated vertical tube

2.3.2 Horizontal Tube For a horizontal pipe, Baker (1954) proposed a flow pattern map as shown in Fig. 6 using the modified gas and liquid mass fluxes, GG/λ and GLλϕ/GG, respectively, where the parameters λ and ϕ are defined,  λ¼

ρG ρL ρGo ρLo

1=2

" # σ μ ρ 2 1=3 o L Lo ϕ¼ σ μLo ρL

(25)

(26)

Mandhane et al.’s (1974) map for horizontal flow of air and water at ambient conditions is shown in Fig. 7. This map was constructed from over 5,900

46

Fundamental Equations for Two-Phase Flow in Tubes 2

Dispersed flow Bubble or froth

Annular flow Wave

2

GG /l (kg/m S)

10

1859

10 Slug Stratified

10 Plug

10−1

1

102

10

103

104

GLlf GG

Fig. 6 Baker’s (1954) flow pattern map for horizontal two-phase flow (Reproduced with permission from Oil and Gas Journal)

DISPERSED FLOW

1.0

ULS (m/s)

Fig. 7 A horizontal flow pattern map by Mandhane et al. (1974) (Copyright 1975, Society of Petroleum Engineers Inc. Reproduced with permission of SPE. Further reproduction prohibited without permission)

BUBBLE, ELONGATED BUBBLE FLOW

SLUG FLOW

ANNULAR, ANNULAR MIST FLOW

0.1 STRATIFIED FLOW WAVE FLOW

0.01

0.1

1.0

10

100

UGS (m/s)

experimental data points in the literature. The effect of fluid properties was considered by Mandhane et al. (1974) to be small, so their flow pattern map can be considered to give an approximate indication of the flow pattern transitions for other fluid systems. Taitel and Dukler (1976a) proposed a flow pattern map for horizontal flow (Fig. 8) using four nondimensional parameters, X, K, F, and T, which are defined,

1860

M. Kawaji 1

ANNULAR - DISPERSED LIQUID (AD) A A 2 K 10

10 B DISPERSED BUBBLE (DB) D

T

D

or

STRATIFIED WAVY (SW)

–1

10

101

C C STRATIFIED SMOOTH (SS)

10–2

10–1

100

F

INTERMITTENT (I)

A

100 –3 10

100

D

B

10–2

A C

101

102

103

10–3 104

Curves A and B: use F vs. X Curve C: use K vs. X Curve D: use T vs. X

X Fig. 8 Taitel and Dukler’s (1976a) map for horizontal tubes (Reproduced with permission from John Wiley & Sons Inc.)

2

3 2

n 2 31=2 dP   1=2 ρL uLS 4CL ρL uLS D  F=  D 2 7 μL 6 6 dx LS 7 X ¼ 4  5 ¼ 44C ρ uGS D m ρG u2 5 G G GS dP F  =dx GS  D 2 μ

(27)

G

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ρG uGS pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ρL  ρG Dg cos α sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ρL uLS D 1=2 ¼ FReLS K¼F μL F¼

2

qffiffiffiffiffiffiffiffiffiffi dPF =dz

(28)

(29)

3

5 T¼4 ðρL  ρG Þg cos α

(30)

where (dpF/dx)LS and (dpF/dx)GS are the friction pressure gradients based on superficial liquid and gas velocities, respectively. Also, note that for the LockhartMartinelli parameter, X, given by Eq. 27, n = m = 0.2 and CG and CL = 0.046 are used for turbulent gas and turbulent liquid two-phase flow. Weisman et al. (1979) proposed a map for a horizontal tube (Fig. 9) which incorporated the effects of fluid properties and tube diameter by using the superficial gas and liquid velocities divided by nondimensional parameters, ϕ1 and ϕ2, respectively. These nondimensional parameters are calculated from the ratios of fluid properties in reference to standard conditions involving air and water as listed in Table 3. The flow pattern maps and transition criteria described above are used to predict two-phase flow patterns for specific flow conditions. The knowledge of the two-phase flow pattern is essential for solving the two-phase flow equations

46

Fundamental Equations for Two-Phase Flow in Tubes 100

1861

(φ1 and φ2 are defined in Table 2.3)

10

Wavy

⁄

1

( ⁄ )

Annular

Dispersed

1 Intermittent

Stratified

0.1

0.01

0.1

/

1.0 2

10

( ⁄ )

Fig. 9 Weisman et al.’s (1979) flow pattern map for a horizontal tube (Reproduced with permission from Elsevier)

Table 3 Parameters f1 and f2 for Weisman et al.’s (1979) flow pattern map (Fig. 9) (Reproduced with permission from Elsevier) Transition to dispersed flow Transition to annular flow Intermittentseparated transition Wavystratified transition

ϕ1 1.0

ϕ2 0:67# 0:16 0:09 0:24 ρL ρLs

0:23 ρGs ρG

Δρ Δρs

0:11 0:11 0:415 σ σs

μLs μL

σ σs

1.0

D Ds

0:45

1.0

D Ds

0:17 1:55 1:55 D Ds

D Ds

μG μGs

ρGs ρG

Δρ Δρs

0:69   σ s 0:69

1.0

σ

“s” denotes standard condition, Ds = 25.4 mm, ρGs = 1.3 kg/m3, ρLs = 1,000 kg/m3, μLs = 0.001 Ns/m2, σ s = 0.07 N/m a According to the text of the paper by Weisman et al. (1979). However, the value of this exponent given in Table 2 of their paper is 0.33

1862

M. Kawaji

described below, in particular, for selecting the constitutive equations for interfacial and wall-to-fluid mass, momentum, and energy transfer terms specific to each flow pattern.

3

Two-Phase Flow Model

3.1

Volume Averaging

In two-phase flow, a mixture of gas and liquid occupies a volume, ΔV, inside a flow channel. If a phase, k, occupies a certain volume, ΔVk, then the volume fraction of phase k is αk = ΔVk/ΔV. A phase average of a variable or property, γ, of a gas or liquid phase can be defined by integrating the variable or property over the volume occupied by that phase. If the averaging volume is limited to only the phasic volume, ΔVk, then an intrinsic phase average value is obtained, while an extrinsic phase average value is obtained if the averaging volume covers the entire volume, ΔV (Faghri and Zhang 2006), Ð k

Intrinsicphaseaverage : hγ k i ¼

ΔV k γ k dV

Ð Extrinsic phase average : hγ k i ¼

ΔV k

ΔV k γ k dV

ΔV

(31)

(32)

They are related by the volume fraction of phase k, hγ k i ¼ αk hγ k ik

(33)

For example, the density of phase k within the space occupied by the same phase is 〈ρk〉k, while the density of phase k over the mixture volume is 〈ρk〉 = αk〈ρk〉k. In the following sections, the variables and properties represent phasic volume averages, 〈γ k〉k.

3.2

Multidimensional Two-Phase Flow Model

3.2.1 Conservation Equations Detailed derivations of the mass, momentum, and energy conservation equations for two-phase flow can be found in many references (e.g., Ishii 1975; Ishii and Mishima 1984; Ishii and Hibiki 2006; Faghri and Zhang 2006). The derivations start with the Reynolds transport theorem. Here, without presenting their derivations, the final forms of the conservation equations for each phase from Ishii and Mishima (1984) are,

46

Fundamental Equations for Two-Phase Flow in Tubes

1863

Continuity Gas :

@ ð α G ρG Þ þ ∇  ð α G ρG V G Þ ¼ Γ G @t

(34a)

@ ðαL ρL Þ þ ∇  ðαL ρL VL Þ ¼ ΓL @t

(34b)

Liquid : Momentum Gas :

  @αG ρG VG þ ∇  ðαG ρG VG VG Þ ¼ αG ∇pG þ ∇  αG τ G þ τ tG @t þ αG ρG g þ VGi ΓG þ MiG  ∇αG  τ i

Liquid :

(35a)

  @αL ρL VL þ ∇  ðαL ρL VL VL Þ ¼ αL ∇pL þ ∇  αL τ L þ τ tL @t þ αL ρL g þ VLi ΓL þ MiL  ∇αL  τ i (35b)

Energy Gas :

  @αG ρG H G þ ∇  ðαG ρG H G VG Þ ¼ ∇  αG qG þ qtG @t DG p þ H Gi ΓG þ q00Gi ai þ ΦG þ αG Dt G (36a)

Liquid :

  @αL ρL H L þ ∇  ðαL ρL H L VL Þ ¼ ∇  αL qL þ qtL @t (36b) DL pL þ H Li ΓL þ q}Li ai þ ΦL þ αL Dt

where ai = interfacial area concentration g = gravitational acceleration Hk = enthalpy of phase k Mik = generalized interfacial drag for phase k pk = pressure of phase k qk = conduction heat flux for phase k qtk = turbulent heat flux for phase k q00ki = interfacial heat flux for phase k Vk = velocity of phase k Vki = interfacial velocity in phase k αk = volume fraction of phase k ρk = density of phase k

1864

M. Kawaji

τ i = interfacial shear stress τ k = average viscous stress for phase k τ tk = turbulent shear stress for phase k Γk = interfacial mass transfer per mixture volume Φk = energy dissipation for phase k The interfacial transfer terms obey the local jump conditions given by, X X

Γ k k

k

¼0

Mik ¼ 0

X  Γk Hki þ q00ki ai ¼ 0 k

(37) (38) (39)

Thus, if the liquid is evaporated and turns into vapor, ΓG is positive and ΓL is negative, so that ΓG = ΓL.

3.2.2 Constitutive Equations In order to solve the two-phase flow equations above, constitutive equations formulated for specific flow patterns are needed. Although constitutive equations for interfacial transfer terms have been developed for many flow patterns, the extent to which they account for detailed flow characteristics varies considerably. For example, churn flow is more difficult to model than bubbly, slug, or annular flow, since the interfacial geometry and local flow structure are more complicated and less amenable to mathematical modeling. For this reason, the constitutive equations for only bubbly and dispersed droplet flows are presented in detail in this section. For bubbly or dispersed droplet flow, the interfacial momentum transfer term for the dispersed phase, Mid, can be expressed as the sum of the steady drag force, lift force, lubrication force, turbulent dispersion force, virtual mass force, and Basset force. In this section, the interfacial momentum transfer terms are presented below based on the works of Ishii and Zuber (1979), Ishii and Chawla (1979), and Ishii and Mishima (1984). A detailed discussion of various constitutive equations is given by Ishii and Hibiki (2006).

Steady Drag Force The steady drag force, MD d , is calculated using a drag coefficient (CD),volume fraction of the dispersed phase (α), density of the continuous phase (ρc), relative velocity between the dispersed and continuous phases (Vr), and projected area (Ad) and volume (Bd) of a typical bubble, liquid droplet, or particle, MD d ¼

αAd C D ρc V r j V r j 2Bd

(40)

46

Fundamental Equations for Two-Phase Flow in Tubes

1865

The drag coefficient includes the effects of both viscous and form drag and has been extensively studied in the past for bubbles in liquid and liquid droplets and solid particles in liquid and gas streams. For a single particle moving in a fluid, a drag coefficient can be given in terms of the particle Reynolds number based on the particle diameter, Db; relative velocity between the particle and fluid, Vr; the fluid density, ρL; and viscosity, μL. However, for dispersed two-phase flows involving many bubbles, droplets, or particles in a continuous fluid, the particle would experience an increased resistance to its motion which appears as if it is moving through a more viscous fluid. This multiparticle effect depends on the volume fraction of the dispersed phase which can be included in the Reynolds number through the use of the viscosity of the two-phase mixture. For example, in a bubbly flow with a void fraction, α, and mean bubble diameter, Db, the Reynolds number and mixture viscosity are, μL Reb ¼ ρL Vμ r Db where μm ¼ 1α m

When the Reynolds number is small, Reb >αLρG.

Inverted annular flow experiments were performed by De Jarlais (1983) and also reported by Ishii and De Jarlais (1986). De Jarlais used water for the liquid, and nitrogen, Freon-12, or helium for the gas, and performed adiabatic experiments in round tubes of 0.90, 1.36 and 1.66 cm ID with different superficial gas and liquid velocities. The interfacial wavelength data were collected for a downward flowing liquid column and gas using a photographic technique. The interfacial wavelength data from De Jarlais’ (1983) experiments are shown by symbols and empirical correlations by a dashed line in Fig. 13. At low Weber numbers or relative velocities, λ/DL was roughly constant at about a value of 5.8. At larger Weber numbers, the wavelength decreased and λ/DL was correlated using a Weber number, 100

/

10

1

0.1

0.01 10–4

De Jarlais (1983) Linear Stability Analysis (Eq. 3.45) 10–3

10–2

10–1

100

101

102

103

⁄

Fig. 13 Comparison of interfacial wavelength between the two-fluid model predictions with De Jarlais’ (1983) data (From Kawaji and Banerjee 1987. Reproduced with permission from ASME)

1878

M. Kawaji

 1=2 λ We ¼ 7:6 2 DL αG

(89)

The analytical expression for λ/DL obtained from the linear stability analysis of unequal-pressure, two-fluid model is also shown by a solid line in Fig. 13. It agrees well with De Jarlai’s data and empirical correlations for both small and large Weber numbers. As the velocity difference, uGuL, is reduced and the surface tension term dominates in the of Eq. 87, λ/DL is predicted to become constant at a pffiffiffi   denominator value of 4.44 ¼ 2π , which is close to the value of 4.508 obtained by Rayleigh for a liquid jet (Lamb 1932). As can be seen in the bubbly flow and inverted annular flow examples described in this section, it is important to account for the phasic pressure difference in the momentum equations of the multidimensional and one-dimensional two-fluid models for accurate predictions of two-phase flow structures.

3.5

Homogeneous Flow Model

The simplest approach to the treatment of gas-liquid two-phase flow in a channel is to treat the two-phase mixture as a homogeneous fluid, with equal velocities assumed for the gas and liquid. The conservation equations for a two-phase flow with a homogeneous mixture density, ρH = ρLαL + ρGαG, are exactly the same as those for a single-phase flow. For flow in a channel of cross-sectional area, A, and inclined at an angle θ to the horizontal, the mass, momentum, and energy equations can be written, @ρH @ ðρH uuÞ ¼0 þ @z @t

(90)

@ ðρH uÞ @ ðρH uuÞ @p τ W PW þ þ ¼  ρH g sin θ @t @z @z A

(91)

Continuity : Momentum :

Energy :

@ ðρH hÞ @ ðρH uhÞ q00W PW @p þ ¼ þ @t @z @t A

(92)

Here, the homogeneous mixture velocity is given by u = G/ρH where G ¼ ðm_ G þ m_ L Þ=A is the total mass flux, and mixture enthalpy is given by h = (1  x)hL + xhG. In the momentum equation, τw is the wall shear stress for the mixture, Pw is the wetted channel perimeter, and A is the channel cross-sectional area. In the energy equation, q00W is the heat flux into the fluid via the channel wall, and h is the enthalpy per unit mass.

46

Fundamental Equations for Two-Phase Flow in Tubes

1879

For steady flow in a constant area channel, the momentum equation simplifies to, 1  dp τW PW 2 d =ρ H ¼ þG þ ρH g sin θ (93) dz A dz The three terms on the RHS represent the frictional, accelerational, and gravitational pressure gradients, and the equation is sometimes written, dp dpF dpa dpg ¼ þ þ dz dz dz dz

(94)

The homogeneous flow model is simple to solve for a variety of two-phase flows in a piping system. Also, the homogeneous flow model can be readily used to estimate pressure drop in complicated passages such as valves and fittings without detailed information about the flow such as the flow pattern, size of gas bubbles, and so on.

4

Void Fraction

4.1

Void Fraction Distribution

Void fraction represents a volume fraction of the gaseous phase in the flow channel; however, there are several different ways of defining void fraction as discussed in Sect. 1.1.1.The local time-averaged void fraction could vary across the channel cross section, especially in bubbly flow. For example, Serizawa et al. (1975) and Liu and Bankoff (1993) both reported wall-peaking and core-peaking void fraction profiles in bubbly flow of air and water in 60 mm and 38 mm ID vertical tubes, respectively, as shown in Fig. 14. Even under the same flow conditions, the void fraction can show wall-peaking or core-peaking behavior depending on how the gas is injected to the tube, surface tension of the fluid system, and bubble size. The primary purpose of this section is to present various methods for predicting area or volume-average void fraction using the basic parameters defined in the preceding sections. The application of one-dimensional models such as the homogeneous flow model and drift flux model will be described along with several empirical correlations. Finally, models that are specific to certain flow regimes such as stratified flow, bubbly flow, and slug flow will be described.

4.2

Homogeneous Flow Model

From the parameter definitions given in Sect. 1, the slip velocity and void fraction are related to each other by, S¼

ρL xαL ρG ð1  xÞα

(95)

0.3

0.4

–1.0

Bubble Flow

Bubble Flow

Bubble Flow

Bubble Flow

Slug Flow

Slug Flow

0 Radial Position r / R Serizawa et al. (1975)

Z/D = 30

Vo = 1.03 m/s

X (%) 0.0085 0.0170 0.0258 0.0341 0.0427 0.0511

1.0

0.5

0 72

0 –10

–0.5

0 r/R

0

0.2

0.4 0.6 r/R

(b2) J0 = 0.347 m s–1

(b1) J0 = 0.180 ms–1 Jf (ms–1) 0.376 0.535 0.753 1.391

Liu and Bankoff (1993)

1.0

24

a1 (%)

12

24

36

15

0.112 0.347

a1 (%)

48

(a2) J1 = 1.087

ms–1

Jg 0.027 0.230

(ms–1)

(a1) J1 = 0.753 ms–1

30

0 45

18

36

54

b

0.8

1.0

a1 (%)

a1 (%)

Fig. 14 Radial void fraction profiles in bubbly flow (Reproduced with permission from Elsevier). (a) Serizawa et al. (1975). (b) Liu and Bankoff (1993)

0

0.1

0.2

Local Volume Fraction

a

1880 M. Kawaji

46

Fundamental Equations for Two-Phase Flow in Tubes

α¼

QG jG ¼ SQL þ QG SjL þ jG

1881

(96a)

The void fraction can also be given in terms of the flow quality, x, and slip ratio, S, α¼

ρL x SρG ð1  xÞ þ ρL x

(96b)

For a homogeneous gas and liquid mixture, the velocities of gas and liquid are assumed to be equal, i.e., uG = uL and S = 1. Thus, from Eq. 96a, α = β, where β is the ratio of the gas volumetric flow rate to the total flow rate and called homogeneous void fraction or volumetric quality, β¼

QG jG ¼ QG þ QL jG þ jL

(97)

Although the homogeneous flow model assumes mechanical equilibrium between the phases (i.e., same velocities), the gas and liquid phases normally flow at different velocities, so that S 6¼ 1 and α 0, jL>0) and downward flow ( jG 10 mm (Reproduced from Sadatomi et al. 1982, with permission from Elsevier) Shape Rectangular

Annulus

Dimensions (mm) 17  50 10  50 7  50 7  20.6 20 , h = 55 Di = 15, Do = 30

Co 1.20 1.24 1.16 1.21 1.34 1.30

Circular

D = 26

1.25

Triangular

 Co ¼ 1:2  0:2ð1  xÞ

ρG ρL

0:5 (113)

and rectangular channels,  Co ¼ 1:35  0:35

ρG ρL

0:5 (114)

For noncircular channels with a hydraulic diameter greater than 10 mm, Sadatomi et al. (1982) recommended the Co values listed in Table 4. For the mean drift velocity, Ishii and Zuber (1979) and Wallis (1969) recommended the following expression for bubbly flow,     σgðρL  ρG Þ 0:25 V Gj ¼ 1:4 ρ2L

(115)

For slug flow, Collier (1972) recommended,     gðρL  ρG ÞD 0:25 V Gj ¼ 0:35 ρL

(116)

For churn-turbulent flow, Zuber and Findlay (1965) recommended,     σgðρL  ρG Þ 0:25 V Gj ¼ 1:18 ρ2L

(117)

where σ is the surface tension. For noncircular channels, Sadatomi et al. (1982) and Sato and Sadatomi (1986) showed that the mean drift velocity, 〈〈VGj〉〉, can be given in terms of an equi-periphery diameter, De, which is defined as the total wetted perimeter divided by π,

46

Fundamental Equations for Two-Phase Flow in Tubes 20

20

17×50

20°,h = 55

10

jL m/s 0.1 0.3 0.5 1.0 2.0

j G / α (m/s)

j G / α (m/s)

10

1885

1

jL m/s 0.1 0.3 0.5 1.0 2.0

1

jG/α = 1.34(jG + jL) + 0.214 0.2 0.1

1

10

20

jG /α = 1.20(jG + jL) + 0.216 0.2 0.1

1

jG+jL (m/s)

10

20

jG+jL (m/s)

Fig. 15 Correlation of void fraction data by Sadatomi et al. (1982) (Reproduced with permission from Elsevier)

Table 5 Drift flux model parameters for horizontal flow (Franca and Lahey 1992) Flow pattern Plug Slug Wavy stratified and annular

Co 1.0 1.2 1.0



V Gj



¼ ð0:31 ~ 0:35Þ

〈〈VGj〉〉(m/s) 0.16 m/s 0.20 m/s 0.2 m/s (for jL = 0.005 m/s) ~ 2.7 m/s (for jL = 0.27 m/s)

pffiffiffiffiffiffiffiffi gDe

(118)

For an annular channel, De = Di + Do where Di and Do are the inner and outer diameters, respectively. Hasan and Kabir (1992) recommended the following equation for two-phase flow in a vertical annulus, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   V Gj ¼ ½0:345 þ 0:1ðDi =Do Þ gDo ðρL  ρG Þ=ρL

(119)

The void fraction data obtained by Sadatomi et al. (1982) for two-phase flow in triangular and rectangular channels are correlated well using the drift flux model as shown in Fig. 15. For horizontal flow, Franca and Lahey (1992) correlated their void fraction data obtained in a 19 mm I.D. tube using the drift flux model. Their experiments were conducted with air-water and covered plug, slug, stratified-smooth/stratified-wavy, and annular flow patterns. The values of the distribution parameter and drift velocity that best correlated the data are summarized in Table 5. Additional expressions for Co and 〈〈VGj〉〉 are also available in the literature for various flow patterns in vertical and horizontal flows through round and noncircular tubes.

1886

M. Kawaji

When the weighted mean drift velocity is zero, i.e., 〈〈VGj〉〉 = 0, the drift flux model is equivalent to the variable density, single-fluid model of Bankoff (1960), α ¼ Kβ

(120)

where the flow parameter, K, corresponds to the inverse of the distribution parameter, Co. Armand (1946) experimentally determined that, α ¼ 0:833β

(121)

would be suitable for air-water flows.

4.5

Empirical Correlations

Since the drift flux model requires parameter values specific to a given flow pattern, a general void fraction correlation applicable over a wide range of flow conditions would be more convenient. One of the oldest but popular empirical correlations for void fraction was proposed by Lockhart and Martinelli (1949). They defined a parameter, X2, as the ratio of the single-phase friction pressure drop of liquid, ΔpFL , to that of gas, ΔpFG , when each phase is assumed to flow alone in the channel, .

ΔpFL

X2 ¼

.Δz

ΔpFG

(122)

Δz

At given gas and liquid flow rates, the friction pressure drop for each phase can be calculated using a laminar or turbulent friction factor as described in the next section. For both turbulent phases, the Lockhart-Martinelli (L-M) parameter is,       1  x 0:9 ρG 0:5 μL 0:1 Xtt ¼ x ρL μG

(123)

where x is the flow quality. Lockhart and Martinelli (1949) proposed a graphical correlation of void fraction in terms of the L-M parameter considered to be applicable to low-pressure systems. A widely used equation proposed by Chisholm and Laird (1958) to represent the L-M correlation applicable to pipes and channels with hydraulic diameters greater than about 10 mm is,   C 1 1 α¼1 1þ þ 2 X X

(124)

46

Fundamental Equations for Two-Phase Flow in Tubes

1887

Quality % by wt. 0.1 Pressure 1.0 Bar (psia)

0.8

10

1

100 1.0

1·01 (14·7)

0.8

Void fraction a

6·89 (100)

0.6

0.6 34·4 (500) 68·9 (1000)

0.4

0.4 103 (1500) 138 (2000) 0.2

0.2

172 (2500) 207 (3000)

221·2 (3206) 0 0.001

0 0.01

0.1

1

Mass quality x

Fig. 16 Void fraction correlation by Martinelli and Nelson (1948) for steam-water systems (Reproduced with permission from ASME)

The value of C ranges from 5 to 20 depending on whether each phase is in laminar or turbulent flow: C ¼ 5 for laminar gas, laminar liquid ðsubscript vvÞ

(125a)

¼ 10 for laminar gas, turbulent liquid ðsubscript vtÞ

(125b)

¼ 12 for turbulent gas, laminar liquid ðsubscript tvÞ

(125c)

¼ 20 for turbulent gas, turbulent liquid ðsubscript ttÞ

(125d)

For high-pressure steam-water systems, Martinelli and Nelson (1948) introduced a pressure correction to the Lockhart-Martinelli correlation and presented their correlation in a plot of void fraction vs. quality as shown in Fig. 16. Their correlation has also been used for fluids other than water at the same ratio of system pressure p to critical pressure pc. Many void fraction correlations published in the past have been compared with a large number of experimental data available in the literature. For example, Friedel and Diener (1998) compared available void fraction correlations with 24,000 data points obtained with water-steam and R12 vapor-liquid flow as well as air-water

1888

M. Kawaji

flows in horizontal and vertical tubes. Woldesemayat and Ghajar (2007) made comparisons of 68 void fraction correlations with 2,845 experimental data points of which 900 were for horizontal, 1,542 were for inclined, and 403 were for vertical pipes. They reported that most of the correlations were very restricted in handling a wide variety of data sets, and the five best correlations were based on a drift flux model. From their study, Woldesemayat and Ghajar (2007) proposed an improvement to the correlation of Dix (Coddington and Macian 2002), so the following correlation would be capable of handling all data sets regardless of flow patterns and inclination angles, 2 0 α ¼ jG 4jG @1 þ



jL jG

0:1 1 ρG ρL

A þ 2:9

n

gDσ ð1þcosθÞðρL ρG ρ2L

o0:25

31 ð1:22 þ 1:22sinθÞ

patm psystem

5

(126)

4.6

Models for Stratified Flow

Applications of a separated flow model to the prediction of void fraction or liquid level in stratified flow will be discussed below. In the first example, the liquid level is assumed to be constant along the horizontal pipe, while an interfacial level gradient (ILG) is accounted for in the second example to illustrate the importance of ILG in achieving accurate predictions.

4.6.1 Stratified Flow Without an Interfacial Level Gradient A theoretical basis for the Lockhart-Martinelli correlation was provided by Taitel and Dukler (1976a, b) for application to stratified flow in a tube with an angle of inclination, θ, from horizontal. The liquid flows at the bottom of the tube, and the gas flows over the liquid. Ignoring the effects of acceleration and interfacial level gradient (ILG) in the liquid phase, Taitel and Dukler (1976a) combined the one-dimensional momentum equations for the liquid and gas phases,  τWG

     lG lL 1 1 þ  τWL þ τi li  ðρL  ρG Þg sin θ ¼ 0: AL AG AG AL

(127)

where AG and AL are the cross-sectional areas occupied by the gas and liquid, respectively; SG and SL are the lengths of the channel perimeters in contact with gas and liquid, respectively; and Si is the interfacial area per unit axial length. Taitel and Dukler (1976a) expressed the wall and interfacial shear stresses in terms of friction factors,

46

Fundamental Equations for Two-Phase Flow in Tubes

τWL ¼ f L ρL

u2L 2

u2G 2 ð uG  u L Þ 2 τ i ¼ f i ρG 2 τWG ¼ f G ρG

1889

(128)

(129)

where uL and uG are the average velocities of the liquid and gas in cross-sectional areas, AL and AG, respectively. The Darcy friction factors, fL and fG, can be calculated using standard correlations for laminar and turbulent flows. For flows in large diameter pipes, since the Reynolds numbers are usually greater than 2,300, the Blasius relation can be used for turbulent flow, f ¼ f ¼

C1 Re

for Re > 2, 300

0:046 Rem

for Re > 2, 300

(130) (131)

where ReL ¼ ρL uμL DL for liquid, ReG ¼ ρG uμG DG for gas, m is equal to 0.2 for Blasius L G equation, and Cl is the geometry factor dependent on the flow area cross section, e.g., 16 for a round tube. The equivalent diameters, DL and DG, for the liquid and gas phases respectively are, 4AL lL

(132)

4AG ðlG þ li Þ

(133)

DL ¼ DG ¼

The expressions for friction factors and shear stresses can now be substituted into Eq. 123, and the resulting equation can be nondimensionalized. With nondimensional quantities denoted by a tilde (~), the momentum balance reduces to,



X2 u~2L~l L   ~ L n A~L u~L D

u~2G

~ ~l L ~l i  lG þ þ A~G A~L A~G  4Y ¼ 0  m ~G u~G D

(134)

~ L ¼ DL and D ~ G ¼ DG (where D is the tube diameter), and where u~L ¼ uj L and u~G ¼ uj G, D D D L G ~l L ¼ lL , ~l G ¼ lG , ~l i ¼ li , A~L ¼ AL2 , and A~G ¼ AG2 . The parameters, m and n, are the D D D D D exponents in the friction factor equation for gas and liquid, respectively, and take on a value of 1.0 for laminar flow and 0.2 for turbulent flow. X is the LockhartMartinelli parameter, and Y is a parameter defined by Taitel and Dukler (1976a) to represent the effect of channel inclination,

1890

M. Kawaji

Fig. 17 Taitel and Dukler’s (1976a) solutions for nondimensional liquid level in stratified flow (Reproduced with permission from John Wiley & Sons, Inc.)

1.0

–10 –10

–1 –(r L – rG) g sin a = –.01 Y= |(dpF / dz) G|

0.0

–0.05

10–3 10–2 10–1

Y¼

.1 1.0 10 10 0

.5

100 X

101

00

10

TURB-TURB TURB-LAM

102

ðρL  ρG Þg sin θ  F dpG     dz 

103

104

(135)

dpF

where dzG is the frictional pressure gradient for the gas phase flowing alone in the channel. In Eq. 134, all the dimensionless quantities are functions of the dimensionless liquid height, h~L ¼ hDL . For given values of X and Y (corresponding to the known flow conditions), the appropriate value of h~L may be estimated by iteration as the solution for Eq. 134 source not found. The solutions obtained by Taitel and Dukler (1976a) are illustrated in Fig. 17. They found that when expressed in this dimensionless form, the results were remarkably similar whether the gas-phase flow was assumed laminar or turbulent. Of course, the actual liquid height would vary, since, for a given set of flow conditions, X and Y are different for laminar and turbulent gas flows. Once h~L is estimated from the above equations or by interpolation of Fig. 17, the void fraction α in stratified flow may be calculated, α¼

 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi      2 1 cos 1 2h~L  1  2h~L  1 1  2h~L  1 π

(136)

4.6.2 Stratified Flow with an Interfacial Level Gradient The stratified flow with an interfacial level gradient (ILG), which often occurs in large diameter horizontal pipes, was analyzed by Sadatomi et al. (1993). In their analysis, gas and liquid are assumed to flow in the same direction in a circular, horizontal pipe. The interface between the two phases is assumed to be smooth, but the liquid level can vary in the flow direction as shown in Fig. 19. If the liquid level is given by hL, the centroid of the liquid cross section is located at khL below the interface, where k is,

46

Fundamental Equations for Two-Phase Flow in Tubes

li

1891

twG PG

2q khL

PL

(1-k)hL

uG

ti

uL

uG + duG PG+ dPG PL+ dPL

t WL hL Centroid of liquid cross-section

uL + duL

hL + dhL

dx

Fig. 18 Stratified flow with an interfacial level gradient (ILG) in a horizontal pipe analyzed by Sadatomi et al. (1993) (Reproduced with permission from Elsevier)

k¼

½ðπ  θÞ cos θ þ sin θ  ðsin 3 θÞ=3 ½ð1 þ cos θÞðπ  θ þ sin θ cos θÞ

(137)

where the angle, θ, is indicated in Fig. 18. The length of the interface is denoted by li, L and the gas-liquid interface level gradient (ILG) in the flow direction is given by dh dx . The centroid parameter, k, varies with the liquid level, so if an ILG exists, the value of k will change with the axial distance. Using this centroid parameter, the average pressures in the gas and liquid phases are related, pL ¼ pG þ kρL ghL

(138)

The pressures pL and pG are defined as the area-averaged pressures for the liquid and gas phase cross sections, respectively. Differentiation of Eq. 134 with respect to the axial coordinate, x, yields, dpL dpG dðkhL Þ ¼ þ ρL g dx dx dx

(139)

The cross-sectional areas, AL and AG; hydraulic diameters, DL and DG; wetted perimeters, lWL and lWG; and the interfacial area per unit axial length, li, can be represented as simple functions of the pipe inner diameter, D; liquid level, hL; and/or angle, θ, as given previously. The shear stresses at the fluid-wall boundary, τWL and τWG, and at the interface, τi, have been also given above. The momentum equations for the liquid and gas phases can be combined to obtain a mixture momentum equation for stratified flow with ILG,      ρ u2 ρ u2 dhL τWG lWG τWL lWL 1 1  þ τi li þ ¼ ρL g  l i G G þ L L AG AL AG AL dx AG AL

(140)

L It is noteworthy that if the flow is well developed and dh dx is negligible, the righthand side of Eq. 140 becomes zero, and the momentum equation reduces to

1892

M. Kawaji

Eq. 137 given by Taitel and Dukler (1976b) for a zero-ILG stratified flow in a horizontal pipe. In order to show the importance of the ILG term in void fraction prediction in stratified flow, comparisons of the predicted and measured void fractions are shown in Fig. 19. In air-water experiments, Koizumi et al. (1990) measured liquid levels at 7.4 m and 26.8 m downstream of the mixing section in a 30.5 m long, 210 mm I.D. circular pipe. The averages of the two void fractions are shown as a function of jG in Fig. 19a. Sadatomi et al. (1993) reported void fractions measured at two locations in air-kerosene experiments in a 50.8 mm high and 101.6 mm wide rectangular duct. Further, liquid level data were obtained in high-pressure steam-water flow experiments at 3, 7.5, and 12 MPa in a 180 mm I.D. pipe (Kawaji et al. 1987). In all cases, the liquid level in the exit tank was kept below the horizontal test section. In Fig. 19, the predictions of Eq. 140 are indicated by solid curves and are seen to be in good agreement with the experimental data. The prediction with a zero-ILG shown by a broken line in Fig. 19a deviates away from the data as jG is reduced. Thus, incorporating the ILG term in the momentum equations for stratified flow is essential for predicting the void fraction (or liquid level) in horizontal stratified flow.

5

Two-Phase Pressure Drop

As described in Sect. 3.4 and given by Eq. 141, there are three components of pressure drop for steady two-phase flow in a tube due to friction, acceleration, and gravity, dp dpF dpa dpg ¼ þ þ dz dz dz dz

(141)

In adiabatic flows in tubes of constant cross-sectional area, the acceleration pressure drop can be neglected if the flow is well developed and void fraction does not change in the flow direction. However, in boiling flows, the quality and void fraction both change along the flow channel, so the acceleration component must be accounted for. In this section, practical methods to calculate the two-phase pressure drop per unit length or the pressure gradient, dp/dz, will be described.

5.1

Gravitational Pressure Drop

The gravitational pressure drop is important for vertical and inclined channels and can be calculated from the pressure gradient using the channel average void fraction, α,

46

Fundamental Equations for Two-Phase Flow in Tubes

1893

a 1.0 0.8 Koizumi el al.’s Data

0.6 α

Eq. 14 with ILG = –0.00056 Eq. 14 with ILG = 0

0.4 0.2

AIR-WATER STRATIFIED FLOW D=210mm, p=0. 1MPa, JL=0.072m/s

0 0.01

0.1

10

1 jG m /s

b 1.0 EXP.(JL=0. 074m/s, JG=0. 34m/s) EXP.(JL=0. 074m/s, JG=0. 68m/s) EXP.(JL=0. 074m/s, JG=1. 03m/s) CAL.

0.8

α

0.6 0.4 0.2 Air-kerosene flow, Rectangular duct 0

0

3 4 1 2 5 DISTANCE FROM CHANNEL EXIT m

6

c 1.0 0.8

α

0.6 2

EXP. (G=115kg/(m ·s), X=0. 052) EXP. ( 114 , 0.093) EXP. ( 42.8 , 0.122) CAL.

0.4 0.2

Steam-water flow at 7.6 MPa, D=180 mm 0

0

2

4

6

8

10

DISTANCE FROM CHANNEL EXIT m

Fig. 19 Comparison of void fractions predicted by Eq. 140 with stratified flow data from (a) Koizumi et al. (1990), (b) Sadatomi et al. (1993), and (c) Kawaji et al. (1987) (Reproduced with permission from Elsevier)

1894

M. Kawaji

Δpg ¼ ½ρL ð1  αÞ þ ρG αg sin θ Δz

(142)

Here, Δz is the distance in the flow direction and θ is the angle of channel inclination from horizontal.

5.2

Frictional Pressure Drop

In one-dimensional two-phase flow, the two-phase frictional pressure gradient is given by, 

dpF τw Pw ¼ dz A

It can be related to a single-phase pressure gradient

(143)



dpF dz L

or



dpF dz Gby

assuming

the gas phase or liquid phase to be flowing alone in the channel and using a two-phase friction multiplier,ΦG and ΦL, 

    dpF dpF dpF ¼ ϕ2L ¼ ϕ2G dz dz L dz G 

where



dpF dz

 ¼ L

2f L G2 ð1  x2 Þ DρL

  dpF 2f G2 x2  ¼ G dz G DρG sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dpF =dz ϕL ¼ ðdpF =dzÞL sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dpF =dz ϕG ¼ ðdpF =dzÞG

(144)

(145a)

(145b)

(146a)

(146b)

For evaporating flows, it is often more convenient to relate the two-phase frictional pressure gradient to the frictional pressure gradient calculated for a single-phase flow at the same total mass velocity and with the physical properties of the liquid phase, namely, (dpF/dz)LO, 

  dpF dpF ¼ ϕ2LO dz dz LO

(147)

46

Fundamental Equations for Two-Phase Flow in Tubes

where ϕLO

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dpF =dz ¼ ðdpF =dzÞLO

1895

(148)

The single-phase pressure gradients can be calculated from the standard equations, 

dpF dz

 ¼ LO

2f LO G2 DρL

(149)

where G is the total mass flux and D is the hydraulic diameter. The friction factor is related to the Reynolds number through standard equations and charts. For laminar flow (Re4,000), the following Blasius equation is used, f ¼

0:079 Re0:25

(150)

The Reynolds number for the friction factor is defined differently for fG, fL, and fLO, GxD μG

(151a)

Gð1  xÞD μL

(151b)

GD μL

(151c)

ReG ¼ ReL ¼

ReLO ¼

where μG is gas viscosity and μL is liquid viscosity.

5.2.1 Homogeneous Model In the homogeneous model, the two-phase frictional pressure gradient is calculated using a two-phase friction factor, fTP, dpF 2f TP G2 ¼ dz DρH

(152)

The friction factor fTP can be calculated using the two-phase Reynolds number, ReTP ¼ GD μH , where ρH is the homogeneous density and μH is the two-phase viscosity for which different formulas have been proposed. The homogeneous density can be calculated using the following formula which ensures the density to equal the liquid density at x = 0 and gas density at x = 1,

1896

M. Kawaji

 ρH ¼

 x 1  x 1 þ ρG ρL

(153)

Idsinga et al. (1977) evaluated the following formulations for the two-phase viscosity by comparing the pressure drop predictions with 2,220 adiabatic and 1,230 diabatic steam-water pressure drop data obtained in round and rectangular tubes, annuli, and a tube bundle: Owens (1961) μH ¼ μL

(154a)

1 1 μ1 H ¼ xμG þ ð1  xÞμL

(154b)

μH ¼ xμG þ ð1  xÞμL

(154c)

μH ¼ βμG þ ð1  βÞμL

(154d)

McAdams et al. (1942)

Cicchitti et al. (1960)

Dukler et al. (1964)

Another expression was proposed by Beattie and Whalley (1982) based on a mixing length theory and by considering the flow regime effects implicitly, μH ¼ βμG þ ð1  βÞð1 þ 2:5βÞμL

(154e)

For evaporating flows, the use of the above expressions for the two-phase viscosity results in the following equations for the two-phase friction multiplier given in Eq. 147: 

ϕ2LO



ρ ¼ 1þx L 1 ρG





ϕ2LO

  ρ 1þx L 1 ρG ¼h

i0:25 μL 1þx μ 1

(155a)

(155b)

G



ϕ2LO



ρ ¼ 1þx L 1 ρG



 0:25 μG 1þx 1 μL

(155c)

46

Fundamental Equations for Two-Phase Flow in Tubes

     0:25 ρ μ ϕ2LO ¼ 1 þ x L  1 β G þ ð1  β Þ ρG μL       0:25 ρ μ ϕ2LO ¼ 1 þ x L  1 β G þ ð1  βÞð1 þ 2:5βÞ ρG μL

1897

(155d)

(155e)

According to Hewitt (1982), the homogeneous model tends to underestimate the two-phase frictional pressure gradient sometimes by a large factor but could give reasonable predictions at high pressures and high mass fluxes.

5.2.2 Separated Flow Model In the separated flow model, the frictional pressure gradient for two-phase flow is calculated using Eqs. 144 or 147, but the two-phase friction multiplier is calculated differently from those given in the previous sections. The most popular correlation used widely to calculate the two-phase friction multipliers for low-pressure two-phase flows was developed by Lockhart and Martinelli (1949). Lockhart and Martinelli (1949) related the friction multipliers ϕL and ϕGto the Lockhart-Martinelli parameter X previously defined in Eq. 122 and showed the relationships graphically as shown in Fig. 20. Different curves are shown for laminar or turbulent flow of each phase, and the subscripts, v and t, are used to indicate this. For example, the friction multiplier ϕLvt indicates the case in which the liquid phase flowing alone in the channel is in laminar flow and the gas phase flowing alone is turbulent. A simple and accurate analytical representation of the Lockhart and Martinelli’s graphic relationships for the friction multipliers was given by Chisholm and Laird (1958), ϕ2L ¼ 1 þ

C 1 þ X X2

ϕ2G ¼ 1 þ CX þ X2

(156a) (156b)

where C is a dimensionless parameter whose value depends on the viscous or turbulent nature of each phase. By comparing Eqs. 124 and 156a, void fraction and two-phase friction multiplier for the turbulent-turbulent case can be related, α¼

ð ϕ L  1Þ ϕL

(157)

Martinelli and Nelson (1948) revised the Lockhart-Martinelli correlation to account for the pressure effect and proposed a friction multiplier correlation as shown in Fig. 20. This correlation was fitted to data for steam-water mixtures over

1898

M. Kawaji 1000

Bar (psia) 1.01 (14.7)

6.89 (100) 100

2

34.4 (500) 68.9 (1000) 103 (1500)

10

138

(2000)

172

(2500)

207 221 2 1

0

(3000) (3206) 60

40

20

80

100

Quality, x %

Fig. 20 Two-phase friction multiplier correlation for steam-water by Martinelli and Nelson (1948) (Reproduced with permission from ASME)

a wide range of pressures. However, since Martinelli and Nelson’s (1948) correlation does not account for surface tension, Fig. 20 should not be used for fluids other than steam-water. In addition, many steam-water data sets (e.g., Cicchitti et al. 1960) showed the effect of mass flux on friction multipliers. When such data are compared with the Martinelli and Nelson correlation and homogeneous flow model, the Martinelli-Nelson curve is approached at low mass fluxes, while the homogeneous model fits the data more closely at high mass fluxes. In boiling flows, both the flow quality and two-phase friction multiplier change along the flow direction. If the quality increases from 0 to x linearly over a length, L, the overall two-phase friction pressure drop can be calculated by integrating the two-phase friction pressure gradient over the pipe length L or quality from 0 to x. Since dpdzF is constant, LO

ðL ΔpF ¼ ϕ2LO 0



dpF dz



2

dz ¼ ϕLO ð0 ! xÞΔpLO LO

(158)

46

Fundamental Equations for Two-Phase Flow in Tubes 1 1000

2

5

10

20

50

1899 100

200 Bar

100% Exit quality 90 80 70 60 50

(0

→

)

100

2

40 30 20 10

10

5 1

1 1

2

5

10

20

50

100

200 Bar

Pressure (bar) Fig. 21 Average two-phase friction multiplier for quality varying from zero to the exit quality (Martinelli and Nelson 1948) (Reproduced with permission from ASME)

2

The average friction multiplier, ϕLO ð0 ! xÞ, can be evaluated from the MartinelliNelson (1948) correlation as shown in Fig. 21. If the quality varies linearly from x1 to x2, the average friction multiplier can be evaluated, 2

ϕLO ðx1 ! x2 Þ ¼

i 1 h 2 2 x2 ϕLO ð0 ! x2 Þ  x1 ϕLO ð0 ! x1 Þ x2  x1

(159)

5.2.3 Other Correlations There are many other two-phase friction pressure drop correlations proposed in the past, for example, by Thom (1964), Baroczy (1965), Chisholm (1973), Friedel (1977, 1979), and Müller-Steinhagen and Heck (1986), among others. Friedel (1977, 1979) proposed the following empirical correlation for ϕ2LO based on a large database,

1900

M. Kawaji

ϕ2LO ¼ E þ

3:24FH Fr 0:045 We0:035 H L

(160)

where FrH, E, F, H, and WeL are, Fr H ¼

G2 gDρ2H

E ¼ ð1  x Þ2 þ x 2

ρL f GO ρG f LO

F ¼ x0:78 ð1  xÞ0:224  H¼

ρL ρG

0:91 

μG μL

0:19   μG 0:7 1 μL

WeL ¼

G2 D σρH

(161) (162) (163) (164)

(165)

The homogeneous density, ρH, in the Weber number is given by Eq. 153. Friedel’s (1977, 1979) correlation is applicable to both vertical upward flow and horizontal flow and is known to work well when the viscosity ratio μL/μG < 1000, which applies to most working fluids.

5.3

Acceleration Pressure Drop

Acceleration pressure drop is significant if there is any spatial acceleration of the fluid in the flow channel. In single-phase flow, any change in the flow channel cross section results in acceleration or deceleration of the fluid, and there will be acceleration pressure drop. This is also true in two-phase flow, but additionally, the two-phase mixture can change its velocity when there is phase change due to heat transfer and pressure changes.

5.3.1 Homogeneous Flow Model The acceleration pressure drop for homogeneous two-phase flow in a constant area flow channel is,   dpa dð1=ρH Þ dvH ¼ G2 ¼ G2 dz dz dz

(166)

where vH is the specific volume of the homogeneous mixture, vH ¼ ð1  xÞvL þ xvG

(167)

46

Fundamental Equations for Two-Phase Flow in Tubes

1901

At saturation conditions, the specific volume of the homogeneous mixture is a function of quality and pressure or temperature. Thus, the specific volume of the homogeneous mixture can change along the flow path due to heat addition (vaporization) or heat removal (condensation) and changes in static pressure, dvH @v dp dx þ vfg ¼ @p dz dz dz

(168)

where vfg = vG  vL. Thus, the acceleration pressure drop for a homogeneous flow at thermal equilibrium is,   dpa dx 2 dvH 2 @v dp þ vfg ¼G ¼G @p dz dz dz dz

(169)

The axial change in quality can be determined by knowing the rate of heat addition/removal and changes in static pressure,    dhfg dp dx q0 1 dhL ¼ þx  _ fg dz mh hfg dp dp dz

(170)

where q0 is the linear heat rate, m_ is the mass flow rate, hL is the enthalpy of saturated liquid, and hfg is the latent heat of vaporization. The derivatives of liquid enthalpy and latent heat with respect to pressure along the saturation line are available from tables of thermodynamic properties. If the pressure drop is small compared to the system pressure, then the second term on the right-hand side of Eq. 170 can be neglected, and the quality change is due to heat addition or removal only.

5.3.2 Separated Flow Model The acceleration pressure drop term derived from the separated flow equations can be given by, " # 2 2 dpa d ð 1  x Þ x ¼ G2 þ dz ρL ð1  αÞ ρG α dz

(171)

The term in the brackets can be regarded as the momentum-weighted-specific volume, v0, where, v0 ¼

ð1  x Þ2 x2 þ ρL ð1  αÞ ρG α

(172)

Then, if we integrate the above equation along the flow path from location 1 to location 2, the acceleration pressure drop is,

1902

M. Kawaji

  Δpa ¼ G2 v02  v01 ¼ r 2 G2

(173)

where r 2 ¼ v02  v01 is the acceleration multiplier evaluated using the quality and void fraction at locations 1 and 2, respectively. For steam-water two-phase flow with zero inlet quality and exit quality, xe, Martinelli and Nelson (1948) presented a graph for the acceleration multiplier, r2( p, xe), at different system pressures as shown in Fig. 22.

1

2

5

10

20

50

100

200

Bar

1000 100% Exit quality 90 80 70 60 50 40 30

100

20 10 )

5

2(

,

1 10

1

0.1

1

2

5

10

20 50 Pressure (bar)

100

200 Bar

Fig. 22 Acceleration multiplier by Martinelli and Nelson (1948) (Reproduced with permission from ASME)

46

6

Fundamental Equations for Two-Phase Flow in Tubes

1903

Summary

In this chapter, several aspects of two-phase flow have been described starting with the basic parameters, followed by descriptions of two-phase flow patterns, flow pattern maps, and transition criteria. The fundamental equations for two-phase flow prediction were then presented starting with multidimensional and one-dimensional mass, momentum, and energy conservation equations referred to as a two-fluid model. Appropriate constitutive equations for specific flow patterns that are needed to solve the two-fluid model equations were also covered, focusing mostly on a bubbly flow. In the rest of this chapter, two-phase flow correlations and models available for predicting void fraction and pressure drop in circular and noncircular tubes were described, including homogeneous, drift flux, and separated flow models. Although two-phase flow has been studied and steady progress made over more than a half century, more work is still needed to advance the predictive capability of multidimensional two-phase flow models.

7

Cross-References

▶ Boiling and Two-Phase Flow in Narrow Channels ▶ Compact Heat Exchangers ▶ Flow Boiling in Tubes ▶ Internal Annular Flow Condensation and Flow Boiling: Context, Results, and Recommendations ▶ Two-Phase Heat Exchangers

References Antal SP, Lahey RT Jr, Flaherty JE (1991) Analysis of phase distribution in fully developed laminar bubbly two-phase flow. Int J Multiphase Flow 17(5):635–652 Armand AA (1946) Resistance to two-phase flow in horizontal tubes. Izv VTI 15(1):16–23 Baker O (1954) Simultaneous flow of oil and gas. Oil Gas J 53:185–195 Banerjee S, Chan AMC (1980) Separated flow models – I analysis of the averaged and local instantaneous formulations. Int J Multiphase Flow 6:1–24 Bankoff SG (1960) A variable density single-fluid model for two-phase flow with particular reference to steam-water flow. J Heat Transf 82:265–272 Baroczy CJ (1965) A systematic correlation of for two-phase pressure drop. Chem Eng Prog Symp Ser 62(44):232–249 Basset AB(1888) On the motion of a sphere in a viscous liquid. Philos Trans Royal Soc London, Ser A Math Phys Sci 179:43–63; also A treatise on hydrodynamics, 1961, Dover, New York, Chap. 22 Beattie DRH, Whalley PB (1982) A simple two-phase frictional pressure drop calculation method. Int J Multiphase Flow 8:83–87

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Bergles AE, Roos JP, Bourne JG (1968) Investigation of boiling flow regimes and critical heat flux. NYO-3304-13 Cheng L, Ribatski G, Thome JR (2008) Two-phase flow patterns and flow-pattern maps: fundamentals and applications. Appl Mech Rev 61(5):050802-050802-28. https://doi.org/10.1115/1.2955990 Chichitti A, Lombardi C, Silvestri M, Soldaini G, Zavattarelli R (1960) Two-phase cooling experiments – pressure drop, heat transfer and burnout measurement. Energ Nucl 7(6):407–425 Chisholm D (1973) Pressure gradients due to friction during the flow of evaporating two-phase mixtures in smooth tubes and channels. Int J Heat Mass Transf 16:347–358 Chisholm D, Laird ADK (1958) Two-phase flow in rough tubes. Trans ASME 80(2):276–286 Coddington P, Macian R (2002) A study of the performance of void fraction correlations used in the context of drift-flux two-phase flow models. Nucl Eng Design 215:199–216 Collier JG (1972) Convective boiling and condensation. McGraw Hill, London Collier JG, Thome JR (1994) Convective boiling and condensation. Oxford University Press, New York De Jarlais G (1983) An experimental study of inverted annular flow hydrodynamics utilizing an adiabatic simulation. NUREG/CR-3339, ANL-83-44 Drew DA, Lahey RT Jr (1987) The virtual mass and lift force on a sphere in rotating and straining inviscid flow. Int J Multiphase Flow 13:113–121 Dukler AE, Taitel Y (1977) Flow regime transitions for vertical upward gas liquid flow: a preliminary approach through physical modeling. Progress Report No. 1, NUREG-0162 Dukler AE, Wicks M, Cleveland RG (1964) Frictional pressure drop in two-phase flow: an approach through similarity analysis. AICHE J 10:44–51 Faghri A, Zhang Y (2006) Transport phenomena in multiphase systems. Elsevier, Burlington Franca F, Lahey RT (1992) The use of drift-flux techniques for the analysis of horizontal two-phase flows. Int J Multiphase Flow 18(6):787–801 Friedel L (1977) Momentum exchange and pressure drop. In: Whalley PB (ed) Two-phase flows and heat transfer. Oxford University Press, Oxford Friedel L (1979) Improved friction drop correlations for horizontal and vertical two-phase pipe flow. Paper E2 presented at the European Two-phase Flow Group Meeting, Ispra Friedel L, Diener R (1998) Reproductive accuracy of selected void fraction correlations for horizontal and vertical up flow. Forsch im Ingenieurwes 64:87–97 Godbole PV, Tang CC, Ghajar AJ (2011) Comparison of void fraction correlations for different flow patterns in upward vertical two-phase flow. Heat Transf Eng 32(10):843–860 Govier GW, Aziz K (1972) The flow of complex mixtures in pipes. Van Nostrand Reinhold, New York Hasan AR, Kabir CS (1992) Two-phase flow in vertical and inclined annuli. Int J Multiphase Flow 18(2):279–293 Hewitt GF (1982) Flow regimes. “Pressure drop” and “void fraction”, sections 2.1–2.3. In: Hetsroni G (ed) Handbook of multiphase systems. McGraw-Hill, New York Hewitt GF, Roberts DN (1969) Studies of two-phase flow patterns by simultaneous X-ray and flash photography. UKAEA Report AERE-M2159 Hubbard MG, Dukler AE (1966) The characterization of flow regimes for horizontal two-phase flow. In: Saad MA, Miller JA (eds) Proceedings of the 1966 heat transfer and fluid mechanics institute, Stanford University Press, Palo Alto, pp 100–121 Idzinga W, Todreas N, Bowring R (1977) An assessment of two-phase pressure drop correlations for steam-water systems. Int J Multiphase Flow 3:401–413 Ishii M (1975) Thermo-fluid dynamic theory of two-phase flow. Eyrolles, Paris Ishii M (1977) One-dimensional drift-flux model and constitutive equations for relative motion between phases in various two-phase flow regimes. ANL Report ANL-77-47 Ishii M, Chawla TC (1979) Local drag laws in dispersed two-phase flow. ANL-79-105, NUREG/ CR-1230 Ishii M, De Jarlais G (1986) Flow regime transition and interfacial characteristics of inverted annular flow. Nucl Eng Des 95:171–184

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Ishii M, Hibiki T (2006) Thermo-fluid dynamics of two-phase flow. Springer US. 10.1007/ 978–0–387-29187-1. http://www.springer.com/us/book/9780387283210 Ishii M, Mishima K (1980) Study of two-fluid model and interfacial area. Argonne National Laboratory Report, ANL-80-111, NUREG/CR-1873 Ishii M, Mishima K (1984) Two-fluid model and hydrodynamic constitutive relations. Nucl Eng Des 82:107–126 Ishii M, Zuber N (1979) Drag coefficient and relative velocity in bubbly, droplet or particulate flows. AICHE J 25:843–855 Ishii M, Kim S, Uhle J (2002) Interfacial area transport equation: model development and benchmark experiments. Int J Heat Mass Transf 45(15):3111–3123 Ishii M, Kim S, Kelly J (2005) Development of interfacial area transport equation. Nucl Eng Technol 37(6):525–536 Jones OC, Zuber N (1975) The interrelation between void fraction fluctuations and flow patterns in two-phase flow. Int J Multiphase Flow 2:273–306 Kawaji M, Banerjee S (1987) Application of a multifield model to reflooding of a hot vertical tube, part 1. Model structure and interfacial phenomena. J Heat Transf 109(1):204–211 Kawaji M, Anoda Y, Nakamura H, Tasaka T (1987) Phase and velocity distributions and holdup in high-pressure steam/water stratified flow in a large diameter horizontal pipe. Int J Multiphase Flow 13(2):145–159 Kim S, Ishii M, Sun X, Beus SG (2002) Interfacial area transport and evaluation of source terms for confined air water bubbly flow. Nucl Eng Des 219(1):61–65 Kocamustafaogullari G, Ishii M (1995) Foundation of the interfacial area transport equation and its closure relation. Int J Heat Mass Transf 38(3):481–493 Koizumi Y, Yamamoto N, Tasaka K (1990) Air/water two-phase flow in a horizontal large-diameter pipe (1st Report, Flow regime). Trans. JSME 56(532, B):3745–3749 Lahey RT Jr, Lopez de Bertodano M, Jones OC Jr (1993) Phase distribution incomplex geometry conduits. Nucl Eng Des 141:117–201 Lamb H (1932) Hydrodynamics, 6th edn. Cambridge University Press, Cambridge, UK Liu TJ, Bankoff SG (1993) Structure of air-water bubbly flow in a vertical pipe – II. Void fraction, bubble velocity and bubble size distribution. Int J Heat Mass Transf 36:1061–1072 Lockhart RW, Martinelli RC (1949) Proposed correlation of data for isothermal two-phase, two-component flow in pipes. Chem Eng Prog 45:39–48 Mandhane JM, Gregory GA, Aziz K (1974) Critical evaluation of holdup prediction methods for gas–liquid flow in horizontal pipes. J Pet Technol 27:1017–1026 Martinelli RC, Nelson DB (1948) Prediction of pressure drop during forced-circulation boiling of water. Trans ASME 70:695–702 McAdams WH, Wood WK, Bryan RL (1942) Vaporization inside horizontal tubes: II, benzene-oil mixtures. Trans ASME 64:193–200 Mei R, Adrian RJ, Hanratty J (1991) Particle dispersion in isotropic turbulence under stokes drag and Basset force with gravitational settling. J Fluid Mech 225:481–495 Michaelides EE (1997) Review-the transient equation of motion for particles, bubbles and droplets. J Fluids Eng 119:233–247 Mishima K, Ishii M (1984) Flow regime transition criteria for upward two-phase flow in vertical tubes. Int J Heat Mass Transf 27(5):723–737 Müller-Steinhagen H, Heck K (1986) A simple friction pressure correlation for two-phase flow in pipes. Chem Eng Process 20:297–308 Nakoryakov VE, Kashinskii ON, Koz’myenko BK, Goryelik RS (1986) Study of upward bubbly flow at low liquid velocities. Izv Sib otdel Akad nauk SSSR 16:15–20 Nigmatulin RI (1979) Spatial averaging in the mechanics of heterogeneous and dispersed systems. Int J Multiphase Flow 4:353–385 Noghrehkar GR, Kawaji M, Chan AMC (1999) Investigation of two-phase flow regimes in tube bundles under cross-flow conditions. Int J Multiphase Flow 25:857–874

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Oshinowo T, Charles ME (1974) Vertical two-phase flow: part 11. Holdup and pressure drop. Can J Chem Eng 56:438–448 Owens WL (1961) Two-phase pressure gradient. ASME Int Develop Heat Transf Part II 363–368 Rouhani SZ, Axelsson E (1970) Calculation of void volume fraction in the sub cooled and quality boiling regions. Int J Heat Mass Transf 13:383–393 Rouhani SZ, Sohal MS (1983) Two-phase flow patterns: a review of research results. Prog Nucl Energy 11(3):219–259 Saadatomi M, Sato Y, Saruwatari S (1982) Two-phase flow in vertical non-circular channels. Int J Multiphase Flow 8(6):641–655 Sadatomi M, Kawaji M, Lorencez CM, Chang T (1993) Prediction of liquid level distribution in horizontal gas-liquid stratified flows with interfacial level gradient. Int J Multiphase Flow 19 (6):987–997 Sato Y, Sadatomi M (1986) Two-phase flow in vertical non-circular channels. In: Cheremisinoff NP (ed) Encyclopedia of fluid mechanics, vol 3. Gulf Publishing, Houston, pp 651–664 Serizawa A, Kataoka I, Michiyoshi I (1975) Turbulence structure of air-water bubbly flow, part II: local properties. Int J Multiphase Flow 2:235–246 Stuhmiller JH (1977) The influence of interfacial pressure on the character of two-phase flow model equations. Int J Multiphase Flow 3:551–560 Taitel Y, Dukler AE (1976a) A model for predicting flow regime transition in horizontal and near horizontal gas-liquid flow. AICHE J 22:47–55 Taitel Y, Dukler AE (1976b) A theoretical approach to the Lockhart-Martinelli correlation for stratified flow. Int J Multiphase Flow 2:591–595 Taitel Y, Bornea D, Dukler AE (1980) Modelling flow pattern transitions for steady upward gas-liquid flow in vertical tubes. AICHE J 26(3):345–354 Thom JRS (1964) Prediction of pressure drop during forced circulation boiling of water. Int J Heat Mass Transf 7:709–724 Tomiyama A, Kataoka I, Zun I, Sakaguchi T (1998) Drag coefficients of single bubbles under normal and micro gravity conditions. JSME Int J, Ser B 41(2):472–479 Tomiyama A, Tamai H, Zun I, Hosokawa S (2002) Transverse migration of single bubbles in simple shear flows. Chem Eng Sci 57:1849–1858 Wallis GB (1969) One-dimensional two-phase flow. McGraw-Hill, New York Wang X, Sun X (2010) Three-dimensional simulations of air–water bubbly flows. Int J Multiphase Flow 36:882–890 Weisman J, Duncan D, Gibson J, Crawford T (1979) Effects of fluid properties and pipe diameter on two-phase flow patterns in horizontal lines. Int J Multiphase Flow 5:437–462 Woldesemayat MA, Ghajar AJ (2007) Comparison of void fraction correlations for different flow patterns in horizontal and upward inclined pipes. Int J Multiphase Flow 33:347–370 Wu Q, Kim S, Ishii M, Beus SG (1998) One-group interfacial area transport in vertical bubbly flow. Int J Heat Mass Transf 41(8–9):1103–1112 Zuber N (1964) On the dispersed flow in the laminar flow regime. Chem Eng Sci 19:897–917 Zuber N, Findlay JA (1965) Average volumetric concentration in two-phase flow systems. J Heat Transf 87:453–468 Zun I (1980) The transverse migration of bubbles influenced by walls in vertical bubbly flow. Int J Multiphase Flow 6:583–588

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Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Characterization of Flow Boiling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Overview of Phenomenology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Flow Regimes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Heat Transfer Regimes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Multiscale Phenomena in Flow Boiling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Approaches of Flow Boiling Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 One-Dimensional Modeling of Flow Boiling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Conservation Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Flow Regimes and Heat Transfer Regimes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Wall Friction Closures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Wall Boiling Closures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Volume Fraction and Relative Velocity Closures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Multidimensional Modeling of Flow Boiling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Conservation Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Interfacial Dynamics Closures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Wall Boiling Closures (Multidimensional) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Phenomenological Model for DNB Prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Phenomenological Model for Dryout Prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Open Issues and Perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Open Issues for Current Modeling Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Advanced Modeling Approach: Direct Numerical Simulation . . . . . . . . . . . . . . . . . . . . . . 6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Y. Liu (*) · N. Dinh North Carolina State University, Raleigh, NC, USA e-mail: [email protected]; [email protected] # Springer International Publishing AG, part of Springer Nature 2018 F. A. Kulacki (ed.), Handbook of Thermal Science and Engineering, https://doi.org/10.1007/978-3-319-26695-4_47

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Abstract

Flow boiling in tubes is a highly efficient heat transfer regime, which is used for thermal management in various engineered systems with high energy density, from power electronics to heat exchangers in power plants and nuclear reactors. Flow boiling can occur in different two-phase flow patterns under a wide range of flow conditions, including transient and developing flows. Thus, analysis of flow boiling in tubes is built upon a broad knowledge base on related processes in two-phase flow and heat transfer mechanisms that have been a subject of numerous experimental, theoretical, and computational investigations over many decades. The main quantities of interest for design, operation, and safety of such systems are boiling heat transfer and the limit of coolability associated with boiling crisis and burnout that occur at critical heat fluxes. This chapter provides an overview of a wide range of phenomena that govern heat transfer in flow boiling in tubes, highlighting the multiscale complexity of flow boiling. The main content of the chapter discusses approaches to modeling of flow boiling, including traditional one-dimensional models and an emerging class of multidimensional treatments. Associated issues, remaining uncertainties, and perspectives are also discussed. Nomenclature

A d cp cv D e f fTP g G h k p q Q S t T u ugs uls v x z

(tube) cross-sectional area; interfacial area bubble diameter heat capacity at constant pressure heat capacity at constant volume tube inner diameter internal energy friction factor friction factor for two-phase flow gravitational acceleration mass flux enthalpy; heat transfer coefficient thermal conductivity; turbulent kinetic energy pressure heat flux flow rate per unit periphery of the liquid film in annular flow slip ratio time temperature velocity gas superficial velocity liquid superficial velocity specific volume thermodynamic quality flow direction

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Greek Symbol

α Γ δ e θ μ ρ σ ϕ Φ2fo

phasic fraction by volume interfacial mass change rate film thickness turbulent dissipation rate angle dynamic viscosity density surface tension representation of a general scalar quantity / contact angle two-phase frictional multiplier

Superscripts

c e q t

convective heat transfer component evaporation heat transfer component quenching heat transfer component turbulence

Subscripts

b f g i l r sat sub sup w

bubble g property difference between fluid and gas (both in saturation state) gas interface liquid relative motion saturation state subcooled state superheat state wall

Nondimensional Groups

Nu Pr Re Reb We

1

Nusselt number (=hD/k) Prandtl number (=cpμ/k) Reynolds number (=ρvD/μ) Bubble Reynolds number (=ρl(vg
vl)d/μl) Weber number (=ρv2d/σ)

Introduction

This chapter provides considerations of physics, modeling, and prediction of flow boiling in tubes. Flow boiling is comprised of two efficient modes for heat transfer, namely, convection (convective heat transfer) and phase change (boiling heat

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transfer). Both heat transfer modes have been extensively investigated over the past century, while flow boiling has found its application in thermal management of a broad range of industrial systems and equipment. Notably, different systems use different fluids as coolant (although water is the more common coolant), different heating methods, different heater materials, different heater surface treatments, etc. The flow channel geometry and heating characteristics in various industrial systems vary widely, from the simplest (circular tube with uniform heat input) to more complex geometries such as fuel rod bundles in the nuclear reactor core, that include spacer grids and mixing vanes, designed to enhance heat transfer under certain system conditions. Furthermore, the performance of flow boiling can vary significantly in vertical, inclined, and horizontal tubes, on flow directions (e.g., upward, downward), and heater arrangement (e.g., partial area heating). The significance of channel geometry and heater characteristics is rooted in geometric, forcing (gravity), and physicochemical sensitivity of multiphase flow phenomena at different scales (e.g., flow regimes, bubble dynamics, nucleation) that govern thermal hydraulics of flow boiling in heated channels. The sensitivity mentioned above results in a great variety of behaviors in flow boiling. Phenomena of flow boiling in tubes discussed in this chapter are characteristic for only a subset of systems that involve flow boiling. For engineering design and analysis of systems with flow boiling, the two main quantities of interest (QOI) are pressure drop over the boiling channel and heat transfer coefficient from the heater to the coolant. The former affects the hydraulic design, while the latter (heat transfer) governs the thermal design, namely, temperature in the heater’s structural materials. Notably, in design and operation of high-power-density systems such as thermal and nuclear power plants, thermal safety margins often dictate temperature control requirements and, as a result, the required heat transfer rate. The purpose of this chapter is to provide an overview of flow boiling phenomenology and discussion of modern approaches for modeling of flow boiling in tubes. The chapter is structured to include a qualitative characterization of flow boiling (Sect. 2), one-dimensional modeling of flow boiling as applied in engineering system-level analysis (Sect. 3), an emerging development of advanced multidimensional modeling and simulation of flow boiling (Sect. 4), and discussion of open issues and perspective (Sect. 5). Conclusion is given in Sect. 6.

2

Characterization of Flow Boiling

2.1

Overview of Phenomenology

Figure 1 provides an overview of the phenomenology of flow boiling in tubes, comprising of two main components: two-phase fluid dynamics and boiling heat transfer. Although the two components are tightly coupled in flow boiling, their decomposition was necessary to allow the components be investigated separately and brought together for analysis.

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Flow Boiling

Two phase flow

Bulk flow characteristic

Interfacial dynamics

Boiling heat transfer

Turbulence

Nucleation

Wall heat transfer

Bubble dynamics

Fig. 1 Phenomena decomposition of flow boiling

The two-phase flow dynamics is governed by bulk flow characteristics, interfacial dynamics, and fluid turbulence. The bulk flow is characterized by macroscale quantities including the flow regime, phasic density, velocity, and volume fraction. The interfacial dynamics is driven by interfacial topology (e.g., bubble deformation, breakup, and coalescence), interfacial momentum exchange (e.g., drag force, lift force), and interfacial mass transfer (e.g., vapor condensation in subcooled liquid region or evaporation in superheated liquid region). The fluid turbulence is central to both bulk flow and interfacial dynamics, involving turbulence-interface interactions (e.g., bubble-induced turbulence). Boiling heat transfer is a complex phenomenon. In the context of flow boiling, boiling heat transfer is governed by nucleation of vapor bubbles on the heater wall surface, mechanisms of wall heat transfer, and near-wall bubble dynamics (e.g., bubble growth, departure, sliding). The wall heat flux in flow boiling can be decomposed into convective heat transfer (characteristically to liquid in pre-burnout/ pre-dryout regimes and to vapor in post-dryout regimes) and latent (evaporation) heat transfer.

2.2

Flow Regimes

An important characterization of two-phase flow in tubes is the interfacial structures between liquid and vapor. A distinctive structure is called flow regime (or flow pattern). The most commonly studied flow regimes in tubes are upward vertical flow and horizontal flow under normal gravity and isothermal conditions. Figure 2 shows representative flow patterns in upward flow in vertical tubes (including bubbly flow, slug flow, churn flow and annular flow) and in horizontal tubes (including bubbly flow, plug flow, stratified flow, wavy flow, slug flow and annular flow). • Bubbly Flow: Bubbles are dispersed in a continuous liquid phase. This regime is observed in relatively low gas volume fraction flow. • Plug Flow: Bubble population includes elongated bubbles. • Stratified Flow: This regime occurs at relatively high gas volume fractions at low flow velocities.

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Fig. 2 Typical flow regimes in two-phase flow (a) in upward vertical flow, (b) horizontal flow

• Wavy Flow: This regime occurs under high gas volume fractions similar to that of stratified flow, while the phase interface experiences instability due to relatively high flow velocities. • Slug Flow: Bubble population includes large bubbles of the size in the order of tube’s diameter. These bullet shape bubbles formed as a result of bubble coalescence. • Churn Flow: Gas phase includes large bubbles of irregular shapes like unstable slug. • Annular Flow: The gas phase forms a continuous core with liquid droplets, while a liquid film flows along the tube’s walls.

Flow regime maps have been studied in the past. Notable are the researches by Taitel and his co-workers for flow regime identification in vertical upward flow (Taitel et al. 1980) and horizontal flow (Taitel and Dukler 1976). They used simplified mechanistic treatments to develop flow regime transition criteria. However, these criteria are obtained for steady, fully developed, and adiabatic flow. Thus, application of these maps for transient, developing, and boiling flow involves significant uncertainty. For vertical upward flows, phasic superficial velocities as well as tube geometry are the main controlling variables, which will be discussed in Sect 3.2. For horizontal flows, Taitel and Dukler (1976) devised other dimensionless parameters to determine flow regimes. The detailed form of equations and their derivations can be found in the references. It also needs to point out that there are also other criteria proposed by other researchers, such as the flow regime map proposed by Kaichiro and Ishii (1984). The research was also performed on flow regime under micro-gravitational condition (Zhao and Rezkallah 1993), inside helically coiled tubes (Liu and Masliyah 1993) or microchannels (Kawahara et al. 2002).

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Heat Transfer Regimes

Flow boiling in tubes characteristically involves the following main heat transfer regimes: single-phase (liquid) convective heat transfer, nucleate boiling, transition boiling, and film boiling. As a subcooled liquid is heated up in a tube, the wall surface temperature increases and exceeds the fluid saturation temperature. As the boundary layer’s liquid becomes superheated, vapor bubbles are nucleated on the heated surface causing onset of nucleate boiling. As a microscopic physical phenomenon, heterogeneous nucleation is known to be dependent on characteristics of heater surface such as surface material, morphology, and wettability. Nucleation and growth of vapor bubbles serve as mechanisms for efficient cooling of the superheated fluid layer and hence heat removal from the heated wall. Depending on bulk flow characteristics, nucleate boiling can occur under subcooled or saturated bulk flow regimes. In subcooled flow boiling, heat transfer from interface to subcooled liquid may cause vapor bubbles to condense, while suppressing them to the near-wall region. As the fluid heats up and evaporates, nucleate boiling evolves from partial to fully developed and eventually reaches a boiling crisis and transition to film boiling. Depending on flow conditions (e.g., mass flow rate, wall heat flux, inlet vapor quality, etc.), the boiling crisis can occur as a departure from nucleate boiling (DNB) or liquid film dryout in bubbly or annular flow regimes, respectively (Fig. 3). Notably, boiling crisis is characterized by critical heat flux (CHF), which is paramount to the design and safety analysis of heat transfer systems involving flow boiling in tubes. The DNB occurs in a system with high wall heat flux and low vapor quality, while the dryout occurs in a system with relatively low wall heat flux and high vapor quality. For systems with a given wall heat flux, the boiling crisis causes a dramatic deterioration of heat transfer leading to a rapid heat up and possible burnout of the heater structural material. For systems with a given heated temperature, the boiling crisis leads to a sharp decrease of heat flux. Mechanisms of boiling crisis that governs the CHF have been a subject of intensive investigations over the past five decades. A significant body of knowledge about boiling crisis was obtained in the study of pool boiling, e.g., Helmholtz instability mechanism of the vapor column (Zuber 1958), dryout under large vapor mushrooms due to bubble coalescence (Haramura and Katto 1983). More recent studies provided new insights that highlight the significance of heater surface on boiling heat transfer and burnout. For flow boiling, the effects of bulk flow conditions on boiling crisis were investigated and captured in different mechanistic models, for example, the near-wall bubble crowding and vapor blanketing (Weisman and Pei 1983) and the dryout of liquid layer beneath vapor blanket due to Helmholtz instability at the sublayer-vapor interface (Lee and Mudawwar 1988). A detailed review of boiling crisis mechanisms can be found in Collier and Thome (1996) and Tong and Tang (1997). While uncertainty exists, experiments and analysis performed to date provide a technical basis for determining CHF in a flow boiling in tubes for a range of conditions of applications of interest. In the post-CHF regime, film boiling can occur in inverted annular flow regime with liquid core surrounded by a vapor film on the heater wall surface, or in

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Fig. 3 Flow regimes and wall boiling heat transfer regimes in flow boiling in a heated vertical tube under (a) DNB and (b) dryout boiling crisis modes

dispersed flow regime with liquid droplets in vapor flow (Fig. 3). In this deficient heat transfer regime, heat removal is governed by convective heat transfer to vapor and radiative heat transfer due to surface overheating in post-dryout/burnout conditions. When the heat flux exceeds the CHF, the boiling crisis can cause an unstable vapor film to form partial film boiling (or transition boiling) with degraded heat transfer. In this regime, liquid droplet can still be able to wet the heated surface under a moderate surface temperature. As the heated surface temperature increases and exceeds the so-called Leidenfrost temperature, the liquid ceases to wet the surface, rendering a stable film boiling regime. As the vapor quality increases, the liquid core diminishes into and a dispersed droplet flow. This liquid-deficient regime combines vapor convective heat transfer and droplet evaporation.

2.4

Multiscale Phenomena in Flow Boiling

Treatment of flow boiling in tubes thus requires understanding and prediction of multiscale (practically, multi-physics) phenomena. The phenomena include

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convective heat transfer in turbulent flow, boiling physics, and the interactions of convection and boiling. The combination of those phenomena results in an array of two-phase flow regimes that vary greatly with flow direction, flow geometry, fluid properties, and heating/heater characteristics. A list of representative phenomena and factors of importance in flow boiling in tubes is provided in Table 1 below. Also, given in the third column are comments on the scale of the phenomena and their treatment in more commonly used mechanistic models.

2.5

Approaches of Flow Boiling Modeling

There are two types of modeling approach of flow boiling which can serve for engineering purpose, the mixture model and two-fluid-model (aka segregated flow model). The major difference between these two types is their treatment of the two phases. The mixture model treats two phases as a mixture whose property is averaged from vapor and liquid. Thus, only one set of conservative equations for mass, momentum, and energy is needed. The mixture model is further divided, based on their treatment of mechanical nonequilibrium (the relative velocity between the two phases). The homogeneous equilibrium model assumes there is no relative velocity between the phases. The slip factor model uses empirical correlations for the slip ratio (which is defined by the ratio of vapor velocity to liquid velocity). The drift flux model uses kinematic constitutive equations to describe the relative flow. Theoretically, all three models can be applied to both one-dimensional cross-sectional averaged problem and multidimensional problems. In practice, however, these models are mainly used for one-dimensional cross-sectional averaged problems, such as in system analysis and engineering calculations. The two-fluid model treats two phases by separate sets of field conservation equations. Based on the treatment of interactions between the two phases, the two-fluid model can be further divided into different models. For one-dimensional cross-sectional averaged formulation, the interfacial interactions are modeled in a correspondingly coarse manner. Notably, the effect of local interactions, e.g., wall boiling heat transfer, is coarsened as a source term that affects the cross-sectional averaged parameters of flow dynamics. In contrary, the multidimensional formulation requires a much larger set of closure relations needed to provide detailed modeling of interfacial interactions and wall heat transfer. The model type and required closures are summarized in Table 2 which highlights main characteristics of the models used for flow boiling calculations and key elements addressed in this chapter.

3

One-Dimensional Modeling of Flow Boiling

Because of its importance to engineering and safety analysis of thermal and nuclear power plants, flow boiling prediction is central to so-called system thermal hydraulics codes such as RELAP-5 (Ransom et al. 1982) and TRACE (Bajorek 2008) and

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Table 1 Multiscale phenomena of importance in flow boiling in tubes Categories Two-phase flow patterns

Phenomena, mechanisms Flow regime, phase distribution, local topology

Two-phase fluid dynamics

Interfacial area density transport (interfacial breakup, coalescence) Condensation in subcooled fluid

Interfacial forces

Turbulence

Droplet entrainment and deposition (in disperse-annular flow) Liquid turbulence Bubble-induced turbulence

Boiling heat transfer

Vapor bubble nucleation

Bubble dynamics (growth, sliding, detachment) Thin liquid film dynamics and evaporation Effect of external flow convection on boiling heat transfer

Effects of heater surface Departure from nucleate boiling

Dryout

Scale and treatment in models Macroscale, treated in one-dimensional homogeneous mixture models and separate flow model Mesoscale, treated in multidimensional, two-fluid model Mesoscale, treated in one-dimensional separate flow model and multidimensional, two-fluid model Mesoscale Treated by simplified empirical model in 1-D separate flow model Treated by detailed mechanistic model in multidimensional, two-fluid model Treated by equivalent body force in DNS Mesoscale, treated in one-dimensional separate flow model and multidimensional, two-fluid model Mesoscale, treated in multidimensional, two-fluid model Mesoscale, treated in multidimensional, two-fluid model Mesoscale The averaged collective behavior is treated in multidimensional two-fluid model Detailed information is treated in DNS Mesoscale Treated in multidimensional two-fluid model Microscale Treated in direct numerical simulation (DNS) Macroscale Treated in one-dimensional homogeneous mixture models and separate flow models (by different heat transfer correlations) Microscale, no proper treatment yet Complex multiscale phenomena Treated in multidimensional two-fluid model based on mesoscale phenomena Complex multiscale phenomena Treated in one-dimensional separate flow model based on macroscale parameters Treated in multidimensional two-fluid model based on mesoscale phenomena (continued)

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Table 1 (continued) Categories

Phenomena, mechanisms Vapor film boiling

Geometry and boundary conditions

Effects of flow channel geometry, flow direction, heating (input heat distribution)

Scale and treatment in models Mesoscale Treated in one-dimensional homogeneous mixture models and separate flow model based on macroscale parameters Macroscale Treatment varied by cases

sub-channel codes such as COBRA-TF (Salko and Avramova 2014). In those engineering practices, the one-dimensional cross-sectional averaged two-phase models are implemented for its computational efficiency. The use of these models is also dictated by the quality of data that can be used to support the model development, calibration, and validation. Both types of models discussed in the previous section have been widely used in this area.

3.1

Conservation Equations

The mixture model treats the two-phase flow as a mixture; the physical properties are represented by the averaged value of the mixture. In its simplest form, the conservation equations are the same as single phase, assuming equal velocity, temperature, and pressure in both phases, with both liquid and vapor at saturation. In reality, the velocities of the two phases are not the same for most cases. Thus, the slip ratio term, or the drift flux model (developed by Zuber and Findlay 1965), needs to be introduced as an improvement for the homogenous model: Mass A

@ρ @ρuA þ ¼0 @t @z

(1)

Momentum
 @ρu @ρuuA @p @ αl αg ρl ρg 2 A þ ¼
A
τw Pw þ ρgA sin θ
ur @t @z @z @z ρ

(2)

Energy A


   @ρe @ρeuA @uA @ αl αg ρl ρg  þ ¼
p þ qw P w
eg
el vg
vl A @t @z @z
@z ρ    @ α l α g ρl ρg  ug
ul vg
vl A
p @z ρ

(3)

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Table 2 Summary of two-phase flow models, required equations, and closures Number of field Models equations Homogeneous 3 equilibrium model Slip model

3

Drift flux equilibrium model

4

Two-fluid 6 model (single pressure)

Number of state variables Required closures 1 pressure One-D Wall boiling, wall 1 temperature friction 1 velocity Multi-D Wall boiling, turbulence 1 pressure One-D Wall boiling, wall 1 temperature friction, slip factor 2 velocity Multi-D Wall boiling, turbulence, slip factor 1 pressure One-D Wall boiling, wall 1 temperature friction, drift velocity, 2 velocity distribution parameter Multi-D Wall boiling, wall friction, drift velocity, distribution parameter 1 pressure One-D Wall boiling, wall 2 temperature friction, drag force, 2 velocity interfacial mass/ energy transfer

Multi-D Wall boiling, turbulence, bubbly size, multiple interfacial forces, interfacial mass/ energy transfer

Application domain Engineering analysis and system modeling approach, with good performance in operational regimes in thermal and nuclear power plants (characteristically, high flow rate and therefore well mixed and high pressure therefore low density ratio)

Engineering analysis, used in industrial system thermal hydraulics modeling. Particularly suitable for high nonequilibrium regime, such as in reduced pressure, accident regimes (low flow rate, segregated flow patterns) More mechanistic description of two-phase flow and heat transfer mechanisms, requiring a large number of closure relations Correspondingly, this model requires experimental data from highly resolved separate-effect tests and controlled combined-effect tests to support qualification, calibration, and validation of the closure relation models

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where A denotes the channel’s flow cross-sectional area and Pw stands for the perimeter. The mixture property can be calculated as, ρϕ ¼ αl ρl Φl þ αg ρg Φg

(4)

where ϕ can be 1, u, or e. The segregated (two-fluid) model considers conservation equations for two-phase flows in two fields separately. Thus, conservative equations are written for each fluid or field: Mass @αl ρl @αl ρl ul Ax þ ¼
Γi @t @z @αg ρg @αg ρg ug Ax þ ¼ Γi A @t @z A

(5)

Momentum @αl ρl ul @αl ρl ul ul Ax @p þ ¼
Γi ^u i
αl A
τwl Pwl þ τi Pi þ αl ρl g sin θA @z @t @z @αg ρg ug @αg ρg ug ug Ax @p þ ¼ Γi ^u i
αg A
τwg Pwg
τi Pi þ αl ρl g sin θA A @z @t @z A

(6)

Energy @αl ρl el @αl ρl el ul Ax @αl ul A @αl Ax
p þ þ Γi ^h i ¼
p þ qwl Pwl þ qil Pi @z @t @z @t (7) @αg ρg eg @αg ρg eg ug Ax @αg ug A @αg Ax
p þ
Γi ^h i ¼
p þ qwl Pwl þ qig Pi A @z @t @z @t A

where Γi is the interfacial mass transfer rate (evaporation/condensation), ^v i is the interface velocity, ^h i is the interfacial enthalpy due to phase change, and Pi is the interfacial perimeter. In addition, the above system of equations is complemented by equations of state for each phase, thermophysical properties, and equation for volume fractions. While derivation of mathematical models (averaged field equations) is beyond the scope of the present chapter, it is instructive to note that each model requires a corresponding set of constitutive laws (or so-called closure relations) to describe interphasic exchanges. The latter are dependent largely on flow regimes and local flow conditions. To sum up, both the mixture model and the two-fluid model require wall friction shear and heat transfer closures for solving the conservation equations. The mixture models also require other closure terms to describe the relative motion between phases, while the segregated models require closure relations for interphasic mass, momentum, and heat exchanges. As a rule, these closure relations are dependent on flow regimes and heat transfer regimes, which vary greatly in flow boiling along the heated channel. This section discusses several representative correlations, excepting the interphasic exchange terms to be discussed in Sect. 4.2. Section 3.2 introduces

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the criteria for determining flow regimes and heat transfer regimes. Sections 3.3 and 3.4 discuss the wall friction and wall heat transfer, respectively. Section 3.5 discusses correlations for describing the relative motion between phases in mixture models and their relationships to phasic volume fractions.

3.2

Flow Regimes and Heat Transfer Regimes

3.2.1 Flow Regime Transitions Two-phase flow regimes are determined using flow regime maps, which are partitioned into a number of areas, each corresponding to a flow regime. Figure 4 depicts a typical flow regime map for a vertical upward flow obtained in Taitel et al. (1980). Characteristically, the maps are governed by phasic superficial velocities uls and ugs (an artificial flow velocity calculated as if the given phase or fluid were the only one flowing or present in a given cross-sectional area), fluid properties, gravity, and flow channel geometry. The transitions between flow regimes are determined by the corresponding flow regime transition criteria, which are obtained through extensive experimental and analytical studies. In some cases, the flow regime transition criteria are determined simply by a critical phasic volume fraction. For the flow regime map given in Fig. 4, the flow regime transition criteria are given by Eqs (8–12),  31=4 2  g ρ l
ρg σ 5 uls ¼ 3:0ugs
1:154 ρ2l

Fig. 4 Generalized flow regime map for upward vertical tube: I. non-dispersed bubbly flow; II: finely dispersed bubbly flow; III: slug flow; IV: churn; V: annular flow

(8)

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8 30:446 9 2  > > =

> ρl ; : ðμl =ρl Þ h i l  pffiffiffiffiffiffi E 40:6 uls þ ugs = gD þ 0:22 ¼ D

(9)

(10)

1=2

h

ugs ρg

 i1=4 ¼ 3:1 σg ρl
ρg

(11)

α ¼ 0:52

(12)

In practice, flow regime maps are implemented in computer simulation codes with smoothing functions across transition regions in parameter space. Such a practice was necessary to avoid discontinuity that could pose challenge to the numerical solution of the partial differential equations. It should be noted that such a treatment is artificial and can introduce significant errors in the prediction of flow regimes in two-phase system in general and flow boiling in particular.

3.2.2 Heat Transfer Regime Transitions For heat transfer, the most distinctive transitions are the onset of nucleate boiling and the departure from nucleate boiling or boiling crisis. Onset of nucleate boiling marks the transition from single-phase heat transfer to boiling heat transfer. This phenomenon is governed by heterogeneous nucleation of vapor bubbles on superheated heater surface, which is known to be sensitive to characteristics of the heater surface, particularly surface roughness (cavity size) and surface wettability (Collier and Thome 1996). Davis and Anderson (1966) developed an analytical solution for relating wall superheat and heat flux, q¼



kl hfg ρg ðTw
Tsat Þ2 8σT sat

(13)

which can be used to calculate the superheat needed for onset of nucleate boiling (ΔTONB). In a more recent work, Basu et al. (2002) related ΔTONB to the available cavity size on heater surface, ΔT ONB ¼

4σT sat Dc ρg hfg

(14)

where the available cavity size Dc is, Dc ¼ FðΦÞ  D0c

(15)

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where F(ϕ) is a correction factor based on contact angle ϕ and D0c is the cavity diameter corresponding to the tangency condition. D0c is obtained by applying the minimum superheat criterion, D0c

" #1=2 8σT sat kl ¼ ρg hfg qw

(16)

Combining the three equations, ΔTONB can be obtained, #1=2 pffiffiffi " 2 σT sat qw ΔT ¼ FðΦÞ ρg hfg kl

(17)

Based on available data for different fluids and surfaces, correlation for F(ϕ) was proposed,   FðΦÞ ¼ 1
exp
Φ3
0:5Φ

(18)

Boiling crisis. Because of the importance of critical heat flux in engineering analysis and the strong effects of flow conditions and geometry, there exist a large number of empirical and semiempirical correlations as well as look-up tables for CHF; see (Collier and Thome 1996) and (Tong and Tang 1997). Most notable is the CHF look-up table by Groeneveld et al. (1986, 1996, 2007), covering a wide range of flow conditions of practical interest. The table is based on a normalized data bank for a vertical 8 mm water-cooled tube containing more than 30,000 data points at 24 pressures, 20 mass fluxes, and 23 qualities. The table can be used to predict the CHF as a function of the coolant pressure (P), mass flux (G), and local thermodynamic quality (x). For tubes with other diameters, an approximate scaling relationship is,

1=2 8 qCHF ¼ qCHF, 8mm (19) Dtube ½mm The look-up table data include four regions represented by different colors in the original paper. The first region was derived directly from the experimental data, for which the CHF values can be calculated with high confidence. The second region includes CHF values predicted using selected models that provide reasonable estimates of CHF under the neighboring conditions. The CHF predictions in this region are considered to have moderate uncertainty. The third region represents conditions where CHF experimentation was limited or unavailable (e.g., critical flow area). The fourth region is called “limit quality region,” where CHF was observed to change rapidly with quality. The CHF values in the last two regions have large uncertainty. For applications that require high confidence level, look-up table values in the first and second regions are interpolated to obtain the required CHF values.

47

3.3

Flow Boiling in Tubes

1923

Wall Friction Closures

For one-dimensional cross-sectional averaged modeling, wall shear force is essential for pressure drop prediction. This wall shear force is expressed in terms of the friction factor f, 2

ρu τw ¼ f (20) 2 For engineering applications, the friction factor f is used to describe wall friction. A method developed by Lockhart and Martinelli (1949) can be used for determining the two-phase flow wall friction factor fTP by modifying the friction factor for single-phase flow with a two-phase frictional multiplier. For mixture models, the two-phase frictional multiplier Φ2fo relates the two-phase friction pressure drop (TP) to that of liquid-only flow ( fo),

dp dz

fric, TP



dp ¼ Φ2 dz fric, fo fo

(21)

An extensive database was developed for Φ2fo values under different conditions and flow patterns. One of the commonly used equations for Φ2fo is, Φ2fo

!#
0:25
" μfg vfg ¼ 1þx 1þx vf μf

(22)

where the mixture properties ( μ ) can be determined using different functional expressions of phasic properties and quality (x), 1 x 1
x ¼ þ μf μ μg μ ¼ xμg þ ð1
xÞμf h i μ ¼ ρ xvg μg þ ð1
xÞvf μf

(23) (24) (25)

For two-fluid segregated models, Lockhart and Martinelli proposed that the ratio of two-phase frictional pressure gradient to that of single-phase flow is a function of fraction and shape of the flow area occupied by this phase. Thus, the two-phase frictional multipliers for liquid phase and gas phase are,



dp 2 dp ¼ Φg dz friction, TP dz friction, g

(26)

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Table 3 Value of C under different flow conditions Liquid Turbulent Turbulent Viscous Viscous

Gas Turbulent Viscous Turbulent Viscous

Φ2l



dp dz

friction, TP

¼ Φ2l



dp dz friction, l

C 20 10 12 5

(27)

Lockhart and Martinelli further suggested that the two multipliers are related by a so-called Martinelli parameter (Xtt), " dp #1=2 Φg dz friction, l ¼ dp Xtt ¼ Φf dz friction, g

(28)

Then, relations between Xtt and the two multipliers were derived from the experimental data, Φ2l ¼ 1 þ

C 1 þ Xtt Xtt 2

Φ2g ¼ 1 þ CXtt þ Xtt 2

(29) (30)

where parameter C depends on the flow regime as listed in Table 3 (Collier and Thome 1996). The Martinelli parameter can also be estimated as a function of thermodynamic quality and the thermophysical properties of the two phases, !0:1



1
x 0:9 ρg 0:5 μl Xtt  (31) x ρl μg

3.4

Wall Boiling Closures

As a major QOI, there are numerous correlations for heat transfer in flow boiling. While in this section, a subset of representative correlations are discussed, readers are referred to Tong and Tang (1997) and Collier and Thome (1996) for a comprehensive review of heat transfer correlations in flow boiling.

3.4.1 Pre-CHF Heat Transfer For nucleate boiling heat transfer, a widely used correlation was proposed by Jens and Lottes (1951) and improved by Thom et al. (1965) for water-cooled systems that relates the wall superheat to the heat flux,

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ΔT sup ¼ 22:5q0:5 w expð
p=87Þ

(32)

where units used for ΔTsup, q, and p are K, MW/m2, and bar, respectively. Another widely used correlation was developed by Chen (1966) in the form of heat transfer coefficient, hNB ¼ F  hFC þ S  hPB

(33)

The terms in the right-hand side represent the forced convective heat transfer and boiling heat transfer. The heat transfer coefficients hFC and hPB are determined for single-phase flow and pool boiling, respectively. Coefficients F and S represent the effect of boiling-induced convection enhancement and suppression of nucleate boiling in flow under increasing mass flux and quality. Chen developed a graphical relationship between F and 1/Xtt to obtain the convective heat transfer component. For pool boiling heat transfer, the correlation by Forster and Zuber (1955) is hPB. As a result, the boiling heat transfer component can be obtained, hPB

! k0:79 Cp0:45 ρ0:49 g0:25 l l l ¼ 0:00122 ðT w
T l Þ0:24 ½PðT w Þ
PðT sat Þ0:75 0:24 0:24 σ 0:5 μ0:29 h ρ g fg l

(34)

while S can be obtained graphically depending on Reynolds number and F. This correlation proposed for saturated nucleate boiling region can also be applied for the subcooled boiling region using F = 1 and the following correlation for S (Butterworth 1979), S¼

1 1 þ 2:53  10


6



Rel F1:25

1:17

(35)

Another group of correlations, called pure convection correlations by Rohsenow et al. (1998), are based on the assumption that the heat transfer coefficient is independent on heat flux. Yet, another form of empirical correlations for convective boiling heat transfer called power type was proposed by Kutateladze (1961) and formalized by Churchill and Churchill (1975) to describe the transition between heat transfer regimes. Steiner and Taborek (1992) generalized the form,  1=n h ¼ hnFC þ hnNB

(36)

where n = 3 for flow boiling. For the annular flow heat transfer, interface evaporation is the dominant pre-CHF heat transfer mode which significantly differs from nucleate boiling (Hewitt 1970). In the interface evaporation regime, the heat is conducted through the liquid film, thus governed by the liquid film thickness (Fig. 5). A simplified case is discussed here for demonstration purpose. Assuming laminar flow in the film, the liquid film flow rate (per unit periphery), Q, can be calculated,

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Fig. 5 Parameters of annular flow in channel

  Gð1
xÞð1
EÞ πD2 =4 GDð1
xÞð1
EÞ ¼ Q¼ πD 4

(37)

where E is the fraction of liquid in the form of droplets. Assuming the liquid film thickness small compared to the tube diameter, the interfacial shear stress τi can be calculated,

D dp ρg þ τi ¼
(38) 4 dz where the pressure gradient, dp/dz, can be estimated from wall friction and ρc is the mixture density of the vapor/liquid core. As a result, the local liquid film velocity can be calculated, ðy τi τi y u¼ dy ¼ (39) μl 0 μl Consequently Q ¼ ρl and

ðδ 0

udy ¼ ρl δ2 τi =2μl



2Qμl 1=2 δ¼ τ i ρl

(40)

(41)

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Flow Boiling in Tubes

1927

The resulting heat transfer coefficient of the laminar liquid film can be determined, h¼

kl ¼ δ

2 1=2 k l τ i ρl 2Qμl

(42)

3.4.2 Post-CHF Heat Transfer For the partial film boiling (transition boiling), because of its unstable nature, the averaged heat transfer coefficient is used in engineering analysis. A widely used correlation for transition boiling heat flux qTB was developed by Tong and Young (1974), "

qTB

#

ΔT sup ð1þ0:0029ΔT sup Þ x2=3 ¼ qNB exp
0:001 ðdx=dzÞ 55:5

(43)

where qNB is the nucleate boiling heat flux taken at CHF. For the stable film boiling, Bromley (1950) proposed the following correlation,

hFB

  2 3 0:172 k3 ρ ρ
ρ h g 1=4 fg g l g g D 4 5 ¼ 0:62 Dμg ΔT sup λ

(44)

where λ is the critical wavelength, which can be calculated, λ ¼ 2π

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi σ   g ρl
ρg

(45)

For the liquid-deficient region, the following heat transfer correlation was proposed by Groeneveld (1973) for flow boiling in both tubes and annuli, 
 b ρg Prcg, w Y d Nug ¼ a Reg x þ ð1
xÞ ρl

(46)

where Prg, w represents the vapor Prandtl number at wall temperature and Y can be expressed, !0:4 ρl Y ¼ 1
0:1
1 ð1
xÞ0:4 (47) ρg For tubes, the coefficients are a = 1.09x10
3, b = 0.989, c = 1.41, and d =
1.15.

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3.5

Y. Liu and N. Dinh

Volume Fraction and Relative Velocity Closures

As discussed in Sect. 2.5, the mixture model can be further divided based on the treatment of relative motion between phases. The slip factor model uses empirical correlations for the slip ratio (which is defined by the ratio of vapor velocity to liquid velocity). The drift flux model uses kinematic constitutive equations to describe the relative flow. With the closures for relative motion and the thermodynamic quality (x), the volume fractions can be obtained. The cross-sectional volume fractions used in one-dimensional two-phase model are related to the mean liquid and gas velocities, vl ¼

ð1
xÞG αl ρl

(48)

xG α g ρg

(49)

vg ¼

where G is the mass flux, ρ is the density, and x is the thermodynamic quality. For slip ratio model, by dividing the two equations and defining the slip factor (S) as the ratio of the vapor and liquid velocity, the gas volume fraction expression can be obtained,

1
x ρg
1 α¼ 1þ S x ρl

(50)

Table 4 provides representative expressions for volume fractions based on different relations for slip ratio. For drift flux model proposed by Zuber and Findlay (1965), the void fraction depends on the distribution parameter C0 and the drift velocity vgj,

 ρg ugj
1 1
x ρg α ¼ C0 1 þ þ x ρl xG


(51)

Table 4 Selected models for void fraction in slip factor model Model Momentum flux

Correlation
 1=2 
1 ρg α ¼ 1 þ 1
x ρ x

Applicable condition Separate flow

l

Smith (1969)




 0:78 ρg 0:58 α ¼ 1 þ 0:79 1
x x ρ


1

Separate flow

l

Zivi (1964) Chisholm (1973)

α¼

 1 þ 1
x x

h

ρg ρl

 i1=2
1 1
x 1
ρρl

Annular flow

g


 2=3 
1 ρg α ¼ 1 þ 1
x ρ x l

Annular flow

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Flow Boiling in Tubes

1929

Table 5 Selected correlations of drift flux model for relative motion closures Model Zuber and Findlay (1965)

Correlation C0 ¼ 1:13
0:25 σgðρl
ρg Þ ugj ¼ 1:41 ρ2

Applicable condition High-pressure steam

Wallis (1969)

C0 ¼ 1:0
0:25 σgðρl
ρg Þ ugj ¼ 1:53 2 ρ

Vertical upward bubbly flow

C0 ¼ 1:0 " # ρ l
ρ g μ v ð 1
αÞ l l ugj ¼ 23 ρg D ρl

Annular flow

l

l

Ishii et al. (1976)

Ishii and Hibiki (2010) showed that the drift flux model could be used without referencing to the specific flow regime. However, the drift flux model should be only applied when the drift velocity is significantly larger than the sum of phasic superficial velocities. Table 5 shows a selected set of drift flux models and their distribution parameters and drift velocities.

4

Multidimensional Modeling of Flow Boiling

4.1

Conservation Equations

The Eulerian-Eulerian two-fluid model framework is the state-of-the-art method to model the two-phase flow with boiling in complex industrial applications. It builds upon the increasing affordability of computing power, the success of computational fluid dynamics (CFD), and the emerging development of computational multiphase fluid dynamics (CMFD) method. The basis of this method is to average the local instantaneous conservation equations, thus eliminating the need for tracking interfaces to achieve computational efficiency. The system of averaged conservation equations needs to be solved numerically, commonly using a finite volume or finite element method. The convergence and accuracy of the solution depend on numerical techniques and temporal and spatial resolutions needed to capture the dynamics and scales of governing physical processes. In multidimensional two-fluid model, the k-phasic mass conservation equation can be written, @ ðαk ρk Þ þ ∇  ðαk ρk Uk Þ ¼ Γki
Γik @t

(52)

where the two terms on the left-hand side represent the rate of change and convection and the two terms on the right-hand side represent the rate of mass exchanges between phases due to condensation and evaporation.

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The k-phasic momentum equation is,

  @ ðαk ρk Uk Þ þ ∇  ðαk ρk Uk Uk Þ ¼
αk ∇pk þ ∇  αk τk þ τtk þ αk ρk g @t þ Γki Ui
Γki Uk þ Mki

(53)

where i represents the interphase between two phases and Mki represents the term of averaged interfacial momentum exchange, which can be modeled by a set of interfacial force correlations. The k-phasic energy conservation equation in terms of specific enthalpy can be given,


 @ ðαk ρk hk Þ μ Dp þ ∇  ðαk ρk hk Uk Þ ¼ ∇  αk λk ∇T k
kt ∇hk þ Γki hi þ αk @t Dt Prk
Γik hk þ Qwall, k

(54)

where the terms on the right-hand side represent heat transfer in phase k, work done by pressure, enthalpy change due to evaporation and condensation, and heat flux from the wall. The wall boiling heat transfer is modeled by a set of closure relations. The readers are referred to Ishii and Hibiki (2010) for detailed derivation of the two-fluid model’s conservation equations. It is instructive to note that, while the above described formulation is general for two-phase flow, the CMFD has been mostly applied to dispersed flow regime (e.g., bubbly flow, droplet flow), for which the physics of phasic interactions is better understood and can be calculated by semi-mechanistic models. The latter are possible because of simplifying assumptions on phase topology and interfacial exchanges. In current practices, most closure correlations were developed for dispersed flow regime. Figure 6 depicts the closures and their connections in a typical CMFD solver. Those closures represent a range of phenomena at the meso- and microscales, including (i) momentum and heat-mass exchanges between phases (at the phase interfaces), e.g., drag force; lift force; virtual mass force; bubble deformation, coalescence, and breakup; and bubble condensation, and (ii) mechanical and thermal interactions on the heating wall, e.g., nucleation, bubble growth and departure, evaporation, and wall heat transfer affected by it. Due to the difference in time and length scales between these physics and averaged dynamics of flow, the effects of small-scale physics on the average flow are normally approximated using so-called closure models. Knowledge of the closure models is derived from theoretical constructions supported by observations and data from separate-effect tests (SET). The development and application of empirical/semiempirical closure models and correlations can be seen as a data assimilation/integration process. Uncertainty in the modeling of flow boiling mostly arises from the necessary use of these data-dependent closure models, which are normally the target of the whole model calibration/validation exercise. A critical issue in this approach is that some important closure models are dependent on “unobservable responses” and deficient

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U

Near wall heat transfer and evaporation

Dry area fraction

Wall boiling

U

α

α

U h Turbulent viscosity Turbulent heat flux

Condensation

Bubble induced turbulence

Turbulence

Interfacial condensation

Momentum exchange

Interfacial forces

Turbulent viscosity Vapor heat transfer

Bubble departure frequency

Evaporation heat transfer Singlephase convective heat transfer Quenching heat transfer

Drag force

Wall function

Nucleation

Bubble departure diameter Active nucleation site density

Turbulence viscosity force Wall lubrication force

Bubble size

Bubble breakup

Bubble coalescence

Lift force

Fig. 6 Closure models in flow boiling and their relationship to the conservation equations

data support to characterize relationships between the closure models. This leads to large model form uncertainties and biases, making the models difficult to validate. It should be noted that the closure models for CMFD solver is a hot research topic, new closures could be proposed in near future that have better predictive capability than current closures discussed in this chapter.

4.2

Interfacial Dynamics Closures

As noted above, the closure relations are most developed for dispersed flow regimes. The closure models can be classified into the following groups: turbulence model, bubble/droplet dynamic model, interfacial mass/momentum/heat transfer model, and wall boiling model. While these models are treated as independent entities and developed on the base of separate-effect tests and studies, these groups are tightly coupled in flow boiling. It is noted that the physics decomposition much needed for a mechanistic treatment of complex phenomena also introduces significant biases and

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sources of model form uncertainty. In the remaining of this section, representative models for closure relations and calculation procedure are discussed.

4.2.1 Turbulence Model In two-fluid modeling, a turbulence model is needed to calculate the turbulent kinematic viscosity of the continuous phase. Technically speaking, all turbulence models developed for single-phase flow can be used for two-phase flow simulation, while in current practices, often the k-e model is used. The k-e model solves two differential transport equations in order to determine the turbulent kinetic energy kb and the turbulent dissipation eb for the continuum phase. The standard k-e model is originally developed by Launder and Spalding (1974) for single-phase flow. In two-phase flow, this model is applied for the continuous phase, with additional terms Se and Sk taking into account the effect of the dispersed phase on the turbulence, " # υeff @ ðk b Þ b þ ðUb  ∇Þkb ¼ ∇  ∇kb þ G
eb þ Sk @t σk

(55)

" # υeff @ ðeb Þ eb b þ ðUb  ∇Þeb ¼ ∇  ∇eb þ ðCe1 G
Ce2 eb Þ þ Se @t σk kb

(56)

where G stands for the production of turbulent kinetic energy. Then, the turbulent kinematic viscosity is, υtb ¼ Cμ

k2b eb

(57)

When taking into account the bubble-induced turbulence, the turbulent kinematic viscosity can be calculated as (Sato and Sekoguchi 1975), υtb ¼ Cμ

k2b 1 þ Cμb Ds αjUa
Ub j eb 2

(58)

Then, the effective liquid kinematic viscosity is defined, t υeff b ¼ υb þ υb

(59)

and the thermal diffusivity is defined, κeff b ¼

λb υt þ bt ρb cpb Pr b

(60)

The turbulence of vapor phase is assumed to be dependent on that of the liquid phase through a turbulence response coefficient Ct, defined as the ratio of the root

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mean square velocity fluctuations of the dispersed phase velocity Ua and of the 0 continuous phase velocity Ub (Rusche 2003). This value is suggested to be unity, 0

Ct ¼

Ua 0 Ub

(61)

Thus, the effective viscosity of the dispersed phase can be calculated as, 2 t υeff a ¼ υa þ Ct υb

(62)

ka ¼ C2t kb

(63)

and the thermal diffusivity can be calculated in the same way, K eff a ¼

λa υt þ at ρa cpa Pr a

(64)

Wall functions for turbulence modeling. In order to predict the velocity profile of flow in the near-wall region where a viscous sub-layer and log-layer profile can be observed, it appears natural to adapt the wall functions developed for k-e model in single-phase flow to the two-fluid model. However, it is noted that such an adaptation involves potential errors, whose magnitude has yet to be evaluated. The uncertainty is largely due to agitation dynamics of interfaces (e.g., nucleation, growth, and departure of bubbles) in the boundary layer region and the slip boundary effect at the bubble dome-liquid interface.

4.2.2 Interfacial Force Model Table 6 presents a sample of equations for calculating interfacial momentum exchange due to different forces in a typical bubbly flow. Table 6 Interfacial momentum exchange due to different forces

Force type Drag force

Expression

Lift force

MLa ¼ CL ρb αðUa
Ub Þ  ðr  Ua Þ   2 1 1 1 ðUa
Ub Þ ! x  nr MWL a ¼ 2 CWL ρb αDS y2
ðD
yÞ2

Wall lubrication force Turbulent dispersion force Virtual mass force

3 MD a ¼
4

CD Ds

3 MTD a ¼
4

ρb αkUa
Ub kðUa
Ub Þ

t C D υb Ds Pr tb

ρb kUa
Ub krα

1 1þ2α MVM a ¼
2 ρb 1
α α

DU

a

Dt

b
DU Dt



Representative model for coefficient Schiller and Naumann (1935) Tomiyama (1998) Antal et al. (1991) Gosman et al. (1992) Rusche (2003)

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4.2.3 Bubble Size Model In bubbly flow modeling, different approximations were proposed to determine the local mean bubble diameter varying from a fixed value to the use of more complex treatment of the interfacial area. For flow boiling in a channel, Anglart et al. (1997) proposed the thermal model,     dref , 1 Tsub
T sub, 2 þ d ref , 2 Tsub, 1
T sub d¼ T sub, 1
T sub, 2

(65)

where Tsub, 1 = 13.5K,Tsub, 2 = 5K, dref, 1 = 0.1 mm, and dref, 2 = 2 mm are empirical constants. Hibiki and Ishii (2002) developed the interfacial area concentration transport equation for calculating the interfacial area concentration corresponding to the area of the dispersed phase per unit volume,

@ ð Aa Þ 2 Aa @α þ ∇  ð Aa U a Þ ¼ þ ∇  ðαUa Þ þ ΦBB þ ΦBC þ ΦNUC @t 3 α @t

(66)

The first term on the right-hand side refers to the contribution of phase change and expansion due to pressure-density change. ΦBB, ΦBC, and ΦNUC represent the source and sink term induced by breakup, coalescence, and nucleation, respectively. Several representative experimental correlations for breakup and coalescence are presented in Table 7. For monodispersed bubbly flow, the spherical bubble diameter can be calculated, Aa ¼

6α d

(67)

4.2.4 Interfacial Mass and Heat Transfer In a subcooled flow boiling, once bubbles are nucleated and moved toward the bulk flow, the bubbles become surrounded by the subcooled liquid causing vapor condensation. The interfacial mass transfer related to condensation of vapor bubbles in the bulk coolant can be described by, Γlg ¼

hlg ðTsat
Tl ÞAs hfg

(68)

where hlg can be calculated using relevant empirical correlations: Wolfert et al. (1978) sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi   1 Ug
Ul  kl 1 hlg ¼ ρl cpl 4 d ρl cpl 1 þ ktl =kl

(69)

Yao and Morel (2004)

Hibiki and Ishii (2002)

Model Wu et al. (1998)

1=3




1=2





6αe ΓTI Wecr Wecr   1
exp
ΦBB ¼ 1=3 We We πd11=3 αmax
α1=3 Wecr ¼ 2:0, αmax ¼ 0:8, ΓTI ¼ 0:18

6ΓTI αð1
αÞe1=3 2 ΦBB ¼ 11=3   exp
K TI We πd αTI, max
α K TI ¼ 1:59, αTI, max ¼ 0:741, ΓTI ¼ 0:021 " #

αe1=3 ΓTI ð 1
αÞ Wecr pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi exp
ΦBB ¼ 11=3 We d 1 þ K TI ð1
αÞ We=Wecr Wecr ¼ 1:24, KTI ¼ 0:24, ΓTI ¼ 1:6

Breakup

Table 7 Correlations for breakup and coalescence source terms " !# 1=3 36ΓBC αe1=3 αmax α1=3   1
exp
K BC 1=3 ΦBC ¼ 1=3 αmax
α1=3 π 2 d11=3 αmax
α1=3 ΓBC ¼ 0:056, K BC ¼ 3:0, αmax ¼ 0:8 rffiffiffiffiffiffiffi! 3ΓBC α2 e1=3 We ΦBC ¼ 11=3   exp
K RC 2 πd αTI, max
α K BC ¼ 1:29, αTI, max ¼ 0:741, ΓBC ¼ 0:0314 2 3 1=3 ΓBC e1=3 4 α2 αmax 5  ΦBC ¼ 11=3  1=3 1=3 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d αmax
α1=3 þ K BC, 1 αmax α We=Wecr rffiffiffiffiffiffiffiffiffiffi

We exp
K BC, 2 Wecr ΓBC ¼ 2:86, K BC, 1 ¼ 1:922, K BC, 2 ¼ 1:017, Wecr ¼ 1:24, αmax ¼ 0:52

Coalescence

47 Flow Boiling in Tubes 1935

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Ranz and Marshall (1952) hlg ¼

4.3

 kl  1=2 2 þ 0:6Reb Pr 1=3 d

(70)

Wall Boiling Closures (Multidimensional)

The wall boiling model is the construct developed to provide a consistent treatment of phenomena that govern heat transfer in boiling, namely, heterogeneous nucleation of vapor bubbles, vapor generation source term, and wall superheat. The general approach is term “heat partitioning,” which decomposes the wall heat flux into several components representing corresponding heat transfer mechanisms. A wall boiling model was first introduced by Kurul and Podowski (1991), including three components: single-phase convective heat transfer, quenching heat transfer, and evaporation heat transfer. This model is now called “Generation-I” model as many new models are proposed based on it. It can be expressed, qw ¼ qcw þ qew þ qqw

(71)

Each component is described by a corresponding set of sub-closure models. Evaporation heat transfer, for example, depends on the nucleation site density (Table 8), bubble departure diameter (Table 9), and bubble departure frequency, π qew ¼ d3 ρv f d N a hfg 6

(72)

It is noted that some of the important phenomena in flow boiling were not adequately accounted for in the Generation-I boiling model such as the bubble sliding effect and the nucleation site interaction under high heat flux boiling. In order to improve the model performance, modifications were made including bubble influence area limitation (Krepper et al. 2007). Moreover, several sub-models discussed above are based on empirical correlations derived from pool boiling experiments, which could result in a large model for uncertainty (Tables 8, 9, 10).

4.4

Phenomenological Model for DNB Prediction

Prediction of boiling crisis in general, and DNB in particular, presents a formidable challenge. The modern approach to boiling crisis prediction is centered on a premise of scale separation that burnout is governed by thermo-fluids in the vicinity of the heater wall. This notion does not contradict the observation of the effect of system parameters such as system pressure, mass flow rate, and bulk flow subcooling on CHF. Instead, the effect of system parameters is effectively represented by parameters of two-phase flow in the near-wall region. Notably, high subcooling and high

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Table 8 Selected models for nucleation site density Model Lemmert and Chawla (1977) Wang and Dhir (1993) Yang and Kim (1988) Hibiki and Ishii (2003)

Correlation for Na, m
2 Na = (aΔTsup)b, a = 210, b = 1.805

Condition Pool boiling

N a ¼ 5  10
31 ð1
cos θÞR
6:0 / ΔT 6:0 c w N a ¼ N a ϕðβÞexpð
CRc Þ

Pool boiling, p = 1 bar Pool boiling

! #

 " 0 θ2 λ gðρþ Þ N a ¼ N a 1
exp
2 exp
1 Rc 8θ

Pool and flow boiling, p ~ [1–198] bar




0

ρþ ¼ logðρ Þ, θ ¼ 0:722 rad, λ ¼ 2:5  10
6 m

Table 9 Selected models for bubble departure diameter Model Cole and Rohsenow (1969) Tolubinsky and Konstanchuk (1972) Kocamustafaogullari (1983) Zeng et al. (1993a, b)

Correlation for d, m qffiffiffiffiffiffiρ c T 5=4 σ l pl sat d ¼ 1:5  10
4 gΔρ ρ hfg g

d = min[0.0006 exp(
ΔTsub/45), 0.0014] d ¼ 1:27  10
3



 ρl
ρg 0:9 dref ρg

Mechanistic model bubble departure/lift off based on force balance analysis

Condition Pool nucleate boiling Subcooled flow boiling Pool and flow boiling, p1–142 bar Pool and flow boiling

Table 10 Selected models for bubble departure frequency Model Cole (1967) Kocamustafaogullari and Ishii (1995) Podowski et al. (1997)

Correlation for f, s
1 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4gðρl
ρg Þ f ¼ 3dρ
f ¼ 1:18 d

l

σgðρl
ρg Þ ρ2l

0:25

Mechanistic models account for waiting time and bubble growth time

Condition Pool nucleate boiling near CHF Subcooled _flow boiling Subcooled flow boiling

mass flow rate tend to reduce accumulation of vapor in the near-wall region through condensation and liquid supply, thus increasing the system’s resistance to burnout. Within the multidimensional two-fluid model framework, DNB prediction model is based on the concept of critical value of the near-wall void fraction (αdry), first proposed by Weisman and Pei (1983), who suggested the value of αdry to be 0.82, while other researchers have used different values ranging from 0.8 to 0.95 for different conditions and mesh sizes. In this subcooled boiling model, initially the wall is fully wetted. However, as the vapor volume fraction in the near-wall region increases, access of liquid to the wall and its heat removal become restricted. This overcritical voiding can trigger an irreversible process of burnout and heater’s overheating.

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Denoting Kdry as the dry patch factor, the wall heat flux combines heat removal to the liquid (wetted) and vapor (dry) areas,    qw ¼ qcw þ qew þ qqw 1
K dry þ K dry qv

(73)

For example, the boiling model in STAR-CCM+ code uses the expression of Kdry as a function of near-wall void fraction (αδ),  K dry ¼

0 αδ αdry f ðβÞ αδ > αdry

f ðβÞ ¼ β2 ð3
2βÞ β¼

αδ
αdry 1
αdry

(74) (75) (76)

Other forms of this function were used in other codes such as in NEPTUNE (Mimouni et al. 2016). It is noted that care must be taken in the implementation of this approach in the multidimensional two-fluid model. The exceedance of local void fraction over the critical value may occur unintentionally during numerical iterations toward a converged solution. This numerical error might trigger the irreversible escalation leading to a premature burnout.

4.5

Phenomenological Model for Dryout Prediction

The basic phenomenological model of dryout in a heated channel is based on one-dimensional treatment of the liquid film flow that solves mass, momentum, and energy conservation equations and takes into account droplet deposition, entrainment, and film evaporation to determine the location of dryout. In such an approach, the dryout is assumed to occur when the liquid film flow rate or corresponding film thickness diminishes to zero or below a threshold value. Pioneered in the 1960s by Hewitt and his co-workers (1965), the approach has been further developed by many research groups since then. Similar to boiling crisis in DNB regime, phenomena that influence the occurrence of dryout are active at several different scales (Bestion 2010). The overall flow is governed by channel-wide parameters, such as system pressure, mass flow rate, disturbance waves, overall flow quality, cross-flows, and flow obstacles. At the mesoscales are droplet and liquid film dynamics, as characterized by droplet size distribution, droplet concentration distribution, and rate of droplet entrainment and deposition. Finally, at the microscopic scale, 10 μm or less, are phenomena that govern liquid film breakup into rivulets, creation and stability of dry patches, dynamic contact line motion, evaporation on film surface, and nucleation and growth of vapor bubbles in the film. Figure 7 shows a phenomenological decomposition of annular flow boiling, separating liquid film and gas core flow. Theoretically, each component can be

U h

h

Source terms

Momentum

δ Energy

δ

Momentum exchange

Drag force

Droplet size

Lift force

Interfacial forces

U α U h Energy

α

Turbulent heat flux

h

Wall function

Turbulence

Turbulent Viscosity

Momentum

Fig. 7 Closure models in annular flow boiling and their relationships to the conservation equations.

Droplet deposition

Droplet entrainment

Interfacial evaporation

Critical liquid film thickness

Dryout identification

U

Mass

Mass Mass momentum and energy exchange

Averaged Conservative Equations for gas core flow

Averaged Conservative Equations for liquid film

Wall droplet heat transfer

Wall gas heat transfer

Source term for post dryout

47 Flow Boiling in Tubes 1939

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treated in one-dimensional and multidimensional descriptions. The averaged conservation equation for liquid film is one-dimensional, while the gas core dispersed (droplet) flow can be treated in CMFD framework as discussed earlier in this section. The method for dryout prediction discussed in this section can be applied in different models. In system thermal hydraulics codes and sub-channel analysis codes, which are based on the segregated multi-field two-phase flow model, the one-dimensional phenomenological models are used for dryout prediction. The liquid film on the wall and liquid droplets in the core flow are treated as separate fields. Readers are referred to for more details of models of this type (Hewitt 1970). Whalley et al. (1973) developed the basic equation of liquid film flux for one-dimensional segregated flow. At a steady state, the film dryout is assumed to occur when the flow rate diminishes, dGlf 4 _e
m _ ev Þ _d
m ¼ ðm D dz

(77)

where subscripts lf, d, e, and ev represent liquid film deposition, entrainment, and evaporation, respectively. In the multidimensional two-fluid model solver, the liquid film thickness δ is usually described by a two-dimensional field which only has value on the near-wall cell; thus, the liquid film thickness can be regarded as a local variable. A set of mass, momentum, and energy conservation equations for liquid film is introduced in addition to the bulk flow. Such treatment is incorporated in the CMFD framework and takes into account droplet deposition, entrainment, and film evaporation to determine the location of dryout. Taking the mass conservation equation as an example, @ ðρδÞ _d
m _e
m _ ev þ ∇  ðρδuÞ ¼ m @t

(78)

The film thickness field is coupled with the bulk two-phase flow equations in the solver; this means the closure terms are correlated to the bulk flow variable such as vapor velocity, pressure, etc. Readers are referred to Li and Anglart (2016) for more details. This treatment makes the calculation of liquid film thickness compatible with transient problems. The dryout location prediction method can be extended to complicated geometries such as rod bundle with spacers or mixing vanes. Both one-dimensional and multidimensional methods require closure relations for liquid film deposition, entrainment, and evaporation. The evaporation term equals, m_ ev ¼ qw =hfg

(79)

while for the droplet deposition and entrainment rates, the closures were developed based on hydrodynamic analysis and experiment data. Readers are referred to Hewitt (1970) for more details. One of the widely used correlations for entrainment rate of upward annular flow is proposed by Paleev and Filippovich (1966),

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Table 11 Reference closure models for liquid film dynamics Models Deposition rate Entrainment rate Critical liquid film thickness Wall gas heat transfer Wall droplet heat transfer

References Hewitt and Govan (1990) de Bertodano et al. (1998) Chun et al. (2003) Jayatilleke (1966) Guo and Mishima (2002)

_ e ¼ 0:41 m

k s τ i μ l vg σ2

(80)

where ks is the roughness factor which is dependent on the film thickness. The interfacial shear (τi) is related to the interfacial friction factor for the liquid film. τi ¼ f fric ρg v2g

(81)

One of the widely used correlations for deposition rate for upward annular flow was proposed by Cousins et al. (1965), m_ d ¼ k0 ΔC

(82)

where k0 is the mass transfer coefficient whose value is suggested by Whalley (1977) as a function of surface tension and ΔC is the concentration gradient that is dependent on the droplet density and vapor/droplet fraction, α e ρl ΔC ¼ (83) αe þ αg A comprehensive discussion on phenomenological modeling of disperse-annular flow and dryout can be found in a recent study (Li and Anglart 2016); the models used in their work are summarized in Table 11.

5

Open Issues and Perspective

5.1

Open Issues for Current Modeling Approaches

The capability of current models of flow boiling to describe and predict quantities of interest in a range of conditions of importance for a number of engineering applications, including in design and analysis of thermal and nuclear power plants, has been demonstrated in practice. It is noted that closure models used in analysis have been extensively calibrated and validated against a body of measured data from a set of appropriately selected separate-effect tests and integral-effect tests. Calibration notwithstanding, there remains uncertainty that hampers the predictive capability of this class of models. The sources, magnitude, and impact of the uncertainty vary in different applications.

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The first group of uncertainty stems from basic assumptions that underlie the multiphase modeling approach. This includes: (i) The averaging procedure to obtain conservation equations, particularly the use of cross-sectional averaged variables in one-dimensional description of flow boiling. For different flow regimes, phase distributions across the channel can vary greatly and not well characterized for flow boiling in different transient scenarios. (ii) Another assumption is scale separation that decouples physics between “global” scale of fluid dynamics (given by the field equations’ advection and diffusion terms) and “sub-grid-scale” local interactions (given by source terms). The source terms typically contain neither time nor spatial functional dependence. The scale separation allows source terms (e.g., interfacial exchanges, wall heat transfer, friction) in conservation equations to be determined via local conditions, which – in one-dimensional model of flow boiling – are axially local and cross-sectional averaged. (iii) Interfacial exchange terms and wall terms are typically decomposed into components, which are considered independent, and whose effect on “global” fluid dynamics is additive. For example, the effect of wall heat transfer is averaged over the cross-sectional flow. Note that (ii) and (iii) reflect “divide-to-conquer” strategy popular in mechanistic modeling of various engineered systems. Their applicability (of scale separation and physics decomposition) is limited by the complexity (nonlinearly coupled/multiscale nature) of two-phase flow, in general, and flow boiling in particular. The second group of uncertainty stems from closure models, which are largely empirical correlations, relying on historical data and published experiments; both the correlations and the data were generated decades ago, long before rigorous procedures for uncertainty analysis became required. Thus, the uncertainty comes from following sources: (iv) Common for legacy experiments is the lack of detail and accurate description of facility, measurement techniques, flow geometry, heater surface characteristics, and inlet and boundary conditions. Consequently, it is not possible to evaluate the experimental data uncertainty, which is instrumental for determining uncertainty of models and model parameters. For safety analysis, conservative assumptions were usually invoked in selecting parameters of empirical correlations. (v) Calibration of models involves a body of data, although not all data are born equal in their relevance to the conditions of interest. Value of information of data varies between experiments, tests (in an experiment), and types (even locations) of measurements (in a test). (vi) A major contribution to data uncertainty is the use of models outside the experimental domain, both interpolation and extrapolation. Even though various scaling techniques were devised and applied, complexity of two-phase

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flow and boiling heat transfer makes it difficult, if not impossible, to satisfy different scaling requirements. (vii) As a special case of scaling distortions, note that empirical correlations were mostly designed for steady- or quasi-steady state and fully developed flow situations. However, there are many problems in two-phase and boiling flow that are intrinsically unsteady or transient. Furthermore, developing flow is dominant in industrial systems. The effect is particularly notable for flow regime criteria, as discussed in the previous section. The third group of uncertainty stems from numerical solution of the model’s partial differential (conservation) equations. A major, and hard-to-evaluate, source of uncertainty results from the combined effect of discretization errors and errors due to approximation (integration) of closure relations over the numerical solution’s finite difference or control volume. The errors increase with variability of different flow characteristics over the control volume.

5.2

Advanced Modeling Approach: Direct Numerical Simulation

Unlike the methods discussed in the previous section which is based on averaged conservative variables, the method of direct numerical simulation (DNS) of two-phase flow is based on one-fluid formulation, aiming to provide a “first-principle,” time- and space-resolved description of two-phase flow system. Following the development and practice of CFD methods for single-phase flow, the DNS-type method is designed to deliver the “numerically exact” solution of the Navier-Stokes equations. For two-phase flow with phase changes, the DNS is ever more challenging, attempting to capture flow turbulence, heat transfer, dynamics of interface, and phase change (evaporation and condensation) by solving the Navier-Stokes and energy equations on both liquid and gas phases. In such an approach, the evolution of phase interface is described using a method for interface capturing or interface tracking (ITM), such as level-set function (Mulder et al. 1992), volume-of-fluid (Scardovelli and Zaleski 2000) or front-tracking methods (Tryggvason et al. 2001). Notably, the surface tension effect is treated as an effective body force using the so-called the continuum surface tension method of Brackbill et al. (1992). It should be noted that over the past decade, thanking to computing power and advances in solution algorithms, significant progress has been made in the numerical simulation of two-phase bubbly and droplet flow; see, for example, the work of Tryggvason et al. (2001) and Bolotnov et al. (2011). Applied to boiling processes, notable achievements were reported in the pioneering work of Dhir and co-workers (Aktinol and Dhir 2012) and, more recently, by Sato and Niceno (2015), both in simulation of pool boiling. Formidable challenges remain in applying methods of DNS to the investigation of pool and flow boiling processes, due to tightly coupled thermo-fluid dynamics at the fluid-solid interface. This multiscale nature of boiling limits the “DNS” to fluid dynamics, while still requiring a detailed modeling of microscale interfacial phenomena, e.g., heterogeneous nucleation, phase changes,

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turbulence-interface interactions, and interactions of fluid interface with solid surface, including treatment of wettability, contact line motion, and dynamics of evaporating thin liquid film. These phenomena are greatly affected by heater surface characteristics, e.g., material chemistry, coolant chemistry, chemical deposition, surface aging, and surface micro-/nano-morphology; readers are referred to references in boiling physics, e.g., Theofanous et al. (2002) and Theofanous and Dinh (2006). With high-performance computing resources becoming increasingly affordable to both research and industry establishments, the DNS methods will become a critical instrument in developing a fundamental understanding of physics of flow boiling. While assumptions are inevitable in the treatment of microscale phenomena, sensitivity analysis can be performed to identify the trend and significance of impact of various surface characteristics on quantities of interest for engineering applications.

6

Conclusion

Flow boiling as a heat transfer regime is present in a broad variety of heat exchanger equipment in industrial systems and power plants. Efficient heat transfer in nucleate boiling regime utilized in such system is limited by the critical heat fluxes upon which the heat transfer is greatly deteriorated causing equipment failure. Thus, the primary quantities of interest in flow boiling include boiling heat transfer coefficient and the critical heat flux. Other quantities of interest are characteristics of two-phase flow in the tube including distribution of phases and their velocities. For example, flow boiling heat transfer governs thermal safety margins in nuclear reactor core in light water reactors. Furthermore, boiling can cause deposition of particulate and soluble substances on the heated surfaces, which in turn have a paramount effect on boiling processes. Traditionally, phenomenology of flow boiling in channels is considered in two areas, namely, wall boiling heat transfer and two-phase flow dynamics. However, it had become recognized that flow boiling is a multiscale, multi-physics phenomenon. The underlying complexity includes a large number of phenomena over a broad range of time scale and length scale. Hierarchically, they include microscopic phenomena at molecular scale (e.g., nucleation, wettability), micro-hydrodynamic scale (e.g., meniscus evaporation, thin film dynamics), continuum mechanics with interface tracking (e.g., bubble dynamics, bubble coalescence, interfacial instability and breakup), flow turbulence (e.g., production, dissipation, bubble-induced turbulence), interphase exchange (evaporation, condensation, drag force, lift force), two-phase flow pattern (e.g., vapor volume fraction distribution), and flow mixing due to gradient across channels (e.g., effect of spacer grids and mixing vane in fuel rod bundles). More importantly, processes at different scales are tightly coupled leading to nonlinear interaction and complexity. In practical situations, chemical and particulate substances present in coolant further complicate boiling phenomena. For example, deposition of chemicals and particulates on heater surfaces affects nucleation and triple contact line motion that govern the onset of and departure from nucleate boiling.

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Over the past century, flow boiling in tubes has been a subject of extensive investigations due to its importance for operation and safety of a wide range of power engineering systems and industrial applications. Typically, flow boiling experiments in tubes employed electrical Ohmic heating to simulate practical situations in which the tube is heated externally by convection or radiation. In such a case, visual observations of flow regimes are inhibited due to opaque tube wall. Also, measurements of fluid temperature, wall temperature, flow velocity, and vapor volume fractions were limited to few locations. Application of flow regime maps obtained for air-water flow has been instrumental for simulation of flow boiling in tubes. Description of flow regimes represents a source of major uncertainty and potential error in analysis of flow boiling in tubes. With the increase of gas flow rate in vertical tubes, the four distinct flow patterns observed are bubbly flow, slug flow, churn flow, and annular flow. In addition, the stratified flow regime is present in horizontal and inclined tubes. Most notably, the well-established flow regime maps represent flow patterns in quasi-steady-state and fully developed flows. The latter is not attainable in flow boiling in tubes for which the flow regime typically changes from singlephase convection to subcooled boiling and to developed nucleate boiling, where vapor production occurs on the heated wall. In nuclear reactor core fuel assembly, two-phase flow regimes are affected by the flow cross-sectional geometry and design of spacer grids that govern fluid mixing including bubble-turbulence interaction in bubbly flow and entrainment and deposition of liquid droplets in disperse-annular flow. Another source of major uncertainty in predicting boiling heat transfer and burnout in flow boiling in tubes is related to the effect of heater surface characteristics. This effect has been widely studied and established for pool boiling under atmospheric condition. It is expected that the effect of surface nano/micromorphology and chemistry on nucleation and wettability phenomena remains applicable under flow boiling conditions. However, it remains uncertain about the ultimate effect of surface characteristics on integral parameters such as boiling heat transfer coefficient and critical heat flux. Analysis of engineering system with flow boiling in channels has been performed primarily with the one-dimensional system-level thermodynamic models. These include different treatments of two-phase flow ranging from homogeneous models to two-fluid models. Increasingly, methods of computational multiphase fluid dynamics (CMFD) are being advanced to capture flow boiling processes in higher level of details. These CMFD methods provide a potentially higher predictive capability of flow boiling by using a three-dimensional two-fluid treatment. In addition, methods of direct numerical simulation with interface tracking have become a research tool to investigate time- and space-resolved multiphase fluid dynamics. The state of the art and the practice in this area have been shaped by several factors: • First, and historically, past studies of flow boiling in tubes were largely empirical, with integral measurements especially for flow boiling under prototypic equipment operating conditions, characteristically high pressure and high heat fluxes. Furthermore, understanding of two-phase flow in flow boiling is based on

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•

•

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knowledge derived from studies of two-phase flow in adiabatic isothermal conditions, often underestimating the effect due to vapor generation in the near-wall region on the flow pattern of flow boiling. Second, understanding and modeling of physical mechanisms in flow boiling have relied heavily on observations, data, and insights obtained in visualization amenable experiments in pool boiling, atmospheric pressure, and low heat fluxes. In particular, flow pattern in high heat flux boiling might not be adequately captured by the bubble centric modeling approach. Third, the recent decades witness an extraordinary progress in the experimentation and computation of two-phase flow and boiling heat transfer, thanking to advances in data-intensive methods in flow and thermal diagnostics and by highresolution numerical simulation with interface tracking methods. Notably, the new data indicate complex stochastic nature of boiling processes over previous time and space averaged measurements and characterizations. Fourth is the increasing capability of computational multiphase fluid dynamics (CMFD) as a potential method tool for engineering analysis and prediction of two-phase flow and boiling heat transfer in the tubes, including critical heat fluxes and post-dryout processes. The CMFD models required treatment of microphysical processes, for which applicable measurements are not available for separate effect testing and model calibration. Fifth and not least are the emerging capabilities provided by advances in data analytics, data assimilation, and data collaboration that help bring together the above factors to advance understanding and prediction of flow boiling in tubes.

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Steiner D, Taborek J (1992) Flow boiling heat transfer in vertical tubes correlated by an asymptotic model. Heat Transf Eng 13(2):43–69 Taitel Y, Dukler AE (1976) A model for predicting flow regime transitions in horizontal and near horizontal gas-liquid flow. AICHE J 22(1):47–55 Taitel Y, Bornea D, Dukler AE (1980) Modelling flow pattern transitions for steady upward gas-liquid flow in vertical tubes. AICHE J 26(3):345–354 Theofanous TG, Dinh T (2006) High heat flux boiling and burnout as microphysical phenomena: mounting evidence and opportunities. Multiph Sci Technol 18(3):251–276 Theofanous TG, Tu JP, Dinh AT, Dinh T (2002) The boiling crisis phenomenon: part I: nucleation and nucleate boiling heat transfer. Exp Thermal Fluid Sci 26(6):775–792 Thom J, Walker WM, Fallon TA, Reising G (1965) Boiling in subcooled water during flow up heated tubes or annuli. In: Symposium on boiling heat transfer in steam generating units and heat exchangers, Manchester Tolubinsky VI, Konstanchuk DM (1972) The rate of vapour-bubble growth in boiling of subcooled water. Heat Transf-Sov Res 4(6):7–12 Tomiyama A (1998) Struggle with computational bubble dynamics. Multiph Sci Technol 10(4):369–405 Tong LS, Tang YS (1997) Boiling heat transfer and two-phase flow, 2nd edn. Taylor & Francis, London Tong LS, Young JD (1974). Phenomenological transition and film boiling heat transfer correlation, Tokyo Tryggvason G et al (2001) A front-tracking method for the computations of multiphase flow. J Comput Phys 169(2):708–759 Wallis GB (1969) One-dimensional two-phase flow. McGraw-Hill, New York Wang CH, Dhir VK (1993) Effect of surface wettability on active nucleation site density during pool boiling of water on a vertical surface. J Heat Transf 115(3):659–669 Weisman J, Pei BS (1983) Prediction of critical heat flux in flow boiling at low qualities. Int J Heat Mass Transf 26(10):1463–1477 Whalley PB (1977) The calculation of dryout in a rod bundle. Int J Multiphase Flow 3(6):501–515 Whalley PB, Hewitt GF, Hutchinson P (1973) Experimental wave and entrainment measurements in vertical annular two-phase flow. Technical Report Wolfert K, Burwell MJ, Enix D (1978) Non-equilibrium mass transfer between liquid and vapor phases during depressurization processes in transient two-phase flow. In: Proceedings of 2nd CSNI specialists meeting, Paris Wu Q, Kim S, Ishii M, Beus SG (1998) One-group interfacial area transport in vertical bubbly flow. Int J Heat Mass Transf 41(8):1103–1112 Yang SR, Kim RH (1988) A mathematical model of the pool boiling nucleation site density in terms of the surface characteristics. Int J Heat Mass Transf 31(6):1127–1135 Yao W, Morel C (2004) Volumetric interfacial area prediction in upward bubbly two-phase flow. Int J Heat Mass Transf 47(2):307–328 Zeng LZ, Klausner JF, Bernhard DM, Mei R (1993a) A unified model for the prediction of bubble detachment diameters in boiling systems—II. Flow boiling. Int J Heat Mass Transf 36(9):2271–2279 Zeng LZ, Klausner JF, Mei R (1993b) A unified model for the prediction of bubble detachment diameters in boiling systems—I. Pool boiling. Int J Heat Mass Transf 36(9):2261–2270 Zhao L, Rezkallah KS (1993) Gas-liquid flow patterns at microgravity conditions. Int J Multiphase Flow 19(5):751–763 Zivi SM (1964) Estimation of steady-state steam void-fraction by means of the principle of minimum entropy production. J Heat Transf 86(2):247–251 Zuber N (1958) On the stability of boiling heat transfer. Trans Am Soc Mech Engrs 80:711–720 Zuber N, Findlay J (1965) Average volumetric concentration in two-phase flow systems. J Heat Transf 87(4):453–468

Boiling and Two-Phase Flow in Narrow Channels

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Contents 1 2 3 4 5 6 7

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Microchannel Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Progression of Flow Boiling Research in Minichannels and Microchannels . . . . . . . . . . . . Onset of Nucleate Boiling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Explosive Boiling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Flow Boiling Instability in Microchannels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pressure Drop During Flow Boiling in Microchannels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Single-Phase Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Two-Phase Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Two-Phase Frictional Pressure Drop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Acceleration Pressure Drop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5 Gravitational Pressure Gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6 Total Pressure Gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Flow Boiling Heat Transfer in Microchannels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Critical Heat Flux in Microchannels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Enhanced Flow Boiling in New Microchannel Configurations . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1 Pin Fins and Nanowires Within Microchannels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Vapor Extraction Through Hollow Open Fins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3 Vapor Removal Through a Hydrophobic Membrane Cover . . . . . . . . . . . . . . . . . . . . . . 10.4 Tapered gap Microchannels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5 Radial Microchannels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Cross-References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1954 1954 1955 1956 1957 1958 1959 1959 1960 1960 1961 1961 1962 1962 1963 1966 1966 1966 1967 1967 1968 1969 1969 1970

S. G. Kandlikar (*) Rochester Institute of Technology, Rochester, NY, USA e-mail: [email protected] # Springer International Publishing AG, part of Springer Nature 2018 F. A. Kulacki (ed.), Handbook of Thermal Science and Engineering, https://doi.org/10.1007/978-3-319-26695-4_48

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Abstract

As the channel dimensions become smaller, the heat transfer surface area and the resulting heat transfer rate per unit flow volume become larger. This is further aided by the higher heat transfer coefficients during single-phase flow in the laminar flow region. Boiling, being an efficient transfer process, could also potentially benefit from this reduction in channel hydraulic diameter. The heat transfer mechanisms and flow instabilities resulting from the boiling process are now generally well understood in minichannels and microchannels. Recent advances in this field indicate that it is possible to achieve excellent heat transfer performance with little pressure drop penalty through appropriate design of the fluid flow passages. Nomenclature

A Ac a, b a1. . .a5 Bo Ca Co D FF1 f G g h hLV K2 m_ Nu P p Po Pr q00 Re rc,min, rc,max T u

Flow area, m2 Cross-sectional area, m2 Sides of a rectangular channel, m Coefficients, dimensionless Boiling number, dimensionless, Bo = q00 /(GhLV) Capillary number, Ca ¼ μρL Gσ L Convection number, dimensionless, Co = [(1 – x)/x]0.9 [ρV/ρL]0.5 Diameter, m Fluid-surface parameter accounting for the nucleation characteristics of different fluid surface combinations, dimensionless single-phase friction factor, dimensionless Mass flux, kg/m2s Acceleration due to gravity, m/s2 Heat transfer coefficient, W/m2-K Latent heat of vaporization, J/kg Ratio of evaporation momentum to surface tension forces at the liquid–vapor Mass flow rate, kg/s Nusselt number, dimensionless, Nu = hDh/k Wetted perimeter, m Pressure, Pa Poiseuille number, Po = f  Re Prandtl number, dimensionless, Pr = μcp/k Heat flux, W/m2 Reynolds number, dimensionless, Re = GD/μ Minimum and maximum cavity radii respectively at ONG, m Temperature, K Velocity, m/s

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Boiling and Two-Phase Flow in Narrow Channels

ν V We x z

1953

Specific volume, ν = 1/ρ, m3/kg Flow volume, m3 Weber number, dimensionless, We = ρVS2D/σ Mass quality, dimensionless length from the channel entrance, m

Greek Symbols

α1; α2; α3 αc Δp ΔTsat ΔTsub δt θ θ μ ρ σ

Coefficients for the pressure distribution along a plane microchannel, dimensionless (Chapter “Microchannel definition”) Channel aspect ratio, dimensionless, αc = a/b (between 0 and 1) Pressure drop, Pa Temperature difference between wall and saturation, K Temperature difference between saturation and subcooled liquid, K Thermal boundary layer thickness, m tube inclination angle to horizontal, degrees Contact angle, degrees Dynamic viscosity, kg/ms Density, kg/m3 Surface tension, N/m

Subscripts

A CBD CHF Ent F G h L LV LO m NBD r SP sat TP t V

Accelerational Convective boiling dominant Critical heat flux Entry Frictional Gravitational Hydraulic Liquid Latent Entire flow as liquid Mean Nucleate boiling dominant Receding Single-phase Saturated condition Two-phase Thermal Vapor

1954

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S. G. Kandlikar

Introduction

Narrow channels, including minichannels and microchannels, are employed in diverse applications such as microfluidics, microreactors, sensors, and high heat flux dissipation systems. In a closed fluidic heat exchange system, heat transfer occurs at the channel surface, while fluid flow occurs through the channel crosssection. Thus, surface area to volume ratio of the channel plays an important role in determining the thermal performance of the system. Smaller hydraulic diameters correspond to higher surface area to volume ratios for the flow channels and will result in higher heat transfer coefficients and more compact heat exchangers. In general, shorter lengths are implemented to reduce the high pressure drop encountered in these systems. Microchannels employing small diameter channels are thus highly suitable for high performance heat exchangers and heat transfer systems. Boiling is an efficient process and combining flow boiling with microchannel flows is seen as an attractive pathway for further improving the system performance. This chapter covers fluid flow and heat transfer performance of microfluidics and thermal systems employing microchannels.

2

Microchannel Definition

Classification of microchannels is primarily intended to serve as a guide to understand the scale of the fluid flow channels being employed in a particular application. Considerations regarding the mechanisms, such as intermolecular distances during gas flow, or flow configurations during two-phase flow and their behavior with channel dimensions are important and may provide specific guidance in a particular field. With this in mind, Kandlikar and Grande (2003) provided the following channel classification scheme that serves as a broad guidance. In two-phase flow, the influence of channel dimension on the flow cannot be easily defined. The nucleating bubbles in flow boiling conditions are influenced mainly by the surface tension and inertia forces in addition to the gravitational force. The role of gravitational force becomes secondary as the channel dimensions become smaller and the surface tension and inertia forces dominate. The demarcation between the gravity-influenced flow and inertia and shear induced flow depends on the fluid properties and flow rate. In addition, forces introduced by evaporation at the liquid-vapor interface also influence the flow morphology especially at higher heat fluxes. In view of the complexity involved, a simpler definition given in Table 1 provides a broad classification while considering single-phase, two-phase, and evaporating/condensing flows. The surface area to volume ratio is an important consideration. For circular channels, the area to volume ratio is given by 4/D. As the channel dimension becomes smaller, this ratio becomes larger. Table 2 gives the channel classification based on this criterion. The channels with minimum channel dimension of 3 mm are considered as conventional channels. For these channels, the conventional large channel flow

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Table 1 Channel classification scheme, Kandlikar and Grande (2003) Channel classification Conventional channels Minichannels Microchannels Transitional microchannels Transitional nanochannels Nanochannels D: Smallest channel dimension

Dimensions > 3 mm 3 mm  D > 200 μm 200 μm  D > 10 μm 10 μm  D > 1 μm 1 μm  D > 0.1 μm 0.1 μm  D

Table 2 Channel classification scheme based on surface area to volume ratio Channel classification Conventional channels Minichannels Microchannels Transitional microchannels Transitional Nanochannels Nanochannels A/V: Surface area to volume ratio for a circular channel

Surface area/volume, m1 A/V < 1.33  103 1.33  103  A/V > 2  104 2  104  A/V > 4  105 4  105  A/V > 4  106 4  105  A/V > 4  107 4  107  A/V

modeling can be applied. The flow is considered continuum in both phases and Navier-Stokes equations are applicable.

3

Progression of Flow Boiling Research in Minichannels and Microchannels

Early work on flow boiling until late 1990s focused on channels of 1 mm and larger hydraulic diameters. Extensive work on flow boiling of refrigerants in 3-mm diameter channels with R-11 was reported by Chawla (1966). The results indicated that the flow boiling in 3-mm diameter tube, which falls on the boundary of the minichannels and conventional channels according to Table 1, is similar to that in conventional channels. Similar observations were made by Lazarek and Black (1982) with flow boiling of R-113 in 3.1-mm diameter tubes and Wambsganss et al. (1993) with R-113 in 2.92-mm diameter tubes. The general flow boiling correlation for conventional tubes by Kandlikar (1990) was able to represent these data quite well. Cornwell and Kew (1992) reported flow patterns with R-113 observed in two sets of parallel minichannels 1.2 mm  0.9 mm and 3.25 mm  1.1 mm in dimension and identified the flow patterns as isolated bubble, confined bubble, and annular-slug regions. In the region of isolated bubbles, the dependence of heat transfer coefficient on heat flux was similar to that seen in nucleate pool boiling. This observation was

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further explored by Kandlikar (2010a) and the striking similarities between flow boiling in microchannels and nucleate pool boiling was noted. Serizawa et al. (2002) studied the two-phase flow in microchannels with air-water in 20-, 25-, and 100-μm diameter and water-steam in 50-μm diameter circular tubes. They observed the flow patterns and compared the pressure drop and flow patterns with larger diameter tubes. They noted that the flow behavior in these tubes did not substantially depart from those seen in conventional diameter tubes. Flow boiling instabilities were identified by Kandlikar et al. (2002a, b) as the basic issue limiting their implementation in practical flow boiling systems. Flow reversal was specifically studied by Steinke and Kandlikar (2004) and Kandlikar et al. (2001) the flow boiling heat transfer mechanisms were identified (Kandlikar 2004). Different strategies were developed by researchers to mitigate the flow boiling instabilities, e.g., by Kandlikar et al. (2005, 2006) and Kosar et al. (2005a, 2006). Subsequently, it was noted by researchers that the heat transfer performance in microchannels was rather poor, with a critical heat flux (CHF) of similar value to the pool boiling CHF on enhanced surfaces. This led to the development of some of the high performance configurations which will be described in greater detail in Sect. 10.

4

Onset of Nucleate Boiling

Bubble nucleation during boiling occurs over a surface heated above its saturation temperature from the existing cavities on a surface. While the surface is submerged in liquid, some of the cavities retain undissolved gases and provide the initial nucleus required for bubble nucleation and growth. However, for nucleation to occur, certain conditions related to cavity size, shape, and thermal field around it prior to nucleation need to be satisfied. These conditions are referred to as ONB criteria and have been extensively studied both in pool and flow boiling. Hsu and Graham (1961) and Hsu (1962) provided the basic framework for the nucleation criterion for ONB in pool boiling using the cavity mouth diameter and the shape of the bubble at departure. The effect of contact angle and flow conditions was subsequently incorporated to extend this model to flow boiling inside channels. A detailed derivation for the ONB criteria under different conditions is given by Kandlikar et al. (2014). Following the extension provided by Davis and Anderson (1966), the range of nucleation cavities radii that will nucleate under a given wall superheat condition is given by:
 r c, min, r c, max ¼

  δt sin θr ΔT Sat  2ð1 þ cos θr Þ ΔT Sat þ ΔT Sub

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! 8σT Sat ðΔT Sat þ ΔT Sub Þð1 þ cos θr Þ 1 1 ρV hLV δt ΔT 2Sat

(1)

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1957

where ΔTSat = TW  TSat, δt is the thermal boundary layer thickness prior to nucleation given by the ratio of liquid thermal conductivity and the single-phase heat transfer coefficient – kL/hL, ΔTSub is the liquid subcooling given by TLTSat, θr is the receding contact angle on the heater surface, and rc,min and rc,max are the minimum and maximum radii of cavities that will nucleate under the given conditions.

5

Explosive Boiling

Equation 1 provides the range of cavity radii where nucleation can occur on the heater surface corresponding to the local wall superheat and local liquid subcooling at a given location along the flow. If cavities in this range of radii are not available, then nucleation will not occur at this section. It will be delayed until the local liquid subcooling reduces and/or the local wall superheat increases as the liquid continues to get heated along the flow length. When nucleation is initiated at higher wall superheats, the vapor bubble grows rapidly in the highly superheated liquid environment. The bubble then fills the entire cross-section of the microchannel and begins to grow due to the excessive pressure build-up inside the bubble causing the bubble interface to expand in both upstream and downstream directions. Figure 1 shows the pressure distribution in a microchannel from inlet to outlet prior to nucleation as described by Kandlikar (2006). At the onset of nucleation at high wall superheats, a bubble is formed and the pressure inside the bubble corresponds to the saturation pressure pV corresponding to the superheated liquid temperature. If this pressure is higher than the inlet pressure, and if the liquid inertia forces are low, the bubble will expand in both forward and backward directions. The

Fig. 1 Pressure build-up upon nucleation at high wall superheats in microchannels (Figure redrawn from Kandlikar (2006))

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liquid behind the interface then gets pushed back toward the inlet. If the nucleation occurs close to the inlet and if the pressure build-up is severe, then the bubble will go into the inlet manifold and cause significant flow disruption leading to unstable operation. The growth of a bubble in a microchannel was numerically simulated by Mukherjee and Kandlikar (2004, 2005, 2009). They showed that the high growth rate could lead to the backflow, the extent of which depends on the resistance to flow offered by the channel in the upstream and downstream directions.

6

Flow Boiling Instability in Microchannels

Flow boiling in microchannels sometimes encounters instabilities in its operation depending on its operating conditions. When the flow is introduced as subcooled liquid, and the liquid undergoes boiling in the parallel microchannel passages, then it may experience one of the following types of instabilities: (a) Instability due to explosive boiling – The sudden expansion of a nucleating bubble leads to flow reversals in individual microchannels. The flow of vapor in the inlet manifold causes severe flow maldistribution and induces unstable flow. The instability could be reduced or avoided by reducing the wall superheat at the onset of nucleate boiling by providing nucleating cavities that meet the nucleation criteria on the walls of the microchannels. Presence of inlet restrictors also helps in reducing this type of flow instability by preventing the backflow (Kandlikar et al. 2005, 2006; Kosar et al. 2006; Kandlikar and Mukherjee 2004, 2005, 2009). (b) Ledinegg (or Flow Excursion) Instability – This type of flow boiling instability is well studied for conventional size channels. It results when the negative value of the slope of the pressure drop-mass flux demand curve becomes greater than the negative value of the pump supply curve. This type of instability occurs in many flow boiling systems due to the presence of two-phase flow and increasing pressure drop with quality in the channels. Peles (2012) provides a detailed analysis of this type of instability related to flow boiling in microchannels. The presence of inlet restrictors also helps in reducing this type of flow instability. (c) Upstream Compressible Volume Instability – This type of instability is the result of compression and expansion of a gas or vapor volume present in the upstream portion of the flow loop in a microchannel system. Peles (2012) provides a detailed explanation of the phenomena. As noted by Bergles and Kandlikar (2005), the presence of dissolved gases in the system may lead to formation of gas pockets in the upstream pipes and induce upstream compressible volume instability. This type of instability is also induced in conjunction with the explosive boiling instability described above. (d) Parallel Channel Instability – In a microchannel evaporator with multiple parallel channels, flow boiling in one channel might interact with other parallel channels in a way similar to the upstream compressible volume instability.

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This coupled with the explosive boiling in the channels may lead to a very complex and severe instability. This can be mitigated by providing inlet restrictors as these will reduce the flow reversal in the channels and thereby largely eliminate the communication between the parallel channels. A detailed mathematical treatment of this instability is provided by Peles (2012).

7

Pressure Drop During Flow Boiling in Microchannels

7.1

Single-Phase Flow

Flow boiling in microchannels may encounter an initial region with single-phase flow prior to onset of nucleation. Similarly, single-phase vapor flow may be encountered toward the exit section when all liquid has evaporated. Pressure drop during the single-phase flow can be calculated using the well-established frictional pressure drop equations for conventional sized tubes available in standard fluid flow textbooks. Frictional pressure drop during fully developed flow is given by ΔpSP ¼

2f SP ρ u2m L D

(2)

where ΔpSP is the single-phase pressure drop for a fluid having a density ρ flowing in a channel with a hydraulic diameter D over a length L at a mean velocity um. The friction factor fSP for the single-phase flow depends on whether the flow is laminar or turbulent. For fully developed laminar flow, the friction factor is given by f SP ¼

Po Re

(3)

where Re is the flow Reynolds number and Po is the Poiseuille number which depends on the channel geometry. For a circular tube, Po = 16 and for a square channel, Po = 14.23. For other rectangular microchannel configurations, the following equation by Shah and London (1978) is recommended.   Po ¼ 24 1  1:3553αc þ 1:9467α2c  1:7012α3c þ 0:9564α4c  0:2537α5c

(4)

where αc is the channel aspect ratio and is between 0 and 1. For fully developed turbulent flow, the friction factor fSP in Eq. 2 may be calculated using the Blasius equation given below: f SP ¼ 0:0791Re0:25

(5)

The entrance region in a microchannel flow offers considerably higher pressure drop. The length Lh of the entrance region is given by:

1960

S. G. Kandlikar

Lent ¼ 0:05 Re Dh

(6)

Pressure drop in the entrance region can be estimated using apparent friction factor. Details of the calculation methodology for microchannel flows may be found in Kandlikar et al. (2014).

7.2

Two-Phase Flow

For two-phase flow in constant area microchannels, the total pressure drop consists of the sum of frictional, acceleration, and gravitational terms. Expressing the pressure drop at any section in terms of the local pressure gradient, the total pressure gradient is expressed in terms of the three components. dp dp dp dp ¼ þ þ dz T dz F dz A dz G

(7)

These are described further in the following sections.

7.3

Two-Phase Frictional Pressure Drop

The presence of the two phases introduces interactions between the two phases in addition to the respective phase interactions with the walls of the channel. The homogeneous flow model provides a simple way to consider the presence of the two phases by considering it as the flow of a single-phase fluid having appropriately averaged properties depending on their significance in the flow structure. Thus, the two-phase friction pressure drop is given by: 

dp 2f ρ u2 ¼ TP TP m dz F D

(8)

where fTP is the two-phase friction factor, ρTP is the average density of the two-phase mixture, and um is the average flow velocity calculated using the average density. The negative sign arises because the pressure decreases along the flow length. The two-phase friction factor is related to the two-phase Reynolds number through the Poiseuille equation similar to Eq. 3. The average properties can be estimated different mixture properties rules, but the following equations suggested by McAdams et al. (1942) have been very successful in predicting the two-phase frictional pressure drop. 1 x 1x ¼ þ ρTP ρV ρL

(9)

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Boiling and Two-Phase Flow in Narrow Channels

1 x 1x ¼ þ μTP μV μL

1961

(10)

where x is the mass quality (vapor fraction) and μ is the dynamic viscosity. A recent study by Dario et al. (2016) reported that the homogeneous flow model using McAdams’ average viscosity equation was able to predict their refrigerant R134a flow boiling pressure drop data in 770-μm diameter channels within less than 10% error. Kalani and Kandlikar (2015a) also reported similar results using water with their 181 μm wide and 400 μm deep microchannels and a 127-μm gap above the channels. Although the flow structure at high qualities differs significantly from the uniform flow of the two-phases assumed in the homogeneous flow model, the model was still able to predict the pressure drop quite well.

7.4

Acceleration Pressure Drop

As the liquid evaporates into vapor, the flow velocity increases due to the lower density of the vapor phase. This in turn introduces the acceleration pressure drop caused by the change in flow momentum. Applying the homogeneous flow model, Collier (1981) presented the following equation for estimating the acceleration pressure drop.

  dp dx dvV dp 2  ¼ G vLV þ x dz A dz dp dz

(11)

where v is the specific volume, and the subscript LV refers to the latent quantity, i.e., the difference between vapor and liquid property values. The first term within the parenthesis on the right hand side accounts for the rate of change in quality, which is affected by the imposed heat flux. These are related through the following equations for a channel with perimeter P, and applied heat flux of q00 , mass flux m_ with a fluid having a latent heat hLV. dx q00 P ¼ _ LV dz mh

(12)

Note that the second term in Eq. 11 within the parenthesis on the right hand side represents the effect of vapor compressibility and includes the total pressure gradient.

7.5

Gravitational Pressure Gradient

There is an additional pressure gradient term introduced due to the channel inclination with respect to gravity. For a channel that is inclined at an angle θ with respect to a horizontal plane, then the gravitational pressure gradient is given by:

1962

S. G. Kandlikar

dp g sin θ  ¼ dz G vTP

(13)

where vTP is the average specific volume (=1/ρTP).

7.6

Total Pressure Gradient

The total pressure gradient is obtained by substituting Eqs. 8, 11, and 13 into Eq. 7. Since the total pressure gradient also appears on the right hand side, of Eq. 11, further rearrangement is needed to get the final expression in the following form.

  2f TP G2 vL vLV dx g sin θ

i 1þx þ G2 vLV þ h dz vL 1 þ x vLV = Dh vL vV dp    ¼ dv dz V 1 þ G2 x dp

(14)

For calculating the total pressure drop across a microchannel undergoing flow boiling, the above equation needs to be integrated from inlet to outlet of the microchannel. Alternatively, the flow length could be subdivided into discrete elements and the pressure gradient within each element can be estimated using the mean flow conditions in the element. In case of the subcooled liquid entry, the pressure drop in the liquid flow region can be estimated using the single-phase flow equation, Eq. 2.

8

Flow Boiling Heat Transfer in Microchannels

Flow boiling heat transfer has been extensively studied in literature. However, the data obtained in many cases are influenced by the instability introduced by explosive bubble growth and other mechanisms discussed in Sect. 5. Therefore, large scatter is observed in the experimental data and comparison to the correlations. The following correlation by (Balasubramanian and Kandlikar 2005) was developed using the available microchannel flow boiling data. It is based on the extension of the flow boiling correlation developed by Kandlikar (1990) for conventional sized tubes. The smaller diameters in microchannels and minichannels result in a low Reynolds number ReLO with all flow as liquid. The resulting laminar single-phase flow heat transfer coefficient hLO, is used to normalize the flow boiling heat transfer hTP. For ReLO > 100:  hTP ¼ larger of

hTP, NBD hTP, CBD

(15)

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Boiling and Two-Phase Flow in Narrow Channels

1963

where the subscripts NBD and CBD denote the nucleate boiling dominant and convective boiling dominant regions. The heat transfer coefficients in these respective regions are given by: hTP, NBD ¼ 0:6683Co0:2 ð1  xÞ0:8 hLO þ 1058:0Bo0:7 ð1  xÞ0:8 FF1 hLO

(16)

hTP, CBD ¼ 1:136Co0:9 ð1  xÞ0:8 hLO þ 667:0Bo0:7 ð1  xÞ0:8 FF1 hLO

(17)

where Co is the convection number given by Co = [(1 – x)/x]0.8(ρV/ρL)0.5 and Bo is the boiling number given by Bo = q00 /GhLV. The single-phase all-liquid flow heat transfer coefficient hLO is given by: for 104  ReLO  5  106 hLO ¼

for 3000  ReLO  104 hLO ¼

ReLO PrL ðf =2ÞðkL =DÞ

2=3 1 þ 12:7 PrL  1 ðf =2Þ0:5

(18)

ðReLO -1000ÞPrL ðf =2ÞðkL =DÞ

2=3 1 þ 12:7 PrL  1 ðf =2Þ0:5

(19)

NuLO k Dh

(20)

for 100  ReLO  1600 hLO ¼

In the transition region between Reynolds numbers of 1600 and 3000, a linear interpolation is suggested for hLO. For ReLO  100, the nucleate boiling mechanism is dominant, and hTP is given by the following NBD region correlation. For ReLO  100, hTP ¼ hTP, NBD ¼ 0:6683Co0:2 ð1  xÞ0:8 hLO þ 1058:0Bo0:7 ð1  xÞ0:8 FFl hLO

(21)

The single-phase heat transfer coefficient hLO is calculated from Eqs. 18, 19, and 20 depending on the ReLO range. The fluid surface parameter FFl in the above equations represents a fluid-surface dependent parameter. Table 3 gives the values of this parameter for various fluid and heater surface combinations. The value of FFl for a new fluid-surface combination may be obtained from the respective experimental flow boiling data using regression analysis.

9

Critical Heat Flux in Microchannels

Critical heat flux in microchannels is reached as a result of extended dry-out conditions occurring when a vapor bubble or a vapor slug occupies the channel walls over an extended period of time as compared to the bubble ebullition cycle

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Table 3 Fluid-surface parameter FFl (fluid surface parameter) in flow boiling correlation, Eqs. 15, 16, 17, 18, 19, 20, and 21

Fluid Water R-11 R-12 R-13B1 R-22 R-113 R-114 R-134a R-152a R-32/R-132 R-141b R-124 Kerosene HFE 7000

FFl 1.00 1.30 1.50 1.31 2.20 1.30 1.24 1.63 1.10 3.30 1.80 1.00 0.488 2.0

FFL = 1 is recommended for stainless steel heater surface

time. The dry-out of the liquid film under a vapor slug also leads to the CHF condition Kandlikar (2002a, b). The flow instability greatly influences the CHF as the liquid supply to dry regions is greatly hampered under unstable operating conditions. Kandlikar (2010c) developed a pool boiling model based on a force balance on a nucleating bubble. It was postulated that the CHF was reached when the forces driving the retaining surface tension forces for a hydrophilic walls at the liquid interface near the contact line region over the bubble base were overcome by the evaporation momentum force resulting from rapid evaporation. In case of flow boiling, the additional force due to liquid inertia needs to be considered. Following a detailed scale analysis of an evaporating interface in a microchannel flow (Kandlikar 2010b), Kandlikar (2010c) developed the following CHF model. Three nondimensional groups were introduced to represent the evaporation momentum, inertia, and surface tension forces, respectively.  K2, CHF ¼

qCHF hLV

2 

 Dh ρV σ

(22)

We ¼

G2 Dh ρm σ

(23)

Ca ¼

μL G ρL σ

(24)

The flow conditions were divided into low inertia region (LIR) and high inertia regions (HIR) based on the Weber number. Further, each region was

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Boiling and Two-Phase Flow in Narrow Channels

1965

subdivided into a low CHF region (LC) and a high CHF region (HC). Specific CHF equations resulting from the force balance analysis are obtained as follows. • Low Inertia Region, LIR: We < 900: • High CHF Subregion: LIR-HC - L/D  140 K 2, CHF ¼ a1 ð1 þ cos θÞ þ a2 Weð1  xÞ þ a3 Cað1  xÞ

(25)

• Low CHF Subregion: LIR-LC - L/D  230 K 2, CHF ¼ a4 ½a1 ð1 þ cos θÞ þ a2 Weð1  xÞ þ a3 Cað1  xÞ

(26)

• High Inertia Region, HIR: We  900: • High CHF Subregion: HIR-HC - L/D < 60 K 2, CHF ¼ a1 ð1 þ cos θÞ þ a2 Weð1  xÞ þ a3 Cað1  xÞ

(27)

• Low CHF Subregion: HIR-LC - L/D  100 K 2, CHF ¼ a4 ½a1 ð1 þ cos θÞ þ a2 Weð1  xÞ þ a3 Cað1  xÞ

(28)

Since the changes between HC and LC subregions in both LIR and HIR regions are step-wise, they represent a change in the mechanism. CHF does not vary linearly in this region, and it is suggested that a conservative value be used based on the two limiting L/D ratio values. Improved Constant a4 in the HIR-LC Region, HIR: We  900: A slightly improved correlation was obtained in the HIR-LC region by slightly changing the set of constants as given below.  n 1 ½a1 ð1 þ cos θÞ þ a2 Weð1  xÞ þ a3 Cað1  xÞ (29) K 2, CHF ¼ a5 We Ca Table 4 gives the values of the constants in the above equations.

Table 4 Values of constants in the CHF model given by Eqs. 25, 26, 27, 28, and 29, Kandlikar (2010c)

Constant a1 a2 a3 a4 a5 n

Value 1.03  104 5.78  105 0.783 0.125 0.14 0.07

1966

10

S. G. Kandlikar

Enhanced Flow Boiling in New Microchannel Configurations

CHF in microchannels has been noted to be quite low, often comparable to the pool boiling CHF over a plain surface. Such low performance is not adequate to justify using a flow boiling system in high heat flux dissipation applications. The singlephase liquid flow has been shown to be quite effective, albeit at a high pressure drop penalty as far back as in 1981 by Tuckerman and Pease (1981). The pressure drop during flow boiling is also quite high. To mitigate the high pressure drop and low CHF, a number of enhancement techniques have been proposed in literature. Kandlikar (2016) provides an overview of the recent developments in this field. A brief overview of some of them is presented in this section.

10.1 Pin Fins and Nanowires Within Microchannels Krishnamurty and Peles (2008) introduced staggered micro pin-fins of 250 μm in length, 100 μm in diameter, and with a pitch-to-diameter ratio of 1.5. They obtained a heat flux dissipation of 20–350 W/cm2 (0.2 to 3.5 MW/m2) with a pressure drop in the range 60–75 kPa. Although the CHF is higher than that in a plain microchannel, around 130–150 W/cm2 (1.3-a.5 MW/m2), the pressure drop is quite high. Kosar et al. (2005) report a pressure drop of 20 kPa to 110 kPa with single-phase water. The flow boiling pressure drop is expected to be significantly higher than that. In addition to the pumping power considerations, the variation in the saturation temperature along the flow length due to the large pressure drop introduces temperature nonuniformity that is of concern in electronics cooling application. Placing nanowires and nanotubes on the channel walls is expected to increase the wettability of the surface and lead to an increase in CHF. Several researchers, including Khanikar et al. (2009), Shenoy et al. (2011), and Morshed et al. (2012) incorporated carbon nanowires in the channel and noted modest increases in CHF with a modest increase in the pressure drop. Further research in this area is needed to understand the heat transfer mechanism and exploit it for further enhancements. Zhu et al. (2014) employed short fins on the base of a 500-μm square microchannel. The channel base provided the wicking structure. This configuration resulted in a CHF of 14.4 MW/m2 at a wall superheat of 45  C and a pressure drop of 38 kPa.

10.2 Vapor Extraction Through Hollow Open Fins The presence of vapor in the microchannels is seen as the major reason for early CHF noted in microchannels. As a means to reduce the vapor fraction in the flow, Woodcock et al. (2014) incorporated hollow micropin-fins in the shape of a tear drop with frontal opening on the flat front side in the flow. The top cover provided outlet from the hollow fins. The reduced vapor flow provided a heat transfer

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Boiling and Two-Phase Flow in Narrow Channels

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improvement. The geometry is being further refined to provide optimal heat transfer performance while reducing pressure drop as a result of reduced vapor flow through the micropin-fin arrays.

10.3 Vapor Removal Through a Hydrophobic Membrane Cover Another technique to remove gas from a liquid in a microchannel was proposed by Xu et al. (2010) by covering the microchannels with a hydrophobic membrane to provide a preferential pathway for the gas. David et al. (2011) employed a polytetrafluoroethylene (PTFE) membrane in an evaporator to remove generated vapor. Fazeli et al. (2015) employed the hydrophobic membrane in a capillary microfabricated pillar structure that provided wicking from the irrigating channels providing liquid. Combination of meniscus evaporation from the capillary structure and vapor removal provided a dual mechanism of bubble formation with vapor venting and meniscus evaporation with vapor venting. This extended the heat flux range to 380 W/cm2 (3.8 MW/m2).

10.4 Tapered gap Microchannels Kandlikar et al. (2013) provided a gap above the microchannel passages to remove the vapor surrounding the heat transfer surface. The heat transfer performance increased and the pressure drop reduced. To reduce the pressure drop further, the gap was tapered from the inlet to outlet. The increasing area in the flow direction caused pressure recovery and extremely low pressure drops were achieved. The effect of varying taper and gap was investigated by Kalani and Kandlikar (2014). Figure 2 shows a schematic of open microchannels with manifold (OMM) and a tapered gap. The microchannels were 217 μm wide, 162 μm deep with 160 μm wide Fig. 2 Open Microchannels with Manifold and Tapered Gap configuration, Redrawn from Kandlikar et al. (2013)

1968

S. G. Kandlikar

fins. The test chip was 25 mm square. Various gap sizes were tested from 127 μm to 1 mm and the shorter gap was found to give the best performance. A taper of 4–6% was seen to provide extremely low pressure drop. The pressure recovery term due to area change (taper) is given by the following equation (Kalani and Kandlikar 2015a):

  dp  ¼ dz taper, area

v i dA 2G2 vL h LV 1þx Ac L   dz dv V 1 þ G2 x dp

(30)

As the area changes along the taper direction, the mass flux G also changes. The above equation needs to be integrated from the inlet section to the outlet section with varying quality x and area A. In order to further improve the performance, liquid inertia force was employed by increasing the flow rate to remove bubbles from the microchannels (Kalani and Kandlikar 2015b). Significant enhancement in CHF was obtained. With an inlet flow Reynolds number of 1095, a CHF of 1.07 kW/cm2 (10.7 MW/m2) was obtained. The corresponding pressure drop was 30 kPa and the heat transfer coefficient was 295 kW/m2  C. Further improvement in performance is expected through optimization of taper and gap. One of the advantages of the above OMM design is that it can be scaled up by incorporating multiple headers side-by-side. The inlet and outlet streams could be combined and large surface area could be effectively cooled using this design.

10.5 Radial Microchannels Another geometry that has attracted attention is the radial microchannels with a central feed. Schultz et al. (2015) tested cooling of a 0.04 mm2 hot spot on a 15.5 mm2 background heated die. Several such heated areas were placed on a 20 mm  20 mm square chip that was cooled with radial microchannels. They could dissipate 20 W/cm2 background heat flux with a localized hot spot of 350 W/cm2. The pressure drop in the system was extremely high at around 340 kPa at the highest heat flux tested. Recinella et al. (2016) implemented the radial microchannel configuration with an open gap manifold. This dramatically reduced the pressure drop and improved heat transfer performance. Radial microchannels as well as radially oriented offset strip fins as shown in Fig. 3 were tested. They dissipated a maximum heat flux of 900 W/cm2. This geometry is seen to have significant potential in dissipating high heat fluxes relevant in computer chip cooling application.

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Fig. 3 (a) Radial microchannels and (b) radially oriented offset strip fins, on a 10 mm square chip cooler footprint, Recinella and Kandlikar (2016)

11

Conclusions

Flow boiling in microchannels has been investigated exhaustively for the past two decades. The basic phenomena of two-phase flow and heat transfer are well understood and predictive equations and models are available for estimating pressure drop and heat transfer coefficient. The research indicates that flow boiling in microchannels leads to unstable operation with a low CHF and high pressure drop. New configurations are being currently developed to address these issues. A heat flux of over 1 kW/cm2 has been achieved with a pressure drop of less than 30 kPa and a wall superheat of around 30 C with water boiling at atmospheric pressure. It is expected that future research will continue in better understanding the boiling mechanism in some of these novel configurations. This is currently an active area of research.

12

Cross-References

▶ Boiling on Enhanced Surfaces ▶ Flow Boiling in Tubes ▶ Fundamental Equations for Two-Phase Flow in Tubes ▶ Heat Pipes and Thermosyphons ▶ Nucleate Pool Boiling ▶ Single- and Multiphase Flow for Electronic Cooling ▶ Two-phase Heat Exchangers

1970

S. G. Kandlikar

References Balasubramanian P, Kandlikar SG (2005) An experimental study of flow patterns, pressure drop and flow instabilities in parallel rectangular minichannels. Heat Tran Eng 26(3):20–27 Bergles AE, Kandlikar SG (2005) On the nature of critical heat flux in microchannels. J Heat Transf 127(1):101–107 Chawla JM (1966) Warmeubergang und Druckabfall in Waagerechten Rohren bei der Stromung von verdampfenden Kaltemitteln, VDI-Forschungschefte, vol 523. VDI-Verlag, Düsseldorf Collier JG (1981) Convective boiling and condensation. McGraw-Hill International Book Co., New York Cornwell K, Kew PA (1992) Boiling in small parallel channels. In: Proceedings of CEC conference on energy efficiency in process technology. Elsevier Applied Sciences, Athens, pp 624–638. Paper 22 Dario ER, Passos JC, Simón ML, Tadrist L (2016) Pressure drop during flow boiling inside parallel microchannels. Int J Refrig 72:111–123. https://doi.org/10.1016/j.ijrefrig.2016.08.002 David MP, Miler J, Steinebrenner JE, Yang Y, Touzelbaev M, Goodson KE (2011) Hydraulic and thermal characteristics of a vapor venting two-phase microchannel heat exchanger. Int J Heat Mass Transf 54(25–26):5504–5516 Davis EJ, Anderson GH (1966) The incipience of nucleate boiling in forced convection flow. AICHE J 12(4):774–780 Fazeli A, Mortazavi M, Moghaddam S (2015) Hierarchical biphilic micro/nanostructures for a new generation phase-change heat sink. Appl Therm Eng 78:380–386 Hsu YY (1962) On the size range of active nucleation cavities on a heating surface. J Heat Transf 84:207–216 Hsu YY, Graham R (1961) An analytical and experimental study of the thermal boundary layer and ebullition cycle in nucleate boiling, Nasa TN-d, vol 594. National Aeronautics and Space Administration, Washington, DC Kalani A, Kandlikar SG (2014) Evaluation of pressure drop performance during enhanced flow boiling in open microchannels with tapered manifolds. J Heat Transf 136(5), 7 pages. https://doi. org/10.1115/1.4026306 Kalani A, Kandlikar SG (2015a) Effect of taper on pressure recovery during flow boiling in open microchannels with manifold using homogeneous flow model. Int J Heat Mass Transf 83:109–117 Kalani A, Kandlikar SG (2015b) Combining liquid inertia with pressure recovery from bubble expansion for enhanced flow boiling. Appl Phys Lett 107:181601. https://doi.org/10.1063/ 1.4935211 Kandlikar SG (1990) A general correlation for saturated two-phase flow boiling heat transfer inside horizontal and vertical tubes. J Heat Transf 112(1):219–228. https://doi.org/10.1115/1.2910348 Kandlikar SG (2002a) Fundamental issues related to flow boiling in minichannels and microchannels. Exp Thermal Fluid Sci 26:389–407. https://doi.org/10.1016/S0894-1777(02)00150-4 Kandlikar SG (2002b) Two-phase flow patterns, pressure drop, and heat transfer during flow boiling in minichannel flow passages of compact evaporators. Heat Tran Eng 23(5):5–23 Kandlikar SG (2004) Heat transfer mechanisms during flow boiling in microchannels. J Heat Transf 126:8–16 Kandlikar SG (2006) Nucleation characteristics and stability considerations during flow boiling in microchannels. Exp Thermal Fluid Sci 30(5):441–447 Kandlikar SG (2010a) Similarities and differences between flow boiling in microchannels and pool boiling. Heat Tran Eng 31(3). https://doi.org/10.1080/01457630903304335 Kandlikar SG (2010b) Scale effects of flow boiling in microchannels: a fundamental perspective. Int J Therm Sci 49(7):1073–1085 Kandlikar SG (2010c) A scale analysis based theoretical force balance model for critical heat flux (CHF) during saturated flow boiling in microchannels and minichannels. J Heat Transf 132 (8):0181501-1–018150113

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Kandlikar SG (2016) Mechanistic considerations for enhancing flow boiling heat transfer in microchannels. J Heat Transf 138(2):02150. https://doi.org/10.1115/1.4031648. (16 pages) Kandlikar, SG, Steinke, ME, Tian, S, Campbell, LA (2001) High-speed photographic observation of flow boiling of water in parallel minichannels. Paper presented at the ASME national heat transfer conference, ASME, Anaheim, 10–12 June 2001, NHTC01-11262 Kandlikar SG, Grande WJ (2003) Evolution of microchannel flow passages – thermohydraulic performance and fabrication technology. Heat Tran Eng 24(1):3–17 Kandlikar, SG, Willistein, DA, Borrelli, J, and ASME (2005) Experimental evaluation of pressure drop elements and fabricated nucleation sites for stabilizing flow boiling in minichannels and microchannels. In: Proceedings of the 3rd international conference on microchannels and minichannels, ASME, 13–15 June, pp 115–124. https://doi.org/10.1115/ICMM2005-75197 Kandlikar SG, Kuan WK, Willistein DA, Borrelli J (2006) Stabilization of flow boiling in microchannels using pressure drop elements and fabricated nucleation sites. J Heat Transf 128(4):389–396. https://doi.org/10.1115/1.2165208 Kandlikar, SG, Widger, T, Kalani, A, and Mejia, V (2013) Enhanced flow boiling over open microchannels with uniform and tapered gap manifolds (OMM), 75th Anniversary Issue. J Heat Transf 135(6):061401 (9 pages) Kandlikar SG, Garimella S, Li D, Colin S, King MR (2014) Heat transfer and fluid flow in minichannnels and microchannels. Elsevier, Oxford, UK Khanikar V, Mudawar I, Fisher T (2009) Flow boiling in a microchannel coated with carbon nanotubes. IEEE Trans Compon Packag Technol 32(3):639–649 Kosar A, Kuo C-J, Peles Y (2005a) Boiling heat transfer in rectangular microchannels with reentrant cavities. Int J Heat Mass Transf 48(23–24):4867–4886. https://doi.org/10.1016/j. ijheatmasstransfer.2005.06.003 Kosar A, Kuo CJ, Peles Y (2006) Suppression of boiling flow oscillations in parallel microchannels by inlet restrictors. J Heat Transf 128(3):251–260. https://doi.org/10.1115/1.2150837 Krishnamurthy S, Peles Y (2008) Flow boiling of water in a circular staggered micro-pin fin heat sink. Int J Heat Mass Transf 51:1349–1364 Lazarek GM, Black SH (1982) Evaporative heat transfer, pressure drop and critical heat flux in a small diameter vertical tube with R-113. Int J Heat Mass Transf 25(7):945–960. https://doi.org/ 10.1016/0017-9310(82)90070-9 McAdams WH, Woods WK, Heroman LC (1942) Vaporization inside horizontal tubes II-benzeneoil mixtures. Trans ASME 64(3):193–200 Morshed AKMM, Yang F, Ali MY, Khan JA, Li C (2012) Enhanced flow boiling in a microchannel with integration of nanowires. Appl Therm Eng 32:68–75. https://doi.org/10.1016/j. applthermaleng.2011.08.031 Mukherjee, A, and Kandlikar, SG (2004) Numerical simulation of growth of a vapor bubble during flow boiling of water in a microchannel. In: Proceedings of the second international conference on microchannels and minichannels, Rochester, pp 565–572. (Also published in Microfluidics and Nanofluidics 1(2):137–145) ASME paper no. ICMM 2004–2382 Mukherjee, A and Kandlikar, SG (2005) Numerical study of the effect of inlet constriction on flow boiling stability in microchannels. In: Proceedings of the third international conference on microchannels and minichannels, Toronto, 13–15 June 2005 ASME paper no. ICMM2005–75143 Mukherjee A, Kandlikar SG (2009) The effect of inlet constriction on bubble growth during flow boiling in microchannels. Int J Heat Mass Transf 52(21–22):5204–5212 Peles Y (2012) In: Kandlikar SG (ed) Contemporary perspectives on flow boiling instabilities in microchannels and minichannels, series in contemporary perspectives in emerging technologies. Begell House, Redding Recinella, A, Kalani, A,and Kandlikar, SG (2016) Enhanced flow boiling heat transfer using radial microchannels. In: ASME 2016 14th international conference on nanochannels, microchannels, and minichannels, Washington, DC, 10–14 July 2016, paper no. ICNMM2016–7975, https:// doi.org/10.1115/ICNMM2016-7975

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Schultz M, Yang F, Colgan E, Polastre R, Dang B, Tsang C, Gaynes M, Parida P, Knickerbocker J, Chainer T (2015) Embedded two-phase cooling of large three-dimensional compatible chips with radial channels. J Electron Packag 138(2). https://doi.org/10.1115/1.4033309. (5 pages) Serizawa A, Feng A, Kawara Z (2002) Two-phase flow in microchannels. Exp Thermal Fluid Sci 26(6–7):703–714. https://doi.org/10.1016/S0894-1777(02)00175-9 Shah RK, London AL (1978) Laminar flow forced convection in ducts, supplement 1 to advances in heat transfer. Academic, New York Shenoy S, Tullius JF, Bayazitoglu Y (2011) Minichannels with carbon nanotube structures surfaces for cooling applications. Int J Heat Mass Transf 54(25–26):5379–5385 Steinke ME, Kandlikar SG (2004) An experimental investigation of flow boiling characteristics of water in parallel microchannels. J Heat Transf 126(4):518–526 Tuckerman DB, Pease RFW (1981) High performance heat sink for VLSI. IEEE Electron Devices Lett EDL 2(5):126–129 Wambsganss MW, France DM, Jendrzejczyk JA, Tran TN (1993) Boiling heat transfer in a small diameter tube. ASME J Heat Transfer 115(4):963–972 Woodcock C, Houshmand F, Plawsky J, Izenson M, Hill R, Phillips S,and Peles Y (2014) Piranha pin-fins (PPF): voracious boiling heat transfer by vapor venting from microchannels–system calibration and single-phase fluid dynamics. In: 14th IEEE ITherm conference, Orlando, 27–30 May, pp 282–289 Xu J, Vaillant R, Attinger D (2010) Use of a porous membrane for gas bubble removal in microchannels: physical mechanisms and design criteria. Microfluid Nanofluid 9:765–772. https://doi.org/10.1007/s10404-010-0592-5 Zhu Y, Antao DS, Chu K-H, Hendricks TJ, and Wang EN (2014) Enhanced flow boiling heat transfer in microchannels with structured surfaces. In: 15th international heat transfer conference, paper no. IHTC15–9508

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Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Why Liquid Cooling? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Indirect Cooling and Direct Immersion Cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Single-Phase Convection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Microchannel Arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Micropin-Enhanced Microgaps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 System Configuration and Design Consideration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Thermal/Electrical Codesign of Microfluidic Cooled 3D ICs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Flow Boiling in Microgaps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Characteristics of Pure Fluid Flow Boiling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Flow Boiling Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Flow Boiling of Fluid Mixtures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 System Configuration and Design Consideration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Cross-References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

Liquid cooling is gaining critical importance for the continuing development of high-density and high-power microelectronic system. In this chapter, both singlephase and two-phase cooling are reviewed with a focus on various passive and active heat transfer enhancement techniques including microchannels, wavy channels, protrusions, dimples, bifurcating manifold, pin fin arrays, carbon

Y. Joshi (*) · Z. Wan George W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA, USA e-mail: [email protected]; [email protected] # Springer International Publishing AG, part of Springer Nature 2018 F. A. Kulacki (ed.), Handbook of Thermal Science and Engineering, https://doi.org/10.1007/978-3-319-26695-4_49

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nanotube, surface coatings, fluid mixture, and jet array. Although liquid cooling demonstrates high heat transfer performance, its integration to the system presents great challenges in terms of cost and reliability issues. The difference of system design between single-phase and two-phase cooling is described. Liquid cooling is especially important for 3D stacked ICs. The codesign of architecture and liquid cooling by considering the hot spot location, leakage power, and routing is required for a successful implementation of liquid-cooled 3D IC system. Nomenclature

A CHF Dp f G h Hp N Nu ΔP R Sl SL ST SV t tc T Re V x*

Area [m2] Critical heat flux [W/m2] Pin diameter [m] Friction factor [
] Maximum mass flux at the smallest cross-sectional area [kg/m2s] Heat transfer coefficient [W/m2K] Pin height [m] Number of channels [
] Nusselt number [
] Pressure drop [Pa] Thermal resistance [K/W] Liquid source term [kg/m3] Longitudinal spacing [m] Transverse spacing [m] Vapor source term [kg/m3] Time [s] Tip clearance [m] Temperature [K] Reynolds number [
] Volumetric flow rate [m3/s] Nondimensional streamwise position [
]

Greek Symbols

α β ρ

Channel aspect ratio Channel width to pitch ratio Density

Subscripts and Superscripts

cond cr f fan l L

Condensation Cross section Fluid Fanning Liquid Longitudinal

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tot sat v

1

1975

Total Saturation Vapor

Introduction

The pursuit of multi-functionality and higher performance for computing, consumer, and military microsystems, such as servers, tablets, radio-frequency (RF) transceivers, solid-state lasers, and light emitting diodes (LED), continues to push up the power consumption and surface area and volumetric power densities at the chip and system levels. For example, gallium-nitride (GaN) high-electron-mobility transistors (HEMT) have power densities many times higher than other power transistor technologies. The output power density of an N-polar GaN HEMTs with a drawn gate length of 0.7 μm and a gate drain spacing of 0.8 μm can be 12.1W/mm (Kolluri et al. 2011), resulting in a heat flux of 1.5  106 W/cm2. A pulse of sub-picosecond duration and petawatt power was first reached at the Lawrence Livermore National Laboratory in 1996, and the Jan USP laser developed there has a peak power intensity of 2  1020 W/cm2 (Salamin et al. 2006). Another common example of a high heat flux device is the microprocessor for which “Moore’s law” (Moore 1998) has been the driving force for development (Mack 2011). The feature size has become smaller and transistor count increased with each technology node to improve integrated circuit (IC) functionality and performance, while decreasing costs (Intel 2016). A recent development in microelectronics is 3D stacked ICs. Compared to conventional 2D ICs, the chips are stacked vertically, which greatly reduces the interconnection length, a promising feature for the continuation of Moore’s law. However, heat removal becomes a key challenge, due to the increased volumetric heat generation and shrinking surface area.

1.1

Why Liquid Cooling?

For decades, air cooling has been used for cooling of various microsystems like desktop computers, mobile phones, and LED systems. For mobile devices such as smartphones, tablets, and smart watches, due to their compact size, natural convection air cooling is the most common cooling method. In higher-power computing systems such as desktops, a copper or aluminum heat sink and fan are commonly employed. Figure 1 shows a typical package cross section with backside air cooling. Due to the low thermal conductivity of the underfill, substrate, and PCB, most of the heat from the die (chip) is conducted through TIM1, lid, TIM2, and heat sink and finally rejected to ambient air with the help of the fan. The junction-to-ambient thermal resistance includes the die, TIM1, lid, TIM2, heat sink base, and convection. Key advantages of this cooling method are its low cost and high reliability.

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Fig. 1 Typical package stack with backside air cooling

Although air cooling is a cost-effective and reliable thermal management method, the maximum allowable heat dissipation is limited. Webb (2005) suggested a practical limit of thermal resistance value 0.336 K/W for a 16-mm2 heat source, which gives 104 W heat rejection for impinging flow. Recent developments in applications such as Intel server and high-performance computing have already led to much higher heat dissipation (Intel 2016) beyond the capability of air cooling. If not effectively removed, the high heat flux would result in high chip temperatures, which will increase the parasitic or leakage power exponentially, degrade the computational performance, and possibly even accelerate failure of the device (Bakir and Meindl 2008). Liquid cooling offers heat transfer coefficients that are one to two orders larger than in air cooling, allowing for continued development of high-performance microsystem. Thermal management is also extremely important for 3D IC development. As the number of chips in a 3D stack increases, the package heat flux based on the top surface area increases. The thermal resistances of the interior chips also increase due to the stacking. Tier-based liquid cooling offers the promise of stacking an arbitrarily large number of chips.

1.2

Indirect Cooling and Direct Immersion Cooling

For higher-power applications, the air-cooled heat sink can be replaced by a coldplate with internal liquid flow, as shown in Fig. 2. The liquid circulates through the coldplate, removing the heat from the package. The warm liquid is then pumped via a remote heat sink, where it rejects the heat to the ambient, and returns cooled liquid to the chip package. A variety of liquid coolants, including water, dielectric fluids, or refrigerants, can be used. Since the typical heat transfer coefficient for liquid flow is much higher than for airflow for similar velocities and flow passage

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Fig. 2 Typical package stack with coldplate cooling

Fig. 3 Cross-sectional view of the assembly of the coldplate on the flip chip BGA and the corresponding thermal resistance

dimensions, the size of coldplate can be reduced compared to air-cooled heat sink. Although a fan is not required for the coldplate, a flow loop including pump, reservoir, and liquid-air heat sink is needed. The liquid-air heat sink may be natural convection cooled for lower power levels and fan cooled for higher powers. Zhang et al. (2003) reported the thermal performance of a flip chip ball grid array packages (FBGAs) with high heat dissipation (Fig. 3). Single-phase coldplate was used for cooling with deionized water as the coolant. To further reduce the overall thermal resistance, the package lid was removed, and the coldplate was placed directly on the chip with a thermal interface material. The measured total thermal resistance ranged from 0.44 K/W to 0.32 K/W for the 12-mm chip case and from

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0.59 to 0.44 K/W for the 10-mm chip case, both under flow rates ranging from 1.67  10
6 m3/s to 1.67  10
5 m3/s. A detailed layer-by-layer thermal resistance analysis revealed that the convection resistance Rba for both cases was decreased by 55%, as the flow rate increased, while the spreading resistance Rsp was decreased by 13% for the 12-mm chip case and 21% for the 10-mm case. This decreased the total thermal resistance by 22% and 19%, respectively. One interesting finding is that the convection thermal resistance Rba decreases quickly at low flow rate. But at higher flow rate, the decrease of Rba became slow and saturated eventually with the further increase in flow rate. However, the pressure drop increased quadratically with the increase of flow rate. On the other hand, the chip resistance Rchip and TIM resistance RTIM are independent of flow rates. With the decrease in coldplate heat sink resistance, RTIM became more significant, amounting to as much as 50% of the total thermal resistance. So instead of continuing to increase the flow rate, further reduction in the thermal resistances can be achieved through the improvement of the TIM performance. Considerable work has been done to improve the TIM performance by thinning TIM thickness and using high conductivity TIM materials with conductive particles in addition (Prasher 2006). While the bulk thermal resistance of the TIM can be reduced by improving the material properties, the contact resistance can still be significant, and degradation of TIM material occurs due to aging. So the TIM performance has been a bottleneck of the cooling of high-power chips. To reduce the TIM overall thermal resistance, on-chip microfluidic cooling can be directly integrated with the chip (Dang et al. 2010), as seen in Fig. 4. The lid and TIM are completely removed from the architecture in this approach. Fluid flows across the electrically inactive side of the chip for heat removal. Usually a microchannel array, or a microgap with surface enhancement structures, is etched on the back of the chips as the flow passage. The advantages of on-chip microfluidic cooling are (1) the overall conduction resistance is minimum; (2) the size of the cooling device is much smaller than the coldplate and air-cooled heat sink solutions; and (3) heat transfer coefficient is increased due to the reduced passage size, also requiring a smaller liquid inventory. However, reliability concerns for on-chip microfluidic cooling result from the pressure drop associated with liquid flow. For coldplate, the liquid does not contact the chip directly, and structurally robust materials, such as copper, can be used to

Fig. 4 On-chip microfluidic cooling

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Table 1 Comparison between three cooling solutions

Backside air cooling Coldplate cooling On-chip microfluidic cooling

Cost Low

Heat transfer performance Low

Reliability High

Size Large

System complexity Low

Medium High

Medium High

Medium Low

Medium Low

Medium High

fabricate the coldplate. For on-chip microfluidic cooling, high pressure drop can pose a danger to the semiconductor chip. Fluid leakage risk is also a concern. Overall, although microfluidic cooling has superior thermal performance, there are numerous challenges associated with this new technology. The move from air cooling to microfluidic cooling also requires a significant change in the design and manufacturing processes. A key challenge is to integrate the microfluidic cooling at low cost, without adversely impacting reliability. A closed flow loop with heat exchanger, pump, and reservoir is needed for microfluidic cooling. These components need to be packaged compactly, or the benefits of microfluidic cooling cannot be fully realized. Possible fluid leakage and lack of mechanical strength can cause failure of the whole system and need to be addressed in a successful design. Table 1 shows a comparison between the three cooling solutions. Liquid cooling is especially useful for future high-power electronics, especially 3D stacked ICs. However, before it can be employed, its characteristics need to be fully understood. Significant work has been done during the past three decades to characterize the single-phase and two-phase thermal performance of single microchannel and microchannel arrays. This work has been reviewed by Kandlikar (2005), Steinke and Kandlikar (2006), Agostini et al. (2007), Kandlikar and Bapat (2007), Khan and Fartaj (2010), Ebadian and Lin (2011), Mudawar (2011), Kandlikar (2012, 2015), Adham et al. (2013), and Wu and Sunden (2014). With the increasing interest in thermal management of emerging chip architectures, such as three-dimensional (3D) stacked ICs, and interposer-based hybrid processor and memory modules, microfluidic cooling is becoming of significant interest. Research on cooling geometries, such as microgaps with surface enhancement structure, such as micropin fin array, is relatively new, and the related work is limited. In this chapter, the most recent developments in single-phase and two-phase microfluidic cooling using microchannel and micropin fin-enhanced microgaps during the past decade are reviewed in Sects. 2, 3, and 4. Focus will be on the various thermal enhancement techniques. The system requirements and challenges for both singlephase and two-phase cooling systems are identified. The inter-tier liquid cooling is especially useful and compatible for 3D stacked ICs. So the current development of 3D ICs with inter-tier microfluidic cooling and electrical/thermal codesign of architecture and microfluidic cooling is reviewed. Section 5 concludes the chapter with a summary of the existing work and identification of gaps and future needs in microfluidic cooling to accelerate its adoption.

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Single-Phase Convection

Single-phase liquid cooling of microsystems has received extensive attention due to its high thermal performance. This section reviews the characterization of thermal transport within microchannels and microgaps with micropin fin array enhancements. It also introduces various techniques to further improve the thermal performance. The challenges in implementation of single-phase liquid-cooling system are also discussed.

2.1

Microchannel Arrays

The microchannel heat sink cooling concept was first introduced by Tuckerman and Pease (1981). A rectangular microchannel heat sink in a 1  1-cm2 silicon wafer was fabricated. A Pyrex top cover was attached using anodic bonding to create a flow path. The microchannels had a width of 50 μm and a depth of 302 μm and were separated by a 50-μm-thick walls. Using deionized water as cooling fluid, the microchannel heat sink was capable of dissipating 790 W/cm2 with a maximum substrate temperature of 71  C above the water inlet temperature and a pressure drop of 214 kPa. Due to their inherent advantages, microchannel array heat sinks have received considerable attention (Samalam 1989; Harms et al. 1999; Rahman 2000; Ochende et al. 2007; Naphon and Khonseur 2008) since Tuckerman and Pease’s pioneering study. The thermal and hydraulic performance of microchannels can be studied by experiments, analysis, and numerical modeling. Numerical modeling is also an important tool for optimization of microchannels, as part of the evaluation of new designs. Qu et al. (2006) investigated flow development and pressure drop both experimentally and computationally for adiabatic single-phase water flow. The single rectangular microchannel was 222 μm wide, 694 μm deep, and 12 cm long, and Reynolds numbers ranged from 196 to 2,215. A microparticle image velocimetry system was used to measure velocity field at Reynolds number of 196 and 1,895 at several locations along the channel. A 3D computational model was constructed to provide a detailed description of liquid velocity in both the developing and fully developed regions. At high Reynolds numbers, sharp entrance effects produced pronounced vortices in the inlet region that had a profound influence on flow development downstream. The prediction of the computational model showed very good agreement with the measured velocity field and pressure drop. Therefore, the conventional Navier-Stokes equations can accurately predict liquid flow in microchannels and is a powerful tool for the design and analysis of microchannel heat sinks intended for electronic cooling. Lee et al. (2005) also validated the continuum theory with microchannel widths ranging from 194 to 534 μm and depth being five times the width in each case. The microchannel was made of copper and contained ten microchannels in parallel and deionized water was used as the coolant. The results showed an average difference of 5% between experiments and modeling for the Reynolds number ranging from approximately 300 to 3,500.

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1981

The effects of the channel height, width, and fin width on the heat transfer coefficient and pressure drop have been studied and well established for microchannels. So the focus in this section is on the various thermal enhancement techniques. Kou et al. (2008) built a 3D numerical model of a half microchannel to study the heat transfer characteristics for various channel heights and widths. The flow was assumed to be steady, incompressible, and fully developed. Constant fluid properties and uniform heat flux boundary condition were applied. Figure 5 shows the effects of the channel width on the thermal resistance when the pumping power is fixed at 0.01 W and 0.001 W. An optimum width of the channel can be observed. The optimum width is different for different pumping powers. The sensitivity to channel width is more prominent at low pumping power. Figure 6 shows the effect of microchannel height on the thermal resistance at pumping powers 0.001 W and 0.01 W. The total heat sink height is fixed in this study. Increasing the height of the microchannel decreases the substrate thickness. It

Fig. 5 Effects of channel width on thermal resistance when pumping power is 0.01 W and 0.001 W (Adapted from Kou et al. (2008))

Fig. 6 Effects of channel height on thermal resistance when pumping power is 0.01 W and 0.001 W (Adapted from Kou et al. (2008))

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was found that the thermal resistance monotonically decreases with the increase of the channel height. By using simulated annealing method, the global minimum thermal resistance for a given pumping power can be found. Another optimization method is the surrogate model optimization (Husain and Kim 2008), as seen in Fig. 7. The first step is to define the design function and design variables. Usually, the thermal resistance is the objective function. Two design variables, ratio of channel width to height and ratio of fin width to height, were selected for the optimization. Within the design variables’ range, a total of 16 design points chosen from four-level full factorial design were used to construct the surrogates. By solving Navier-Stokes and heat conduction equations, the thermal resistances at specified design points can be obtained and optimized using surrogate models. Three surrogate models, response surface approximation (RSA), Kriging (KRG), and radial basis neural network (RBNN), can be applied to predict the optimal design point. The assumption of fully developed flow in microchannel might not be valid. As noted by Lee et al. (2005), the entrance effects should not be neglected. Experimental data were compared with both fully developed and developing laminar flow prediction and significant deviations were seen. The chip width limits the total number of channels. The previous optimization work did not include these effects in their optimization. Wang et al. (2011) optimized the thermal resistance of a microchannel heat sink with bottom surface area 10  10 mm. Constant heat flux of 100 W/cm2 was applied in their model under developing flow conditions. Thermal resistance was defined as a function of number of channels, N; channel aspect ratio, α; and the ratio of channel width to pitch, β. The multi-parameter optimization approach evaluated the gradients of the thermal resistance function and established a conjugate direction for the updated search variables in each iteration. Under a constant pumping power of 0.05 W, the optimal design values are N = 71, α = 8.24, and β = 0.6, with a minimum overall thermal resistance of 0.144 K/W. One important finding was that the optimized geometry depends on the pumping power. As can be seen in Fig. 8, as pumping power increases from 0.005 to 1 W, the optimal number of channels, N, and optimal channel aspect ratio, α, increases, and the optimal ratio of the channel width to pitch, β, decreases. Increase in heat removal ability of the microchannel heat sink can be achieved either by increasing the heat transfer area or the heat transfer coefficient. To achieve

Fig. 7 Surrogate model optimization procedure (Adapted from Husain and Kim 2008)

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Fig. 8 Optimum geometric parameters for various pumping powers (Adapted from Wang et al. (2011))

Fig. 9 Double-layer microchannel heat sink

higher heat transfer area, multiple layers of microchannels can be stacked vertically (Xie et al. 2013b). Figure 9 shows a double-layer microchannel. The flow direction in each layer can be the same or in the opposite direction. A three-dimensional model was built to study the heat transfer and pressure drop characteristics of such double-layer microchannels under uniform heat flux dissipation. Compared to a single-layer microchannel, the heat transfer area of a double-layer microchannel is doubled for the same single-layer microchannel size. So under the same volumetric flow rate, the inlet velocity is half of a single-layer microchannel. Therefore, the pressure drop along the channel is reduced and thus the total pumping power. The modeling results showed that to achieve the same thermal resistance, the

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pressure drop of double-layer microchannel, either in parallel flow or counter flow, is much smaller than the single-layer microchannel. This indicated that double-layer microchannel can provide much better thermal performance than single layer. Parallelflow layout is better for heat dissipation at low pumping power, while counterflow layout is better at high pumping power. This is because for counter flow at low flow rate, the outlet fluid temperature in the first layer can be even higher than the solid temperature at the second layer. So heat can be conducted from the liquid to the solid and deteriorate the thermal performance. One advantage of the counterflow layer is the reduced temperature gradient due to the mutual heat transfer between the top and bottom channels. The channel height in each layer can be different. By keeping the total height of the heat sink constant, while adjusting the height in each layer, it was found that the thermal resistance also changes. This enables further optimization of double-layer microchannel. In a microchannel, the boundary layer develops as the fluid flows along it. Within laminar developing flow regime, the heat transfer coefficient increases with Re. But at the downstream region, the flow becomes fully developed, reaching a constant value of the Nusselt number. Techniques to further enhance heat transfer include increasing the flow mixing and promoting turbulence. Both passive and active flow enhancement techniques have been investigated. Passive heat transfer enhancement techniques include adding dimples, protrusions, and pins on the channel surface and using wavy channels, instead of straight channels. Figure 10 shows rectangular microchannel with dimpled bottom surfaces. Each microchannel is 50 μm deep and 200 μm wide (Wei et al. 2006). The in-line dimples with a depth of 20 μm and footprint diameter of 98 μm are etched at the bottom of the channel. A 3D CFD model was built to model one unit cell. Fully developed periodic velocity and temperature boundary conditions were applied at the inlet and outlet. Numerical modeling result revealed recirculating flow, and secondary flow patterns, and their development along the flow direction. Heat transfer augmentation (relative to a channel with smooth walls) is observed both on the bottom dimpled surface and on the sidewalls of the channel. The pressure drops in the laminar microscale flow were found to be either equivalent to or less than that in a smooth channel with no dimples. Therefore the dimples, as an effective passive heat transfer augmentation for macroscale channels, can be used for heat transfer enhancement inside microchannels.

Fig. 10 Microchannel with dimples

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1985

Flow characteristics and heat transfer performance in a rectangular microchannel with combination of dimples and protrusions (Fig. 11) were also studied with water (Lan et al. 2011). The results showed that incorporating dimples/protrusions in a microchannel has the potential to enhance heat transfer, with low pressure drop penalty. The normalized Nusselt numbers ranged from 1.12 to 4.77 and the corresponding normalized friction factors from 0.94 to 2.03. The thermal performance values showed that the dimple+protrusion cases perform better than the smooth surface with dimples only. On the upstream portion of dimple, a lower heat transfer region was produced because of flow separation. But on the downstream portion of the dimple and in the dimple wake, high heat transfer was observed because of flow reattachment and the vortices. Contrary to the dimples, a high heat transfer region occurred on the upstream portion of the protrusion, while low heat transfer region emerges on the downstream portion of the protrusion. In addition, the cases with smaller streamwise pitches perform better. The staggered cases perform better than the in-line ones. Another passive heat transfer enhancement technique in a rectangular straight microchannel heat sink is a bifurcation flow arrangement as shown in Fig. 12. The bifurcating flow increases the heat transfer by breaking up the boundary layer

Fig. 11 Microchannels with dimple/protrusion

Fig. 12 Straight microchannel heat sink with bifurcation flow arrangement

1986

Y. Joshi and Z. Wan

growth. Four different configurations with different bifurcation and length ratio were studied (Xie et al. 2013a). Results showed that the channel with bifurcation flow performed better than that of the corresponding straight microchannel. The microchannel with larger bifurcation ratio and length ratio provided better thermal performance with higher pressure drop. So by proper design of the bifurcation length ratio and number of channels, the overall thermal performance of the liquid-cooling microchannel heat sinks can be maximized. Adding pin fins within a microchannel also enhances heat transfer by increasing the surface area and enhances flow mixing. The performance of a microchannel with offset pin fins inside was compared with a typical microchannel with no pins (Shafeie et al. 2010). It was shown that using offset pin fins increases heat removal rate. However, the pressure drop is also significantly increased at the same time, which deteriorates the overall coefficient of performance for microchannel with this type of microstructure. The heat removal and pressure drop increase 22% and 83%, respectively. Wavy microchannel (Fig. 13) has also been studied due to the secondary flow and vortices, leading to chaotic advection, which can greatly enhance the convective fluid mixing and thus the heat transfer performance (Sui et al. 2010). A comparison with straight microchannels for the same cross section showed that wavy microchannels provide significant improvement in performance. In addition, the pressure drop penalty of the wavy microchannels is much smaller than the heat transfer enhancement. Furthermore, the wavy microchannel design is flexible in terms of relative wave amplitude and waviness. The wave amplitude of the microchannels can be varied along the flow direction for various practical purposes, without compromising their compactness and efficiency. By increasing the waviness along the flow direction, higher heat transfer performance can be achieved, increasing the temperature uniformity across the devices. Lastly, high waviness can be designed in the high heat flux regions to mitigate hot spot. One of the active heat transfer enhancement techniques is an impinging jet (Robinson 2009). The coolant is injected onto the hot surface from an orifice. The heat transfer performance of an impinging jet array was compared with microchannel (Robinson 2009). Both heat sinks were required to dissipate 250 W/cm2 while being maintained at a temperature of 85  C (Fig. 14). The modeling results showed that both the impinging jet array and microchannel heat sinks can meet the cooling requirements with pumping power less than 0.1 W. Microchannels achieve this cooling target at high pressure drop and low volumetric flow rate. On the contrary, impinging jet array heat sink requires a lower pressure drop and higher volumetric flow rate. For a cooling system, lower pressure drop is favorable with the concern of system reliability. Higher flow rate can reduce the bulk fluid temperature rise and thus increase the temperature uniformity across the device. Another benefit of impinging jet array is that the orifices can be concentrated on the high heat flux Fig. 13 Wavy microchannel

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1987

Fig. 14 Schematic of microchannel and impinging jet array

area, while microchannel is a global cooling strategy. From a practical engineering point, the cost of impinging jet array heat sink can be lower since the orifice plates can be manufactured at a reduced cost due to its simplicity. So impinging jet array heat sink can be the choice of future high-power electronic devices. The benefits of microchannels with high surface area and impinging jet arrays with high heat transfer performance can be combined to produce a hybrid cooling module in which a series of micro-jets deposit the coolant into each microchannel in a heat sink (Sung and Mudawar 2008). The coolant absorbs the heat and is expelled through both ends of the microchannel (Fig. 15). Three microjet patterns were studied: decreasing jet size (relative to center of channel), equal jet size, and increasing jet size. The heat transfer performance was evaluated with dielectric fluid HFE 7100 as the coolant. 3D numerical simulation using the standard k-e model predicted the experimentally measured wall temperatures well. The modeling results showed that the hybrid cooling module presented complex interactions between impinging jets and microchannel flow. When increasing the coolant flow rate, the heat transfer performance was enhanced, and thus wall temperature is decreased. However, this benefit is achieved at the penalty of greater wall temperature gradients. The comparison of the three patterns showed that decreasing-jet-size pattern yields the highest convective heat transfer coefficients and thus lowest wall temperatures, while the equal-jet-size pattern provides lowest temperature gradient. The increasing-jet-size pattern resulted in greater wall temperature gradients. This is because the impingement from larger jets near the channel outlets blocks the spent fluid flow. In the previous hybrid cooling module, high-temperature gradient occurs as the fluid flows to the two ends. A modified high-performance ultrathin manifold microchannel heat sink is shown in Fig. 16 (Escher et al. 2010). The heat sink consists of impinging liquid slot jets on a structured surface fed with liquid coolant by an overlying two-dimensional manifold. The cool liquid is injected into the slot and returned immediately to the upper layer through nearby slots, which significantly reduce the bulk fluid temperature rise. It was found that a design with 12.5 manifold systems and 25-μm-wide microchannels as the heat transfer structure achieved

1988

Y. Joshi and Z. Wan

Fig. 15 Hybrid cooling module. (a) Equal jet size. (b) Decreasing jet size. (c) Increasing jet size

Liquid

Liquid

Fig. 16 Schematic of a manifold microchannel heat sink

Liquid

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1989

optimum performance. The chip size is 2 cm and 2 cm and water is the coolant. The thermal test chip is 500 μm thick and the microchannel is 300 μm deep, while the total height of the heat sink does not exceed 2 mm. With a volumetric flow rate of 1.3 l/min, a total thermal resistance between the maximum heater temperature and fluid inlet temperature of 0.09 cm2K/W was achieved with a pressure drop of 0.22 bar. So maximum temperature difference between the chip and the fluid inlet is 65 K when the power density is more than 700 W/cm2. The high performance of the manifold structure allows for elevated inlet temperature of up to 70  C. So for a flow rate of 1 l/min and heat flux 100 W/cm2, the maximum chip temperature can be maintained at 81  C. The low thermal resistance and the tolerable high inlet temperatures promise the possibility of eliminating the need for chillers. In addition, the rise of the inlet temperature to 70  C enhances the overall efficiency of the heat sink by more than 40%. Hence, not only the energy for the chiller can be saved but also decrease the total pressure drop across the heat sink and thus increase the overall performance of the cooling system. The performance of the manifold heat sink can be further enhanced by lowering the hydrodynamic resistance of the heat sink. This can be achieved by decreasing the nozzle length and by increasing the manifold channel height. The reduction in the hydrodynamic resistance of the manifold results in higher numbers of manifold systems, reducing the flow path length through the heat transfer structure. Therefore, narrower channel widths can be designed and further decrease the total thermal resistance of the heat sink. Another modified design of the microjet array is parallel inlet and outlet manifolds with a distributed return (Brunschwiler et al. 2006). The nozzle to heater gap size ranges from 3 to 300 μm. Pressure drops less than 0.1 bar at 2.5 l/min flow rate have been obtained with a hierarchical treelike double-branching manifold. An optimal heat-removal rate of 420 W/cm2 using water as a coolant was achieved through parametric study. For a near-optimal design with a gap to inlet diameter ratio of 1.2, a heat transfer coefficient of 8.7 W/cm2 K and a junction to inlet fluid unit thermal resistance of 0.17 Kcm2/W were obtained. This allows for 370 W/cm2 cooling performance at a junction to inlet fluid temperature rise of 63  C, a pressure drop of 35 kPa, and a flow rate of 2.5 l/min. High heat transfer performance in a microchannel can also be achieved by using high conductivity substrate material. Due to the superior thermal conductivity of individual carbon nanotube (CNT) in the axial direction, CNT arrays have been explored as an effective material for the thermal management of MEMS-based devices by introducing CNT into the microchannel (Ekstrand et al. 2005). A reduction of thermal resistance of the microchannel from 0.98 K/W to about 0.43 K/W was obtained when CNT array fins were used. One concern about the CNT is its strength. By utilizing a metal-enhanced CNT transfer technique, the interface between the CNTs and the chip surface was improved by minimizing the thermal contact resistance and promoting the mechanical strength of the microfins. A heat flux of 1,000 W/cm2 with water was projected due to the increased heat exchange area of the CNT microfins (Fu et al. 2012).

1990

Y. Joshi and Z. Wan

The bulk fluid temperature rise in microchannel results in high chip temperature nonuniformity. To reduce the bulk fluid temperature rise, a split-flow channel (Fig. 17) was proposed (Tan et al. 2011). The advantages of the split-flow arrangement over the single-pass arrangement are (1) for the same pumping power, the total flow rate can be doubled, while the pressure drop halved due to the reduced flow path, and (2) the bulk fluid temperature rise along the microchannel is also halved if a uniform heat flux is assumed. By designing the heat sink with a supply plenum tapering downstream, an even flow distribution for removing the heat away from the die can be obtained to further improve the thermal performance. With such a microchannel array arrangement, a thermal resistance of 0.15 K/W and lower temperature variation on die can be obtained. The lower pressure drop due to the shorter flow length allows integrated pumps to operate either at smaller sizes or higher flow rates. Kandlikar and Upadhe’s experiment with the split microchannel showed dissipating heat fluxes beyond 300 W/cm2 using water as the coolant with a pressure drop of around 35 kPa (Kandlikar and Upadhye 2005). Tree-shaped microchannel patterns (Fig. 18) have been pursued to improve the chip temperature uniformity (Wang et al. 2007). The effect of the bifurcation angles in the constructal nets on the fluid flow and heat transfer characteristics of such network were studied using a three-dimensional computational fluid dynamics approach. Results show that the bifurcation angle is an important factor for such cooling nets. Surface temperature distributions and pressure drop along the flow paths are analyzed and compared. For the same boundary conditions, a lower temperature and pressure variation are observed at lower bifurcation angles.

Fig. 17 Split-flow design

Fig. 18 Bifurcating treeshaped microchannel

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Fig. 19 Bifurcating manifold

The hydrodynamic and thermal performance of a bifurcating (Fig. 19) and a parallel channel network branching from a single manifold channel were compared (Escher et al. 2009). Although it has larger temperature nonuniformity for the parallel microchannel, it has also several advantages: (1) the parallel channel is much more densely packed than the bifurcating manifold and distributes the coolant more efficiently to a larger heat transfer surface area; (2) for a constant flow rate, the parallel channel network achieves more than 5 higher coefficient of performance than the bifurcating manifold, while almost 4 more heat can be removed for a constant pressure gradient; and (3) parallel channel design is less complicated and cheaper compared to the bifurcating manifold. The parallel channel is integrated into a single plane, while the bifurcating design needs a second plane for fluid return.

2.2

Micropin-Enhanced Microgaps

Significant temperature variations across the chip persist for conventional singlepass parallel-flow microchannel heat sink, since the heat transfer performance deteriorates in the flow direction in microchannels, as the coolant heats up. Recent advancement in micromachining techniques allows more complex 3D microsize geometries to be fabricated directly into the high thermal conductivity materials that can be used as the substrates for miniature heat sinks. This makes it possible to explore structures that may be more effective in heat transfer enhancement than the parallel microchannels and can provide better temperature uniformity. One such enhanced structure is micropin fin arrays (Fig. 20), with pin characteristic dimensions of tens to hundreds of micrometers and height (Hp) to diameter (Dp) ratio from 0.5 to 8. The flow disruptions provided by the separated pin fins increase the flow mixing and can also serve to break up the boundary layer (Steinke and Kandlikar 2004). Thermal design and performance assessment of a micropin fin heat sink require a fundamental understanding and accurate prediction of flow and heat transfer in

1992

Y. Joshi and Z. Wan

Fig. 20 Schematic of staggered micropin fin array

microsize short pin fin arrays. The thermal-hydraulic performance of micropin fin arrays and microchannel was compared in terms of total thermal resistance (Rtot) by Peles et al. (2005). At an inlet pressure of two atmospheres, the minimum Rtot was 0.0389 K/W with water, which corresponded to 7.8  C maximum wall temperature raise for 200 W/cm2 heat flux, and 30.7  C at 790 W/cm2, compared to 71  C for microchannel cooling (Tuckerman and Pease 1981). A recent study by Jasperson et al. (2010) showed that the flow rate was a factor in determining whether microchannels or micropin fin arrays have better performance. In their study, microchannels and micropin fin arrays of same height and width (670 μm and 200 μm) were made of copper. Under a mass flow rate from 30 to 90 g/min, the pressure drop of micropin fin heat sink was always higher than that of microchannel heat sink, the difference increasing with the flow rate. The convection thermal resistance, Rconv, of micropin fin heat sink was higher than that of microchannel heat sink at flow rate less than 60 g/min and lower above 60 g/min. The micropin fin arrays can be arranged in in-line or staggered configurations. Kosar et al. (2005) experimentally studied and compared the heat transfer coefficient associated with the forced flow of deionized water over staggered and in-line circular micropin fins. The Dp was 100 μm, Hp was 100 μm, and longitudinal (SL) and transverse (ST) spacings were both 150 μm. Under the same Re, the staggered configuration resulted in higher heat transfer coefficient than the in-line configuration. A more recent comparison study by Brunschwiler et al. (2009) showed that under the same flow rate, in-line pin fin showed lower heat transfer coefficient and higher thermal resistance than staggered pin fin. Another interesting finding was that in-line pin fin presented a flow regime transition, which manifested itself as an abrupt pressure gradient change and a local heat transfer maximum. The transition moved toward the inlet at increasing flow rate. Due to the superior performance of staggered micropin fin arrays, considerable research has been done to study its thermal and hydraulic characteristics, including the effects of Re, ST, SL, Hp, pin shape, and tip clearance (tc).

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2.2.1 Effect of Re Due to the small pin fin dimensions at the microscale, the flow regime is expected to be predominantly laminar. For a specific pin fin configuration, the heat transfer coefficient (h) and friction factor increase with Re. The nondimensional friction factor (f) could be defined as (Kosar and Peles 2007): f fan ¼

2ΔPρf N L G2

(1)

where ρf is the fluid density, NL is the number of pins in the longitudinal direction, and G is the maximum mass flux at the smallest cross-sectional area. The friction factor is comprised of two components, one accounting for the drag due to flow separation and the other stemming from shear stress (Kosar and Peles 2007). As the fluid flows across the micropin fins, a thin boundary layer is formed at the pin surface. As Re increases, the boundary layer becomes thinner, and flow separation is enhanced (Kosar and Peles 2007). Kosar et al. (2005) experimentally studied the pressure drop and f of circular micropin fin arrays. Their results showed that f decreased with increasing Re, which was also observed by Prasher et al. (2007), Short et al. (2002b), and Tullius et al. (2012). A change in the relationship of f to Re was observed by both Kosar et al. (2005) and Prasher et al. (2007), showing that f was very sensitive to Re for Re < 100 and was less sensitive to Re for Re > 100. It was believed that a flow pattern transition occurred around Re = 100. The Nusselt number (Nu) increased with Re, which was also observed by Prasher et al. (2007), Tullius et al. (2012), and Short et al. (2002a). Similarly, two distinct regions of the Nu dependency on the Re separated by Re = 100 have been identified (Kosar et al. 2005; Kosar and Peles 2007). Kosar and Peles (2006a, b) attributed such dependency to two factors: endwall effects and a delay in flow separation. Flow separation was assumed to control the transition Re.

2.2.2 Effect of Spacing The effect of longitudinal spacing was evident when comparing the friction factors of micropin fin array with SL = 150 μm and SL = 350 μm (Prasher et al. 2007). Device with SL = 150 μm had larger f than device with SL = 350 μm, which suggested that densely populated pin fins lead to higher f. This was because that wakes behind pin fins formed due to flow separation. For densely populated pin fins, the wake generated downstream of a pin fin interacted more strongly with pin fins in the following row. As a result, f were higher for closely packed objects. This was also observed by Prasher et al. (2007) and Short et al. (2002b). However, Tullius et al. (2012) showed an opposite trend. h of device SL = 150 um were greater than for device SL = 350 μm over Re ranging from 30 to 112 (Kosar and Peles 2007). This may be due to the pronounced wake-pin fin interaction in SL = 150 μm. Because of the dense spacing, the wake formed downstream a pin fin may interact with the pin fins in the following row, so that mixing and heat transfer were enhanced. This was reflected as higher h in SL = 150 μm. Since at low Re, the wake-pin fin interaction is moderated, deviations between Nu of the two devices diminished. Results of Prasher

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Y. Joshi and Z. Wan

et al. (2007) agreed with observation in (Kosar and Peles 2007). However, Both Tullius et al. (2012) and Short et al. (2002a) showed that sparse pins had larger h. Very little work studied the effect of ST on f and Nu. Prasher et al. (2007) and Short et al. (2002b) reported an increasing f with decreasing ST, while an opposite trend was presented by Tullius et al. (2012). Both Tullius et al. (2012) and Short et al. (2002a) found that Nu increased with increasing ST.

2.2.3 Effect of Aspect Ratio (Hp/Dp) The ratio of Hp to Dp has a significant effect on the f. It was reported that the micropin fin arrays with lower HP/DP ratio produced higher f at the same pin densities and Re (Kosar et al. 2005). However, The HP/DP ratio effect reduced with increasing Re (Kosar et al. 2005). This was attributed to the endwall effect. Small aspect ratio devices were more affected by the hydrodynamic boundary layer imposed by the endwalls (wall pin interaction), resulting in increased viscous shear forces, and therefore larger f were obtained (Kosar et al. 2011). The effect of pin fin Hp to Dp aspect ratio was significant for low Re and diminished at larger Re because of a thinning hydrodynamic boundary layer. Therefore, with the increase in Re, the deviation between smaller and larger Hp to Dp pillars moderated. The observation in (Kosar et al. 2005, 2011) agreed with Prasher et al. (2007) for Re < 100 and Short et al. (2002b). However, f was reported to be increased with aspect ratio for Re > 100 (Prasher et al. 2007; Tullius et al. 2012). The dependence of the Nu on the Re was considerably more notable than for long pin fin as a result of endwall effects (Kosar et al. 2005) since the endwall effects were significant at a low Re flow over short pin fins (1/2 < HP/DP < 6). Two intrinsically coupled physical factors adversely affected h at 10 < Re < 100 for flow over micropin fins: the thermal and hydrodynamic boundary layer established at the endwall and a delay in flow separation to higher Re. The suppression of h was amplified with the reduction of H/D ratio and was more evident at low Re (Kosar and Peles 2006a). The results of Short et al. (2002a) showed that Nu decreased with increased aspect ratio. However, Koz et al. (2011) did a parametric study on the effect of endwalls on heat transfer and found that Nu increased with Re but not necessarily with HP/DP. Nu of various HP/ DP ratios can show different trends when plotted against Re. Tullius et al. (2012) found that Nu increased with aspect ratio.

2.2.4 Effect of Tip Clearance The clearance (tc) between the pin tip and wall was believed to affect the overall hydraulic and thermal performance of a given cooling system (Sparrow et al. 1980). Introducing tip clearance resulted in a decrease in form drag and an increase in shear stresses at the walls (Rozati et al. 2008). The balance between the two determined the overall increase or decrease in f. A slight increase in f with tc/DP = 0.1 at high Re was observed (Sparrow et al. 1980). As the tc increased further to tc/DP = 0.3 and 0.4, f was decreased. The reduction in f with increasing tc was much more evident at

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higher Re. Moores et al. (2009) also found that the overall pressure drop is initially increased with the introduction of tip clearances for tc/DP < 0.1 and then decreased. The pin fins tend to have relatively low height to diameter ratio due to the high h. So the pin tip area can represent a considerable portion of the total area of the array. Exposing the tips of the pin fins to the cooling fluid can increase the total heat transfer area. Also the tip clearance introduces a three-dimensional behavior into the flow field around the pins and may change the local heat transfer rate. Moores and Joshi (2003) reported an increase of heat transfer rate with the introduction of tc and attributed the enhancement primarily to the additional surface area exposed to the cooling fluid. Rozati et al. (2008) believed that tc affected h by eliminating viscositydominated endwall effects on the pin, by eliminating the pin wake shadow on the endwalls, by inducing accelerated flow in the vicinity of the top wall and the pin top, by reducing or impeding the development of the recirculating wakes, and by redistributing the flow along the height of the channel. For a tip gap of tc/DP = 0.1, Nu decreased sharply. As tc/DP continued to increase, the Nu increased (Rozati et al. 2008).

2.2.5 Effect of Pin Shape Advanced microfabrication technologies enable researchers to explore the characteristics of micropin fins with different shapes besides circular pins. The pins can be divided into two categories: streamlined shapes, such as hydrofoils, and unstreamlined ones including circular, square, diamond, cone, triangle, hexagon, and ellipse. Heat transfer characteristics of square micropin fins were experimentally studied by Siu-Ho et al. (2007) and Liu et al. (2011). They found that both heat transfer coefficient and pressure drop increased with increasing Re. Also, h was higher near the heat sink inlet and decreased along the flow direction. This trend may be caused by entrance and streamwise heat conduction effects (Siu-Ho et al. 2007). A friction factor transition phenomenon appeared at Re ~ 300 (Liu et al. 2011). Micropin fins having sharp edges generated higher h than streamlined pin fins (Kosar and Peles 2007). This was associated with separation effects mitigated by sharp edges, as well as extended wake regions, which increased the mixing and heat transfer. However, sharp-edged pins also resulted in larger f, at the same Re. The sharp edges enhance wake-pin interactions, causing larger pressure drops. A comparison between circular, diamond, and square micropin fins with same hydraulic diameter, spacing, and height by Kosar et al. (2005) and Mita et al. (2011) showed that for the same Re, circular micropin fins had smallest f. Hydrofoil pin fins with zero angle of attack resulted in considerably lower f compared to circular pins, especially at high Re (Kosar et al. 2011), around 7.5 times lower at a Re of 1,000. The difference in f continued to amplify as the Re increased above 1,000, primarily because crossflow over circular-shaped fins transitioned to turbulent flow, where the friction factor is less dependent on Re, while flow over the hydrofoil fins was well within the laminar flow regime. Therefore, the merits of using micropin fin device are dependent on the performance evaluation criterion used, as well as on the hydrodynamic conditions. In general, for a fixed pressure drop and pumping

1996

Y. Joshi and Z. Wan

power, utilizing streamlined pin fin heat sink is favored at moderate pressure drops and flow rate, while for very high and lower pressure drops and flow rates, pin fins promoting flow separation should be favored. For fixed mass flow rate, streamlined pin fins provided inferior performance.

2.3

System Configuration and Design Consideration

Attention was focused on the performance of the microchannel itself in the previous section. However, the successful integration of the microchannel heat sink in the overall cooling system is equally important but has received limited attention. Figure 21 shows a typical liquid-cooling system. It consists of pump, controlled temperature bath, flow meter, filter, metering valve, pressure transducer, thermocouple, microchannel heat sink, air-cooled heat sink, and reservoir. The pump circulates the fluid through the microchannel heat sink. A filter is used to keep the inlet water clean and prevent clogging of the microchannels. An air-cooled remote heat sink is used to reject the heat to the ambient. The reservoir is used to store excess fluid inventory to make up any deficit over extended operation periods. Based on the thermal and pressure drop performance of the microchannel heat sink, and the chip power dissipation, the necessary flow rate within the cooling system can be determined. Next, the remote heat sink, pump, and reservoir can be selected based on the flow rate. The component materials must be compatible with the coolant, which can be either water, often with additives, or dielectric fluid, depending on the applications and the dissipated heat. If the microchannel heat sink is used as an externally attached coldplate, metals such as copper can be used to fabricate it, and it can withstand higher pressures than a silicon microchannel heat

Fig. 21 Typical liquid-cooling system

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sink. Typically, water which has excellent convective heat transfer characteristics can be used as the coolant. For on-chip microchannels, dielectric fluid may often be required, to mitigate any danger of electrical shorts. Before the coolant is charged into the flow loop, it needs to be degassed, and the flow loop vacuumed. Noncondensable gas in the loop can cause malfunction of the pump and decrease the flow rate. It can also degrade the convective heat transfer coefficient. A chip such as a microprocessor may work continuously for several years. Therefore, a highly reliable cooling system is required. First, the flow loop should be leakage-free. Any leakage of the liquid will reduce the amount of the coolant in the flow loop and eventually lead to failure of the processor. Secondly, the flow loop should be very clean. A filter is usually placed upstream of the microchannel heat sink. If the pore size of the filter is too small, the filter can be blocked by contaminant particles in the loop, reducing the flow rate. Thirdly, the microchannel heat sink should be designed to withstand the mechanical stresses resulting from the fluid flow. High pressure can cause cracking of the microchannels in semiconductor materials such as Si. So the microchannel heat sink should be carefully designed to reduce the pressure drop. In addition, a compact cooling system is required. Although the size of the microchannel heat sink is much smaller than conventional air-cooled heat sink due to its high performance, the other components typically increase the size and complexity of the whole system. Lastly, the cost of the liquidcooling system must be considered. A liquid-cooling system is usually more expensive than an air-cooled heat sink. The performance benefit gained by the liquidcooling system must exceed the extra cost due to the liquid-cooling system. Among many, the most prominent but often ignored challenge is the long-term reliability of the system. In order to be practical, the thermal management system must maintain its cooling efficiency for the entire duration of service time. Any physical or chemical degradations of the coolant itself or the components of the system can adversely impact the cooling efficiency and, therefore, must be prevented (Chang et al. 2006). In this regard, corrosion is particularly problematic for liquidbased thermal management for microsystems with an integrated microscale heat exchange system. While water as a coolant provides superior heat transfer properties, it tends to corrode components of the thermal management system, particularly in the presence of additives such as antifreeze. Additives must not seriously impair the desirable thermal transport properties of water, such as low viscosity and high heat capability, while protecting it from freezing and posing no serious health hazard. Majority of antifreeze additives are based on ionic salts, most notably calcium chloride, potassium acetate, and sodium formate. Their addition increases the chemical activity of the coolant, thereby the probability of corrosion. Secondly, the ratio of surface area to volume can reach as high as 1cm2/ml, which means that even a slight amount of corrosion can seriously affect the basic properties of the coolant and result in an acceleration of corrosion. Finally, corrosion products can significantly endanger the system reliability by causing clogging and thus rapidly decreasing the cooling efficiency. It is very important that a complete assessment of the corrosion risk for a chosen coolant is conducted before implementation of liquid-cooling system. Equally

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Y. Joshi and Z. Wan

important is to find suitable coolant additives that can effectively prohibit or retard the corrosion.

3

Thermal/Electrical Codesign of Microfluidic Cooled 3D ICs

3D ICs can significantly increase the processor performance by vertical stacking of chips. However, when the chips are stacked, the heat flux per unit surface area is increased, and the interior layers in the stack are susceptible to overheating. On-chip liquid cooling is a promising choice for thermal management of 3D ICs. The heat from the chip is removed by individual liquid-cooling layers and therefore the number of chips stacked is not limited. A lot of work has been done to study the thermal performance of 3D ICs with liquid cooling, experimentally and numerically.

3.1

Experiments

Sekar et al. (2008) presented a 3D IC technology with integrated microchannel cooling (Fig. 22). Fluidic interconnect network fabrication proceeds at the wafer level, is compatible with CMOS processing and flip-chip assembly, and requires four lithography steps. The electrical signal vias are embedded in the fins of microgaps. Experimental results show that the integrated microchannel heat sinks enable cooling of power density of more than 100 W/cm2 of each chip tier. A key characteristic of this platform is the ability to assemble chips with electrical and fluidic I/Os and seal fluidic interconnections at each stratum interface (King et al. 2010).

Fig. 22 3D ICs with integrated microchannel

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1999

Fig. 23 Integrated cooling solution for a compact 3D silicon module

Tang et al. (2010) proposed an integrated cooling solution for a compact 3D silicon module (Fig. 23), which spreads the heat over a larger coldplate heat exchanger, where conventional air-cooling suffices to reject the heat. The system is designed such that no external fluidic interconnects are required to assemble the package on the mother board. Two silicon carriers with embedded fluidic microchannels are stacked vertically, and each carrier is mounted with a chip dissipating 100 W. At a high flow rate (400 ml/min) for a single carrier, the maximum pressure drop in the carrier is 90 kPa. The sealing technique for the fluidic path worked well and no leakage was observed. The pressure drop can be further reduced by the optimized design and a decrease of 30–50% was obtained, as compared to a samesized convectional carrier. The simulation results showed that the thermal resistance of the optimized carrier and the conventional carrier are each about 0.18 K/W. However, the variation of the junction temperature for the optimized carrier is 8.1 K, which is much lower than 14.1 K for the conventional carrier.

3.2

Modeling

Numerical modeling enables rapid evaluation of thermal performance, and therefore a significant work has been done to characterize the performance of 3D ICs with integrated liquid cooling. Koo et al. (2005) presented a 3D circuit network to study the effect of logic and memory layout on the thermal performance. Their results show that integrated microchannel cooling can be scaled up to multiple tiers. Aravind presented 3D ICE (Sridhar et al. 2010), a compact transient model for the thermal simulation of 3D ICs with multiple inter-tier microchannel liquid cooling.

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The model is compatible with existing thermal CAD tools for ICs and demonstrated significant speedup (up to 975) over typical commercial computational fluid dynamics (CFD) simulation tool, while preserving reasonable accuracy. Alfieri et al. (2010) developed a multiscale heat transfer model for integrated water cooling of a four-tier chip stack with a footprint of 1 cm2. The cooling structure, which consists of microgaps with cylindrical pin fin array, can be directly integrated with the device layout in multilayer chips. The total power in the stack is 390 W, corresponding to about 1.3 kW/cm3 volumetric heat dissipation. A temperature gradient from maximal junction temperature to liquid inlet was 60 K. The computational resource and complexity of detailed computational fluid dynamics (CFD) and heat transfer modeling in full stacked chips with integrated cooling can be expensive. Therefore, a porous medium approach was utilized to model the 3D structure. The model parameters of heat transfer and hydrodynamic resistance are derived by averaging the results of the detailed 3D-CFD simulations of a single streamwise row of fins. Variable properties due to temperature gradients are considered. The porous medium approach with a local thermal nonequilibrium, as well as orthotropic heat conduction and hydrodynamic resistance, is combined to capture the heat transfer in the cooling sublayers. The improved porous media model reproduces the temperature measured in the stack within 10% while lowering the computing costs for a 3D chip stack nearly 8,000 times compared to those estimated for modeling the detailed geometry containing 20,000 pins in the chip stack with four cooling layers. In 3D chip stacks, the chip architecture design may produce a variety of different hot spot regions. The hot spot can limit the overall performance of 3D ICs. The micropin fin array offers the flexibility of various pin diameters for different heat flux regions. Figure 24 shows the schematic of 3D chip stack package with integrated cooling, which consists of a row of pins with various diameters and regions with hot spot (Alfieri et al. 2013). It was found that the local Nusselt number

Fig. 24 Schematic of integrated liquid cooling of 3D ICs with hot spots

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(Nu) profiles along the channel were affected by the hot spot significantly. The positive peaks of Nu at the position of the hot spot were observed and revealed a local heat transfer enhancement, which indicates that the sudden rise in the heat flux reduces the temperature difference between the liquid and the wall. This effect is similar to the reason that causes the Nu to be very high at the inlet, where the local temperature difference between the wall and fluid is very small. For the same reason, as the heat flux decreases from high to low, the Nu also decreases. The results also show that at the interface between hot spot and background heating, the local Nu is even lower than that of only a background heat flux. As the flow develops at the new imposed conditions, Nu then slowly approaches the corresponding fully developed value again. Since the hot spot can enhance the Nu, the total heat transfer can be improved by placing the hot spot near the outlet where the Nu for background heat flux is usually low due to the fully developed flow. On the other hand, one might choose to place the hot spot near the inlet to take the advantage of high Nu and thus limit the hot spot temperature. The effects of nonuniform pin diameter are also studied. By placing the larger pin at the hot spot region, the heat transfer area is increased, and also the flow around the pin is accelerated. Result shows that increasing the pin diameter 2 at the positions where the hot spot was operating resulted in a rise in the Nusselt number of up to ~30%, depending on the chosen configuration. 3D chip stack can also mitigate the hot spots on individual dies by strong interlayer heat conduction if the relative position of the hot spots is selected carefully to result in a heat load and flow which are well balanced laterally (Madhour et al. 2014). To manage hot spots, several techniques have been proposed: (1) nonuniform distribution of microchannels (Shi et al. 2011); more channels were placed above the hot spot area, while fewer channels above the low-power area. In this case, the pumping power efficiency can be increased. (2) Workload dynamic management (Coskun et al. 2009), both reactive and proactive methods, can be employed. In fixed reactive, the workload scheduler directs the incoming job to the coolest core on the die. On the other hand, fixed proactive method forecasts future temperature to project hot core and cool core and, then based on this moves workload from projected hotter cores to cores that are projected to be cooler. Due to the delay of temperature response, proactive management can reduce and balance the temperature on the die more effectively compared to fixed reactive. (3) Dynamic flow rate control (Coskun et al. 2009); closed-loop control was used to adjust the coolant flow rate to meet the temperature characteristics of the system. The flow rate can be adjusted based on the maximum temperature observed during the last measurement interval or based on the forecast. Heat removal and power delivery are two major design challenges in 3D stacked IC (Lee and Lim 2008). These thermal and power/ground interconnects, together with those used for signal delivery, compete for routing resources, including various types of through-silicon vias (TSVs). The routing with various interconnects in 3D was studied. By increasing the microfluidic channel (MFC) depth, mass flow rate and thus cooling capability can be increased. However, the die thickness is also increased, and with the fixed aspect ratio of TSVs, the diameter of signal TSVs gets

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increased proportionally. Thus, the routing capacity between dies, aggravating signal net routability, is decreased. Increasing the MFC width also improves mass flow rate and cooling capability, but the space available for signal TSVs will be decreased. By increasing the MFC pitch, total heat transfer area between the channel wall and the coolant will decrease, and thus the cooling performance is degraded, but the routability of TSVs will be improved. The thickness of integrated microchannel can affect the performance of the electrical TSVs (Zhang et al. 2011a). The TSV capacitance increases linearly with the thickness of the die. For example, assuming aspect ratio limited to 20:1, Ctsv for a 200 μm micropin fin design is 272.6 fF, while that of a 400 μm micropin fin design, as optimized, is 847.2 fF. If the TSV aspect ratio keeps increasing, the TSV capacitance continuously decreases. A heat sink design without consideration of TSV performance can greatly diminish the advantages of 3D ICs. Leakage power of the transistors amounts to be a significant part of the total power consumption but it does not contribute to the computing. The leakage power has been measured and compared under three different cooling conditions: natural air cooling, forced convection cooling, and liquid cooling (Wan et al. 2014b). The liquid cooling is proved to be effective to reduce the leakage power significantly. Wan et al. (2014a) built a simulation framework which can simulate the power map and temperature map simultaneously under real applications. Their modeling results also confirmed that the leakage power needs to be included in the simulation.

4

Flow Boiling in Microgaps

A key limitation of single-phase cooling is the bulk fluid temperature rise along the flow direction due to the sensible heating, which results in temperature nonuniformity on the chip. Flow boiling is an alternative thermal management approach for which the bulk fluid temperature depends on the saturation pressure. It can achieve higher heat transfer coefficients than in single-phase flow, so reduced fluid flow rates may be required (Bakir and Meindl 2008). The resulting smaller pressure drop can result in higher surface temperature uniformity. Two-phase microgap heat sinks that utilize arrays of microsize pin fins as internal heat transfer enhancement structures have recently emerged as a promising alternative to the popular two-phase microchannel array heat sinks to meet the future high heat flux electronic cooling needs (Wan et al. 2014a). This section will briefly introduce the mechanism of flow boiling in microchannel, with a focus on flow boiling, and the various flow boiling enhancement techniques.

4.1

Characteristics of Pure Fluid Flow Boiling

The regimes of nucleate boiling and two-phase forced convection govern saturated flow boiling heat transfer (Qu and Siu-Ho 2009). Nucleate boiling is associated with bubbly and slug flow. In this regime, liquid near the heated surface is superheated to a sufficient degree to sustain nucleation. The heat transfer coefficient is dependent

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upon heat flux, but fairly independent of mass velocity and vapor quality. The general trend is increasing h with increasing heat flux due to intensification of nucleation. The regime dominated by two-phase forced convection, on the other hand, is often associated with annular flow. In this regime, nucleation is suppressed along the heated surface, and heat is transferred mainly by conduction across the liquid film and evaporation at the liquid-vapor interface. As such, h is dependent upon mass velocity and vapor quality, but less sensitive to heat flux. The general trend is increasing h with increasing mass velocity and vapor quality due to reduction in liquid film thickness along the heated surface. One big disadvantage of flow boiling compared to single-phase liquid cooling is the instability (Muwanga et al. 2007). Qu and Mudawar conducted flow boiling experiments with two Si microchannels (Qu and Mudawar 2004). A high-speed camera was used to determine dominant flow regimes and characterize hydrodynamic instabilities. When gradually increasing the heat dissipation to the incipient boiling heat flux, few nucleation sites were observed within the microchannels. When further increasing the heater power, the flow abruptly transited to intermittent two-phase flow. At this regime, the boiling two-phase boundary, interface between the single-phase liquid region and two-phase mixture region, of all the channels oscillated back and forth between the channel inlet and outlet. For some severe cases, this oscillation can even result in vapor entering the inlet plenum. During the oscillation, the inlet and exit pressures, as well as the heat sink temperatures, fluctuated significantly, and the pump could not deliver a constant flow rate. This instability was classified as pressure drop oscillation, which is very undesirable for two-phase microchannel heat sink operation, since it not only leads to large amplitude of pressure and temperature fluctuations but also reduces critical heat flux (CHF). Premature CHF appeared at a much lower heat flux for instable two-phase boiling compared to the CHF at steady flow boiling. Fortunately, their experiments proved the pressure drop oscillation can be suppressed by throttling the control valve situated immediately upstream of the test module, which helped increase the system’s stiffness (Qu and Mudawar 2004). Another issue with flow boiling is the nonuniform heating (Bogojevic et al. 2011). The boiling usually starts first at high heat flux area. Since the flow resistance of the vapor region is much higher than that of single-phase liquid region, the fluid flow will bypass the high heat flux region, which further deteriorates the cooling of the high heat flux region. The location of the hot spot affects the flow pattern, pressure, and temperature distribution. When the power map changed from one pattern to another, the flow boiling pattern, pressure, and temperature also changed, which further increased the instability of flow boiling. To prevent the instability, including a throttling valve and inserting an orifice upstream of the microchannel are proposed. The general idea is to produce a high pressure drop immediately upstream of the microchannel. This increases the stiffness of the system and damps the instability. The validity of inlet orifices on saturated CHF in multi-microchannel heat sinks was studied (Park et al. 2009). Two different multi-microchannel heat sinks made in copper were tested: one had 20 parallel rectangular microchannels of 467  4,052 μm (width  depth), and the other had

2004

Y. Joshi and Z. Wan

29 channels of 199  756 μm (width  depth). Three low pressure refrigerants (R134a, R236fa, R245fa) were used as the coolant. Flow visualization was conducted with and without an orifice insert at the inlet of microchannels, which demonstrated the existence of flow fluctuation, backflow, and nonuniform distribution of flow among the channels when there is no orifice insert. When the boiling starts and bubbles are generated and grow, they occupy the microchannel, and the pressure inside the channel increases quickly and pushes the adjacent fluid toward both downstream and upstream. If there is not a high pressure gradient at the inlet of the channels, the flow may easily go back into the inlet plenum. Afterward, the bubbles are observed, to be merged in the inlet plenum, and broke up again to flow back into the channels. With the inlet orifice, instabilities such as backflow and thus premature CHF were successfully prevented. But the inlet orifice affected the flow behavior inside the channels by creating a jet flow. When boiling starts, the bubbles generated grow and push the neighboring fluid. While a part of the flow goes downstream, another part of fluid goes back toward the upstream of the channel. The inlet orifice will block the backflow due to its high pressure gradient and thus the backflow will merge with the continuing incoming flow due to the vapor generation. At the same time, the jet flow from the inlet orifice encounters the backflow, forming a recirculation loop, which was quite stable and did not create flow maldistribution between channels or premature CHF. When continue to increase the heat flux, the recirculated bubbles at the channel inlets started to merge into elongated bubbles. The liquid from the inlet orifice is injected to the vapor-trapping zone near the channel inlets and broke up downstream and merged with the liquid-vapor mixture in the boiling region. If inlet flow has a very low subcooling, the pressure of the fluid drops significantly across the orifice so that it can be partially flashed. In this case, the liquid entering the channel will be in the form of a two-phase mixture. In general, without an inlet orifice, the fluid needs to be superheated over its local saturated temperature to initiate boiling. With the flashing effect induced by the orifice, bubbles were already generated in the incoming flow when they enter the microchannel. Therefore, two-phase flow can be observed even in the unheated entrance region of the channels. As the orifice results into an additional pressure drop, subcooling of the fluid is reduced at the actual inlet of the channel. Therefore, boiling starts earlier at a lower heat flux. This has an advantage of reducing wall-temperature overshoot and preventing instability. Without the inlet orifices, the wall temperature increased to values over 100  C at heat flux of 140 W/cm2. However, in the case with the orifices, it was only 60  C. The flow boiling curves suggest that the orifices can effectively suppress instabilities and maldistributions. Instead of producing a high pressure drop upstream of the microchannel, the suppression of instabilities can also be achieved by reducing the downstream flow resistance (Kandlikar et al. 2013). A new tapered manifold design with open microchannels has been proposed (Fig. 25). The microchannels have an open area above them with a gap, which provides increased flow area for the fluid. The gap increases downstream, decreasing the slope of the pressure gradient along the flow channel and increasing the flow stability.

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Fig. 25 Tapered manifold design with open microchannels

Fig. 26 Pressure drop fluctuations over a 20-s period at different heat fluxes for uniform (UM) and tapered (TM) manifolds, S = 0.127 mm and V = 80 mL/min (Adapted from Kandlikar et al. 2013)

The effect of the manifold taper on flow stability was evaluated by comparing the inlet pressure fluctuations at steady-state conditions for both uniform manifold (UM) and tapered manifold (TM) blocks, with a plain chip and an entry spacing of S = 0.127 mm. For each heat flux, the pressure drop data over a 20-s period at 5-Hz frequency are shown. The UM and TM tests were run with a flow rate of V = 80 mL/ min. UM shows pressure fluctuations between 80 and 160 kPa, as seen in Fig. 26 for heat fluxes in the range of 50–250 W/cm2. TM in the same heat flux range shows pressure fluctuations below 20 kPa. This indicates that TM can effectively suppress the flow boiling instability experienced with the UM. The TM and open microchannel design also helps removal of the vapor from the heat flux area and supply the liquid to the nucleation sites. The vapor generated in the microchannel can be removed easily by the manifold above the microchannel since extra space downstream is available for the vapor to flow away from the boiling surface. Liquid flow is favored inside the microchannel region due to capillary forces and thus dryout is delayed. Figure 27 shows the results of the heat transfer performance for the TM and the UM. The tests were conducted with two different volumetric flow rate: 225 and 40 mL/min. The depth of the tapered manifold is increased by 0.18 mm over the

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Fig. 27 Comparison of boiling performance for the tapered (TM) and uniform (UM) manifold blocks with V = 40 mL/min and S = 0.127 mm (Adapted from Kandlikar et al. (2013))

Fig. 28 Microchannel with micropillars

length of the microchannel region. At V = 40 mL/min, the TM reached 315 W/cm2 at ΔT_sat = 17.2 C, while the UM achieved 308 W/cm2 at ΔT_sat = 20.5 C. Both manifolds did not reach the CHF limits. The enhanced heat transfer performance of the TM is believed to be due to its flow stabilization effect. By suppressing the flow instability, the coolant has an uninterrupted flow passage and a continuous contact with the heated region. The experiments results also showed that reducing the manifold depth may provide additional heat transfer enhancement by increasing flow velocity and by further preventing backflow. Surface microstructures in two-phase microchannels can also be modified to suppress flow instabilities and enhance heat transfer (Zhu et al. 2016). By fabricating silicon micropillar arrays on the bottom of the microchannel wall (Fig. 28), capillary flow for thin film evaporation can be enhanced, while facilitating nucleation only from the side walls. The single microchannel has dimensions of 10 mm in length, 500 μm in width, and 500 μm in height. The micropillar on the bottom surface of the microchannel has diameters of 5–10 μm, pitches of 10–40 μm, and heights of 25 μm. These designs of the micropillar geometries were based on the following reasons: (1) the fabrication

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of the microchannel is easy and controllable using the standard Si etching process, (2) the capillary pressure is high enough to manipulate the flow behavior, and (3) the surface structures are mechanically strong and will not break or deform as the liquid evaporates. Performance between the smooth surface and structured microchannels at mass flux 300 kg/m2 s was compared. At low heat flux q00 < 400 W/cm2, both smooth surface and the structure surface microchannels show similar and small fluctuations. When the heat flux increased to q00 = 430 W/cm2, the smooth surface microchannels show temperature spikes and increased pressure drop fluctuations. However, small temperature (5  C) and pressure drop fluctuations (300 Pa) are observed for all the structured surface microchannels. When continuing to increase the heat flux, the smooth surface microchannel shows large amplitude (>20  C) periodic dryout, while the structured surface microchannel showed stable temperature and pressure drop at the same heat flux. The smooth surface could not supply liquid to the surface and maintain this liquid film and thus the dryout area expanded. For the structured surface, wicking capability of the microstructures helps wet the surface and maintain the liquid film. Visualization of the flow pattern and the dryout process demonstrates that the micropillar surface can enhance capillary flow and increase flow stability by maintaining a stable annular flow and thin film evaporation. The structured surface microchannel contributes to an enhanced heat transfer coefficient and CHF (maximum 57%) compared to a smooth surface microchannel. In addition, the additional pressure drop introduced by the micropillar structures was negligible. Silicon nanowire coating has also been investigated to suppress flow instability and enhance heat transfer (Li et al. 2012). Figure 29 shows the boiling curve for the plain surface and nanowire-coated surface. At single-phase flow region for both mass fluxes, the heat transfer characteristics are almost the same for both plain-surface and nanowire-coated

Fig. 29 Flow boiling curves for plain-surface and nanowire-coated microchannels for mass fluxes at G = 476 kg/m2 s. Tests are conducted at Tin = 60  C (Adapted from Li et al. (2012))

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microchannels. However, for the latter, an earlier ONB is observed at a lower heat flux. The visualization study revealed abundant and much more uniformly distributed bubbles in the nanowire-coated microchannels. Similar to pool boiling, this was attributed to the larger number of nucleation sites in the nanowire-coated microchannel. After ONB, the onset of flow oscillation (OFO) was immediately observed, which resulted in a very limited stable flow boiling condition for plain-surface microchannels. On the other hand, stable flow boiling was observed for a much larger range of heat fluxes for nanowire-coated microchannels, which indicates that nanowire-coated microchannels could significantly suppress flow instability and delay the OFO. Figure 29 also shows that the nanowire-coated microchannels can significantly enhance heat transfer coefficient compared to plain microchannels. Such enhancement can be attributed to the unique properties of nanowire coatings, including high nucleation site density, large surface area, large capillary force, and superhydrophilicity. The previous stabilization methods are passive but come usually at the cost of high pressure drop penalty. Active control methods can suppress inherent flow instabilities, especially in transient applications, thus improve the coefficient of performance. Based on the theoretical analysis of oscillatory flow boiling, a set of active control schemes are developed and studied to suppress flow oscillations and to increase the CHF. With the available control devices – inlet valve and supply pump – different active control schemes are studied to improve the transient two-phase cooling performance (Zhang et al. 2011b). For effective transient electronic cooling, the main control objectives are to maintain high heat transfer performance of two-phase flow inside the heated channel and to avoid pressure drop and flow oscillations, even under large transient heat load changes. Two control elements are available for feedback: the inlet valve of the boiling channel and the supply pump in the system. Inlet valve-based control methods are widely used in practice. In the pump-based flow control system, the manipulated variable – inlet flow rate – is linearly dependent on the positive displacement pump voltage. Therefore, inlet flow rate is not fixed but changes with heat load, and the change rate can affect the flow behavior of both the tank and the boiling channel. Inlet valve-driven feedforward control: a control valve before the heated channel can be used to suppress flow boiling instability. However, to control valve induces higher pressure loss and potentially higher supply pumping power compared to a system without an inlet valve. So active valve control strategies instead of fixed inlet valve for the suppression of two-phase flow instabilities are more desirable. The most straightforward method is to keep the valve fully open when the superheated or subcooled flow is stable, while reducing the valve opening position when the flow is not stable. Modeling results of transient control responses showed that the oscillatory behavior of mass flux and pressure is eliminated with the valve controller even when the heat load changes. Supply pump-driven feedback control: although the inlet valve suppresses the upstream compressible flow instability, it increases pressure loss and pumping power. Alternatively, the inlet-positive displacement pump can be used to regulate the downstream flow conditions while removing the inlet restrictor (valve) so that no

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additional pressure loss is induced. It was found that with the pump-driven active feedback control scheme, the flow oscillation can be successfully suppressed. The controller is implemented. The inlet refrigerant flow is not kept constant; instead, it will change with transient flow and heat load conditions. One important parameter with flow boiling is the critical heat flux (CHF), which determines the maximum heat flux the microchannel can dissipate before the device fails. After CHF is reached, a stable liquid film cannot be formed, vapor occupies the channel, and heat is transferred to the vapor mainly by conduction. So, one method to increase the CHF is to remove the vapor immediately after it is generated. A novel two-phase microchannel cooling device that incorporates perforated side walls (Fig. 30) was proposed for potential use as an embedded thermal management solution for high heat flux semiconductor devices (Warrier et al. 2014). In this cooling device, perforated side walls are used to form the channels. Each liquid channel is separated from the two adjacent vapor channels. While flowing in the liquid channels, the liquid is transported through the perforations by capillary force and evaporates into the vapor channels. In this case, both the liquid and vapor flow in the single-phase regime. Evaporation only occurs in the arrays of micro-perforations in the side walls of high thermal conductivity. This design enables one to circumvent flow instability and dryout. The unique perforated channel walls of high aspect ratio have two benefits: it can serve as heat transfer surface and separate the vapor and liquid. Thermo-fluid modeling results for one specific implementation of the device concept predicted pressure drop of T sat ðvaporizationÞ T sat

(5)

T sat
T v ; Sv ¼
Sl ; if T v > T sat ðcondensationÞ T sat

(6)

Sl ¼
λl αl ρl Sl ¼ λv αv ρv

VOF methods for flow boiling simulation are promising due to the intrinsic mass conservation achieved, unlike with other Eulerian methods. However, VOF uses fixed-grid approaches to estimate the interface profile, which may lead to unphysical flows due to the numerical error in estimating the interface surface tension. In order to suppress the unphysical flow, evaporative heat and mass source terms are calculated

Fig. 35 Porous media model of flow boiling in microchannel

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using a saturated-interface-volume phase change model which fixes the interface at the saturation temperature at each time step to achieve stability (Pan et al. 2015). This modified model can eliminate numerical oscillation of the evaporation source terms. With the help of a non-iterative time advancement scheme, the computational cost is reduced. The reference frame is set to move with the vapor slug to artificially increase the local velocity magnitude, which reduces the influence of numerical errors from calculation of the surface tension force, and thus suppress the development of spurious currents. So nonuniform meshes can be used to efficiently resolve high aspect ratio geometries and flow features and significantly reduce the overall numerical expense. The mass and heat transfer source terms in the VOF modeling depend on the specific flow boiling regime, since the interface dynamics and heat transfer depend on the specific regime. Harirchian and Garimella (2012) identified four regimes for flow boiling in microchannel: slug, confined annular, bubbly, and alternating churn/annular/wispy-annular flow. Separate models must be used to accurately model the flow boiling in different regimes. VOF approach with a two-zone model is used to model the slug flow, which is characterized by elongated bubbles separated by liquid slugs (Magnini and Thome 2015). As shown in Fig. 36, the vapor bubble and liquid slug pair are decomposed into liquid film, dry vapor plug, and liquid slug zones within this model. In the full vapor and liquid regions, single-phase convection is assumed to govern heat transfer. Heat transfer coefficient in these zones is evaluated by single-phase correlations for hydrodynamically and thermally developing flow. In the liquid film region, steadystate heat conduction is assumed to dominate heat transfer. The liquid film dryout process is assumed to be triggered by the local evaporation of liquid, exclusively. The liquid film thickness varies as it evaporates along the microchannel and is estimated based on the local bubble velocity and acceleration by implementing a correlation for circular channels. A submodel which accounts for the actual liquidvapor slip was used to calculate the bubble velocity and acceleration. Superheating of the liquid within the slug can also be properly evaluated. The liquid film dryout is not included in the boiling model as it does not occur under the simulation setup. In the VOF model, the interface is implicitly captured by the volume fraction in each control volume. Therefore, it can handle the topology change of the interface. However, the computation is limited to a fluid with small liquid-vapor density ratio (Ansys 2016). To model fluid with high liquid-vapor density ratio in microchannel, a

Fig. 36 Schematic of the two-zone model

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level-set (LS) method can be used (Lee and Son 2008). The liquid-vapor interface is determined by the LS function, which is defined as a signed distance from the interface. Since the distance function and its spatial derivatives are smooth and continuous, the LS method can be used to compute an interfacial curvature more accurately than the VOF method. Although the inhomogeneous model provides better spatial and temporal resolution than homogeneous model, and enables understanding of the detailed bubble dynamics, the homogeneous model has several advantages. First, it does not need to solve complex, flow pattern-dependent, interfacial transport terms required for multi-fluid models. Since these transport terms are largely unknown or not well validated for microchannel two-phase flow, the added accuracy of using an inhomogeneous model would be questionable. Second, accurate or widely applicable void fraction correlations or models are not yet available for flow boiling of microchannel. On the other hand, homogeneous void fraction matches well with many experimental void fraction measurements in microchannels. Therefore it is uncertain that using more complex models, such as a drift-flux model, or a diffusion model, would increase the accuracy (Saenen and Thome 2015). Finally, a typical microchannel evaporator consists of 20–100 microchannels. Modeling all these microchannels with a 2D or 3D interface tracking model would be computationally very expensive. Also, the model would need to be able to accurately simulate a wide variety of flow patterns and conditions in microchannel, which is not straightforward to accomplish.

4.3

Flow Boiling of Fluid Mixtures

Fluid mixtures have been a research topic for boiling enhancement and studied extensively. A large body of work has focused on pool boiling of mixtures, and the enhancement mechanisms have been studied. Van Wijk et al. (1956) studied the mixtures of water with acetone, alcohols, ethylene glycol, and methyl ethyl ketone. A CHF enhancement was achieved at an optimum concentration. They concluded that this enhancement in CHF was due to reduction in the bubble departure diameters. McGillis and Carey (1996) investigated pool boiling of mixtures of water with ethylene glycol, methanol, and 2-propanol. Addition of small amount of alcohol to water enhanced the CHF. The mixtures were classified into positive (more volatile component having lower surface tension) and negative (less volatile component having lower surface tension) mixtures. Due to the differences in fluid volatility, preferential evaporation of one component occurred along the liquid-vapor interface of a binary mixture. The variation in concentration along the liquid-vapor interface resulted in a surface tension gradient due to the Marangoni effect. If the surface tension of the more volatile component was less than that of the less volatile component, the concentration gradient would generate a force that pulled the liquid toward the heated wall. If the surface tension of the more volatile component was greater than that of the less volatile component, a force that pulled the liquid away from the heated surface could be generated. The study by Hovestreijdt (1963) and Fujita and Bai (1997) attributed the CHF enhancement to the Marangoni effect.

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Kandlikar and Alves (1999) performed pool boiling experiments using mixtures of water with ethylene glycol at low concentrations (1–10 wt. %). The effects of surface tension gradients were negligible at low mixture concentrations, and they attributed the observed improvement in heat transfer coefficient to the changes in contact angle and wetting characteristics of the mixture. Arik and Bar-Cohen (2010) observed significant CHF enhancement using mixture of FC-72 and FC-40. They attributed the enhancement to the improvement in thermal properties of the mixture. There exist a few studies on flow boiling of mixtures. Peng et al. (1996) and Lin et al. (2011) studied the flow boiling of water-methanol mixtures in microchannels. The CHF increased at low concentration but decreased as the concentration of methanol in water increased. The enhancement was attributed to the Marangoni effect. However, heat transfer degradation was also observed by other studies. Bennett and Chen (Bennett and Chen 1980) observed a significant reduction in heat transfer coefficient for mixtures of water and ethylene glycol and attributed it to mass transfer effects. Kandlikar and Bulut (2003) studied the flow boiling of ethylene glycol and water. The heat transfer performance deteriorated as ethylene glycol concentration increased. They also attributed the degradation to the mass transfer. Sathyanarayana (2013) conducted flow boiling experiment with 20 wt. % mixture of HFE 7200 – methanol in a microgap channel. The CHF enhancement was attributed to the smaller bubble departure diameter.

4.4

System Configuration and Design Consideration

System design considerations are drastically different from single-phase flow. The properties of the coolant are very important for the design of a flow boiling cooling system. The pressure-dependent saturation temperature determines the operation temperature of the chip. Surface tension affects CHF and determines the maximum heat dissipation rate. For example, water has much higher thermal conductivity and latent heat of vaporization than typical refrigerants and dielectric coolants, such as R-134a and HFE-7200. However, water has much higher saturation temperature at atmosphere pressure. Therefore, for electronic cooling, subatmospheric system pressure is required with water to ensure a sufficiently low operating temperature. Table 2 shows the thermophysical properties of several selected coolants (Mohapatra 2006 and Ellsworth 2006). Table 2 Thermophysical properties of selected coolants (Mohapatra 2006 and Ellsworth 2006) Coolant FC-77 R134a R245fa Water HFE-7200

Dynamic viscosity (kg/m s) 0.0011 0.00001 0.00001 0.00089 0.00067

Thermal conductivity (W/mK) 0.06 0.014 0.01405 0.61 0.068

Specific heat (J/kg K) 1,100 854 920.5 4,180 1,214

Density (kg/m3) 1,800 4.23 5.675 997 1,424

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Instability is a significant issue with flow boiling. Fluctuations of flow rate, pressure drop, and temperature are not desirable and may cause damage or failure of the device to be cooled. Throttling valve or inserting orifice can be used to suppress the instability, but at the penalty of higher pressure drop. As the vapor generates and expands in the microchannel, the system pressure is elevated significantly. Thus, device reliability and safety are a big concern. Dryout must be avoided since the associated excessive temperature rise can burn the device. Both direct and indirect cooling systems (Fig. 37) have been explored (Lee and Mudawar 2009). In the first direct refrigeration configuration, the cooling module of the electronic device is incorporated as an evaporator in a vapor compression cycle, and the refrigerant serves as coolant. The alternative indirect refrigeration cooling

Fig. 37 (a) Direct cooling system. (b) Indirect liquid-cooling system

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configuration uses two fluid loops. Heat from the device is dissipated to the coolant in the primary flow loop and then rejected through a heat exchanger to the coolant flowing in a separate vapor compression cycle. In the direct refrigeration cooling system, microchannel heat sink serves as an evaporator. So the coolant’s operating conditions need to conform to those of a vapor compression cycle. First, for most refrigeration compressors, the refrigerant entering the microchannel evaporator should be a two-phase mixture and exits as saturated or superheated vapor. Saturated or superheated vapor conditions are favored for safe operation of the compressor. This requires the use of a secondary heater or phase separator downstream of the microchannel. For indirect cooling system, using a separate loop for the primary coolant enables attaining the desired microchannel heat sink’s inlet conditions. Most importantly, the coolant is not required to be maintained in a near-saturated or superheated state, as required by the compressor in the direct cooling configuration. In the primary loop, the inlet fluid of the heat sink can be highly subcooled. Compared to saturated boiling, subcooled boiling increases the CHF significantly, which is beneficial for high heat flux cooling. A secondary flow loop is used to dissipate heat to the ambient by using a fluid-to-fluid heat exchanger. With the two-loop structure, distributed multiple heat loads could be handled with simple and small pump loops, all of which could be coupled to a centralized chiller (secondary flow loop). However, highly subcooled boiling of two-phase cooling systems suffers from various flow boiling instabilities. Flow boiling oscillations can modify the hydrodynamics of the flow, can generate acoustic noise, and can endanger the structural integrity of the system. Most importantly, flow instability can lead to premature initiation of the CHF condition.

5

Summary

Liquid cooling, as a promising technology, can handle very high heat flux compared to air-cooled heat sinks. With microfluidic cooling, small high heat flux components, such as microprocessors, can be kept at lower operating temperature, displaying improved performance. For large facilities such as data centers, significant amounts of energy can also be saved due to the leakage power reduction from chips, as well as the reduction in cooling power. Understanding of characteristics of liquid cooling, both under single-phase and two-phase conditions, is essential for its successful applications. In single-phase liquid cooling using microchannel arrays, the heat transfer performance is compromised by saturation of the heat transfer coefficient, and the bulk fluid temperature rises. Various heat transfer enhancement techniques such as wavy channels, adding pin fins and incorporating microjet array, have been proposed to enhance the flow separation and mixing, which can increase heat transfer coefficient. However, the increase of heat transfer is usually accompanied by increased pressure drop. Liquid cooling is an attractive cooling technology for 3D ICs. However, the integration of 3D ICs with liquid cooling is not well demonstrated and requires the

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consideration of signal, power, and liquid cooling, since they compete for the available space and may interact with each other. In addition, not all the liquid-cooling methods are suitable for integration into 3D IC applications. For example, microjets might not be easy to adapt to 3D ICs, since it requires the fluid to be injected to the microchannel from the top. Compared to single-phase cooling, flow boiling in pin fin-enhanced microgaps can achieve even higher heat transfer performance. However, new challenges come from flow instabilities and dryout. Inserting orifice and throttling valve can suppress the instability. However, they also produce additional pressure drop. The inserted orifice is usually small and clogging is a significant concern. By depositing or growing nanostructures on the microchannel surface, flow boiling can be enhanced. However, the strength of these nanostructures to withstand the flow force at high flow rates is of concern. Modeling of flow boiling is challenging due to the complex mechanisms involved. Flow boiling can be divided into different flow regimes with different heat transfer and bubble dynamics characteristics. Accurate interface heat and mass transfer models are required. Before microfluidic cooling can be used widely, cost and reliability issues need to be considered and addressed. Due to the high efficiency and heat removal capability of microfluidic cooling, the cooling system can be more compact compared to air cooling. Liquid-cooling system is usually complicated and requires a flow loop consisting of pump, heat exchanger, filter, and reservoir. Liquid leakage and clogging need to be addressed. Integrating liquid cooling into an application such as microprocessor cooling can increase the cost of fabrication, as it would require some modification of the existing chip fabrication and assembly procedures. A few recommendations for the future research direction include: 1. Optimization of the liquid-cooled heat sink design to further enhance heat transfer performance, reduce temperature nonuniformity, and decrease pressure drop 2. Codesign of cooling, power, and signal delivery for 3D ICs. Dynamic control of power and cooling 3. Exploration of alternative ways to suppress flow boiling instability and enhance CHF 4. Further integration and miniaturization of the liquid-cooling system

6

Cross-References

▶ Boiling and Two-Phase Flow in Narrow Channels ▶ Boiling on Enhanced Surfaces ▶ Design of Thermal Systems ▶ Electrohydrodynamically Augmented Internal Forced Convection ▶ Enhancement of Convective Heat Transfer ▶ Evaporative Heat Exchangers ▶ Mixture Boiling ▶ Numerical Methods for Conduction-Type Phenomena

2024

Y. Joshi and Z. Wan

▶ Single-Phase Convective Heat Transfer: Basic Equations and Solutions ▶ Single-Phase Heat Exchangers ▶ Transition and Film Boiling ▶ Two-Phase Heat Exchangers

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Tan SP, Toh KC, Khan N, Pinjala D, Kripesh V (2011) Development of single phase liquid cooling solution for 3-D silicon modules. IEEE Trans Components Packag Manuf Technol 1(4): 536–544. https://doi.org/10.1109/TCPMT.2010.2100710 Tang GY, Tan SP, Khan N, Pinjala D, Lau JH, Yu AB, Vaidyanathan K, Toh KC (2010) Integrated liquid cooling systems for 3D stacked TSV modules. IEEE Trans Components Packag Technol 33(1):184–195. https://doi.org/10.1109/TCAPT.2009.2033039 Tuckerman DB, Pease RFW (1981) High performance heat sinking for VLSI. IEEE Electron Device Lett 2(5):126–129. https://doi.org/10.1109/EDL.1981.25367 Tullius JF, Tullius TK, Bayazitoglu Y (2012) Optimization of short micro pin fins in minichannels. Int J Heat Mass Transf 55:3921–3932 Van Wijk WR, Vos AS, Van Strallen SJD (1956) Heat transfer to boiling binary liquid mixtures. Chem Eng Sci 5:68–80 Wan Z, Xiao H, Joshi Y, Yalamanchili S (2014a) Co-design of multicore architectures and microfluidic cooling for 3D stacked ICs. Microelectron J 45(2):1814–1821 Wan Z, Yueh W, Joshi Y, Mukhopadhyay S (2014b) Enhancement in CMOS chip performance through microfluidic cooling. In: Proceedings of 2014 20th international workshop on thermal investigations of ICs and systems (THERMINIC), pp 1–5 Wang XQ, Mujumdar AS, Yap C (2007) Effect of bifurcation angle in tree-shaped microchannel networks. J Appl Phys 102:073530. https://doi.org/10.1063/1.2794379 Wang ZH, Wang XD, Yan WM, Duan YY, Lee DJ, Xu JL (2011) Multi-parameters optimization for microchannel heat sink using inverse problem method. Int J Heat Mass Transf 54 (13–14):2811–2819 Warrier GR, Kim CJ, Ju YS (2014) Microchannel cooling device with perforated side walls: design and modeling. Int J Heat Mass Transf 68:174–183 Webb RL (2005) Next generation devices for electronic cooling with heat rejection to air. J Heat Transf 127:2–10. https://doi.org/10.1115/1.1800512 Wei XJ, Joshi YK, Ligrani PM (2006) Numerical simulation of laminar flow and heat transfer inside a microchannel with one dimpled surface. J Electron Packag 129(1):63–70. https://doi.org/ 10.1115/1.2429711 Wu Z, Sunden B (2014) On further enhancement of single-phase and flow boiling heat transfer in micro/minichannels. Renew Sust Energ Rev 40:11–27 Xie G, Chen Z, Sunden B, Zhang W (2013a) Numerical analysis of flow and thermal performance of liquid-cooling microchannel heat sinks with bifurcation. Numer Heat Transf Part A: Appl 61(11):902–919. https://doi.org/10.1080/10407782.2013.807689 Xie G, Liu Y, Sunden B, Zhang W (2013b) Computational study and optimization of laminar heat transfer and pressure loss of double-layer microchannels for chip liquid cooling. J Therm Sci Eng Appl 5:011004. https://doi.org/10.1115/1.4007778 Yang F, Li X, Li W, Li C (2015) Integrate monolithic nanostructures in microchannels to enhance flow boiling heat transfer of HFE-7000. In: Proceedings of IPACK, pp V002T06A006 Zhang HY, Pinjala D, Joshi YK, Wong TN, Toh KC (2003) Development of liquid cooling techniques for flip chip ball grid array packages with high heat flux dissipations. IEEE Trans Components Packag Technol 28:127–135. https://doi.org/10.1109/TCAPT.2004.843164 Zhang Y, King CR, Zaveri J, Kim YJ, Sahu V, Joshi Y, Bakir MS (2011a) Coupled electrical and thermal 3D IC centric microfluidic heat sink design and technology. In: Proceedings of ECTC, pp 2037–2044 Zhang T, Wen JT, Peles Y, Catano J, Zhou R, Jense MK (2011b) Two- phase refrigerant flow instability analysis and active control in transient electronics cooling systems. Int J Multiphase Flow 37(1):84–97 Zhang H, Liu JJ, Li Y, Yao SC (2013) Porous media modeling of two-phase microchannel cooling of electronic chips with nonuniform power distribution. In: Proceedings of IMECE, pp V08BT09A071 Zhu Y, Antao DS, Chu KH, Chen S, Hendricks TJ, Zhang T, Wang EN (2016) Surface structure enhanced microchannel flow boiling. J Heat Transf 138(9):091501. https://doi.org/10.1115/ 1.4033497

Film and Dropwise Condensation

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John W. Rose

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Interface Temperature Discontinuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Film Condensation on Plates and Tubes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Natural Convection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Forced Convection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Effect of Vapor Superheat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Effect of Presence of a Noncondensing Gas in the Vapor . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Film Condensation on Low Integral-Finned Tubes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Condensation in Microchannels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Dropwise Condensation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Experimental Investigations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Theory of Dropwise Condensation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Cross-References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2035 2036 2039 2039 2040 2042 2043 2046 2051 2053 2053 2060 2069 2070 2070 2070

Abstract

The chapter covers four main areas of condensation heat transfer. The process at the vapor-liquid interface during condensation is first discussed. In many cases it is adequate to assume equilibrium at the interface but in dropwise condensation and condensation of metals the interface temperature discontinuity plays and important role. The traditional problems of laminar film condensation on plates and tubes are covered in some detail including natural and forced convection problems, the effect of vapor superheat and of the presence of non-condensing

J. W. Rose (*) School of Engineering and Materials Science, Queen Mary University of London, London, UK e-mail: [email protected] # Springer International Publishing AG, part of Springer Nature 2018 F. A. Kulacki (ed.), Handbook of Thermal Science and Engineering, https://doi.org/10.1007/978-3-319-26695-4_50

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J. W. Rose

gases in the vapor. The specific problems of condensation on finned surfaces and in microchannels are treated in some detail. An extensive section covers dropwise condensation and incudes both experimental investigations and theory. Nomenclature

A A(r) b D d d0 dr F Fx f ff fs G g h hv hfg K K1 K2 K20 K21 K3 k L L0 L3 Mv Mg m mx Nu Nud Nux N(r) P Pv

Cross-sectional area of channel Distribution function; see Eq. 65 Spacing between fin flanks at fin tip Vapor-gas diffusion coefficient Diameter of tube Tube diameter measured to fin tip Tube diameter measured to fin root Defined in Eq. 23 Defined in Eq. 17 Fraction of surface area covered by drops with base radius greater than r Defined in Eq. 35 Defined in Eq. 36 Dimensionless quantity defined in Eq. 12, mass flux of vapor in channel Specific force of gravity Radial height of fin Effective vertical height of fin; see Eqs. 40 and 41 Specific latent heat of evaporation Defined in Eq. 19 Constant in Eq. 47 Constant defined in Eq. 60 Ratio of base to curved surface area of drop; see Eq. 52 Defined in Eq. 62 Constant in Eq. 69 Thermal conductivity of condensate Height of condensing surface Defined in Eq. 57 Defined in Eq. 70 Molar mass of vapor Molar mass of noncondensing gas Interface mass flux, condensation mass flux Local condensation mass flux Mean Nusselt number Nusselt number for condensation on horizontal tube Local Nusselt number Distribution function; see Eq. 66 Pressure of vapor-gas mixture Vapor pressure

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Film and Dropwise Condensation

p Psat(Tv) Psat(T0) n Q Q1 Q2 Q21 q, q qb qi qNu q* R Rex Red ~ x Re ~ d Re r rc rmax rmin Sc Sp s Tv Tw Tsat T1sat T0 T* Tt Tf Ts t tp U1 u vf vg vfg W1 W0

2033

Perimeter of channel Saturation temperature at Tv Saturation temperature at T0 Constant in Eq. 64 Heat flux Defined in Eq. 58 Defined in Eq. 59 Value of Q2 for steam at Tsat = 373.15 K, i.e., Q21 = 2.556 GW/m2 Heat flux, mean heat flux for surface Mean heat flux at base of drop Mean heat flux at curved surface of drop Heat flux given by Nusselt theory Dimensionless heat flux defined in Eq. 72 Specific ideal-gas constant Reynolds number, U1ρvx/μv Reynolds number, U1ρvd/μv Two-phase Reynolds number, U1ρx/μ Two-phase Reynolds number, U1ρd/μ Base radius of drop Radius of curvature of condensate surface, radius of curved surface of drop Effective mean base radius of largest drop Base radius of smallest viable drop Schmidt number, μv/ρv D Defined in Eq. 26 Spacing between fin flanks at fin root Vapor temperature Wall temperature Saturation temperature 373.15 K Vapor-liquid interface temperature Reference temperature Defined in Eq. 37 Defined in Eq. 38 Defined in Eq. 39, saturation temperature Fin thickness at tip Promoter layer thickness Vapor or vapor-gas mixture free stream velocity Condensate streamwise velocity Specific volume of saturated liquid Specific volume of saturated vapor vg  vf Mass fraction of noncondensing gas in the bulk Mass fraction of noncondensing gas at the interface

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X x y z

J. W. Rose

Defined in Eq. 25 Coordinate along channel normal to streamwise direction Coordinate normal to surface Streamwise coordinate

Greek Symbols

α αz β

βx γ δ ΔP ΔT ΔTc ΔTi ΔTp ΔTσ Δρ ζ eΔT θ θ0 λ λl λp μ μv v ξ ρ ρv ρg ρf ρfg e ρ σ τι ϕ

Heat-transfer coefficient q/ΔT Local (averaged around perimeter) heat-transfer coefficient at distance z along channel Constant in Eq. 6, half angle at fin tip in Eqs. 33, 34, 35, and 36, contact angle, channel inclination to vertical in Eq. 43. Defined in Eq. 30 Ratio of principal specific heat capacities of vapor Local condensate film thickness Difference between vapor pressure and saturation pressure at interface temperature Vapor-surface temperature difference Temperature difference attributable to conduction in drop Temperature difference attributable to interphase matter transfer Temperature difference across promoter layer Temperature difference attributable to surface curvature ρf  ρg Defined in Eq. 31, defined in Eq. 45 Enhancement ratio Celsius temperature, dimensionless temperature difference defined in Eq. 73 Defined in Eq. 74 Thermal conductivity of liquid Thermal conductivity of liquid Thermal conductivity of promoter layer Viscosity of condensate Viscosity of vapor or vapor-gas mixture μ/ρ Constant in Eq. 1, function defined in Eq. 42 Density of liquid, condensate Density of vapor or vapor-gas mixture Density of saturated vapor Density of saturated liquid ρf  ρg ρ – ρv Surface tension Streamwise vapor shear stress at condensate surface Retention angle measured from top of tube

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Film and Dropwise Condensation

χ ψ ω

1

2035

Vapor quality Angle between normal to channel surface and Y coordinate (see Fig. 1 of Wang and Rose 2005) Defined in Eq. 29

Introduction

The following topics are covered: (a) The vapor-liquid interface. The fact that equilibrium conditions do not prevail during condensation results in a temperature drop between the bulk vapor and the liquid in the immediate vicinity of the interface. This is generally small in comparison with the temperature drop in the condensate between the interface and the solid condensing surface. However, the interface temperature drop is important for condensation of metals owing to their high liquid thermal conductivity and in dropwise condensation where much of the heat-transfer takes place through extremely small droplets. Tb = 378 K 393 K 405 K

60

420 K

433 K ΔT/K

40

448 K 463 K 40 493 K

0

0.4

0.8

1.2

Q /(MW m–2)

Fig. 1 Condensation of mercury at different pressures. Vapor-surface temperature difference versus heat flux (Niknejad and Rose 1981). Tb bulk vapor temperature

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J. W. Rose

(b) Film condensation on smooth and finned surfaces and in small channels. Condensate films are generally very thin and film Reynolds numbers are often low so that laminar flow analysis covers a wide range of practically important problems. Effect of the presence of a noncondensing gas in the vapor is also discussed. (c) Dropwise condensation. Dropwise condensation has recently seen a resurgence of interest and is treated in some detail. There now exists a reliable experimental heattransfer database, and the theory is well understood. Heat-transfer coefficients are much higher than those for film condensation. However, to date no method for sustaining the dropwise mode for sufficiently long time intervals has been found, and dropwise condensation has not yet been successfully employed in practice.

2

Interface Temperature Discontinuity

In most applications, the interface temperature discontinuity is negligible in comparison with the temperature drop across the condensate but is of crucial importance in dropwise condensation and in film condensation of metals. It is also significant in condensation on low-finned tubes in cases where the condensate thermal conductivity is relatively high as in condensation of steam. Phenomena at the vapor-liquid interface during condensation and evaporation have been investigated for many years. In kinetic theory, the liquid is regarded as having a sharply defined mathematical surface from which molecules are emitted and incident molecules from the vapor are absorbed; the problem is to obtain an (approximate) solution of the Boltzmann equation for the vapor in the immediate vicinity (a few vapor mean free paths) of the liquid surface. Account may be taken of the possibility of reflection of incident vapor molecules at the liquid surface by incorporating a condensation coefficient, the fraction of vapor molecules incident on the liquid surface which remains in the liquid phase. If all incident molecules remained in the liquid phase, the condensation coefficient would be unity. If in addition, and at equilibrium, the assumed Maxwellian velocity distribution in the bulk vapor persists up to the interface, then the emitted flux must also be (half) Maxwellian and may be readily calculated. If the condensation coefficient is less than unity, the evaporative flux is less than the Maxwellian value, and an evaporation coefficient may be defined as the ratio of the evaporative flux to the Maxwellian value. Evidentially the evaporation and condensation coefficients are equal at equilibrium. For net evaporation or condensation, a Maxwellian distribution corresponding to the interface temperature is used for emitted molecules, and in these circumstances, the velocity distribution for the vapor immediately adjacent to the interface cannot be Maxwellian. In many approaches, the two coefficients are taken to be unity for net evaporation and condensation. For near-equilibrium conditions, and when the evaporation/condensation coefficient is taken as unity, somewhat different kinetic theory approaches for monatomic molecules give virtually identical results (see Rose 1998a) and do not differ widely from each other for significant departure from equilibrium.

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Film and Dropwise Condensation

2037

Experimental investigations of both evaporation and condensation have led to a wide range of reported values of evaporation/condensation coefficients. Comparisons with kinetic theory have indicated condensation coefficients ranging from around 0.03 to unity. Knudsen (1915) reported experiments on evaporation of mercury which first indicated an evaporation coefficient of 0.0005, but, after continually increasing the purity of the mercury, a value of unity was eventually obtained with a stated experimental error of 1% (see Schrage 1953). For low condensation rates, kinetic theory results of several investigators (Labuntsov 1967; Labuntsov and Muratova 1969; Sone and Onishi 1973; Labuntsov and Kryukov 1979; Ytrehus and Alvestad 1981; Rose 2000) indicate that the interface temperature drop and condensation mass flux are related by Psat ðT v Þ  Psat ðT 0 Þ ¼ mðRT 0 Þ1=2 =ξ

(1)

where ξ depends on condensation coefficient and, when the condensation coefficient is taken as unity, varies between 0.66 and 0.67 according to the different approaches (see Rose 1998a). For condensation of mercury the calculated interface temperature drop is much larger than that across the condensate film. Measurements of Niknejad and Rose (1981) for vapor pressures in the approximate range 50 Pa to 4 kPa are shown in Fig. 1. The calculated temperature difference across the condensate film ranged from around 0.1 K at the lowest pressure to around 3 K at the highest pressure and heat flux. It is evident that in all cases, the interface temperature drop significantly exceeds that across the condensate film. The accuracy, with which the interface temperature drop is determined by subtracting the calculated temperature drop across the condensate film from the observed vapor-surface temperature difference, is clearly little affected by the accuracy with which the condensate temperature drop is calculated, particularly at the lower pressures. These data may be used to determine values of ξ. It may be seen from Fig. 2 that values of ξ determined from these measurements may reasonably be extrapolated, for m ! 0, to a value around 0.66–0.67. These results provide strong evidence for the general validity of the kinetic theory model and a value of condensation coefficient near unity, at least for low condensation rates. Note that Eq. 1 applies strictly to monatomic molecules. An “intuitively and phenomenologically” derived correction for polyatomic molecules has been given by Le Fevre (1964): Psat ðT v Þ  Psat ðT 0 Þ ¼

ð γ þ 1Þ mðRT s Þ1=2 =ξ 4ð γ  1Þ

(2)

where γ is the ratio of the principal specific heat capacities for the vapor. Equation 2 evidently agrees with Eq. 1 for monatomic molecules (γ = 5/3) and gives an increased interface temperature difference for more complex molecules with

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J. W. Rose

Fig. 2 Condensation of mercury at different pressures. ξ versus condensation mass flux (Niknejad and Rose 1981). Note – m here denotes condensation mass flux

1.1

Tb = 378 K

1.0

393 K

0.9

405 K

0.8

420 K

0.7

433 K 448 K

x

0.6

463 K

0.5

493 K

0.4 0.3 0.2 0.1 0.0 0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

–m/kg/(m2 s)

smaller values of γ. Taking ξ as 2/3 the Clausius-Clapeyron equation with the approximation (dP/dT)sat = (ΔP/ΔT)sat when (Tv-T0)  Tv and q = m hfg, Eq. 2 gives T v  T 0 ¼ 3qvfg T v ðRT v Þ1=2 ðγ þ 1Þ=8ðγ  1Þhfg 2

(3)

More recently molecular dynamic simulation approaches (Nagayama and Tsuruta 2003; Wang et al. 2003; Meland et al. 2004; Ishiyama et al. 2004a, b, 2005; Tsuruta and Nagayama 2005) have been used to assess the validity of kinetic theory with reported values of evaporation/condensation coefficients. It is important to note that that these coefficients are essentially concepts adopted in the kinetic theory model to account for discrepancies between (mostly inaccurate) experiments and the model. In molecular dynamic simulation, as in reality, there is no abrupt interface, and subjective judgment (based on density distribution) is necessary to select the position for determination of an “interface temperature” appropriate for comparison with the kinetic theory model. Systematic methods adopted in molecular dynamic simulations for selecting the effective interface position have led to conclusions that the condensation coefficient is temperature dependent, falling from a value near unity to somewhat smaller values at higher temperatures. Similar conclusions have been reached in molecule tracking treatments where subjective decisions are needed on what are considered to be reflected, absorbed, and emitted molecules. Overall it seems that little more can be said with certainty than that molecular dynamic simulations lend general support to the kinetic theory model with condensation/evaporation coefficient not far from unity. For practical calculations where the interface temperature drop is thought to be important, Eq. 3 is recommended for its determination/estimation.

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Film and Dropwise Condensation

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3

Film Condensation on Plates and Tubes

3.1

Natural Convection

The Nusselt (1916) treatment of laminar film condensation of a saturated vapor has proved remarkably successful. The key approximations are laminar boundary layer flow of the condensate film, inertia and convection terms neglected, shear stress and temperature drop at the condensate-vapor interface neglected, pressure in condensate film at given depth equal to that in the bulk vapor, and properties taken as uniform. This leads to the well-known results:  1=4  1=4 g ρ Δρ hfg L3 q L 4 g ρ Δρ hfg L3 ¼ Nu ¼ ¼ 0:943 ΔT k 3 4μ k ΔT μ k ΔT

(4)

for the vertical plate with uniform temperature. It may be noted that the Nusselt theory predicts an infinite local heat-transfer coefficient at the top of the plate. In cases where the coolant-side thermal resistance is substantially greater than that on the condensing side, a more suitable approximation may be that the heat flux rather than the surface temperature is uniform. This gives the same expression (Eq. 4) with uniform q and mean ΔT. For the horizontal cylinder, and with the additional approximation that the condensate film thickness is much smaller than the radius of the cylinder, numerical integration is needed to obtain: 

g ρΔρ hfg d3 Nu ¼ 0:728 μ k ΔT

1=4 (5)

(The leading constant in Eq. 5 is (8/3)(2π)1/2Γ(1/3)9/4 = 0.728 018. . . (Rose 1998b). It is interesting to note that Nusselt used a planimeter for integration and obtained results which give a remarkably accurate value of 0.725 for the constant in Eq. 5.) The fact that the calculated film thickness approaches infinity near the bottom of the cylinder does not lead to significant error since the heat flux there is very small. A solution for the uniform heat flux case (Fujii et al. 1972c) gives a leading constant 0.695 in Eq. 5 with a mean value of ΔT. However, in this case, the result is less accurate since the mean value of the vapor-surface temperature difference is significantly affected by the erroneous values toward the lower part of the cylinder. Measurements show that the temperature variation around the tube surface is quite well represented by a cosine function (Memory and Rose 1991) and when this is used the same expression as Eq. 5 is found with mean heat flux and mean temperature difference. Negligible effect was seen when taking account of two-dimensional conduction in the condensate film due to variation of tube surface temperature (Zhou and Rose 1996). With the advent of digital computers, the Nusselt approximations (neglect of inertia and convection terms, effect of surface shear stress, variable properties) have

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J. W. Rose

been examined by Sparrow and Gregg (1959), Koh et al. (1961), Chen (1961a, b), and others. These show the Nusselt approximations to be remarkably accurate and that errors are likely to be smaller than other uncertainties in practical problems. Various reference temperatures for property evaluation have been suggested, typically of the form: T  ¼ T w þ βðT 0  T w Þ

(6)

with β in the range 0.1–0.3 according to fluid and conditions. However, in practice the choice of reference temperature used for property evaluation is not critical. The following, obtained by taking the k and ρ to vary linearly with temperature across the condensate film and ln(μ) to vary linearly with reciprocal temperature, are suggested: k ¼ fkðT 0 Þ þ kðT w Þg=2

(7)

ρ ¼ fρðT 0 Þ þ ρðT w Þg=2

(8)

μ ¼ μðT  Þ where T  ¼ ð3=4ÞT w þ ð1=4ÞT 0

(9)

hfg ¼ hfg ðT 0 Þ

(10)

with

3.2

Forced Convection

As for free convection condensation, the uniform property laminar boundary layer equations for both vapor and condensate film, with matching conditions for velocity and shear stress at the interface, may be solved exactly using similarity transformations. For forced convection condensation, it is implicit that the shear stress from the flowing vapor on the condensate surface (insignificant in the free convection case) be included. For a horizontal flat plate, where gravity is not involved, solutions with differing degrees of approximation have been given by Cess (1960), Koh (1962), and Shekriladze and Gomelauri (1966). These are discussed in detail by Rose (1988a). For the most general solution due to Koh (1962), the results have been summarized by Rose (1989): ( ~ x 1=2 Nux Re where

1:508

1 ¼ 0:436  3=2 þ G 1 þ kΔT=μ hfg    k ΔT ρ μ 1=2 G¼ μ hfg ρv μ v

)1=3 (11)

(12)

50

Film and Dropwise Condensation

2041

In the low condensation rate limit (G ! 0), Eq. 11 becomes ~ x 1=2 ¼ 0:436 G1=3 Nux Re

(13)

as obtained by Cess (1960) and in the high condensation rate limit (G ! 1)   ~ x 1=2 ¼ 0:5 1 þ k ΔT=μ hfg 1=:2 (14) Nux Re as obtained by Shekriladze and Gomelauri (1966). k ΔT/μ hfg is generally small (note that ΔT is the temperature drop across the condensate film and not the difference between the remote vapor and wall temperatures when the vapor is superheated or contains a noncondensing gas or the interface temperature drop is significant as for condensation of metals) so that in most cases ~ x 1=2 ¼ 0:5 Nux Re

(15)

is adequate for high condensation rates. Note that in the above expressions, the heattransfer coefficient varies as x1/2 so that the mean value over distance L is twice the local value at L. For vertical surfaces with vertical vapor downflow, both vapor shear stress and gravity play significant roles, and there is no similarity solution. Numerical solutions by Shekriladze and Gomelauri (1966) for the case of high condensation rate give: ~ x Nux Re

1=2

( )1=2 1 1 þ ð1 þ 16Fx Þ1=2 ¼ 2 2

(16)

where Fx ¼

μhfg gx kΔTU1 2

(17)

measures the relative importance of gravity and vapor velocity. Approximate integral solutions by Fujii and Uehara (1972) for the more general case (covering both high and low condensation rates) were summarized by:   ~ x 1=2 ¼ K 4 þ Fx =4 1=4 Nux Re

(18)

 1=3 K ¼ 0:45 1:2 þ G1

(19)

where

In the relatively rare circumstances where k ΔT/μ hfg is not small, Rose (1988a) proposed that K be amended to

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J. W. Rose

0

11=3

1:508 1C B K ¼ 0:436@n o3=2 þ A G 1 þ kΔT μhfg

(20)

For forced convection condensation on a horizontal tube, solutions have been proposed by Shekriladze and Gomelauri (1966), Denny and Mills (1969), and Fujii et al. (1972b). Fujii et al. (1972b) give an expression which may be written: 8
q}w
CHF1
U ) are more likely. The main point of the above discussions is to distinguish between CHF mechanisms of pool boiling (Collier and Thome 1994; Dhir 1998) and flow boiling despite obvious similarities. The analogies, as given in Collier and Thome (1994) through three-dimensional qualitative plots involving {q}w , Tw ðxÞ, XðxÞ} axes or the projected curves in the q}w and Tw(x) planes, can be improved – though it is not done here – with the help of above-reported figures and three-dimensional qualitative plots involving {hx , ΔT(x) , X(x)} or {q}w , Tw ðxÞ, XðxÞ} axes. Clearly one needs to seek Nux correlations of the type given by Eqs. (13) and (14) – provided one is also able to keep wall heat-flux q}w values below estimated threshold values of q}CHF1 and q}CHF2 .

2.3

Segmented Flow-Regime Dependent Nux Correlations

The earlier flow-regime specific discussions (in Sect. 2.2) regarding other indirect variables influencing key variables in the arguments list presented for Nux correlations suggest that they may also indirectly influence the very form of the Nux function – in ways such that the best forms of the correlations may significantly differ from one flow-regime to another. This suggests that it may be more convenient to obtain the sought-for and more accurate Nux correlations in Eqs. (8) and (9) by an

51

Internal Annular Flow Condensation and Flow Boiling: Context, Results, and. . .

2101

approach that restricts the correlations to one regime at a time. That is, if “regime - i” denotes a particular flow-regime (such as bubbly, plug-slug or annular flow-regimes, etc.), and its neighboring upstream and downstream regimes in Figs. 1, 2, and 3 are, respectively, marked as “regime - (i–1)” and “regime - (i + 1)” – with corresponding flow-regime transition criteria in terms of quality (see Eqs. (17) and (18)) denoted as Xcrj(i
1) !i and Xcrji!(i+1) – one could seek correlations of the type:   Ja ρV μV , , ,... Nux ¼ Nuxjregime
i X, ReT , PrL ρL μL  or,  ρ μ Nux ¼ Nuxjregime
i X, BI, ReT , V , V , . . . ρL μ L

(20)

where X ð^ x Þ is in the range Xcrjði
1Þ!i  X ð^x Þ  Xcrji!ðiþ1Þ

(21)

and nondimensional ^x  x=Dh .

2.4

Underlying One-Dimensional Modeling Approach to Obtain Spatial x–Variations of Flow Variables That Are Known or Correlated in Terms of Quality X and Other Parameters

For the flow-boiling and flow-condensation realizations in Figs. 1, 2, 3, and 4, onedimensional energy balance can be applied to the control volume between any two arbitrary locations “x” and “x + Δx” (see Fig. 6). It is easy to see that, in the limit of Δx! 0, the energy balance yields:  }  q ðxÞ∙PH dXðxÞ hx jTw ðxÞ
Tsat ðp0 Þj∙PH ffi w ¼

_ in hfg ðp0 Þ dx G∙A∙hfg M

(22)

The “+” and “–” signs in Eq. (22) are, respectively, for flow boiling and flow condensation. Using the definitions given earlier, PH being the heated perimeter (potentially a fraction frP  PH/PF, where PF is the wetted – by liquid or vapor – Fig. 6 A schematic of a control volume between “x” and “x + Δx.” The heat-flux arrows, as shown, are positive for boiling. The reversed direction negative values are for flow condensation

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_ in  G∙A, and the relevant nondimensional variables and numbers as perimeter), M defined following Eq. (9), Eq. (22) can be nondimensionalized as: dXð^x Þ Ja 1 μL ¼ 4:frP ∙Nux : : :θw ðxÞ d^x PrL ReT μV

(23)

for known wall temperatures specifying the “method of heating/cooling.” For known heat-flux values specifying the “method of heating/cooling,” Eq. (22) is nondimensionalized as: dXð^x Þ ¼ 4:frP ∙BI:Ψq ðxÞ d^x

(24)

2.4.1 Use of Nux Correlations Covering All Flow Regimes For known wall temperatures specifying the “method of heating/cooling,” the nonlinear ordinary differential equation (ODE) in Eq. (23) may be solved over 0  ^x  L=Dh, in conjunction with a reliable Nux correlation (covering all flow-regimes of saturated flow boiling or flow condensation in Figs. 1 and 2) given in the form of Eq. (11), or its equivalent, and subject to initial condition. ( Xð0Þ ¼

0, for saturated flow
boiling 1, for saturated flow
condensation

(25)

For a known heat-flux value specifying the “method of heating/cooling,” the ODE in Eq. (24) yields linear X ð^x Þ variation if it is solved for a uniform heat-flux prescription (i.e., for Ψq(x)  1) – over 0  ^ x  L=Dh subject to the initial conditions in Eq. (25). If the overall Nux correlation in Eq. (12) (or its equivalent) is known, it allows evaluation of hx given values of q}w , Bl, and computed X ð^x Þ variations. The use of this hx in the defining relationship of Eq. (4) then yields the temperatures Tw(x) over 0  x  L. A more interesting and a relatively difficult case is when the Nux correlation is available from known heat-flux-based measurements and associated correlations in the form of Eq. (12) – as in the example given later on in Sect. 3.1.1 – and one wants to use/solve Eq. (23) to make X(x) and q}w ðxÞ predictions for a known wall temperature Tw(x) case. In this case, preliminary reasonable guesses of q}w and Ψq(x) ffi 1 are employed to express Nux in Eq. (12) as a function of quality X for use in the solution of Eq. (23). The solution of Eq. (23) yields associated X(x) variations. Thesenew X(x) variations are used in Eq. (12) and Eq. (24) to obtain Bl ∙ Ψq(x) or q}w ðxÞnew . This yields new guesses of q}w and Ψq(x) toward iterative evaluation of Nux in Eq. (12) as a function of quality X followed by solution of Eq. (23). The process is repeated until converged values of X(x), q}w , and Ψq(x) are obtained.

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2.4.2 Use of Flow-Regime-Specific Nux Correlations Alternatively, for the flow-regime-specific Nux correlations given in flow-regimespecific forms of Eq. (20) or Eq. (21) along with known wall temperature “method of heating/cooling” cases, the related nonlinear ODE in Eq. (23) – with or without the need for combining it with Eq. (24) – may be solved over x i  x  x iþ1 ; the distances over which flow-regime “i” is realized. The initial condition for these flow-regime-specific solutions of X(x), for a guessed value of x i , then becomes:
 X x i ¼ Xcrjði
1Þ!i

(26)

where the Xcrj(i
1) ! i correlation is available from Eq. (18) and
theguessed value of x i remains unknown until X(x) is obtained from x = 0 up to X x i , through similar considerations of all the prior flow-regimes. Current knowledge of segmented and flow-regime-specific Nux correlations may be approximate and reasonable for annular flow realizations, but the knowledge of the flow-regime boundaries, as sought in nondimensional forms (such as Eqs. (17) and (18)), is poor for the flows in Fig. 4. The flow-regime transition boundaries, which occur for flows in Figs. 1, 2, and 3, are also poorly known. Alternatively for cases where “method of heating/cooling” is specified by known heat-flux values specifying the cases and a reasonable flow-regime-specific Nux correlation is available (as in Eq. (20)), the solution of the ODE in Eq. (24) over x i  x  x iþ1 may be obtained – and X ð^x Þ will vary linearly with ^x for the initial assumption of Ψq(x)  1. With the help of the initial condition in Eq. (26) and “regime-i”-specific Nux correlation in Eq. (20), hx can be evaluated for given values of q}w, Bl, and Xð^ x Þ variation. The use of this hx in the defining relationship of Eq. (4) then yields temperatures Tw(x) over the x-locations x i  x  x iþ1 for regime-i. The ability to correctly place the physical x-location of this “regime-i” at a suitable distance form is possible only if one knows the x ¼ x i location in Eq. (26) through similar considerations of all the prior flow-regimes between x = 0 and x ¼ x i .

3

Overview of Available Correlations for Direct and Indirect Variables of Interest

The correlations that are relevant to the focus of this review – annular flow boiling and flow condensation – correspond typically to laminar liquid film flows, laminar or turbulent vapor flows, and negligible entrainment rates. These correlations (for heat transfer coefficient and pressure-drop, etc.) are related to key variables (dimensional or nondimensional) that define a specific flow realization, namely, hydraulic diameter Dh, mass-flux G, relevant fluid properties, imposed heat-flux q}w or wall temperature Tw , length L, quality X, and inlet quality Xin (for annular flow boiling only) in the horizontal tube configuration of the flows in Figs. 1, 2, and 3. Though horizontal tube considerations are sufficient for most inner diameters Dh and massflux G values of interest to this review, downward tilted (gx > 0) flow-boilers and

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flow-condensers for some macroscale diameters (5 mm  Dh  15 mm) and low mean mass-flux (G = 10 to 30 kg/m2 . s) may also be of interest and are briefly discussed here. Effects of nonuniform heating/cooling methods have been discussed by Naik et al. (2016), Naik and Narain (2016), and Ranga Prasad et al. (2017) from a theoretical point of view. Regardless of whether the correlations assume knowledge of wall temperature Tw(x) or heat-flux q}w ðxÞ prescriptions, one can establish (with the help of CFD simulations) equivalence between the two prescriptions (see Naik et al. 2016 and Ranga Prasad et al. 2017) for convective boiling and flow condensation. Next, relevant correlations for heat transfer coefficient hx, flow-regime maps, pressure-drop, and void fraction are reviewed. Furthermore, additional available information on CHF issues (for flow boiling) as discussed in Sect. 2.2 and related to the annular flows in Figs. 1, 2, and 3 are also discussed.

3.1

Local Heat Transfer Coefficient hx from Nusselt Number Nux Correlations

3.1.1 Flow Boiling Known results from the flow-boiling experiments that are cited here typically use the fact that the mean wall heat-flux q}w and key flow defining variables are known, and Nux, for traditional boiling operations, can be correlated in the structural form (or its equivalent) indicated in Eq. (12). Several local heat transfer coefficient correlations covering a range of experiments (in mini-/micro-channels for both single and multi-channel configurations, as given in Table 1) have been recently considered by Kim and Mudawar (2013c), and an order of magnitude curve fit for flow boiling – covering all the flow-regimes in Fig. 1 – has been proposed. Though these experiments, and hence associated correlations, also cover the annular regime results, which are of particular interest to this review, the experimental data underlying the correlations’ data are more biased toward regimes marked I–III in Fig. 1b. The “more general” HTC correlation of Kim and Mudawar (2013c) – given in Eqs. (27), (28), (29), and (30) below – has a built in, but ad hoc, breakup of total HTC into its nucleate and convective components:  0:5 hx ¼ h2xjnb þ h2xjcb

(27)

where, "

hxjnb and

#    PH 0:7 0:38
0:51 0:8 0:4 kL  2345 Bl PR ð1
XÞ 0:023ReL PrL PF Dh

(28)

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Table 1 Previous saturated flow-boiling heat transfer correlations or experimental data considered by Kim and Mudawar (2013c). The parenthetical notation (V/V-u/V-d, H; C/R) under “Remarks,” respectively, represent considered flow directions (V vertical-up or vertical-down not stated, V-u vertical-up, V-d vertical-down, H horizontal) and following the semicolon (C/R), respectively, represent considered cross-sections (C circular and/or R rectangular) Authors Remarks Recommended for macro-channels 6000 data points for nucleate pool boiling Cooper (1984b)a Gungor and Winterton D = 2.95–32 mm, water, R11, R12, R113, R114, R22, ethylene (1986) glycol, 4300 data points – (V-u, V-d, H; C) Liu and Winterton (1991) Same data as Gungor and Winterton (1986) – (V, H; C) Shah (1982) Experiments are mostly in horizontal tubes of circular or rectangular cross-sections D = 6–25.4 mm, water, R11, R12, R113, cyclohexane, 780 data points – (V, H; C) Recommended for mini/micro-channels Agostini and Bontemps Dh = 2.01 mm, 11 parallel channels, R134a – (V-u; R) (2005) Bertsch et al. (2009) Dh = 0.16–3.1 m, water, refrigerants, FC-77, nitrogen, 3899 data points– (V, H; C, R) Ducoulombier et al. (2011) D = 0.529 mm, CO2 – (H; C) Lazarek and Black (1982) D = 3.15 mm, R113, nucleate boiling dominant – (V; C) Li and Wu (2010) Dh = 0.16–3.1 m, water, refrigerants, FC-72, ethanol, propane, CO2, 3744 data points – (V, H; C, R) Oh and Son (2011) D = 1.77, 3.36, 5.35 mm, R134a, R22 – (H; C) Tran et al. (1996) D = 2.46, 2.92 mm, Dh= 2.40 mm, R12, R113, nucleate boiling dominant – (H; C,R) Warrier et al. (2002) Dh = 0.75 mm, 5 parallel channels, FC-84 – (H; R) Yu et al. (2002) D = 2.98 mm, water, ethylene glycol, nucleate boiling dominant – (H; C) a

The Cooper (1984b) correlation was developed for nucleate pool boiling

"

hxjcb

   0:94  0:25 #  PH 0:08
0:54 1 ρV 0:8 0:4 kL  5:2 Bl WeL0 þ 3:5 0:023ReL PrL e tt PF ρL Dh X (29)

The parameters in the above definitions of hx are: Bl 

μ CpL q}w p Gð1
XÞDh GDh , PR ¼ o , ReL  , ReL0  , PrL  L , G:hfg pcr μL μl kL

WeL0

 0:1
  0:5 G2 Dh ~ 1
X 0:9 ρV , X tt ¼ μμL  X ρL V ρL σ

(30)

where PF is the wetted perimeter (in case a tube/channel is not wetted on all its periphery P) and PH is the heated perimeter (and includes cases for which a tube/

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channel is not heated on all its periphery). This is also the perimeter where q}w ðxÞ is replaced by q}w. The correlation in Eqs. (27), (28), (29), and (30) covers all the saturated flowboiling regimes depicted in Fig. 1, i.e., as the quality X increases and the flow goes from tube diameter determined nucleate boiling (nb) dominated zone I to the macroscale convective boiling (cb) controlled regime III. The use of the Martinelli e tt in Eq. (29) is typically not indicative of both phases being turbulent – it parameter X is simply a correlation parameter. The uncertainty in the predictions from a correlation, such as the one in Eqs. (27), (28), (29), and (30), can be high as it allows for many experimental and conceptual uncertainties. Some of the significant sources of uncertainty are associated with: _ v ðxÞ=M _ in inferred from the expression of X(x) ffi (i) Replacing XðxÞ ¼ M Xth(x) where Xth(x) is a thermodynamic vapor quality obtained from experimentally assessing (with some uncertainty) an “x = 0” location where the fluid is at close to saturation conditions. And then applying energy balance to the flow-boiling control volume, between the identified “x = 0” location and one at an “x > 0” location, with knowledge of the experimentally measured values Ðx } (with its own uncertainties) of heat input (q½0, x ¼ 0 qw ðxÞ:PH :dx) between the two locations. (ii) Uncertainties in measuring ΔT(x) (  |Tw(x)
Tsat(p0)|) and q}w ðxÞ (due to the limitations in accuracy of thermocouple and other instruments – more so with older experiments with less accurate sensors) and then evaluating hx  q}w ðxÞ =ΔTðxÞ. This error is also related to occasional assumption, without verification, that the wall temperature Tw(x) is nearly uniform and one can replace ΔT (x) by the average temperature difference ΔT. (iii) Uncertainties associated with the current practice of combining data obtained for different cross-sectional geometries, e.g., from channel flows (i.e., high aspect ratio rectangular cross-sections) not significantly influenced by curvature effects on the flow field as well as surface tension effects with those obtained from mm-scale tubes, where the flow, interfacial shear, and interfacial pressure differences are affected by curvatures and surface tension effects. Similarly, significant uncertainties arise from combining data from experiments conducted at different tube inclinations – particularly data for low to moderate mass-flux G values. (iv) There are inherent curve-fitting inaccuracies associated with developing a single correlation that cover all the flow-regimes (as in Eqs. (11) and (12)) as opposed to developing segmented correlations specific to different flowregimes (as in Eq. (20)). (v) A rather ad hoc splitting of total HTC hx in its macroscale convective (hxjcb) and macroscale nucleate (hxjnb) boiling parts (see Sect. 6, Ranga Prasad et al. 2017, and Gorgitrattanagul 2017). Despite the aforementioned uncertainties and weaknesses, correlations – such as the one in Eqs. (27), (28), (29), and (30) – can provide useful order of magnitude

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estimates until more accurate flow-regime and flow-physics specific correlations are developed in appropriately classified ranges of parameter space of interest to the user. For design purposes, as in the sample example of Sect. 4, one can use a range of estimates based on the results in Eqs. (27), (28), (29), and (30). A crude estimate of uncertainty of the correlation in Eqs. (27), (28), (29), and (30) is 0:5hxjEq ð27Þ
ð30Þ  hx  2hxjEq ð27Þ
ð30Þ

(31)

Correlation of Kim and Mudawar (2013c) in Eqs. (27), (28), (29), and (30) also covers annular flows – involving both laminar and turbulent flows of the vapor and the liquid phases. Direct numerical simulation (DNS) approach (see Sect. 5 for further discussions) has been recently used by Ranga Prasad et al. (2017) to propose local heat transfer coefficient (hxjcb) correlations for annular boiling cases over the length of the boiler (Fig. 3a) – but the cases were limited to sufficiently thin and laminar liquid film flows and laminar-to-turbulent vapor flows. Such hxjcb values, particularly after its semi-theoretical extension to cover laminar liquid film flows and full range of turbulent vapor flows, can be effectively combined with experimental hx values for annular flow boiling, and this will yield superior estimate of microscale nucleate boiling contributions, where hxjnb = hxjnb
micro – as is expected for region III and part of region IV in Fig. 1b (see Sect. 6 for further considerations and uses of this aspect of annular flows). A sample correlation, for low values of imposed heat-flux q}w ðxÞ as well as low mass-flux G values, is obtained by CFD solutions, and their correlation is presented in Ranga Prasad et al. (2017) for channel flows (of gap “h” and a reference length e h ¼ 4h) – with CFD employing laminar liquid and scale of “h” changed here to D laminar vapor assumptions but the resulting heat transfer correlations (given in Eq. (32) below) continuing to apply to moderately turbulent vapor phases as well (ReV < 40,000). The correlation provided in Ranga Prasad et al. (2017) is for “method of heating” specified by known wall temperatures, with the considered/ specific parameter space corresponding to flow situations in Table 2, and is given by:  
0:0583  
0:399  0:454 eh hxjcb ∙D Ja ρV μV 1:61 0:128 0:0284 ¼ 21:46 X Xin ReT PrL kL ρL μL

(32)

Table 2 Ranges of raw variables and fluid flow conditions considered for the development of the correlation given in Eq. (32) Working fluids Inlet pressure, p0 (kPa) Channel height, h (mm) Mass flux, G (ρVU) (kg/m2s) Transverse gravity, gy (m/s2) Average inlet vapor speed, U (m/s) Temperature difference, ΔT ( C)

FC-72 105.1 2 7–35
9.81 0.5–2.5 5–25

R113 105.1–200 2 3.8–56.3
9.81 0.5–4 5–25

R123 105.1 2 3.4–26.8
9.81 0.5–4 5–25

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where, 0.5  Xin  0.86, 2466  ReT  39,524, 0.0048  Ja/PrL  0.0424, 0.00466  ρV/ρL  0.0097, 0.0216  μV/μL  0.0295. Correlations such as Eq. (32) can also be developed with the help of simulations for low heat-flux q}w ðxÞ and mass-flux G values for cases where “method of heating” is specified by known heat-flux values. These issues, along with forthcoming methodologies for obtaining hxjcb for a higher range of q}w ðxÞ and G values – typically associated with laminar liquid and turbulent vapor flows in Fig. 3 – are also discussed in Ranga Prasad et al. (2017) and Gorgitrattanagul (2017). For such cases, vapor flows can be laminar or turbulent, and thin liquid film flows can be controlled to be in the laminar regime (at least for devices in Fig. 3); the forthcoming semi-empirical CFD approaches cannot only yield reliable hxjcb correlations for a higher range of G and other parameters, but they can also be supplemented with experimental measurement-based correlations for the total HTC hx values – in the expected presence of significant microscale nucleate boiling contributions hxjnb
micro (region III of Fig. 1b) through an n = 1 relation in Eq. (33) below. Thus, following the superposition approach in Eq. (33), one can also propose annular flow-boiling correlations in the presence of microscale nucleate boiling. Such correlation proposals – which employ n = 1 (Eq. (33)) – are being studied for compatibility with experiments dealing with the flow in Fig. 3a (Gorgitrattanagul 2017; Sepahyar 2018) and are expected to be compared with results obtained from other available correlations of the following approximate forms: hx ¼





hxjnb

n


n 1=n þ hxjcb , n ¼ 1, 2, 3, . . .

(33)

Here n = 1 is preferred if hxjcb is given by the direct suppressed nucleation assumption-based CFD approaches, similar to the one leading to Eq. (32). A variation of n = 1 approach – such as those of Chen (1966), Kenning and Cooper (1989), and Gungor and Winterton (1986) – takes a suitable hxjcb empirical correlation (typically for region III in Fig. 1b) and adds it to a suitable macroscale hxjnb correlation (after multiplying it by a suppression factor S < 1) for pool boiling regimes – this is typically done for both vertical and horizontal in-tube flowboiling operations in regions I and II of Fig. 1b. These authors often employ a pool-boiling correlation for hxjnb. Lately pool-boiling correlations of Cooper (1984a), or Cooper (1984b), or Gorenflo (1993) are preferred over the earlier Forster and Zuber (1955) correlation, although their relative merits depend on many factors – even for pool boiling (see Jones et al. 2009). The popular Cooper (1984b) correlation is given by: ð0:12
0:2∙log10 Ra Þ

hxjnb ¼ C PR


0:67 ð
log10 PR Þ
0:55 M
0:5 q}w

(34)

where PR is the reduced pressure as defined in Eq. (30), M is the molecular weight of the working fluid in kg/kmol, heat-flux is measured in W/m2, Ra is a specific roughness measure in μm divided by 1 μm, and although C is fluid-solid pair specific – often  C ffi 55(kg/kmol)0.5 ∙ (W/m2)
0.67 ∙ (W/m2 C) is recommended if specific information is not available.

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It should be reiterated that the use of the separate terms hxjcb and hxjnb in Eq. (33) – through popular correlations such as Cooper (1984b) – only focuses on how macronucleation effects in regions I–II of Fig. 1b become subservient to the controlling dominance of hxjcb in region III of Fig. 1b. However, the focus of this chapter is on thin film (about 300–200 μm thick) annular flow-boiling devices of Fig. 3 – where micro-nucleation effects of hxjnb
micro needs to be more specifically modeled (for regions III-IV of Fig. 1b) in the decomposition of the local HTC hx through hxjnb = hxjnb
micro, hxjcb = hxjcb
ann, and hx = hxjnb
micro + hxjcb
ann.

3.1.2 Flow Condensation Known results from flow-condensation experiments that are cited here assume that the mean wall temperature Tw along with some key problem defining variables are known and Nux values that can be correlated in the structural form (or its equivalent) indicated in Eq. (11). Table 3 shows several “local” heat transfer coefficient hx correlations (that included consideration of annular regime in the flow-condensation experiments from which data were used) considered by Kim and Mudawar (2013a) before they Table 3 Previous annular flow-condensation heat transfer correlations or data considered by Kim and Mudawar (2013a) involved experiments that employed macro- and mini-/micro hydraulic diameter straight ducts of circular and rectangular cross-sections - and experienced annular flows as the dominant flow-regime. The parenthetical notation (V/V-u/V-d, H; C/R) under “Remarks,” respectively, represents considered flow directions (V vertical-up or vertical-down not stated, V-u vertical-up, V-d vertical-down, H horizontal) and following the semicolon (C/R), respectively, represent considered cross-sections (C circular and/or R rectangular) Author (s) Remarks Recommended for macro-channels Akers and Rosson D = 19.05 mm R12, propane   0:5 (1960) > 20; 000, reL > 5000 – (H; C) reV μμV ρρL L

Cavallini and Zecchin (1974) Dobson and Chato (1998) Haraguchi et al. (1994) Moser et al. (1998) Shah (1982)

V

R12, R22, R113, 7000  reLO  53,000 – (V; C) D = 3.14–7.04 mm; R12, R22, R134a, R32/R125 – (H; C)

D = 8.4 mm; R22, R123, R134a – (H; C) D = 3.14–20 mm; R11, R12, R125, R22, R134a, R410a – (H; C) D = 7–40 mm; water, R11, R12, R22, R113, methanol, ethanol, benzene, toluene, trichloroethylene (V, H, I; C) Recommended for micro-channels Bohdal et al. (2011) D = 0.31–3.30 mm; R134a, R404a – (H; R) Huang et al. (2010) D = 1.6, 4.18 mm; R410a, R410a/oil – (H; C) Koyama et al. (2003) Dh= 1.46 mm; R134a; multi-channel – (H; C) Park et al. (2011) Dh= 1.45 mm; R134a, R236fa, R1234ze (E); multi-channel – (V-d; R) Wang et al. (2002) Dh= 1.46 mm; R134a; multi-channel – (H, R) The experiments mostly employ straight ducts and annular flows are the dominant flow-regimes. Both circular and rectangular cross-section ducts have been considered.

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proposed their own order of magnitude curve fit correlation given below in Eqs. (35), (36), (37), and (38): Nux 

  hx D h 0:34 Φg ¼ 0:048Re0:69 Pr L L e tt kL X

(35)

e tt , turbulent-turbulent Lockhart-Martinelli parameter, is used merely as a where X known function and defined as: e tt ¼ X



μL μV

   0:1  1
X 0:9 ρV 0:5 X ρL

(36)

The parameter Φg in Eq. (35) is the two-phase multiplier defined as: Φ2g ¼ 1 þ CXm þ X2m

(37)

where C is a Lockhart-Martinelli coefficient defined in Kim and Mudawar (2013a) and in Table 6a of this review and Xm is a Lockhart-Martinelli parameter defined as:  ðdp=dxÞL 1=2 Xm ¼ ðdp=dxÞV where (dp/dx)L and (dp/dx)V represent the frictional pressure gradients of liquid and vapor phases, respectively, flowing alone in the pipe and are computed using the following equations:
dp
2f L G2 ð1
XÞ2 Gð1
XÞD ReL   dx L μL ρL D
dp
2f V G2 X2 GXD (38) ReV  dx V  μV ρV D f L ¼ BRe
n L

f V ¼ BRe
n V

The friction factor fL and fV are defined as above for vapor and liquid phases, with laminar flows’ (ReL/V < 2000) values of B = 16 and n = 1 and turbulent flows’ (ReL/V > 2000) values of B = 0.0079 and n = 0.25. Computational fluid dynamics (CFD) which becomes a direct numerical simulation (DNS) approach for laminar/laminar case has been recently used by Narain et al. (2015), Naik et al. (2016), and Naik and Narain (2016) (also see Sect. 5 for further discussions) to propose local heat transfer coefficient hx correlations for annular flow-condensation cases (Fig. 3b) that involve sufficiently thin laminar liquid flows and low values of both heat-flux q}w ðxÞ and mass-flux G. A sample Nux correlation, presented in Eq. (39) below, is for laminar/laminar e h ¼ 4 h), and its validity range channel flows (of gap “h” and reference length D extends to reasonably turbulent vapor phases (ReV < 40,000) as well. The

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Table 4 Range of raw fluid variables and flow conditions considered for the development of the correlation given in Eq. (39) Working fluids Inlet pressure (kPa) Saturation temperature ( C) Hydraulic diameter (h = 0.001–0.003 m in Fig. 3b) Transverse gravity, gy (m/s2) Mass flux, G (kg/m2s) Temperature difference, ΔT ( C)

FC 72 100 55.94 4h

R113 25 11.1 4h

0  |gy|  g 0  |gy|  g 4.55–127.4 4.2–115.3 2.93–12.30 8.69–36.50

R113 225 73.86 4h

R134a 150
17.15 4h

0  |gy|  g 5.51–154.1 2.25–9.45

0  |gy|  g 4.1–113.5 3.45–14.45

correlation is for “method of cooling” specified by known wall temperatures, with the parameter space restrictions given below, immediately following Eq. (39):   0:3  
0:73  0:069  eh hx :D Ja ρV μV   Nux  ¼ 0:02 ð1
XÞ
0:59 Re0:122 (39) T e PrL kL ρL μL Dh

where 3200  Rein  92,000, 0.0058  Ja/PrL  0.021, 0.0013  ρV/ρL  0.011, and 0.012  μV/μL  0.034. These parameter restrictions arose from considering the range of flow conditions given in Table 4. It was shown by Narain et al. (2015) that, for low heat-flux q}w and mass-flux G cases, the channel flow DNS yields hx values much higher than the corresponding predictions from the Kim and Mudawar correlation in Eqs. (35), (36), (37), and (38). The in-tube (as opposed to channel) shear-dominated condensing flow predictions for hx from Naik et al. (2016) and Naik and Narain (2016) yield values closer to the one in Kim and Mudawar (2013a), whereas the channel predictions yield much higher values. This suggests that high curvature effects associated with mm-scale in-tube flows may degrade the heat transfer performance relative to condensing flows in channels (i.e., high aspect ratio rectangular cross-sectional ducts). As discussed in Sect. 5 of this review, the CFD/DNS approach can be extended to yield reliable hx values for a higher range of parameters (particularly mass-flux G values, etc.) than those indicated in Table 4.

3.2

Flow-Regime Maps/Correlations

Flow-regime maps are needed, and recommendations exist (see Ghiaasiaan 2007; Kim and Mudawar 2013a) for identifying flow-regimes associated with traditional operations of boiling (Fig. 1) and condensing (Fig. 2) flows.

3.2.1 Annular Adiabatic Cases Flow-regime maps also exist for adiabatic flows (Carey 1992; Ghiaasiaan 2007) which may also be applicable and useful in the inlet region, just past the splitter

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plates in the innovative annular operations shown in Fig. 3. In this region, the flow behavior will approach adiabatic flows because it would not have been exposed to significant amounts of externally imposed heating/cooling (see Fig. 3). Precise nondimensional transition maps, as suggested by correlations of the type indicated in Eqs. (17) and (18), currently do not exist – even for adiabatic two-phase flows. For the adiabatic case, following certain earlier larger diameter (>5 mm) studies (Baker 1953; Hewitt and Roberts 1969), a map employing raw variables was proposed by Mandhane et al. (1974). This map uses superficial velocities of vapor and liquid phases but covers a significant range of the parameter space, which includes conditions of interest to this review (i.e., millimeter-scale horizontal ducts with cocurrent flows).

3.2.2 Annular Flow Boiling This regime is of primary interest here (Fig. 3a), and it occurs at locations further downstream of the plug-slug regime as quality X increases in the traditional saturated flow-boiling cases (Fig. 1). For known uniform heat-flux values specifying the “method of heating,” Harirchian and Garimella (2012) recommended the following criteria for micron-scale Dh: f > 160 Bo
0:5 :Re  
0:258 (40) X:PH ρL
ρV 0:5 f Re : > 96:65 Bo ρV D2 pffiffiffiffi where PH = heated perimeter, D  A, Bo  g(ρL
ρV)D2/σ, Bl is as defined in f  G:D=μL . They assume heating levels and flow rates as discussed Eq. (9), and Re for Fig. 1 realizations, and therefore annular flows are expected only over downstream distances “x” satisfying: Bl •

 
0:258 f x > xA  96:65 Bo0:5 Re Bl
1

ρV A ρ L
ρ V PH

(41)

or quality X satisfying: X > Xcr ¼ Xðx ¼ xA Þ

(42)

where X (x) has been obtained from the approach described in Sect. 2.4 along with use of an appropriate Nux correlation, such as the one in Eqs. (27), (28), (29), and (30), and estimates being approximate (as in Eq. (31)). The parameter ranges for validity of Eq. (41) are as given in Harirchian and Garimella (2012), but it does not include sufficient experimental data (even in nondimensional terms) that would cover the larger mm-scale Dh values of interest to this review. For identifying critical transition quality for annular flow boiling, Kim and Mudawar (2013a) also recommend another criterion. This criterion is similar to the one for flow condensation and, therefore, is described instead in Sect. 3.2.3. The effectiveness of these empirical

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correlations is limited and may provide only some order of magnitude estimates. More accurate, nonlinear stability analysis based transition quality (plug-slug to annular) correlations for Eqs. (17) and (18) can also be obtained by a synthesis of CFD/DNS approach (see Ranga Prasad et al. 2017) and new especially designed experiments – but such estimates are currently limited to low mass-flux and heat-flux cases of annular flow boiling in channels under hypothetical pure convective boiling scenarios (i.e., suppressed nucleation cases - real or assumed, as in Eq. (32)).

3.2.3 Annular Flow Condensation This regime is of interest here (Fig. 3b), and it occurs upstream of the slug and plug regimes (Fig. 2), as quality X decreases, in traditional flow-condenser operations. For “method of cooling” being specified by known/assumed wall temperatures, Kim and Mudawar (2013a) recommend the following criteria: e 0:2 We > 7:X tt

(43)

e tt is as defined in Eq. (30) (or Eq. (36)) and: where X We 

2:45ReV ðxÞ0:64  0:4 for ReL ðxÞ  1250 e 0:039 Su0:3 1 þ 1:09 X tt vo

(44)

or e tt 0:85Rev X We   0:4 e 0:039 Su0:3 1 þ 1:09 X tt vo 0:157



"   #0:084 μ V 2 ρL for ReL ðxÞ > 1250 μL ρV

(45)

Note that in Eqs. (43), (44), and (45), SuVO  ρV σDh =μ2V , and the range of parameter space for these correlations is as given in Kim and Mudawar (2013a). By plotting the above criteria on an X
ReT plane, the results in Eqs. (43), (44), and (45) can be presented in the form of Eq. (17). Again, effectiveness of the abovedescribed type of correlations is expected to be limited and, at best, is meant only to provide order of magnitude estimates for circular and rectangular cross-sections (of aspect ratio near unity) tubes. For specific fluids and parameter ranges (inlet pressure, etc.), the correlations in Eqs. (43), (44), and (45) can be compared with flow-regime maps of Coleman and Garimella (2003), etc. For low mass fluxes (G) and channel flow condensation in Fig. 3b (aspect ratio ~ 0), the correlation in Eqs. (43), (44), and (45) can even be compared with accurate nonlinear stabilitybased transition quality (annular to plug-slug) correlation given by Naik et al. (2016) and Naik and Narain (2016). The correlation proposed by Naik et al. (2016) and Naik and Narain (2016) for flow condensation in a channel yields the distance from x = 0 (where X(0) = 1) to the onset of plug-slug regime (x = xA|lg) in the horizontal channel flow configurae h ¼ 4h (instead of “h” used as characteristic tions of Fig. 3b. Replacing Dh by D

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length in Naik et al. 2016 and Naik and Narain 2016), it is recalled here that (see Narain et al. 2015):  
2:17  1:03  1:64 Ja ρ μ 0:85 ^ x A jlg ffi 0:237 ðReT Þ : V : V (46) PrL ρL μL and x Þj^x ¼^x A jlg Xcr ¼ Xð^ ffi1


5:41∙^x j0:73 ^x ¼^x A jlg

ðReT Þ


0:64



Ja PrL

1:02  
0:33  
0:86 ! ρ μ : V : V ρL μL

(47)

The parameter ranges over which Eqs. (46) and (47) are valid are the same as the ones given for Eq. (39). These low heat and mass-flux parameter ranges are more limited than the ones given in Kim and Mudawar (2013a) for Eq. (43). There is an order of magnitude agreement in the flow-regime transition boundaries obtained from Eqs. (43), (46), and (47), (see Narain et al. 2015) for 16,000  ReT  32,000. Note that there is no surface-tension dependence in Eqs. (46) and (47) for modest mass-flux thin-film annular flows (unlike Suratman number SuVO dependence in Eqs. (43), (44), and (45)), and this is consistent with the specific physics of channel flows. Eq. (43), being a curve fit, loses accuracy but gains in the parameter space ranges over which order of magnitude estimates are valid.

3.3

Void Fraction (ϵ) and Quality (X) Correlations

Void fraction (ϵ) can be defined “locally” (Carey 1992; Ghiaasiaan 2007), i.e., as a variable whose value depends on the location of a point and the instant of time in a given two-phase flow field. However, for the “steady-in-the-mean” in-tube flows of interest (such as the ones in Figs. 1, 2, and 3), void fraction ϵ definition simplifies (Ghiaasiaan 2007) to: AV ðxÞ ϵ (48) A where AV(x) is the cross-sectional area occupied by the gas phase at any location “x” (see Figs. 1, 2, and 3) for two-phase flows in a tube of cross-sectional
area “A.” _ V ðxÞ=M _ in Þ, It is expected that dependence of void fraction ϵ, on quality XðxÞ  M density ratio ρV/ρL, viscosity ratio μV/μL, etc., need to be correlated. This is important in assessing the significance of actual mean gas-phase speed UV(x) and the mean liquidphase speed UL(x) – as opposed to uniform superficial speeds (Carey 1992) jV  G/ρV _ V ðxÞ=ðρV :AV ðxÞÞ ¼ G:XðxÞ=ðρV :ϵðxÞÞ and jL  G/ρL. This is because UV ðxÞ  M _ L ðxÞ=ðρL :AL ðxÞÞ ¼ G:ð1
XðxÞÞ=ðρL :ð1
ϵðxÞÞÞ . As a result of and UL ðxÞ  M this importance, several such correlations have been developed and are used in   developing correlations for total pressure gradient
@@ px , interfacial shear T

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correlations, film thickness correlations, and heat transfer rate correlations for high mass-flux annular flows (Cavallini et al. 2006; Kosky and Staub 1971; Thome 2004). The acceleration component of the total pressure gradient is very important in high heat-flux boiling because UV(x) rapidly increases with “x” (Kim and Mudawar 2014; Thome 2004). Most of these void-fraction correlations have been developed by considering adiabatic flows and using various homogeneous, separated, and drift-flux modeling hypotheses (Ghiaasiaan 2007) – suitably correlated for agreement with adiabatic flow experiments in limited contexts. These correlations are not to be taken, however, as ones that model the “physics” of the phase-change flows. For example, for low mass flux laminar/laminar flows in a channel, ϵ
X relationships can be obtained by a combination of exact analysis or equivalent, DNS, for: annular adiabatic, certain suppressed nucleation annular flow-boiling (see Ranga Prasad et al. 2016), and for annular flow-condensation cases (Naik et al. 2016; Naik and Narain 2016; Narain et al. 2015). The comparisons of exact results for a representative situation (involving ρV/ρL = 0.086, μV/μL = 0.0235, Ja/PrL = 0.034, ReT = 4 . Reh = 9616) are plotted in Fig. 7 – along with results from some well-known correlations in literature – on an ϵ
X plane. Clearly, laminar liquid/laminar vapor nearly exact ϵ
X relationship of adiabatic flows does not match similar exact CFD-/DNS-based accurately modeled “flow physics” results obtained for flow condensation and flow boiling. Despite the “physics” issues associated with using ϵ
X adiabatic correlations for phase-change flows, two such popularly used engineering correlations are also plotted for laminar/laminar situations in Fig. 7. These popular correlations are as follows: • Zivi (1964): ϵ¼

Fig. 7 Comparisons of exact solutions (analytical and computational) for laminar/ laminar annular flow – adiabatic flows, flow condensation (Ja/PrL = 0.034), and flow boiling (Ja/PrL = 0.034). For order of magnitude comparison purposes, Zivi (1964) and Rouhani and Axelsson (1970) correlations are also plotted for the same parameters (ρV/ρL = 0.086, μV/μL= 0.0235, ReT = 4 . Reh= 9616)

1

  1
X ρv 2=3 1þ X ρL

(49)

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• Rouhani and Axelsson (1970) (with Steiner 1993 modifications for horizontal tubes): ( X ϵ¼ ρV

)
1   X 1
X 1:18 gσ ðρL
ρV Þ 1=4 ½1 þ 0:12ð1
XÞ þ ð1
XÞ þ ρV ρL G ρL 2 

(50) Besides quality X, adiabatic flow ϵ
X curve depends on ρV/ρL, μV/μL, and ReT. Condensing and boiling flows additionally depend on Ja/PrL (or Bl). The main value and use of adiabatic flow correlations, such as the ones in Eqs. (49) and (50), for flow boiling and flow condensation are not their ability to capture flow physics. It lies in the fact that their order of magnitude correctness (see, e.g., comparisons in Fig. 7) are used over a large range of parameters (G values, liquid-vapor Reynolds numbers, etc.) that are not currently accessible by proper flow-physics based analyses. Also correlations such as Eqs. (44) and (50) allow development of other “curve fit” correlations for HTC and pressure-drop that cover a similarly large range of parameters in their experimental data – such as those in Kim and Mudawar (2013c) and Kim and Mudawar (2013d).

3.4

Pressure-Drop Correlations

The engineering approach (as opposed to CFD simulations approach) is to use results obtained from an integrated momentum balance (see Carey 1992) for the control volume between any two arbitrary locations “x” and
“x +Δx” in Fig. 6 for an a priori decomposition of the total pressure gradient
@@ px T for two-phase flows into three parts (frictional, gravitational, and acceleration/momentum) as:         @p  @p @p @p
¼
þ
þ
@x T @x fric @x g @x acc

(51)

The subsequent step consists of defining/modeling the three parts on the right side of Eq. (51) separately and then assembling the three terms over the tube length (0  x  L) of interest. Denoting the total pressure-drop, or rise, as ΔpT ( pin
pout = p(0)
p(L)) – and allowing ΔpT to be negative (as pressure rise is possible for some condensing flow cases) – one obtains:    ðL  ðL  ðL  @p @p @p ΔpT 
:dx þ
:dx þ
:dx @x fric @x g @x acc 0

0

 ðΔpÞfric þ ðΔpÞg þ ðΔpÞacc

0

(52)

For each of the three pressure gradient terms, particularly the frictional pressure gradient, there are several modeling approaches (Ghiaasiaan 2007; Kim and

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Mudawar 2014; Thome 2004). For the acceleration and gravity terms, the following definitions arise from adding 1-D form of vapor and liquid momentum balance equations (see derivation and definitions in Carey 1992): " #   2 @p ð1
XðxÞÞ2 2 d X ð xÞ
¼G þ @x acc dx ρV∙ ϵ ρL ð1
ϵ Þ

(53)

  @p
¼ ½ϵρV þ ð1
ϵÞρL :g sin φ @x g

(54)

In obtaining Eqs. (53) and (54), mass-balance equations and definitions of void fraction ϵ, mass-flux G, and quality X are also used. In Eq. (54), “φ” measures the  angle between the tube axis and the horizontal, and hence
@@ px ffi 0 for g

horizontal tubes, as φ = 0. For integrating Eq. (53), one may choose a void-fraction model – such as Zivi’s in Eq. (49) or the one in Eq. (50). For the frictional pressure gradient in Eq. (52), one may choose any one of several adiabatic pressure gradient calculating models by replacing “X(x)= constant” situation for the adiabatic cases with genuine X(x) variations with x – as obtained by integrating the ODEs (Eqs. (23) and (24) of Sect. 2.3) under use of appropriate flow-boiling or flow-condensation Nux correlations. Samples of such Nux correlations are given in Eqs. (27), (28), (29), and (30) of Sect. 3.1.1 or in Eqs. (35), (36), (37), and (38) of Sect. 3.1.2. Two well-known frictional pressure gradient models for Eq. (51) are as follows. Lockhart-Martinelli Model     @p @p
¼
Ø2 @x fric @x L L

(55)

where ØL is one of the several two-phase multipliers which depends on a certain e MF through the following relationships: Martinelli factor X  


@p @x

 L





2f L G2 ð1
XðxÞÞ2 ρL Dh

2f V G2 XðxÞ2 V ρV Dh C 1 Ø2L ¼ 1 þ þ ~ ~ X MF X MF ð @p=@x ÞL 2 ~  X MF ð@p=@xÞV


@p @x



(56)

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The following quantities, with subscript “k” – where “k = L” or “k = V,” are needed for evaluating the terms in Eqs. (55) and (56). These quantities include ReL(x)  G(1
X(x))Dh/μL and ReV(x)  G . X(x)Dh/μV. For phase “k” to be laminar, Rek < 2000, and for it to be turbulent, Rek > 2000. The constant C in Eq. (56) is given in Table 5. The friction factors in Eq. (56)1–2 are: f k ¼ 16Re
1 k   for  Rek < 2000, f k ¼ 0:079Re
0:25   for  2000  Rek  20, 000, and k f k ¼ 0:046Re
0:2   for  Rek  20, 000 k Friedel Model 


@p @x

 ¼ fric

  @p
Ø2 @x L Friedel

(57)

where while retaining the remaining definitions in Eq. (56), Ø2Friedel is defined (see Friedel 1979) as:

Ø2Friedel  E þ

3:24 FH Fr0:045 ∙We0:035 H L

(58)

The terms in Eq. (58) are:

E ¼ ð1
XÞ2 þ X2 :

ρL f V : ρV f L

F ¼ ð1
XÞ0:224 • X0:78  0:91  0:19  0:7 H ¼ ρρL : μμV : 1
μμV V

L

L

G2 FrH ¼ gDρ2H h i
1 ρH ¼ ρX þ 1
X ρ V

(59)

L

G2 Dh WeL ¼ σ:ρH Table 5 Values of C for Liquid Lockhart-Martinelli model Turbulent Laminar Turbulent Laminar

Gas Turbulent Turbulent Laminar Laminar

C 20 12 10 5

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Table 6 Values of C for (a) adiabatic and condensing flows; (b) boiling flows (a) Liquid Turbulent

Gas Turbulent

Laminar

Turbulent

0:36 0:19 0:0015Re0:59 LO SuVO ðρL =ρV Þ

Turbulent

Laminar

0:29 0:5 8:7  10
4 Re0:17 LO SuVO ðρL =ρV Þ

Laminar

Laminar

0:48 0:5 3:5  10
5 Re0:44 LO SuVO ðρL =ρV Þ

Cnon
boiling 0:35 0:1 0:39Re0:03 LO SuVO ðρL =ρV Þ

(b) Cboiling

ReL  2000 ReL < 2000

  0:78 PH Bo Cnon
boiling 1 þ 60We0:32 LO PF   1:09 PH Bo Cnon
boiling 1 þ 530We0:52 LO PF

Kim and Mudawar (2014) report poor comparisons of experimental pressuredrop (considering a large set of data) for flow boiling and flow condensation with those obtained from the above-described procedures employing correlations by Lockhart and Martinelli (1949) and Friedel (1979). Kim (2013d), after introducing ReLO( GDh/μL) and
and Mudawar  SuVO  ρV σDh =μ2V , recommend continued use of Lockhart and Martinelli model in Eqs. (55) and (56) with replacements for the constant C (appearing in Eq. (25)) in Table 5 being given, for adiabatic and condensing flows, as in Table 6a below. For boiling flows, calculation of constant C = Cboiling involves use of Cnon
boiling (calculated for adiabatic and condensing flows) in Table 6a through relationships given in Table 6b. Grönnerud Model This popular pressure gradient model, available in Grönnerud (1972), is used here – but not reviewed for brevity.

3.5

Available CHF Considerations and Correlations

As discussed in the Introduction and a subsection (Flow and CHF-Related Instabilities) of Sect. 2.2, crossing certain threshold values of critical heat-flux while progressively increasing the values of a uniform heat-flux impositions on the flow-boilers (most conservative “method of heating” for identifying this phenomena) leads to vapor blanketing of certain downstream parts of the boiling surface. This is followed by sustained unsteady instabilities and subsequently – in some cases – runaway unsteady rise in the temperature of the vapor-blanketed boiling surface. There are several mechanisms for such flow-boiling instabilities – with only a qualitative relationship to the extensively studied CHF mechanisms for pool boiling.

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The pool-boiling CHF, as discussed for the boiling curves in the “q}w
ΔT” plane (Collier and Thome 1994; Dhir 1998), relates to emission of bubbles with increasing and then plateauing bubble departure diameters, significantly increasing bubble departure frequency as well as nucleation site density with increasing heat-flux (see, e.g., McHale and Garimella 2010). At certain limiting values of these millimeter-scale departure diameters and plateauing bubble departure speeds, the bubbles ram into one another and coalesce – creating both vertical and lateral “connected vapor zones.” Thus, the heated surface loses direct contact with liquid (except for a solid-like adsorbed liquid layer) and develops, instead, a poor but direct thermal contact with the vapor of much lower thermal conductivity. This poolboiling CHF threshold has been studied from the point of views of instability (Kelvin-Helmholtz) of vapor columns of jets – issuing in the direction normal to the boiling surface – as CHF heat-flux level is approached from lower values of heatflux (Dhir 1998; Dhir and Liaw 1989; Haramura and Katto 1983, etc.). This has also been studied from the point of view of instability (Rayleigh-Taylor) of vapor film blankets as post-CHF heat-flux values are reduced (Linehard and Dhir 1973; Zuber 1959). These and other related studies (Katto 1994) have been reviewed, and their relationship to ebullition cycle of nucleating bubbles below CHF and nucleation site density growth have been and are being explored/studied (Dhillon and Buongiorno 2017; Gerardi et al. 2009; McHale and Garimella 2010; Phan et al. 2009; Zeng et al. 1993a, b; Jones et al. 2009, etc.). Such investigations need to be better related to CHF. These pool boiling works relate to flow-boiling CHF only for the limited G ffi 0 zone in Fig. 5. The flow-boiling CHF mechanisms at higher G values – as discussed through Figs. 1, 4, and 5 – and associated dry-out heat-flux and dry-out-related CHF values (Fig. 1b) are available in Kim and Mudawar (2013b). Other correlations such as Wojtan et al. (2006), Katto and Ohno (1984), Bowring (1972), and Zhang et al. (2006) can also be used with discrimination – as discussed in Basu et al. (2011). These are the CHF estimates (CHF1 in Fig. 1b) that are of interest to this chapter. This is because the onset of inverted annular flows related CHF (CHF2 of Fig. 4 and Fig. 5) are typically higher for the innovative designs of interest – which involve millimeter-scale ducts, controlled liquid film thickness annular boiling, and avoidance/minimization of liquid entrainment up to the exit. If the heat-flux relative to mass flux is higher than a certain threshold (Fig. 5), at lower qualities – through departure from nucleate boiling (DNB) and associated inverted annular flows – CHF is achieved (Fig. 4), and its value can be estimated. The estimates can be based on CHF for subcooled flows at the inlet in the limit of the subcooling going to zero. Such CHF results are available in Collier and Thome (1994) and Qu and Mudawar (2004). In the subsection (Flow and CHF-Related Instabilities) of Sect. 2.2, it was argued that inverted annular flow CHF typically begins somewhere in the macro-nucleate boiling region I or plug-slug region II of Fig. 1b. However, there are exceptions. For example – at the connected inlet of multiple parallel channels – the instability may begin right at the inlet (Qu and Mudawar 2004), or for microscale ducts, it may begin at large local qualities

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associated with the wispy annular regimes (see experimental photographs in Kuo and Peles 2009). Also, for microscale ducts – which are not of primary interest here – CHF values get affected by surface tension and surface texture (reentrant cavities, etc.) in a different way than the larger-scale ducts (see Kuo and Peles 2009). Also, for microscale duct flow boiling, CHF mechanisms get affected when vapor bubbles grow to the size of the micron-scale ducts (see Kumar and Kadam 2016). Since the threshold defining critical curves in Fig. 5 have not yet been established with any degree of consensus, it is not uncommon for researchers to report that, at moderate mass flux, CHF could manifest itself through either of the two mechanisms discussed above (Shah 2015a). As conjectured in Das et al. (2012), CHF arises from interactions between macroscale instability far (in the normal direction) from the heated surface and a more universal microscale phenomena near the heated surface. Near the heated surface, at CHF, replenishment of fluid into the micro-layer (see Raghupathi and Kandlikar 2016 and Sect. 6) present on the wetting surface – between the wet and dry sides – is not allowed from the wet side. With the macro-layer instability and micro-layer, phenomena working together, a dynamic CHF process is often triggered. Therefore, this CHF process has as many quantitative characterizations as there are varieties of macroscale flows (see multitudes of CHF characterizations in Katto 1994). The dynamic CHF process results in a situation where only an adsorbed layer of fluid in the vapor-blanketed domain is allowed while the time-averaged area associated with this vapor blanketing becomes larger and sustained. In low mass-flux and low heat-flux cases, the dry-out instability as heat-flux values are increased for the annular flow in Fig. 1b is also indicated by a large surplus of mechanical energy that is supplied from the liquid film to the adjoining vapor flow, while the thin liquid film near dry-out zone consumes less and less viscous dissipation energy – see CFD studies of Ranga Prasad et al. (2017). This surplus of mechanical energy, typically, has to go to the vapor flow near the dry-out zone (see streamline patterns in Ranga Prasad et al. 2017). As higher heat-flux values are approached, this surplus energy can no longer be absorbed by either the steady flow of the vapor or the steady flow of the liquid (on the wetting surface) near the dry-out zone – and this is likely one of the contributing reasons that lead to dry-out instabilities. This is also a likely reason as to why dry-out instabilities can often be delayed (i.e., CHF values increased) by enhancing the liquid retaining wetting characteristics near the exit zone of the flow-boiler in Fig. 3a. These liquid retaining or liquid supplying techniques require sustaining the liquid flow in the liquid microlayer in a way that it resists formation of nearby dry adsorbed layer through passive material alteration leading to super-hydrophilicity of the surface near the boiler exit or use of a porous and wetting exit zone boiling surface with additional and independent liquid supply into the pores or by active enhancement of wettability by use of suitable electric field impositions underneath a dielectric (near the exit) boiling surface, etc. For design purposes of flow-boilers in Fig. 3a (as discussed in the next section), one can tentatively use a relevant and existing CHF threshold value estimates and/or simply ensure – by flow control (also see Sect. 6, as this is possible through proper

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design of innovative annular flow-boilers) – that the exiting liquid layer is kept sufficiently thick (>10–20 μm or so).

4

Above-Reviewed Two-Phase Flow Correlations in the Context of their Use in the Design of Innovative Devices Experiencing Annular and Steady Flow Boiling and Flow Condensation

The correlation structure discussed in Sect. 2 supported the specific correlations reported in Sect. 3. The specific correlations in Sect. 3 were either developed with the help of data obtained from experiments (often flow-regime transition and pressure gradient correlations used adiabatic flow-regime maps) or computational/analytical solutions or by a combination of the two. The structure of correlations reported in Sect. 2 were developed with the aid of theory underlying nondimensionalization processes and some understanding of the physics that underlie these phase-change flows. Design of new experiments or new systems for a given working fluid and results obtained for a range of operating conditions can help develop such correlations. This, in turn, requires preliminary – but mutually consistent – “order of magnitude” estimates for values of variables (such as liquid and vapor flow rates, wall heat-flux or wall temperature values, inlet pressure, length and hydraulic diameter of the device, heat transfer coefficient values, pressure-drop values, liquid film thickness values, etc.) which will make device operations possible. To begin with, until better correlations or experimentally obtained refinements for the existing correlations become available, correlations that are presently available need to be judiciously used to define the experimental and/or design operating conditions and associated instrumentation requirements. Need and use of existing correlations are motivated here with the help of two specific examples. The first example is that of a preliminary design of a millimeter-scale flowboiler – operating in a steady annular/stratified regime with thin liquid film flows (Fig. 3a) – either in the presence of nucleation (at submicron-scale bubble diameters, as in regimes III and IV of Fig. 1b) or under suppressed nucleation conditions (as in regime Vof Fig. 1b). The second example is that of a preliminary design of a millimeterscale flow-condenser for a steady annular/stratified film-wise condensation on a hydrophilic surface (see Fig. 3b). In both examples, it is assumed here that the fluid flows through a horizontal duct of rectangular cross-section with high aspect ratio (adequately modeled as a channel) – with the heat-exchange surface being the bottom plate.

4.1

Design of Millimeter-Scale Annular Flow Boilers: An Example Illustrating Use of the Reviewed Correlations

The next subsection discusses some of the desired specifications and constraints for the design of an innovative flow-boiler operating in the annular flow-regime.

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4.1.1 (i) (ii)

(iii)

(iv)

2123

Desired Specifications, Constraints, and Information/Knowledge Needed for Meeting the Requirements A pure working fluid should be chosen such that it has a saturation temperature in 30–90  C range for operating inlet pressures in 100–110 kPa range. The inlet quality (at x = 0 in Fig. 3a) should be higher than the critical quality of transition from non-annular to annular flow-regimes – ensuring annular flow realization. For this, it is important to know, approximately, the quality at which the flow-regime transition occurs with estimates coming from different flow-regime transition maps. The required scientific structures for this have been discussed in Sect. 2, and the status of available knowledge has been discussed in Sect. 3. These estimates are sought to be within a reasonable range of values that would suffice, with trial-and-error experimental adjustments, for present purposes. It is hoped that, in the future, such estimates could be further improved (i.e., made more accurate) through a proper synthesis of experiments and modeling. The inlet film thickness Δin, the liquid film thickness just downstream of the splitter plate in Fig. 3a, is desired to be around 300 μm. This is, presumably, neither too thin nor too thick, and it is important for subsequent and possible transitioning of the steady operations to pulsatile operations – the benefits of which have been alluded to through curves I1
(ii) and I2
(ii) in Fig. 1b and are further discussed in Sect. 6 (see experimental results given by Kivisalu et al. 2014). The control of inlet liquid film thickness by adjusting the recirculating vapor flow rate is essential because, otherwise, the liquid film thickness may change abruptly – as it exits the splitter plate in Fig. 3a – to an undesirable range of values. Thickness, at or below 300 μm, is needed to keep the films very stable – with or without micron/submicron diameters nucleating bubbles – even when large amplitude standing waves are superposed on the interface. For such a design, a correlation for inlet film thickness with dependence on inlet quality is required as input. Since this is a stratified/annular flow through a rectangular channel, with nearly adiabatic self-seeking free-surface locations immediately downstream of the inlet splitter plate (see Fig. 3a), a set of voidfraction correlations for adiabatic flows (as discussed in Sect. 3) could be used to obtain a good range of appropriate inlet quality and associated inlet film thickness values. To avoid dry-out-related CHF instabilities at or near the exit, the exit quality should be less than 1 and the heat-flux at the exit should be less than the available order of magnitude estimates for CHF associated with dry-out instability ( q}wjexit < q}CHFjdry
out ). For this, an estimate of a dry-out-related CHF, obtained from correlation(s), such as the ones presented by Qu and Mudawar (2004), may be used. However, instead, the following more conservative constraint is utilized here. As a safety measure, it is required that the film thickness at the exit, estimated in different ways, be equal to or greater than 20 μm or, alternatively, one-fifth of the inlet film thickness (i.e., Δout  Δin /5  O (10 μm)). This constraint is also helpful in ensuring that inlet vapor

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speeds (or qualities Xin ) are not too large, as entrainment of liquid into the vapor needs to be avoided/minimized. Some of the important information that are required for this design are the Reynolds numbers: Reynolds numbers based on total mass-flux ReT (Gh/ μV), liquid Reynolds number ReL ( G(1
X)h/μL), and vapor Reynolds number ReV ( GXh/μV). The first Reynolds number is typically used in evaluating Nusselt number correlations for saturated flow boiling covering annular regimes (over appropriate range of qualities). Meanwhile, the second and third Reynolds numbers are often needed for assessing whether the liquid and vapor flows are laminar or turbulent and, accordingly, for selecting constants in sub-correlations (either directly for Nusselt number correlations or indirectly for related pressure-drop correlations). The knowledge of the laminar/turbulent nature of liquid and vapor flows are also useful in choosing appropriate correlation(s) that are available for different void-fraction models. Since there are separate inlet channels for liquid and vapor, and a splitter plate is used (as in the experiments of Kivisalu et al. 2014) to ensure that the liquid and vapor do not mix before they enter the test section (as shown in the Fig. 3a), the cross-sectional area for vapor flow just before the inlet (locations x < 0) is different from the cross-sectional area for the vapor flow within the test section (locations x  0). Furthermore, because of nonzero vapor inflow rate at the inlet and its subsequent acceleration associated with flow boiling, the vapor speed at the exit of the test section would be higher (much higher for higher heat-fluxes) than it is at the inlet of the test section. To avoid compressibility-related choking effects (see Ghiaasiaan 2007), it is required that the vapor speeds at both the inlet and the exit be, approximately, less than one-third of the speed of sound for saturated vapor operating at pressures that are at or below the inlet pressure value. The inlet pressure pin is required to be above atmospheric pressure such that, despite the pressure-drop along the length of the channel, the exit pressure is also higher than the atmospheric pressure (i.e., pout > patm). This requirement is to make the design simpler, so that the system does not have to exhibit stringent and prolonged tolerance to vacuum pressures. For systems operating below atmospheric pressure, even small leaks of air can, over time, lead to substantial buildup of non-condensable air into the system, and then this system design, based on pure vapor and pure liquid flow-physics assumption and associated correlations, as proposed, will fail. Further, despite the pressure-drop between the inlet and the exit of the channel, mechanical power in the vapor at the exit is likely to be much more than the incoming power at the inlet – i.e., for exiting vapor speeds relative to the inlet, it is expected that PV,out( pout . vV,out . Aout) > PV , in( pin . vV,in . Ain). The net power out (PV,out
PV,in) has to be maximized for high heat-flux cases. This is to ensure that no compressor is needed for steady operations and for start-up, shutdown, and other transients, only minimal additional power is needed for the recirculating vapor flow compressor shown in Fig. 3a. Note that the liquid pump and pulsator options (discussed in Sect. 6) already consume insignificant power.

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(ix) For a known uniform heat-flux specifying the “method of heating,” Eq. (24) yields linear quality variations, and, subsequently, order of magnitude estimate of the heat transfer coefficient hx can be obtained from a Nusselt number correlation – which is only needed for assessing wall temperature Tw(x) variations associated with the boiling surface. It is required that the design be such that the mean wall temperature Tw be not too high and remain below a certain threshold value specified by the application (e.g., Tw  85  C for electronic cooling applications). (x) The key parameters whose desired ranges need to be recommended, or chosen, to propose a design that satisfies the above conditions are massflux G, length of the channel L, height of the channel h, inlet pressure pin, and inlet quality Xin.

4.1.2

Implementation of a Sample Design Methodology: Meeting the Requirements Given in “Section 4.1.1” and Obtaining Results for Steady Annular Flow-Boiler Operations A range of inlet pressures pin, total mass fluxes G, and mean heat-flux q}w (or mean wall temperatures Tw , depending on what is known or preferred assumption about the “method of heating”) are initially chosen and considered. They are then optimized to satisfy most of the constraints mentioned in the above-described design requirements. Note that the length L of the flow-boiler may also be adjusted to satisfy remaining constraints. This is a reasonable approach because of the flexibility to stack these flow-boilers in series and/or in parallel arrangements toward a feasible solution for a new high heat-flux system design that can remove heat from a designated cooling area of interest. Toward achieving a high heat-flux design, to begin with, one particular combination of inlet pressure, total mass-flux, a selected channel height h, and mean heatflux (or mean wall temperature), is defined as a specific initial choice of operating condition for a single flow-boiler element (Fig. 3a) of the required system design. For example, if one is interested in achieving a steady heat removal capability at 1 kW/cm2 from each of the flow-boilers in a new system design, one can investigate and attain an acceptable steady annular operation (say, at a lower imposed heat-flux value of about 20 W/cm2) and a design restriction of Tw(x) < 85  C over the length of the boiler. This can be achieved for suitable initial operating conditions specified for steady operations in Fig. 3a. Next by superposing pulsatile or other innovative operations, on the initial steady annular operating conditions indicated by points A and B and curves I1
(ii) and I2
(ii) in Fig. 1b, c, and controlling the range of external impositions (e.g., pulsation amplitudes at a given frequency, see Sect. 6), different phenomena (e.g., different levels of large amplitude standing waves on the interface) can be realized to obtain a range of very high local heat transfer coefficient hx for the innovative operations. It can be shown that the initial steady design’s q}w ( 20 W/cm2) handling capability can be further increased in two steps by shifting to pulsatile (or similarly new) operations and increasing the local heat transfer coefficient hx by a factor of about 10–30 (over the initial steady realization), provided a good strategy is

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followed. In the first step, pulsatile (or equivalent) operations’ increasing values of imposed controlling variable (e.g., amplitude at a certain frequency, see Sect. 6) can lead to a q}w (200 W/cm2) and Tw(x) < 85  C operations by only slightly adjusting the initial mass flux for the chosen length L. In the second step, pulsatile (or equivalent) operations’ imposed controlling variable’s values are further increased (e.g., amplitudes at the same frequency for pulsation impositions) with concurrent increases in q}w (to 1 kW/cm2) and inlet mass-flux G while retaining the inlet quality from the second step. The second step also requires simultaneous suitable vapor bleeding flow rates from a set number of locations on the top dry plate in Fig. 3a. This is for satisfying incompressible vapor (Mach number < 0.28) and wall temperature Tw(x) < 85  C restrictions. However, the purpose of this review is not to propose a high heat-flux system design. Its focus is limited to elucidating the importance of a steady design methodology for the initial annular steady flow realization – as an enabling component of a high heat-flux system design. For this illustration, toward meeting the design requirements elucidated in Sect. 4.1.1, an example case is chosen for which the channel height is 5 mm, the working fluid is R-123, and wall temperature restriction is Tw(x) < 150  C. Additional tentative choices are inlet pressure pin = 120.1 kPa, total mass-flux G = 300 kg/m2 s, and mean heat-flux q}w = 50 W/cm2 = 500 kW/m2. The flowboiler length L is to be chosen in a way that the requirements given in Sect. 4.1.1 are met by following the steps given below: (i) For any chosen operating conditions, the first step is to find the critical quality for transition from non-annular to annular flow-regimes (XcrjNA
A) through available maps for traditional Fig. 1a operations, so an inlet quality Xin > XcrjNA
A can be chosen for Fig. 3a operations. This is accomplished by using approximate flow-regime transition maps. A few appropriate flow-regime transitions maps should be chosen, based on hydraulic diameter Dh, fluid, orientation (horizontal, in this case) of the channel/tube, etc. Since the flow-regime transition maps are semi-empirical in nature (i.e., correlated using data from either existing experiments or modeling/simulations), as a conservative measure, it is recommended that the critical quality for transition from non-annular to annular flow-regime XcrjNA
A be obtained from at least three different flow-regime transition maps and the highest of those qualities be used. For the flow-boiler design under consideration, flow-regime transition maps proposed by Harirchian and Garimella (2012), Kim and Mudawar (2013a), and Mandhane et al. (1974) were considered here to be approximate and relevant ones for further evaluations. Even though the map by Mandhane et al. (1974) was developed for adiabatic flows, its use in the present scenario is justified because the flow will be approximately adiabatic immediately after it exits the splitter plate at the inlet (Fig. 3a). The transition qualities obtained using the correlations are tabulated in Table 7. It can be noticed that the critical transition qualities obtained from correlations proposed by Kim and Mudawar (2013a) and Harirchian and Garimella (2012) are quite low and less likely to be applicable to the present design estimates. It was

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Table 7 Critical transition qualities from non-annular to annular flow-regime obtained using correlations for the channel height (h = 5 mm; fluid, R-123; and operating conditions, pin = 120 kPa; G = 300 kg/m2 s;
q }w ¼ 50 W/cm2) Flow-regime transition correlations Harirchian and Garimella (2012) Kim and Mudawar (2013a) Mandhane et al. (1974)

Transition quality, XcrjNA
A 0.028 0.0283 0.307

further noticed that, for low values of mass-flux (G < 30 kg/m2 s), Kim and Mudawar (2013a) correlation provided higher values of critical transition quality, while the critical transition qualities obtained from correlation by Harirchian and Garimella (2012) and flow-regime map by Mandhane et al. (1974) were quite low ( 0.3 (as implied by flow-regime map of Mandhane et al. 1974). It should be noted that, typically, there is also an experimentally determined upper bound on Xin so one can avoid entrainment and wispy annular flows. (ii) For approximately realizing the assumed inlet film thickness Δin of around 300 μm, first the vapor Reynolds number ReV (GXh/μ2) is calculated with XcrjNA
A as the initial guess of inlet quality Xin. This is to check if the vapor flow is laminar or turbulent. If the vapor flow is turbulent (and it is ensured that the liquid flows are still laminar), the adiabatic void-fraction correlations proposed by Zivi (1964) and Rouhani and Axelsson (1970) (with modifications for horizontal flows proposed by Steiner 1993) are used to calculate film thickness (note that ϵinjcorrelation = (h
Δin)/h). If the vapor flow is also laminar, void-fraction correlation proposed by the authors (Ranga Prasad et al. 2017) may be used along with the ones mentioned earlier. Although all the abovementioned void-fraction correlations were originally proposed for adiabatic flows, their use in this case is, once again, justified since the flow is approximately adiabatic immediately after the splitter plate at the inlet (Fig. 3a). Using XcrjNA
A as the first guess, the corresponding film thickness values are calculated using the abovementioned void-fraction correlations. Next, the inlet quality value is typically increased to values higher than XcrjNA
A until the mean film thickness calculated from the chosen correlations gives a value of around 300 μm or less. However, since these are semi-empirical correlations with large uncertainties, the mean of the different film thicknesses obtained from the three different correlations above is actually considered the final film thickness value from this void-fraction approach. The final inlet film thickness value, for this example, is estimated to be 299.9 μm, and the corresponding inlet quality is found to be 0.405 (>XcrjNA
A). (iii) For this inlet quality, using the space available for the vapor flow above the splitter plate (3.7 mm for the current case), the vapor velocity and its ratio relative to the speed of sound are calculated. For the current case, it is found to be 0.17. If the ratio of inlet vapor speed to the speed of sound is more than 0.28 (conservatively chosen to be less than 0.33), it is suggested that, initially, the mass-flux G be reduced (which essentially means changing the initially

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Fig. 8 Variation of quality X along the length of the channel

guessed operating condition) and steps (i) to (ii) be repeated. If that is not feasible, the height of the channel may also be increased so that the ratio of the vapor speed to sound speed is reduced. (iv) If the compressibility effect constraint (i.e., Mach number 0.06 m corresponds to annular to non-annular transition and yields critical quality XcrjA
NA correlations as given in Sect. 3.2.2

Fig. 23 Streamline patterns of a steady flow for (a) a horizontal channel and (b) a 2 downward inclined channel for annular condensation. The color map shows the velocity magnitude distributions. The run parameters are same as in Fig. 21

The above-described approach, if further developed, can be used to obtain improved hx and pressure-drop correlations as well as improved understanding of the flow physics.

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High Heat-Flux Advantages of Superposing High-Amplitude Standing Waves on the Interface for Steady-in-the-Mean Innovative Annular Flow-Boiling and Flow-Condensation Operations

In recent experimental works dealing with innovative operations of flow boiling and flow condensation (see Fig. 3 and Kivisalu et al. 2014) at low mass-flux and heatflux values, the thinness of the liquid films (see Figs. 3 and 10) and their improved stability allowed superposition of large-amplitude standing waves on the interface (see Fig. 24c) created by acoustic standing-waves in the vapor-phase (of long wavelengths relative to the device length) that, in turn, were created by pulsations introduced through oscillating flow rates (5-15 Hz) of the incoming vapor (and liquid, if needed) flows. There are other superior options for creating standing interfacial waves, but these are not discussed here. Such large amplitude standing wave realizations, as opposed to Fig. 24a, b realizations associated with steady non-pulsatile cases, are quite different in their micro-convection capabilities. The standing interfacial waves are formed over longer duration, as opposed to the short duration of forward moving interfacial traveling waves, as they result from the vapor-phase’s long wave-length standing acoustic waves (and associated oscillatory interfacial shear) which, in turn, result from the fact that the flow-boiler exit is mostly obstructed in the predominant direction of vapor flow and pulsations-induced vaporphase acoustic-waves, in the flow-direction, reflect from the exit (see Fig. 24c and Fig. 3a). As a result of superposition of such interfacial standing waves on a nearly steady interface location, as troughs got closer to the wetting surface under increasing amplitude of the waves, local time-varying heat-flux measurements indicated very high heat-fluxes (see Fig. 25 and Kivisalu et al. 2014). Consequently, both for annular flow condensation and annular flow boiling, the mean values of the measured heat-fluxes (at a representative location) increased significantly (see Fig. 26a, b). Note that experimentally attained local heat-flux enhancement (100-300%) for the flow-boiling case in Fig. 26a is much lower than the enhancements (up to 800%) observed for some comparable flow-condensation cases depicted in Fig. 26b. The above-described pulsatile flow realization approach consists of: (i) Selecting proper values of: inlet mass-flux, inlet quality, exit quality, filmthickness range, and length of the flow-boiler – as per design discussions in Section 4 – and attain a representative steady annular flow realization indicated by “point” A or “point B” in Fig. 1b, c. (ii) Employing energy efficient ways to introduce suitable amplitudes of inlet pressure or flow-rate pulsations imposed on the incoming vapor flow (and liquid flow, if needed) – ensuring inlet flow rate value (which includes a selected range of recirculating vapor flow rates) always lies within a suitable range. In case of high heat-flux boiling, the extra mechanical power available at the exit of the boiler is used, in the arrangement of Fig. 3a, to maximize the use of by-pass valve and minimize the need for a recirculation-aiding compressor.

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O (10 µm) troughs associated with high heat-flux Fig. 24 (a) Non-pulsatile steady innovative annular flow-boiling operations (Fig. 3a) at high heat and mass fluxes (under use of common refrigerants) is dominated, as per Sect. 4 estimates, by tiny nucleating bubbles (of micron to sub-micron diameters, and represented by “dots”) even for the steady annular regimes involving film thicknesses in the range of 50–300 μm. (b) Non-pulsatile steady annular flow boiling (arising from controlled recirculating vapor flow-rate at the inlet) at low mass and heat-fluxes may experience suppressed nucleation at sufficiently low film-thickness locations realized under conditions of low inlet film thickness Δ0 ~ 100–300 μm. (c) Depiction of long term and large amplitude standing interfacial waves. Large dynamic heat-flux enhancements are experienced at the “troughs,” see Fig. 25

(iii) Using primarily closed end exit condition (Fig. 3a, b) for the axial direction of the vapor flow to reflect acoustic signals (associated with vapor flow pulsations) in the vapor phase to establish standing wave patterns. This is done to transform, over time, forward-moving short duration interfacial waves into standing

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Fig. 25 Time-varying boiling heat-flux without (lower average) and with (higher average) imposed inlet liquid-vapor pulsations, measured by a flush-type heat-flux meter. The dashed lines represent long-term (~40 min) averages

interfacial wave patterns shown in Fig. 24c, and such realizations are observed in the experiments by Kivisalu et al. (2014). For creating large-amplitude standing waves, alternative and more technology-friendly approaches are possible and are being explored. (iv) Using controlled thinness (50–300 μm) of the liquid film flow to stabilize the wavy film. This allows both large amplitude standing wave formation and, concurrently, reduced and insignificant chances of liquid entrainment (even for high mass-flux cases). Under aforementioned conditions of steady-in-the-mean pulsatile flow realizations, the heat transfer enhancements shown in Fig. 26a, b are believed to result from the following hypotheses. First, it is assumed that the flow-boiling enhancements in Fig. 26a may have, at higher heat-fluxes, more explosive nucleating micro-bubble (micron or submicron diameters) presence and more significant heat-transfer contributions through hx|nb-micro, which is already (i.e., for low mass and heat-flux cases considered here) up to 80% of the total HTC given by Eq. (61) – representative calculations supportive this assertion are described in Gorgitrattanagul (2017). Furthermore, the 100300% heat-flux enhancements shown by the solid-curve (for low heat-flux cases) in Fig. 26a may not be realized, as desired and indicated by the dashed curve in Fig. 26a (or the corresponding red curves in Figs. 1b-c), for cases involving higher heat-flux

Internal Annular Flow Condensation and Flow Boiling: Context, Results, and. . .

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Expected experimental results for an efficient and planned nucleation enhancement mechanism

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q”W (x = 40 cm) trend

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Fig. 26 (a) Increase of average heat-flux with amplitude of imposed vapor fluctuations for a representative low mass-flux annular flow-boiling situation (Kivisalu et al. 2014). (b) Increase of average heat-flux with amplitude of imposed vapor pulsations for a representative low mass-flux annular condensing flow situation (Kivisalu et al. 2014)

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impositions. This is expected due to limitations in bubble-departure mechanisms available to the flow at the higher heat-flux imposition levels. Note that higher values of nucleation site densities, increased bubble generation frequencies, and limited ranges of bubble departure diameters are expected – and, as a result, the microbubbles may not be effectively carried away by the explosive levels of nucleation expected for heat-flux impositions in the range of 500-1000 W/cm2. Despite this, the enhancement levels indicated by the dashed curve in Fig. 26a is realizable for high heat-flux cases by using additional supplementary approaches – which are to be described elsewhere. However, at low heat-flux levels, there could be very significant enhancements, as shown in Figs. 26a and b. This is because classical convective contribution hx|cb significantly increases in Eq. (61) for flow boiling and for flow condensation (for which hx = hx|cb) considered here. The reason being convective heat-transfer mechanisms, associated with the thin film steady flow-physics of flow boiling (or flowcondensation sans nucleating bubbles) depicted in Figs. 24a-b, significantly changes to incorporate new mechanisms associated with the new bulk micro-convection shown in Fig. 24c. The mechanisms present for Fig. 24c are described here with the help of Figs. 27a-b (which exhibit presence of some of the known mechanisms present in pool-boiling, which is depicted in Fig. 28 and recalled here for relevant discussions). The proposed new bulk hx|cb-micro enhancement phenomena, as described below, is the sole contributor for the flow condensation (which has no nucleating bubbles) enhancement case in Fig. 26b. It is hypothesized that contactline flow-physics of Fig. 28 (well known for macro-scale pool boiling) appears here – for the micro-scale thin liquid film’s flow-physics – in a modified way to yield high heat-fluxes observed in Fig. 26. For these micron-scale (O(100 μm)) thin liquid film flows, with superposed large amplitude standing waves (such that the liquid thickness at troughs are O(10 μm)), consider the following for Figs. 27a-b:: • The three intersecting circles represent effects of interaction zones of nm-scale, mm-scale, and macro- or mm-scale phenomena. • Between the liquid-vapor interface (formed by the curve at the juncture of blue and pink zones) and the solid-like adsorbed liquid layer (marked yellow), the liquid flow at the troughs experience reduced pressure and slip-like reduced shear that allow enabling micro-convective motion upstream and downstream of the troughs – as well as the requisite through flow at the troughs. The low pressure at troughs is enabled both by the surface-tension and curvature effects near the vapor-liquid interface as well as by the liquid’s interface with the adsorbed (marked yellow) layer through negative tensile stresses (disjoining pressure) between the lower solid-liquid interface and the liquid above. This low pressure at troughs, besides allowing forward liquid motion (straight arrow in Fig. 27a) between the upstream crest-region adjacent to the trough, also enables microcirculation associated convective heat transfer enhancements within neighboring crests (recirculation arrows in Fig. 27a). The micro-circulation in the macroregion of Fig. 28 are also enabled by analogous phenomena. The expected and

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Fig. 27 (a) Near-interface schematic of dynamics of the instantaneous spatial nondimensional film thickness profile (δ  Δ/h) associated with μm-scale wavy film flows (over a wetting surface) encountered in pulsatile flow boiling (or flow condensation). The three intersecting circles in the figure, respectively, represent zones where nm-scale, μm-scale, and mm-scale phenomena interact with one another. (b) Time-varying film thickness profile for location x = x* in Fig. 27a. The film thickness time history at a trough location (x = x ) over an imposed time tP = 1/fp, where, fp is the imposed vapor-phase pulsation frequency, is such that the crests and troughs “dwell” times weighted by instantaneous heat-flux values, are dominated by the contributions of the troughs to the local time-averaged heat-flux (Fig. 25)

requisite low shear at the troughs in Fig. 27 is also hypothesized here to arise from mechanisms similar to the low shear (rather stick-slip behavior) expected near the contact-line rim of the pool-boiling bubbles in Fig. 28. For the pool-boiling case in Fig. 28, low shear at the contact-line rim possibly arises from the fact that the micro-region liquid film in Fig. 28b has intermediate slip enabling solid-like behavior adjacent to the adsorbed liquid region (or, macroscopically, the dry region) and fluid-like no-slip behavior adjacent to the macro-region. This lubricated slip-like behavior (perhaps assisted by temperatures and phase-change in a way that is not completely understood) is a key phenomenon that enables realization of such large bubble departure frequencies (5 – 500 Hz) in the poolboiling case of Fig. 28, where the surrounding stagnant liquid pool’s inertia is large and yet the frequencies keep increasing with increasing heat-flux. • Micro-scale nucleation (of bubbles) in the flows of Fig. 24 and Fig. 27 can be encouraged, diminished, or suppressed over a range of thicknesses for thin film

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y

Liquid or

Vapor Bubble Solid

r x

Heat

Contact line

Adsorbed layer

Motion depending on advancing or receding contact line

Fig. 28 Reduced pressure (disjoining pressure effects) and reduced shear (slip-like conditions) at the contact line of nucleating bubbles enable high heat-flux values at the contact line. This is due to alternating and vigorous recirculating motion – at reasonably high frequency, typically at about 5-500 Hz – of the liquid near the nano-micro-macro layer thick contact-line zone in the presence of phase-change and solid-liquid and liquid-vapor interfaces. Left and right figures are views obtained for the same contact-line zone when the smallest resolvable length scales are, respectively, approximately O(100 μm) and O (100 nm)

annular flow boiling on wetting surfaces (or super-hydrophilic surfaces with nucleation sites) – see regions III-V in Figs. 1b-c. And yet the aforementioned contact-line physics assisted micro-convection advantages in Fig. 27, as in the micro-region of Fig. 28 for mm-scale nucleating bubbles, is approximately retained with the following differences: (i) the wetting surface in Fig. 27 is always wet (except downstream of stable dry-out points, not shown) with an O(10 μm) micro-layer of liquid at the trough (which is absent in Fig. 28 where the vapor is in contact with solid-like adsorbed layer), and (ii) there is continued forward flow through the “tens of microns” thick “layer“ formed at the wave-troughs. The above-described flow physics for pulsatile operations of annular flow-boilers and flow-condensers – along with suitable design modifications for the non-pulsatile flows in Sect. 4 and some hardware modifications for the flow-loop design (see Kivisalu et al. 2014 and Gorgitrattanagul 2017) – enable development of high heatflux devices embedded in proper flow loops that are equipped with suitable flow controls (see, e.g., Fig. 29 for a flow loop being used for science experiments and Sepahyar 2018) that are capable of removing heat from the boiling surface at the desired 500–1000 W/cm2 levels.

7

Summary

This review article accomplishes the following purposes: • The Introduction (Sect. 1) and the subsection “Flow and CHF-Related Instabilities” of Sect. 2.2 provide a coherent and clarifying review of traditional in-duct flow-boiling processes (with some emphasis on flow condensation). These cover

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Pin Min

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Flow Meter (FM2) Controllable Displacement Pump, P3

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Controllable Power & Temperature

V7 Check Valve

Controllable Displacement Pump, P1

V9 Steam Generator

Filter

V6

Controllable Displacement Pump, P2 Sight Glass

Capped Reservoir With Band Heaters

Fig. 29 The broad new flow-loop structure for innovative flow-boiling experiments using water

the more complex issues and physics associated with realization of different flowregimes, local heat transfer coefficient variations, various instabilities, CHF, etc. • The review maintains a direction for readers interested in developing science and technology of high heat-flux thin film (or annular) flow boiling – and flow condensation – that employ innovative millimeter-scale hydraulic diameter devices of suitable lengths and stacking arrangements. This is facilitated by introducing the operating principle of high heat removal innovative devices early on in the article (Fig. 3) and then selecting the review of pertinent items (from the broader subject area of boiling/condensing flows) that are relevant to the physics, modeling, simulations, and experimental issues for such devices. This review also addresses the narrow “window of opportunity” that exists (as discussed through performance curves of typical millimeter-scale devices in Figs. 1, 3, 4, and 5 along with hypotheses and related physics given in Sect. 6), as discussed in recently published literature, for further development of this approach. • Besides the focus indicated above, the other primary purpose of the review was to elucidate existing knowledge base of correlations, and their relevance to the proposed innovations. Toward this, Sect. 2 presents the framework within which current and future knowledge dealing with empirical correlations for heat transfer coefficient, pressure-drop, void fraction, flow-regime transition, etc.,

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should be structured. Sect. 3 attempts to review and summarize the specifics of some of the relevant, better established (including latest and accurate) knowledge that are available in this area. • Sect. 4 makes the review of some apparently scattered topics reviewed in Sects. 2 and 3 more coherent. This is done by showing how all these approximate, quantitative, and correlation-based knowledge in Sects. 2 and 3 become useful when they come together in a design tool for new experiments and new systems that involve innovative flow-boiler and flow-condenser operations, as introduced earlier. • Even as successful new designs and experimental flow loops can be initiated and completed with the help of existing knowledge described in Sects. 1, 2, and 4 and the newer knowledge described in Sect. 6, there remains an ongoing need – both for steady and steady-in-the-mean (with superposed large-amplitude interfacial standing waves) cases – for more refined correlations as tools for assisting more accurate design of innovative systems (such as the one in Fig. 29 or its miniaturized versions) and their flow-and-heating control techniques. The principles and samples for developing such refined correlations are discussed in Section 5. When the innovative devices shift their steady operations to the mode of operation with superposed large-amplitude interfacial standing waves or other enhancement approaches (to secure operational benefits as indicated by curves I1
(ii) and I2
(ii) in Fig. 1b, c), the preliminary experimental design approach for realizing the underlying steady flows are covered by this review and cited references. Development of correlations and discussion of system design and flow-control issues for developing a priori design and assessment tools for this promising operation mode involving superposed large-amplitude interfacial standing waves or other approaches are not part of this review – these constitute an active and developing area of research.

8

Cross-References

▶ Boiling and Two-Phase Flow in Narrow Channels ▶ Boiling on Enhanced Surfaces ▶ Film and Dropwise Condensation ▶ Flow Boiling in Tubes ▶ Heat Pipes and Thermosyphons ▶ Mixture Boiling ▶ Nucleate Pool Boiling ▶ Numerical Methods for Conduction-Type Phenomena ▶ Single- and Multiphase Flow for Electronic Cooling Acknowledgment This work was supported by NSF grant CBET-1402702. All figures, which are schematics in their nature, are contributions of Mr. Patcharapol Gorgitrattanagul, a graduate student at MTU.

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Heat Pipes and Thermosyphons

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Contents 1 2 3 4 5 6 7

Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Principles of Operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Types of Heat Pipes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Working Fluids and Temperature Range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Capillary Wicks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Heat Pipe Heat Transport Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Heat Pipe Thermal Network Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Thermosyphon Thermal Resistance (RTS) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Heat Pipe Thermal Resistance (RHP) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Heat Pipe Analysis and Numerical Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Steady State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Transient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 Frozen Startup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5 Thermosyphons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6 Loop Heat Pipes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.7 Axial Grooved Heat Pipes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.8 Pulsating Heat Pipes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Heat Pipe Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Electronic and Electrical Equipment Cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Energy Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Aerospace and Avionics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4 Heat Exchangers and Heat Pumps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5 Gas Turbine Engines and the Automotive Industry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.6 Production Tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.7 Medicine and Human Body Temperature Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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A. Faghri (*) Department of Mechanical Engineering, University of Connecticut, Storrs, CT, USA e-mail: [email protected] # Springer International Publishing AG, part of Springer Nature 2018 F. A. Kulacki (ed.), Handbook of Thermal Science and Engineering, https://doi.org/10.1007/978-3-319-26695-4_52

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9.8 Ovens and Furnaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.9 Permafrost Stabilization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.10 Transportation Systems and Deicing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2202 2203 2203 2204 2204

Abstract

Heat pipes are highly effective passive devices designed to transfer large quantities of heat through a small cross-sectional area over considerable distances, while operating nearly isothermally. Heat pipes are composed of a sealed container, lined internally with a wick and filled partially with a working fluid. Heat pipes are liquid-vapor phase change devices that can transfer heat from a hot source to a cold source through capillary forces generated by the flow of liquid in a wick or other porous media. To accomplish this, heat pipes take advantage of the latent heat of an internal working fluid to transfer heat. Nomenclature

A Ab Ac,f At Aw cp cv D Dh d Fl Fv f g h hf hc,f,r hfg HP i K k L m_ n N p

Area (m2) Bare condenser/evaporator surface area exposed (m2) Inner surface area of the liquid film in the condenser (m2) Total surface area of the condenser/evaporator section exposed (m2) Cross-sectional area of the wick (m2) Specific heat at constant pressure (J/kg∙K) Specific heat at constant volume (J/kg∙K) Diameter (m) Hydraulic diameter (m) Screen wick wire diameter (m) Liquid fractional coefficient Fl =
μl/(ρl Aw K hfg) Vapor fractional coefficient Fv ¼ fRez, v μv =2R2v Av ρv hfg Ergun coefficient Gravitational acceleration (m/s2) Heat transfer coefficient (W/ m2∙K) Heat transfer coefficient of the liquid film (W/ m2∙K) Heat transfer coefficient of internal liquid film in thermosyphon (W/ m2∙K) Latent heat of vaporization (kJ/kg) Heat pipe Enthalpy (kJ) Wick permeability (m2) Thermal conductivity (W/m∙K) Length (m) Mass flow rate (kg/s) Unit vector Mesh number Pressure (Pa)

52

Heat Pipes and Thermosyphons

patm Pr Δp pe, δ pc, δ pg Q Qa Qboiling Qent Qflooding Qsonic q” R Rb Rc,f,r Re,f,r Ri Rh,w Re Red Rez,v Rg r reff t T T T0 ΔT TS V V v z 〈〉 〈〉f

Atmospheric pressure (Pa) Prandtl number Pressure difference (Pa) Pressure drop due to interfacial evaporation (Pa) Pressure drop due to interfacial condensation (Pa) Pressure change due to gravitational effects (Pa) Heat transfer rate (W) Axial heat flow through the adiabatic section (W) Boiling heat transfer limit (W) Entrainment heat transfer limit for conventional heat pipes (W) Flooding heat transfer limit for thermosyphons (W) Sonic heat transfer limit (W) Heat flux (W/m2) Radius (m)/thermal resistance ( C/W, K/W) Effective bubble radius (m) Radial thermal resistance due to liquid film ( C/W, K/W) Effective internal thermal resistance of the evaporator ( C/W, K/W) Inner radius of the heat pipe (m) Hydraulic radius of the wick surface pore (m) Reynolds number Reynolds number based on bare heat pipe diameter Axial vapor Reynolds Number Specific gas constant (J/kg∙K) Radius (m) Effective pore radius (m) Time (s)/thickness (m) Temperature ( C) Average temperature ( C) Temperature at the evaporator end cap (K) Temperature difference ( C) Thermosyphon Velocity (m/s)/Volume (m3) Velocity vector (m/s) Specific volume (m3/kg) Coordinate direction (m) Averaged over the volume Averaged over the volume of the fluid

Greek

α φ θ μ ρ

2165

Accommodation coefficient Porosity Inclination angle of the heat pipe relative to the horizontal Viscosity (Pa∙s) Density (kg/m3)

2166

ρ0 σ τ ϕ

A. Faghri

Density at the evaporator end cap (kg/m3) Surface tension (N/m) Stress tensor Viscous heating (W/m3)

Subscripts

a ax b c cap cold e eff ex f fg g hot HP i in inter l max o p r s tot TS w wk.

1

Adiabatic Axial Bare (nonfinned HP or TS) Condenser Capillary Cold Evaporator Effective External (outside HP/TS) Film, fin, fluid Liquid-vapor Gravity Hot Heat pipe Inner/inlet Internal (inside HP/TS) Interfacial Liquid Maximum Outer/outlet Liquid pool Radial Solid Total Thermosyphon Wick/wall Wick

Background

Heat pipes are highly effective passive devices designed to transfer large quantities of heat through a small cross-sectional area over considerable distances, while operating nearly isothermally. Heat pipes are composed of a sealed container, lined internally with a wick and filled partially with a working fluid. Heat pipes are liquidvapor phase change devices that can transfer heat from a hot source to a cold source through capillary forces generated by the flow of liquid in a wick or other porous media. To accomplish this, heat pipes take advantage of the latent heat of an internal working fluid to transfer heat (Faghri 2016).

52

Heat Pipes and Thermosyphons

2167

Operational temperatures for heat pipes can range from cryogenic temperatures of less than 271  C to extremely high temperatures in excess of 2200  C. Heat pipes can achieve significantly higher heat transfer though the same cross-sectional area compared to rods made with metals such as copper or other highly conductive materials. Heat pipes effective thermal conductivity can approach 150 kW/mK, compared to 400 W/mK for solid copper. Additionally, when heat pipes are designed properly, they are highly reliable with possible lifetimes over 15 years. The simplicity, lightweight construction, high thermal conductivity, efficient heat transfer, and possibility of varying sizes, shapes, and materials allows heat pipes to be designed for various applications and temperature ranges.

2

Principles of Operation

The operation of a heat pipe, as shown in Fig. 1, is most easily understood with a cylindrical geometry. Though heat pipes can be designed in nearly any shape or size, their principles of operation do not stray far from what can be established with a cylindrical heat pipe. Heat pipes are made with a sealed container, including a pipe wall and end caps, a wick structure (excluding thermosyphons and rotating heat pipes), and a working fluid. Heat pipes can be divided into three sections: an evaporator, condenser, and adiabatic section. A heat pipe can have multiple evaporators and/or condensers, with or without an adiabatic section depending on the application. Heat pipes function when heat is applied to the exterior wall of the container – this heat is transferred through the container wall and the wick, where it vaporizes the working fluid. The resulting vapor pressure drives the vapor towards the condenser,

Container

Evaporator end cap

Vapor flow Liquid flow

Condenser end cap

Fig. 1 Conventional heat pipe configuration

Direction of gravity

2168

A. Faghri

where heat is drawn from the working fluid as it condenses, releasing its latent heat into the heat sink. Capillary pressure pumps the condensate from the condenser section to the evaporator section to continue the liquid circulation process. The process will continue so long as capillary pressure remains sufficient to drive the working fluid back to the evaporator, overcoming the vapor and liquid pressure drops (Faghri 2016). The menisci at the liquid-vapor interface are highly curved in the evaporator section due to the liquid receding into the pores of the wick, as shown in Fig. 2a. Capillary pressure is caused by the surface tension at the liquid-vapor interface, and the curvature of the liquid-vapor interface. The difference in curvature of the menisci causes the capillary pressure to change along the heat pipe. This pressure gradient is the driving force for liquid return, and must overcome the vapor and liquid pressure drops, and any adverse gravitational effects. Vapor and liquid pressure change along the heat pipe is the effect of friction, inertia, blowing (evaporation), and suction (condensation). The liquid pressure drop is mainly the result of frictional forces in the wick. The liquid-vapor interface is flat near the condenser end cap, as there is a zero local pressure gradient at low vapor flow rates. Figure 2b shows the vapor and liquid pressure drops that the capillary force must overcome for low, moderate, and high vapor flow rates.

3

Types of Heat Pipes

There are many types of heat pipes, as heat pipes offer great flexibility in their design and capabilities. Working fluid must be circulated through the heat pipe, which is typically done by capillary forces through a wick; however, gravitational, centrifugal, electrostatic and osmotic forces can also be used. The type of heat pipe used depends on its applications – various types will offer different benefits and drawbacks. Types of heat pipes (Fig. 3) include two-phase closed gravity-assisted thermosyphons, capillary-driven heat pipes, annular heat pipes, vapor chamber heat pipes, rotating heat pipes, loop heat pipes, capillary-loop heat pipes, pulsating heat pipes, micro and miniature heat pipes, and inverted meniscus heat pipes. Heat pipes can have a variety of cross-sections, though the most common are circular and rectangular, depending on the application. A detailed presentation of various types of heat pipes, including performance characteristics, is given by Faghri (2016). The two-phase thermosyphon (Fig. 3a) is a gravity-assisted, wickless heat pipe. The condenser section must be located above the evaporator so that gravity can return the condensate to the evaporator. The most severe limitation in thermosyphons is flooding, the thermosyphons equivalent to the entrainment limit (Faghri et al. 1989a). The entrainment limit is a heat transfer limitation in heat pipes caused by the shear force between the liquid-vapor interface, which tears liquid from the wick. Thermosyphons are more sensitive to the working fluid fill volume – the maximum heat transfer rate will increase with the liquid fill up to a certain value. Heat pipes and thermosyphons differ only in that heat pipes use a wick to return

L

e

Capillary pressure difference

Liquid pressure drop

Vapor pressure drop

L

c

Adverse gravity force

No gravity force

L a Distance Low Vapor Flow Rate

Liquid

Liquid

Vapor

e

Liquid pressure drop

Vapor pressure drop

L c

Adverse gravity force

No gravity force

Distance Moderate Vapor Flow Rate

L a

Liquid

Vapor

Condenser section

Heat sink Lc

Vapor and liquid pressure distributions

L

Adiabatic section

Liquid flow Wall

Vapor flow

Wall Liquid flow

La

Liquid-vapor interface

Liquid

Capillary pressure difference

Evaporator section

Heat source Le

L

e

Liquid

Liquid

Vapor

L a Distance High Vapor Flow Rate

Liquid pressure drop

Vapor pressure drop

L c

No gravity force Adverse gravity force

Capillary pressure difference

Heat Pipes and Thermosyphons

Pressure

Fig. 2 (a) Side view of the liquid-vapor interface, (b) vapor and liquid pressure distributions at different vapor flow rates

Pressure

b

a

Pressure

52 2169

2170

A. Faghri

a Heat output

b

Condenser section

Evaporator section Adiabatic section

Evaporator end cap Vapor space Wick Wall

Condenser section Vapor flow Gravity

Liquid flow (Falling film)

Vapor flow

w

id flo

Adiabatic section

Liqu

Container

Heat source

w

id flo

Liqu

Heat input

Condenser end cap

Evaporator section

c

Sink

Direction of gravity

Heat sink

Heat input

d

Heat out

Liquid flow

Heat output

Insulation

Inner and Outer Heater

f

Liquid flow

Wick Liquid flow Container

Container

Vapor flow

Heat out Axis of rotation Heat out

Optional block of wick meterial

Compensation Chamber Evaporator

Heat source

Heat in

Heat In

Wick Vapor Removal Channel

Heat sink

Vapor Line

Condenser

Vapor flow

Heat source

Heat in

g

Gas

Liquid

h

Vapor flow

Heat Transport

Heat In

e

Vapor flow

Vapor flow

g

Liquid

Heat Out

Evaporator section

Condenser section Reserv

n=1

n=2

n=3

n=4

n=5

n=6

n=7

n=8

Heat input

i

e

ourc Heat s

Heat output

Noncondensible gas

Heat sink

Vapor space

Evaporator section

Condenser section

Fig. 3 Several Different Types of Heat Pipes (a) Thermosyphon (b) Conventional Heat Pipe (c) Annular Heat Pipe (d) Vapor Chamber (e) Rotating Heat Pipe (f) Loop Heat Pipe (g) Gas-Loaded Heat Pipe (h) Pulsating Heat Pipe (i) Leading Edge Heat Pipe

52

Heat Pipes and Thermosyphons

2171

condensate from the condenser to the evaporator via capillary action, while thermosyphons rely on gravitational effects to return condensate. Sometimes wicks have been added for the benefit of promoting more uniform liquid distribution in the thermosyphon, especially in the evaporator section. Thermosyphons, operating without a wick, can act as thermal diodes, as heat can only flow opposed to the gravitational force if condensate is to be returned by gravity. The capillary-driven heat pipes (Fig. 3b) (also known as conventional heat pipes) consist of a sealed container with a wick structure along the inside wall and a small amount of working fluid inside. The wick acts as a capillary pump, returning the working fluid from the condenser to the evaporator regardless of orientation. These are by far the most common heat pipes, due to their relatively simple design, and ability to operate at any orientation. These heat pipes can operate at various orientations both with and opposed to gravitational forces, as well as operate in zero gravity. The annular heat pipe (Fig. 3c) is similar to a conventional heat pipe, the main difference being that the cross-sectional area is annular as opposed to circular (Faghri and Thomas 1989). This enables a wick to be placed on the inner surface of the outer wall and on the outer surface of the inner pipe. This increases the surface area, and allows for a significant increase in heat transfer with no change in the outer diameter. Annular heat pipes offer greater efficiency, in terms of heat transfer rate per cross-sectional area, but can be more difficult to manufacture compared to conventional heat pipes. The vapor chamber (Fig. 3d) is similar to a capillary-driven heat pipe, but has a small aspect ratio, and in a flat configuration (Xiao and Faghri 2008). Additional wick material is placed between the evaporator and condenser to further aid in condensate return. The vapor chamber is commonly used in electronics cooling. Heat flow in a vapor chamber is two- or three-dimensional, unlike a conventional heat pipe that experiences mostly one-dimensional flow. The rotating heat pipe (Fig. 3e) takes advantage of centripetal forces, as opposed to gravitational forces or capillary forces, to return condensate to the evaporator (Harley and Faghri 1995). These heat pipes can be fabricated as either a circular cylinder or a disk. They can be used to cool the rotating parts of motors, metal cutting tools, and mills, and they are proposed for cooling turbine components and automobile brakes. The gas-loaded heat pipe (Fig. 3g) is a type of variable conductance heat pipe (Harley and Faghri 1994b; Faghri 2016). A noncondensing gas is introduced to the vapor space at the condenser end, and will expand and contract with the pressure and temperature changes. This limits the surface area in which the vapor working fluid can contact the condenser, preventing it from transferring heat through that part of the condenser. This allows these heat pipes to maintain a near constant evaporator temperature regardless of heat input (Faghri 2016). Gas-loaded heat pipes are used in applications that require precise amounts of heat to be transferred either in or out. The loop heat pipe (Fig. 3f) is based on the same principles as a conventional heat pipe, and consists of a capillary pump, compensation chamber, condenser, and liquid and vapor lines. A wick structure is only placed in the evaporator and condenser chambers, where a high capillary force is created through use of fine pored wicks, such as sintered metal wicks (Maydanik 2005). The compensation chamber accommodates excess liquid during normal operation. A secondary wick connects this

2172

A. Faghri

chamber to the evaporator to supply the primary wick with liquid. The vapor and liquid pressure drops are lower than other heat pipes with wick throughout the heat pipe, making this an attractive option for space applications and for applications moving heat over long distances. The pulsating heat pipe (Fig. 3h) is made of a single, long capillary tube bent into multiple turns, with the condensers and evaporators located at these turns (Zhang and Faghri 2008). They can be either looped or unlopped, depending on if the two ends are connected. Pulsating heat pipes rely on the pressure differences of liquid slugs and vapor plugs within the working fluid to drive flow, and as such do no use a wick. The entrainment limit has no effect on pulsating heat pipes. The structure of a pulsating heat pipe is simpler and lighter than a comparable conventional heat pipe. The micro and miniature heat pipes: The micro heat pipe is defined as those heat pipes in which the mean curvature of the liquid-vapor interface is comparable in magnitude to the reciprocal radius of the vapor flow channel (Cotter 1984; Peterson 1992; Cao and Faghri 1994; Khrustalev and Faghri 1994). In micro heat pipes the hydraulic diameter is 10–500 μm, while miniature heat pipes have hydraulic diameters ranging from 0.5 to 5 mm (Hopkins et al. 1999). These heat pipes are used in nano- and microscale applications. The leading edge heat pipe (Fig. 3i) is proposed for applications such as hypersonic aircraft and reentry spacecraft leading edge cooling (Cao and Faghri 1993a). Leading edge heat pipes must be designed geometrically for their specific application. For aircraft, the heat pipe takes on the geometry of the leading edge of a wing, while for spacecraft it takes the form of a heatshield between the capsule and the atmosphere.

4

Working Fluids and Temperature Range

A good heat pipe working fluid should have a high latent heat of vaporization, surface tension, and thermal conductivity. Table 1 shows various working fluids, and their effective ranges for heat pipes. The effective range typically extends from the point where the saturation pressure is greater than 0.1 atm to less than 20 atm. Temperature ranges are divided into four categories – cryogenic, low, intermediate, and high. Cryogenic heat pipes operate between 4 and 200 K. The amount of heat these heat pipes can transfer is low relative to heat pipes operating at higher temperatures, due to the small heats of vaporization, high viscosities, and small surface tension of cryogenic working fluids. Low temperature range is from 200 to 500 K. Most heat pipe applications fall within this range. Water is the most common working fluid in this range, as it has good thermophysical properties such as a large heat of vaporization and surface tension. The practical range for copper-water heat pipes is 283–560 K. Other working fluids within this range include ammonia and acetone. Intermediate temperature heat pipes operate between 500 and 750 K. The most common working fluids at this range are Dowtherm-A and NaK. Operating ranges for high temperature heat pipes are over 750 K. These heat pipes often use liquid-metal working fluids such as silver, potassium, or sodium. The

52

Heat Pipes and Thermosyphons

2173

Table 1 Working fluids and temperature ranges of heat pipes Melting Point, K at 1 atm

Boiling Point, K at 1 atm

Useful Range, K

Helium

1.0

4.21

2-4

Hydrogen

13.8

20.38

14-31

Neon

24.4

27.09

27-37

Nitrogen

63.1

77.35

70-103

Argon

83.9

87.29

84-116

Oxygen

54.7

90.18

73-119

Methane

90.6

111.4

91-150

Krypton

115.8

119.7

116-160

184.6

150-240

172.0

246.6

198-323

Ammonia

195.5

239.9

208-373

Pentane

143.1

309.2

253-393

Acetone

180.0

329.4

273-393

Methanol

175.1

337.8

283-403

Ethanol

158.7

351.5

273-403

Heptane

182.5

371.5

273-423

Toluene

178.1

383.7

323-473

Water

273.1

373.1

283-560

Naphthalene

353.4

490

408-623

Dowtherm A

285.1

527.0

423-668

NaK

260.9

1058

780-1080

Cesium

301.6

943.0

723-1173

Rubidium

312.7

959.2

800-1275

Potassium

336.4

1032

773-1273

Sodium

371.0

1151

873-1473

Lithium

453.7

1615

1273-2073

Calcium

1112

1762

1400-2100

Silver

1234

2485

2073-2573

Intermediate Temperature High Temperature Range Range

89.9

R134a

Low Temperature Range

Ethane

Cryogenic Temperature Range

Working Fluid

heat transfer rates of liquid-metal heat pipes are generally much higher than those of heat pipes operating at other temperature ranges, due to the very high values of latent heat of vaporization, thermal conductivity, and surface tension. Poplaski et al. (2017) reviewed the past experimental and theoretical studies on the application of nanofluids in various types of heat pipes. Furthermore, Poplaski et al. (2017) developed a full numerical simulation to account for the effect of nanofluids in a conventional heat pipe. The optimal nanofluid concentration of Al2O3, TiO2, and CuO corresponding to the capillary limit for a conventional nanofluid filled heat pipe was determined to be 25% by volume for Al2O3 and

2174

A. Faghri

Table 2 Generalized results of compatibility tests for heat pipes Working fluid Acetone Ammonia

Dowtherm A Helium Heptane Hydrogen Lithium

Compatible material Aluminum, stainless steel Aluminum, cold rolled steel, nickel, stainless steel Haynes, Inconel, niobium, stainless steel, titanium Aluminum, stainless steel, titanium Stainless steel, titanium Aluminum Stainless steel Molybdenum, niobium, tungsten

Methanol Nitrogen Potassium R134a Silver Sodium Water

Copper, stainless steel Aluminum, stainless steel Haynes, Inconel, stainless steel Stainless steel Molybdenum, tungsten Haynes, Inconel, stainless steel Copper, Monel, nickel, titanium

Cesium

Incompatible material Copper, titanium Copper, copper-nickel, Monel Copper, copper-nickel

Inconel, nickel, stainless steel, titanium Aluminum, titanium Copper, Monel Rhenium Titanium Aluminum, Inconel, stainless steel

TiO2, and 35% for CuO. Overall, a maximum decrease in total thermal resistance was observed to be 83%, 79%, and 76% for Al2O3, CuO, and TiO2, respectively. Special attention must be payed to the compatibility of materials used in heat pipes at all interfaces. Compatibility is crucial to reliable long-term use, as incompatible materials can chemically react with one another, leading to failure. Materials selection should attempt to find wicks and container materials with high thermal conductivity for the evaporator and condenser sections while working fluids should have high latent heat of evaporation and beneficial thermal-fluid properties depending on the wick design and temperature range. Common materials include copper, stainless steel, nickel, and aluminum for the container and wick. Water, methanol, ammonia, and sodium are some common working fluids. Wick options can vary greatly depending on the needs of the application, but common wicks include metal screens, grooves, sintered metal powders and felts, and foams (Faghri 2014, 2016). Longevity of a heat pipe is dependent on the selection of a wick and container that are compatible with the working fluid. Compatible container materials and working fluids are presented in Table 2.

5

Capillary Wicks

Most heat pipes will use a wick structure to return condensate from the condenser to the evaporator. Many wick structures have been developed for different applications, and these are broken into two broad categories (Faghri 2016). Homogenous wicks

52

Heat Pipes and Thermosyphons

2175

are made of a single material with a uniform design. This makes them relatively inexpensive and easy to manufacture, but more limited in their application. Composite wicks take advantage of multiple materials and structures in the same wick structure; however, this leads to increased cost and manufacturing issues. The permeability is a property of the wick, characterizing the wick’s resistance to liquid flow for a given pressure drop. In most heat pipes, liquid flow through the wick can be simplified to one-dimensional axial flow (Fig. 2), as the wick structure is typically very thin. Based on this assumption, the permeability can be calculated using Daray’s equation (Faghri and Zhang 2006): dpl μ m_ l ¼ l , dz ρ l Aw K

(1)

where pl is the liquid pressure, μl is the liquid viscosity, m_ l is the liquid mass flow rate, ρl is the liquid density, Aw is the wick cross-sectional area, and K is the wick permeability. Maximum capillary head for a given wick structure can be calculated by the Young-Laplace equation: 2σ Δpcap, max ¼ , (2) r eff where Δpcap , max is the maximum capillary head, reff is the effective pore radius of the wick, and σ is the surface tension. Figures 4 and 5 show several common homogenous and composite wick options. Figure 4 features typical composite and homogenous capillary wick designs for heat pipes. The parameters required in calculating these wick options are also shown in Fig. 4. Figure 5 shows a cross-section view of common wick designs, both homogenous and composite.

a

Screen

t

b

c S W

2R1

W tw Dg d

2R2

e d tw

S W

Screen Dg

Fig. 4 Typical capillary wick structures (a) annular channel, (b) rectangular grooves, (c) square wire mesh screen, (d) sintered felted metal fibers, (e) screen-covered rectangular grooves.

2176

A. Faghri

Fig. 5 Cross-sections of typical wick structures (a) wrapped screen, (b) sintered metal, (c) axial grooves, (d) composite screen, (e) screen covered grooves, (f) composite slab

There are three important wick properties in heat pipe design (Faghri 2016): Effective Pore Radius: The minimum capillary radius should be kept as small as possible to raise the capillary pressure difference. This increased the heat pipe’s ability to pump condensate back to the evaporator. Permeability: This determines the ease with which liquid can flow through the wick. Permeability should be large to reduce the liquid pressure drop. The larger this value, the greater the heat transport capacity. Effective Thermal Conductivity: The thermal conductivity of the wick should be as high as possible in order to provide a small temperature drop across the wick. A high thermal conductivity and permeability are often contradictory towards a low capillary radius in most wick configurations. Therefore, tradeoffs must be made in wick selection to achieve the greatest effectiveness and heat transfer capability. Table 3 provides a comparison of the three primary properties of common wick structures for heat pipes. Table 4 includes expressions for calculating the permeability and effective pore radius of several common wick structures. The Reynolds number values for which these equations hold true are also presented in Table 4.

6

Heat Pipe Heat Transport Limitations

Although heat pipes are very efficient heat transfer devices, under certain conditions their maximum heat transfer rate will encounter limitations. The type of limitation that restricts heat flow through the heat pipe is determined by which limitation has

52

Heat Pipes and Thermosyphons

2177

Table 3 Typical Homogenous and Composite Wicks for Heat Pipes Wick type Wrapped screen

Capillary pumping High

Thermal conductivity Low

Sintered metal

High

Average

Axial grooves

Low

High

Composite screen Screen covered grooves Composite slab

High High

Lowaverage High

High

High

Permeability Lowaverage Lowaverage Averagehigh Average Averagehigh High

Comments Single or multiple wire mesh screens Packed spherical particles, felt metal fibers, or powder Rectangular, circular, trapezoidal, or triangular grooves Two or more layers of homogenous materials Axial grooves covered with one screen Slab of material with circumferential grooves

the lowest value at a specific working temperature. The maximum axial heat transport rate as a function of heat pipe operating temperature for various heat transport limitations are presented in Fig. 6. Detailed methods for predicting these limits under all operating conditions for both heat pipes and thermosyphons are given in Faghri (2016). These limitations are briefly described below, and the appropriate formulas to predict these limits under special conditions are given: Capillary Limit: The pumping ability of a capillary (wick) structure is limited. As this limit is approached, the wick will be unable to return sufficient condensate to the evaporator, limiting heat transport capability. This limit is important when designing a wick structure for a heat pipe. Stable circulation of the working fluid through a heat pipe is achieved through the capillary pressure head developed by the wick structure. In order to achieve maximum heat transfer, the capillary pressure head must be greater than or equal to the sum of all pressure drops along the liquid-vapor path. In order for a heat pipe to function properly, the following pressure balance must be satisfied: Δpcap, max  Δpl þ Δpv þ Δpe, δ þ Δpc, δ þ Δpg

(3)

where Δpl is the pressure drop of the liquid flow through the wick structure due to frictional drag, Δpv is the pressure drop of the vapor flow, Δpg is the pressure change due to gravitational effects, and Δpe,δ and Δpc,δ are the pressure drops due to interfacial evaporation and condensation, respectively. The total vapor pressure drop in a heat pipe can be approximated by (Faghri 2016): Δpv ¼

4Qa μv ðLe þ 2La þ Lc Þ πρv hfg R4v

(4)

W

Rectangular grooves

Wþd 2

Screen covered rectangular grooves

1 2N

Or

D2h φ 2ðfRel, h Þ

AðX2 1Þ X1

2

3

Established for metal powder (mainly copper): 50  106 < D < 3  104 0.27 < φ < 0.66

D2 φ3 150ð1φÞ2

d 2 φ3 122ð1φÞ2

D2h φ 2ðfRel, h Þ

D2h φ 2ðfRel, h Þ

Permeability (K, m2)

8  10 φ X ¼ 1 þ ðBd 1φÞ2 1) is used, dp ¼ 2r  ¼

4σvliq 4σv1 ¼ RTlnS kB TlnS

(15)

65

Synthesis of Nanosize Particles in Thermal Plasmas

2797

In this equation, v1 ¼ N A vliq is the volume of monomer in liquid, NA is the Avogadro number, and kB denotes Boltzmann’s constant (= R/NA). This equation gives the size of nuclei at a certain saturation ratio S (> 1).

2.3.3 Equilibrium Size Distribution We consider a reaction between a g-class cluster and a (g1)-class cluster. The reaction Ag1 + A1 ! Ag happens if condensation occurs by the monomer flux. This reaction rate in unit volume per time is expressed, I g ¼ βsg1 ng1  αs sg ng

(16)

Here ng denotes the density of g-mer, sg is the surface area of g-mer, and β is the flux of condensing monomers with random thermal velocity,  1 1 1 p1 8kB T 2 p1 β ¼ n1 v 1 ¼ ¼ 1 : 4 4 kB T πm1 ð2πm1 kB T Þ2

(17)

where m1 and p1, respectively, denote the monomer mass and the pressure. However, αs is the evaporation flux from g-mer, which is expressed considering a Kelvin relation,
 p r ¼ dp =2 1 α s ¼ ns v 1 ¼ 1 4 ð2πm1 kB T Þ2   ps 4σv1 ¼ , 1 exp dp kB T ð2πm1 kB T Þ2

(18)

(19)

where dp = 2r is the g-mer particle diameter in liquid. For this relation, the following relation was used for the diameter of a particle with g and v1,   4 dp 3 π ¼ gv1 3 2

(20)

For the equilibrium condition Ag1 + A1 $ Ag, i.e., the net flux is zero, Ig = 0, βsg1 ng1 ¼ αs sg ng :

(21)

Substituting Eqs. 17, 19, and 20 into 21, the following relation is obtainable assuming sg1  sg, "  1 # ng1 αs 1 2σv1 4π 3 ¼ ¼ exp S ng β kB T 3gv1

(22)

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Y. Tanaka

Multiplying Eq. 22 varying g from g to 2,

ng1 ng

2 3  13 4π g 2σv ng2 n2 n1 n1 1 6 1 3v1 X 13 7 ... ¼ ¼ g1 exp4 g 5 ng1 n3 n2 ng S kB T g¼2

(23)

If the g is sufficiently large, then the summation could be the integration, g X

g 3  1

ðg 0

g¼2

3 2 1 g3 dg ¼ g3 : 2

(24)

Consequently, the equilibrium distribution of nuclei can be expressed, 2 6 g neq g ¼ ns S exp4

 13 2 3 4π 3 3v1 g 7 5 kB T

3σv1

(25)

Therein, ns = ps/(kBT ) and S = p1/ps. The designation neq g is used for ng obtained for equilibrium state. If the vapor is not saturated, i.e., S < 1, then ng decreases concomitantly with decreasing g. If the vapor is saturated, i.e., S > 1, then there is a local minimum at a certain value,

1 3

g ¼

2σv1

 13 4π 3v1

(26)

kB TlnS

From this relation and Eq. 20, the critical diameter of nuclei dp is, dp ¼

4σv1 : kB TlnS

(27)

This critical diameter of nuclei is extremely important to understand the nucleation process in gas-to-particle conversion. In addition, the number density of nuclei with the critical diameter is therefore expressed by, " ng

¼ n1 exp 

16πσ 3 v21 3ðkB T Þ3 ðlnSÞ2

# (28)

The number density depends strongly on the saturation ratio S. It is noteworthy that Eq. 25 has a large error for small g. Moreover, the critical number of g* is,

65

Synthesis of Nanosize Particles in Thermal Plasmas

g ¼ 36πv21



2σ 3kB TlnS

2799

3 :

(29)

  σs1 2 g 3 neq g ¼ n S exp  s g kB T

(30)

Equation 25 can be rewritten,

Therein, s1 is the surface area of the monomer, neq g represents the number density of g-mer in equilibrium state clearly. To obtain this equation, relation s31 ¼ 36πv21 is used. In addition, Θ signifying the normalized surface tension is, Θ¼

σs1 : kB T

(31)

Using Θ, Eq. 30 is rewritten,  2 g 3 neq ¼ n S exp Θ g s g

2.4

(32)

Gas-to-Particle Conversion: Nucleation, Condensation, and Coagulation

In nanoparticle synthesis, using thermal plasmas, the nucleation, condensation, and coagulation occurs to grow nanoparticles. Here, nucleation, condensation, and coagulation will briefly be mentioned (Friedlander 2000).

2.4.1 Homogeneous Nucleation If condensation occurs, the equilibrium condition is not satisfied. This reaction rate in unit volume per time is again expressed, I g ¼ βsg1 ng1  αs sg ng

(33)

According to the homogeneous nucleation theory by Friedlander (2000), the homogeneous nucleation rate I is written, " I¼2

# p1



1

ð2πm1 kB T Þ2

σv23 1 n1 v 1 kB T

!12

2 3

exp 

16πσ 3 v21 3ðkT Þ3 ðln SÞ2

This can be rewritten using Θ,  I¼

β11 n21

Θ 2π

12

exp 

4Θ3 27ðln SÞ2

! (34)

! (35)

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Y. Tanaka

To get this equation, we used the relation βsg1 = β1,g1n1, where βij is the collision frequency between i-mer and j-mer. This equation is called classical nucleation theory (CNT) and is widely used to estimate the homogeneous nucleation rate. Furthermore, this homogeneous nucleation rate I was improved by Girshick et al. (1990a) and Girshick and Chiu (1990b) to satisfy monomer and without the constrained equilibrium of the CNT, which can be expressed,  I¼

β11 n2s S

Θ 2π

12

exp Θ 

4Θ3 27ðlnSÞ2

! ,

(36)

where S is the supersaturation ratio, S¼

p1 n1 ¼ ps ns

(37)

Subscript 1 and s, respectively, represent the monomer and the saturation state. This equation indicates kinetic nucleation theory and is well adopted to predict the homogeneous nucleation rate for nanoparticle synthesis. The difference between his equation and the CNT is a factor exp(Θ)/S. If the particle diameters are much smaller than the mean free path of the vapor, i.e., dp > vm), the discrete distribution can be treated as the continuous distribution for calculation,

65

Synthesis of Nanosize Particles in Thermal Plasmas

  @nðvÞ @n @ þ ∇  ðnðvÞvÞ ¼ ∇:D∇nðvÞ þ  ðGnðvÞÞ @t @t nucl @v ð 1 v þ βðv~, v  v~Þnðv~Þnðv  v~Þd~ v 2 0 ð1  βðv,~ v ÞnðvÞ nðv~Þd~ v  ∇  ðcnðvÞÞ

2803

(49)

0

where G is the growth rate due to heterogeneous condensation. To solve the evolution in time of characteristics of the particles during nucleation process, mainly two approaches are adopted: a discrete-type method like sectional model or nodal model and a method of moments (MOM). The discrete-type method offers more precise and detailed results but requires complicated procedures and high cost for calculation than MOM (Shigeta and Watanabe 2007, 2010). The MOM provides much lower cost calculation for GDE. The method of moments (MOM) solves GDE for low-order moments of the particle size distribution function (PSDF). The k-order moment Mk for n(v) is defined, Mk ¼

ð1

vk nðvÞdv

(50)

0

Using this, Eq. 49 can be written for steady state,         @Mk @Mk @Mk @Mk u  ∇Mk  þ þ þ @t nucl @t coag @t cond @t diff

(51)

where (@Mk/@t) presents net production rate of k-order moments by nucleation, coagulation, condensation, and diffusion. This set of moment equations is unclosed generally by themselves. With assumption of the form of distribution, i.e., the form of log-normal distribution, Eq. 51 can be closed to be solved. There are two main closure methods for nanoparticles: one is the method of moments with interpolative closure (MOMIC) and the other is the quadrature method of moments (QMOM). MOMIC uses interpolation among known whole-order moments to close fractionalorder moments due to coagulation and particle surface growth. QMOM was proposed by McGraw (1997), a more precise method than MOMIC. QMOM do not require any assumption of PSDF. It uses a quadrature method to approximate the distribution as a set of weighted particles.

2.8

Examples of Modeling Approach for Particle Growth

Girshick et al. (1988) derived a discrete model to calculate nanoparticle growth as coagulation among clusters and monomers. They calculated particle growth for Fe

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and SiC nanoparticles. Girshick and Chiu (1990b) developed a discrete-sectional model by combining their discrete model and the sectional model. This method has been widely used to predict particle growth in different conditions. They successfully predicted MgO nanoparticle growth using the developed model. The MOM has been well adopted to calculate nanoparticle growth considering nucleation, condensation, and coagulation with combination of thermofluid flow calculation (Proulx and Bilodeau 1989; Girshick et al. 1993; Bilodeau and Proulx 1996; Désilets et al. 1997; Cruz and Munz 1997, 2001; Murphy 2004; MendozaGonzalez et al. 2007a, b; Shigeta and Watanabe 2008). Proulx and Bilodeau (1989) and Girshick et al. (1993) successfully used the MOM to predict Fe nanoparticle growth with thermofluid dynamics for induction thermal plasma synthesis. Bilodeau and Proulx (1996) also obtained two-dimensional thermofluid flow and also particle growth taking into account diffusion, convection, and thermophoresis. Desilets et al. (1997) made a model monomer production from chemical reaction for Si nanoparticles synthesis. Cruz and Munz (1997) used MOM to simulate nucleation and growth of AlN nanoparticles in a transferred arc with Ar/NH3/H2 + Al vapor. Titanium nanoparticle formation from Ar–H2 thermal plasmas with TiCl4 injection was predicted by Murphy (2004) using MOM. He considered 14 chemical reactions with reaction rates in MOM and studied influence of quench rate. Figure 1 indicates the dependence of titanium yield on quench rate for different initial mixtures of x(TiCI4) = 0.10, x(Ar) = 0.90 x(TiCI4) = 0.01, x(Ar) = 0.99 x(TiCI4) = 0.01, x(Ar) = 0.49, x(H2) = 0.50 100

x(TiCI4) = 0.01, x(H2) = 0.99

80

Yield (%)

60

40

20

0 1

10

100

-dT/dt (K μs-1) Fig. 1 Dependence of titanium yield on quench rate for different initial mixtures of TiCl4, Ar, and H2 (Murphy (2004))

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Synthesis of Nanosize Particles in Thermal Plasmas

2805

TiCl4, Ar, and H2. This calculation was done for high initial temperature to ensure complete dissociation of TiCl4. It was found from this figure that almost 100% titanium yield could be obtained for sufficiently long residence times at a high temperature and then sufficiently rapid quench rates.

2.9

Experimental Setups for Nanoparticle Synthesis Using Thermal Plasmas

The thermal plasma has a unique feature of high gas temperature above 10,000 K with high power density. This results in a benefit of high enthalpy which facilitates evaporation of injected feedstock into atoms and molecules in gas phase. The evaporated vapor is cooled down by some methods to create nanoparticles. The thermal plasma is thus recognized as an effective heat source for evaporation of the feedstock. In addition, the reactive molecular gas injection into a thermal plasma offers a reactive field. This may provide composite nanoparticles and core-shell structured nanomaterials. Thermal plasmas of various types such as arc plasmas and inductively coupled thermal plasmas have been adopted for nanoparticle synthesis. Heating mechanism for each of arc plasmas and inductively coupled thermal plasma is so different. Each type of thermal plasmas has a unique feature for thermal plasma field for nanoparticle synthesis. In addition, cooling method for synthesized vapor markedly affects nanoparticle synthesis like size distribution, morphology, composition, phase, etc. Thus, control of heating method and cooling method is essential to control nanoparticle synthesis. In this section, some results will be introduced for nanoparticle synthesis using thermal plasmas. Nanoparticle synthesis using plasmas including thermal plasmas is well summarized by Fauchais et al. (1997), Ostrikov and Murphy (2007), Shigeta and Murphy (2011), and Seo and Hong (2012). Many researchers have made great efforts to develop some nanoparticle synthesis techniques for these three decades. Nowadays, thermal plasma synthesis is adopted for industrial production of nanoparticles by some companies, for example, Tekna in Canada, Nisshin Seifun in Japan, and Tetronics in the UK, using thermal plasmas. In the following subsection, the basic principles of a DC arc plasma and an RF inductively coupled thermal plasma are briefly introduced. Some examples of nanoparticle syntheses will be described.

2.9.1 DC Arc Plasma Technique DC arc plasma torches Direct current (DC) arc plasmas are widely used for nanoparticle synthesis. A DC arc plasma is established between the electrodes by DC electric current at several hundred to several thousand amperes. The DC arc plasma is characterized by high gas temperature about 6000–20,000 K, with high gas flow velocity of about 100–1000 m/s at the center axis. The arc voltage reaches ten to several tens of volts. Such high gas temperatures and high gas velocities of arc plasmas are realized especially using arc plasma torches with a gas flow nozzle. Two

2806 Fig. 2 Configurations of (a) transferred arc and (b) non-transferred arc

Y. Tanaka

a

b

types of arc plasma torches are usually adopted to form the DC arc plasma for nanoparticle synthesis: the transferred arc plasma torch and the non-transferred arc plasma torch. Figure 2 presents the configuration of the transferred arc torch and the non-transferred arc torch. Both torches have a gas flow nozzle and a cathode inside the torch. The nozzle is cooled by cooling water inside. This arrangement of the cooled nozzle and the cathode produces strong gas flow to pinch the arc plasma by convection loss. In addition, the arc plasma is further pinched by Lorentz force attributable to the arc current and self-induced magnetic fields. Such pinching phenomena elevate the current density. Then they elevate the arc temperature and arc gas flow velocity. This high temperature and high gas flow velocity are useful for nanoparticle synthesis. The transferred arc plasma is formed between the cathode inside the plasma torch and the anode outside the plasma torch. In this case, the material to be heated for nanoparticle synthesis is used as the anode. Consequently, just the electrically conductive material is used for the anode corresponding to raw material for nanoparticle synthesis. The anode material is heated by heat transfer from the arc and also by input power to the electrons in electrode fall voltage region. However, the transferred arc plasma is used as an anode. Joule heating occurs only in the arc plasma between the cathode and anode spot. From this arc plasma, the plasma jet is ejected from the nozzle. This plasma jet is used for evaporation of feedstock located outside the torch. In this case, the material by the non-transferred arc plasma or the plasma jet is less efficiently evaporated than by the transferred arc plasma. However, the non-transferred arc presents the important advantage that it can heat electrically nonconductive materials. As other ways, powder feedstock is often used for injection to the plasma jets. Examples Nanoparticle synthesis has been accomplished at low cost using the transferred arc plasma with high-efficient evaporation of materials. Here, examples are presented of nanoparticle synthesis using a transferred arc and a nontransferred arc. A transferred arc is often used to produce various types of nanoparticles by many researchers (Uda et al. 1983; Ohno and Uda 1984; Fudoligh et al. 1997; Araya et al. 1988; Watanabe et al. 2001; Lee et al. 2007; Tanaka and Watanabe 2008; Watanabe et al. 2013 etc.). Uda et al. (1983) developed a method to use a transferred arc with

65

Synthesis of Nanosize Particles in Thermal Plasmas

2807

reactive gas like hydrogen to produce metallic nanoparticles. In this method, raw material metal bulk was placed on the anode, which was irradiated by a 50%Ar–50%H2 transferred arc at atmospheric pressure. The raw metal material is molten by the arc, and then from the molten metal, the metallic nanoparticles were produced. This method uses hydrogen–metal interaction with a high production rate about 0.9 g/min for iron nanoparticles. This rate was found 15 times higher than the simple vaporization method of iron bulk. They also synthesized nitride nanoparticles by a transferred arc with nitrogen gas (Uda et al. 1987). Watanabe et al. (2001) produced Ti nanoparticles using 6 kW Ar–H2 transferred arc plasmas with a tungsten cathode. In the paper, they mentioned that the H2 gas inclusion elevates the evaporation rate of Ti bulk anode. The evaporation rate of Ti bulk anode was estimated as 0.2 g/min and found that 5% of the input power was used for the evaporation. Tanaka and Watanabe (2008) also use an Ar–H2 transferred arc to synthesize Sn–Ag nanoparticles. Figure 3 shows TEM images of Sn–Ag nanoparticles synthesized from 70%Sn–30%Ag ingot. Panel (a) is the result by 100%Ar arc, while panel (b) shows the result by 50%Ar–50%H2 arc. Nanoparticles synthesized in both conditions have spherical shape. However, the mean diameter was estimated as 19.8 nm and 56.5 nm, respectively, for 100%Ar arc and 50%Ar–50%H2 arc. This may be due to higher density of metallic vapor arising from higher evaporation rate of the raw material in case of 50%H2 inclusion. Attention has been paid to carbon nanomaterials from discovery of carbon nanotube by using transferred arcs (Ebbensen and Ajayan 1992; Iijima and Ichihashi 1993; Ando et al. 2000; Shi et al. 2000; Kim et al. 2006, 2007, 2009, 2017). Ando et al. (2000) developed a mass production method for single-wall carbon nanotube (SWCNT) with a production rate of 1.24 mg/min by using a DC arc plasma with two graphite electrodes (Figs. 4 and 5) (Ando et al. (2000)). The AlN nanoparticles were synthesized by a transferred arc in Cruz and Munz (1997). They separated a reaction section from the plasma reactor, and the Al vapor is generated by

Fig. 3 TEM images of nanoparticles prepared from 30wt%Ar–70%Sn ingot raw material. (a) 100%Ar arc, (b) 50%Ar–50%H2 (Tanaka and Watanabe 2008)

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Y. Tanaka

Fig. 4 A schematic diagram of the DC arc plasma jet (APJ) apparatus (Ando et al. 2000)

the transferred arc, and it was mixed with Ar/NH3 gas to synthesize AlN nanoparticles. For this method, they also developed a numerical model for nucleation and particle growth. Cubic boron nitride nanoparticles are now also receiving attention because of their high hardness and high thermal conductivity. Recent example of nanoparticle synthesis using transferred arc are as follows: Ko et al. (2015) synthesized cubic boron nitride nanoparticles by a transferred Ar–N2 thermal plasma at 13.5 kW at atmospheric pressure. They used boron oxide (B2O3), melamine (C3H6N6), and NH3 as raw materials. Then, they synthesized c-BN nanoparticles with sizes smaller than 150 nm. Stein and Kruis reported optimal process conditions for a single transferred arc electrode pair for pure metal nanoparticle synthesis with a high production rate (Stein and Kruis 2015, 2016). They also have done a scaled-up approach with

65

Synthesis of Nanosize Particles in Thermal Plasmas

2809

Fig. 5 A characteristic SEM image of cottonlike carbon soot produced by APJ method (60A, He 500 torr) (Ando et al. 2000)

parallelization of multiple transferred arcs and synthesized copper nanoparticles with mean size of 79 nm with a production rate of 69 g/h. Kulkarni et al. (2009) reported nanoparticle synthesis of Al2O2, AlN, and FexOy using a DC transferred arc plasma, indicating the effect of pressure on their crystalline phases. Non-transferred arcs are also used for nanoparticle synthesis (Oh and Park 1998; Tong and Reddy 2005, 2006). One feature of the non-transferred arc involves raw material injection to arc jet to evaporate it. Another feature of this method is a less erosion of electrodes compared to transferred arcs. Choi et al. (2006) developed a continuous production method of classical nucleation theory using non-transferred arc jet with methane. Figure 6 indicates a developed arc jet plasma reactor for CNT synthesis by decomposition of CH4. The cathode was made of tungsten, and a copper nozzle is used as an anode. Catalyst Ni and Y powder was directly injected into arc plasma jet. They successfully synthesized CNTs in continuous mode. They found multiwalled CNTs which were mainly produced in high purity with a few other structures of nanotubes observed. Similar continuous synthesis method has been reported by Bystrzejewski et al. (2008). On the other hand, Kim et al. prepared AlN nanoparticles using non-transferred arc (Kim et al. 2013). A pellet of microsized aluminum (Al) powders was evaporated using an argon–nitrogen thermal plasma flame. Also, ammonia (NH3) was introduced into the reactor as a reactive gas. They studied effect of NH3 gas flow rate on the size and the crystallinity of the synthesized AlN particles. Supersonic thermal plasma expansion has been used for nanoparticle synthesis (Goortani et al. 2006; Kakati et al. 2008; Bora et al. 2010, 2012, 2013). Bora et al. (2013) studied synthesis of nanostructured TiN using a supersonic thermal plasma expansion. Figure 7 shows the reactor configuration for nanoparticle synthesis using thermal plasma expansion process. The supersonic thermal plasma can avoid problems of broad particle size distribution and serious interparticle agglomeration that conventional thermal plasmas have encountered. The supersonic thermal plasma expansion provides narrower size distribution because of less agglomeration due to the directed supersonic velocity of the particles

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Y. Tanaka

Fig. 6 Schematic diagram of an arc jet plasma reactor for CNT synthesis by decomposition of CH4 (Choi et al. 2006)

embedded in the plasma. They successfully synthesized mono-crystalline TiN nanoparticles with mean diameter of 15 nm from TiCl4 and NH3 as reactants.

2.9.2

Radio-Frequency (RF) Inductively Coupled Thermal Plasma Technique Principle of RF inductively coupled thermal plasma Inductively coupled thermal plasma (ICTP) or induction thermal plasma (ITP) is established by electromagnetic coupling without electrode. Because there is no electrode, the thermal plasma is almost entirely uncontaminated by impurities. This lack of contamination from the electrodes is extremely useful to synthesize pure nanoparticles. Figure 8 shows the typical RF ICTP torch configuration. The RF ICTP torch is usually composed of a dielectric cylindrical tube and the coil surrounding the tube. To this coil, the radio-frequency coil current is supplied to generate an alternative magnetic field in axial direction and alternative electric field in the azimuthal direction. This electric field can accelerate an electron present in the torch. Then the accelerated electron collides with the gas species. If the kinetic energy exceeds

65

Synthesis of Nanosize Particles in Thermal Plasmas

2811

Fig. 7 Reactor configurations for injection of reactants (a) at the hot zone just after the anode and (b) at the colder tail zone of the plasma flame, where the torch has two additional floating rings after the anode (Bora et al. 2013)

Fig. 8 Configurations of radio-frequency inductively coupled thermal plasma

the ionization potential of gas species, then the gas is ionized to produce the thermal plasma inside the torch. Some feedstock in solid powder, liquid, or solution and suspension is used for nanoparticle synthesis. The feedstock is injected from the head of the torch. Usually, a water-cooled powder feeding probe is inserted into the

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Y. Tanaka

thermal plasma to avoid feeding powder in backflow there. The injected feedstock is heated by the thermal plasma to increase the feedstock temperature. This increase in the temperature results in melting and evaporation of the feedstock. The important benefits of the ICTP are that it is (1) free from contamination, (2) with a large diameter thermal plasma with high gas temperature, and (3) with slow gas flow velocity of several tens of meters per second compared to a DC arc plasma and a plasma jet. This slow gas flow velocity results in long residence time of the feedstock. Long residence time of the order of 10 ms supports sufficient evaporation of the feedstock and also sufficient reactions. Another benefit is that any reactive gas is useful for chemical reaction without concern related to electrode erosion. However, molecular gas injection involves dissociation in thermal plasma, which takes power from the thermal plasma. In addition, as well known, Yoshida et al. (1983) developed a unique plasma torch: the hybrid plasma torch in which an RF inductively coupled thermal plasma with a DC arc plasma jet (Fig. 9). This technique is greatly useful to the stable operation of an RF induction thermal plasma with injection of feedstock and efficient heating of the particles around the center axis of the torch with wider high-temperature area. The hybrid plasma torch has been widely used for nanoparticle synthesis as well as plasma spraying. Examples The ICTP is widely used to synthesize nanoparticles of various kinds such as metallic, oxide, nitride, carbide, and other compounds because of the advantages of ICTP. Nanoparticle synthesis using an RF induction thermal plasmas has continuously been developed from pioneering works (Yoshida et al. 1979; Guo et al. 1995). Following the works, many researchers used radio-frequency (RF) inductively coupled thermal plasma torch to synthesize various kinds of nanoparticles: nitrides (Yoshida et al. 1979; Lee et al. 1990; Soucy et al. 1995; Cruz and Munz 2001, etc.), carbides (Eguchi et al. 1989; Guo et al. 1995; Gitzhofer 1996; Leparoux et al. 2005; Ishigaki et al. 2005; Thompson et al. 2013, etc.), oxides (Oh and Ishigaki 2004; Goortani et al. 2006; Li et al. 2007; Ishigaki and Li 2007; Marion et al. 2007; Zhang et al. 2014, etc.), and some metals (Yoshida and Akashi 1981; Harada et al. 1985; Anekawa et al. 1985; Son et al. 2002; Boulos et al. 2006; Cheng et al. 2014; Tanaka et al. 2016; Sone et al. 2016, etc.). Doped nanoparticles, intermetallic nanoparticles, and core-shell nanoparticles are also synthesized using inductively coupled thermal plasmas. Yoshida et al. (1979), Guo et al. (1995), and Gitzhofer (1996) contributed syntheses of SiC nanoparticles or ultrafine powders using an induction thermal plasma at a high power above 40 kW. They investigated experimentally feedstock size and feeding probe position on the evaporation rate of Si feedstock and then numerically studied heat and mass transfer in the RF induction thermal plasma with reactive gas CH4. These results have become a base of nanoparticle synthesis using RF induction thermal plasmas. Ishigaki and Li (2007) investigated cooling effect of thermal plasma by injection of quenching gas for nanoparticle synthesis. The cooling of the vapor and gas is essential for high efficiency and control of nanoparticle synthesis. It also influences the composition of synthesized nanoparticles. Figure 10 indicates experimental

65

Synthesis of Nanosize Particles in Thermal Plasmas

2813

M DC POWER SUPPLY (+)

(-)

Ar

MSiCI

4+Ar

M

Ar + H2

WATER OUT RF COIL WATER IN

MNH3

THERMOCOUPLES

WINDOW PYREX TUBE

TO VACUUM PUMP

WATER IN WATER OUT

TO EXHAUST SYSTEM

Fig. 9 Schematic view of a hybrid plasma reactor chamber designed for the preparation of ultrafine Si3N4 (Yoshida et al. 1983)

setups with different quenching gas directions: (a) transverse swirl-flow injection and (b) counterflow injection. The direction of the quenching gas injection influences cooling effect of vapors. Figure 11 shows the calculated streamlines and temperatures for Ar quenching gas injection with the above directions. The predicted trajectories of a test particle are also presented. The Ar/O2 RF induction thermal plasma is assumed for TiO2 nanoparticle synthesis, and liquid precursor was adopted. Titanium dioxide has high photocatalyst ability for UV light, and it is used for photocatalyst, photonic crystal, photo-electrochemical cells, and various sensors. As seen, when Ar quenching gas is injected counter to the plasma plume, the temperature field and also the trajectories of the particle are profoundly affected. The authors found that a test particle experiences high-temperature zone in a shorter

2814

Y. Tanaka

a

b quench gas

gas

quench gas

quench gas

gas

quench gas supply gas

quench gas

gas gas

gas

Fig. 10 Experimental setups for transverse swirl-flow injection (a) and counterflow injection (b) of quench gases (Ishigaki and Li 2007)

0 1 2 3 4 5 6 7 8 9 10 11

z (mm)

200

400

600

800

b

T (103 K)

0

0 1 2 3 4 5 6 7 8 9 10 11

200

400

600

800

1000 −100 −50

0

r (mm)

50

100

1000 −100 −50

c

T (103 K)

0

0 1 2 3 4 5 6 7 8 9 10 11

200

z (mm)

T (103 K)

0

z (mm)

a

400

600

800

0

50

r (mm)

100

1000 −100 −50

0

50

100

r (mm)

Fig. 11 Streamlines and temperature distribution for (a) no quench gas, (b) transverse swirl-flow injection of Ar at 100 slpm, and (c) counterflow injection of Ar at 100 slpm (Ishigaki and Li 2007)

duration and that it favors the formation of finer particles. They also investigated synthesis of phase-controlled TiO2 nanoparticles (Li and Ishigaki 2004) and the synthesis of metallic ion-doped TiO2 nanoparticles using an RF induction thermal plasma (Wang et al. 2005; Zhang et al. 2010, 2014). Metallic ion-doped TiO2 nanoparticles like Eu3+-doped TiO2, Fe3+-doped TiO2, and Eu3+–Nb5+-co-doped TiO2 were successfully synthesized using the RF induction thermal plasmas. The

65

Synthesis of Nanosize Particles in Thermal Plasmas

2815

impurity doping markedly improves photocatalyst efficiency for visible light (Barolo et al. 2012). Metallic nanoparticles have been synthesized using an RF inductively coupled thermal plasmas. Kim et al. reported synthesis of nickel nanoparticles using inductively coupled thermal plasma with a Ni(OH)2 micro-powder feedstock (Kim et al. 2016). Nickel nanoparticles are intended for use as internal electrodes of multilayer ceramic capacitors (MLCC). They studied the dependence of process parameters on the size and morphology of synthesized nanoparticles. They also synthesized core-shell structured composite powders using induction thermal plasmas (Kim et al. 2017). Recently, attention is paid to nanomaterials for energy issue solution. One of them is nanomaterials for electrical batteries. Si nanoparticles are promising elements which can be used for anode material in lithium ion batteries (LIB). Reaction chamber geometries for Si nanoparticle synthesis with an RF thermal plasma were studied by numerical thermofluid simulation by Colombo et al. (2012, 2013). They also investigated gas injection along the wall and found the advantage of the conical chamber geometry on the high yield of nanoparticles. Tanaka et al. (2016) and Sone et al. (2016) reported synthesis of lithium metal oxide composite nanoparticles by induction thermal plasmas. Lithium metal oxide composite nanoparticles are expected as cathode materials for solid oxide fuel cell. They synthesized Li–metal oxide nanoparticles of various shapes in Li–Mn, Li–Cr, Li–Ni, and Li–Co systems with Ar/O2 induction thermal plasma. They used solid mixture powder feedstock of Li2CO3 and MnO2. Figure 12 shows an SEM image of the as-prepared nanoparticles. As seen, synthesized LiMnO2 has a tetradecahedral crystalline structure. They found that such spinel-structured LiMn2O4 nanoparticles were successfully obtained by controlling the admixture ratio of O2 to Ar in the thermal plasma. They also studied the formation mechanism of Li–Mn oxide nanoparticles theoretically. They found that in the Li–Mn oxide nucleation system, MnO nucleation is first initiated, and then MnO and Li2O vapors co-condense on the formed nucleus, then forming LiMn2O4.

Fig. 12 Representative SEM image of the LiMn2O4 nanoparticles prepared under an O2 flow rate of 2.5 L= min. The powder feed and carrier gas flow rates used in this experiment are 0.4 g=min and 3 L=min, respectively (Sone et al. 2016)

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Off-time On-time

HCL LCL

Coil current

Fig. 13 Modulated coil current and modulation parameter definitions (Kodama et al. 2014)

0

Time

2.9.3 New Technology: Modulated Induction Thermal Plasmas Methodology of a large-scale nanopowder synthesis system Recently, a unique method for nanoparticle synthesis at a high production rate was developed by Tanaka et al. (2010, 2012): the PMITP–TCFF method. The PMITP is the pulse-modulated induction thermal plasma. The TCFF is time-controlled feeding of feedstock. Here, details of the developed method are introduced for synthesis of large amounts of Al3+doped TiO2 nanopowders (Kodama et al. 2014). The developed PMITP–TCFF method uses the pulse-modulated induction thermal plasma (PMITP), which is sustained with the coil current of approximately several hundred amperes, the amplitude of which is modulated into a rectangular waveform (Kodama et al. (2014)). Such modulation of the coil current can repetitively produce a high-temperature field during the “on-time” and a low-temperature field during “offtime” in thermal plasmas. Figure 13 portrays the coil current modulated into a rectangular waveform along with the definition of modulation parameters. As presented in Fig. 13, the on-time is the time period with the higher current level (HCL); although the off-time is the time period with the lower current level (LCL). We have also defined a shimmer current level (SCL) as a ratio of LCL to HCL. The duty factor (DF) has also been defined as the ratio of on-time in one modulation period. A condition of 100%SCL or 100%DF corresponds to the non-modulation condition. A lower SCL condition is equivalent to a condition with a larger modulation degree. Figure 14 presents the methodology developed for the synthesis of large amounts of nanopowder using the PMITP–TCFF method. The PMITP can repetitively produce a higher temperature field and a lower temperature field according to the coil current modulation. To this PMITP, the feedstock solid powder is supplied through a powder feeding tube from a powder feeder along with Ar carrier gas from the top of the plasma torch head to the PMITP. Furthermore, a high-speed valve is installed on the tube between the powder feeder and the plasma torch. Setting the open and close timing of the valve can control the actual timing and the time length of the powder feeding. The right-hand side of Fig. 14 shows a timing chart of the coil current modulation, switching signal of the solenoid valve, and actual powder feeding. For synthesis of

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plasma Rapid & complete evaporation

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td +7 ms Timing of feeding feedstock

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Quenching gas Quenching gas PMITP + TCFF method Fig. 14 Method for large-scale nanopowder synthesis using pulse-modulated induction thermal plasma with time-controlled feedstock feeding (PMITP–TCFF) (Kodama et al. 2014)

large amounts of nanopowder, heavy-load feeding of feedstock is necessary without extinction of the plasma or incomplete evaporation of the feedstock. In the developed PMITP–TCFF method, the feedstock powder is controlled to be fed intermittently and synchronously only during the high-temperature period in the on-time of the PMITP. This synchronized powder feeding can be executed easily by controlling the delay time td for the opening timing of the valve in reference to the pulse modulation signal of the PMITP. The injected feedstock powder with heavy load is evaporated rapidly, completely, and efficiently in high-temperature plasma during the on-time of the PMITP because of higher power injection to the PMITP. The feedstock injection is stopped by closing the solenoid valve during the successive off-time. During the off-time, the evaporated feedstock material is cooled rapidly because the thermal plasma temperature decreases as a result of the decreased input power to the PMITP. This rapid cooling might promote particle nucleation from evaporated atomic materials in vapor in the PMITP. Nucleated particles are transported downstream of the PMITP torch with particle growth. Downstream of the PMITP torch, the quenching gas is injected in the radial direction. Such a quenching gas injection cools the evaporated material further to restrain the synthesized particle growth. Then, in the successive on-time, the input power increases to rebuild high-temperature thermal plasma for the subsequent powder injection. In this way, the PMITP–TCFF method described above can create effective vaporization of the feedstock and support effective cooling of the evaporated material. It enables synthesis of large amounts of nanopowder.

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Fig. 15 FE–SEM images for (a) feedstock and synthesized powder with conditions (b) 80%SCL, gpow = 12 g min1, (c) 70%SCL, gpow = 12 g min1, and (d) 60%SCL, gpow = 19 g min1 (Kodama et al. 2014)

Results and discussions Figure 15 presents FE–SEM images of feedstocks and the synthesized nanopowder collected at the filter under different SCL conditions (Kodama et al. 2014). The averaged input power and the pressure were 20 kW and 300 torr, respectively, and 95wt%Ti–5%Al was used. The feedstock Ti powder has mean diameter of approximately 27 μm with various shapes. However, most synthesized particles were found to have size of 100 nm or less in any of the three SCL conditions in the present experiments. Therefore, nano-sized particles were produced despite heavy-load feeding of the feedstock using our developed method. The synthesized particle shapes were observed to be almost spherical for these three conditions, which implies that nanoparticles were synthesized and grown in gas phase rather than on the filter or wall surface. It was also found that the mean particle diameter can be controlled by SCL. Based on these results and weight measurements of the synthesized powder, the production rates of nanopowder were estimated as higher than 400 g h1 using the 20-kW PMITP. This production rate value is 10–20 times higher than those obtained using the conventional thermal plasma method reported in the literature (Tsai et al. 2012; Lee et al. 2004). Figure 16 presents BF–TEM images and TEM/EDX mapping results of synthesized nanopowder collected at the filter. Panel (a) presents results for the condition of 80%SCL with a powder feed range gpow of 12 g min1, whereas panel (b) shows

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Fig. 16 BF–TEM and TEM/EDX mapping for (a) 80%SCL, gpow = 12 g min1, (b) 70%SCL, gpow = 12 g min1, and (c) 60%SCL, gpow = 19 g min1 (Kodama et al. 2014)

results for the condition 70%SCL and gpow = 12 g min1. Finally, panel (c) shows the results for the condition 60%SCL and gpow = 19 g min1. Each panel has a BF–TEM image and Ti, O, and Al element distributions. Magnifications of TEM images and TEM/EDX mapping differ among panels. From the BF–TEM images, results show that spherical nanoparticles without mesopores were synthesized for these three SCL conditions. The TEM/EDX mapping shows that Al was detectable and that it was distributed almost uniformly in the synthesized particles similarly to Ti and O for 80%SCL and 70%SCL. That result, considered together results of XRD and XPS analysis, as described later, means that elemental Al might be distributed almost uniformly on the synthesized TiO2 nanoparticles by our developed PMITP–TCFF method with heavy-load feeding of feedstock. After other analyses using XRD, XPS, and optical property measurements, the synthesized nanoparticles were found to be Al3+-doped TiO2.

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In this way, results showed that the PMITP–TCFF method can synthesize Al3+doped TiO2 nanopowder in large amounts. Effects of the coil current modulation degree are found to offer a high evaporation rate and efficient nucleation of nanoparticles, with the control of the mean particle diameter synthesized. The final production rate was estimated as higher than 400 g h1. Spectroscopic observation of feedstock evaporation and precursor formation A key point to enhancing production efficiency is to elucidate nanoparticle formation mechanisms in an induction thermal plasma torch. Recently, Tanaka et al. have investigated the mechanism using a two-dimensional spectroscopic observation technique during TiO2 NP synthesis. Results show that the two-dimensional spectroscopic observation system can produce temporal variation of two-dimensional radiation intensity distributions of Ti I and TiO spectra in the PMITP (Kodama et al. 2016). The intermittent injection of the feedstock was conducted using a high-speed solenoid valve. The open (topen)/close time (tclose) of the solenoid valve was regulated at 8/22 ms (27%DFvalve). This condition corresponds to an almost single-shot feeding of feedstock, which can avoid interaction between single powder feedings during intermittent feeding. Figure 17a again shows the observation result of Ti I intensity variation for a valve opening time topen of 8 ms; Fig. 17b is the observation result of TiO intensity variation for topen of 8 ms. These were observed simultaneously using the two-dimensional spectroscopic imaging system. However, Fig. 17c–e, respectively, a

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Fig. 17 Understanding of Ti feedstock evaporation and precursor TiO molecules in the torch: (a) Observation result for Ti I in valve open condition of 8 ms; (b) observation result for TiO in valve open condition of 8 ms; (c) estimated timing of feedstock injection into the torch; (d) estimated temporal variation of temperature; and (e) estimated timing of formation of precursor TiO molecules (Kodama et al. 2016)

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show our estimated time chart showing Ti feedstock injection into the torch, the temperature variation in the torch, and the temporal variation in formation of precursor TiO molecules. In addition, Fig. 18 shows the physical picture with comments inferred for the temporal and spatial distribution of Ti atomic density and TiO molecular density in the case of 20%DFvalve. In this figure, the green region shows the distribution of dense Ti atom; the red region shows the distribution of dense TiO molecule. Figure 18a is the inferred temperature distribution before injection of Ti feedstock into the torch. Figure 18b shows the introduction of Ti feedstock to the torch, Ti evaporation, and initiation of TiO formation at t  7 ms. Figure 18c depicts the mixing of Ti and O atoms and formation of TiO at t ~ 14 ms. Figure 18d shows the transport of TiO to the downstream of the torch at t ~ 18 ms. Figure 18e presents the decrease in TiO density and increase in Ti atomic density attributable to dissociation reaction of TiO!Ti+O by temperature recovery at t ~ 20 ms. Figure 18f shows the decrease in Ti density attributable to convective transport to the downstream of the torch at t ~ 22 ms.

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Fig. 18 Temporal and spatial variation of Ti atomic density and TiO molecular density in the ICTP torch estimated from two-dimensional spectroscopic observation results for 27%DFvalve: (a) Before feedstock injection into the ICTP torch (t = 0 ms); (b) Ti feedstock feeding, Ti evaporation, and TiO formation (t ~ 7 ms); (c) formation of high-density Ti–O mixture and TiO molecules (t ~ 14 ms); (d) TiO transport and TiO dissociation attributable to the increased temperature by stopping feedstock feeding (t ~ 18 ms); (e) ormation of high-density Ti atoms attributable to dissociation reaction of TiO!Ti+O (t ~ 20 ms); and (e) convective transport of Ti atoms (t ~ 22 ms) (Kodama et al. 2016)

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The feedstock feeding starts by opening the valve. The feedstock powder from the valve arrives at the tip of the water-cooled tube into the thermal plasma within 7–8 ms, as presented in Figs. 17c and 18b. The injected feedstock powder begins to evaporate very quickly in 0.5 ms to generate Ti atoms just under the outlet of the feeding tube because of high temperature of the ICTP around 5000 K. The feedstock evaporation offers dense Ti atoms. The generated Ti atoms are transported to the axial direction downstream by flow of the carrier gas. The Ti atoms are also diffused in the radial direction because of high gradient in Ti atomic density formed by feedstock evaporation. At the same time, the on-axis temperature decreased rapidly to 2000–5000 K because of the cool carrier gas flow and also by the feedstock evaporation shown in Fig. 18b. Therefore, the time variation is estimated in the temperature just under the tube as presented in Fig. 17d. Because of this temperature decrease to 2000–5000 K, Ti and O are associated to produce precursor TiO molecules around the on-axis region according to the calculated equilibrium composition. This formation reaction occurs almost simultaneously with Ti feedstock evaporation: within 0.5 ms. Therefore, high-density Ti atomic gas is present rather in the off-axis region although high-density TiO molecular gas is present around the on-axis region. Increasing Ti atomic density between t = 8–10 ms promotes the TiO molecule formation reaction, resulting in the high-density TiO molecules at t = 8–10 ms, as presented in Fig. 17e. In t = 10–16 ms, the time variation of Ti atomic and TiO molecular densities becomes almost steady because of the balance between feedstock evaporation and the diffusive transportation of Ti atoms and TiO molecules. Therefore, the spatial distributions of Ti atom and TiO molecule are obtainable as shown in Fig. 18c. When the feedstock feeding and evaporation are finished, the temperature just under the outlet of the feeding tube recovers to higher values of 9000 K in the torch, as shown in Fig. 17d. In this case, TiO molecules are dissociated, leading to decreasing TiO molecular density, as shown in Fig. 18d. At the same time, Ti atomic density is maintained, although the TiO molecular density is decreased because Ti atoms are regenerated by a dissociation reaction of TiO molecules, as presented in Fig. 18e. These TiO and Ti are transported downstream of the plasma torch and then to the reaction chamber. As described above, high-density Ti atoms are created in the off-axis region. High-density TiO are produced in the on-axis region inside the plasma torch. They are then transported to the reaction chamber by gas flow. Such high-density Ti atoms can produce Ti clusters by homogeneous nucleation. Then they can be oxidized to create TiOx clusters, i.e., the nuclei of TiO2 NPs. However, the high-density TiO can also become TiOx clusters by homogeneous nucleation with association. These clusters can grow by inhomogeneous processes including chemical vapor deposition from Ti, O, and TiO in gas phase. Then, coagulation and agglomeration occur to form NPs in the reaction chamber. The observation results provided fundamental information related to evaporation of the Ti feedstock and transport of Ti atoms, in addition to the subsequent formation

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and transport processes of TiO. These data are expected to be useful to assess highly efficient nanoparticle synthesis using inductively coupled thermal plasmas.

3

Conclusions

This chapter is devoted to description of the fundamentals and basics of nanoparticle synthesis using thermal plasmas. As described, thermal plasma is an effective heat source, a powerful tool to evaporate raw material feedstock and then to cool the evaporated material for formation of nanoparticles as a high-temperature gradient source. The thermal plasma method offers great potential to contribute to the construction of future society.

4

Cross-References

▶ Analytical Methods in Heat Transfer ▶ Droplet Impact and Solidification in Plasma Spraying ▶ Electrohydrodynamically Augmented Internal Forced Convection ▶ Evaporative Heat Exchangers ▶ Full-Coverage Effusion Cooling in External Forced Convection: Sparse and Dense Hole Arrays ▶ Heat Transfer in Arc Welding ▶ Heat Transfer in DC and RF Plasma Torches ▶ Heat Transfer in Plasma Arc Cutting ▶ Heat Transfer in Rotating Flows ▶ Heat Transfer in Suspension Plasma Spraying ▶ Macroscopic Heat Conduction Formulation ▶ Numerical Methods for Conduction-type Phenomena ▶ Phase Change Materials ▶ Plasma Waste Destruction ▶ Plasma-Particle Heat Transfer ▶ Radiative Plasma Heat Transfer ▶ Radiative Properties of Gases ▶ Radiative Properties of Particles ▶ Radiative Transfer Equation and Solutions ▶ Single-Phase Convective Heat Transfer: Basic Equations and Solutions ▶ Single-phase Heat Exchangers ▶ Thermal Transport in Micro- and Nanoscale Systems ▶ Thermophysical Properties Measurement and Identification ▶ Turbulence Effects on Convective Heat Transfer ▶ Two-phase Heat Exchangers

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Shigeta M, Watanabe T (2007) Growth mechanism of silicon-based functional nanoparticles fabricated by inductively coupled thermal plasmas. J Phys D Appl Phys 40:2407–2419 Shigeta M, Watanabe T (2008) Numerical investigation of cooling effect on platinum nanoparticle formation in an inductively coupled thermal plasma. J Appl Phys 103:074903 Shigeta M, Watanabe T (2010) Growth mechanism of binary alloy nanopowders for thermal plasma synthesis. J Appl Phys 108:043306 Smoluchowski M (1916) Drei Vortrage uber Diffusion, Brownsche Molekularbewegung undKoagulation von Kolloidteilchen. Z Physik Z 17(557–571):585–599 Son S, Taheri M, Carpenter E, Harris VG, McHenry ME (2002) Synthesis of ferrite and nickel ferrite nanoparticles using radiofrequency thermal plasma torch. J Appl Phys 91:7589 Sone H, Kageyama T, Tanaka M, Okamoto D, Watanabe T (2016) Induction thermal plasma synthesis of lithium oxide composite nanoparticles with a spinel structure. Jpn J Appl Phys 55 (7S2):07LE04 Soucy G, Jurewicz JW, Boulos MI (1995) Parametric study of the plasma synthesis of ultrafine silicon nitride powders. J Mater Sci 30(8):2008–2018 Stein M, Kruis FE (2015) Scale-up of metal nanoparticle production. NSTI: Adv Mater Tech Connect Briefs 1:203–206 Stein M, Kruis FE (2016) Optimization of a transferred arc reactor for metal nanoparticle synthesis. J Nanopart Res 18(9):258 Tanaka M, Kageyama T, Sone H, Yoshida S, Okamoto D, Watanabe T (2016) Synthesis of lithium metal oxide nanoparticles by induction thermal plasmas. Nano 6(4):60 Tanaka Y, Nagumo T, Sakai H, Uesugi Y, Nakamura K (2010) Nanoparticle synthesis using highpowered pulse-modulated induction thermal plasma. J Phys D Appl Phys 43:265201 Tanaka Y, Tsuke T, Guo W, Uesugi Y, Ishijima T, Watanabe S, Nakamura K (2012) A large amount synthesis of nanopowder using modulated induction thermal plasmas synchronized with intermittent feeding of raw materials. J Phys D: Conf Ser 406:012001 Tanaka M, Watanabe T (2008) Vaporization mechanism from Sn-Ag mixture by Ar-H2 arc for nanoparticle preparation. Thin Solid Films 516(19):6645–6649 Thompson D, Leparoux M, Jaeggi C, Buha J, Pui DYH, Wang J (2013) Aerosol emission monitoring in the production of silicon carbide nanoparticles by induction plasma synthesis. J Nanopart Res 15:12 Tong L, Reddy RG (2005) Synthesis of titanium carbide nano-powders by thermal plasma. Scr Mater 52:1253 Tong L, Reddy RG (2006) Thermal plasma synthesis of SiC nano-powders/nano-fibers. Mater Res Bull 41:2303–2310 Tsai YC, Hsi CH, Bai H, Fan SK, Sun DH (2012) Single-step synthesis of Al-doped TiO2 nanoparticles using non-transferred thermal plasma torch. Jpn J Appl Phys 51:01AL01 Uda M, Ohno S, Hoshi T (1983) Process for producing fine metal particles. US Patent 4376740 Uda M, Ohno S, Okuyama H (1987) Process for producing particles of ceramic. US Patent 4642207 Wang XH, Li JG, Kamiyama H, Katada M, Ohashi N, Moriyoshi Y, Ishigaki T (2005) Pyrogenic iron (III)-doped TiO2 nanopowders synthesized in RF thermal plasma: phase formation, defect structure, band gap, and magnetic properties. J Am Chem Soc 127:10982–10990 Watanabe T, Itoh H, Ishii Y (2001) Preparation of ultrafine particles of silicon base intermetallic compound by arc plasma method. Thin Solid Films 390(1–2):44–50 Watanabe T, Tanaka M, Shimizu T, Liang F (2013) Metal nanoparticle production by anode jet of argon-hydrogen dc arc. Adv Mater Res 628:11–14 Yoshida T, Akashi K (1981) Preparation of ultrafine iron particles using an RF plasma. Trans Jpn Inst Metals 22:371–378 Yoshida T, Kawasaki A, Nakagawa K, Akashi K (1979) The synthesis of ultrafine titanium nitride in an rf plasma. J Mater Sci 14:1624–1630 Yoshida T, Tani T, Nishimura H, Akashi K (1983) Characterization of a hybrid plasma and its application to a chemical synthesis. J Appl Phys 54:640–646

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Zhang C, Li JG, Uchikoshi T, Watanabe T, Ishigaki T (2010) (Eu3+-Nb5+)-codoped TiO2 nanopowders synthesized via Ar/O2 radio-frequency thermal plasma oxidation processing: phase composition and photoluminescence properties through energy transfer. Thin Solid Films 518:3531–3334 Zhang C, Uchikoshi T, Li JG, Watanabe T, Ishigaki T (2014) Photocatalytic activities of europium (III) and niobium (V) co-doped TiO2 nanopowders synthesized in Ar/O2 radio-frequency thermal plasmas. J Alloys Compd 606:37–43

Plasma Waste Destruction

66

Milan Hrabovsky and Izak Jacobus van der Walt

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Waste Destruction Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Incineration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Pyrolysis and Fast Pyrolysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Gasification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Plasma Arc Gasification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Characteristics of Plasma Treatment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Plasma Gasification Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Chemistry of Organics Gasification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Plasma-Material Heat Transfer and Gasification Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Mass and Energy Balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Syngas Production from Organic Waste . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Plasma Waste to Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Energy Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Process Layout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Influence of Waste Type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4 Financial Viability (Financial Modeling) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2830 2833 2833 2834 2834 2836 2840 2845 2848 2853 2857 2864 2870 2871 2873 2874 2876 2881 2882

M. Hrabovsky (*) Institute of Plasma Physics ASCR, Prague, Czech Republic e-mail: [email protected] I. J. van der Walt R&D Plasma Development, The South African Nuclear Energy Corporation, Plelindaba, North West Province, South Africa e-mail: [email protected] # Springer International Publishing AG, part of Springer Nature 2018 F. A. Kulacki (ed.), Handbook of Thermal Science and Engineering, https://doi.org/10.1007/978-3-319-26695-4_32

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M. Hrabovsky and I. J. van der Walt

Abstract

This chapter analyzes the physical and technological aspects of waste treatment using thermal plasma technology. The waste treatment problems are briefly characterized and an overview of methods of waste treatment is presented. Main industrial-scale plasma waste treatment units installed in the world are described. Basic principles of plasma waste treatment systems are presented and processes of waste-to-energy conversion by plasma gasification of organics are analyzed. Potential advantages of using plasma for gasification are summarized, fundamental chemistry of the plasma gasification process as well as basic thermodynamics, the energy balance, and the kinetics of the process are described. Examples of results of thermal plasma gasification of various organic wastes are presented. Produced gas compositions for different feed materials are shown, including sawdust, wooden pellets, brown coal, polyethylene, and pyrolytic oil produced from car tires. The results show that it is possible to produce syngas of high hydrogen and carbon monoxide content, with low levels of contaminants, and with a high calorific value. Plasma gasification is a process with huge potential for converting low-value materials to a high-value fuel, syngas. The process offers at the same time a means to convert electrical energy to chemical energy, which is particularly relevant to the storage of renewable energy.

1

Introduction

The saying is: One man’s trash is another man’s treasure. This is becoming a serious reality in the modern world. With the world energy crisis, the strict environmental laws and the pressure on industry to aim for zero waste-to-landfill, there is a great need to re-evaluate “waste.” Globally there is a renewed effort to look at alternative energy sources. Various countries employ an integrated energy plan which allows for almost all electricity sources to be exploited. This energy mix distribution for a number of countries is depicted in Fig. 1. It is clear from this figure that coal is still one of the major electricity producers worldwide especially in countries like the EU and USA. There is also a significant contribution from nuclear power plants in these countries. In special cases like Austria and Sweden hydro-, and in France, nuclear power, overshadow the electricity production landscape. To a certain extent, these countries could be at risk because of their high dependency on a single electricity source. Dependable resources and the associated technologies render this viable. The USA, EU Mix, and Germany are good examples countries with a good electricity mix. It can also be observed that 1–10% of the electricity supply is from alternative technologies using biomass as feedstock. More data about the status in India and China may be found by visiting the following links, in the reference list (International Energy Agency 2015) and (World Energy China Outlook 2014).

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Fig. 1 Electricity generation mix data for different countries in 2014 (Anon 2014)

Electric energy supplied in TWh p.a. 500 PV CSP Wind Hydro Nuclear

400

300

Renewable TWh`s in 2030 (14%)

CCGT/CCGT

Carbon free TWh`s in 2030 (34%)

200 Coal 100

0 2010

2015

2020

2025

2030

Fig. 2 Electric energy distribution planned for 2030 in SA (DoE 2011)

The historical and planned energy mix information for South Africa is shown in Fig. 2. This includes a 14% renewable energy (including energy from biomass) component by 2030. South Africa as a developing country depends mostly on coal as the bulk electricity source. Only about 10% of the country’s electricity is currently produced from other sources. This is planned to increase to 34% by 2030. The world energy, and especially the renewable energy mix, consists of several technologies with biomass the largest contributor, as indicated in Fig. 3.

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Fig. 3 The total world energy consumption in 2013 (IEA Renewable energy working party 2002)

Renewable energy, by definition, is obtained from a source that is not depleted by use, such as wind or solar power. Clearly, all renewable energy comes either directly or indirectly from the sun. Many ways have been proposed to utilize sunlight and most of them will probably find a place where they are the most appropriate. Several technologies are considered for electricity production from renewable energy sources. These are solar energy, wind power, geothermal energy, hydroelectric, and hydrodynamic (Union of concerned 2016). Municipal Solid Waste (MSW) was not initially considered a renewable energy source. This was based on bad experiences with incineration facilities producing toxic emissions as well as these plants using recyclable components as feed material (Environmental and energy study institute 2009). This view has changed and MSW was reclassified as a renewable energy source due to the growing environmental, climate, and energy concerns as well as technological advances and regulatory changes (ASME SWPD 2007). These, along with the growing need for landfill space, make MSW a viable source of energy. The treatment of municipal solid waste is problematic in many countries. It is currently a running expense to local municipalities due to logistics, and in many areas, they cannot adequately cope with this problem, causing severe pollution of water sources and land. Turning these wastes into fuel and energy can turn a problem into an opportunity. Sale of the energy (like electricity) can generate income to help finance collection and sorting of the waste contribute to job creation and to clean up of the environment. A completely different scenario would be associated with, for instance, a more affluent chicken or pig farmer who has to dispose of animal wastes in order to cut on fuel or electricity costs. More examples are medical, pharmaceutical, and hazardous wastes. In these cases, the extremely high conventional disposal cost requires an effective and cheap alternative disposal/destruction method. The environmental and social hazard must also be addressed.

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The economic and environmental drivers in each of these cases would be entirely different and so it is impossible to do a general requirement analysis. Different waste streams would require different treatment methods or variations thereof such as incineration, torrefaction, pyrolysis, conventional gasification, plasma gasification, and sometimes a combination of these as discussed in Sect. 2

2

Waste Destruction Methods

2.1

Incineration

Incineration or burning has historically been used to reduce the volume of household and garden waste in local dumps and is still being done in rural areas with the resultant air pollution and health risks due to largely uncontrolled conditions. Incineration or “mass burn” is an exothermic process carried out under controlled conditions in the presence of excess air. Organic matter is completely converted to heat, carbon dioxide (CO2), and water. Depending on the composition of the fuel (e.g., MSW) and the temperature, the off-gas can contain fly ash and minor amounts of products of incomplete combustion (PIC’s), nitrogen oxides (NOx), sulfur dioxide (SO2), chlorinated dibenzo-dioxins (PCDDs), dibenzofurans (PCDFs), and hydrochloric acid (HCl). The latter compounds can be attributed to the presence of chlorine-containing matter in the feedstock, such as polyvinyl chloride (PVC). PCDD and PCDF formation have been attributed to inorganic chlorides in char-containing ash (Steiglitz and Vogg 1988). The flue gas, therefore, has to be cleaned up by various means (quenching, scrubbing, and filtration) before releasing to the atmosphere. The majority of inorganic matter is recovered as bottom ash. Depending on its composition (e.g., heavy metals content), the ash may be classified as hazardous material requiring further treatment. In Japan, fly ash and bottom ash from incinerators are routinely vitrified in plasma furnaces to diminish the leachability of hazardous components, e.g., in the EBARA process (Suzuki 2007) being applied in 15 plants around the country. In the densely populated areas in developed countries, MSW incineration has been used for many years due to the lack of suitable space for land-filling. Waste heat from these facilities is used for district heating or in cogeneration plants. As an example (Ramazzini et al. 2012), five plants in Italy situated at Brescia, Acerra, Milan, Corteolona, and Bergamo incinerate a total of 2 mega ton per annum (Mt/a) of municipal waste. The Acerra plant has a throughput of 1650 ton per day (t/d) (68.75 ton per hour (t/h)) and produces about 92 megawatt electrical (MWe) or about 1.34 megawatt per ton (MW/t) nett. In this case, 38% of the waste stream was recycled and composted, another 38% disposed of in landfill, and 24% incinerated in 2011. The 2010 averages for the EU were 40%, 38%, and 22% respectively. Figure 4 presents a flow diagram of a typical incineration plant.

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Air Emissions Gas Cleanup

By-products Such as Sulphur and Acid Gases

Feedstock Municipal Solid Waste (MSW)

Preprocessing

Fluid Bed Boiler or Moving Grate Reactor Air/O2

Recyclables

Power Generation: _Electrical Energy and _Steam

Electricity to Grid

Ash & Metals

Fig. 4 Process flow diagram of a typical incineration plant (Rajvanshi 1986)

2.2

Pyrolysis and Fast Pyrolysis

Is the thermal decomposition of material in the absence of oxygen (Ramboll and Whitford 2007; Young 2010). It is the fundamental precursor of both combustion and gasification. Pyrolysis naturally occurs in the first 2 s (BioEnergyConsult n.d.). The pyrolysis process consists of both simultaneous and successive reactions. Thermal decomposition of the organic components in biomass starts at 350–550
C and goes up to 700–800
C. The long chains of carbon, hydrogen, and oxygen compounds in biomass break down into smaller gaseous molecules, condensable vapors (tars and oils), and solid charcoal. The rate and extent of decomposition of each of these components depend on the process parameters, such as reactor temperature, biomass heating rate, pressure, reactor configuration, and feedstock (Lin et al. 1999; Zafar 2015) A typical flow diagram of a pyrolysis plant is presented in Fig. 5.

2.3

Gasification

Gasification is carried out under partial oxygen feed, with or without steam addition, and produces syngas consisting mainly of carbon monoxide (CO), hydrogen (H2), and nitrogen (N2) together with varying amounts of methane (CH4), carbon dioxide (CO2), and water vapor (H2O) (Higman and Van der Burgt 2008). In the case of MSW and coal gasification (SASOL process), some sulfur compounds and acid

66

Plasma Waste Destruction

2835 Air Emissions Syngas Cleanup

Synthesis Gas (SYNGAS)

Feedstock Municipal Solid Waste (MSW)

Preprocessing

By-products Such as Sulphur and Acid Gases

Pyrolysis Reactor

Power Generation: _Electrical Energy and _Steam

Recyclables Ash, Carbon Char, & Metals

Electricity to Grid

Pyrolytic Oil

Fig. 5 Flow diagram of a pyrolysis plant (Rajvanshi 1986)

gases like HCl, NOx, HF, etc. will also be present. Gasification is a combination of endothermic and exothermic processes, depending on the composition of the feed material. The amount of ash produced depends on the inorganic content of the feed material. An excellent introduction to the subject of gasification is available on the website of the NETL (Laboratory 2016). There are many different proprietary designs for gasification reactors, but, broadly speaking, they fall into two classes, i.e., fixed-bed and fluidized-bed gasifiers (Kumar 2009). Fixed-bed gasifiers can further be divided into updraft (countercurrent) or downdraft (cocurrent) designs. The ash may be problematic in fluidized-bed applications where high alkali content leads to fluxing and clinker formation in the bed. In general, a fixed-bed downdraft gasifier is a good choice if the syngas is to be used for electricity generation because it produces the gas at high temperatures and with little tar and condensable content. Typical gasification reactions are illustrated in Sect. 4, Eqs. 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10, together with the corresponding reaction enthalpies at 25
C, although the benchmark temperature for the proposed plasma gasification is 1000
C. The composition and calorific value of the product stream from the gasifier vary according to the initial feed composition, moisture content, air feed, final processing temperature and the amount of inert gas it contains. The net energy recovery from MSW depends on the thermal efficiency of the process. By way of illustration, the energy content (calorific value) of various fuels is compared in Table 1. The heating value for MSW derived syngas quoted here was obtained in a 10 t/h pilot plant (Carabin and Holcroft 2005).

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Table 1 Illustrative calorific values of various fuels (Engineering_toolbox 2016) Higher calorific value Fuel Anthracite Bituminous coal Butane Charcoal Coke Diesel Hydrogen Methane Natural gas Paraffin Petrol Propane Town gas Vegetable oils Wood (dry) Syngas (MSW derived, plasma process)

(Gross calorific value – GCV) MJ/kg 32.5–34 17–23.25 49.5 29.6 28.0–31.0 44.8 141.8 55.5 46.0 48.0 50.4

Lower calorific value (Net calorific value – NCV) MJ/kg MJ/Nm3

45.8

43.4 121.0 50.0

133.0

13.0 39.8 43.0

41.5 46.4

101.0 18.0

39.0–48.0 14.4–17.4 4.0–5.1

Most of the commercially available equipment at present is of advanced, wellproven conventional design and aimed at direct heating/steam generation and cogeneration applications. The term “conventional” is understood here as a process in which the required energy for gasification is provided by partial combustion of the waste material. For cogeneration purposes, the syngas product (H2 + CO) is cleaned to the required standard (as specified by the supplier). This gas can also be utilized for chemicals or liquid fuels production as evidenced by, e.g., the application of the well-known Fischer-Tropsch synthesis (hydrocarbons) or biochemical processes (alcohols) (Coskata Inc. 2016; Coskata Inc. et al. 2016). Figure 6 presents the flow diagram of a conventional gasification process (Rajvanshi 1986).

2.4

Plasma Arc Gasification

In plasma gasification a gas stream, typically air or nitrogen, is heated by an electric arc to very high temperatures (5000
C or more) to supply the required energy to the process. Current systems are designed to meet varying, site-specific user requirements ranging from technology demonstration units and vitrification of MSW incinerator bottom ash to the production of syngas for electricity generation. Figure 7 presents the flow diagram of a typical plasma arc gasification process.

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Plasma Waste Destruction

2837 Air Emissions Syngas Cleanup By-products Such as Sulphur and Acid Gases

Synthesis Gas (SYNGAS)

Feedstock Municipal Solid Waste (MSW)

Preprocessing

Conventional Gasification Reactor

Air/O2

Recyclables

Power Generation: _Electrical Energy and _Steam

Electricity to Grid

Ash, and/or Slag & Metals

Fig. 6 Flow diagram of a conventional gasification plant

Within the context of European Union (EU) energy policy and sustainability in waste management, recent EU regulations demand energy efficient and environmentally sound disposal methods of MSW. Data published by Eurostat show that in the period 1995–2014, annual per capita waste generation in the EU has declined steadily since peaking at about 522 kg in 2006 to 2008 to 475 kg in 2014, the same level as in 1995. At present 98% of municipal waste is being treated, with about 82% being equally divided ( 27%) between recycling, landfill, and incineration/energy recovery. About 16% is composted or digested. Within the waste management hierarchy, thermal disposal especially incineration is a viable and proven alternative. However, the dominant method, mass-burn grate incineration has drawbacks as well particularly hazardous emissions and harmful process residues. In recent years, pyrolysis and gasification technologies have emerged to address these issues and improve the energy output. In some systems, such as the plant at Trail Road, Ontario, Canada (Plasco 2016), gasification is achieved by conventional means and the syngas finally “polished” in a plasma process where any residual hydrocarbons, tars, and soot are converted to carbon monoxide and hydrogen as seen in Fig. 8. The syngas from the process fuels Jenbacher gas engine/generator sets, producing electricity which is fed into the local grid. Heat recovered from the process is used to provide steam for additional electricity generation. The plant can generate about 22 MW of electricity. Noncombustible waste is recovered as a nonleaching glassy slag aggregate suitable for construction purposes. The plant is designed to handle 110,000 t/a of postrecycled MSW from the city of Ottawa over a period of 20 years.

Recyclables

Preprocessing

Vitrified Slag & Metals

Plasma Arc Gasification Reactor

Fig. 7 Flow diagram of a plasma arc gasification process (Rajvanshi 1986)

Feedstock Municipal Solid Waste (MSW)

Synthesis Gas (SYNGAS)

Air/O2

Plasma Torches

By-products Such as Sulphur and Acid Gases

Syngas Cleanup

Power Generation: _Electrical Energy and _Steam

Air Emissions

Electricity to Grid

2838 M. Hrabovsky and I. J. van der Walt

Recycled Process Heat Solid Residue Melter

AGGREGATES

Solid Particulate

∞

H2 (SYNGAS)

∞ CO

CYCLONE

Plasma Torch

Plasma Torches

CARBON RECOVERY VESSEL

CONVERSION CHAMBER (CRUDE SYNGAS)

RECOVERED METALS

REFINING CHAMBER

Water

Gas Cleaning & Cooling

Water Treatment

CLEAN SYNGAS

Heat Recovery Power Generator

Residuals

POWER

WATER FOR REUSE

Engines

Steam Turbine

Exhaust

Plasma Waste Destruction

Fig. 8 Plasco MSW gasification process

Inerts

Front End Seperation

Post Recycled MSW

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The largest plasma waste gasifier to date (970 t/day, 300,000 t/a) was built by Westinghouse Plasma for a waste-to-energy plant erected by Air Products at Billingham near Stockton-on-Tees in the UK (AlterNRG 2016; AirProducts 2016a, b). The plant (Tees Valley 1) has a 50 MW electrical generating capacity (combined cycle) and can treat 350,000 t/a of nonrecyclable waste. It was expected to be commissioned in 2015. Construction of a second, similar plant (TV 2) was started in April 2014 but was suspended in Nov 2015 due to technical difficulties with the commissioning of TV 1 (Cambridge Companies 2016). Air Products have recently announced their decision to withdraw from the waste-to-energy business (Air Products 2016). A process block diagram and typical process equipment are illustrated in Figs. 9 and 10, respectively (AlterNRG 2013). When MSW is treated, additional gas purification is required to remove hazardous and toxic contaminants, such as chlorine, mercury, zinc, lead, cadmium, and sulfur. Coke and/or lime is fed together with the MSW to render the gasification process more robust and to aid heating of the waste. Westinghouse Plasma has also built and commissioned (2002) a plasma MSW gasifier in Japan, which serves the towns of Mihama and Mikata, treating 20 t/day of MSW and 4 t/day of sewage sludge. The syngas is converted to heat for the drying of the sewage sludge prior to gasification. The Hitachi Metals MSW treatment plant (2  110 t/day units) at Utashinai in rural Hokkaido, Japan, is based on Westinghouse technology. The plant was commissioned in 2003 but has since been closed due to lack of feedstock. Lessons learnt during commissioning and subsequent operation have been incorporated into present Westinghouse designs. A demonstration plasma gasification plant has been built by Europlasma at Morcenx, France, and is operated by CHO Power (Power 2016). The plant has a design capacity of about 35,000 t/a of industrial waste and 15,000 t/a of biomass, producing 12 MWe. It has been announced that the plant reached FAR (Final Acceptance with Reservation) status in Nov 2015, i.e., has been technically validated (Europlasma 2015). The economic and environmental optimization remain before final acceptance without reserve.

3

Characteristics of Plasma Treatment

The recent focus on resource recovery in the waste industry, i.e., material and energy recovery, has triggered the search for more advanced waste treatment technologies. Among these, thermal plasma treatment has been recognized as promising and offering specific performance characteristics. Heat supplied by a plasma is used for gasification of organic substances and for melting and vitrification of inorganic residues. While decomposition of waste and dangerous materials in thermal plasmas has been intensively studied in the last decade (Heberlein and Murphy 2008; Gomez et al. 2009; Ruj and Ghosh 2014), and industrial-scale systems for treatment of various types of waste have been

Basic Waste Water Treatment

MSW Shredder

Feed System (MSW)

Magnetic Separator

Slag System

Cooling Tower

Quench (Scrubber / Spray Tower)

Mercury Removal

Sulphur Removal

POWER ISLAND

SUPPORT SYSTEMS

SYNGAS CLEANUP SYSTEM

SLAG SYSTEM

FEED SYSTEM

LEGEND

WESP

Fig. 9 Block diagram of Alter NRG/Westinghouse WTE plasma process (Plasco_Energy_Group 2016)

Receiving & Storage

O2 Plant

Steam Turbine Generator

HRSG

Gas Turbine

66 Plasma Waste Destruction 2841

Plasma Torches

For Sale to Market i.e. Aggregate

Stag: 250 tpd

No metal content in this waste source

Quench

GAS COOLING

Stag & Recovered Metals

Air or Oxygen

PLASMA GASIFICATION, SLAG & METAL RECOVERY

Fig. 10 Typical Alter NRG/Westinghouse plasma WTE facility

Total:1181 tpd

Coke: 40 tpd

Other Inputs: Flux Material: 141 tpd

Inorganic Material: 21%

Moisture Content: 25%

Municipal Solid Waste: 1000 tpd

FEED HANDLING

SULPHUR REMOVAL

Landfill for Sludge to disposal or recycle Landfill back into gasifier

Cd: 0.5-10 kg/d Cr: 20-200 kg/d Pb: 80-800 kg/d Hg: 0.1-1.3 kg/d

For Safe to Market

Fine Particulate Sulphur Removed: Coarse 0.1-1 tpd Particulate Matter & Heavy Metals Removed: Matter: 20 tpd 20 tpd

FINE COARSE PARTICULATE PARTICULATE & HEAVY METALS REMOVAL REMOVAL

For Sale to Market

Electricity via Combined Cycle: 49 MW Gross

Electricity

Steam Turbine

HRSG

Gas Turbine

Clean Syngas

END PRODUCT(S)

2842 M. Hrabovsky and I. J. van der Walt

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Plasma Waste Destruction

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installed, waste-to-energy processes by plasma gasification of organics is a relatively new application. The principal goal of gasification is the production of fuel gases, principally a mixture of carbon monoxide and hydrogen (syngas), of high heating value and purity. Thermal plasmas enable decomposition of organics in the absence of oxygen (pyrolysis), or with a stoichiometric amount of added oxygen, i.e., with equal molar fractions of C and O in input reactants (gasification). Plasma process leads to the production of high-quality syngas, with high content of hydrogen and carbon monoxide and minimum of other components. Depending on the waste composition and moisture content, additional oxygen or carbon may be required to optimize production of CO. Given that the main goal of this technology is the production of fuel gas, the energy balance of the process and syngas quality and high heating value are much more important than in the case of hazardous waste treatment, where the principal goal is material decomposition. A number of nonplasma systems have been developed for the production of syngas from organic waste and biomass (Boerrigter and vanderDrift 2005; Lesmasle and Marcelin 2001; Surisetty et al. 2012). In a widely used gasification process based on partial feedstock oxidation, a solid or liquid carbonaceous material containing chemically bound carbon, hydrogen, and oxygen reacts with air or oxygen. The oxidation reactions are sufficiently exothermic to produce a gaseous product containing mostly CO, H2, CO2, H2O (steam), and a small amount of hydrocarbons. In an autothermal process, the heat required for the process is supplied by the exothermal gasification reactions in the reactor. In practice, heat from external sources can be supplied to the reactor to control the process and the reaction temperature. In an allothermal process, gasification takes place under the action of externally supplied heat. If the material is heated without additional oxidant, pyrolysis occurs. Even if energy is supplied from external sources, it is advantageous to balance the amount of carbon and oxygen atoms to reach a maximum CO yield. If the oxygen content in the feedstock is insufficient to achieve this, it is convenient to add some oxidizing agent (O2, CO2, H2O, air) to prevent the formation of soot. The principal limitations of autothermal (partial oxidation) gasification technologies are the low heating value of the syngas and tar formation by complex molecules of hydrocarbons created during the process at lower temperatures. The syngas from low-temperature gasification typically contains only 50% of the available energy in the syngas, while the remainder is contained in CH4 and higher aromatic hydrocarbons (Boerrigter and vanderDrift 2005). The possibility of controlling syngas composition in autothermal processes is limited. The need for production of clean syngas with controlled composition results in the use of technologies based on external energy supply. Energy can be carried into the gasification reactors by solids (sand), or hot gases provided by combustion or the use of thermal plasmas. Plasma pyrolysis and gasification for the production of syngas is an alternative to traditional methods for organic waste and biomass treatment. A thermal plasma is the medium with the highest energy density, and therefore substantially lower gas flow rates are needed to supply a sufficient amount of energy, compared to other media

2844

M. Hrabovsky and I. J. van der Walt

used for this purpose. The result is minimum contamination and dilution of the produced syngas by plasma gas. The process also acts as energy storage – electrical energy is transferred into plasma enthalpy and then stored in the produced syngas. The main advantages are a better control over the composition of the gas produced, its higher calorific value, and reduction of undesired contaminants like tar, CO2, water vapor, CH4, and higher hydrocarbons. Another advantage of plasma processing is the wide choice of potentially treatable material. Since the energy for the process is supplied by the plasma and chemical reactions in the reaction products are not the primary source of energy, the process can be applied to a wide range of organic materials and biomass. These advantages of plasma technology together with its overall thermal inefficiencies and energy consumption must be taken into account when evaluating the technical and economic feasibility of plasma treatment. Plasma treatment offers a better control over the temperature of the process, higher process rates, lower reaction volume, and especially an optimum composition of produced syngas. The process exploits the thermochemical properties of plasma. Decomposition of the material is achieved by the action of the reaction kinetics of plasma particles, which is extremely high due to high temperatures of the plasma. In addition, the presence of charged and excited species makes the plasma environment highly reactive, which can catalyze homogeneous and heterogeneous chemical reactions. The main advantage of plasma consists in much higher enthalpy and temperature compared to those of gases used in nonplasma methods. Thus, substantially lower plasma flow rates are able to carry sufficient energy for the process, and the composition of the produced syngas is not much influenced by the plasma gas. Moreover, substantially less energy is required to heat the plasma to the reaction temperature. Compared to nonplasma methods, the advantages of plasma waste treatment can be summarized as follows (Heberlein and Murphy 2008; Fabry et al. 2013): • The energy for gasification is supplied by plasma rather than by energy liberated from combustion, and therefore it is independent of the substances used, providing flexibility, fast control over the process, and more options in the chemistry of the process. A broad range of organic waste and biomass feedstock, including biodegradable fractions of waste, can be used for gasification. • No combustion gases generated in conventional autothermal reactors are produced. • The temperature in the reactor can be easily controlled by controlling plasma power and material feed rate. • Since sufficiently high temperatures and homogeneous temperature distribution can be easily maintained in the entire reactor, production of higher hydrocarbons, tars, and other complex molecules is substantially reduced. • High energy density and high heat transfer efficiency can be achieved, allowing shorter residence times and increased throughputs. • Highly reactive environment and easy control of the composition of reaction products.

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• Low thermal inertia and easy feedback control. • Much lower plasma gas input is required per a heating power unit than in nonplasma reactors, and therefore energy necessary to heat the plasma to reaction temperature is low; also the amount of gases diluting the syngas produced is lower. • High energy densities, lower gas flows, and volume reduction enable the use of plants that are smaller in size than non-plasma reactors.

4

Plasma Gasification Processes

Thermal plasma treatment of waste materials exploits the thermochemical properties of the plasma. The plasma delivers the energy needed to maintain the temperature inside the reactor volume at values needed for dissociation of molecules of gases produced by material decomposition. Due to the high temperature, the inorganic components of treated materials are melted, organic components are volatilized, and complex molecules are dissociated. Molten inorganics are removed from the reactor and after cooling and solidification produce a substance similar to lava. Organic materials, containing mostly chemically bound carbon, hydrogen, and oxygen, are decomposed into syngas that can be utilized as a high-quality fuel or in the chemical synthesis industry. The principal scheme of a plasma waste treatment processes is presented in Fig. 11. Material to be treated is supplied to the reactor together with gases added for control of syngas composition. Plasma produced in plasma torches flows into the reactor where it interacts with feed material; heat transfer to the material causes its destruction and volatilization. Product gases interact further with the hightemperature plasma before flowing into a quenching chamber or heat exchanger where products are rapidly cooled down to prevent unwanted chemical reactions. This preserves the syngas composition corresponding to high temperatures in the plasma reactor. Melted inorganic components flow out of the reactor where they are cooled and solidified.

Fig. 11 Schematics of the processes in the plasma gasifier

2846

M. Hrabovsky and I. J. van der Walt

The role of plasma in waste treatment can be summarized as follows: • Transport of energy needed for endothermic decomposition reactions, melting, and volatilization into the reactor volume. • Energy transfer to treated material. • Control of temperature in the reactor. • Supply of chemicals for control of composition of reaction products. A thermal plasma is generated in a plasma torch by ionization of gas in an electrical discharge. Thermal plasmas for waste treatment are mostly generated in direct current plasma torches either in a transferred or nontransferred arc mode. In the nontransferred mode, the arc is ignited between two water-cooled electrodes inside the plasma torch and the plasma exits the torch through a nozzle, which can be one of the electrodes, usually the anode. For transferred arcs, the material to be treated forms the counter electrode. The arc column is positioned inside the reactor and the material is in direct contact with it. This has the advantage of high heat transfer to the treated material and low power loss to the torch body. Some plasma torch systems can be switched between nontransferred and transferred arc mode. However, the use of transferred arcs depends on an electrically conducting counter electrode, e.g., a molten metal or slag bath. A much wider range of applications exists for nontransferred arc plasma systems that can be applied for any kind of material processing, including organics and biomass, liquids, or gases. Peak plasma temperatures at the exit of plasma torch are typically several thousand Kelvin. Treated material is in contact with the plasma jet exiting the torch that is attached to the reactor wall. The plasma temperature in the jet decreases along its length, but it is still high enough to ensure material decomposition. The temperature inside the reactor can be controlled by the arc power and the material feeding rate. Another option for waste treatment is based on radio frequency plasma discharges, where the plasma is heated by a radio-frequency electric current induced from a working coil surrounding the plasma chamber. Microwave plasmas are used in experimental plasma gasification systems. The main advantage of RF and microwave systems follows from the absence of discharge electrodes and thus electrode erosion is avoided. In arc, plasma torches electrode erosion limits the lifetime and length of continuous operation. Nevertheless, most plasma waste treatment systems are based on arc discharge plasmas, which offer simple material injection, wide choice of materials to be treated, and higher torch heating efficiencies and torch sizes ranging from 10 kW to segmented torches of several MW capacity. Typical power range of commonly available induction plasma torches is 10–100 kW. The scaling up to power several hundred kW up to 1 MW is needed for industrial applications (Mostaghimi 2015). Various plasma gases are used in plasma waste treatment systems, e.g., oxygen, air, nitrogen, argon, steam, or carbon dioxide. A steam plasma offers optimal characteristics for waste treatment, namely high enthalpy (energy content in given

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amount of plasma), high heat transfer to the treated material, and optimal composition (hydrogen and oxygen). An air plasma is the cheapest option, but the product gas is diluted by a large amount of nitrogen. The nitrogen present in the plasma can also contribute to the formation of nitrogen oxides in output gases. The problem of plasma gases containing oxygen, and especially steam, is increased electrode erosion rates in arc torches. Electrode erosion is low for an argon plasma, but the plasma has low enthalpy and low thermal conductivity, and consequently low capacity of energy transport into the reactor and low rate of heat transfer from plasma to treated material. It is important to ensure good mixing of the plasma with the reactor atmosphere and efficient heat transfer to materials surface. The residence time of gaseous decomposition products must be high enough for their complete dissociation and for the formation of syngas. The feed rate of material into the reactor must be chosen with respect to the rate of decomposition of the material and the rate of chemical reactions in the product gases. Processes of plasma treatment of organic waste are principally shown in Fig. 12. The three processes differ in the amount of externally added oxygen to the reaction volume. In plasma pyrolysis, only plasma and material are supplied into the reactor. All energy needed for destruction and volatilization of material and for chemical reactions between produced gases and plasma gas is supplied by plasma. Heating of material leads to its volatilization, high temperature in the reactor volume causes dissociation of molecules of gas produced by volatilization, and syngas is produced

Fig. 12 Plasma assisted pyrolysis, gasification, and combustion

2848

M. Hrabovsky and I. J. van der Walt

with CO and H2 as the main components and a small amount of other molecules like CH4, CO2, and others, depending on the composition of feed material and plasma gas. If the total number of carbon atoms supplied to the treated material and plasma is higher than the number of oxygen atoms, solid carbon is produced together with syngas in the form of char or carbon powder. In the case of gasification, a certain amount of oxygen is supplied into the reactor to balance the total molar fractions of carbon and oxygen in input reagents. Oxygen molar fraction can be increased by an addition of air, oxygen, carbon dioxide, steam, or water. Most of the energy for the endothermic process of material gasification is supplied by plasma. In the case of usage of air or oxygen, an additional amount of energy comes from the oxidation of the volatilized material. In this case, the process is similar to partial oxidation, but for most of the organic materials, including biomass, some energy must be added to plasma to achieve complete gasification. Oxygen can be added in the form of carbon dioxide or steam as gasifying media. In these cases, additional energy is needed for their dissociation. In an ideal gasification process, all carbon atoms are bound in CO molecules and produced syngas is composed only of carbon monoxide and hydrogen. The ratio of molar fractions of hydrogen to carbon monoxide, which is an important parameter for syngas utilization, can be controlled by combination of added oxidation agents. Gasification and partial combustion are similar and differ only in the amount of oxygen added. The combustion process mainly differs from gasification by the addition of excess oxidizer, thereby releasing additional energy which will decrease the plasma power requirement. The plasma is used mainly for control of temperature in the reactor volume and as the process initiator and sustainer. However, the calorific value of the produced syngas is lower than in the first two processes, as part of heating value of material is spent for the production of syngas. Moreover, quality/purity of syngas is lower due to the higher content of carbon dioxide and other products of combustion, including higher hydrocarbons. Partial combustion is used in waste treatment systems with high throughputs of material, where the destruction of waste is the principal goal and quality and heating value of syngas is less important. The choice of the process must result from analysis of the demands for energy efficiency on the one hand, and on the quality of produced syngas (composition, purity, and heating value) on the other.

5

Chemistry of Organics Gasification

Conversion of organic waste to syngas involves a number of chemical reactions. High temperature in the gasification reactor leads to heterogeneous gas-solid phase reactions when solid materials are volatilized due to heat transfer from the plasma and hot reactor atmosphere. Basic homogeneous gas phase reactions take place in

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the reactor atmosphere in gases produced by the volatilization process. These equations are presented below with enthalpies calculated at 20
C: Oxidation: C þ O2 Ð CO2 ðΔH ¼ 393:65 kJ=molÞ

(1)

H2 þ ½ O2 Ð H2 OðΔH ¼ 241:09 kJ=molÞ

(2)

CO þ ½ O2 Ð CO2 ðΔH ¼ 283:09 kJ=molÞ

(3)

CH4 þ 2 O2 Ð CO2 þ 2 H2 OðΔH  802:29 kJ=molÞ

(4)

Partial oxidation: C þ ½ O2 Ð CO ðΔH ¼ 110:56 kJ=molÞ

(5)

CH4 þ ½ O2 Ð CO þ 2 H2 ðΔH ¼ 35:66 kJ=molÞ

(6)

Boudouard reaction: C þ CO2 Ð 2 COðΔH ¼ þ131:2 kJ=molÞ

(7)

Water-gas reaction: C þ H2 O Ð CO þ H2 ðΔH ¼ þ172:52 kJ=molÞ

(8)

Water gas-shift reaction: CO þ H2 O Ð CO2 þ H2 ðΔH ¼ 41:18 kJ=molÞ

(9)

CO þ 3H2 Ð CH4 þ H2 OðΔH ¼ 206:23 kJ=molÞ

(10)

Methanation:

Steam reforming: CH4 þ H2 O Ð CO þ 3H2 ðΔH ¼ þ206:23 kJ=molÞ

(11)

Dry reforming: CH4 þ CO2 ! 2CO þ 2H2 ðΔH ¼ þ246:93 kJ=molÞ

(12)

Other chemical reactions producing more complex molecules can take place if the temperature in the reactor is not sufficiently high. Plasma generates a number of other species, namely radicals and ionized species (atoms and molecules). The high temperature and presence of radicals and ions lead to an increase in reaction rates. Conditions in the reactor volume must lead to complete mixing of all components

2850

M. Hrabovsky and I. J. van der Walt

and uniformly high temperature to ensure the proper composition of reaction products. Optimally the conditions in the reaction zone of the reactor volume should be close to thermodynamic equilibrium, and therefore, the final composition of the produced gas is determined by the temperature. The main objective of the gasification process is the production of syngas. In principle, all carbon and hydrogen atoms from treated organics can be used for syngas production if the material and produced gases are heated to sufficiently high temperature. Maximum material to syngas conversion efficiency is achieved if all carbon is oxidized to CO. If the carbon molar content in the feedstock is higher than its oxygen content an oxidizing agent is added to prevent the production of solid carbon and ensure complete carbon gasification. This is usually done by an addition of oxygen, air, steam, or CO2. The following three processes represent common gasification routes: Gasification by reaction with oxygen Mþ

ð nC  nO Þ O2 ) nC CO þ nH2 H2 2

(13)

Gasification by reaction with steam M þ ðnc  n0 Þ H2 O ) nc CO þ ðnH2 þ nc  nO Þ H2

(14)

Gasification by reaction with CO2 M þ ðnC  nO Þ CO2 ) ð2nC  nO Þ CO þ nH2 H2

(15)

where nC = c/MC, nH2 = h/2MH, and nO = O/MO are molar concentrations of carbon, hydrogen, and oxygen in treated material M with mass fractions of carbon, hydrogen, and oxygen equal to C, H, and O, respectively. The resulting composition of reaction products is determined by the composition of treated material, plasma, and added gases, and by conditions inside the reactor, especially by temperature. Conditions for thermodynamic equilibrium can be established in plasma reactors due to high plasma temperatures and high enthalpies. The resulting composition of produced gases can then be determined by thermodynamic computations. Figure 13 presents the temperature dependence of composition in a system containing mass fractions of carbon, hydrogen, and oxygen for wood. The equilibrium composition of this heterogeneous system was calculated using the method described in a paper by Coufal (Coufal, Composition of the reacting mixture SF6 and Cu in the range from 298.15 to 3000 K and 0.1–2 Mpa. 1994). The input data for calculations of standard reaction enthalpy and standard thermodynamic functions of system components were taken from a database published by Coufal et al. (2005). Molar fractions of the gas phase components are shown for gas components H2, H2O, CO, CO2, CH4, and C2H2. For solid carbon (C(C)), the ratio of the number of solid carbon moles to the number of all moles in the gas phase is given. It can be seen

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Mol Fraction [mol/g]

1 H

0,8

H2 H2O

0,6

CO CO2

0,4

CH4 C2H2

0,2 0 300

C(c)

1300

2300

Temperature (K)

Fig. 13 Composition of products of wood pyrolysis. The mass ratios of components in wood: c = 0.511, h = 0.064, o = 0.425

1

Mol Fraction [mol/g]

Ar

0,8

H H2

0,6

H2O CO

0,4

CO2 CH4

0,2

C2H2 C(c)

0 300

1300

2300

Temperature (K)

Fig. 14 Composition of products of wood gasification. Wood 47 kg/h, humidity 6.5%, CO2 115 slm, oxygen 30 slm, steam plasma 18 g/min, argon 13.5 slm. The mass ratios of components in wood: c = 0.511, h = 0.064, o = 0.425

that wood is decomposed into hydrogen, carbon monoxide, and solid carbon with a small amount of other components at temperatures above 1200 K. The presence of solid carbon, which in gasification reactors contributes to formation of char, can be suppressed by addition of a gas containing oxygen. To maintain high concentrations of CO and H2 in the produced gas, it is advantageous to use oxygen, carbon dioxide, or steam as oxidizing media as is illustrated by Eqs. 13, 14, and 15. Figure 14 shows the composition of products of gasification of wood with the addition of CO2 and oxygen in an experimental gasification reactor using a steam/

2852 Fig. 15 Calculated and measured composition of syngas. Wooden pellets (10–15 mm) – 30 kg/h, water 79.6 g/min, steam plasma 18 g/min. Theoretical – Eq. 14

M. Hrabovsky and I. J. van der Walt

60.00 50.00 40.00 30.00 20.00 10.00 0.00 CO % H2 % O2 % CO2 % CH4 % Theoretical Measurements

Thermodynamic equilibrium

argon plasma torch. The amount of added oxygen and carbon dioxide was determined so as to gasify all carbon contained in the wood. It can be seen that the optimum composition of syngas with high concentration of H2 and CO and without the presence of solid carbon and other gas molecules can be reached at temperatures higher than 1200 K. The small amount of solid carbon produced could be reduced by the addition of more oxidizing admixtures (O2, CO2, water, or steam). Thus, application of a plasma offers the possibility to produce syngas with a composition optimized for specific application by the choice of input reactants. The process can be applied for large choice of input materials. In real conditions in gasification reactors the composition of the product gases can be close to the theoretical composition. As an example. Figure 15 presents a comparison of data calculated from thermodynamic equilibria and composition corresponding to Eq. 3, and experimental data measured for gasification of wood in a plasma reactor with a steam/argon plasma torch (Hrabovsky 2017). The composition calculated by thermodynamic modeling and the composition determined from Eq. 14 are almost the same. The measured data are close to calculated values. The experimentally obtained concentrations of CH4 and O2 were lower than 1%, and that of CO2 was about 4%. The small deviations of measured composition from the theoretical one result from complex conditions in a relatively large-scale experimental reactor. As these deviations are really small, it can be concluded that it is possible to reach conditions close to the equilibrium situation even in large scale plasma reactors. In this chapter, only reactions of basic components of organic materials, i.e., carbon, hydrogen, and oxygen were considered. Organic materials can contain also small amounts of other elements; possible chemical reactions of these additional elements must be analyzed for each individual case.

66

6

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Plasma-Material Heat Transfer and Gasification Rate

The principal processes of heat transfer from plasma to material are convection and radiation. The basic relationship for heat transfer is,
 q ¼ hS T p  T s þ Qr

(16)

where q is the heat per unit time, S is the area of material surface, h is the heat transfer coefficient, Tp is plasma temperature, Ts material surface temperature, and Qr is power transfer by radiation. Radiation transfer plays an important role for plasma temperatures above 10,000 K. In most cases, the plasma temperature inside the reactor is lower and heat transfer is dominated by convection. The heat transfer coefficient h is a complicated function of many plasma properties and characteristics of plasma flow. It is often given, h ¼ k=D f ðRe, Pr, . . .Þ

(17)

Fig. 16 Dependence of thermal conductivity on temperature for plasmas of steam, nitrogen, and mixture of argon with hydrogen (mixture 33 slm of argon and 10 slm hydrogen)

Thermal conductivity [W/m.k]

where k is thermal conductivity of plasma, D is the linear dimension of the solid body, and f is a function of characteristics of plasma flow around the material, which are described by dimensionless criteria of the flow field like Reynolds number (Re) and Prandtl number (Pr). The thermal conductivity (k) is an important plasma property for heat transfer from plasma to the material surface. The dependence of thermal conductivity on temperature for three common plasma gases is shown in Fig. 16. It can be seen that a steam plasma has the highest thermal conductivity values for all plasma temperatures. Moreover, the peak in thermal dependence of k, corresponding to the dissociation of hydrogen molecules, is at relatively low temperatures, below 4000 K, which correspond to temperatures in waste treatment reactors in the surrounding of treated material. A steam plasma is the optimal choice for waste treatment technology due to its high thermal conductivity, very high

7 H2O

6

N2

Ar

5 4 3 2 1 0

0

5000

10000 Temperature [K]

15000

20000

2854 Fig. 17 Particle of gasified material with sheath of gas produced by volatilization

M. Hrabovsky and I. J. van der Walt

Plasma flow Heat transfer to material

volatilization Gas sheath

Gas flow Dissociation of gas molecules

enthalpy and optimal plasma composition (hydrogen, oxygen). On the other hand, the abovementioned steam plasma characteristics are also responsible for high erosion rates of electrodes in electric discharge plasma sources. The rate of the plasma gasification process is dependent on material volatilization rate, which is controlled by heat transfer to the material surface, and on the kinetics of chemical processes in the gas phase produced by volatilization. Due to rapid material volatilization, a gas sheath is created between the material surface and the plasma. Heat transfer through this sheath substantially reduces heat transfer rate to the material and reduces volatilization rate. At elevated temperatures, the time constants of chemical reactions in the gas phase within the reactors are substantially lower than time constants of volatilization, and therefore, the gasification rate is limited by the heat transfer process between plasma and material surface. Figure 17 illustrates conditions around material particle in a plasma flow. An exact theoretical description of plasma gasification process should be based on a fluid dynamic model of plasma-material interaction, a model of heat transfer to the material surface, material heating, and volatilization, as well as on a description of the kinetics of chemical reactions in the reactor. The model should describe the interaction of the material particles with the plasma flow, the effect of the particles’ shape and size, the interaction between the particles, and mixing of the plasma flow with produced gas. To illustrate the effects of material properties, particle size, and temperature in the reactor, the following simple gasification model is presented for an individual particle. The modeling computations result in an analytical equation describing the relations between particle size, material properties, reactor temperature, and gasification rate. The model is based on a solution of the Arrhenius equation describing volatilization of material at a given temperature combined with equations describing heat and mass transfer between reactor atmosphere and the surface of the treated material. The rate of material volatilization (ṁ) is commonly described by the Arrhenius equation:

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E m_ ¼ Aexp  RT s

 (18)

which determines the dependence of volatilization rate ṁ on temperature of material surface Ts. The frequency factor, A, and activation energy, E, were determined for various biomass materials and R is the universal gas constant. The film model (Bird et al. 2002) is used to describe the heat transfer to the particle. The heat flux through the gas sheath surrounding a spherical particle with surface temperature Ts is given by, q0 ¼

_ p ðT r  T s Þ mC mC _

e h  1

(19)

where ṁ is volatilization rate, Tr is temperature in plasma flow outside the sheath, Cp is specific heat, and h heat transfer coefficient, both corresponding to local conditions in the sheath. The heat transfer coefficient can be approximated by a relation for heat transfer to the sphere in a flowing fluid (Bird et al. 2002), h¼

 k:Nu k  1 1 ¼ 2 þ 0:6 Re2 Pr2 D D

(20)

where Nu is Nusselt number, Re Reynolds number, Pr Prandtl number, D diameter of the sphere, and k thermal conductivity within the sheath. The relation between the volatilization rate and heat flux is given by the energy balance equation, q0 ¼ m_ ΔH gas

(21)

where ΔHgas is the energy needed for material gasification. From Eqs. 18, 19, 20, and 21, the following relation between the mass gasification rate m_ and the difference of temperatures in plasma flow around the particle Tr and at the particle surface Ts is obtained, m_ ¼

  h Cp ln ðT r  T s Þ  1 Cp ΔH gas

(22)

Solving Eqs. 17, 18, 19, 20, 21, and 22 leads to the dependence of the volatilization rate and surface temperature Ts on the temperature in the reactor Tr and the particle diameter D. Dependences calculated for various diameters of spherical wood particles are presented in Figs. 18, 19, and 20. The computations were made using characteristic parameters for input wood (C = 0.511, H = 0.064, O = 0.425, A = 7.7106 s1, E = 1.11105 J/mol) and sheath values of transport and thermodynamic coefficients k, h, and Cp corresponding to a mixture of hydrogen and CO with the volume ratio of 1:1. Zero relative velocity between a particle and the surrounding gas, and an averaged sheath temperature were assumed.

Fig. 18 Surface temperature of wood particles in dependence of reactor temperature for various wood particle diameters

M. Hrabovsky and I. J. van der Walt

Particle surface temperature [K]

2856

Particle diameter D= 1 mm D= 5 mm D= 10 mm D= 50 mm

1000

900

800

700

600 1000

1200

1400

1600

1800

Reactor gas temperature [K]

0.25

Gasification rate [kg/m2/h]

Fig. 19 Gasification rate of wood particles in dependence of reactor temperature for various wood particle diameters

0.2

Particle diameter D= 1 mm D= 5 mm D= 10 mm D= 50 mm

0.15

0.1

0.05

0 1000

1200

1400

1600

1800

Reactor gas temperature [K]

T sheath ¼

ðT r  T s Þ 2

(23)

The representation of wood gasification kinetics by a set of parameters A and E is a simplification of the real conditions. The values corresponding to high gasification rates in plasma cannot be found in the literature, and therefore for these calculations, the values representing gasification of lignin were used (Miller and Bellan 1997).

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Fig. 20 Ratio of volume occupied by wood particles to feeding rate which corresponds to calculated gasification rate

Particle Diameter D= 50 mm D= 10 mm D= 5 mm D= 1 mm

V/M feed [m3h/kg]

1

0.1

0.01

0.001

0.0001 1000

1200

1400

1600

1800

Reactor gas temperature [K]

Figures 18 and 19 show the effect of the reactor temperature on the particle’s surface temperature and gasification rate. The graphs show that both parameters are substantially influenced by the particle diameter. An increase of the diameter results in reduced heat transfer to the particle, as the particle is more intensively shielded by the gas sheath formed from volatilized material. In Fig. 20, the ratio of the total volume occupied by particles to the gasification rate is plotted in dependence of the reactor temperature. The relations between particle volume, sizes, and process rate should be taken into account when selecting appropriate reactor volume, process temperature, and particle size for a specific material. A minimum reactor volume required for a given material throughput can be determined from these dependences, assuming that to ensure good heat transfer, reactor volume should be several times higher than the volume occupied by the particles. Fig. 20 shows that the required reactor volume increases substantially with particle size.

7

Mass and Energy Balance

The primary role of plasma in the waste treatment process is transporting energy needed for material decomposition into the plasma reactor and transfer of energy to the treated material. The power delivered into the reactor by the plasma is described by the integral of plasma enthalpy flow over the cross section of the plasma torch exit nozzle, ðR Fh ¼ 2π ρvHrdr 0

(24)

2858 500 Enthalpy [MJ/kg]

Fig. 21 Dependence of plasma enthalpy on temperature for three plasma gases.

M. Hrabovsky and I. J. van der Walt

400 Ar

H2O

N2

300 200 100 0

0

5000

10000 15000 Temperature [K]

20000

where ρ is plasma density, H enthalpy and v flow velocity, and r is radius of the torch exit nozzle. Total enthalpy flux in the exit nozzle of the torch is determined by the plasma torch power W and power loss Ptorch to the torch body, Fh ¼ W  Ptorch

(25)

Plasma enthalpy in waste treatment systems is often characterized by mass averaged enthalpy Hav, which can be evaluated from easily measured data, H av ¼

W  Ptorch G

(26)

where G is total plasma mass flow rate from the torch exit nozzle. Total energy input to the reactor carried by plasma is thus, Fh ¼ GH av

(27)

In Fig. 21, the enthalpies of three commonly used plasma gases are compared. As can be seen, plasma produced from steam has the highest ability to accumulate energy, argon has very low enthalpy. Advantage of argon is given by very low erosion rate of electrodes. Argon can be mixed with other gases for an enthalpy increase. Electrode erosion is the main problem of systems operated with steam plasma. The use of nitrogen may be a compromise. For gases with low plasma enthalpy, higher mass flow rates must be used, and produced syngas is then diluted by a higher amount of plasma gas. Steam plasma offers not only high plasma enthalpy but also high thermal conductivity and thus high heat transfer from plasma to the treated material. Moreover, the composition of plasma (hydrogen and oxygen) does not contribute to the dilution of syngas by other gases, which is a problem especially with plasma torches operated with high flow rates of air or nitrogen. Plasma gasification systems utilize mostly arc plasma torches, typical power losses in arc plasma torches are 10–40% of electric power W. Generally, higher efficiency and thus lower power loss can be achieved in plasma torches with high plasma gas flow rates G, as plasma temperatures are lower and the high flow rate of

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2859

the colder gas surrounding the high temperature arc column reduces heat transfer from the plasma to the arc chamber walls and electrodes. On the other hand, high plasma gas flow rates lead to higher power needed for heating the plasma gas to the temperature inside the reactor, which must be kept high enough to ensure proper conditions for chemical reactions. As can be seen in Figs. 13 and 14, this temperature must be higher than 1200 K for syngas production from biomass. Similar temperature values could be found for other organic materials. Plasma enthalpy available for material processing ΔH is given by the difference of input total plasma enthalpy out H in av and plasma enthalpy H av corresponding to the temperature of plasma gas leaving the reactor, out ΔH ¼ GH in av  GH av

(28)

The enthalpy Hout av , which is not used for material treatment corresponds to the temperature of gases leaving the reactor, can be higher if complete mixing and heat transfer from plasma to material is not ensured and some amount of plasma leaves the reactor at higher temperature. The efficiency of the utilization of plasma enthalpy for gasification process is then given by the equation, ηR ¼


 out G H in av  H av GH in av

¼1

H out av H in av

(29)

Efficiency ηR will be higher for lower values of ratio of output plasma enthalpy to the input enthalpy. Thus, high values of efficiency can be achieved using plasma torches with high plasma temperatures and enthalpies, which are achieved with low plasma gas flow rates. This compensates the fact that plasma torches with high plasma gas flow rates are characterized by lower plasma torch losses. Energy balance of processes in the reactor volume should include material heating, melting, and volatilization, chemical reactions in produced gases, and energy loss due to the reaction products outflow from the reactor. Plasma interacts with treated material in the reactor volume, heat transferred to material causes melting of inorganic and volatilization of organic material components. Plasma is mixed with produced gases, and due to high temperature in the reactor volume, chemical reactions take place between gases produced by organic volatilization, plasma gas, and added gases. These reactions can be exothermic or endothermic depending on composition of reacting gases, especially on an amount of added oxygen. The temperature in the reactor volume must be high enough to produce syngas with optimal composition. As can be seen in Figs. 13 and 14, the temperature for production of syngas with optimal composition with maximum content of hydrogen and carbon monoxide and minimum presence of other components should be higher than about 1200 K for most of organic materials. Power balance of processes in the reactor volume can be written, volat chem out out GH av ¼ Qmelt inorg þ Pinorg þ Qorg þ Qorg þ Porg þ Preactor þ Pplasma

(30)

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M. Hrabovsky and I. J. van der Walt

volat where Qmelt power for organics inorg is power spent for inorganics melting, Qorg chem out volatilization, Qorg power needed for chemical reactions, Pout inorg , Porg are power losses carried out of the reactor by products of treatment of inorganics and organics, Pplasma power of plasma not transferred to material, and Preactor power loss to the reactor walls. All terms on the right hand side of Eq. 30 with the exception of Qchem org are always positive and represent power consumption. The term Qchem org corresponding to chemical reactions in the gas phase can be positive or negative depending on organics composition and on the amount of added oxygen. In plasma systems with high material throughputs, a large amount of oxygen or air is added and energy is produced by partial combustion in the reactor volume, the term Qchem is then org negative. Plasma is in these cases is used especially for control of temperature distribution inside the reactor; power input by plasma enthalpy creates only part of the power spent for material treatment. Temperature in the reactor volume is controlled by the energy balance Eq. 30, and it is thus determined by power supplied by plasma GHav and power consumption represented by the terms in the right-hand side of Eq. 30. The energy balance is of primary importance mainly for applications where production of syngas by plasma gasification of organic materials is the main goal of the material treatment. The analysis of energy balance of plasma gasification of biomass (wood) is presented here, realized through reactions with oxygen, steam, and CO2 described by Eqs. 13, 14, and 15 i.e., for reactions with equal molar fractions of C and O in input reactants. All three reactions are for most organic materials endothermic. Energy needed for realization of the reactions has to be supplied by the plasma. This includes the energy needed for the production of gases by volatilization of material and energy for reactions in the gas mixture of the volatilized material, added oxidizing reagents and plasma. Evaluation of the reaction enthalpy from enthalpies of formation of reaction components is often impossible as the data is not known for most of treated materials. The evaluation can be made on the basis of heats of combustion of materials that are usually known. Energy ΔHr needed for realization of reactions represented by the Eqs. 13, 14, and 15 to can be evaluated from known heats of combustion on the basis of scheme presented in Fig. 22. This scheme corresponds to the gasification of biomass by reaction (13). The heat of gasification, i.e., heat for a production of syngas with composition nc CO + nH2 H2, is calculated as the difference between the heat of combustion ΔHc,net of biomass and the heat of combustion of syngas ΔHc,syng

ΔHgas ¼ ΔH c, net  ΔH c, syng

(31)

The heat of combustion of syngas produced by the reaction (13) is, ΔH c, syng ¼ nðH2 Þ ΔH 2 þ nC ΔH3

(32)

where ΔH2 = 241.09 kJ/mol is the enthalpy of reaction) and ΔH3 = 283.09 kJ/ mol is the enthalpy of the reaction). The heat of combustion ΔHc, net is known for many materials that are used as fuels.

66

Plasma Waste Destruction

2861 DHgas = DHc ,net –DHc,syng

Fig. 22 Scheme of reactions for determination of reaction heat for biomass gasification.

O2

O2

DHc,syng

CO +H 2

Biomass Gasification

Combustion

CO2+H2O

Combustion

O2

DHc ,net = 13.23 r0 [kJ / g]

The heat of combustion of cellulosic materials, like wood or other biomass materials, can be calculated from (Dietenberger 2002), ΔH c, net ¼ 13:23 r 0 ½MJ=kg

(33)

where r0 is the external oxygen mass fraction needed for complete combustion, r 0 ¼ ð8=3ÞC þ 8ðH Þ  O

(34)

The heat of combustion of syngas (LHV) produced by complete gasification of biomass by reaction (13) can be expressed,
 ΔH c, syng ¼ nC Δf H o ðCO2 Þ  Δf H o ðCOÞ þ nðH2 Þ Δf H o ðH2 OÞ

(35)

where Δf H o are the heats of formation of individual molar components. For the reactions (14) and (15), the heats of reaction also include heats of dissociation of H2O and CO2, respectively. Corresponding values of ΔHc,syng can be calculated according to Eq. 35, where nH2 and nC are replaced by corresponding numbers of moles of H2 and CO in syngas that are given by Eqs. 14 and 15. The gasification reactions (13), (14), and (15) can be realized only at elevated temperature Tr. The total heat needed for gasification is then given by the sum, ΔH r ¼ ΔH gas þ ΔH T

(36)

where ΔHgas is given by Eq. 31 and ΔHT is heat needed for heating of components on the right-hand side of Eqs. 13, 14, and 15 from standard temperature to the reaction temperature Tr. Figure 23 shows components of energy balance for 1 kg of dry wood calculated by solving Eqs. 31, 32, 33, 34, 35, and 36 for the process of reaction of wood with oxygen Eq. 13 and for reactions with steam (14) and CO2 (15). The amounts of oxidizing agents needed for complete gasification are shown for the three processes. Figure 23 presents the energy needed for volatilization of wood and its gasification ΔHc,net, heating of all components in the reactor to 1200 K, dissociation of added water and carbon dioxide, and the sum of all reaction enthalpies. The lower heating

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M. Hrabovsky and I. J. van der Walt

Amount of gasifying medium per 1 kg of wood: Oxygen: 0.832 Nm3 Carbon dioxide: 1.664 Nm3 Water: 1.339 l O2

20,40%

41,35% 79,60%

Mass ratios CO2

22,38%

H2O 77,62%

58,65%

Reaction enthalpy [MJ]

Reaction enthalpies for 1 kg of wood 25 20

O2

15

CO2 water

10 5 0 wood volatilization

gas heating

CO2 and total reaction LHV syngas enthalpy H2O dissociation + water evaporation

Fig. 23 Mass and heat balances for the process of dry wood gasification by reactions with oxygen, water, and carbon dioxide. Mass ratios of components in wood: c = 0.511, h = 0.064, o = 0.425

values (LHV) of produced syngas, i.e., heating values without the heat produced by steam condensation, are shown in the last columns in Fig. 23. The calculated values show that when oxygen is used as oxidizing agent, the ratio of syngas heating value to the total enthalpy required by the process is 7.7. When water or carbon dioxide are used this ratio is approximately 3 in both cases. The ratios would be 7.9 and 3.4 if the energy obtained by cooling the syngas from reaction temperature 1200–300 K was also utilized. The ratio of energy obtained by combustion of syngas (LHV) to the energy needed for its production is plotted in Fig. 24 against wood humidity for gasification by reaction with oxygen, and in Fig. 25 for the CO2 process. It can be seen that LHV of syngas produced from gasification of dry wood with oxygen is 7.7 times higher than the heat spent for its production, for wood with 10% humidity this ratio is 5. If a torch efficiency of 60–90% is considered and the sum of power losses defined in Eq. 30 is about 10% of the torch power, the ratio of LHV of syngas to total energy needed for its production is 3.8–6.2 for dry wood and 2.5–4.0 for wood with 10% humidity, at a reaction temperature of 1200 K. As all power losses are losses to the cooling water of the torch and the reactor, the real power gain after recovery of heat in a cooling system could be higher.

Plasma Waste Destruction

Fig. 24 Dependence on reaction temperature of the ratio of LHV of syngas to total reaction enthalpy for gasification of wood by reaction with oxygen. Mass ratios of components in wood: C = 0.511, H = 0.064, O = 0.425

2863

8

wood humidity

0% 10% 20% 6 ΔHc/Δhr

66

4

2 1200

1400

1600

1800

Reaction temperature [K]

4

wood humidity 0%

3.6

10% 20%

ΔHc/Δhr

Fig. 25 Dependence of ratio of LHV of syngas to total reaction enthalpy for gasification of wood by reaction with carbon dioxide on reaction temperature. Mass ratios of components in wood: C = 0.511, H = 0.064, O = 0.425

3.2

2.8

2.4

2 1200

1400

1600

1800

Reaction temperature [K]

The ratio of the heating value of produced syngas to the energy needed for its production can be increased if the amount of oxygen added is higher than needed for balancing carbon and oxygen molar concentrations. Part of the carbon is then oxidized to CO2, producing energy and lowering the additional process energy

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M. Hrabovsky and I. J. van der Walt 10

mass ratio

Fig. 26 Ratio of mass of gas carrying energy ΔHr for complete gasification of wood, to the mass of wood, in dependence on gas temperature

nitrogen oxygen steam

1

0.1

0.01

4000

8000

12000

16000

Temperature [K] requirement. Syngas heating value is then reduced due to the CO2 produced by partial combustion and a large amount of nitrogen introduced with the air. However, the ratio of output to input energy is increased. In large plasma waste treatment and gasification systems, oxygen is often added by the introduction of a large amount of air, either directly into the reactor and/or in the form of air plasmas. The substantial advantage of plasma treatment is in the reduction of mass flow rate of the gasifying medium compared to the flow rate of gases used for nonplasma gasification. The produced syngas is less diluted by gas supplied into the reactor and has a higher heating value. Also, the power losses associated with the heating of the gasifying medium to the reaction temperature are reduced. The plasma mass flow per unit mass of wood required to supply the energy for complete gasification (ΔHr) is plotted vs plasma temperature in Fig. 26 for nitrogen, oxygen, and steam plasmas. The curves were calculated from thermodynamic equilibrium enthalpies of the three gases (Boulos et al. 1994; Krenek 2008) and from the total energy of gasification determined above. For temperatures lower than 3000 K the ratio is close to 1. Thus, for gasification with hot air (closely approximated by the values for nitrogen), almost half of the weight of produced syngas is air and thus syngas is diluted by approximately 39% m/m of nitrogen. In comparison, for a steam plasma with input temperature 16,000 K, this ratio is less than 0.02 and thus virtually undiluted syngas is produced.

8

Syngas Production from Organic Waste

Plasma gasification processes can be utilized for conversion of solid, liquid, or gaseous organic material to synthesis gas. Feed material properties and size range may influence the rate of the process and energy efficiency. The composition of produced syngas can be controlled by the selection of an additional oxidizing agent.

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Common oxidizing agents used for organics gasification are oxygen, air, CO2, steam, or water. Oxygen and air gasification lead to the highest energy efficiency, as partial oxidation of material produces additional energy. A higher amount of oxygen results in an increase of CO2 content in syngas, and, in the case of air, the produced syngas is diluted by a high amount of nitrogen. Steam and CO2 plasmas can be used for the production of pure syngas containing high amounts of hydrogen and CO and minimum other components, but the energy efficiency is reduced as additional energy is needed for dissociation of CO2 and evaporation and dissociation of water. Syngas with the highest calorific value is produced if molar concentrations of carbon and oxygen in all input reagents, i.e., in treated material, oxidizing agents, and plasma, are balanced. Examples of syngas composition obtained by gasification of several solid and liquid materials in steam plasmas are given in Table 2. All materials were treated under the same conditions in a reactor with inner volume 0.2 m3 (Hrabovsky 2009, 2017). The power of the plasma torch was between 110 kW and 140 kW. The torch efficiency was 65%, thus the total plasma enthalpy flow into the reactor volume was 72–91 kW. The mass flow rate of steam was 18 g/min and the temperature measured at the reactor walls was 1100–1300 K. The following materials were treated: • • • • • • •

Fir saw dust (humidity 12% m/m). Wooden pellets 5 mm diameter, 10 mm long (humidity 7.0% m/m). Sunflower seeds skins. Soft brownish coal (Lignite) powder (humidity 45% m/m). Polyethylene pellets, diameter 3 mm. Shredded waste plastics from bottles, 2–10 mm particles. RDF (refuse-derived fuel) processed from waste excavated from landfill sites Composed of municipal solid waste (59%) and industrial waste (41%). • Low-temperature pyrolysis oil from used tires. (>21 wt % of water, HV39.5 MJ/ kg, molecular composition C5H8O). It contained a number of complex hydrocarbons including harmful and dangerous ones.

Various combinations of oxidizing media (H2O, CO2, O2) were added. The ratio of molar concentrations C/O was close to 1 for all cases. No additional oxidizing agent was added in case of lignite with high humidity, as water content in material supplied enough oxygen to ensure carbon/oxygen balance. Material feed rates and flow rates of added oxidizers are given in carbon yield (Cyield) is the ratio of the carbon content in product gases to carbon content in all input reagents. It can be seen that syngas with high concentrations of H2 and CO was produced for all materials. Concentrations of CH4 and O2 were very low in all cases. For all materials, the content of tar and higher hydrocarbons in the product gas was substantially below 10 mg/Nm3. This content is lower than in most non-plasma gasifiers, where the tar content for various types of reactors varies from 10 mg/Nm3 to 100 g/Nm3. Composition of the syngas was close to the one determined by thermodynamic equilibrium computations as illustrated for wood gasification (Fig. 27). The syngas

Inputs Material Wood Wood Wood Wood pellets Seed skins Coal Polyethylene Plastics RDF RDF Pyrolysis oil Pyrolysis oil

Feed rate kg/h 41 41 60 60 95 60 11 11 30 30 11 22

CO2 slm – 125 86 248 120 – 210 300 216 – – 200 O2 slm 64 – 66 – 30 – 80 – 118 – 89 89

H2O g/min 18 18 18 18 18 18 18 18 18 403 18 18 1.0 0.7 0.83 0.85 1.0 0.8

Syngas Cyield 1.0 0.9 1.0 0.8 H2% 45 42 41 42 76 61 35 42 30 56 45 34

Table 2 Input reagents, composition of syngas, and its heating value for several materials CO % 39 42 52 53 15 25 42 50 48 30 48 44

CO2% 15 15 5 4 3 13 22 7 20 10 2 16

CH4% 1 1 1 0 5 1 0 0 2 4 1 3

O2% 0 0 1 1 0 0 1 1 0 0 4 3

LHV MJ/m3 10.1 10.2 11.1 11.2 10.0 10.1 9.1 10.8 10.0 11.3 11.2 10.1

2866 M. Hrabovsky and I. J. van der Walt

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Fig. 27 Composition of syngas produced by saw dust gasification (60 kg/h, water 159 g/min). Comparison of measured concentrations with results of thermodynamic equilibrium calculations and solution of Eq. 14 (Theoretical)

2867 60,00 50,00 40,00 30,00 20,00 10,00 0,00 CO %

H2 %

Theoretical

O2 % CO2 %

CH2 %

Measurements

Thermodynamic equilibrium

1

Molar Fractions

0,8 CO

0,6

H2 H2O CO2

0,4

C2H4

0,2 0 600

1000

1400 1800 Temperature [K]

2200

Fig. 28 Thermodynamic equilibrium composition of products of polyethylene gasification (Reaction of 1 mole of CO2 with 14 g of polyethylene)

composition is compared with the thermodynamic equilibrium composition and the composition calculated from Eq. 14. Small deviations of measured concentrations from the theoretical ones result from complex conditions in the experimental reactor. It should be noted that higher value of measured hydrogen concentration does not mean that total hydrogen production was higher than the theoretical one. In Fig. 28, the composition of syngas determined by thermodynamic equilibrium computations is presented for the reaction of polyethylene with CO2. Syngas composed of H2 and CO with minimum content of other components is produced

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at high temperatures. As in the case of other organic materials, this optimum syngas composition is obtained for temperatures higher than about 1200 K. In experiments with polyethylene and plastics higher concentrations of CO2 were found, but concentrations of other gases did not exceed 1%. The efficiency of carbon gasification of materials presented in Table 2 was high and varied between 0.7 and 0.9. The heating values of syngas are high due to high content of hydrogen and carbon monoxide. Higher content of CO2, and consequently lower syngas heating values, were obtained for some conditions. The energy balance of gasification of 1 kg dry wood, with oxygen and CO2 at a ratio of oxygen/carbon moles = 1 and a reaction temperature of 1200 K is shown in Table 3. If CO2 is used as oxidizing medium, the ratio of heating value of syngas to energy needed for syngas production is lower than in the oxygen process, as additional energy is needed for CO2 dissociation. The ratio of syngas LHV to energy consumed is higher in the oxygen process if a higher amount of oxygen is added. Part of the carbon is oxidized to CO2 and additional energy is produced in the reactor by this oxidation. LHV of produced syngas is then reduced for the same energy due to an increase in CO2 concentration. This is used in large gasification units for higher material throughputs to reduce power requirements of plasma generators. The total energy efficiency, i.e., the ratio of heating value of syngas to total energy input (which is equal to the sum of energy consumed for the process and the heating value of wood) is almost the same for both processes, as an increase of reaction energy due to CO2 dissociation is compensated by an increase of the heating value of syngas for the same value. The energy needed for the process is delivered to treated material by plasma with some power losses (sum of the power loss in the plasma torch and the power loss in the reactor). Energy can also be lost if heat transfer from the plasma to material is not complete. Efficiency of plasma torches is usually between 60% and 80%. Power loss in the reactor depends on its geometry and thermal insulation. The efficiency of utilization of plasma generator electrical energy for the process may be thus close to 50%. The total energy efficiency of the system would be in this case 82% for the oxygen process and 72% for the CO2 process. Mass and energy balances of gasification of pyrolytic oil by oxygen, steam, and CO2 processes for molar ratio O/C = 1 are shown in Fig. 29. The following equations describe the oil gasification processes: C5 H8 O þ 2O2 ! 5CO þ 4H2

(37)

C5 H8 O þ 4H2 O ! 5CO þ 8H2

(38)

C5 H8 O þ 4CO2 ! 9CO þ 4H2

(39)

As in the case of other organic materials, the optimum syngas composition is obtained for temperatures higher than about 1200 K. In experiments with polyethylene and plastics, higher concentrations of CO2 were found, but concentrations of other gases did not exceed 1%.

0.92 0.92

20 24.6

O2 CO2

7.8 3.2

Syngas LHV Energyinput

Syngas LHV Energyconsumption

Syngas low heating value

19.2 19.2

Total energy consumption 2.58 7.6

Heating to T = 1200 K 1.97 2.39

Reaction wood + O ! CO2 + H2 0.61 0.61

Energy consumption for 1 kg of wood [MJ] Oxidizing gas Reaction CO2 ! CO + O O2 – CO2 4.6 Energy balance of 1 kg wood gasification Oxidizing gas Wood heat of combustion[MJ]

Table 3 Energy balance of gasification wood

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Amount of gasifying medium per 1 kg of oil: Oxygen: 0.832 Nm3 Carbon dioxide: 1.664 Nm3 Water: 1.339 l Mass ratios oil 23%

oil 43%

oil 40% O2 60%

water 57%

CO2 77%

Reaction enthalpies for 1 kg of pyrolitic oil Reaction enthalpy [MJ]

60 50 40 O2 CO2

30

water

20 10 0 oil volatilization

gas heating

CO2 and H2O dissociation + water evaporation

total reaction enthalpy

LHV syngas

Fig. 29 Mass and energy balances of gasification of pyrolytic oil

9

Plasma Waste to Energy

In the search for alternative, renewable energy sources to supplement fossil fuels and nuclear technology, much work has been done on biomass, especially in Europe and North America. There is an overwhelming amount of literature available, and this summary attempts to provide an overview of examples of available technology. Most of the commercially available equipment at present is of advanced, wellproven conventional (nonplasma) design aimed at direct heating/steam generation or cogeneration applications. Cogeneration is achieved either by steam turbines or by gas engines and gas turbines. In the latter case, the biomass is pyrolyzed and gasified in multistage reactors and the product gas (H2 + CO) cleaned up to an acceptable standard. This gas can also be utilized for chemical or liquid fuel production as evidenced by the emerging application of Fischer-Tropsch synthesis. Fuel sources for biomass-derived energy include agricultural and forestry wastes, such as timber chips, sawdust, bark chips, straw, grass, animal manure, fruit stones, sugar cane bagasse, and any cellulosic material. These materials can be combined with fossil-fuel wastes such as coal fines for, e.g., industrial heating and steam generation.

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The net energy yield of biomass is determined to a large extent by the amount of moisture it contains. The heat of combustion (or calorific value) is similar to that of some grades of coal and typically ranges from 15 to 20 MJ/kg, whereas the CVs of petroleum-based fuels range between 30 and 50 MJ/kg (Engineering_toolbox 2016). The CV of RDF varies widely (8–18 MJ/kg) depending on prior sorting and preparation. Experiments with various types of biomass proved that when different feed materials are gasified by a thermal plasma, the product mixture can be changed by manipulating the plasma conditions in order to generate a good quality syngas. The compressed syngas is purified, and, in one integrated process, Fischer-Tropsch (FT) synthesis can be used to synthesize solid (wax), liquid and volatile hydrocarbon products. Coal gasification via the FT synthesis process is a well-known process and economically viable on a large scale. Small-scale portable plasma/FT systems are being envisaged for developing countries such as South Africa. Economics of scale usually favors big systems, but the need arose for microscale systems situated closer to the end user, or in rural areas. These systems could range in size from 1 to 100 ton per day (tpd) of biomass feed. The relatively high-cost contribution of instrumentation and reduced thermal efficiency of small systems limit the degree of downscaling and their economic feasibility. However, this can be addressed by intelligent system integration.

9.1

Energy Products

Syngas, also known as synthesis gas, synthetic gas, or producer gas, can be produced from a variety of different materials containing carbon. These can include biomass, plastics, coal, municipal waste, or similar materials. Town gas produced from bituminous coals was historically supplied to many residences in Europe and other industrialised countries from the early nineteenth to mid-twentieth century.

9.1.1 Electricity Production Electricity can be generated from syngas by various means, depending on the application. The most common are internal combustion gas engines. External combustion engines such as steam engines, Stirling engines, as well as steam and gas turbines are also used. Internal combustion (IC) engines are widely applied commercially in similar processes and are available in a wide range of sizes. However, these engines require a certain quality and composition of syngas for proper operation. Where available, LPG or natural gas could be used to augment calorific value of the syngas stream. Although external combustion (Sterling) engines are not as sensitive to syngas composition as the IC type, they are still relatively small in comparison. IC equipment is available in capacities up to 4 MW. The world’s largest Stirling engine based biomass-to-energy plant was commissioned in Thuringia, Germany, in 2011. Fuelled by locally sourced wood chips, it produces up to 4000 MWh of heat and 1000 MWh of electricity annually (Wade_News_Service 2011).

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Syngas-fuelled gas engines can be configured for combined heat and power delivery to maximize the efficiency of the system. The composition of syngas is highly dependent upon the inputs to the gasifier. A number of the components of syngas, such as tars, hydrogen levels, and moisture, pose challenges which must be addressed at the outset. Hydrogen burns much more quickly than methane, which is the normal energy source for gas engines. Under normal circumstances this faster combustion in the engine cylinders would lead to the potential of preignition, knocking, and engine backfiring. In order to counter this, the engine has a number of technical modifications and its output is reduced to between 50% and 70% compared to when running on natural gas. For example, a 1063 kW engine running on natural gas will deliver a maximum of 730 kW on synthetic gas (Wade_News_Service 2011).

9.1.2 Advantages of Fueling Gas Engines with Syngas • Independent power supply • Reduced energy costs, and greater predictability and stability (compared to conventional bulk electricity supply • Efficient and economic combined heat and electricity supply (Energy.gov 2016) • Best suited for an electrical output range of a few hundred kW up to 20–30 MW • Low gas pressure required • Alternative disposal of a problem gas while simultaneously harnessing it as an energy source • Substitute to conventional fuels • Environmental benefits by greenhouse gas reduction 9.1.3 Cogeneration and CHP Cogeneration (cogen) through combined heat and power (CHP) is the simultaneous production of electricity with the recovery and usage of reject heat. Cogeneration is an efficient form of energy conversion and it can achieve energy savings of up to 40% compared to the purchase of electricity from the national electricity grid and a gas boiler for onsite heating (Clarke_Energy 2016). Combined heat and power plants are generally placed close to the user and therefore help reduce line losses and thereby improving the overall performance of the electricity transmission and distribution network. 9.1.4 Fischer-Tropsch Synthesis The FT process was developed almost a hundred years ago in Germany and commercialized by companies such as SASOL to produce a series of hydrocarbons, e.g., synthetic fuels, waxes, or gases. In the Fischer-Tropsch process, crude syngas (CO + H2) produced from carboncontaining material (e.g., biomass, coal) is filtered, stripped of acid gases (carbon dioxide and sulfur compounds). The composition is adjusted to the CO/H2 molar ratio required for the main product synthesis. The syngas is then reacted at elevated temperature and pressure in the presence of a catalyst such as iron to produce a crude mixture of liquid (syn-crude) and gaseous hydrocarbons and waxes. Several reactor design types are available, of which the single-pass slurry-phase processes can achieve up to 80% conversion to liquid products (Kreutz et al. 2008). A whole

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range of saturated and unsaturated hydrocarbons, waxes, and alcohols can be produced, depending on the choice of catalyst and operating conditions.

9.1.5 Biotechnology Ethanol can be produced directly from syngas in a fermentation reactor, e.g., the COSKATA process (He et al. 2012). The company has demonstration facilities at Madison (Pennsylvania). Their “Lighthouse” facility has 15,000 operational hours converting natural gas, woodchips, and simulated waste to ethanol using their proprietary micro-organisms. The company had a successful multiyear demonstration program at the Westinghouse Plasma demonstration center where syngas produced by Westinghouse from biomass and MSW was converted to ethanol. 9.1.6 Methane and Alcohol Production Production of methane requires CO and H2 in the presence of a catalyst at an H2/CO ratio of 3, and a water-gas shift (WGS) reaction is necessary (Krumpelt et al. 2002). Reduced nickel can be used to catalyze the gasification of cellulose to form methane and carbon dioxide (Minowa et al. 1994). On average 1 kg of biomass produces about 2.5 m3 of combustible gas (methane) at STP (Rajvanshi 1986). The use of syngas has better control for power generation when compared to the use of solid fuel, and this paves the way for cleaner and more efficient usage of energy sources. Ethanol could be produced in large quantities by gasification of biomass to syngas (CO + H2), followed by catalytic conversion. Both homogeneous and heterogeneous catalytic processes have been investigated. The homogeneous catalytic processes are relatively more selective for ethanol. However, the need for expensive catalyst, high operating pressure, and the tedious workup procedures involved for catalyst separation and recycling make these processes unattractive for commercial applications. The heterogeneous catalytic processes for converting syngas to ethanol suffer from low yield and poor selectivity due to slow kinetics of the initial C–C bond formation and fast chain growth of the C2 intermediate (Subramani and Gangwal n.d.). Enerkem Inc. has built a plant to produce ethanol from syngas generated from MSW in Edmonton, Alberta, Canada (Enerkem 2000). ICC-CAS (Institute of Coal Chemistry, Chinese Academy of Sciences) has been working on mixed alcohol synthesis over heterogeneous catalysts and gained some interesting results in the catalyst preparation and process engineering. This paper thus briefly introduced the recent research progresses at ICC-CAS (Fang et al. 2009).

9.2

Process Layout

The process layout for a typical plasma waste to energy facility will look similar to a conventional gasification, pyrolysis, or incineration facility (Sect. 2). All of the abovementioned facilities need a feed preparation section for sorting, size reduction, and intermediate storage. The waste is then transferred to feed hoppers before being introduced to the gasification reactor. In the reactor, the heat from a heat source converts the solid waste into a gas fraction and a solid fraction (the composition of

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M. Hrabovsky and I. J. van der Walt

Fig. 30 Different plasma torches from different suppliers in the world

which depends on the treatment process). Heat for the process could be supplied by a variety of different sized plasma torches, as presented in Fig. 30. The hot product gas exiting the reactor is quenched and cleaned up to comply with the specification of the downstream process and emissions requirements. Quenching is conventionally done in either by a wet or dry process. After quenching, the gas is filtered to remove particulate matter and scrubbed to reduce the halogen content and other contaminants to below the prescribed limits. At this point, the gas is ready for further processing. A block diagram of a typical plasma waste to energy gasification facility is presented in Fig. 31 (Westinghouse_Plasma 2016). Depending on the waste type the design can change in order to cater for additional chemicals like chlorine or a high ash content.

9.3

Influence of Waste Type

Various types of C-H-O containing waste can be used as gasification feedstock. These can include MSW, plant material, sewage, coal fines, used tires, medical and pharmaceutical waste, and many more. The chemical species in the abovementioned wastes vary widely. Plant material, for instance, consists of cellulose, lignin, varying

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Receiving, preparing storage system of waste (Option)

Control and Monitoring System Plasmatrons

Waste Feed-in Sytem

Water Supply Plasma Furnace

Fly Ash Feed-in System Waste Feed-out Sytem

Gas Supply Outgas

Receiving, storage system of inert slag (Option)

After burning chamber

Power Supply

Quenching Cyclone

Steam Turbine (Option)

Fly Ash

Steam Boiler (Option)

Heat-Exchanger

Fly Ash

HRSG (Option)

Cyclone with Dry Filter

Gas Turbine (Option)

Water Scrubber

Gasholder (Option) Syngas

Water pool Fly Ash collector

Filter

Fly Ash

Fig. 31 Schematic diagram of HTTC process

amounts of water, and ash-forming components, while coal fines waste consists almost exclusively of carbon, varying amounts of volatile compounds and adsorbed water. MSW is a mixture of paper, plastics, food, and plant material with inorganic contaminants in the form of metals and glass. Plastics like PVC also contain significant amounts of chlorine. Medical waste resembles MSW with the added biological hazard. Table 4 presents the elemental analysis of different waste types as analyzed by Pelindaba Analytical Lab, Necsa, South Africa.

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Table 4 Generalized elemental analysis of different waste types MSW (RDF) Bio waste (pits) Tyre waste Mixed plastic waste Pharmaceutical waste

Ash 15–20 0.5 8.7 5.3 25

Moisture 12 – 16 9 0.85 13 35

C 55 54.7 78.4 76.3 57

H 8 7.3 6.8 11.5 7.6

O 34 3.7 4.4 33

S 0.4 – 1.6 0.2 –

Cl 0.7 – – – –

F 0.014 – – – –

The ash content of the waste has a direct influence on the economic viability of a plasma waste-to-energy (WTE) process since it may occur in a significant concentration. Ash does not contribute to the energy content of the final product and actually requires energy to be heated up to gasification temperature. This reduces the overall energy efficiency of the process significantly. For a low ash content waste like plant material, 1 ton of waste can be converted into 1500–2000 m3 of syngas and then into ~55 kW of electricity. As the ash content increases, the yield of syngas per kg of waste is correspondingly lower, meaning that much less energy can be produced from 1 ton of waste. It is therefore beneficial to sort the waste prior to gasification. The removal of ash from gasification reactors is an art with various options. The simplest option is the removal of fly ash by filtration downstream from the reactor. This is an option for low ash content feedstock only. For high ash content feedstock, the ash is accumulated in the bottom of the reactor as melt or clinker, depending on temperature. Sometimes the gasification reactor is not operated at the melting temperature of the ash and a flux like CaCO3 is added to lower the melting point. The moisture content in the waste plays a very important role in the conversion of the carbon into CO and H2. Not only is energy required to vaporize the water, but the gasification reaction is endothermic and will also influence the CO/H2 ratio in the syngas. When the moisture content is too high, the syngas becomes diluted with CO2 and water. The moisture content can be optimized by a combination of drying and blending with dry material. As can be seen in Table 4, the C, H, and O content varies significantly. In order to convert the waste effectively into good quality syngas, one needs to consider the chemistry. In general, the H2/CO should be about 2 for FischerTropsch applications and 0.5 for electricity generation in an internal combustion engine. The product composition can be varied accordingly by mixing waste streams or by adding specific materials (e.g., carbon or steam) to the feed mixture.

9.4

Financial Viability (Financial Modeling)

In general, it is good practice, even for small projects, to perform a financial viability evaluation before the project is started to determine the profitability. The first step in such a study is to estimate the capital cost of the project. This cost is a combination of the equipment cost (fixed capital cost), the direct and indirect costs. The accuracy of the capital project cost estimation may vary depending on the technology readiness level of the project. An “order-of-magnitude” cost estimation is

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generally based on cost data for a similar project and will yield an accuracy of >  30%. This costing process is inexpensive and can give a good first order idea of the economic viability of a project. A “factored estimate” is based on a knowledge of the major equipment and this will give a <  30% estimate. The cost of this process increases significantly and is directly related to the time spent on the engineering design and specification documentation for the main equipment. A “preliminary estimate” has a  20% accuracy and the detailed estimate 5%. In order to increase the accuracy of the cost estimation the amount of work and time needed increases significantly. This cost increases from 0.15% of the capital project cost for the “order-of-magnitude” estimate, to 10% of the capital project cost for the “detailed estimate” (Peters and Timmerhaus 1991). Generally, a “preliminary cost” estimation is fairly simple and manageable for small projects. An acceptable method for determining the direct and indirect cost is by using the “percentage of fixed capital cost” method. For simple, small, and medium size plants, the following fractions of delivered equipment cost will apply for the direct and indirect project costs and is presented in Table 5. Depending on the size and complexity of the project, higher factors could be used. Larger and more complex projects will use a higher factor, but within the set limits. This information is then used as an input to the financial model. A high-level evaluation of the economic viability of a business relating to the technology in question, requires calculation of the internal rate of return (IRR), net present value (NPV) and the payback period. The robustness of these values can be determined by performing a sensitivity analysis (Peters and Timmerhaus 1991). The capital expenditure includes all costs associated with the plant as-built and ready for start-up, as determined above. The variable cost includes the cost of utilities,

Table 5 Factor ranges for direct and indirect project costs Direct costs (DC) Installation costs Instrumentation Valves and piping Electrical Buildings Incl. Offices, stores, ablution, etc. Yard improvements Service facilities Land Indirect costs (IC) Engineering and supervision Construction expenses Additional costs Contractors fees (% of DC + IC) Contingency (% of DC + IC)

Factor range (% of purchased equipment) 20–50 8–30 10–60 6–15 12–30 8–15 15–80 4–8 20–40 16–40 8–24 15–45

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reagents, royalties (if any), feed material, etc. The fixed costs include costs for labor, factory equipment, business and production overheads, and repair and maintenance. The income streams must be taken into account and balanced against the expenses. Generally, the ideal situation would be if the income is greater than the expenses. Income may be from the profitable sale of product/s or cost savings over a previously employed process. For instance, the cost of disposing medical waste may be as high as $3.5 /kg. If a plasma gasification facility is employed to destroy this waste at source and with the additional possible benefit of producing steam, hot water, or electricity, this cost could be cut significantly. A discounted cash flow model can be developed at the conceptual design stage of the project using the available market information and conservative estimates for the unknown financial inputs and factors. The financial model could be improved as the design and market information are updated to a higher level of accuracy as the project unfolds. The following major assumptions are usually made: • A 10-year project lifetime. • A one-year plant construction period with all cash flows in this year. • Ramp up to full-plant capacity in year two with production of 85% of capacity in the first year of operation. • Plant location should be close or adjacent to one of the sources of raw materials. • The debt/equity ratio for project financing is taken into account. • The discount factor must be calculated with reference to the client company policy. • Product selling prices must be conservatively selected and the projected revenue streams are therefore conservative. The scenario discussed below is for a financial viability model to determine the minimum economical size for a plasma gasification plant treating Refuse Derived Fuel (RDF). Arbitrary inputs, not taking development expenses into account, are presented in Table 6: Results from viability modeling are presented in Table 7. The data in Table 7 show unfavorable economic viability for the 1 and 10 ton per day plants. As a general rule plants with a medium-to-low risk factor (like this one) require an internal rate of return (IRR) >20% for economic viability. The net present value (NPV) of the plant at the end of its lifetime is also lower than the original plant cost for the 1 and 10 tpd cases. In addition to the unfavorable IRR and NPV, the payback period for these two plants is greater than the upper limit of 5 years. All of these economic factors are favorable for plants with waste-feed capacities of 20, 50, and 100 tpd. Once the economic viability model is developed and accurate enough, a sensitivity analysis can be done in order to determine if the plant economics are sensitive to fluctuations in the product value, fixed and variable costs, capital cost, interest rates, tipping fees, feedstock composition, etc. In this example, the picture changes significantly depending on the feed material properties (e.g., ash content, moisture

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Table 6 Arbitrary inputs for economic viability modeling Feed material Desired product Conventional tipping fee (saving) Ash content Moisture content Electricity cost Process heat sold Plant cost

Refuse Derived Fuel (RDF) Electricity $35/ton 5% 10% $0.12/kWh 50% of the electricity cost 1 tpd ➔ $1 mil 10 tpd ➔ $2.8 mil 20 tpd ➔ $4 mil 50 tpd ➔ $11 mil 100 tpd ➔ $18 mil 1 tpd ➔ $310/a 10 tpd ➔ $3,900/a 20 tpd ➔ $8,000/a 50 tpd ➔ $21,000/a 100 tpd ➔ $118,000/a 1 tpd ➔ $104,000/a 10 tpd ➔ $890,000/a 20 tpd ➔ $890,000/a 50 tpd ➔ $890,000/a 100 tpd ➔ $996,000/a

Variable costs (excl. Feed income)

Fixed costs

Table 7 Economic viability model results Plant capacity (tpd) IRR (%) NPV ($Mil) Payback (y)

1 1.6 0 19

10 14 4 6.9

20 38 19 2.8

50 47 72 2.3

100 58 158 1.2

content) and tipping fees. For this reason, it is recommended that each process be individually evaluated in early stages of the testing and evaluation phase as well as in the eventual feasibility study. Industrial Example

A plasma gasification test facility was constructed in the City of Ottawa, Canada, by the Plasco Energy Group (Young 2010). This facility can treat 94 t/d of garbage from the Trail Road Landfill and produces electricity and a vitrified road-filler material. Figure 32 presents the block flow diagram of the facility. The feed material is pretreated by shredding and metal recovery before being fed to the gasification reactor. Here the waste is preheated to 700
C before reaching the 1000
C gasification section. The crude syngas is extracted from the top, treated in a plasma reactor, cooled and impurities like sulfur and acid gases

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M. Hrabovsky and I. J. van der Walt

Syngas Cooling

MSW Feedstock

Plasma Arc Gasification reactor

Gas cleanup

Air emissions instrumentation

By-products

Vitrified Slag

Electrical power generation

Electricity for sale

Fig. 32 Process block flow diagram of the Trail Road facility

Table 8 Input parameters for Economic analysis of Ottawa Trail road Plasma Gasification facility (Young 2010) Total capital (including grant)(US$) Energy value (kWh/ton) Energy to grid (kWh/ton) Plant capacity (tons/day) Operational cost (15 operators) (US$/h) Variable cost (US$/y) Capital budget reserve (US$/y) Vitrified slag (0.2 ton slag/1 ton MSW) (US$/ton) Tipping fee (US$/ton) Electricity selling price (US$c/kWh)

2 828,900 1277 1021 94 28 078600 328,300 15 32.68 9.91

removed. The syngas is used for electricity generation via internal combustion engines and steam turbines. All off-gases are passed through air emission control equipment before being released to the atmosphere. The bottom ash is vitrified in a plasma and recovered from the bottom of the reactor. The power plant produces electricity to sustain the plant as well as additional electricity to sell into the grid. An economic analysis was done on this plant with the inputs in Table 8 showed a net positive cash flow (Young 2010). Air emissions on this plant were closely monitored and the results exceeded expectations on all levels. Concentrations significantly lower than the air emissions limits of all of the measured compounds were observed. The emissions data from the generator set are presented in Table 9.

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Table 9 Air emissions data from the electricity generator (Young 2010) Parameter HCl SOx Organic matter

10

Unit ppmv ppmv ppmv

Air emission limits 18 21 225

Plasma plant data particles

2.8

fluidparticles

10–3

10–1

Φp

four way

dense suspension

Other Forces (Saffman and Magnus Lift, Charging)

The Saffman and Magnus forces are caused by the presence of the particle in a strong velocity gradient and the rotation of the particle, respectively. The reader is referred to the reference of Crowe and his collaborators (Crowe 2012) for a complete description of these forces which may have an effect in very narrow regions, for example, the impact of a particle on a surface in plasma spraying. Small particles can become emitters of electrons at high temperatures, but for small particles at lower temperatures in a thermal (weakly ionized) plasma, their charge results from the attachment of electrons or ions to them. The particles are therefore charged, and consequently their behavior changes. This is a very important factor for aerosols in the earth atmosphere, as is often observed. The process is very different for dielectric and metallic particles. For metallic particles the charges can move through the entire particle, which is not the case for dielectric particles. The electric force on the charged particle is, !

Felec ¼ qP E þ

q2p 16πϵ 0 aT 2

(16)

where the Coulomb force appears as the first term and the second term is only significant when the particle is very close to the surface of the target, which

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Plasma-Particle Heat Transfer

2897

represents the so-called image force. ϵ 0 is the permittivity of vacuum, qp is the particle charge, E is the electric field, and a is the distance between the particle and, for example, a biased substrate. The particle-charging mechanism is a complex phenomenon that includes field charging by ion bombardment on the particles and diffusion charging by ion diffusion. The field charging theories were initiated by Pauthenier and Moreau-Hanot (1932) as well as Rohman (1923), while diffusion charging modeling was developed first by Arendt and Kallman (1926) and Fuchs (1947). Recently Zhengwei and Qiang (2010) evaluated the different available mathematical models for charging of submicron-sized particles in electrostatic precipitator modeling, where the charges are generated from a corona discharge. They divided the type of models in categories depending on the type of charging (field or diffusion) and the hypothesis of constant or transient charging. The particle motion Eq. 1 in this case takes the form, !

! ! ! dup ¼ F drag þ F g þ F elec dt !

(17)

!

3Eq

with F elec ¼ 4ρ dp3 p p

The charging model was proposed by White (1951), assuming the particle charge is constant, and evaluated from field charging theory, qs ¼ π

2ϵ 0 K p Ed2 Kp þ 2 p

(18)

where qs is the particle charge, Kp is the particle dielectric constant, and E is the local electric field. From the numerous models evaluated by Zhengwei and Qiang (2010), it appears that the model developed by Lawless (1996) has the best performance for the prediction of submicron particle trajectories (from 0.2 to 4 μm). The calculation of particle trajectories including the effect of lift forces and charging effect has been the subject of studies involving the use of cold spray guns coupled with nonequilibrium plasma discharges for particle charging in order to accelerate the particle in the vicinity of the target with a superimposed electric field (Tanaka et al. 2008; Jen et al. 2006; Ye et al. 2002). The conclusions of Tanaka et al. (2008) were that the Saffman lift became more important behind the bow shock for larger particles (12 μm), while the charging effect was effective for smaller particles (~1 μm) ahead of the bow shock and enabled smaller particles to be deposited.

3

Single-Particle Heat Transfer

When a single particle is suddenly immersed in a thermal plasma, the steady-state temperature and velocity profile in the boundary layer surrounding the particle are established very rapidly. Bourdin et al. (1983) and Vaessen (1984) demonstrated,

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using CFD, that the characteristic time justifies the use of the steady macroscopic Nusselt and drag coefficient in the calculation of the particle movement and thermal history. Simplifying considerably the analysis, assuming constant properties without convection, the analytic solution for the transient thermal boundary layer around a single spherical particle can be obtained. The transient Nusselt number for pure conduction becomes formulated, dp Nu ¼ 2 þ pffiffiffiffiffiffiffi παt

(17)

Using typical values of thermal diffusivity for commonly used plasmas, this expression can be used to estimate the relaxation time of the particle boundary layer (Proulx 1987). It has also been suggested by Konopliv and Sparrow (1972) that the thermal relaxation time for a Stokesian flow can be evaluated from the approximate expression, τ

1 u21 k 3 ρCp D4p

. For plasma temperatures in the 10,000 K range, this relaxation time is approximately 0.05 μs.

3.1

Convective Heat Transfer

The dimensionless convective heat transfer (Nusselt) can be used to determine the thermal history of a single particle in the same way that the drag coefficient is used for the calculation of the trajectory. Use of the lumped parameter approach (uniform temperature inside the particle) which is valid when Biot number is Ar, Alumina 30 microns -40 m/s radial injection

6

Temperatures

5 4

no corrections

3 2 1

2903

Trajectories

7

0

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95

7

Ar plasma --> Ar, Alumina 60 microns -20 m/s radial injection

3500

3500 6 3000 5 2500 4 2000 3 1500 1000 2

Young-Pfender correction 500

0

ArH2 plasma --> ArH2, Alumina 30 microns -40 m/s radial injection

3000 2500 2000 1500 1000

1

500

0

0

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95

67

ArH2 plasma --> ArH2, Alumina 60 microns -20 m/s radial injection

3500 7

6

3000

5

2500

4

2000

3

1500

3500

6

3000

5

2500

1500

2

1000 2

1000

1

500

1

500

0

0

0

0

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95

2000

3

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95

4

Fig. 4 Effect of the properties corrections on the trajectories and temperature histories of particles injected into a DC plasma jet. 30 and 60 μm alumina particles injected into argon plasma discharging into an argon environment and an argon-hydrogen plasma (90–10% volume) discharged into an argon-hydrogen environment

Fig. 5 Turbulent dispersion of 20 μm iron particles injected into an argon plasma DC jet. The particles are 20 μm with a standard deviation of 1 μm. Right vertical axis, diameter of the particle. Left vertical axis, radial position

Ar plasma --> Ar, Iron 20 microns turbulent dispersion

5

20 4 15 3 10

2

5

1 0 0

20

40

60

80

0 100

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P. Proulx

1 @ 2 @T @T r ¼α r 2 @r @r @t

(23)

which is completed by the plasma-particle boundary condition, hðT 1  T R Þ ¼ k

@T @r

(24)

where α and k are the thermal diffusivity and thermal conductivity of the particle and h is the heat transfer coefficient. Vaessen (1984) used similar methods to solve the same problem, while Lee (1988) proposed analytical solutions of the heat conduction inside the particles injected into a plasma flow. The calculation of the plasma-particle interaction for larger Biot number becomes significantly more computationally extensive since the solution of trajectories and temperature histories of each particle is coupled to the solution of the differential equation for internal conduction, leading to another level of iterative process. The Lagrangian PSI-Cell model which may demand the solution of tens of thousands of individual trajectories in order to have reasonable accuracy is augmented by the necessity to solve the internal conduction equation at each time step for each trajectory. With modern computers this is a feasible but yet very heavy task.

3.5

Melting and Evaporation

The field of thermal spray deals with melting of precursor particles, and evaporation of some of the material is often unavoidable. The same can be said from the application to spheroidization of material. The evaporation of material is predominant in ICP spectrochemistry where the emission from the analyte atoms/ions has to be measured. The same basic problem of internal conduction versus external convection can be qualified using Biot number. As Biot number becomes larger than unity, the importance of internal heat transfer becomes larger, and the lumped parameter approach cannot be used. If Biot number is small and the lumped parameter approach can be used, the problem of melting and evaporation can be simplified assuming the evaporation occurs only at the boiling point as in Proulx et al. (1985), Q¼

ρπd 3p Cps dT p 6 dt

Tp < Tm

(25:1)

Q¼

ρπd 3p Cpl dT p 6 dt

Tm < Tp < Tb

(25:2)

Q¼

ρπd3p Hm dXp 6 dt

T ¼ Tm

(25:3)

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Plasma-Particle Heat Transfer

Q¼

2905

ρπd3p H v dd p 2 dt

T ¼ Tb

(25:4)

where Q is calculated using Eq. 16, corrected for properties, rarefaction, and radiation as previously mentioned,    Q ¼ hA T  T p  σϵA T 4p  T 4a

(25:5)

with the subscript a corresponds to the particle environment, i.e., the plasma reactor walls. This formulation appears deceptively simple since the evaporation of the material from the particle can involve drastic changes in the particle’s surrounding plasma properties, transport properties, as well as radiative heat transfer. In the spectrochemical analysis using ICP, aerosol particles are evaporated in order to measure the radiation from the analytes. This process has been studied in depth by numerous researchers (Horner and Hieftje 1998; Horner et al. 2008; Benson et al. 2001, 2003; Shan and Mostaghimi 2003; Aghaei and Bogaerts 2016). The use of single-particle analysis in ICP has enabled researchers to measure the effect of a single evaporating droplet on the plasma radiation (Chan and Hieftje 2016). Suspension/solution plasma spray has brought attention more than a decade ago. This technology involves the injection of the material under suspension form, often as nanoparticles, and/or solvents with solutions of material. This involves very complex steps that can involve liquid droplet breakup, nanomaterial agglomeration and subsequent heating, and the presence of solvent species and material vapors in the plasma to name a few. The recent review of Vardelle et al. (2014) draws a very accurate description of the problems encountered in the understanding of the suspension plasma spray technology and concludes: “After nearly 15 years of intensive studies, suspension and solution plasma spraying are not sufficiently developed to meet industrial standards....”

3.6

Charging

In the free molecular regime, the heat transfer regime changes significantly, and the charging of the particles affects the heat transfer to the particles. The heat flux from the plasma is expressed as the sum of the different species contribution. For example, in the argon plasma, the total flux of the plasma to the particle is, qp ¼ qAr þ qþ Ar þ qe-

(26)

In this sum, the charge of the particles does not affect the flux from the neutral argon atoms, but this is not true for the ions and electrons. Without relative flow, the first of these fluxes is written,

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P. Proulx

qAr

  1 1 2 ¼ nVa mV  2kT 4 2

(27)

where n, V, m, and T are the number density, mean velocity, thermal accommodation, and mass species of argon. For the ions,

  1 qArþ ¼ ni V i ai eϕF þ 2kT  2kTp 4

(28)

and for the electrons,    qe- ¼ ai eϕF þ 2kT  2kTp þ EI

(29)

where ϕF is the floating potential and EI is the ionization energy. From these expressions it can be seen that the particle charge affects the heat transfer.

4

Plasma-Particle (Loading) Interaction Effects

Two-phase flow modeling has been evolving rapidly with the use of direct numerical simulation (DNS) and large eddy simulation (LES) for the solution of turbulent flow fields including one-way and two-way coupling. Lattice-Boltzmann (LB) methods which derived from the lattice-gas automata developed in the 1970s have gained wide popularity in their use in multiphase flows and have been used to model plasma jets (Sun and Dang 2010; Zhang and Wang 2007) and plasma-particle loading effect (Djebali et al. 2011). These methods, along with the relatively new smoothed particle hydrodynamics (SPH) Shadloo et al. (2016), show great potential and will surely attract more and more attention in the near future. The availability of massively parallel computer is very well suited for methods like LB and SPH. In the present work, mostly the work that has been done using classical CFD methods found in most modern commercial of open-source packages readily available to engineers and researchers is referred. These methods are based on Reynolds-averaged Navier-Stokes (RANS) modeling of turbulence and two-phase Eulerian-Lagrangian PSI-Cell modeling of the two-phase flow. The modern available CFD packages enable one to calculate the two-phase flow-particle coupling by specifying few parameters such as the initial particle injection pattern and the particulate flow rates. Both one-way coupling (flow affects the particles) and two-way coupling (flow affects particles, and particles affect flow) can be calculated with relative ease, although the calculation of the heat and mass transfer to the particles has to take into account the specific effects mentioned in the previous section. The first attempts at mathematical modeling of the treatment of particulate materials in thermal plasmas date from the late 1970s. The early studies found in the scientific literature on the behavior of small particles injected in a thermal ICP as well as DC plasma jets were mainly concerned with understanding the trajectories and temperature histories of the particles injected in the high temperature core of the plasma,

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Plasma-Particle Heat Transfer

2907

without taking into account the two-phase loading effect (Boulos and Gauvin 1974; Yoshida et al. 1979, 1983; Bhattacharyya and Gauvin 1975; Yoshida and Akashi 1977, 1981). These studies dealt with the so-called one-way effect, where particles are considered to be dilute enough to have no influence on the temperature and flow fields of the plasma. The development of the PSI-Cell algorithm (particle source in cell) of Crowe and his collaborators (1977) enabled researchers to use a CFD method that coupled the Eulerian equations for the plasma flow to be coupled with the Lagrangian trajectories of the particles. This technique is well suited for plasmaparticle flows since it assumes that the particle occupies a negligible fraction of the total volume. The basic PSI-Cell algorithm used in thermal plasma modeling is: 1. Solution of the non-perturbed flow equations, @ϕ ! ! ! ! þ ∇ ρu ϕ ¼ ∇ Γϕ  ∇ ϕ @t

(30)

2. Solution of the trajectories and thermal histories of the particles in the nonperturbed flow. Calculate the exchange of momentum, thermal energy, and mass in each control volume: Sϕp .These source terms are evaluated from Proulx (1985): Thermal energy source term   Δ mp Ep E SP ¼ C (31:1) τ Momentum source term !

SM P

  Δ mp ! up ¼C τ

(31:2)

  Δ mp ¼C τ

(31:3)

Mass source term SωP

where C is the steady-state number concentration of particles associated with a particle trajectory across a control volume and τ is the associated time of flight of the particle in the control volume. The steady-state local number concentration of the particles injected in the plasma field has been described by Proulx (1983,1985) for an axisymmetric RF plasma as N kl τkl

ij ij Ckl ij ¼ V ij where I and j refer to the control volume cells and k and l refer to the particle distributions.1 N is the number of particle circulating on a given trajectory, τ is the residence time in a given cell, and V is the volume.

1

For example, particle size, injection position, etc.

2908

P. Proulx

3. Solution of the perturbed flow equations, @ϕ ! ! ! ! þ ∇ ρ u ϕ ¼ ∇ Γϕ  ∇ ϕ þ SPϕ @t

(32)

where the term Sϕp corresponds to the exchange between the particles and the plasma for the corresponding dependent variable ϕ, typically momentum, enthalpy, and mass. In turbulent flow the modulation of turbulence from the particulate phase is also modeled through source terms. Several models have been proposed since the early work of Chen and Wood (1985). See Crowe et al. (2012) for a recent review of the RANS two-phase turbulence models. 4. Loop steps 2 and 3 until this so-called super-iteration converges. The application of thermal plasma technology to particulate material treatment has been based largely on three types of plasma torches: RF inductively coupled plasma direct current DC plasma torches, and free-burning or transferred arc plasma torches. Some examples of the combination of transferred and DC torches and DC torches and ICP have also been used in a few instances. Typical flow and temperature as well as electromagnetic fields in these torches are widely different and present characteristics that are of the utmost importance when dealing with particulate material thermal treatment. A brief discussion of the specific importance of particle loading in these types of torches follows.

4.1

Loading Effects in RF Inductively Coupled Plasma (ICP) Torches

The industrial interest in the treatment of particulate materials in RF ICPs for the production of nanoparticles and spheroidization has for many years brought interest in the evaluation of the plasma-particle interaction under dense loading. Proulx (1984) was the first to report results of the mathematical modeling of an inductively coupled plasma with high loading of particles. A fine copper powder was injected centrally in the discharge and carried by the so-called probe flow. They used two powder size normal distributions with the same mean powder diameter of 70 μm, one with a standard deviation of 30 and one with a standard deviation of 3. The mass flow rate of powder of copper was varied between 1.5 and 12 g/min, and the plasma power was 3 kw in an argon plasma torch 25 mm in diameter. Figure 6 shows the temperature, stream lines, and mass fraction for a mass loading of 7.5 g/min; a significant decrease of the centerline temperature is clearly noted. The results shown in Table 1 show that the evaporation fraction decreases with increasing mass flow rate and that the total evaporation even decreases as the loading becomes more important. Proulx et al. (1985, 1987) studied extensively the effect of loading from varied particulate materials and took into account the effect of the evaporated material on

67

Plasma-Particle Heat Transfer

2909

Fig. 6 Temperature, flow, and concentration fields in an RF inductively coupled plasma with injection of 7.5 g/min of copper (From Proulx (1984))

Table 1 Plasma-particle modeling results, argon plasma and copper particles, from Proulx (1984)

Particle size range (microns) 10–130 10–130 10–130 10–130 64–76

Powder feed rate (g/min) 1.5 4.5 7.5 12 4.5

Evaporation rate (g/min) 0.58 1.12 1.26 1.2 2.8

Power absorbed by the particulate phase (W) Particles Vapor 113 23 257 36 340 35 426 29 328 47

% of the plasma power absorbed 3.8 8.6 11.3 14.2 10.9

2910

P. Proulx 10000 900

10000

Al2O3

800

NO PARTICLES

80000

Ni

700 60000 QT [W]

Tp [K]

600 W

40000

400

Ni

20000

Al2O3 0 0.00

0.05

0.10

W 500

300 0.15

0.20

0.25

200

Z [m] 100 0

0

10

20

30

40

50

• m P [g/min]

Fig. 7 Centerline particle temperatures showing the importance of high particle loading (50 g/min) and total heat transfer to the particles as a function of particle loading (From Proulx et al. (1987))

the transport properties of the plasma. The loading effect is illustrated clearly on Fig. 7 where the centerline particle histories are plotted for a loading of 50 g/min of alumina, nickel, and tungsten particles, and the total heat absorbed by the same types of material injected in the ICP is shown as a function of the loading. These illustrate clearly that refractory material such as alumina rapidly attains a plateau and that adding more material results in very poor heating. As can be expected, the evaporation of easily ionizable material such as copper in the discharge zone causes important two-phase coupling because of the effect of the copper vapor on the electrical conductivity (Mostaghimi and Pfender (1984)). This is illustrated in Fig. 8 from the work of Proulx et al. (1985). The importance of the central gas injection velocity was found to be critical since the presence of even minute amounts of copper in the discharge zone affects the electromagnetic coupling. The importance of loading and plasma-particle interaction was further studied by Proulx et al. (1991) by including the effect of the radiative losses from the copper vapor in the energy balance of the plasma. The authors also showed that discrepancies in data from different sources for the radiative losses by the inductively coupled plasma were significant. As a matter of fact, since the early model used by Boulos (1976), mathematical models of the ICP have used either measured values for the emissive losses from the plasma volume or results of calculations based on the net emission coefficient from Lowke (1974). Often in the literature the term “optically thin” is used to describe the radiative losses of the plasma; however this is not the case since, for argon plasma, for example, most of the radiation from the resonance lines is reabsorbed in the plasma, while a fraction of the excited lines escapes from the plasma volume. The net emission coefficient calculation is calculated as a

2911

–.1

Plasma-Particle Heat Transfer

–.1

67

9.0x103

9.6

[K]

.5

.5

9.0 7.5

7.5x103 .4

[K]

3.0 4.5

.2

.03

6.0

.2

.03

.4

6.0

4.5 3.0

Fig. 8 Importance of the copper vapor properties, in particular, electrical conductivity, on the ICP modeling. Left: no particles; right, 0.5 g/min of copper (From Proulx et al. (1985))

function of the plasma geometry; typically the losses are expressed as a function of the radius of an isotherm column of plasma (Essoltani et al. 1990). Figure 9 shows the temperature fields predicted by the mathematical model using the radiative losses obtained from Mensing and Boedeker (1969), for temperatures below 10,000, and Evans and Tankin (1967) for temperatures higher that 10,000 K. These are compared to the predictions using the radiative net emission coefficient calculated by Cram (1987). The comparison of the two fields shows a difference of almost 1000 in the heat of the plasma flame. Table 2 presents the results obtained when copper particles were injected in the plasma, and the effect of the copper vapors on the transport properties but also on the radiative emission of the plasma as calculated by Cram (1987) was taken into account. The amount of vaporization of the copper particles is seen to be drastically reduced when the radiative losses of the copper vapor are taken into account. The temperature fields of the plasma from the first two lines of the Table 2 (particle loading 1 g/min) are shown in Fig. 10. The left part of the figure shows that the loading of 1 g/min has a small effect on the temperature fields, while the right half of the figure shows a decrease of several thousands of degrees along the centerline. While the presence of the vapors decreases significantly the temperature fields, this

2912

P. Proulx

Fig. 9 Temperature field predicted by the mathematical model (Proulx et al. 1991) using data from: (a) Mensing and Boedeker (1969) and Evans and Tankin (1967) (b) Cram (1987)

a Z LT

b .8

.4

00

1

0

.4

.8

1

9

.2 T = 10 [103 K]

.4

.6

8 9

8

7

7 .8 6

5

6

4

5

3

2

4 3

2 1

Table 2 Effect of the copper vapor on the heat load on the plasma (From Proulx et al. (1991)) Powder feed rate g/min 1

10

W/o copper vapor radiation With copper vapor radiation W/o copper vapor radiation With copper vapor radiation

Power absorbed by the particles (W) 90

% mass of the powder evaporated 100

74

42

529

15

367

1.7

decrease is due to an important increase in the radiative fields. The results showing the 50,000 W/m3 contour of the local emission coefficient in the plasma for the same conditions shown in Fig. 10 are shown in Fig. 11. This illustrates clearly that although the temperature is drastically reduced, the plasma flame itself appears much larger and appears “hotter” or at least more luminous because of the higher rate of heat losses due to increased radiation. The modeling works cited above dealt with laminar conditions in the plasma flame where plasma-particle loading is expected to be very important. When turbulent mixing is present, the temperature gradients will decrease significantly, and therefore coupling should be less important. However, even in industrial conditions

67

Plasma-Particle Heat Transfer

Fig. 10 Temperature fields predicted for a pure argon plasma (a) and in the presence of copper powder injection (b) (From Proulx et al. (1991)). The effect of the copper vapors on the radiative losses of the plasma is calculated using the data from Cram (1987)

2913

a

b .8

.4

0

1

9 Z LT

.4

0

.8

1

9 8 67

8

5

7

4 6

5

4 2 3

3 2

Fig. 11 Volumetric radiation from the plasma in the presence of the injection of copper vapors. (a) Without radiation from the vapor and (b) with radiation from the vapor (From Proulx et al. (1991))

a Z 0 LT

0

b .4

.2

.4

QR ≥ 105 W m3

.6

.8

1

.8

1

0

.4

.8

1

2914

P. Proulx

for typical applications such as spheroidization and nanoparticle productions where turbulence is expected, there are regions of the plasma flame where very high temperatures lead to low local Reynolds numbers and low turbulence intensity. In the ICP torch, for example, the core of the plasma remains mostly laminar, and this region is therefore subject to strong plasma-particle loading effect. Turbulence occurs mostly in after the plasma torch, a zone often called the “reactor,” or at the colder edges of the ICP in the plasma torch itself. Recent modeling work by Lopes (2016) on an ICP torch and reactor used to produce metallic nanoparticles from micrometric powders illustrates the very strong loading caused by the radiative losses from metallic vapors. She used the opensource CFD package OpenFOAM to model a pure argon ICP torch and reactor used for the production of metallic nanoparticles. The model included the injection of precursor particles in the plasma torch, their evaporation, nucleation, nanoparticle creation, and the resulting population balance solution describing the nanoparticle size and morphology. The results presented on Fig. 12 show a strong two-way coupling in the torch evaporation at relatively low loading rates (0.4 and 0.6 g/min) due to evaporation and the effect of the aluminum vapor on the transport, thermodynamic, and particularly radiation of the plasma. The importance of the plasma-particle loading effect in laminar ICP torches at low particle flow rates is probably illustrated best by two recent publications. Aghaei and Bogarts (2016) modeled the injection of copper particles from 1 ng/s to 0.5 mg/s. The cooling of the plasma flame is illustrated on Fig. 13 as the mass loading of particles reaches 100 mg/s. Figure 9 presents in detail the rate of mass evaporation as a function of loading; the loading effect does not in fact translate into a drastic reduction of the mass evaporation rate, since even at higher loadings the maximum evaporation rate

Fig. 12 Effect of particle loading on the plasma temperature at the centerline in a nanoparticle production reactor (From Lopes (2016))

67

Plasma-Particle Heat Transfer

2915

Fig. 13 Cooling of the ICP spectrochemical torch from the injection of 1 μm copper particles, 1 ng/s, 100 mg/s, and 500 mg/s (From Aghaei and Bogaerts (2016))

scales quite well with the loading. It rather affects the position at which the evaporation occurs, a fact which can be observed in connection with Fig. 14. Chan and Hieftje (2016) demonstrated experimentally the plasma-particle loading using single-particle analysis in a spectrochemical ICP. They conclude that: The introduction of a single micrometer-sized droplet causes a significant perturbation in an ICP. . .we observed plasma shrinkage due to a thermal pinch, significant cooling of the plasma downstream from a vaporizing droplet, and plasma reheating upstream from the droplet. For a 50-μm diameter droplet, this local cooling effect is strong and extends 6 mm from the physical location of the droplet... if the microdroplet introduction rate is higher than 100 Hz, the induced perturbation in plasma impedance is unlikely to return to a steady-state level before arrival of the next droplet

The experimental work of Fan et al. (1998) in such plasma torch and reactor system statistically demonstrates the effect of two-way coupling in ICP

2916

P. Proulx

a 1 ng/s

b 10 ng/s

Max. = 8,6 x 10–13 kg/s

Max. = 8,7 x 10–12 kg/s

d 1 μg/s

c 100 ng/s

Max. = 9.0 x 10–10 kg/s

Particle mass source (kg/s)

0e

+0 0 1e 0 –0 1 2e 3 –0 1 4e 3 –0 1 5e 3 –0 1 6e 3 –0 1 7e 3 –0 1 9e 3 –0 1 1e 3 –0 12

Max. = 7,7 x 10–11 kg/s

f 50 μg/s

e 10 μg/s

Max. = 8.39 x 10–9 kg/s

Max. = 3.6 x 10–8 kg/s

h 500 μg/s

g 100 μg/s

Max. = 1,1 x 10–7 kg/s

0e +

00 1e 0 –0 0 2e 9 –0 0 4e 9 –0 0 5e 9 –0 0 6e 9 –0 0 7e 9 –0 0 9e 9 –0 0 1e 9 –0 08

Max. = 4.1 x 10–8 kg/s

Particel mass source (kg/s) Fig. 14 Evaporation rates in the ICP spectrochemical torch from the injection of 1 μm copper particles as a function of mass loading (From Aghaei and Bogaerts (2016))

67

Plasma-Particle Heat Transfer

2917

spheroidization of alumina particles in the 10–100 μm size range. Similarly, the experimental and modeling studies of Ye et al. (2004), based on the 2-D, axisymmetrical plasma model developed by Mostaghimi et al. (1985) and Proulx (1984) and on the spheroidization of alumina for argon-hydrogen and argon-nitrogen plasmas, show the same trends, although the agreement was only satisfactory at low particulate flow rates. Colombo and Ghedini (2007) modeled the same process using the full 3D induction plasma torch and reactor they developed using Fluent’s CFD commercial package and have also shown significant two-way coupling effect. The threshold of particle to plasma gas ratio where one can safely assume that there is only one-way interaction depends heavily on a number of parameters: geometry of the torch and reactor configuration, flow conditions, plasma gas composition, particle injection conditions, and particle properties. Tong et al. (2016) reported recently using commercial CFD to model the spheroidization of particles in an RF ICP, and comparison to their experimental results shows remarkable agreement at negligible particle loading rates. In this case, the radiation from the aluminum cloud (Essoltani et al. 1994) produced by the aluminum evaporation proves to be a dominant factor in the two-way coupling effect that cooled the plasma even at particle loading around 0.05 g-particle/g-gas in a 30 kw ICP torch.

4.2

Particle Heating in DC Plasma Jets

The DC plasma jet two-phase coupling is very complex and shows very strong coupling that can be modeled using the Lagrangian-Eulerian PSI-Cell technique. The base coupling of mass, momentum, and thermal energy has to be extended to include two-phase turbulence coupling. Pfender and his collaborators (Lee et al. 1981; Pfender 1985, 1989; Chen and Pfender 1982, 1982, 1983, 1983; Chyou and Pfender 1989) published extensive modeling studies of the behavior of small particles injected in turbulent plasma DC jets. A few conclusions of the milestone contributions have been largely confirmed in later studies for conditions typical of DC jet plasma-particle injection. • The Basset term and added mass are mostly negligible. • Thermophoresis plays a minor role except for particles of the order of 1 μm and less. • Turbulent dispersion of the particles is important for particles in the 10 μm and lower size range. • The strongly varying properties due to the strong temperature and concentration gradients around the particle have to be taken into account. • Non-continuum (Knudsen) effect is controlled by the individual contributions of the species present in the plasma. The importance of the plasma-particle loading effect in DC plasma jets has been studied experimentally by Bokhari (1987) in the spheroidization of nickel and

2918 Fig. 15 Importance of particle loading in DC plasma jets. (a) Argon plasma. (b) Argon hydrogen plasma (90–10%)

P. Proulx

a

Ar plasma --> Ar, Copper 30 microns diameter and trajectory Loading 10 and 50 g/min

7 10 g/min 50 g/min

6

30

5

25

4

20

3

15

2

10

1

5

0

0 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95

b

ArH plasma --> ArH, Copper 30 microns diameter and trajectory Loading 10 and 50 g/min

7 6

10 g/min 50 g/min

30

5

25

4

20

3

15

2

10

1

5

0

0 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95

alumina microparticles, as a function of the loading of particles in a nitrogen DC plasma jet. The results presented over a wide range of particle feed rate (10–100 g/ min) and plasma power between 20 and 30 kW show that the spheroidization efficiency changes drastically from almost 100% at lower particle loading to less than 20% when the particle feed rates are increased to 100 g/min. Comparison with the mathematical model of Proulx (1987) was in good agreement. Figure 15 shows the loading effect on argon and argon-hydrogen DC plasma jet. The conditions showed on this figure correspond to the two-way coupling region of Elgobashi’s map (Fig. 1) where the plasma flow affects the particles and the particles affect the plasma flow. Both the trajectories and the temperature histories are affected, the right axis showing the average trajectories and diameters of particles injected with identical initial conditions.

67

Plasma-Particle Heat Transfer

5

Cross-References

2919

▶ A Prelude to the Fundamentals and Applications of Radiation Transfer ▶ Droplet Impact and Solidification in Plasma Spraying ▶ Heat Transfer in DC and RF Plasma Torches ▶ Heat Transfer in Suspension Plasma Spraying ▶ Macroscopic Heat Conduction Formulation ▶ Radiative Plasma Heat Transfer ▶ Radiative Properties of Particles ▶ Synthesis of Nanosize Particles in Thermal Plasmas

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Heat Transfer in Suspension Plasma Spraying

68

Mehdi Jadidi, Armelle Vardelle, Ali Dolatabadi, and Christian Moreau

Contents 1 2 3 4

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Suspension Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Plasma Jet Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Suspension/Plasma Jet Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Suspension/Liquid Breakup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Droplet/Particle Phase Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Heat Transfer to the Substrate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Particle Trajectories Around the Substrate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Cross-References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

Suspension plasma spraying (SPS) is an emerging thermal spray processes. Its specific feature is to use a suspension that is a heterogeneous mixture of very fine particles in an aqueous or organic solvent, to achieve finely structured coatings. The latter have a great potential for demanding applications like solid-oxide fuel cells or thermal barriers in gas turbines. The suspension feedstock is injected radially or axially into a DC plasma jet in the form of a spray of droplets with size of 20–200 μm or a continuous jet. The interaction of the suspension drops/jet with the high-temperature high-velocity plasma jet results in their fragmentation M. Jadidi (*) · A. Dolatabadi · C. Moreau Department of Mechanical and Industrial Engineering, Concordia University, Montreal, QC, Canada e-mail: [email protected]; [email protected]; [email protected] A. Vardelle European Ceramic Center, Laboratoire Sciences des Procédés Céramiques et de Traitements de Surface, University of Limoges, Limoges Cedex, France e-mail: [email protected] # Springer International Publishing AG, part of Springer Nature 2018 F. A. Kulacki (ed.), Handbook of Thermal Science and Engineering, https://doi.org/10.1007/978-3-319-26695-4_30

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into small droplets and vaporization of solvent in a few microseconds. Subsequently, the solid particles are accelerated, heated, and melted by the plasma jet and rapidly flatten and solidify after impact on the substrate. Although this process has been investigated for about 20 years, the effect of process operation parameters and mechanisms are not fully understood yet. In this chapter, the phenomena involved in suspension plasma spraying process, the basic aspects in the simulation of this process, as well as the main technological challenges are described in detail. Different correlations for suspension properties, liquid penetration height in gaseous crossflow, effervescent atomizers, droplet/particle drag coefficient, and Nusselt number, as well as the importance of coating particles’ Stokes number are discussed. Moreover, various methods for plasma jet modeling, the specific volume of fluid approach for modeling the liquid-plasma jet interaction, the commonly used secondary breakup models, and an approach for modeling suspension droplet based on a multicomponent assumption are reviewed.

1

Introduction

Conventional plasma spraying uses powders of particle size between 10 and 100 μm to produce coatings with a characteristic microstructural dimension in the micrometer range. This dimension corresponds to the thickness of the lamellae formed by the impact of the molten particles on the substrate. If it were possible to spray nanoand submicron-sized particles, the resulting coatings would have a smaller characteristic length scale, and this would result in improved wear resistance, enhanced thermal insulation and thermal shock resistance, and superior superhydrophobicity and catalytic behavior (Killinger et al. 2011; Fauchais et al. 2011; Aghasibeig et al. 2014). However, the use of nano- and submicron-sized particles in conventional plasma spray techniques is challenging for the following reasons: 1. Possible clogging of the feed line because of the natural tendency of fine particles to agglomerate 2. The difficulty of injecting fine particles in the core of a high-temperature highvelocity gas jet because the injection force of particles has to be of the same order of magnitude as the force imparted by the gas flow (S.ρ.v2, where S is the cross section of the injected particle, ρ the specific mass of the plasma, and v the particle velocity) 3. The low inertia of the fine particles that may result in their following the gas streamlines instead of impacting on the substrate (Jadidi et al. 2015a) There are two techniques called suspension and solution precursor plasma spray used to address the two first issues stated above (Fauchais et al. 2011; Pawlowski 2009). The basic idea of these techniques is to use a conventional plasma spray system but replace the powder carrier gas by a liquid whose density is about 1000 times greater than that of the gas.

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Suspensions are a combination of nano
/submicron-sized particles, a liquid carrier (e.g., water or alcohol), and usually a dispersant (Pawlowski 2009). Solutions are made by dissolving metal salts, organometallic precursors, or liquid metal precursors in a solvent so that the submicronic particles are formed in flight. This chapter will be limited to suspension plasma spraying that, a priori, does not involve chemistry in the in-flight treatment of the suspension particles. Detailed information about solution precursor plasma spray can be found in the following review papers (Gell et al. 2008; Killinger et al. 2011; Jordan et al. 2015; Vardelle et al. 2016). Suspensions are commonly injected into the plasma flow issuing from a plasma torch in the form of fine droplets by using spray atomization (Fig. 1a) or in the form of a continuous liquid jet by using mechanical injection (Fig. 1b). In this second injection method, the high-velocity plasma crossflow causes the atomization of the liquid jet. Depending on the type of plasma torch used, the liquid feedstock can be injected into the plasma jet either radially or axially (Fauchais et al. 2008; Curry et al. 2014). The suspension droplet evolution in a high-temperature high-velocity gas flow is shown schematically in Fig. 2 (Pawlowski 2009). When the droplets or liquid jet

Fig. 1 Radial external injection of a suspension into a high-temperature high-velocity plasma flow: (a) spray atomization and (b) mechanical injection (Fauchais et al. 2008; Curry et al. 2014)

Fig. 2 A schematic of a suspension droplet evolution in a high-temperature high-velocity gas flow (Pawlowski 2009)

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penetrate into the plasma flow, they are subjected to a strong shear stress by the plasma flow and a very high heat flux. The former causes the fragmentation of the liquid feedstock and the latter its vaporization. However, the characteristic time scales of fragmentation and vaporization differ by at least two orders of magnitude: the liquid feedstock is first atomized by the jet (primary and secondary breakup) and then the liquid evaporation becomes dominant freeing the fine solid particles or agglomerates which are then accelerated, heated up, and melted by the gas flow before impact on the substrate (Fazilleau et al. 2006; Pawlowski 2009; Fauchais et al. 2010). It is worth mentioning that, the suspension vaporization includes complex phenomena and different scenarios such as shell formation, and micro-explosion may happen as comprehensively discussed in Jadidi et al. (2015a). The microstructure of the coating and its properties depend on the distributions of temperature, velocity, and size of the particles impacting on the substrate which in turn depend on the trajectory of the suspension droplets in the plasma jet and therefore on the primary and eventually secondary breakup of the suspension (Fazilleau et al. 2006; Pawlowski 2009). These phenomena strongly depend on the injector type, and angle and location of the injector. Moreover, the suspension properties and injection velocity combined with the characteristics of the plasma jet control the penetration of the suspension into the gas flow and ensuing breakup and evaporation phenomena (Jadidi et al. 2015a). In the following sections, the suspension properties, primary and secondary breakup, and suspension evaporation phenomena are explained in more details.

2

Suspension Properties

The key properties of a suspension are its density, viscosity, surface tension, specific heat, thermal conductivity, and heat of evaporation of the solvent. They control to a great extent, the behavior of the suspension in the plasma jet. These properties depend on the solid particle concentration, type, size and shape, solvent type, surfactant concentration and composition, suspension acidity (pH), and temperature (Schramm 1996; Litchfield and Baird 2006; Gadow et al. 2008; Yu et al. 2008; Ghadimi et al. 2011; Tanvir and Qiao 2012). The most used solvents are distilled/deionized water, alcohols, or water-alcohol mixtures. Empirical correlations as well as a few theoretical equations allow the prediction of suspension properties. However, these correlations are usually developed for particular conditions, for example, for a given solid particle concentration and size (Schramm 1996; Litchfield and Baird 2006; Yu et al. 2008; Ghadimi et al. 2011; Tanvir and Qiao 2012). The most prominent correlations for predicting the suspension properties are presented below. The density (ρ) and specific heat (C) are defined,
 ρ ¼ 1
αp ρl þ αp ρp

(1)

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 C ¼ 1
αp Cl þ αp Cp
 ρC ¼ 1
αp ρl Cl þ αp ρp Cp

(2) (3)

where αp, ρp, and Cp are the solid particle volume fraction, density, and specific heat, respectively, and, ρland Cl are the liquid density and specific heat, respectively (Ghadimi et al. 2011; Fan and Zhu 1998). Equations 2 and 3 are both widely used in the literature to calculate the suspension specific heat. The suspension viscosity is generally higher than the solvent viscosity. Moreover, a Newtonian-base liquid can exhibit a non-Newtonian behavior at low to high particle concentrations (i.e., the suspension is usually pseudoplastic at low to moderate particle concentration and can be dilatant at high particle concentration). However, when the base liquid is Newtonian and particle concentration is very low (i.e., dilute suspension), the suspension behaves as a Newtonian fluid (Schramm 1996). The following correlations are used to calculate the suspension viscosity (μ) as a function of the solid particle volume fraction (αp) and base liquid viscosity μ0 (Schramm 1996; Litchfield and Baird 2006),
 μ ¼ μ0 1 þ 2:5αp

(4)



 μ ¼ μ0 1 þ 2:5αp þ 10:5αp 2 þ 0:00273exp 16:6αp

(5)

For anisometric particles,
 μ ¼ μ0 1 þ aαp =1:47b

(6)

where a and b are the major and minor dimensions of the particle. Equations 4–6 are applied to dilute Newtonian suspensions (i.e., αp < 1%) where the particle size is in the range of submicron and micron and no strong electrostatic interactions exist between particles. To estimate the viscosity of dilute and dense suspensions, the Kitano et al.’s (1981), Krieger-Dougherty, and the modified Krieger-Dougherty equations are generally used (Krieger and Thomas 1957; Chen et al. 2007), 

2 1 μ ¼ μ0 1
αp =αpm  ηαpm 1 μ ¼ μ0 1
αp =αpm μ ¼ μ0

1 1
αp ðra =rÞ1:2 =αpm

(7)

(8)

!2:5αpm (9)

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where αpm is the maximum packing volume fraction (it generally ranges between 0.495 and 0.54 and can reach 0.605 and 0.3 for spherical and rodlike particles, respectively), η is the intrinsic viscosity (which is equal to 2.5 for monodispersed hard spherical particles), and r and ra are the radii of primary and aggregated particles, respectively (Pabst 2004; Mishra et al. 2014). Using αpm, the particle properties such as size distribution, shape, and porosity can all be considered (Pabst 2004; Litchfield and Baird 2006). Although the above correlations are applicable for a wide range of αp, they are not accurate when the particle size is less than 100 nm (Mishra et al. 2014). The thermal conductivity of a suspension is generally greater than that of the solvent. The Maxwell correlation can be used to predict the suspension thermal conductivity (Kleinstreuer and Feng 2011), κ ¼ κl þ κl


 3αp κp
κ l
 κp þ 2κl
αp κp
κl

(10)

where κ l and κp are the thermal conductivity of the liquid and solid phases, respectively. However, this correlation is only appropriate for dilute suspensions (i.e., αp < 1%) of relatively large particles. If the particle size is less than 100 nm, other empirical correlations should be used (Yu et al. 2008; Ghadimi et al. 2011). The Hamilton-Crosser model makes possible to take the particle shape into account (Tavman et al. 2008),
 κ κp þ ðn
1Þκl
ðn
1Þ κl
κp αp
 ¼ , κl κ p þ ðn
1Þκl þ κl
κp αp

n¼

3 ψ

(11)

where ψ is a sphericity factor (i.e., ψ = 1 for spherical particles). When the particle size is less than 100 nm, the Hamilton-Crosser model is modified as follows (Chen et al. 2009), κ κa þ ðn
1Þκl
ðn
1Þðκ l
κa Þαp ðr a =r Þ1:2 ¼ κl κa þ ðn
1Þκl þ ðκ l
κa Þαp ðr a =r Þ1:2

(12)

where κa is the aggregate thermal conductivity calculated with the following correlation, 8 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi9  2 κa 1 < κp κp κp = , ¼ ð3αin
1Þ þ ð3ð1
αin Þ
1Þ þ ð3αin
1Þ þ ð3ð1
αin Þ
1Þ þ 8 κl 4 : κl κl κl ;

1:2 αin ¼ rra

(13) The suspension surface tension depends on the solid particle volume fraction, material and size, and surfactant type and concentration, base liquid, and temperature

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(Tanvir and Qiao 2012). The experimental data proposed in the literature present inconsistencies regarding whether it decreases or increases compared to that of the base liquid (Brian and Chen 1987; Tanvir and Qiao 2012; Bhuiyan et al. 2015; Chinnam et al. 2015) and therefore, the suspension surface tension should be experimentally determined. Although the effect of particles on the suspension surface tension is not clear yet, it was found that adding surfactant resulted in a reduction of surface tension compared to that of the pure-liquid value (Kihm and Deignan 1995; Tanvir and Qiao 2012). In addition, the surface tension of surfactant solution is time dependent because the surfactant molecules require a certain time to reach the gas-liquid interface and change the surface tension. Depending on the concentration and composition of surfactants, this time can vary from a few milliseconds to days. Therefore, the static (equilibrium) and dynamic surface tensions are generally defined for surfactant solutions (Kihm and Deignan 1995; Jadidi et al. 2015a). By increasing the concentration of surfactant, the static surface tension decreases and reaches its minimum value at a certain surfactant concentration named as the critical micelle concentration (CMC) (Kihm and Deignan 1995). As the surfactant concentration goes further than the CMC, the static surface tension remains relatively constant. The dynamic rather than the static surface tension is involved in the breakup of suspensions (Kihm and Deignan 1995). To determine the dynamic surface tension, a simple, accurate, and inexpensive method is measuring the characteristics (e.g., interface shape and wavelength) of an oscillating free jet emerging from an elliptical injector, and inputting the experimental data in an analytical model (Ronay 1978; Bechtel et al. 1995, 1998, 2002; Howell et al. 2004). Finally, an important characteristic of the suspension is the enthalpy of vaporization of the solvent as the liquid vaporization cooled down the plasma jet according to its heat of vaporization; for example, the latter is 2260 kJ/kg for water and 841 kJ/ kg for ethanol.

3

Plasma Jet Modeling

The suspension plasma spray (SPS) process can be considered to consist of the following subsystems: plasma jet formation in the plasma torch, liquid feedstock injection in the plasma jet, suspension treatment and formation of the coating on the substrate. Conventional plasma spray uses a direct current nontransferred plasma torch to produce a high temperature and high-velocity plasma jet by conversion of electrical energy into chemical and thermal energy. The most used plasma torches consist of a rod-shaped doped-tungsten cathode with a conical tip and a concentric water-cooled copper anode. The arc issuing from the tip of the cathode attaches to the anode by at least one high-temperature, low-density gas column through the cold gas boundary layer that develops on the water-cooled anode wall. The cold gas flow in the boundary layer exerts a pulling down drag force on the hot column while the Lorentz forces may act in the same or opposite direction depending on the curvature of the arc attachment column. Under the combined actions of these forces but also effect of

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Fig. 3 Classification of DC plasma arc modes based on the time-evolution of the voltage difference between the electrodes (Δϕ) (Trelles et al. 2006)

thermal and acoustic phenomena, the arc root moves on the anode surface. This movement brings about variations in the arc length and, therefore, in arc voltage that result in variations in the enthalpy input to the gas, velocity of the plasma jet and the way it mixes with the ambient gas when issuing from the torch. The plasma jet produced by a conventional DC plasma torch has a specific enthalpy between 5 MJ/kg and 35 MJ/kg. Its maximum temperature at the nozzle exit is around 10,000–12,000 K and its maximum velocity between 400 m/s and 2600 m/s. The plasma-forming gas is either a gas with a high-atomic weight (Ar, N2) or a mixture of these gases with a gas of higher thermal conductivity (H2, He) or viscosity (He). The use of diatomic gases (N2, H2) results in an increase in the plasma enthalpy with the addition of molecule dissociation energy. The time-evolution of the voltage difference between the electrodes (Δϕ) makes it possible to investigate the arc behavior inside the plasma torch. The arc operation mode is commonly referred as steady, takeover, and restrike mode depending on the transient features of Δϕ (Trelles et al. 2006; Meillot et al. 2009). These modes are schematically depicted in Fig. 3. The steady mode is characterized by a relatively small voltage difference between the electrodes (Δϕ) with negligible time fluctuations. The takeover mode is characterized by larger values of Δϕ and presents periodic or quasi-periodic voltage fluctuations, while the restrike mode is characterized by more chaotic, large amplitude voltage fluctuations with a characteristic saw-tooth shaped voltage evolution (Trelles et al. 2006; Meillot et al. 2009). The steady arc mode is characteristic of a stable plasma jet issuing from the nozzle but causes rapid erosion of the anode, whereas the restrike mode results in a highly fluctuating plasma jet which enhances turbulence development in the plasma jet and its mixing with the surrounding cold gas. This mode is promoted by the use of diatomic gases and high gas flow rate. The ensuing fluctuations of the plasma jet issuing from the torch affect the injection and processing of the suspension and in particular when the suspension is injected radially in the plasma jet. Indeed, the low

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density of the liquid and small inertia of the fine particles makes them very sensitive to the space- and time-variations of the plasma jet. The modeling of the suspension plasma process is generally based on the modeling of the plasma jet issuing from the plasma torch and the treatment of the suspension within the plasma jet. However, at least two questions arise: (i) is it necessary to model the plasma formation in the plasma torch and (ii) does the plasma jet model need to take into account the time and space fluctuations of the plasma jet? The prediction of arc behavior and plasma jet formation in the plasma torch is based on the simultaneous solution of the Navier-Stokes equations (gas mass, momentum, and species), energy conservation equation (gas temperature), and the Maxwell equations (electric and magnetic fields). The more advanced models take into account the nonlocal thermal equilibrium (NLTE) that prevails close to the electrodes by using two energy equations, for electrons and heavy species, respectively (Trelles et al. 2007). The set of equations is described in details in Trelles et al. (2006, 2009) and Moreau et al. (2006). If the current arc models are useful tools for plasma torch parametric studies and geometry improvement, their predictions should be carefully validated against experimental data obtained under well-defined torch operating conditions and geometry as they are not fully predictive. A fully predictive model of plasma spray torch operation that can reproduce the effect of the process parameters on the arc behavior without the need of adjustable model parameters requires further developments that involve the use of chemical and thermodynamic nonequilibrium plasma model (NCTE) and inclusion of the electrodes in the computational domain along with the sheath model (Vardelle et al. 2015; Chazelas et al. 2017). If a complete arc model is not necessary for works dealing with plasma torch operation and geometry improvement, simpler models can be used to set the plasma jet conditions at the exit of the torch nozzle for investigation of the suspension in the plasma jet whose predictions of interest are the characteristics of the particles and eventually droplets impacting on the substrate. The first model consists in solving the Navier-Stokes equations in a domain upstream of the nozzle exit (lower part of the nozzle) and adds a source term in the energy equation to take into account the conversion of electrical energy to gas enthalpy. For a nonfluctuating arc, this volumetric heat source can be expressed, P¼

ηt ϕI V

(14)

where P, ηt,ϕ, I, and V are the volumetric heat source, torch thermal efficiency, arc voltage, current, and, anode volume, respectively (Pourang et al. 2016). For a fluctuating plasma flow (i.e., arc operating in the takeover and restrike modes), the energy conversion can be considered as effective in a volume whose dimensions vary with time in order to model the time-variation of the arc length (Mariaux and Vardelle 2005; Meillot et al. 2008; Dalir 2016). For example, this volume can consist of a cone (V1) followed by a cylinder (V2) as shown in Fig. 4 (Meillot et al. 2008; Dalir 2016); the volume of the conical shape is constant, while the length of the cylinder L2 varies with time according to the following expression (Dalir 2016),

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Fig. 4 Example of volumes used to model the conversion of electrical energy in enthalpy inside the plasma torch (Dalir 2016)

L2 ðtÞ ¼ aϕðtÞ þ b

(15)

where the constant a and b are determined from the actual minimum and maximum values of the arc voltage ϕmin and ϕmax. The heat sources P1 and P2 in V1 and V2 respectively are then calculated by, P1 ¼

Pm L1 Pm L2 ðtÞ ¼ constant ¼ constant, P2 ¼ Lm V 1 Lm V 2 ðtÞ

(16)

where Lm is the average length of V2 drawn from the comparison of the predicted torch thermal efficiency in steady-state calculations with the actual thermal efficiency corresponding to the mean electric power Pm (Meillot et al. 2008; Dalir 2016). A second model consists in using transient boundary conditions for the velocity and temperature (or enthalpy) profiles imposed at the nozzle exit assuming their time-variation is modeled on arc fluctuations (Dussoubs et al. 1999). For example, for a plasma torch operating in the restrike mode, the local velocity and temperature of the gas can be assumed to vary continuously about a mean value over a time period corresponding to the typical time τ of an arc attachment spot. The instantaneous gas velocity at the nozzle exit can be then expressed,  ut ¼ us

β sin ωt 1þ 100

 (17)

where the subscript s represents the stationary value drawn from steady-state calculations or experiments, and ω = 2πf, where f is the arc fluctuation frequency, t the time, and β the arc fluctuation percentage. The most common assumptions of the model of the plasma jet issuing from the torch are the following: • The local thermodynamic equilibrium (LTE) prevailed in the whole calculation domain even though departures from equilibrium may occur in regions with steep temperature gradients. • The medium is considered as a continuum.

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Fig. 5 Thermophysical and transport properties of plasma gases (Boulos et al. 1994)

• The fluid is Newtonian. • The plasma is optically thin. • No demixion and chemical reactions occurred in the gas phase. The thermodynamic (specific heat and specific enthalpy) and transport properties (viscosity and thermal conductivity) are expressed in terms of temperature and composition. Figure 5 shows the temperature-variation of these properties for argon, hydrogen, and helium (Boulos et al. 1994; Remesh et al. 2003; Trelles et al. 2009; Fauchais et al. 2014).

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For a compressible gas flow, the continuity, momentum, species mass fractions (Yk), energy, and state equations are expressed (Echekki and Mastorakos 2010; Jadidi et al. 2015a): Continuity equation @ρ þ ∇  ρu ¼ 0 @t

(18)

where u is the gas velocity vector, ρ is the gas density, and t is the time. Momentum equation ρ

N X @u þ ρu  ∇u ¼
∇P þ ∇  τ þ ρ Ykf k @t k¼1

(19)

where P is the pressure, τ is the tensor of viscous shear stress, and fk is the body force related to the kth species. Species conservation equation (k = 1,. . .,N) ρ

@Y k þ ρu  ∇Y k ¼ ∇  ð
ρV k Y k Þ þ ω_ k @t

(20)

where ω_ k is the kth species reaction rate and Vk is the diffusive speed of the kth species. Energy equation ρ

N X @e þ ρu  ∇e ¼
∇  q
P∇  u þ τ : ∇u þ ρ Ykf k  Vk @t k¼1

(21)

where e and q stand for the gas mixture internal energy and the heat flux, respectively. State equation under ideal gas assumption P ¼ ρRT ¼ ρRu T

 N  X Yk k¼1

Wk

(22)

where Wk stands for the kth species molecular weight. The tensor of viscous shear stress for a Newtonian fluid is given by  h i 2 μ
γ ð∇  uÞI τ ¼ μ ð∇uÞ þ ð∇uÞT
3

(23)

where μ, γ, and I are the dynamic viscosity, bulk viscosity, and identity matrix, respectively. The term ρYkVk in Eq. 20 shows the species transport by the molecular diffusion phenomenon. Using Fick’s law, Vk is related to the molar (Xk) or mass fraction gradients (Echekki and Mastorakos 2010; Jadidi et al. 2015a),

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Xk V k ¼
Dm k ∇X k

(24)

where Dm k stands for the mixture-averaged mass diffusivity (mass diffusion coefficient) for species k. Furthermore, the term q in Eq. 21 consists of conduction heat transfer, radiation heat transfer, and heat diffusion by the mass diffusion of the various species (Echekki and Mastorakos 2010; Jadidi et al. 2015a), q ¼
λ∇T þ qrad þ ρ

N X

hi Y i V i

(25)

i¼1

The rates of diffusion of heat, momentum, and species in the plasma flow are determined to a great extent, by the level of turbulence and the way this turbulence is modeled. The arc fluctuations enhance the development of the turbulence that starts at the jet border where it comes into contact with the ambient gas at rest. Three approaches are commonly used to model gas flow turbulence: direct numerical simulation (DNS), large eddy simulation (LES), and Reynolds-averaged Navier-Stokes (RANS) (Echekki and Mastorakos 2010; ANSYS 2011; Jadidi et al. 2015a). DNS consists in solving the Navier-Stokes equations on a fine grid using a small time-step and so it captures all eddy sizes, including the smallest turbulence scales. If the predictions are accurate, the overall computation cost is proportional to Re3 and therefore the model is not suited to multiphase plasma flows with the current computing power. Large eddy simulation (LES) retains only the largest and most important eddies and predicts their motion uncoupled from the small eddies that are modeled using a subgrid-scale model (Garnier et al. 2009; Echekki and Mastorakos 2010; Jadidi et al. 2015a). The RANS approach is based on the Reynolds decomposition and the time averaging of Navier-Stokes equations. In this approach, further terms including the Reynolds stresses and fluxes exist in the transport equations and all the turbulence scales are simulated (Echekki and Mastorakos 2010; ANSYS 2011; Jadidi et al. 2015a). To simulate the Reynolds stresses and fluxes, two approaches namely the eddy viscosity models and the Reynolds stress model (RSM) are commonly used. In the eddy viscosity models, an eddy-viscosity constitutive relation is used to make a correlation between the Reynolds stress term and mean velocity profiles (e.g., Boussinesq equation). In this relation, a parameter called eddy viscosity, μt, exists that should be calculated. To predict μt, many models such as standard k-ε and RNG k-ε (where k is the turbulent kinetic energy and ε is the viscous dissipation rate of turbulent kinetic energy) are frequently used (Echekki and Mastorakos 2010; ANSYS 2011; Jadidi et al. 2015a). In the RSM, each component of Reynolds stress tensor has an individual transport equation and a scale-determining equation (e.g., viscous dissipation rate of turbulent kinetic energy (ε) equation) is considered. Compared to the eddy viscosity models, the RSM usually predicts more accurate results, particularly for the swirling flows and flows with strong streamline curvature (ANSYS 2011; Jadidi et al. 2015a). To model the thermal spray processes, the eddy viscosity as well as the RSM models available in commercial software packages

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such as ANSYS Fluent and open source CFD softwares as Code_Saturne (2017) are extensively used. Using ANSYS Fluent (2011), Jabbari et al. (2014) showed that, under the conditions of their study, the RSM model predicted the plasma jet behavior more accurately than the eddy viscosity model.

4

Suspension/Plasma Jet Interaction

4.1

Suspension/Liquid Breakup

Figure 6 illustrates the radial injection of a suspension in the form of a liquid jet (mechanical injection) into a plasma jet issuing from a dc plasma spray torch. The spray field can be divided into three zones: (1) liquid column jet, (2) disintegration of liquid column subjected to the action of the plasma crossflow and formation of ligaments, and (3) droplets (Marchand et al. 2008). When the liquid is injected in the form of drops (spray injection), only ligaments and droplets are present in the plasma flow (see Fig. 1). After fine droplets are produced, the liquid evaporation becomes dominant. The interactions between the plasma flow and liquid control the final size of droplets, their trajectories and heating and acceleration in the plasma flow, and finally the coating quality. Therefore, the penetration height of the suspension in the gas flow, droplet size distribution, and momentum flux should be carefully 14 12

Zone of liquid jet

10 8

4

[mm]

6

Contact point between liquid jet and plasma jet

2 Zone of disintegration of liquid column and formation of ligaments

0 –2 –4

Zone of dropletes

–6 –10.0 –7.5

–5.0

–2.5

0.0 [mm]

2.5

5.0

7.5

10.0

Fig. 6 The suspension atomization in plasma crossflow is divided into three zones: (1) cylindrical liquid jet, (2) disintegration of liquid column and formation of ligaments, (3) droplets (Marchand et al. 2008)

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controlled. In the following sections, liquid jet breakup in the plasma crossflow and droplet breakup phenomena are discussed in more details.

4.1.1 Fundamental Studies on Breakup Phenomena The breakup is defined as a process in which the surface-to-volume ratio of liquid augments (Dumouchel 2008). It happens when the liquid flow interacts with a highmomentum gas flow. The breakup process is generally divided into three parts namely as liquid flow ejection, primary breakup, and secondary breakup. In general, after initiation of liquid flow from the orifice, the deformations/perturbations appear on the liquid-gas interface. By the growth of the perturbations in space and time, liquid fragments eject from the main flow eventually. The initial flow perturbations and subsequent liquid fragment formation are known as the primary breakup mechanism. The secondary breakup mechanism results from the distortion of the liquid fragments and their breakdown into smaller elements. At last, stable drops are formed when the surface tension force is strong enough to ensure the liquid fragment cohesion (Dumouchel 2008; Jadidi et al. 2015a). Different atomizers and various techniques for injecting the suspension in the plasma jet are designed and applied in industry. They involve either cylindrical liquid jets interacting with the plasma crossflow or two-fluid atomizers (Ashgriz 2011a). Primary Breakup Liquid Jet in Crossflow

As shown in Fig. 6, after cylindrical jet formation, the jet interacts with the plasma crossflow (zones 2 and 3). The liquid column bends in the direction of the plasma gas flow due to the dynamic pressure exerted by the gas flow (see Fig. 7). The perturbations and instabilities that develop on the liquid column bring about the formation Fig. 7 Breakup of liquid jets in transverse subsonic gas flow (Jadidi et al. 2015a)

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of ligaments and drops (i.e., column breakup). The ligaments break up further into smaller drops. The location where the liquid column ceases to exist is called the column breakup point (CBP). On the lower surface of the liquid jet (leeward), waves with short wavelengths are detected. As a result, drops are sheared off the leeward surface, which is referred to as surface breakup. The size of drops formed by surface breakup is smaller than drops produced from ligaments (Wu et al. 1997; Inamura and Nagai 1997; Inamura 2000; Becker and Hassa 2002; Cavaliere et al. 2003; Lubarsky et al. 2012; Jadidi et al. 2015a, 2016b) In general, six nondimensional numbers are defined to characterize the behavior of liquid jets in crossflows: the gaseous Reynolds number, ReG; liquid Reynolds number, ReL; gaseous Weber number, WeG; momentum flux ratio, q (or liquid Weber number, WeL); liquid-to-gas density ratio; and ratio of duct width (L ) to orifice diameter (D) (Jadidi et al. 2015a), ReG ¼

ρ U2 D ρG U G L ρ UL D ρ U2 ρ D (26) , ReL ¼ L , WeG ¼ G G , q ¼ L L2 , ε ¼ G , δ ¼ μG μL L σ ρL ρG U G

where ρG and ρL are the density of the gas and liquid phase, respectively, μG and μL are the viscosity of the gas and liquid phase, respectively, and σ the liquid surface tension. Wu et al. (1997) showed that the gaseous Weber number and momentum flux ratio are the main parameters that control the spray characteristics. They also established a map based on these nondimensional numbers to evaluate the breakup regimes of liquid jets in crossflows (see Figs. 8, and 9) (Wu et al. 1997; Sallam et al. 2004; Ashgriz 2011b). As shown in Figs. 8, and 9, based on the gaseous Weber number, four breakup regimes named as enhanced capillary, bag, multimode, and shear breakup are observed. In addition, the map shows that for low values of WeG and q, the drops are formed from the ligaments and the column breakup mechanism is dominant. However, for high values of WeG and q, drops can be sheared off the leeward surface and the surface breakup becomes important as well. In other words,

Fig. 8 Different modes of liquid jets breakup in subsonic gaseous crossflows: (a) enhanced column breakup, (b) bag breakup, (c) bag/shear (multimode) breakup, (d) shear breakup (Sallam et al. 2004; Ashgriz 2011b)

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Fig. 9 Breakup regime map of liquid jets in subsonic crossflows (Wu et al. 1997)

as the Weber numbers increase, the surface breakup extent grows gradually (Wu et al. 1997; Sallam et al. 2004; Tambe et al. 2005). In addition to the mentioned parameters, the injection angle and injector internal design have significant influence on the liquid jet behavior in crossflows. Fuller et al. (2000) and Costa et al. (2006) found that the injection angle can affect the spray characteristics (e.g., breakup mechanisms and spray trajectory) even more than q. Ahn et al. (2006) indicated that cavitation and hydraulic flip of the injector internal flow cause the breakup mechanisms and the observed trajectories to be considerably different from the ones reported by Wu et al. (1997). In the suspension plasma spray process, the gas Weber number is high and the surface/column breakup regime can be reached (Fazilleau et al. 2006). The spray trajectory or liquid penetration height, and the location of liquid column breakup point (CBP) are considered as the main parameters that characterize the near field behavior of liquid jets in crossflows (Lubarsky et al. 2012). In general, the liquid penetration height increases as q increases and is independent of the gaseous Weber number (noting that in a few studies, it is reported that the liquid penetration height slightly decreases when WeG increases) (Wu et al. 1997; Becker and Hassa 2002; Lubarsky et al. 2012). Furthermore, the liquid penetration height decreases with the increase of liquid viscosity, liquid temperature, and gas temperature (Stenzler et al. 2003; Lakhamraju 2005). In the literature, there are many experimental and theoretical studies that attempt to link the liquid penetration height and CBP location with q, WeG, and gas temperature and pressure. Although numerous correlations have been developed, their results are significantly different from each other due to dissimilar injector shape, different turbulence intensity, and measurements methods (Lubarsky et al. 2012). The most used correlations for

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spray trajectory and CBP location are the following (x and y show the crossflow and liquid jet directions, respectively): Spray trajectory proposed by Wu et al. (1997) for nonturbulent liquid jets in uniform crossflow: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi y ¼ 1:37 qðx=DÞ D

(27)

CBP height for nonturbulent liquid jets in uniform crossflow (Wu et al. 1997): pffiffiffi yb ¼ 3:44 q D

(28)

CBP axial distance for nonturbulent liquid jets in uniform crossflow (Wu et al. 1997): xb ¼ 8:06 D

(29)

Lee et al. (2007) found that for turbulent liquid jets in uniform crossflow, xb/ D = 5.2 and the spray trajectory correlation was similar to Eq. 27. In the following correlation, the effect of gas temperature, T1, on the spray trajectory is included (Lakhamraju 2005),   x T 
0:117 y 1 ¼ 1:8444q0:546 ln 1 þ 1:324 , D D T0

T 0 ¼ 294 K

(30)

In the SPS process, the plasma crossflow is fluctuating both in length and width; it also exhibits steep radial gradients of properties (enthalpy, density, and velocity) which are time dependent. The effect of air crossflow oscillation on the liquid jet penetration was fundamentally studied by changing the crossflow oscillation frequency between 0 and 450 Hz, and it was found that the strength of the response of the liquid jet decreases as the oscillation frequency of the crossflow increases that is the spray oscillation was reduced by the increase of crossflow frequency (Sharma 2015). In addition, it was found that the crossflow velocity gradient (i.e., nonuniform/shear-laden crossflow) had significant effects on the spray trajectory (Tambe et al. 2007; Tambe 2010). Jadidi et al. that the spray trajectory p(2017) ffiffiffiffiffiffiffiffiffiffiffiffiffiffifound ffi (y/D) in nonuniform crossflow is equal to A qðx=DÞ where A is a function of the slope of gas velocity profile, and q is defined from the gas average velocity. Furthermore, when the crossflow is swirling, it was observed that the liquid jet follows a path close to the crossflow helical trajectory. In the case of swirling crossflow, the liquid penetration is also dependent on the swirl number (Tambe 2010; Tambe and Jeng 2010). Two-Fluid Atomizers

Most practical atomizers are of the pressure, rotary, or twin-fluid type. The latter use the kinetic energy of a gas flow to break a liquid into ligaments and then drops. This

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Fig. 10 A schematic of a typical effervescent nozzle (Roesler 1988; Roesler and Lefebvre 1988; Sovani et al. 2001)

type is generally used in suspension plasma spraying and essentially includes airblast, air-assist, and effervescent atomizers (Kassner et al. 2008; Esfarjani and Dolatabadi 2009; Fauchais and Montavon 2010; Aubignat et al. 2016). They differ in the amount of gas employed and its flow velocity. Figure 1-a shows the injection of a suspension in the plasma jet using an effervescent atomizer. This type of atomizer is shown in Fig. 10. It involves four components: the liquid supply port, gas supply port, mixing chamber, and an exit orifice (Sovani et al. 2001). By injecting the atomizing gas (the gas supply pressure is slightly higher than the liquid supply pressure) into the liquid, a bubbly two-phase mixture is formed upstream of the exit orifice (see Fig. 11). Then, this bubbly mixture is ejected through the discharge orifice (Roesler 1988; Roesler and Lefebvre 1988; Sovani et al. 2001). It is worth mentioning that, compared to the single phase flow, a two-phase flow through a nozzle chokes at considerably lower velocity. In other words, a two-phase flow with relatively low injection pressures and low flow velocities can experience a steep pressure jump at the nozzle exit (Sovani et al. 2001). In effervescent atomizer, the bubbles experience a sudden pressure relaxation and expand quickly on leaving the nozzle. Due to the presence of expanding bubbles, the liquid is shattered into drops (see Fig. 11a) (Roesler 1988; Roesler and Lefebvre 1988; Sovani et al. 2001). In addition, as shown in Fig. 11b, the liquid can form an annular sheath within the orifice of the atomizer when gas/liquid mass ratio (GLR) is high. In this case, the rapidly expanding gas core causes the liquid breaks up into thin

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Fig. 11 Schematic representation of the atomization mechanism in effervescent nozzle: (a) bubbly flow (Roesler 1988; Roesler and Lefebvre 1988), (b) annular flow (Buckner et al. 1990; Santangelo and Sojka 1995)

ligaments (Buckner et al. 1990; Lund et al. 1993; Santangelo and Sojka 1995; Sutherland et al. 1997; Sovani et al. 2001). At high gas/liquid mass ratios (GLRs > 0.4), drops suspended in the atomizing gas discharge from the orifice (Chin and Lefebvre 1993; Sovani et al. 2001). In general, the drop sizes obtained from effervescent atomizer are smaller than those generated by other conventional atomizers. The injection pressures and the gas flow rates are much lower than those required by other twin-fluid atomizers. Using this atomizer, the clogging problem can be alleviated and the orifice erosion can be reduced (Lefebvre 1989; Sovani et al. 2001). Furthermore, the effect of liquid viscosity on the mean drop size is relatively negligible (Buckner and Sojka 1991; Lund et al. 1993; Sutherland et al. 1997). However, the mean drop size is significantly affected by the length/diameter ratio of the orifice. The mean drop size decreases as the length/diameter ratio reduces (Chin and Lefebvre 1995). As the gas to liquid ratio (GLR) increases from zero to around 0.03, the Sauter mean diameter (SMD), i.e., the mean diameter of spheres that have the same volume/ surface area ratio, decreases rapidly. For higher GLR, the mean diameter reduces slightly (Whitlow and Lefebvre 1993). In general, as the GLR increases, the drop velocity and the spray momentum flux increase, and the spray cone angle widens (Chen and Lefebvre 1994; Bush et al. 1996; Panchagnula and Sojka 1999).

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Numerical Modeling of Primary Breakup of the Liquid jet In order to identify the location of the liquid phase and topology of the liquid-gas interface, the interface tracking methods are usually applied. These methods are based on the Eulerian approach and can be categorized into two classes: fixed-grid methods and moving-grid methods (Jiang et al. 2010; Jadidi et al. 2015a). In fixedgrid methods, the grid is predefined and does not move with the interface. The fixedgrid methods are the most commonly used due to their simple description and ease of programming. Among various fixed-grid methods, the volume of fluid (VOF) method is perhaps the most commonly used interface tracking to simulate the primary atomization of the liquid jet (Jiang et al. 2010; Jadidi et al. 2015a). In VOF method, an indicator function (called also as volume fraction and color function) is defined to implicitly calculate the liquid-gas interface location, its curvature and normal. The indicator function, α, denotes the fractional volume of the cell occupied by liquid: α = 0 and α = 1 correspond to a cell full of gas and a cell full of liquid, respectively. In addition, a cell with α value between zero and one represents the location of the liquid-gas interface (see Fig. 12). In this method, the set of the Navier-Stokes equations is solved for both gas and liquid phases as well as the volume fraction advection equation (Hirt and Nichols 1981; Prosperetti and Tryggvason 2009; Tryggvason et al. 2011). Furthermore, the surface tension force is usually included in the momentum equation and it is determined by the interface curvature (Brackbill et al. 1992). The main advantages and disadvantages of the VOF methods are the mass conservation and grid dependency, respectively (Jadidi et al. 2015a; Prosperetti and Tryggvason 2009; Tryggvason et al. 2011). To model the liquid primary breakup in the crossflow plasma jet, a specific compressible VOF method was recently developed (Vincent et al. 2009; Meillot et al. 2013). It assumes that the plasma behaves like a perfect gas, and the conservation equations of continuity, momentum, volume fraction, energy, and species (Eqs. 31–35) are formulated (Vincent et al. 2009; Meillot et al. 2013),

Fig. 12 Distribution of the indicator function, α, which represents the fractional volume of the cell occupied by liquid (Jadidi et al. 2015a)

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@P 1 þ ∇u¼0 @t χ T  

 @u τ ρ þ ρu  ∇u ¼ ρg
∇ P
∇:u þ ∇  μ ∇u þ ∇T u þ FTS @t χT @α þ ð∇  uÞα ¼ 0 @t   @T þ ð∇  uÞT ¼ ∇  ðλ∇T Þ þ Φrad ρCp @t @ψ i þ u  ∇ψ i ¼ ∇  ðDi ψ i Þ @t

ρ ¼ αρL þ ð1
αÞ ψ a ρa þ ð1
ψ a Þρp

μ ¼ αμL þ ð1
αÞ ψ a μa þ ð1
ψ a Þμp

λ ¼ αλL þ ð1
αÞ ψ a λa þ ð1
ψ a Þλp

Cp ¼ αCpL þ ð1
αÞ ψ a Cpa þ ð1
ψ a ÞCpp

D ¼ αDL þ ð1
αÞ ψ a Da þ ð1
ψ a ÞDp FTS ¼ σκni δi

(31)

(32) (33) (34) (35) (36) (37) (38) (39) (40) (41)

where P, ρ, u, g, μ, and T stand for pressure, density, local velocity vector, gravitational acceleration, viscosity, and temperature, respectively. χ T = (@ρ/@P)/ρ is the adiabatic compressibility, τ the characteristic time assumed to be equal to the numerical dynamic time step, Cp the specific heat, λ the thermal conductivity, ψ i the concentration of species i, Di the mass diffusion coefficient of species i, and Φrad represents the radiative effects. FTS, σ, κ, ni, and δi are the surface tension force, surface tension coefficient, local interface curvature, interface normal, and Dirac function, respectively. In addition, the index L, a, and p stand for liquid, air, and plasma, respectively (Vincent et al. 2009; Meillot et al. 2013). The results of this method will be discussed in the next paragraphs. The interaction of a train of spherical water droplets with a plasma flow was numerically studied by Meillot et al. (2013). The droplet diameter was assumed to be 360 μm. Moreover, a new droplet was injected every 0.1 μs. Figure 13 shows the behavior of the first three droplets in a steady Ar/H2 plasma crossflow as well as the temperature isotherms. Predictions show that droplet deformation began as soon as it reached the plasma flow. Furthermore, in actual plasma condition the droplet fragmentation started faster and large fragments were quickly destroyed (Meillot et al. 2013), compared to the case of a uniform temperature plasma flow (gas temperature was fixed to 3200 K).

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Fig. 13 Behavior of 360-μm water droplets injected in a steady Ar/H2 (45/15 SLM) plasma flow (the lines represent the gas isotherms) (Meillot et al. 2013)

Meillot et al. (2013) also numerically studied the primary breakup of continuous water jets in Ar/He and Ar/H2 plasma crossflows exhibiting different fluctuation levels: the Ar/He plasma jet was supposed to fluctuate according to the take-over mode and the Ar/H2 jet according to the restrike mode. For the Ar/He plasma flow, the arc current, mean voltage, and gas flow rates were 700 A, 40 V, and 30/30 SLM, respectively, while for the Ar/H2 flow, they were 500 A, 76 V, and 45/15 SLM, respectively. The liquid (water) jet velocity and diameter, right before the interaction point with the plasma flow, were assumed equal to 22 m/s and 200 μm, respectively. Figure 14 shows the penetration and breakup of the continuous water jet injected in a steady Ar/H2 plasma flow: the surface waves are developed on the liquid column, and arm-shaped filaments are formed and break up into droplets. Predictions showed that the water penetration in Ar/He is much less than in Ar/H2. The model was also applied to the breakup of a continuous liquid jet injected into a time-dependent Ar/H2 plasma jet. Figure 15 compares the liquid behavior at two different times (60 and 70 μs) and confirms that the liquid penetration height depends on time since the gas velocity and temperature fields change with time (Meillot et al. 2013).

Secondary Breakup As the drops enter the disruptive flow field, a secondary breakup process may initiate. An unequal pressure distribution is created around the drop by the acceleration of the ambient fluid and, the initial spherical drop is deformed. The viscous and interfacial tension forces resist the deformation. However, when the aerodynamic

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0.003 Velocity (m/s) 850 767 684 601 518 435 352 269 186 103 20

0.015

0

Fig. 14 Interaction of a continuous water jet with a steady Ar/H2 (45/15 SLM) plasma flow (Meillot et al. 2013) Jet inje

Jet inje

ction (2

ction (2

2m/s)

2m/s)

Temperature (K)

0 m/s

0 m/s

plasm

plasm

a limit

a limit

0.003

0.003

Plasma zone

Plasma zone

420

0.015

plasm

a torch

Flow velocity (m/s)

460 m/s

0.015

plasm

a torch

axis

0

1000 943 886 829 772 715 658 601 544 487 430 373

axis

0

850 765 680 595 510 425 340 255 170 85 0

Fig. 15 Behavior of water jet in a time-dependent Ar/H2 (45/15 SLM) plasma flow at 60 μs (left image) and 70 μs (right image) (Meillot et al. 2013)

forces are large enough, the drop fragmentation happens (Guildenbecher et al. 2009). The following nondimensional numbers are used in secondary breakup analysis, We ¼

ρG U 20 d0 μL ρ U0 d0 ρ μ U0 , Oh ¼ pffiffiffiffiffiffiffiffiffiffiffiffi (42) , ε ¼ L , N ¼ L , Ma ¼ , ReG ¼ G σ μG ρG μG c ρL d 0 σ

where d0 is the initial diameter of the drop, U0 the initial relative speed between the gas and drop, and c the speed of sound (Shraiber et al. 1996; Guildenbecher et al. 2009).

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Fig. 16 A schematic of the secondary breakup regimes of a Newtonian drop. From top to bottom: vibrational, bag, multimode, sheet-thinning, and catastrophic breakup regimes (Guildenbecher et al. 2009)

In secondary breakup analysis, the Weber number, We, and Ohnesorge number, Oh, are the most effective parameters. As We increases, the tendency toward disintegration enhances. In contrast, the tendency toward disintegration decreases when Oh increases. A schematic of different breakup regimes for Newtonian drops is shown in Fig. 16. From top to bottom, the figure illustrates vibrational, bag, multimode, sheet-thinning, and catastrophic breakup regimes (Guildenbecher et al. 2009). The transition between two breakup regimes depends on We and Oh (see Fig. 17). When Oh < 0.1, the transition Weber numbers are constant and reported in Table 1. Figure 17 shows that the transition Weber numbers increase as Oh increases (Hsiang and Faeth 1995). During vibrational breakup, a few fragments with sizes comparable to the size of the parent drop are generated. Pilch and Erdman (1987) explained that the vibrational breakup does not always happen and does not result in small final fragment sizes. Therefore, most authors ignore the vibrational breakup and consider bag breakup as the first secondary breakup regime. In the bag breakup regime, a thin hollow bag attached to a thicker toroidal rim is observed. The bag breaks up first, producing a large number of small drops. Then the toroidal rim breaks up generating a small number of large fragments. As We increases, a stamen oriented anti-parallel to the drop motion direction is added to bag breakup. This breakup regime is known as multimode. After bag disintegration, the rime and the stamen breakup. In sheet-

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103

ρf/ρG = 580–12.000

Theory

Weber number

Shear breakup

102

Deformation < 20% Multimode breakup Bag breakup

101

Oscillatory deformation Deformation < 5%

10% < deformation < 20%

10

0

10–4

5% < deformation < 10%

10–3

10–2

10–1 100 Ohnesorge number

101

102

103

Fig. 17 Map of secondary breakup regimes (Hsiang and Faeth 1995)

Table 1 Secondary breakup regimes of a Newtonian drop as a function of Weber number when Oh < 0.1 (Pilch and Erdman 1987; Hsiang and Faeth 1992; Guildenbecher et al. 2009) Vibrational Bag Multimode Sheet-thinning Catastrophic

0 < We < ~11 ~11 < We < ~35 ~35 < We < ~80 ~80 < We < ~350 We > ~350

thinning regime, a film is unceasingly eroded from the drop surface and breaks up after being detached. Consequently, a small drop plethora, and sometimes a core with a size that is comparable to the size of the parent drop, is generated. In the catastrophic breakup regime, large amplitude waves with long wavelengths cause the drop surface to corrugate. As a result, a small number of large fragments are formed. Then, these fragments disintegrate into smaller units (Guildenbecher et al. 2009). The breakup regime in the SPS process strongly depends on the plasma flow properties at the instant the drops penetrate the gas flow, drop size, and injection velocity. In other words, in plasma spray condition, due to time and space variations of the plasma flow properties, temperature, and velocity, the drops may experience different breakup regimes (Vardelle et al. 2008). Figure 18 shows the images of 250-μm-diameter water drops penetrating a transverse Ar/H2 (45/15 NLM) plasma jet. As clearly shown, depending on the time when the drops penetrate the gas flow, different breakup regimes can be observed (Vardelle et al. 2008).

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Fig. 18 Various secondary breakup regimes of 250-μm-dimeter water drops penetrating the transverse Ar/H2 plasma jet (Vardelle et al. 2008)

4.2

Droplet/Particle Phase Modeling

Today, the Eulerian-Lagrangian approach is mostly used to model multiphase spray flows. In this approach, the gas phase is modeled as a continuum while Lagrangian particle tracking (LPT) models are used to evaluate the trajectory, velocity, and temperature of droplets/particles. This approach is appropriate for simulating dilute flows. One-way coupling and two-way coupling are the main assumptions in the Eulerian-Lagrangian approach. In one-way coupling, the effect of the gas phase on droplets/particles is only considered and there are no reverse effects contrarily to the two-way coupled assumption (Crowe et al. 1998; Fan and Zhu 1998; Jiang et al. 2010; Jadidi et al. 2015a).

4.2.1 Droplet/Particle Motion The overall mass conservation of the droplet/ particle is expressed, dmd ¼
ρs wS dt

(43)

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where md is the droplet/particle mass, ρs is density at droplet/particle surface, w is the gas velocity at the droplet/particle surface, and S is the droplet/particle surface area (Crowe et al. 1998; Fan and Zhu 1998; Jiang et al. 2010). The droplet/particle velocity v is calculated by solving the droplet/particle motion equation in the gas flow, F ¼ md

dv dt

(44)

where F is the fluid forces acting on the droplet /particle of mass md. The drag force is the most important force acting on a droplet/particle in plasma spray. It is expressed, 1 FD ¼ ρG CD Aju
vjðu
vÞ 2

(45)

where ρG, CD, A, and u are the gas density, drag coefficient, droplet/particle projected area, and gas velocity, respectively. The drag coefficient, CD, depends on the Reynolds (Re) relating to particle, Mach (M ) and Knudsen (Kn) numbers (Crowe et al. 1998; Fan and Zhu 1998). However, for relatively large particles (micrometer sized) with low Mach number, Oberkampf and Talpallikar (1994) utilized the correlation,

CD ¼

8
1000

(46)

Under plasma condition, a steep temperature gradient exists across the gas boundary layer surrounding the particle, which results in severe variations of the gas transport properties (Fauchais et al. 2014). To account for this temperature gradient, Lewis and Gauvin (1973) recommended that the drag coefficient for a particle immersed in an Ar plasma jet should be corrected,
0:15 CD ¼ CDF νf =ν1

(47)

where CDF is the drag coefficient where the fluid properties are calculated at the arithmetic mean film temperature across the boundary layer (Tf = (Ts + T1)/2 where Ts and T1 are the particle surface temperature and gas stream temperature, respectively), and νf and ν1 are the kinematic viscosities estimated at the mean film temperature and the free stream temperature, respectively. Another correction factor for the plasma condition was proposed by Lee et al. (1981), CD ¼ CDF ðρ1 μ1 =ρs μs Þ
0:45

(48)

where the index “s” stands for the plasma properties at the droplet/particle surface temperature.

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Also, to estimate the transport of relatively small particles (particle diameter less than 10 μm) in plasma flow, another correction factor for the noncontinuum effect (i.e., Knudsen effect) should be taken into account. Chen and Pfender (1983) proposed the following correction for the drag coefficient, 2

30:45

1 5 CD ¼ CDF ðρ1 μ1 =ρs μs Þ
0:45 4
2
a γ  4 1 þ a 1þγ Prs Kn
 Kn ¼ λ =d0 , λ ¼ 2k=ρs vs cp Prs

(49)

where γ = cp/cv, a is the thermal accommodation coefficient, and Prs is the gas Prandtl number at the droplet/particle surface temperature, vs is the average molecular speed at the particle/droplet surface temperature, and k and cp are the average thermal conductivity and specific heat between the true droplet/particle surface temperature and the gas temperature at the droplet/particle surface (due to temperature jump condition). Kn is the Knudsen number based on an effective mean free path length. Another phenomenon which an affect the droplet/particle dynamics, especially when the particle size is small, is thermophoresis. This effect arises due to steep temperature gradients in the gas phase. To estimate the thermophoresis force, FT, the following empirical correlation which is valid for a wide range of thermal conductivity ratios and Knudsen numbers is extensively used, FT ¼


6πμ2G d 0 Cs 1 kG =kd þ 2Ct Kn ∇T 1 þ 6Cm Kn 1 þ 2kG =kd þ 4Ct Kn T ρG md

(50)

where μG is the gas viscosity, kG is the gas thermal conductivity, d0 is the droplet/ particle diameter, and kd is the droplet/particle thermal conductivity. Cs, Ct, and Cm are 1.17, 2.18, and 1.14, respectively (Crowe et al. 1998).

4.2.2 Droplet Breakup In Eulerian-Lagrangian approach, two droplet breakup models named as the Taylor analogy breakup (TAB) and the wave breakup models have been extensively applied (Taylor 1963; Reitz 1987; Patterson and Reitz 1999; Jiang et al. 2010). In general, TAB and wave models are appropriate for low and high Weber conditions, respectively. The TAB model is based on the analogy between the oscillating-distorting droplet and a spring-mass system. In this model, the liquid viscosity, liquid surface tension, and aerodynamic force are similar to the damping force, spring restoring force, and external force in the spring-mass system, respectively (Taylor 1963; Jiang et al. 2010). The following equation is solved to calculate the distortion parameter, y, y€ þ

5μL 8σ 2 ρG ð v
uÞ 2 _ y ¼ y þ ρL r 3 3 ρL ρL r 2 r2

(51)

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where r is the droplet radius. The droplet breaks up when y > 1 (Taylor 1963; Jiang et al. 2010). The unstable growth of Kelvin-Helmholtz waves at the liquid-gas interface is considered in the wave breakup model (Reitz 1987; Patterson and Reitz 1999). A dispersion equation which relates the perturbation growth rate, Ω, to the wavelength, Λ, and physical parameters is obtained from the stability analysis. The following correlations are developed for estimating the maximum growth rate and its wavelength,  Ω

ρL a 3 σ

0:5 ¼

0:34 þ 0:38WeG 1:5
 ð1 þ OhÞ 1 þ 1:4J 0:6



   !2 1 þ 0:45Oh0:5 1 þ 0:4J 0:7 Λ ρL ReL 2 ¼ 9:02 , J ¼ 1=
0:6 a ρG WeL 1 þ 0:87WeG 1:67

(52)

(53)

where WeG and Oh are calculated from the parent droplet radius, a. In this model, the child droplets with radius r are formed from a parent droplet with, r ¼ B0 Λ

where ðB0 Λ  aÞ or h
1=3
2 1=3 i r ¼ min 3πa2 U G =2Ω , 3a Λ=4 where ðB0 Λ > aÞ

(54)

where B0 = 0.61. In this model, the child droplet size is assumed to be proportional to the wavelength of the fastest-growing wave. In recent works, the KelvinHelmholtz Rayleigh-Taylor (KHRT) breakup model which is a combination of wave (Kelvin-Helmholtz) breakup model with the Rayleigh-Taylor breakup model is used. In this model, the mechanism (KH vs. RT) that has the shorter breakup time results in the droplet breakup (Reitz 1987; Patterson and Reitz 1999; ANSYS 2011).

4.2.3

Droplet/Particle Heat Transfer (in-Flight Heating and Cooling of Particles) If the thermal conductivity of the droplet/particle material, kd, is much higher than that of the gas kg and the droplet/particle is homogenous, the particle/droplet energy equation can be calculated by using the lumped capacity method (for droplet Biot numbers less than 0.1) (Crowe et al. 1998; Fan and Zhu 1998), md Cp

dT d ¼ Q_ þ m_ d hfg dt

(55)

where Q_ is the convective and radiative heat transfer, Td is the droplet/particle temperature, Cp is the particle phase specific heat, and hfg is the vaporization latent heat.

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The convection heat transfer is formulated, Q_ c ¼ hSðT
T d Þ

(56)

where h is the heat transfer coefficient, which is usually calculated from the Nusselt number, Nu, by using the Ranz-Marshall correlation, Nu ¼

hd 1=3 ¼ 2 þ 0:6Re1=2 , p Pr kG

Re < 200 and 0:5 < Pr < 1:0

(57)

where Pr and Re are the Prandtl number of the gas phase and Reynolds number, respectively (Crowe et al. 1998; Fan and Zhu 1998). Under plasma conditions, due to the steep temperature gradient across the thermal boundary layer surrounding the particle and the strong variations of the gas thermal conductivity with temperature, a question arises about the temperature at which the gas thermal conductivity should be evaluated (e.g., it can be the film temperature or any other temperature). Based on the theoretical analysis of the problem of pure conduction (assuming a Nusselt number of 2.0), Bourdin et al. (1983) suggested that the gas thermal conductivity can be evaluated as an integrated mean value defined by, 2 3 Tð1 1 6 7 kG ¼ 4 kG ðT ÞdT 5 ðT 1
T s Þ 2 ¼ ¼

1 6 4 ðT 1
T s Þ

Ts

Tð1

Tðs

kG ðT ÞdT
T 300

3 7 kG ðT ÞdT5

T 300

1 ½ I ðT 1 Þ
I ðT s Þ ðT 1
T s Þ

(58)

where I(T ) is the heat conduction potential and is given in the literature (Bourdin et al. 1983). It should be noted
that  if kG is a linear function of temperature, the above equation reduces to kG ¼ kG T f where Tf is the mean film temperature. Lewis and Gauvin (1973) developed the following correlation for plasma flow,  
0:15 Nu ¼ 2 þ 0:515Re1=2 νf =ν1 p

(59)

where Re is evaluated at film temperature. Lee et al. (1981) developed the correlation,  ρ μ 0:6 c 0:38 p1 1 1 1=3 Pr Nu ¼ 2 þ 0:6Re1=2 p ρs μ s cps

(60)

where Re and Pr are evaluated at film temperature. Chen et al. (1982) also proposed the correlation,

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0 B Nu ¼ 2@1 þ 0:63Re1 Pr 0:8 1



0   12 1 0:42  0:52 1
hs 1:1 h1 Pr s ρ1 μ 1 B C C @  2 A A Pr 1 ρs μ s hs 1
h1

(61)

where hs and h1 are the specific enthalpy of plasma at the particle surface temperature and plasma temperature, respectively. To account for noncontinuum effects in plasma conditions, Chen and Pfender (1983) developed the following correlation using the temperature jump concept, 2

3

6 q ¼ ðqÞcont: 6 4

7 1 7    5 2
a γ 4  Kn 1þ a 1 þ γ Prs

(62)

where (q)cont. is the heat flux obtained from continuum assumption, and Kn as well as other parameters are defined in equation (49). If the particle phase is assumed to be a diffuse-gray surface, the radiation heat transfer equation would be,
 Q_ r ¼ Sασ b T 41
T 4d

(63)

where α is the absorptivity (from Kirchhoff’s law ε = α where ε is the emissivity), T1 is the ambient temperature, and σ b = 5.6704  10
8W/(m2 . K4) is the StefanBoltzmann constant (Siegel and Howell 1981; Crowe et al. 1998). When the droplet/particle temperature is between the melting and boiling temperatures (Tmelt  Td < Tboil), the vaporization phenomenon should be modeled. In the case of slow vaporization rate, it can be assumed that the process is only controlled by diffusion (ANSYS 2011; Jadidi et al. 2015a),
 N i ¼ hm Ci, s
Ci, 1

(64)

where Ni is the vapor molar flux, hm is the mass transfer coefficient, Ci, s is the vapor concentration at the droplet surface, and Ci, 1 is the vapor concentration in the bulk gas. hm is usually calculated using the Sherwood number correlation, Sh ¼

hm d 0 ¼ 2 þ 0:6Re1=2 Sc1=3 Di, j

(65)

where Di, j is the binary diffusion coefficient, Re is the droplet/particle Reynolds number, and Sc is the Schmidt number (ratio of kinematic viscosity (i.e., momentum diffusivity) and mass diffusivity, Sc = ν/Di, j). The parameter Ci, s in Eq. 64 is usually calculated by,

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Heat Transfer in Suspension Plasma Spraying

C i, s ¼

Psat Ru T d

2955

(66)

where Psat is the saturated vapor pressure at the droplet/particle temperature, and Ru is the universal gas constant. In addition, Ci, 1 is calculated by solving the transport equation for species. In the case of high vaporization rate, vapor diffusion and convection from the droplet/particle surface to the surrounding gas should be considered. The following correlation is suggested to estimate the vaporization rate (Sazhin 2006), dmd Y i, s
Y i, 1 ¼ hm SρG lnð1 þ Bm Þ, Bm ¼ dt 1
Y i, s

(67)

where Yi, s and Yi, 1 are the vapor mass fraction at the droplet/particle surface and in the bulk gas, respectively, and Bm is the Spalding mass number. When the boiling temperature is reached, the droplet/particle energy equation becomes,



 dmd hfg ¼ hSðT
T d Þ þ Sασ b T 41
T 4d dt

(68)

In the case of high vaporization rate, the convection heat transfer coefficient can be computed by, Nu ¼

 hd lnð1 þ BT Þ  ¼ 2 þ 0:6Re1=2 Pr 1=3 kG BT

(69)

where BT is the Spalding heat transfer number. It is usually assumed that BT = Bm (ANSYS 2011; Jadidi et al. 2015a).

4.2.4 An Approach to Model the Suspension Droplets Jabbari et al. (2014) used an Eulerian-Lagrangian approach with two-way coupling assumption to model the suspension plasma spray processes. The suspension droplets (containing nickel (15 wt.%) and ethanol) were radially injected into an atmospheric plasma jet. The injector diameter was 0.15 mm. Suspension with a given particle concentration (i.e., slurry droplet) was modeled as a multicomponent droplet carrying properties of solid particles and base liquid (see Fig. 19). In other words, one component is the base liquid (e.g., ethanol) and another one includes the particle properties such as density and latent heat of evaporation. The KHRT breakup model was used to simulate the droplet breakup. After completion of the suspension breakup and evaporation, the solid particles were tracked through the computational domain to determine the characteristics of the coating particles (see Fig. 19). To evaluate the particle temperature before its melting point, the specific heat of solid particle was used. The particle melting phenomenon was approximated by the following formula:

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Fig. 19 Suspension droplet evolution in realistic and model cases (Jabbari et al. 2014)

Cp ΔT ¼ H f

(70)

where Cp is the estimated particle specific heat, ΔT is assumed to be 10 K, and Hf is the heat of fusion. After melting, the specific heat of molten particle was used (see also Jadidi et al. 2015b, 2016a; Pourang et al. 2016; Dalir 2016). The temperature evolution of a 40 μm suspension droplet (containing zirconia (10 wt.%) and ethanol) in a steady undisturbed plasma flow (argon and hydrogen (10% volume fraction) jet with mass flow rate of 1.48 g/s and arc current and voltage of 500 A and 65 V, respectively) was analyzed using the abovementioned model (Pourang 2015). The droplet was located at the center of the nozzle exit plane and the breakup phenomenon was not considered. Figure 20 illustrates the evolution of suspension droplet and particle as a function of time while it is traveling inside the plasma flow. It is shown that after ethanol evaporation, the particle temperature increases and then decreases due to its trajectory along the jet centerline. Figure 20b shows that the melting state of the particle can be captured by the above assumption. The velocity, location, and angle of injection of the suspension have an important effect on its penetration in the plasma jet and ensuing particle inflight behavior. For example, Jabbari et al. (2014) showed that the penetration of suspension containing nickel (10 wt.%) and ethanol increases as the injection momentum increases. However, when the liquid/gas momentum ratio was too high, the plasma jet cooled down severely, and the resulting number of high-temperature particles decrease. Figure 21 illustrates the effect of injection angle and location on the suspension penetration and plasma temperature. If the injector is near the torch exit and its angle is toward the torch, the suspension penetrates more efficiently the plasma jet resulting eventually in higher particle velocity and temperature (Jabbari et al. 2014).

5

Heat Transfer to the Substrate

Heat is transferred to the substrate during spraying due to the impingement of the hot gas jet and spray particles. The heat transfer mechanisms are similar in SPS to that in air plasma spray (APS) and other thermal spray processes (Mariaux et al. 2003). However, the heat transfer to the substrate from the hot gas jet is much higher in SPS

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Heat Transfer in Suspension Plasma Spraying

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Fig. 20 Evolution of the overall temperature of a suspension droplet (containing zirconia (10 wt.%) and ethanol) as function of time (Pourang 2015)

Fig. 21 Effects of injector location and angle on the suspension penetration and plasma temperature (the suspension injection velocity is 25.7 m/s) (Jabbari et al. 2014)

than APS because of the shorter spray distances used in SPS (3–5 cm) as compared to APS (7–12 cm). The heat flux brought by the plasma jet is generally higher than 10 MW/m2 in SPS while it is below 2 MW/m2 at the spraying distances used in APS. In all spray processes, the molten or semi-molten particles transfer energy to the substrate from their kinetic energy upon impact, specific heat and heat of

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solidification. Although the size of the spray particles is smaller in SPS than that in APS, the energy transfer is expected to be in the same range as their spray rate and impact velocity are comparable. One distinctive characteristic of SPS is the actual particle trajectories around the substrate as discussed below.

5.1

Particle Trajectories Around the Substrate

In SPS, the diameter of molten particles is in the range of 0.1–4 μm. Due to their low inertia, these particles tend to be decelerated and deflected by the presence of the substrate (Oberste-Berghaus et al. 2005). To provide detailed information on the coating particles upon impact, the influence of flat substrates located at various standoff distances from the torch exit was investigated by Jadidi et al. (2015b) (suspension droplet containing nickel (15 wt.%) and ethanol, and particles in the range of 0.5–3.5 μm were considered). In this study, it was assumed that when particles hit the substrate, they stick on it. It was found that, by adding the substrate in the domain, the Ar plasma temperature near the substrate location increases due to less mixing of air and plasma. Moreover, it was shown that many small particles are decelerated and get diverted by the stagnation region near the substrate location. Noting that the particles that move close to the plasma jet centerline are less affected by the stagnation region (see Fig. 22). The effect of Stokes number on the particle inflight behavior near the substrate was also investigated (see Fig. 23). It was revealed that particles with low Stokes number (St = 0.1) follow the gas streamlines and decelerate dramatically. On the other hand, particles with high Stokes number (St = 2.1) decelerate gradually. They concluded that, particles with low Stokes number coat the substrate at large radial distance from the centerline, with low velocity (Jadidi et al. 2015b). The effect of substrate shape and curvature on particle inflight behavior was studied by Pourang et al. (2016). They developed the works of Jabbari et al. (2014) and Jadidi et al. (2015b) by adding a constant volumetric heat source (Eq. 14) in the plasma flow energy equation. Flat and cylindrical substrates at different standoff distances were considered. They showed that the substrate shape has significant influences on the particles trajectory and velocity near the substrate (see Fig. 24).

Fig. 22 Nickel particle velocity and trajectory at different standoff distances (D) (Jadidi et al. 2015b)

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Heat Transfer in Suspension Plasma Spraying

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Fig. 23 Effects of Stokes number on particle inflight behavior near the substrate (using normalized diameter and velocity, D is the distance between the torch exit and the substrate, d is the distance from the substrate, di = 7.88 mm, and v0 = 1800 m/s) (Jadidi et al. 2015b)

Fig. 24 The effect of cylindrical substrate on Yttria-stabilized zirconia particle temperature and trajectory (standoff distance is 40 mm) (Pourang et al. 2016)

Based on the assumed operating conditions, it was revealed that, in a fixed time interval, particles struck the flat substrate 2.2 times more often than observed on the cylindrical substrate (Pourang et al. 2016). Therefore, they introduced a new parameter called catch rate, which shows the maximum possible deposition rate, as follows:

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Catch rate ð%Þ ¼

mass of landed particles in Δt  100 mass of injected particles in Δt

(71)

Based on the operating conditions assumed in their paper, they explained that catch rates on the flat and cylindrical substrates were 23 and 11%, respectively. In other words, the deposition rate on the curved substrates was found to be lower (Pourang et al. 2016). The model presented by Jabbari et al. (2014) as well as the approach of variable conical/cylindrical zones (Eqs. 15, and 16) was employed by Dalir (2016) to study the effect of plasma fluctuations on the particle trajectory, velocity, and temperature in the suspension plasma spray processes. It was found that the suspension penetration height significantly changes due to the plasma jet fluctuations. It was also shown that as the injection velocity increases, the changes of suspension penetration height become more severe. Moreover, it was found that particle trajectory, temperature, and velocity are strongly dependent on time since the gas temperature and velocity change with time due to plasma fluctuations (Dalir 2016).

6

Conclusion

Suspension plasma spray (SPS) is an innovative process in which a suspension of submicron size particles is injected in a DC plasma jet. The suspension is rapidly atomized by the high speed plasma jet forming suspension droplets that, in turn, are heated by the plasma freeing the suspended particles that are then molten and propelled toward a substrate by the plasma jet. The atomization of the suspension plays a fundamental role in the final spray particle characteristics as it directly influences the droplet size and trajectories in the plasma jet and then the heat and momentum transfer to the spray particles. This chapter focused on the mechanical and thermal interactions of the injected suspension with a DC plasma jet. 3D modeling of these interactions makes it possible to develop a better understanding of the influence of the spray parameters on the spray particle characteristics that combined with the substrate properties (material, surface roughness and cleanliness, temperature, standoff distance, etc.) determine the coating characteristics. This is an ongoing journey as the physical and chemical transformations occurring during spraying are complex.

7

Cross-References

▶ A Prelude to the Fundamentals and Applications of Radiation Transfer ▶ Analytical Methods in Heat Transfer ▶ Droplet Impact and Solidification in Plasma Spraying ▶ Electrohydrodynamically Augmented Internal Forced Convection

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▶ Full-Coverage Effusion Cooling in External Forced Convection: Sparse and Dense Hole Arrays ▶ Fundamental Equations for Two-Phase Flow in Tubes ▶ Heat Transfer in DC and RF Plasma Torches ▶ Macroscopic Heat Conduction Formulation ▶ Plasma-Particle Heat Transfer ▶ Radiative Plasma Heat Transfer ▶ Radiative Properties of Gases ▶ Radiative Properties of Particles ▶ Synthesis of Nanosize Particles in Thermal Plasmas ▶ Thermophysical Properties Measurement and Identification ▶ Turbulence Effects on Convective Heat Transfer

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Droplet Impact and Solidification in Plasma Spraying

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Javad Mostaghimi and Sanjeev Chandra

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Droplet Impact, Spread, and Solidification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Axisymmetric Impact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Droplet Splashing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Photographing Plasma Particle Impact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Splat Shapes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Mathematical Model of Impact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Fluid Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Interface Tracking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Heat Transfer and Solidification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Initial and Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Simulations of Droplet Impact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Smoothed Particle Hydrodynamics (SPH) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Coating Buildup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Porosity Formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Modeling Coating Formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

Plasma coatings are built up by agglomeration of splats formed by the impact, spread, and solidification of individual particles. Coating microstructure is determined by fluid flow and heat transfer during droplet impact. Coating properties such J. Mostaghimi (*) Centre for Advanced Coating Technologies, Department of Mechanical and Industrial Engineering, Faculty of Applied Science + Engineering, University of Toronto, Toronto, ON, Canada e-mail: [email protected] S. Chandra Department of Mechanical and Industrial Engineering, University of Toronto, Toronto, ON, Canada e-mail: [email protected] # Springer International Publishing AG, part of Springer Nature 2018 F. A. Kulacki (ed.), Handbook of Thermal Science and Engineering, https://doi.org/10.1007/978-3-319-26695-4_78

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as porosity, adhesion strength, and surface roughness depend on the shape of splats and how they bond together and to the substrate. The splat shape is dependent on material properties of the powder, impact conditions (impact velocity and temperature), and substrate conditions (substrate roughness, material, temperature, and thermal contact resistance between the droplet and substrate). Coating adhesion strength increases by almost an order of magnitude as surface temperature is raised from room temperature to 650
C. Increasing substrate temperature also changes the shape of splats formed by solidified droplets after impact on the surface. On a cold surface, there is splashing and droplet breakup, while splats on a hot surface are circular. Particle impact dynamics depends on the rate at which a droplet solidifies during impact, which is a function of substrate temperature, thermal contact resistance, and initial droplet temperature. Heating the surface affects droplet impact dynamics by changing thermal contact resistance, decreasing it by removing volatile compounds adsorbed on the surface. The trajectory of particles within the plasma and their residence time in the high-temperature zone determines their state at the point of impact: particles may be fully or partially melted with a few still completely solid.

1

Introduction

Plasma coatings are built up by agglomeration of splats formed by the impact, spread, and solidification of individual particles. Figure 1 shows a schematic diagram of a DC plasma spraying process. Inspection of a plasma coating cross section (Fig. 2) reveals that it is built up of thin lamellae formed by flattened droplets that land on each other and fuse together. These coatings are not fully dense since voids may be present at the interface between splats. To ensure strong adhesion of a thermal spray coating, it is necessary to carefully prepare the substrate. Typically, the surface is roughened by grit blasting. Mechanical interlocking between solidified droplets and protrusions on the substrate produces strong bonding. Coating microstructure is highly dependent on fluid flow and heat transfer during droplet impact and is strongly affected by surface temperature.

Fig. 1 Schematic of a DC plasma spray coating process

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Fig. 2 A typical cross section of nickel-sprayed plasma coating (a), with higher magnification (b)

The effect of heat transfer and substrate condition on adhesion strength of plasmasprayed coatings was demonstrated by Pershin et al. (2003). They sprayed nickel powder onto a stainless steel substrate and found that coating adhesion strength increased by almost an order of magnitude as surface temperature was raised from room temperature to 650
C. Several explanations were offered: heating the surface removes volatile contaminants adsorbed on the surface, improving contact between impinging particles and the substrate, reducing the solidification rate of droplets allows them to flow into surface cavities before freezing, and enhancing mechanical bonding. The most visible effect of increasing substrate temperature, though, was to change the shape of splats formed by solidified droplets after impact on the surface. Figure 3 shows micrographs of splats produced by spraying nickel powder, sieved to give a size distribution of +63–75 μm, onto stainless steel surfaces maintained at either 290
C (Fig. 3a) or 400
C (Fig. 3b). Particle temperature in-flight was measured to be 1600  220
C and impact velocity 73  9 m/s. On the colder surface, there was evidence of splashing and droplet breakup, while splats on the hotter surface were circular. The effect of substrate temperature on splat shape has been well established in a number of studies, reviewed in detail by Fauchais and Fukumoto (2004). Fukumoto and Ohgitani (2004) performed a statistical analysis of splat shapes deposited on a surface and defined a “transition temperature” (Tt) as the substrate temperature where half of the splats on the surface were circular without splashing. Other researchers also observed this change of splat shape and showed that the transition temperature was a complex function of particle and substrate material properties (Zhang and Wang 2001), surface contamination (Li et al. 1998), and surface oxidation (Pech and Hannoyer 2000). Jiang and Wan (2001) plasma sprayed molybdenum onto polished stainless steel coupons and found that increasing impact velocity enhanced splashing; removing adsorbed volatile compounds on the surface reduced splashing. Fukomoto and Huang (1999) suggested that freezing along the bottom of an impinging droplet causes splashing: liquid flowing on top of the solid layer jets off and splashes.

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Fig. 3 Splats formed by spraying molten nickel particles on a stainless steel surface initially at (a) 290
C and (b) 400
C. The particle size distribution was 53 + 63 μm, particle temperature before impact 1600  220
C, velocity 73  9 m/s

Particle impact dynamics depends on the rate at which a droplet solidifies during impact, which is a function of the heat flux from the molten droplet to the substrate. When molten metal comes suddenly in contact with a rough, solid surface, air may be trapped in crevices at the liquid-solid interface, creating a temperature difference between the molten metal and the substrate, whose value depends on surface finish, contact pressure, and material properties. To quantify the magnitude of this effect, the thermal contact resistance (Rc) is defined as the temperature difference between the droplet (Td) and substrate (Ts) divided by the heat flux (q00 ) between the two: Rc ¼

Td  Ts q00

(1)

Droplet solidification rate is therefore a function not just of substrate temperature but also of contact resistance and initial droplet temperature. Heating the surface indirectly affects droplet impact dynamics by changing thermal contact resistance, either decreasing it by removing volatile compounds adsorbed on the surface or possibly increasing it in the case of metallic substrates heated in air, due to the formation of an oxide layer. If nickel particles are plasma sprayed onto a steel surface that is at room temperature, they will splash, but not on a surface that is maintained at 400
C; however, splashing is also suppressed on a surface that is heated to 400
C in air, oxidized, and then cooled (Pershin et al. 2003). Computer simulations of impacting molten metal droplets, Mehdizadeh and Raessi (2004) provide insight into a mechanism for solidification-induced splashing. A spreading drop begins to freeze along its edges, where it first contacts the colder substrate. The solid rim formed obstructs further flow, forcing liquid to jet off the

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surface so that it becomes unstable and breaks up into satellite droplets. Reducing heat transfer from the droplet slows solidification and allows the droplet to spread into a disk before freezing. It was found in simulations that the rate of solidification was much more sensitive to values of thermal contact resistance than substrate temperature. Simulations of impact of nickel particles, Mostaghimi et al. (2002) showed that raising substrate temperature from 290
C to 400
C had little effect on impact dynamics, but increasing thermal contact resistance from 107 to 106 m2K/W diminished heat transfer sufficiently to prevent splashing. An oxide layer or adsorbed contaminants on the surface may, in practice, alter thermal contact resistance. The state of particles at the point of impact is important in the type of microstructure the coating will have, and it is dependent on the trajectory of particles and their residence time within the plasma. Thus, the particles may be fully or partially melted with a few still completely solid. Coating properties such as porosity, adhesion strength, and surface roughness depend on the shape of these splats and how they bond together and to the substrate. The splat shape is dependent on material properties of the powder, impact conditions (e.g., impact velocity and temperature), and substrate conditions, e.g., substrate roughness, material, temperature, and contact resistance. The next few sections will review the dynamics of impact and solidification in thermal spray processes and the effect of impact parameters on the final shape of splats.

2

Droplet Impact, Spread, and Solidification

Individual splats are the building blocks of thermal spray coatings. The shapes of these splats are a function of particle impact conditions, physical properties of the powder and substrate, and substrate temperature, roughness, and chemistry. To better understand coating formation, we need to investigate the following topics: 1. Relationship between the final splat shape and impact parameters, thermophysical properties of the powder, substrate thermal properties, and substrate roughness 2. Splashing and breakup of impacting fully or partially molten droplets 3. Interaction of splats on the substrate Prediction of splat shapes involves numerical simulation of fluid flow and heat transfer of an impacting droplet. In general, this is a three-dimensional, timedependent problem. One challenge is the prediction of rapid and large deformations of impacting droplets and their simultaneous solidification on the substrate.

2.1

Axisymmetric Impact

Consider the isothermal normal impact of a spherical droplet on a smooth, flat surface, as shown in Fig. 4. Furthermore, assume the gas phase is passive and does not influence the impact. The parameters that affect the impact are initial

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Fig. 4 Schematic of droplet impact

V0, D0, T0

droplet diameter D0, impact velocity V0, droplet density ρ, liquid viscosity μ, liquidgas surface tension σ, and liquid-solid contact angle θ. Combining these into nondimensional groups reduces the number of variables to three: contact angle and the Reynolds and the Weber numbers, Re ¼

ρV o D0 μ

,

We ¼

ρV 20 D0 σ

(2)

There have been many successful attempts to derive analytical expressions for the extent of maximum spread, ξmax = Dmax/D0, as a function of process variables (Madejski 1976; Pasandideh-Fard pffiffiffiffiffiffiet al. 1996). In the absence of solidification, and with the condition of We  Re and We  12 , which is normally the case in spray coating process, a simple formulation for the degree of maximum spread is obtained, ξmax ¼ a Reb

(3)

where a = 1.293, b = 1/5 (Madejski 1976), or a = 0.5, b = 1/4 (Pasandideh-Fard et al. 1996). Pasandideh-Fard et al. (1998) developed a simple model to predict the maximum spread diameter of an impacting droplet. In this model, they equated the energy before and after impact, accounting for the energy dissipation during impact. The initial kinetic energy (KE1) and surface energy (SE1) of a liquid droplet before impact are,   1 2 π 3  ρV D KE1 ¼ 2 o 6 o

(4)

SE1 ¼ πD2o σ

(5)

After impact, when the droplet is at its maximum extension, the kinetic energy is zero, and the surface energy (SE2) is approximately,

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π SE2 ¼ D2max σ ð1  cos θa Þ 4

(6)

where θa is the advancing liquid-solid contact angle. The work done in deforming the droplet against viscosity (W ) is approximately, π 1 W ¼ ρV 2o Do D2max pffiffiffiffiffiffi 3 Re

(7)

The effect of solidification in restricting droplet spread is modeled by assuming that all the kinetic energy stored in the solidified layer is lost. If the solid layer has average thickness s and diameter ds when the splat is at its maximum extension, then the loss of kinetic energy (ΔKE) is approximated by, ΔKE ¼

π 4

d 2s s

1 2

 ρV 2o

(8)

ds varies from 0 to Dmax during droplet spread: a reasonable estimate of its mean value is ds ~ Dmax/2. Substituting Eqs. 4–8 into the energy balance, KE1 + SE1 = SE2 + W + ΔKE yields an expression for the maximum spread factor, ξmax

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Dmax u We þ 12 u ¼ ¼u 3 We t Do Wes þ 3ð1  cos θa Þ þ 4 pffiffiffiffiffiffi 8 Re

(9)

where s* is the dimensionless solid layer thickness (s* = s/Do). There are two unknowns in Eq. 9: advancing contact angle (θa) and solidified layer thickness (s*). Liquid-solid contact angles during spreading and recoil of tin droplets on a stainless steel were measured from enlarged photographs by Aziz and Chandra (2000), and the advancing contact angle was found to be almost constant at θa = 140
. The growth in thickness of the solidified layer can be calculated using an approximate analytical solution developed by Poirier and Poirier (1994). The model assumes that heat transfer is by one-dimensional conduction; there is no thermal contact resistance at the droplet-substrate interface; the temperature drop across the solid layer is negligible; the substrate is semi-infinite in extent and has constant thermal properties. The dimensionless solidification thickness was expressed as a function of the Stefan number (Ste = C(Tm  Tw, i)/Hf, where Tm is the melting temperature of the droplet, Tw,I the initial substrate temperature, and Hf the latent heat of fusion), Peclet number (Pe = VoDo/a), and γ = kρC, 2 s ¼ pffiffiffi Ste π 

sffiffiffiffiffiffiffiffiffiffi t γ w Peγ d

(10)

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J. Mostaghimi and S. Chandra 7.0

6.0 maximum spread factor, ξmax

Fig. 5 Calculated (lines) and measured (symbols) variation of maximum spread factor with impact velocity for 2.0mm-diameter tin droplets landing on a stainless steel surface with initial temperature Tw,i

5.0

Tw,i = 240 °C Tw,i = 25 °C

Equation (10) with no solidification

4.0

3.0 Equation (10) with solidification

2.0

1.0 0.0

1.0

2.0 3.0 velocity (m/s)

4.0

5.0

The nondimensional time to arrive at the maximum spread is shown by Pasandideh-Fard et al. (1998) to be approximately t* = 8/3. Substituting Eq. 10 into Eq. 9 gives the maximum spread of a droplet that is solidifying during impact, ξmax

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u We þ 12   rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼u u t 3γ w We We Ste þ 3ð1  cos θa Þ þ 4 pffiffiffiffiffiffi 2πPeγ d Re

(11)

The variation of ξmax with impact velocity predicted by Eq. 11 for droplets falling on a substrate at 25
C is shown in Fig. 5, along with measured values. Predictions of ξmax from Eq. 11, for a droplet spreading without solidifying, are also compared with measurements for droplets impacting a surface at 240
C. Agreement between measured and calculated values is good in both instances. At low impact velocity, Eq. 11 predicts somewhat larger values of ξmax than were measured. To estimate viscous dissipation, the model assumes that there exists a thin boundary layer in the drop which is not true when the droplet is deposited very gently. The effect of solidification on droplet spreading can be estimated from Eq. 11. In thermal spraying, the second term in the denominator of Eq. 11 is negligible. The ratio of the first to last term in the denominator will provide a measure of the importance of solidification on the dynamics of droplet impact. Aziz and Chandra (2000) proposed that if the following nondimensional parameter is much less than unity, solidification effect on the extent of droplet spread is negligible,

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Droplet Impact and Solidification in Plasma Spraying

Ste Φ ¼ pffiffiffiffiffi Pr

rffiffiffiffiffiffiffiffi γw 1 γd

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(12)

The above analytical relations are quite useful in approximately describing the relation between maximum spread and impact variables, and they provide some information about potential breakup upon impact. Modeling breakup and splashing and interaction of splats on the substrate require more detailed numerical models. To better understand the dynamics of impact, spread, and solidification, a number of two-dimensional, axisymmetric models were initially developed. Zhao et al. (1996a, b) studied, both experimentally and numerically, heat transfer and fluid flow of an impacting droplet. Solidification was not considered in this work. Bennet and Poulikakos (1994) and Kang et al. (1994) studied droplet deposition assuming solidification to start after spreading is completed. As discussed above, the validity of this assumption depends on both Prandtl and Stefan numbers. Liu et al. (1993), Bertagnolli et al. (1995), and Trapaga et al. (1992) numerically studied solidification and spreading of the impacting drops. The substrate was assumed to be isothermal. Additionally, the important effect of thermal contact resistance between the drop and the substrate was ignored. The liquid-solid contact angle was also considered to be constant, with an arbitrarily assigned value. Pasandideh-Fard et al. (1996), however, showed that the value of contact angle can have a significant effect on the results. Pasandideh-Fard and Mostaghimi (1996) studied the effect of thermal contact resistance between the droplet and the substrate. They showed that its magnitude could have a dramatic effect on droplet spreading and solidification. Solidification and heat transfer within the substrate were modeled assuming one-dimensional heat conduction. The model was later completed, and a fully two-dimensional axisymmetric model of droplet impact was developed, and impact and solidification of relatively large tin droplets (~2 mm diameter) on stainless steel substrates were studied both numerically and experimentally (Pasandideh-Fard et al. 1998). The model correctly predicted the shape of the deforming droplet. The values of thermal contact resistance were estimated by matching the numerical predictions of substrate temperature with those measured experimentally. While thermal contact resistance should, in principle, vary at different contact points, it was shown that accurate simulations of the impact could be done using a constant value. The results also showed the sensitivity of the predicted maximum spread to the value of thermal contact resistance. A few experimental studies have investigated the impact of molten droplets. Inada and Yang (1994) measured the temperature variation of a plate on which a molten lead droplet was dropped and concluded that the droplet cooling rate was a function of impact velocity. Watanabe et al. (1992) photographed impact of n-cetane and n-eicosane droplets on a cold surface and concluded that in their tests, droplets spread completely before solidifying. Fukanuma and Ohmori (1994) photographed the impact of tin and zinc droplets and also found that freezing had no influence on droplet spread. Inada and Yang (1994) used holographic interferometry to observe droplet-substrate contact during impact of lead droplets on a quartz plate.

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Liu et al. (1995) measured the temperature variation on the upper surface of an impacting metal droplet by a pyrometer and used these results to estimate the thermal resistance under the drop. However, the response time of the pyrometer (25 ms) was longer than the time taken by the droplet to spread, so that their results are applicable to the period after the droplet had come to rest rather than the duration of the impact itself. Pasandideh-Fard et al. (1998) photographed the impact of tin droplets on a stainless steel substrate and measured the changes in substrate temperature during the impact. They showed that the value of the maximum spread is sensitive to the magnitude of thermal contact resistance, which in their case was estimated from the measurements.

2.2

Droplet Splashing

When a droplet impacts a solid surface, it spreads into a thin circular sheet. If the impact velocity is sufficiently large, fluid instabilities create undulations around the edge of the spreading sheet that grow larger and form fingers. The fingers detach and form satellite droplets, a process that is commonly known as “splashing.” The first experimental study of droplet fingering and splashing – in the absence of solidification – was that of Worthington (1876, 1907) which was published over a century ago. He observed that the number of fingers increased with droplet size and impact speed, observed merging of the fingers at or soon after the maximum spread, and found fingering to be more pronounced for fluids that did not wet the substrate. Many researchers have since contributed to the understanding of the fingering and splashing in the absence of solidification. A review of their findings may be found in the works of Bussmann et al. (1999, 2000) and Bussmann (1999). Aziz and Chandra (2000) proposed a simple model based on the Rayleigh-Taylor instability and showed that in the absence of solidification, the number of fingers around the impacting droplet is, K N ¼ pffiffiffi 4 3

(13)

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 u 0 u uWe B C We þ 12 u  C rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi N¼u B @ t 12 3γ w We A We Ste þ 3ð1  cos θa Þ þ 4 pffiffiffiffiffiffi 2πPeγ d Re

(14)

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffi where K ¼ We Re is the so-called splash parameter. The above derivation ffiffiffiffi  1 . These two conditions are satisfied in the assumes that We  12 and pWe Re thermal spray coating process. When solidification is included in the analysis, the number of fingers at the maximum spread is,

Droplet Impact and Solidification in Plasma Spraying

Fig. 6 Calculated (lines) and measured (symbols) variation of maximum spread factor with impact velocity for 2.0mm-diameter tin droplets landing on a stainless steel surface with initial temperature Tw,i (Aziz and Chandra)

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50 45 40 number of fingers, N

69

35

Tw,i = 25 °C Tw,i = 240 °C

Equation (13)

30 25 20 15 Equation (14)

10 5 0 0.0

1.0

2.0 3.0 velocity (m/s)

4.0

5.0

Figure 6 shows a comparison of the predicted number of fingers by Eqs. 13 and 14 and experiments for two substrate temperatures. Bussmann et al. (2000) developed a three-dimensional model for the isothermal impact of a droplet on a solid surface. The model employs a fixed-grid Eulerian approach along with a volume tracking algorithm to track fluid deformation and droplet-free surface. Pasandideh-Fard et al. (2002a) extended this model to include heat transfer and solidification. This model is described in the next section. Some of the difficulty in predicting when splashing will occur can be attributed to uncertainties about surface wettability and the effect of the surrounding atmosphere. However, the term “splashing” has been used to refer to several different mechanisms that lead to breakup of droplets after impact. Rioboo et al. (2001) identified three different types of splashing. Immediately after impact, as the liquid sheet under the droplet spreads out, its edge becomes unstable and fingers around the edge begin to break off and form small droplets. This has been termed “prompt splash” and occurs when the edge of the lamella is still in contact with the surface. The second type of splashing has been termed “corona” splashing: the liquid lamella lifts off the surface; the edge becomes unstable so that fingers grow at regular spaced intervals and their tips break off in the crown-like shape characteristic of splashing drops. Many studies have been devoted to predicting when corona splashes will occur. Mundo et al. (1995) found that droplets splashed only if the so-called splash parameter K = We1/2Re1/4 exceeds a critical value K = 57.7. Cossali et al. (1997) developed an empirical correlation between K, surface roughness Ra, and the liquid lamella thickness h.

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The air film trapped under the impacting droplet plays an important role in creating instabilities. Xu et al. (2005) demonstrated that if the pressure in the atmosphere surrounding an impacting drop is reduced, corona splashes are suppressed. Prompt splashing, however, persists even in the absence of surrounding gas (Xu et al. 2007). The third type of splashing is known as “receding breakup,” in which the droplet remains intact until it has spread to its maximum extent, and then, as surface tension forces pull it back, the fingers formed due to instabilities around its periphery grow longer and begin to break up into smaller droplets. If the liquid-solid contact angle is small, less than 90
, neighboring fingers along the edges of the spreading liquid sheet tend to merge with each other and disappear. However, if the contact angle is large, as is the case with droplets of molten metal, the cylindrical fingers become unstable and disintegrate. Apart from these three mechanisms, there are two others that can cause breakup of impacting droplets. If a droplet impacts on a substrate that is cold enough to cause freezing, the solid layer formed at the liquid-substrate interface acts as a barrier. The spreading liquid hits the solid mass obstructing its path, jets upward, and disintegrates. This is known as freezing-induced splashing (Dhiman and Chandra 2005) and whether it occurs depends on the rate of heat transfer between the droplet and substrate, which is controlled by the substrate temperature, substrate thermal properties, and the thermal contact resistance at the liquid-solid interface. Finally, if impact velocities are very high, the liquid film may become very thin so that air trapped under it breaks through. These holes in the liquid grow larger and can eventually lead to complete disintegration of the droplet (Mehdizadeh et al. 2005).

2.3

Photographing Plasma Particle Impact

McDonald et al. (2006) photographed the impact of plasma-sprayed molybdenum and amorphous steel particles (38–55 μm diameter) during impact (velocity 120–200 m/s) and spreading on a smooth glass surface that was maintained at either room temperature or 400
C. Figure 7 shows a schematic diagram of the experimental apparatus used. A plasma torch was passed rapidly across the glass substrate that was protected from heat by a series of barriers with holes in them through which a few particles could pass. After exiting the third barrier and just before impacting the substrate, the thermal radiation of the particle was measured with a rapid two-color pyrometric system consisting of an optical sensor head which focused the collected radiation on an optical fiber covered with a mask that was opaque except for three slits (see Fig. 8). The two smaller slits (slits b and c in Fig. 8a) were used to detect thermal radiation emitted by particles while they were still in flight, from which their temperature, velocity, and diameter could be calculated. The largest slit (slit e in Fig. 8a) was used to collect thermal radiation of the particle as it impacted and spread on the substrate. With the thermal radiation from this slit, the splat temperature, diameter, and cooling rate were calculated at 100 ns intervals after impact. Figure 8b shows a typical signal captured by a photodetector. The labels, a–f, correspond to the position of a particle (shown in Fig. 8c) as it passes through the fields of view of each of the optical slits. At points a and d, the particle was not in the

Droplet Impact and Solidification in Plasma Spraying

2979

Fig. 7 Schematic of the experimental assembly [G]

69

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a 180 μm

b 30 μm

300 μm

150 μm

Signal (V) 30 μm 30 μm

150 μm

a

bc d

e f

800 μm

Time (μs)

c

In-flight Particle

a b

c

d

e 28°

f

Splat

320 μm

Substrate

Fig. 8 (a) Details of the three-slit mask, (b) a typical signal collected by the three-slit mask, (c) schematic of the optical detector fields of view [G]

optical field of view of any of the slits, so the signal voltage was zero. The two peaks at points b and c were produced by thermal emissions from the particle as it passed through the first two small slits. The droplet average in-flight velocity was calculated by dividing the known distance between the centers of the two slits by the measured time of flight. At point e, the droplet entered the field of view of the third and largest optical slit. This is shown on the thermal signal by a plateau in the profile. Upon impact at f, the signal increased as the particle spread and eventually decreased as the particle cooled down and/or splashed out of the field of view. A CCD camera was used to capture images of the spreading particles from the back of the glass substrate. The electronic shutter of the camera was triggered to open by a signal from the flashlamp of the laser.

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Droplet Impact and Solidification in Plasma Spraying

2981

Figure 9a shows images of molybdenum splats at different times after impact on glass held at room temperature or at 400
C. The figure also shows the corresponding thermal emission signals. For molybdenum, the average droplet diameter was 40 μm, the average impact velocity was 135 m/s, and the average temperature of the particles in flight was 2980
C, well above the melting point (2617
C). The photodetector signal of impact and spread on the glass held at room temperature was subdivided into four intervals (indicated by labels a–e in Fig. 9a), and photographs taken in each of these time periods are grouped together in Fig. 9. The approximate time after impact that corresponds to each interval is shown in the figure. To demonstrate the repeatability of the process, two splat images are shown during each time interval. The a to b range represents splats immediately before or upon achieving the maximum spread diameter of 400 μm. Beyond point b, the liquid portion of the splats begins to disintegrate, initially from the solidified central core, and later, from sites within the liquid film. After point d, the splat is almost totally disintegrated and only a central solidified core remains on the glass. Figure 9b shows the results after impact on a glass substrate at 400
C. There was almost no splat breakup or splashing, unlike that seen in Fig. 9a. The time required for the splat to spread to its maximum diameter after impact was measured starting at the instant the pyrometric thermal emission signals began to increase after the plateau (point f of Fig. 9b) to the maximum voltage on the thermal emission signal profile. For molybdenum on glass held at room temperature, the maximum spread time was 2 μs and on glass held at 400
C, it was 1 μs. The evolution of the liquid temperature during the spreading of molybdenum splats on cold and hot glass is illustrated in Fig. 10. The average slope of the curves (dT/dt) represents the splat cooling rate. The liquid cooling rate on a glass surface held at 400
C is approximately an order of magnitude larger (~108 K/s) than on a surface held at room temperature (~107 K/s) demonstrating that thermal contact resistance between the splat and the cold glass is greater than that between it and the hot glass. The cause of the increased thermal contact resistance on the cold surface is probably a gas barrier, formed after evaporation of adsorbed substances on the substrate beneath the splat. Heating the surface removes the adsorbed substances, producing better contact. McDonald et al. (2007) developed a one-dimensional heat conduction model to estimate the magnitude of thermal contact resistance and estimated it to be 4.9  105 m2K/W for molybdenum splats impacting a glass surface at room temperature, while it was 6.5  107 m2K/W for a glass surface at 400
C. Table 1 summarizes the results of these experiments.

2.4

Splat Shapes

Impacting plasma spray particles may fragment due to two different mechanisms. If the thermal contact resistance under the splat is very low, and cooling is very rapid, it begins to solidify as it spreads. The solid layer obstructs and destabilizes the flow of liquid, leading to fingers being formed around its edges. At the other extreme, if contact resistance is very high, the particle remains liquid and spreads into a very

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Fig. 9 Typical thermal emission signals and images of molybdenum splats at different times after impact on glass held at (a) room temperature and (b) 400
C [G]

thin film that ruptures internally. In this case, the splat is also fragmented, but its shape is different, appearing as a small central core surrounded by a ring. Diskshaped splats are formed if the value of thermal contact resistance lies between these two extremes, so solidification starts after the particle has already flattened out and does not obstruct the liquid flowing outward, but is still sufficiently rapid to prevent the splat from spreading so thin that it ruptures internally.

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Droplet Impact and Solidification in Plasma Spraying

2983

Fig. 10 Typical cooling curves of molybdenum splats on glass held at (a) room temperature and (b) 400
C [G]

Table 1 Average cooling rates of molybdenum and amorphous steel splats Material Molybdenum Amorphous steel

Glass temperature(
C) 27 400 27 400

No. of samples 17 21 12 6

dT dt

 107 ðK=sÞ

3.3 22 5.8 32

   

0.2 1.2 0.8 1.7

Dhiman et al. (2007) proposed a single parameter to estimate the importance of freezing during solidification and predict the likelihood of splat breakup. When a molten droplet lands on a solid surface, it spreads into a thin splat of uniform thickness h. If the substrate is at a temperature lower than the melting point of the droplet, a solid layer of thickness s grows in it during the time it takes to reach its maximum spread. The solidification parameter is defined as the ratio of the solid layer thickness to splat thickness (Θ = s/h). Dhiman et al. (2007) developed an analytical expression to calculate the value of Θ as a function of the droplet impact parameters. The magnitude of Θ can be used to predict what the final shape of the splat will be and what the mechanism of break up, if it occurs, is. Three outcomes are possible during spreading:

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1. A very thin solid layer (Θ1) has no effect on spreading. The splat spreads into a thin sheet of liquid, ruptures internally, and fragments, producing a small central splat surrounded by a ring of debris. 2. If solid layer growth is significant (Θ ~ 0.1–0.3), it will restrain the splat from spreading too far and becoming thin enough to rupture, producing a disk-type shape. 3. If solidification is very rapid (Θ ~ 1), the solid layer obstructs the outward spreading liquid and produces a splat with fingers radiating out from its periphery. Comparison with experimental photographs, Dhiman et al. (2007) showed the value of the solidification parameter gave a reasonably accurate method of predicting the shape of the final splat. Figure 11 shows photographs of splats taken both during and after impact, illustrating the three different modes of droplet impact. In Fig. 11a, for molybdenum and nickel particles landing on substrates at room temperature, thermal contact resistance was high (~105 m2K/W) and Θ ~ 0.01. The splats spread into a thin film that ruptured internally and fragmented. The final splats all showed a central portion at the point of impact that adhered strongly to the substrate, surrounded by a ring. Raising the substrate temperature reduced the thermal contact resistance by an order of magnitude, since it evaporated adsorbed contaminants on the surface. Figure 11b shows impact of zirconia and nickel particles on surfaces heated to 400
C which had Rc ~ 106 m2 K/W and correspondingly Θ ~ 0.1. Solidification occurred near the end of droplet flattening, when the spreading liquid did not have enough momentum to jet over the solid rim and instead came to rest forming a circular splat with smooth edges. If Θ was increased further (Θ ~ 0.4, see Fig. 11c), solid layer growth was sufficiently rapid to obstruct flow of liquid early during spreading. The liquid had enough momentum that it jetted outward, producing fingers radiating out from the central splat. For nickel particles spreading on a steel substrate oxidized by heating to 640
C (Fig. 11c), the splat was intact, and smooth at the center, where solidification was slow. The edges, which solidified very rapidly, have a rough surface since surface tension did not have time to level irregularities before solidification occurred. Molybdenum splats on a glass surface heated to 400
C also have fingers radiating out.

3

Mathematical Model of Impact

3.1

Fluid Flow

Assume that the droplet is spherical at impact, the liquid is incompressible, and flow is Newtonian and laminar, shear at the free surface is negligible. Finally, droplet properties are assumed to be constant.

3.1.1 Governing Equations The equations governing the conservation of mass and momentum in an Eulerian frame of reference are,

69

Droplet Impact and Solidification in Plasma Spraying

Splat Impact Mode

Pictures during impact

2985

Splat images

Fragmentation (a)

(Slow solidification) Θ~0.01 Mo on glass, room temp

Ni on steel, room temp

Disk Splats (b)

(Intermediate rate of solidification) Θ~0.1 ZrO2 on glass, 400°C

Ni on steel 400°C

Freezing induced break-up (c)

(Rapid solidification) Θ~0.4 Mo on glass, 400°C Ni on steel 640°C

Fig. 11 Photographs during and after impact for splats (a) fragmenting during impact, (b) forming disk splats, and (c) undergoing freezing-induced breakup [J]

@ρ þ ∇ ðρVÞ ¼ 0 @t

(15)

@ ðρVÞ þ ∇ ðρVVÞ ¼ ∇p þ ∇ τ þ Fb þ FST @t

(16)

where V is the velocity vector, ρ is the fluid density, p is the pressure,τ is the shear stress tensor, Fb represents body forces such as gravity acting on the fluid elements, and FST is the surface tension force which acts only near the fluids interface (Brackbill et al. 1992). Assuming the fluids are Newtonian, the shear stress tensor is expressed,   τ ¼ μ ∇V þ ∇VT

(17)

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where μ is the dynamic viscosity. Using the one-field approach and having several fluids in the domain, each with velocity field Vk, we may assume that all fluids move with the local center of mass velocity V, Kothe (1998), Vk ¼ V

3.2

(18)

Interface Tracking

There are a number of methods described in the literature to resolve the interface between two immiscible and incompressible fluids. These include the volume of fluid (VOF) method (Hirt and Nichols 1981), level set (LS) method (Osher and Fedkiw 2001), coupled level set volume of fluid (CLSVOF) method, height function (HF) method (Afkhami and Bussmann 2008), and volume of fluid with advecting normal (VOF-AN) method (Raessi et al. 2007). One of the most common and robust approaches for tracking interfaces is the volume of fluid (VOF) approach. In this method, a scalar function f is defined to mark the space where each fluid resides. In the case of two immiscible fluids, the values are assigned zero in one fluid and unity in the second one. Since all fluids are assumed to be incompressible, f is passively advected with the flow and, thus, it satisfies the advection equation, @f þ V ∇f ¼ 0 @t

(19)

where 8   < 1, ! f r ¼ : 0,

!

r  fluid 1

!

r  fluid 2:

(20)

!

r is the position vector. The interface normal and curvature can be calculated from the VOF data by, ^n ¼

∇f j∇f j

κ ¼ ∇ ^n

(21) (22)

The numerically discretized form of f is the fraction of a numerical control volume occupied by fluid 1, i.e., 1 F¼ V

ð f dv V

(23)

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Droplet Impact and Solidification in Plasma Spraying

2987

Thus f = 1 in cells that are fully occupied by fluid 1 and f = 0 for cells that are filled with fluid 2. Thus 0 < F < 1 for those cells that contain both fluids. Eq. (19) is numerically solved using the Youngs algorithm (Bussmann et al. 1999). Based on the volume fraction of each phase, mixture properties in the interface cells are defined, ρ ¼ f ρd þ ð1  f Þρb μ ¼ f μd þ ð1  f Þμb κ1 ¼ f =κ d þ ð1  f Þ=κ b

(24)

where d and b refer to dispersed and bulk phases, respectively. The surface tension force, FST, which is nonzero only at the interface, can be expressed, FST ¼ σ ðT Þκ∇f þ ∇k σ ðT Þ j∇f j ffi σ ðT Þκ∇f

(25)

where κ is the local interface curvature, T is the interface temperature, and ∇k is the tangential surface derivative. The first term on the right-hand side corresponds to the temperature-dependent normal surface tension component, while the second term corresponds to the Marangoni force. The Marangoni convection force is negligible during droplet impact. Thus, Eq. 16 becomes,   @ ðρVÞ þ ∇ ðρVVÞ ¼ ∇p þ ∇ μ ∇V þ ∇VT þ σ ðT ÞK∇f þ Fb @t

3.3

(26)

Heat Transfer and Solidification

We assume that solidification occurs at melting temperature and we neglect viscous dissipation. Densities of liquid and solid phases are assumed to be constant and equal to each other. The energy equation is then written, @h 1 þ ðV ∇Þh ¼ ∇ ðk∇T Þ @t ρ

(27)

Energy equation has two dependent variables; these are temperature T and enthalpy h. The method of Cao et al. (1989) may be employed to transform the energy equation in terms of enthalpy alone. The main advantage of this method is that it solves the energy equation for both phases simultaneously. The transformed energy equation is as follows (Cao et al. 1989), @h 1 1 þ ðV ∇Þh ¼ ∇2 ðβhÞ þ ∇2 ϕ @t ρ ρ

(28)

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J. Mostaghimi and S. Chandra

where in the solid phase, h  0;

β¼

ks , Cs

ϕ¼0

(29)

at the liquid-solid interface, 0 < h < Hf ;

β ¼ 0,

ϕ¼0

(30)

and in the liquid phase, h Hf ;

β¼

kl , Cl

ϕ¼

H f kl Cl

(31)

where ϕ is a new source term, and Hf is the latent heat of fusion. Subscripts l and s refer to liquid and solid properties, respectively. The energy equation has now only one dependent variable, the enthalpy, h. The relationship between temperature and enthalpy is, 1 T ¼ T m þ ð β h þ ϕÞ k

(32)

where Tm is the melting point of the droplet. Heat transfer within the substrate is by conduction only. The governing equation is, ρw Cw

@T w ¼ ∇ ðkw ∇T w Þ @t

(33)

where subscript w indicates the substrate. An adiabatic boundary condition was used at the free surface. Note that, initially, the dominant heat loss from the droplet is due to heat conduction to the substrate and, later on, conduction and convection to the solidified layer. Estimates of heat loss by convection from the droplet surface to the surrounding gas showed that it is three orders of magnitude lower than heat conduction to the substrate. Therefore, the adiabatic condition at the free surface is reasonable. This condition can, however, be easily modified to a convective, radiative, or mixed boundary condition.

3.4

Initial and Boundary Conditions

Initial conditions, i.e., droplet size, impact velocity, substrate and droplet temperatures, liquid-substrate contact angle, and thermal contact resistance, are given along with the thermophysical properties of the droplet and substrate. Heat conduction within the substrate is accounted for.

69

Droplet Impact and Solidification in Plasma Spraying

2989

The incomplete contact between the drop and the substrate results in a temperature discontinuity across the contact surface. The effect can be incorporated in the model via definition of the thermal contact resistance, Rc (see Eq. 1). Values of Rc are provided as an input to this model. Although in principle Rc could vary with time and/or position on the interface, in this analysis, it was assumed to be constant. In practice, Rc typically varies between 107 and 106 m2 K/W. Computation of velocity field has to account for the presence of a moving, irregularly shaped solidification front on which the relevant boundary conditions are applied. The solidified regions are treated by a modified version of the fixed velocity method. In this approach, a liquid volume fraction Θ is defined such that Θ = 1 for a cell completely filled with liquid, Θ = 0 for a cell filled with solid, and 0 < Θ < 1 for a cell containing a portion of the solidification front. Normal and tangential velocities on the faces of cells containing only solidified material are set to zero. The modified continuity and momentum equations are then given by Pasandideh-Fard et al. (2002), ∇ ðΘVÞ ¼ 0

(34)

@ ðΘVÞ Θ Θ þ ðΘV ∇ÞV ¼ ∇p þ Θυ∇2 V þ Fb @t ρ ρ

(35)

@f þ ðΘV ∇Þf ¼ 0 @t

(36)

The modified Navier-Stokes, volume of fluid, and energy equations are solved on an Eulerian, rectangular, staggered mesh in a 3D Cartesian coordinate system. Details of the computational procedure are described in Pasandideh-Fard et al. (2002).

3.5

Simulations of Droplet Impact

The model is first validated by comparing its predictions to experimental measurements of Aziz and Chandra (2000). Relevant properties for nickel, tin, and stainless steel are shown in Table 2. Figure 12 shows the spread factor ratio versus time for the impact of a 2.7 mm tin droplet impacting on a stainless steel substrate at 1 m/s and 513 K. The substrate temperature is 298 K and the melting point of tin is 505 K. As shown in the figure, the predictions are in excellent agreement with the experimental results as the resolution of the numerical calculations is increased to 27 cell per radius (CPR) or higher. Figure 13 shows a comparison between experimental measurements and numerical simulations of the impact of a tin droplet on a previously deposited and solidified tin splat (Ghafouri-Azar et al. 2004). The comparison between the predictions and experiments is again excellent for such a relatively complicated situation.

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Table 2 Properties of nickel, alumina, and stainless steel. For substrate material (stainless steel), the only properties needed are density, thermal conductivity, and specific heat Properties Density [kg/m3] Melting point [oC] Heat of fusion [J/kg] Kinematic viscosity [m2/s]

Liquid thermal conductivity [W/(m.K)] Liquid specific heat [J/(kg.K)] Surface tension [N/m] Solid thermal conductivity [W/(m.K)]

Solid specific heat [J/(kg.K)]

Material Nickel 7.9E3 1453 3.1E5
C 1453 1577 1627 1727 45 444 1.78
C 527 727 927 1227
C 527 727 927 1227

Tin 6.970E3 232 5.81E4 2.756E-7

Stainless steel 7.900E3 – – –

33.6 244 0.69 62.2

– – –
C 127 327 527 727 927
C 127 327 527 727 927 1227

6.7E-7 5.7E-7 5.4E-7 5.0E-7

67.6 71.8 76.2 82.6 210 530 562 594 616

16.6 19.8 22.6 25.4 28.0 515 557 582 611 640 682

3.5.1 Effect of Solidification on Breakup Figure 14 shows the different stages of the normal impact of a 60 μm nickel droplet on a smooth stainless steel substrate at 290
C (Pasandideh-Fard et al. 2002). The impact speed is 73 m/s and the initial droplet temperature is 1600
C. Thermal contact resistance is assumed to be 107 m2K/W. This case corresponds to Re = 7892, We = 1419, Ste = 1.67, and Pr = 0.043; hence, Ste/Pr = 38.3, which indicates the effect of solidification on droplet spreading is important. As droplet starts spreading, instabilities around the rim appear. These instabilities result in generation of a number of fingers as well as breakup of the finger tips into smaller drops (Fig. 14). Examination of the numerical results shows that, for this impact conditions, these instabilities occur due to solidification. This is demonstrated in Fig. 15. As the thermal contact resistance is increased by an order of magnitude (Fig. 15c), solidification occurs at a slower rate, and the splat assumes a circular disk shape. The effect of substrate temperature has been found to be of great importance in affecting the dynamics of the impact on metallic substrates (Dhiman et al. 2007).

Droplet Impact and Solidification in Plasma Spraying

Fig. 12 Spread factor versus time for a tin droplet impacting on a stainless steel substrate. Impact conditions: impact velocity 1 m/s, initial droplet temperature 513 K, substrate temperature 298 K, droplet diameter 2.7 mm, melting point of tin 505 K (Experimental points from Aziz and Chandra (2000)

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Experiments Simulations (54 CPR) (40 CPR) (27 CPR) (17 CPR)

2.5

Spread factor x

69

2

1.5

1

0.5

0 0

2

4 6 Time (ms)

8

10

Fig. 13 Comparison of photographs and computer-generated images of a 2.2-mm-diameter tin droplet landing with a velocity of 2.5 m/s at a point 3.0 mm from the center of a solidified splat (Ghafouri-Azar et al. 2004)

3.5.2 Effect of Surface Roughness on Impact Dynamics Raessi et al. (2005) and Parizi et al. (2007) studied the effect of surface roughness on the dynamics of droplet impact; silicon substrates were etched and patterned with cubes of 1, 2, and 3 um. The distance between the cubes was the same as their height. Figure 16 shows good agreement between the final splat shape of a nickel droplet impacting on the 1 um rough surface and the numerical predictions. For these patterned surfaces, as the roughness increases, the final splat shape is no longer circular (Fig. 17). This effect is particularly important for the case of 3 μm roughness. The effect is due to the fact that solidification rate depends on the direction of the spreading droplet. As shown in Fig. 17b, the calculation of the splat shape in the absence of solidification results in disk-like splat. Figures 18a, b show the contact area of the liquid droplet with the surface of the substrate. As shown, the contact is

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Fig. 14 Simulations showing the impact of a 60-μm-diameter molten nickel particle at 1600
C landing with a velocity of 73 m/s on a stainless steel plate initially at a temperature of 290
C. The contact resistance at the substrate surface was assumed to be 107 m2K/W (Adapted from Ghafouri et al. (2003))

maximum at an angle of 45
. Increased contact results in a faster rate of solidification; hence, the spreading is arrested quickly in the directions with high contact.

3.5.3 Impact of Partially Molten Droplets Wu et al. (2009) and Alavi et al. (2012) studied the impact of partially molten zirconia and partially molten nickel droplets, respectively. The droplets are melted on the outer layer and have a solid core. This situation often occurs in thermal spray

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Fig. 15 Nickel splat shapes on a steel plate initially at 400
C from (a) experiments, (b) numerical model assuming a contact resistance of 107 m2K/W, and (c) numerical model assuming a contact resistance of 106 m2K/W, (McPherson and Shafer 1982)

coating process when a particle is not heated sufficiently and does not fully melt. Insufficient heat transfer may be due to the trajectory of the particle as well as its big size. Figure 19 shows the dynamics of the impact of a fully molten nickel droplet on a smooth, stainless steel substrate. Because of the high substrate temperature, solidification rate is rather slow and no splashing is observed. Upon impact, the drop starts spreading and solidifying. Some splashing is observed after around 1 μs after the impact. The final splat is shaped as a flat disk with raised rims. As the streamlines illustrate, vortices are generated in the gas flow during the particle impingement. These vortices influence the amount of the material detached from the particle during splashing. It may be noted that at 1 μs, in addition to the main vortex flow, another circulation is observed which is caused by the movement of the splashed droplet. Figure 20 shows spreading of a partially molten nickel droplet. Compared to the fully molten case, the presence of the hard core results in a reduction in spreading and less splashing. Furthermore, because of the unmelted core, there is a bump in the center of the final splat. Alavi et al. (2012) show that increase in the impact speed will have no effect on the size of this bump, but will increase splashing and decrease the thickness of the final splat. A larger unmelted core promotes splashing.

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Fig. 16 Effect of surface roughness on spreading of a nickel droplet on a silicon substrate (Monaghan 2012)

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Fig. 17 Comparison between (a) the shape of alumina splats on different surface conditions in the presence of solidification and (b) splat shape on a substrate with 3 μm roughness and an alumina droplet on the same substrate and the corresponding time but without solidification. 40-μm-diameter alumina droplets at 2055
C impacting with a velocity of 65 m/s onto alumina substrates initially at 25
C and at different surface roughness (Mehdizadeh et al. 2005)

Fig. 18 Cross section of the alumina splat on a substrate with 3 μm roughness in the directions shown in Fig. 6b. The cubes on the substrate and the splat are shown in blue and red, respectively (Monaghan 2012)

3.6

Smoothed Particle Hydrodynamics (SPH)

Another promising method to study the impact and solidification of droplets and formation of a coating is the so-called smoothed particle hydrodynamics or SPH. The method was originally introduced and developed by Gingold and Monaghan (1977) and Lucy (1977). In SPH the computational domain is discretized using fluid particles. Each particle has density and mass to represent a lump of fluid moving around with the velocity of the fluid at that location in a Lagrangian manner. Properties of these particles are smoothed over a distance known as the smoothing

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length. This means that the properties of a particle of interest can be calculated from its neighboring particles. The contribution of neighbors is weighted using a kernel function that mostly depends on the distance between neighboring particles. Since its inception, SPH has been extensively used in simulating different physical phenomena in fields like astrophysics, fluid sciences, oceanography, ballistics, etc. One of the major subjects studied in SPH is interfacial flows. Practical studies like tsunami simulations (Liu et al. 2008), simulation of floating bodies like ships (Cartwright et al. 2004), and multiphase studies (Hu and Adams 2006) are among them. In multiphase flows, numerical study of droplets has been of interest to many researchers due to applications in fields like spray coating and inkjet printing. Recently, Farrokhpanah et al. (2015) studied droplet impact on a surface and proposed and implemented a model for contact angle.

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SPH solves Navier-Stokes equations in a Lagrangian framework. In this frame, Eqs. 15 and 16 for an isothermal case are, Dρ ¼ ρ∇ V Dt

(37)

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D where Dt ¼ @@ t þ V: ∇ is the substantial derivative, and Fb represents external body forces such as gravity. The surface tension force, Fst, is approximated based on the continuum surface force (CSF) model of Brackbill et al. (1992).

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The continuity and momentum equations are closed by the equation of state, which calculates pressure using density in the form of (see Monaghan 2012),  γ ρ P ¼ P0 þb ρ0

(39)

where γ = 7 and 1.4 for liquid and gas phases, respectively, b is a background pressure, and P0 represents a reference pressure adjusted to keep maximum density deviations from ρ0 in the order of O(1%). In SPH, the local values of dependent variables are interpolated by an integral interpolant. For example, quantity A, which is a function of spatial coordinate system, may be exactly expressed, ð Aðr Þ ¼ Aðr 0 Þδðr  r 0 Þdr 0

(40)

where r is spatial coordinates, dr0 is the differential volume element, and δ is the Dirac delta function. The above may be approximated by a kernel, W, ð Aðr Þ ¼ Aðr 0 ÞW ðr  r 0 , hÞdr 0

(41)

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The kernel is defined,

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Also, gradient operator and others may be similarly approximated, e.g., ∇Ai ¼

N X  mj  Aj  Ai ∇i W ij ρj j¼1

where N is the total number of particles in the domain. In practice, the summation is limited to a limited number of particles which are in the neighborhood of particle i since W rapidly approaches zero with distance from particle i. The most commonly used kernel is the cubic spline, which, in one dimension, has the following form:

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C stainless steel substrate. Solid line (SPH), dashed line (VOF55), symbols (measurements14)

8 h io 1 > 3 3 > ð 2  q Þ  4 ð 1  q Þ , > < 6 W ðx, hÞ=h ¼ 1 ð2  qÞ3 , > > > 6 : 0

for 0  q  1 for 1  q  2 for q > 2

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where q = |x|/h. For a detailed description of the application of SPH technique to fluid equations, the reader is referred to the review article by Monaghan (2012). Farrokhpanah et al. presented a method for applying contact angle on a horizontal surface during the impact of a drop using SPH. The model is capable of accurately applying contact angle to a stationary and a moving contact line. In the method, the prescribed value of contact angle is used to adjust the interface profile near the triple phase point. This is done by adjusting the surface normally close to the contact line and interpolating the drop profile into the boundaries. Farrokhpanah et al. (2015) developed a parallel, GPU (graphic processing unit)-compatible SPH solver to capture interface evolution during droplet impact. To improve stability and performance of the solver, a customized reduction algorithm is used on the shared memory of GPU. Speedup using a variety of different memory management algorithms on GPU-CPU were studied. The algorithm was validated using the Rayleigh-Taylor instability test. Figure 21 compares the predicted SPH results (Farrokhpanah 2016) for the spread factor with those obtained with VOF-based algorithm (Pasandideh-Fard et al. 2002) and by experimental measurement (Aziz and Chandra 2000). The comparison is very good.

4

Coating Buildup

Thermal spray deposition involves the impingement of a very large number of droplets that first land on a bare substrate and then, as the deposit grows thicker, on previously accumulated splats. The growing mass of the coating material on the

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substrate loses heat by conduction to the substrate and by convection and radiation to the surrounding atmosphere. If incoming droplets add energy to the deposit faster than it is lost, the temperature of the metal will increase and the spray will land in a layer of molten metal. If heat transfer to the surroundings is sufficiently fast to allow droplets to cool down and freeze after impact, they will form solid splats. For a molten droplet to fuse with a solid deposit, it must have enough energy to cause remelting in the material under its impact point, which then solidifies again. Ghafouri-Azar et al. (2003, 2004) studied the coalescence of 2.2-mm-diameter tin droplets deposited in lines on a substrate, each offset by a small amount from the other. Figure 22 shows splats formed by depositing four tin drops along a straight line, with the center of each drop offset by 2.0, 3.0, and 3.0 mm, respectively, from that of the previous one. They used numerical simulations to predict the shapes of splats formed by interacting droplets and to calculate where sufficient remelting occurred for splats to fuse with each other.

4.1

Porosity Formation

During spray deposition, molten droplets fuse together to form a solid layer that is not perfectly dense, but contain pores and cracks that may or may not be desirable, depending on the function of the coating. In general, low porosity is desirable since that increases the strength of the coating and makes it impervious. In some specialized applications, closed porosity may be desirable, such as in thermal barrier coatings, where the insulating properties are improved by the presence of air pockets. Several different mechanisms have been identified that can create porosity: curling up of splats due to thermal stresses, entrapment of gas under impacting particles, and incomplete filling of cavities in the already deposited coating. Protuberances may already exist on a rough substrate, or they may be created during spraying by the presence of unmelted particles in the spray, or as a result of satellite droplets detaching from impacting droplets and solidifying on the surface. Pores formed by gas entrapment are typically very small and found at the interface between splats in thermal spray coatings. Based on transmission electron microscopy of plasma-sprayed coatings, McPherson and Shafer (1982) showed that the interfaces between lamellae consist of regions of perfect contact alternating with gaps of 0.01–0.1 μm which probably arise from absorbed or entrapped gas between impinging droplets and previously solidified layers. Splat curl up is caused by residual stresses in the splat as it cools and shrinks after being sprayed on the surface. The bottom surface of the splat is attached to the substrate and cannot shrink, while the upper surface is free to contract. The resulting stresses are relieved either by the edges of the splat curling up or by generation of cracks. Fukanuma (1994) developed a model for porosity formation during thermal spray coating process by considering deformation of a molten particle and showed that most of the porosity is near the periphery of the splat, starting at a distance from its center of about 0.6 times the spat radius (R). Porosity

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Fig. 22 Splats formed by depositing four 2.2-mmdiameter tin drops along a straight line, with the center of each drop offset by 2.0, 3.0, and 3.0 mm, respectively, from that of the previous one (Ghafouri-Azar et al. 2003)

was sensitive to particle velocity, ambient gas pressure, particle diameter, and molten material viscosity. Xue et al. (2007) studied the impact of molten droplets on a rough surface where liquid is driven into the crevices between asperities on the surface by liquid pressure, while surface tension restrains it from completely filling gaps, leaving voids. An analytical model was developed to calculate the volume of these voids that predicted, within an order of magnitude, the volume of voids measured from experiments.

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Modeling Coating Formation

Thermal spray coatings are formed by the impact and deposition of millions of molten and semi-molten droplets on a substrate. While the impact of many droplets on a surface can be accurately modeled, it is not yet computationally possible to deposit millions of droplets using computational fluid dynamics and predict the microstructure of the coating. In order to develop such a model, based on Monte Carlo approach, Ghafouri-Azar et al. (2003, 2006) developed a three-dimensional model of coating formation. The model was further developed by Xue et al. (2007), (2008), and Parizi et al. (2010). One of the promising modeling approaches for detail simulation of coating formation is smoothed particle hydrodynamics approach, first introduced by Gingold and Monaghan (1977) and Lucy (1977) in 1977. The method is Lagrangian and can handle many droplet impact events simultaneously in a very efficient manner. The next two sections summarize the stochastic approach for predicting microstructure of coatings and the SPH approach, respectively.

4.2.1 Monte Carlo Simulations Ghafouri-Azar et al. (2003, 2006) proposed a three-dimensional, stochastic model of thermal spray coating formation that can predict coating porosity and roughness as a function of spray parameters. The model assigns values of droplet size, velocity, and temperature T, dispersion angle and azimuthal angle to molten droplets on the substrate by generating random values of these properties, assuming that their properties follow known distributions with user-specified mean and standard deviation (Xue et al. 2008) that can be obtained for specific experiments by diagnostic instruments. Once the impact conditions of the individual droplet are selected from these distributions, the splat size is calculated by a simple analytical expression proposed by Aziz and Chandra (2000) (Eq. 9). Interaction of an impacting droplet over a previously deposited splat was considered in the following manner. When a droplet lands overlapping a previously deposited splat, it will not spread into a disk-shaped splat but will assume a shape that depends on its distance from the center of the splat under it. Based on experimental results, and some detailed simulations of sequential droplet impact using a three-dimensional model, Ghafouri-Azar et al. (2003) developed four possible scenarios for the second splat shape formed by two-droplet interactions. To select one of these scenarios, the distances between the droplet impact point and the center points of all previously deposited splats were evaluated. The smallest distance was then used to determine the splat shape according to the rules that were established by approximating detailed numerical simulations of droplet interactions on a substrate and observations of interacting plasma-sprayed splats collected on a surface during experiments. The surface area of noncircular splats was assumed to be the same as it would have been had the splats remained circular. According to Xue et al. (2006, 2007), porosity is formed because of the incomplete filling of the interstices on previously deposited splats, since surface tension prevents molten material from entering small gaps. The model assumes that the

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impacting molten droplet is in contact with a series of uniform hemispherical asperities on the surface along the splat radius. In order to calculate the volume of the voids created between the hemispheres and the liquid layer, the equilibrium profile of the liquid meniscus is calculated using a method in which the total potential and surface energies of the system are minimized. Knowing the shape of the liquid meniscus and the profile of the asperity, one can use some geometrical expressions and then integrate the gap area over the total length of the splat to calculate the volume of the incompletely filled voids. In addition to curl up and incomplete filling of interstices, a third phenomenon may result in the formation of porosity. The small, satellite droplets which are formed when droplets splash and are settled on the surface also promote the formation of porosity. Based on this assumption, the number of satellite droplets, their sizes, and locations are approximated using the theories presented by Xue et al. (2008). These satellite droplets, in turn, create surface roughness and the incomplete filling of the coating layer will create more porosity. Figures 23 and 24 show the result of the stochastic model for an yttria-stabilized zirconia plasma-sprayed coating (Xue et al. 2008). The calculations were based on a

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Fig. 23 Cross sections of three YSZ coating cases from (a) experiments (b) simulations

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Fig. 24 Variations of coating porosity, average thickness, and average roughness with particle size and impact speed

the following spray parameters: average particle diameter and standard deviation of 25 μm and 5 μm, respectively; average particle temperature and standard deviation of 3000 K and 50 K, respectively; average impact speed and standard deviation of

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100 m/s and 10 m/s, respectively; dispersion angle and standard deviation of 3.0 and 0.58
, respectively; and a uniform distribution in the azimuthal direction. The gun standoff distance was 0.12 m, and the average powder feed rate for the cases studied was 0.35 g/s. The gun moved constantly back and forth along the length of the substrate with a speed of 0.6 m/s. The model correctly predicts the effect of different operating parameters on coating porosity, roughness, and thickness (Xue et al. 2008). Stochastic models are very useful in predicting microstructure of coatings as a function of operating conditions and show the dependence of porosity, roughness, and thickness on different parameters.

References Afkhami S, Bussmann M (2008) Height functions for applying contact angles to 3D VOF simulations. Int J Numer Methods Fluids 61:827–847 Alavi S, Pasandideh-Fard M, Mostaghimi J (2012) Simulation of semi-molten particle impacts including heat transfer and phase change. J Therm Spray Technol 21:1278–1293 Aziz S, Chandra S (2000) Impact, recoil and splashing of molten metal droplets. Int J Heat Mass Transf 43:2841–2857 Bennett T, Poulikakos D (1994) Heat-transfer aspects of splat-quench solidification – modeling and experiment. J Mater Sci 29:2025–2039 Bertagnolli M, Marchese M, Jacucci G (1995) Modeling of particles impacting on a rigid substrate under plasma spraying conditions. J Thermal Spray Tech 4:41–49 Brackbill J, Kothe D, Zemach C (1992) A continuum method for modeling surface tension. J Comput Phys 100:335–354 Bussmann M (1999) PhD thesis, Department of Mechanical and Industrial Engineering, University of Toronto, 1999 Bussmann M, Mostaghimi J, Chandra S (1999) On a three-dimensional volume tracking model of droplet impact. Phys Fluids 11:1406 Bussmann M, Chandra S, Mostaghimi J (2000) Modeling the splash of a droplet impacting a solid surface. Phys Fluids 12:3121 Cao Y, Faghri A, Chang W (1989) A numerical analysis of Stefan problems for generalized multidimensional phase-change structures using the enthalpy transforming model. Int J Heat Mass Transf 32:1289–1298 Cartwright B, Groenenboom P, Mcguckin D (2004) Examples of ship motions and wash predictions by smoothed particle hydrodynamics. 9th international symposium on the practical design of ships and other floating structures Cossali G, Coghe A, Marengo M (1997) The impact of a single drop on a wetted solid surface. Exp Fluids 22:463–472 Dhiman R, Chandra S (2005) Freezing-induced splashing during impact of molten metal droplets with high Weber numbers. Int J Heat Mass Transf 48:5625–5638 Dhiman R, McDonald A, Chandra S (2007) Predicting splat morphology in a thermal spray process. Surf Coat Technol 201:7789–8801 Farrokhpanah A (2016) Numerical solution to phase change problem: application to suspension plasma spray. PhD thesis, University of Toronto, 2016 Farrokhpanah A, Samareh B, Mostaghimi J (2015) Applying contact angle to a two dimensional multiphase smoothed particle hydrodynamics model. J Fluids Eng 137:041303–041301 Fauchais P, Fukumoto M (2004) Knowledge concerning splat formation: an invited review. J Therm Spray Technol 13:1–24

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Fukanuma H (1994) A porosity formation and flattening model of an impinging molten particle in thermal spray coatings. J Therm Spray Technol 3:33–44 Fukanuma H, Ohmori A (1994) Behavior of molten droplets impinging on flat surfaces. In: Proceedings of the 7th national thermal spray conference, 563 (1994) Fukomoto M, Huang Y (1999) Flattening mechanism in thermal sprayed Ni particles impinging on flat substrate surface. J Therm Spray Technol 8(3):427–432 Fukumoto M, Ohgitani I (2004) Effect of substrate surface change by heating on transition in flattening behavior of thermal sprayed particles. In: Lugscheider E, Berndt C (eds) Thermal spray: advances in technology and application. ASM International, Materials Park, pp 1–6 Ghafouri-Azar R, Shakeri S, Chandra S, Mostaghimi J (2003) Interactions between molten metal droplets impinging on a solid surface. Int J Heat Mass Transf 46:1395–1407 Ghafouri-Azar R, Mostaghimi J, Chandra S, Charmchi M (2003) A stochastic model to simulate the formation of a thermal spray coating. J Therm Spray Technol 12:54–69 Ghafouri-Azar R, Mostaghimi J, Chandra S (2004) Numerical study of solidification of a droplet over a deposited frozen splat. Int J Comput Fluid Dyn 18:133–138 Ghafouri-Azar R, Mostaghimi J, Chandra S, Charmchi M (2006) Development of residual stresses in thermal spray coatings. Comput Mater Sci 3:13–26 Gingold R, Monaghan J (1977) Smoothed particle hydrodynamics: theory and applications to non-spherical stars. Mon Not R Astron Soc 181:375–389 Hirt C, Nichols B (1981) Volume of fluid (VOF) method for the dynamics of free boundaries. J Comput Phys 39:201–225 Hu X, Adams N (2006) A multi-phase SPH method for macroscopic and mesoscopic flows. J Comput Phys 213:844–861 Inada S, Yang W (1994) Solidification of molten metal droplets impinging on a cold surface. Exp Heat Transfer 7:93 Jiang X, Wan Y (2001) Role of condensates and adsorbates on substrate surface on fragmentation of impinging molten droplets during thermal spray. Thin Solid Films 385:132–141 Kang B, Zhao Z, Poulikakos D (1994) Solidification of liquid-metal droplets impacting sequentially on a solid-surface. ASME J Heat Transfer 116:436–445 Kothe D (1998) Perspective on Eulerian finite volume methods for incompressible interfacial flows. In: Kuhlmann H (ed) Free surface flows. Springer, New York, pp 267–331 Li C, Li J, Wang W (1998) The effect of substrate preheating and surface organic covering on splat formation. In: Coddet C (ed) Proceedings of the 15th international thermal spray conference. ASM International, Materials Park, pp 473–480 Liu H, Lavernia E, Rangel R (1993) Numerical-simulation of substrate impact and freezing of droplets in plasma spray processes. J Phys D: Appl Phys 26:1900–1908 Liu V, Wang G, Matthys E (1995) Thermal analysis and measurements for a molten metal drop impacting on a substrate: cooling, solidification and heat transfer coefficient. Int J Heat Mass Transf 38:1387 Liu P, Yeh H, Costas S (eds) (2008) Advances in coastal and ocean engineering: advanced numerical models for simulating tsunami waves and runup. World Scientific Publishing Singapore 10 Lucy L (1977) A numerical approach to the testing of the fission hypothesis. Astron J 82:1013–1024 Madejski J (1976) Solidification of droplets on a cold surface. Int J Heat Mass Transf 19:1009 McDonald A, Lamontagne M, Moreau C, Chandra S (2006) Impact of plasma-sprayed metal particles on hot and cold glass surfaces. Thin Solid Films 514:212–222 McDonald A, Moreau C, Chandra S (2007) Thermal contact resistance between plasma-sprayed particles and flat surfaces. Int J Heat Mass Transf 50:1737–1749 McPherson R, Shafer B (1982) Interlamellar contact within plasma-sprayed coatings. Thin Solid Films 97:201–204 Mehdizadeh N, Raessi M (2004) Effect of substrate temperature on splashing of molten tin droplets. ASME J Heat Transfer 126(3):445–452

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Mehdizadeh N, Lamontagne M, Moreau C, Chandra S, Mostaghimi J (2005) Photographing impact of molten molybdenum particles in a plasma spray. J Thermal Spray Technol 14:354–361 Monaghan J (2012) Smoothed particle hydrodynamics and its diverse applications. Annu Rev Fluid Mech 44:323–346 Mostaghimi J, Pasandideh-Fard M, Chandra S (2002) Dynamics of splat formation in plasma spray coating process. Plasma Chem Plasma Process 22:59–84 Mundo C, Sommerfeld M, Tropea C (1995) Droplet-wall collisions: experimental studies of the deformation and breakup process. Int J Multiphase Flow 21:151–173 Osher S, Fedkiw R (2001) Level set methods: an overview and some recent results. J Comput Phys 169:463–502 Parizi H, Rosenzweig L, Mostaghimi J, Chandra S, Coyle T, Salimi H, Pershin L, McDonald A, Moreau C (2007) Numerical simulation of droplet impact on patterned surfaces. J Therm Spray Technol 16:713–721 Parizi H, Mostaghimi J, Pershin L, Jazi H (2010) Analysis of the microstructure of thermal spray coatings: a modeling approach. J Therm Spray Technol 19:736–744 Pasandideh-Fard M, Mostaghimi J (1996) On the spreading and solidification of molten particles in a plasma spray process: effect of thermal contact resistance. Plasma Chem Plasma Process 16S: S83–S98 Pasandideh-Fard M, Qiao Y, Chandra S, Mostaghimi J (1996) Capillary effects during droplet impact on a solid surface. Phys Fluids 8:650 Pasandideh-Fard M, Bhola R, Chandra S, Mostaghimi J (1998) Deposition of tin droplets on a steel plate: simulations and experiments. Int J Heat Mass Transf 41:2929–2945 Pasandideh-Fard M, Chandra S, Mostaghimi J (2002a) A three-dimensional model of droplet impact and solidification. Int J Heat/Mass Transf 45(1):2229–2242 Pasandideh-Fard M, Pershin V, Chandra S, Mostaghimi J (2002b) Splat shapes in a thermal spray coating process: simulations and experiments. J Therm Spray Technol 11:206–217 Pech J, Hannoyer B (2000) Influence of substrate preheating monitoring on alumina splat formation in DC plasma process. In: Berndt CC (ed) Proceedings of the 1st international thermal spray conference. ASM International, Materials Park., 2000, pp 759–765 Pershin V, Lufitha M, Chandra S, Mostaghimi J (2003) Effect of substrate temperature on adhesion strength of plasma-sprayed nickel coatings. J Therm Spray Technol 12:370–376 Poirier D, Poirier E (1994) Heat transfer fundamentals for metal casting, 2nd edn. Minerals, Metals and Materials Society, Warrendale, pp 41–42 Raessi M, Mostaghimi J, Bussmann M (2005) Droplet impact during the plasma spray coating process – effect of surface roughness on splat shapes. In: Proceedings of 17th international symposium on plasma chemistry, Toronto, pp 916–917 Raessi M, Mostaghimi J, Bussmann M (2007) Advecting normal vectors: a new method for calculating interface normal and curvatures when modeling two-phase flows. J Comput Phys 226:774–797 Rioboo R, Tropea C, Marengo M (2001) Outcomes from a drop impact on solid surfaces. Atomization Sprays 11:155–165 Trapaga G, Mathys E, Valencia J, Szekely J (1992) Fluid-flow, heat-transfer, and solidification of molten-metal droplets impinging on substrates – comparison of numerical and experimental results. Metall Trans B 23B:701–718 Watanabe T, Kuribayashi I, Honda T, Kanzawa A (1992) Deformation and solidification of a droplet on a cold substrate. Chem Eng Sci 47:3059 Worthington A (1876) A second paper on the forms assumed by drops of liquids falling vertically on a horizontal plate. Proc R Soc London 25:498 Worthington A (1907) The splash of a drop. The Society for Promoting Christian Knowledge, London. 1907 Wu T, Bussmann M, Mostaghimi J (2009) The impact of partially molten YSZ particle. J Therm Spray Technol 18:957–964 Xu L, Zhang W, Nagel S (2005) Drop splashing on a dry smooth surface. Phys Rev Lett 94:184505

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Index

A Abel’s integral equation, 1277 Ablation, 2265, 2491, 2494, 2503, 2505, 2508, 2509, 2512, 2515 Absorptance, 1210 spectral directional, 1031 Absorption absorbed power, 1153 chilling, 1602 cross-section, 1153, 1154, 1165 heat pumps, 1312 mechanisms, 921 pathlength, 1210, 1224 process, 1145 Absorption coefficient, 938, 1210, 2391, 2462, 2606, 2607, 2611, 2621, 2623, 2625, 2627, 2637, 2641 Absorption line blackbody distribution function (ALBDF), 1116 Acceptable design, 222, 248, 254–256 Access2Flow, 1563 Acerra plant, 2833 Acoustics, 1638 Activation energy, 2432, 2436, 2442 Active cooling, 2492, 2508–2514 Active flow rotation, see Flow rotation Active technique, 448, 469 Additives, 1817 for gases, 450, 468 for liquids, 450, 468 manufacturing, 1505, 1508, 1543 Adhesion strength, 2971 Adiabatic covering surface, 569, 571 Adiabatic film effectiveness area-averaged, 434 dense hole array, 439–441 line-averaged, 434, 442, 443 spatially-averaged, 437, 438 Adiabatic inner section, 557

Adjacent horizontal isothermal square surfaces, 566–569 Adjacent narrow plane surfaces, 540, 541 Adsorbed contaminants, 2971 Adsorbed gas discharge, 51 Adsorption, 893, 1842 isotherm, 1835 Adsorption-desorption, 1829, 1831, 1834 Advection, 278 Aerospace and avionics, 2199 Air conditioning systems, 1299, 1475, 1477, 1491 Air coolers, 1625 Air flow rate, 889 Air permeability, 889 Air void, 894 Alfa Laval, 1545 Algebraic eigenvalue problem, 93 Algebraic turbulence model (ATM), 400, 416, 417 Algorithms, 1230 Alifanov’s iterative regularization, 184 Allothermal process, 2843 Alloys basic properties at normal conditions, 1369, 1391 electrical resistivity vs. temperature, 1375 maximum allowable stress vs. temperature, 1372 Poisson’s ratio vs. temperature, 1374 specific heat vs. temperature, 1371 thermal conductivity vs. temperature, 1370 Young’s modulus vs. temperature, 1373 Alternating current (AC), 496 Aluminum, 2914 Aluminum alloy, 1510 Amplitude scattering matrix, 1148–1150 Angular redistribution terms, 954 Animate, 332

# Springer International Publishing AG, part of Springer Nature 2018 F. A. Kulacki (ed.), Handbook of Thermal Science and Engineering, https://doi.org/10.1007/978-3-319-26695-4

3009

3010 Anisotropics, 9 factor, 2470 materials, 1026 metamaterials, 1026 Anisotropic rigorous-coupled wave analysis (RCWA), 1026–1031 Annular flow, 1781, 1784, 1806, 1854, 2051 boiling, 2112–2113 condensation, 2113–2114 heat transfer, 1925 regime, 2079 Annular heat pipe, 2171 Annular-mist flow, 1854 Annular/stratified regimes, 2079 Anode, 2650 attachment, 2738, 2740 boundary layer, 2532 heat transfer to, 2535 Anomalous diffraction approximation, 1166 Aqueous humor, 2383, 2387, 2400–2401, 2405, 2407 Aqueous reagent solutions, 1828, 1829 Arc anode attachment, 2731, 2738, 2739 anode boundary, 2686 attachment, 2530 cathode attachment (cathode spot), 2731, 2743 cathode boundary, 2691 column, 2525, 2527, 2544, 2661 double arcing, 2734, 2750, 2769 electrode interactions, 2686 furnace, 2650 equations for modelling, 2662 ignition, 2731, 2755 pilot, 2731, 2735, 2786 power supply system stability, 2553 radiation, 2651 re-attachment phenomenon, 2548 restrike, 2537 root movement, 2537 stabilization, 2554 stand-off, 2733 termination erosion, 2760 welding, 2658 Area fraction, 335, 336, 338, 347 Area to point, 355 Arithmetic Mean Temperature Difference (AMTD), 1328 Arrhenius equation, 2854 Arrhenius integral, 2473 Artificial neural network, 1281 Artificial nucleation sites, 1783–1784

Index Asymmetry factor, 956 Asymmetry parameter, 1154 Asymptotic fouling, 1627 Atomic lines, 2614, 2621, 2628 Attenuation, large scale turbulence, 404 Augmentation, 448 Augmented heat transfer surface, 1508 Autocovariance function, 200 Autothermal process, 2843 Auxiliary eigenvalue problem, 91 Average Nusselt number, 814 Avogadro number, 613 Axial conduction, 1540 Axial grooved heat pipes, 2195 Axially twisted ducts, 465 Axially twisted tubes, 465 Azeotropic mixtures, 1799, 1801 Azeotropic refrigerants, 1808 B Basset, 2892 Basset force, 1868 Batchelor flow, 673, 674 Batteries, thermal management, 2248–2250 Battery pack, 2249 Bayesian approach, 1281 Bayesian framework, 185 Bayesian inference, 1261 Bayesian priors, 1263 Bayes theorem, 186 Beer-Lambert law, 1210 Beer’s law, 1184, 1192 Bent-strip type turbulators, 461 Bernoulli effect, 1870 Beta distribution, 196 Bian stone, 2337 Biasing method, 1237 Bi-CG method, see Bi-conjugate gradient (Bi-CG) method Bi-conductive surface, 1777, 1778 Bi-conjugate gradient (Bi-CG) method, 161 Bi-directional reflectivity, 1214 Bifurcated configuration, 339 Bifurcating manifold, 1991 Bifurcation(s), 678, 680, 684, 685, 737, 747, 755 Bifurcation theory, 653, 663, 680, 681, 685, 686 Binary alloys, 693–754 Binary gas mixture, 1127 Bioavailability, 2412

Index Biocides, 1635 Bioheat equation, 818, 2471 Bioheat transfer, 2280, 2393–2395 Biological fouling, 1614 Biological tissues, 818 Biomass, 1190, 1192, 2844 Biomaterials, 2280, 2282, 2284, 2286, 2288, 2291, 2301 Biophysical model, 2434, 2440 Biophysical parameter, 2439, 2441 Biophysical process, 2432 Biot, 2898 Biothermal, 2492, 2502 Biot modulus, 836 Biot-Willis coefficient, 835, 836 Bi-philic surface, 1775–1776 Bipolar, 2503, 2505, 2506, 2508, 2509, 2511, 2514 Blackbody, 922, 980, 981, 990, 997, 999, 1000, 1005, 1007, 2604 effect, 2632 emissive power, 2605, 2606 intensity, 2625, 2637 properties, 2603 radiation, 2631 Blasius equation, 731 Blasius’s method, 732 Blasius solution, 370, 371 Bloch-Floquet condition, 1029 Blood perfusion, 2286, 2394, 2472 Blood vessels, 819 Blowing ratio (BR), 427, 444 adiabatic film effectiveness, dense hole array, 437–438 adiabatic film effectiveness, sparse hole array, 433–435 heat transfer coefficients, dense hole array, 438 heat transfer coefficients, sparse hole array, 435–437 variations, 431 Boilers, 1174, 1301–1302, 1475, 1478, 1487 utility, 1193–1194 Boiling crisis, 1922 Boiling curve, 1725, 1731, 2011 hysteresis, 1701 for internal flow, 1737 mass flux and equilibrium quality, 1737 parametric effects, 1707 and pool boiling regimes, 1698 Boiling heat transfer, 1483, 1911 Boiling instabilities, 2022 Boiling limit, 2180

3011 Boiling number, 308, 1809 Boiling on enhanced surfaces bubble nucleation, 1753–1760 concepts, 1775–1776 microchannel flow/convective boiling (see Microchannel flow/convective boiling) pool boiling (see Pool boiling) Boltzmann constant, 613, 922 Boltzmann’s law, 2615, 2628 Boltzmann transport equation (BTE), 279–280, 282 Bond number, 308, 1777, 1778 Bottoming cycle cogeneration, 1596 Boudouard reaction, 2849 Boundary conditions, 9, 945, 2291, 2292, 2494, 2498, 2500, 2505, 2508 Dirichlet, 9, 10 Neumann, 9 Robin, 9, 10 Boundary element methods, 129 Boundary layer, turbulent, 394–396 entrainment, 395 flat plate, 398 laminar, 403–404 Boundary layer development, 1469 Boundary layer thickness, 369 Boundary scattering, 284, 286 Bound-bound transitions, 1077 Boussinesq, 2890 Boussinesq approximation, 696, 710, 717, 811, 1710 Bradymetabolism, 2335 Brazed joints, 1505 Brazing, 1504 Bremsstrahlung, 2610 Brinkman-extended Darcy model, 809 Broadening, 2616 Brownian, 2895 Brownian motion, 610 Bubble departure diameter, 1656, 1660–1663, 1937 Bubble departure frequency, 1937 Bubble dynamics, 1658, 1679, 1680, 1826, 1831, 1842 Bubble growth period, 1683 Bubble induced turbulence, 1932 Bubble injection, 1552–1553 Bubble merger, 1663 Bubble nucleation, 1660, 1753, 1780, 1783, 1785, 1825, 1826, 1956 Bubble point, 1800 Bubble radius, 1811

3012 Bubble release frequency, 1663–1665, 1669, 1671, 1672, 1681 Bubble size model, 1934 Bubbly flow, 1780 Building materials, 2250 Bulges, 1715, 1722 Bulk modulus, 836 Buoyancy force, 605 Burgers’ equation, 85 Burnout heat flux, 1648 C Calorific value, 2844, 2871 Cancer, 817 Capacitative, 2269 Capillary assisted boiling, 1779 Capillary-driven heat pipes, 2171 Capillary forces, 889 Capillary limit, 2177–2180 Capillary pressure, 1751–1753, 1774, 2168 Capillary wicking, 1786 Carbon footprint, 1597 Carbon monoxide, 2843 Carbon nanotube (CNT), 1989, 2010 Cartesian coordinate system, 939, 1218 Catalytic distillation, 1580 selective hydrogenation, 1580–1581 CatalyticFOAM, 1555 Cataract, 2386–2387 Catch rate, 2959 Cathode (electrode), 2650 attachment, 2731, 2743, 2751, 2754 energy balance for, 2531 erosion, 2735, 2749–2762 jet, 2534 modes of operation, 2749 arc termination effect on, 2760 constant current erosion, 2750 cyclic erosion, 2755 effect of swirling gas flow on, 2755 first start erosion, 2760 start erosion, 2758, 2760 region, 2526, 2528, 2529 voltage drop, 2742, 2744 CB&I, 1580 See also CDHydro process CDHydro process, 1581–1585 Centrifugal, 693, 723, 742 acceleration, 732, 735, 737, 740 buoyancy, 720, 729, 733, 755 casting processes, 694 centrifugal body force, 728, 748

Index centrifugal number, 698, 712 centrifugal Rayleigh number, 719, 720, 733, 735, 739, 751 crystal separation processes, 693 dimensionless group, 717 Centrifugally driven convection, 722, 740 Ceramatec, 1541 CFD, see Computational fluids dynamics (CFD) Channels, 333, 350, 351, 355, 356 classification scheme, 1955 inclination, 2052 Chaos, 737 Chaotic solutions, 737 Char, 2848 Characteristic length, 553 Charge conservation, 494 Charging, 2232, 2897 Charging process, 862 Chemical cleaning, 1638 Chemical potential, 1044 Chemical reactions, 223, 227, 231, 238, 244, 2848 fouling, 1613–1614 Chemical stability, 2230 Chemical vapour deposition (CVD), 223, 225, 228, 238, 245, 262, 267 CHF enhancement, 2020 Chillers, 1306 Chilling injury, 2427–2428 Chip, 1975 Choroid, 2384, 2387, 2394, 2395, 2397, 2405, 2407, 2410 blood flow, 2394 melanoma, 2386 pigmentation, 2397 CII, see Colloidal instability index (CII) Circular cylinder, 586 Circular horizontal surfaces, 528 Classical integral transform technique (CITT), 74–82 Classification of exchanger type, 1476 Clausius–Clapeyron equation, 2697 Clausius invariant, 952 Clothed cylinder model, 899–903 Clothing insulation, 912 Clothing ventilation, 897 thin fabric model for, 894 Coagulation, 2799, 2803, 2801–2802, 2804, 2822 Coal, 1188, 1190, 1192, 1193 Co-design, of microfluidic cooled 3D ICs, 1998–2002 Coefficient of performance (COP), 1781 Cogeneration, 2870, 2872

Index Coiled tubes, 449, 465–468 Colburn factor, 1517 Cold cathodes, 2530 Cold fluid, 1316, 1319, 1320, 1322, 1325, 1326, 1329, 1333, 1335, 1338 Cold plate, 1999 Cold protein denaturation, 2339 Collapse pressure, 1366 Collateral damage, 2459 Collision based algorithm, 1221, 1236 Colloidal instability index (CII), 1633 Color schlieren technique, 768 Combined diffusion coefficients, 2679 Combined heat and power plants, 2872 Combined heating, cooling and power (CHCP), 1601 Combined mode heat transfer, 945 Combustion, 2844 air preheating, 1599 products, 1308 Communicating channels, 1509 Compact heat exchangers, 1342, 1453, 1454, 1502, 1538 compactness, 1502–1503 construction, 1503–1510 design problem, 1510–1512 design problem solution procedure, 1513–1519 Compact reactor, 1560–1561 Complex shape, 528 Composition, 2620 Composition gradients, 1799, 1805–1806 Compound enhancement, 449, 469 Compressed fluid, 1358 Compressed sparse block (CSB), 151, 158 Compressed sparse rows (CSR), 151, 157 Compressibility, 836 Compression-sensitive MR elastography, 852 Computational fluid dynamics (CFD), 653, 661, 663, 680, 1185, 1187, 1510, 2000, 2110, 2144 Computations, 2078 COMSOL™ Multiphysics, 103 Concentrated photovoltaics (CPVs), 299 Concentration gradients, 889 Conceptual design, 222, 248 Condensation, 309, 1388, 2797, 2799–2801, 2803 coefficient, 2036 halo, 317 of mercury, 2037 Condenser limit, 2182 Condensers, 1302–1303, 1475, 1477, 1478, 1485, 1487, 1488, 1491, 1494, 2078

3013 Condition number, 1248 Conductance, 332 Conduction, 891, 1298 in drop, 2060–2061 heat (see Heat conduction) pumping, 500, 508–511 Conduction-type phenomena, 129, 166 CVFDMs (see Control-volume finite difference methods (CVFDMs)) CVFEMs (see Control-volume finite element methods (CVFEMs)) discretized equations, solution of, 160 grid-independent numerical solutions and order of accuracy, 163 measures of error, 166 partial differential equations, 132 validation and verification, 162 Conductivity, 2495, 2497, 2501 Conjugated heat transfer, 98 Conjugate gradient (CG) regularization, 1254 Conjugate mechanisms, 238 Conservation of energy, 921 Conservative form, 942 Constant current erosion, 2750–2755 Constructal design, 331, 333–334 Constructal law, 330, 333, 355, 357 Constructal theory, 354–356, 2264 constructal design method, 333 constructal law and vascularization, 332 effect of size, 356–357 flow spacings, 350 heat conduction, 350–354 imperfection, 333 open cavities, configurations for (see Open cavities) thermal resistance, 332–333 Contact angles, 1699, 1701, 1702, 1704, 1708, 1728, 1730, 1733, 1751, 1756, 1757, 1766–1768, 1836 Contact-line flow-physics, 2079, 2081 Contact resistance, 2283 Containment, 2237 Contamination, 1708 Continuum, 2608, 2610, 2611, 2637, 2894 Continuum flow limit, 2181 Continuum models, 2472 Contraction ratio (Cr) acceleration parameter, 432 adiabatic film effectiveness, dense hole array, 437 coolant mass flow rate variation, 432–433

3014 Contraction ratio (Cr) (cont.) heat transfer coefficients, dense hole array, 438–439 and mainstream acceleration, 431–432, 443 Control parameter, 682, 685 Control-volume finite difference methods (CVFDMs), 131 steady multidimensional problems, 144 steady three-dimensional problems, 141 steady two-dimensional planar problems, 133 unsteady multidimensional problems, 158 Control-volume finite element methods (CVFEMs), 131 steady three-dimensional problems, 153 steady two-dimensional axisymmetric problems, 153 steady two-dimensional planar problems, 146 unsteady multidimensional problems, 158 Convection, 10, 27, 45, 891, 1298, 2400, 2853 boiling, 307–309, 1813, 2106 eigenvalue problem, 114 flow, 2670 heat (see Heat convection) heat transfer, 364, 945, 2671 Rayleigh-Benard configuration, 771 in two-fluid layer system, 772–773 Coolant mass flow rate, 432 Coolant Reynolds number (Refc), 428, 444 Coolers, 1306 Cooling, 1299 electronics, 350 towers, 1299, 1522, 1530, 1531 Coordinate systems, 12 Copper, 2911 Coriolis effect, 694, 695, 711, 720, 722, 728, 740, 751, 753, 755 Coriolis forces, 653, 660 Cornea, 2383, 2385–2386 Corning, 1560 Correlated-k (CK), 1176, 1181 Correlated-k distribution method, 1101 Corrosion, 1614 Corrosion risk, 1997 Corrugated fins, 459 Coulomb force, 492 Counter-current heat exchanger, 1477 Counter electrode, 2846 Counter flow, 1316, 1319, 1321, 1327, 1329, 1330, 1337, 1341 Counter flow exchanger, 1515

Index Counter flow heat exchangers, 1451, 1464 Coupled integral equations, 41, 42 Coupled plasma, 2571 Courant–Friedrichs–Lewy (CFL) condition, 159 Covariance matrix, 183 Covering surface, 528 CPA, see Cryoprotective additives (CPAs) CPVs, see Concentrated photovoltaics (CPVs) Crank-Nicolson scheme, 159 Creep response, 833 Critical heat flux (CHF), 305, 1648, 1703, 1707, 1708, 1725, 1728, 1733, 1735, 1737, 1738, 1740, 1758, 1760, 1763, 1764, 1769, 1772, 1774, 1778, 1779, 1785, 1787 definition, 1781 Kandlikar bubble momentum force-balance model, 1765 mechanisms, 1785 prediction on micro/nanostructured surfaces, 1772 for rough surface, 1773 Zuber hydrodynamic model, 1763 Critical micelle concentration (CMC), 1829, 2929 Critical overlap concentration, 1835, 1843 Critical points, 662, 678, 681–685, 1358 Critical polymer concentration, 1834 Cross flow, 1325, 1331, 1341, 1342, 1347 Cross-flow heat exchangers, 1452, 1466, 1478 Cross-flow unmixed-unmixed flow arrangement, 1513 Cross-sectional flow area, 1809 Cryogenic, 2285 Cryogenic gases density vs. temperature, 1392 dynamic viscosity vs. temperature, 1396 graphs of, 1410 latent heat of evaporation vs. temperature, 1400 Prandtl number vs. temperature, 1399 specific enthalpy vs. temperature, 1395 specific heat vs. temperature, 1394 surface tension vs. temperature, 1401 thermal conductivity vs. temperature, 1393 thermal diffusivity vs. temperature, 1398 volumetric expansivity vs. temperature, 1397 Cryogenic heat pipes, 2172 Cryomedicine, 2340–2342 Cryomicroscopy, 2430, 2432, 2433

Index Cryopreservation, 2280, 2289, 2291, 2293, 2340, 2428, 2430, 2434, 2439, 2440, 2443, 2444 Cryoprotectants, 2280, 2282, 2284, 2285, 2287, 2297, 2300, 2301 Cryoprotective additives (CPAs), 2421, 2422, 2424, 2427, 2440, 2443 Cryosurgery, 2281, 2290, 2292, 2340, 2491 Crystal growth from solution, 773–775 interferograms, 774 schlieren images, 775 shadowgraph images, 776 Crystallization, 2237 Crystallization fouling, 1614 CSB, see Compressed sparse block (CSB) CSR, see Compressed sparse rows (CSR) Cumulative distribution function (CDF), 1207 Current continuity equation, 2663 Current density, 494, 2747–2749 Current-density field, 1035 Current–voltage characteristic (VAC), 2552 Curtis-Godson approximation, 1108 Cut energy balance, 2735 metal piercing, 2733 quality of, 2781 components, 2736 dross, 2782 squareness, 2784 striations, 2783 Cutting, 2648, 2650 CVD reactor, see Chemical vapour deposition (CVD) CVFDMs, see Control-volume finite difference methods (CVFDMs) CVFEMs, see Control-volume finite element methods (CVFEMs) Cyclic erosion, 2755–2760 Cylinders, 1719, 2773, 2776 Cylindrical bodies, 528 Cylindrical coordinate system, 941, 954 Cylindrical wavy surfaces, 581, 582 Cytoplasm, 2428, 2433, 2434, 2438 D Darcy equation, 711, 737 Darcy friction factors, 1889 Darcy law, 694, 695, 712, 723, 833, 2402 Darcy model, 711, 722, 746, 753, 754, 864 Darcy number, 711, 713, 720, 751 Data centers, 221, 223, 233, 256, 297 Dean flow, 466

3015 Dean vortices, 1543 Debye length, 506 Dehumidifying chamber, 1299 Dehydration, 2224, 2423, 2428, 2429, 2433 Demixing, 2677 Denaturation, 2339, 2389 Density gradients, 2239 Departure from nucleate boiling (DNB), 1736, 1737, 1913 Desiccation, 2444 Design margins, 1617 Design of thermal systems, see Thermal systems Design optimization, see Optimization, thermal systems DETCHEM, 1556 Deteriorated heat transfer, 1373 Dew point, 1800 Dew point evaporative cooling, 1522, 1526 Dew point temperature, 1524 Diabetic retinopathy, 2386, 2391 Diamonds basic properties at normal conditions, 1369, 1391 specific heat vs. temperature, 1371 thermal conductivity vs. temperature, 1370 Diamond scheme, 961 Diatomic molecules rotations, 1079–1080 vibrations, 1081–1082 Dichotomy, 353 Dielectric fluids, 1976 Dielectric function, 1029 Dielectrophoretic force, 492 Diesel engine, 1189, 1194 Differencing schemes, 961 Differential scanning calorimetry (DSC), 2284, 2432, 2433 Diffraction orders, 1029 Diffuse attachment, 2531 Diffuse reflection, 1215 Diffuse surface, 1214, 1229 Diffusion, 2237, 2397 Diffusion-adsorption kinetics, 1835 Diffusion approximation, 957 Diffusive and convective fabric models, 892–893 Diffusivity, 2426, 2427 Dimensional analysis, 654 Dimethyl sulfoxide (DMSO), 2280, 2290, 2427 Dimples, 1984, 1985 Direct compact type, 1476

3016 Direct current (DC), 493 DC torch designs, 2550 Direct evaporative cooler, 1528, 1529 Direct evaporative cooling, 1524, 1525, 1527, 1528 Directional-hemispherical reflectance (R), 1031 Directional-hemispherical transmittance (T ), 1031 Directionally dependent properties, 1214 Directional surfaces, 1229 Direct methods, 161 Direct numerical simulation (DNS), 607, 663, 664, 672, 677, 1175, 1178, 1943, 2110 Direct transfer heat exchanger, 1449 Direct type heat exchangers, 1299 Discharging, 2232 Discharging process, 862 Discrete Dipole Approximation, 1162 Discrete ill-posed problem, 1248 Discrete ordinates equations, 958 Discrete ordinates method (DOM), 957, 1176, 2636, 2637 Discrete Picard criterion, 1248 Discretized equations, 137, 140, 142, 149, 157, 159, 160 Discretized system, 1205, 1231 Dish/Stirling systems, 2198 Disjoining pressure effects, 2154 Dispersed-droplet film boiling, 1735 Dispersion relation, 1033 Displaced enhancement devices, 449, 461 Dissipated power, 2571 Dissociation, 2845 DistMesh, 148, 155, 168 Dittus–Boelter correlation, 1367 Diversity, 332, 354 DMSO, see Dimethyl sulfoxide (DMSO) DOM, see Discrete ordinates method (DOM) Domestic solar system (DSS), 2252 Doppler effect, 2616, 2618 Doppler profile, 1089 Double arcing (DA), 2734, 2769–2771 Double-pipe heat exchanger, 1316, 1487 Drag, 2891 Drag coefficient, 2950 Drift flux model, 1885, 1928 Droplets, 310, 311, 1189, 1193 breakup, 2951–2952 detachment, 2701 interactions, 3002 heat transfer, 2952–2955

Index particle motion, 2949 transfer, 2707 Drop size distribution, 2064–2069 Dropwise, 309 Dropwise condensation, on horizontal tubes, 2069 Dropwise to filmwise condensation transition, 2069 Dross, 2782 Drude model, 1033 Drug delivery, 2383, 2401–2404 intravitreal injections, 2407 subconjunctival injections, 2406 systemic administration, 2406 topical administration of drug, 2405–2406 transscleral drug delivery (see Transscleral drug delivery) Drug distribution, 2402 Drug-eluting contact lens, 2412 Drug mimic, 2404 Dry air, 1523, 1524 Dry and evaporative resistance of fabrics, 890–891 normal flow, 891–892 Dry bulb temperature, 1524, 1525, 1527 Dry cooling, 2263 Dryers, 1310 Dryout, 1735, 1737, 2005 DSC, see Differential scanning calorimetry (DSC) DSM fine chemicals, 1561 Dufour effect, 34, 35 Dullenkopf and Mayle correlation, 407 Duty cycle, 2269 Dyadic Green’s functions, 987, 988, 995, 1002, 1004, 1008, 1011, 1013 Dynamic flowrate control, 2001 Dynamic surface tension, 1833, 1834, 1836, 1841, 1844 E EBARA process, 2833 ECM, see Extracellular matrix (ECM) Economizers, 1625 ECT, see Electrochemotherapy (ECT) Ectothermy, 2335 Edematous tissue, 854 Edge effects, 534 Edge flow, 534 Effective decay length, 411 Effective heat and mass transfer coefficients, 895

Index Effectiveness–NTU method, 1461, 1510, 1513 Effective Poisson’s ratio, 846 Effective Prandtl number, 717, 737 Effective thermal conductivity, 862 Effect of heat of fusion, 2775–2780 Effervescent atomizer, 2941 EGT, see Electrogenetherapy (EGT) EHD pumping technique, 511 Eigenfunctions, 66, 70 Eigenmodes, 1039 Eigenvalue problems, 105 Eigenvalues, 66, 70 Einstein coefficients, 1084 Ekman-Bödewadt flow, 674 Ekman number, 695, 696, 698, 700, 712, 720, 724, 742 Elasticity of flexible wall, 814 Elastography, 833 Electrical, 2490, 2491, 2494, 2495, 2504, 2506, 2507, 2510, 2511 charge, 2493 conductivity, 282, 283, 289, 2496, 2497, 2500, 2501, 2503, 2509 heating, 1299 potential distribution, 2495 properties, 2496 pulse parameters, 2491, 2502 Electrically-resistant heaters, 1299 Electric current attachment (anode spot), 2738 Electric double layer (EDL), 505 Electric fields, 2492, 2494, 2497 Electricity generation, 2831 Electrochemotherapy (ECT), 2491, 2515 Electrode(s), 2494, 2497, 2499, 2501, 2503, 2514, 2648 Electrode erodes, 2735 Electrogenetherapy (EGT), 2491, 2515 Electrohydrodynamics (EHD), 491 Electrokinetics, 1826, 1839 Electromagnetic, 2889 Electromagnetic field, 1029, 2443 Electromagnetic waves, 1026 propagation and scattering, 920 spectrum, 921 Electromagnetism, 2492 Electron emission cathode voltage drop, 2742, 2744 current density, 2747 Schottky correction, 2741 thermionic and field, 2741 work function, 2741 Electronic and electrical equipment cooling, 2196–2197

3017 Electronics cooling, 281, 296–301, 1475, 1508 Electrophoretic force, 492 Electroporation, 2490 Electroquasistatic approximation, 2493 Electrostatic fields, 450, 1553 Electrostriction force, 492 Elenbaas–Heller equation, 2547, 2672 EM43, 2476 Emission, 981, 2911 coefficient, 2607, 2611 mechanism, 921, 2741–2742 Emissive power, 2604 Emissivity, 997–1002 spectral directional, 1031 Encapsulated PCM, 2243 Encapsulation, 2237 Enclosure problems, 1217 Endothelium, 819 Endothermy, 2335 Energy, 221, 223, 225, 227, 230, 238, 243, 263, 2490, 2492, 2493, 2495, 2498, 2500, 2502, 2512 density, 982, 984, 986, 988, 990, 995, 997, 1004, 1007 efficiency, 2848, 2865, 2868 equation, 867 generation, 7 intensity, 1595 saving, 1602 security, 1595 storage medium, 2221 systems, 2197 Energy balance, 8, 2684, 2857–2864 cutting, 2735–2740 Energy conservation, 944 equation, 2663, 2664 law, 867 systems, 1598 Engines, 920 Enhanced convergence, 106 Enhancement, 594 Enhancement factor, 1808 Enhancement ratio, 2048 Enhancement techniques, 448 Enthalpy, 1502, 2190, 2440, 2844 method, 2290 rate, 1511 vaporization, 2929 wet air, 1524 porosity theory, 862 Entrainment, boundary layer, 395 Entrance length, 376 Environment, 221, 225, 238, 246

3018 Equation of energy conservation, 2679 Equation of gas conservation, 2678 Equation of state, 1801 Equilibrium composition, 2609, 2610, 2614 Equivalent heat capacity method, 863 Escape factor, 2627 Estimation of the uncertainty, 1238 Ethanol, 2873 Ethylene glycol, 2427 Eulerian, 2895 Eulerian-Eulerian two-fluid model framework, 1929 Eulerian form, 939 Eulerian–Lagrangian approach, 2949 Evanescent modes, 980, 981, 986, 988–994, 996, 997, 1004, 1007 Evanescent wave, 929, 990, 991 Evanescent wave coupling, 929 Evaporation coefficient, 2036 Evaporation momentum force, 1964 Evaporation of feedstock, 2793–2794, 2806 Evaporative condenser, 1532 Evaporative heat flow, 891 Evaporators, 1303–1307, 1475, 1478, 1481, 1487, 1491, 1495 Even-parity formulation, 947 Evolution, 330, 331, 350, 358 Exchange factor formulation, 1204 Excimer laser, 2385, 2386, 2388 Exhaustive search, 331, 334 Experimental correlations, 2231, 2235 Experiments analog to light scattering, 1169 laboratory scattering and transmission, 1169–1170 Explicit scheme, 159 Explosive boiling, 1957–1958 instability due to, 1958 Extended surface heat exchangers, 1491 Extended surfaces, 448, 449, 454–461 External combustion engines, 2871 External convective heat transfer, turbulence effects flat plate, 394 flat plate layer heat transfer, 397 laminar layer heat transfer (see Laminar layer heat transfer) Extinction, 1152 cross-section, 1153 power, 1153 Extinction coefficient, 1213 Extracellular ice formation, 2428, 2432

Index Extracellular matrix (ECM), 2433 Eye anatomy of, 2383–2384 drug development, drug bioavailability, 2410–2412 laser-based eye treatment (see Laser eye therapies) mass transfer (see Mass transfer processes, eye) ocular cryotherapy, 2398–2399 Eye therapy, 2402 F Fabric density, 891 Fabric regain, 896 Factored cost estimation, 2877 Fakner-Skan wedge flows, 374 False scattering, 971 Fanning factor, 1517 Far-field, 981, 988, 991, 1003, 1005, 1007 Fassel torch, 2581 Fatty acids, 2221 Favre-averaged, 1175 Feasibility, 251–253 Feasible design, see Feasibility Femtosecond lasers, 2388 Ferrofluid, 798 Fictitious gases approach, 1109 Field emission, 2741 Filler metal, 1507 Film boiling on cylinders, 1719 on horizontal, flat surface, 1717 on spheres, 1721 thermal radiation in, 1723 on vertical flat surfaces, 1709 Film condensation, 1483 Film cooling test, 428–430 plates, 430 Filmwise, 309 Filter, 79 Filtered problem, 79 Filtering schemes, 78–82 Filtration, 1635 Fin, 38, 39 Fin Analogy number (Fa), 1331, 1332 Fin efficiency, 1456 Finite difference methods (FDMs), 129, 130 See also Control-volume finite difference methods (CVFDMs) Finite-difference time-domain (FDTD), 1027

Index Finite element formulation, 813 Finite element method (FEM), 129, 130, 809, 963, 1027 See also Control-volume finite element methods (CVFEMs) Finite volume methods (FVMs), 961 CVFDMs (see Control-volume finite difference methods (CVFDMs)) CVFEMs (see Control-volume finite element methods (CVFEMs)) Finned heat exchangers, 1452 Finned surfaces, 454 Fin waviness, 461 Fire, 920, 1189, 1190, 1192, 1193 First construct, 339–345, 352 First start erosion, 2760 Fischer Tropsch, 1564, 1566, 1567, 1570, 1581 Fischer-Tropsch synthesis, 2872 5-year survival rate, 2456 Fixed-bed gasifiers, 2835 Flat plate, interaction of turbulence, 394–396 Flat plate boundary layers, 370–372 Flexible circulating fluidized bed (CFB) boiler, 1302, 1304 Flow and CHF related instabilities, 2094 Flow arrangement, 1296, 1316, 1327, 1336, 1476, 1484, 1487 Flow boiling, 2084 enhanced flow boiling in microchannels, 1966–1968 heat transfer, 1962–1964 instability, 1958–1959 pressure drop during, 1959 research progression, 1955–1956 Flow boiling in tubes direct numerical simulation, 1943 flow regimes, 1911 heat transfer regimes, 1913–1914 multidimensional modeling, 1929–1933 multiscale phenomena, 1914–1917 one-dimensional modeling, 1915–1929 phenomenology, 1910 Flow condensation, 312, 2084 Flow distribution, 518–521 Flow instabilities, 1781, 1782, 1785 Flow quality, 1853 Flow regime, 1911–1912 Flow-regime dependent Nux correlations, 2100, 2101 Flow-regime transition criteria, 2093 Flow rotation, 1550–1551 Flow separation, 662, 678, 680, 681, 685 Flow system, 331, 332, 334, 356

3019 Flow turbulence on plasma fields, 2573 Fluctuating currents, 981, 985–987, 1010 Fluctuational electrodynamics, 929, 981, 982, 985, 986, 998, 999, 1005 Fluctuation-dissipation theorem, 981, 982, 986, 988, 1012 Fluid, 1360 Fluid instabilities, 2976 Fluidised bed, 1191 Fluid mixtures, 2019 Fluid on saturation line basic properties, 1402, 1414 density vs. temperature, 1407 dynamic viscosity vs. temperature, 1410 explosion, fire-hazard and toxicity, 1406 freezing properties, 1404 latent heat of evaporation vs. temperature, 1414 metals and alloys, 1405 Prandtl number vs. temperature, 1413 specific heat vs. temperature, 1409 surface tension vs. temperature, 1415 thermal conductivity vs. temperature, 1408 thermal diffusivity vs. temperature, 1412 thermophysical properties, 1403 volumetric expansivity vs. temperature, 1411 Fluid–structure interaction (FSI), 809, 810 non-Darcian effects, 809–818 in tissue during hyperthermia, 817–825 Fluid vibration, 450, 469 Fluorinert electronic cooling fluids, 2087 Flux, 1003, 1004 Food processing, 1300 Forced convection internal flows, 481 condensation, 2044 enhancement, 1496 Forchheimer-Darcy law, 863 Forchheimer-extended Darcy model, 809 Forensic medicine, 2343 Forward Monte Carlo method, 1230 Forward ray tracing, 1230 Fouling, 1611 Fouling factor, 1455, 1612 Fourier, 5 continuum models, 2346 heat equation, 2394 Law, 8, 62, 279–282 number, 16 Fourier transform infrared spectroscopy (FTIR), 1281, 2433 Four-layer model, 819

3020 Fractal models, 2351 fractional differential equations, 2352–2353 vascular network structure, 2351 Fractional blackbody emissive power (FBEP), 1074 Fredholm intergral equation, 1266 Free-burning arcs, 2646 Free convection, 529, 1385 Free convection condensation, 2043 Freedom, 334, 340, 346, 348–350, 353, 355 Free flow area, 1509 Free molecular flow, 2894 Free-running generator circuit, 2592 Free stream temperature, 365 Freestream velocity, 373 Freezing injury, 2428–2429 Frequency, of electromagnetic wave spectrum, 921 Frequency-domain method, 1027 Fresnel reflection coefficients, 995 Fresnel transmission coefficient, 998 Friction coefficient, 668, 670 Friction factor, 377, 1468, 1993 Froude number, 698, 711, 717 Frozen start-up, 2194 limit, 2181 FTIR, see Fourier transform infrared spectroscopy (FTIR) Fuelling gas engines, 2872 Full spectrum correlated-k (FSCK), 1176 Full-spectrum k-distribution (FSK) method, 1232, 1234 Fully developed boiling, 1708, 1733 Fully developed laminar flow, circular tubes, 483–492 Fully developed velocity profile, 1467 Fully-implicit scheme, 159 Furnaces, 1308, 2601 FZK, see DSM fine chemicals G Galerkin FEM, 965 Galerkin method, 809 Galerkin scheme, 965 Galileo number, 1718 Gallium, 2512 Gamma distribution, 196 Gas at atmospheric pressure basic properties, 1381, 1404 density vs. temperature, 1382 dynamic viscosity vs. temperature, 1386

Index in nuclear, thermal-power and industries, 1390 Prandtl number vs. temperature, 1389 specific enthalpy vs. temperature, 1385 specific heat vs. temperature, 1384 thermal conductivity vs. temperature, 1383 thermal diffusivity vs. temperature, 1388 volumetric expansivity vs. temperature, 1387 Gas entrapment, 1650, 1651 Gasification, 2834, 2843 organics, 2848–2852 plasma arc, 2836 Gasification rate, 2854 Gas–liquid interface, 1831 Gas/liquid mass ratio, 2941 Gas–liquid-solid interfaces, 1825 Gas-loaded heat pipe, 2171 Gas turbine combustor, 1191 Gaussian distribution, 194, 1216 Gaussian Markov random field, 197 Gaussian measurement errors, 183 Gaussian prior density model, 187 Gaussian smoothness prior, 197 Gauss’ Law, 494 Gauss-Newton method, 1257 Generalized diffusive form, 112 Generalized integral transform technique, 87 Generalized Sturm-Liouville problem, 70 General linear diffusion problem, 74 Genetic algorithm, 331, 334, 1260 Geometrical optics approximation, 1165–1166 for spheres, 1168 Geometric optics, 1271 Glass furnace, 1192 Glass transition, 2288, 2289 Glaucoma, 2405 Glycerol, 2280, 2289, 2427 Gold-silica nanoshells, 2459–2460 Gradient index media, 951 Gradient index medium RTE (GRTE), 952 discrete ordinates equation, 968 Graphene, 1025 optical and radiative properties, 1044 ribbon, 1046 surface plasmon, 1045 Graphic Processing Unit, 2999 Grashof number, 369, 657 Gravity, 697, 702, 711, 716, 718, 722, 728, 735, 736, 745–748–753 driven convection, 746, 748, 751 driven thermal convection, 746 related Rayleigh number, 719, 720

Index Gray gas absorption coefficients, 1129 Gray gas model, 1111 Green’s second identity, 76 Grid-independent numerical solutions, 163 GRTE, see Gradient index medium RTE (GRTE) Guarded hot plate method, 2308 Gyrotactic microorganisms, 619 H Hamilton-Crosser model, 2928 Hanging drop technique, 775–778 Hartmann number, 613 HB parameters, 398, 400, 402 Healthy tissues, 818 Heat and moisture transfer, 888 convective, 888 model equations, 904 transient diffusion, 892 two-stage adsorption model, 893 Heat capacity, 867 Heat capacity rate ratio, 1513 Heat conduction, 6, 62, 279, 281–283 with adsorption, 33 Boltzmann transport equation, 279 composite media, 21 conjugate, 27 dimensionless, 15 in drying, 29 effect of chemical reaction on, 2735 effect of metal melting on, 2777 extended formulations, 6, 19–36 formulation, 6 Fourier’s law, 280 heat transfer inside the kerf, 2772 heterogeneous media, 19 hyperbolic, 26 with mass diffusion, 34 moving heat source (MHS), 2772, 2775 with phase-change, 23 cylindrical MHS, 2772, 2773, 2775 linear MHS, 2772, 2773 tilted MHS, 2775 thermal interface materials, 291–293 thermoelectricity, 284 volt equivalent of the heat, 2738 Heat convection, 293–294 condensation, micro/nanotextured surfaces for, 309 electronics cooling, 296

3021 equations and dimensionless numbers, 294–295 micro heat pipe, 303 microscale boiling, 305 nanoengineered icing, 312–318 Heated horizontal surface, 548 Heated thermocouple method, 2311 Heater, 356 Heater size, 1685, 1686 Heat exchange distillation (HIDiC), 1573–1574 Heat exchange equipment, 1296 Heat exchanger, 1598, 2253 applications, 1300 challenges, 1336 classification, 1296, 1297, 1450 compact, 1341–1345 constructions, 1474 design/sizing problems, 1300, 1456 effectiveness, 1321, 1325, 1461–1466, 1513 efficiency, 1328–1331 heat exchangers networks, 1335 heat transfer elements, 1450 LMTD, 1320 log mean temperature difference, 1456 microchannel, 1350 mixed and unmixed fluids, 1464 multi-pass, 1338–1341 networks, 1334–1336 NTU equations, 1463 overall heat transfer coefficient, 1454–1458 pressure drop determination, 1345, 1467 problems, 1456–1466 rating problem, 1332 reactors, 1563 sizing problem, 1332 stepping-off method, 1336 types, 1298–1300, 1449 Heat flux, 4, 6, 9, 11, 13, 14, 16, 27, 30, 33, 34, 210, 484, 1976, 2384 at base of drop, 2062–2064 Heating, 1299, 2089, 2773 duration, 2467 protocols, 2457 speed, 819 value, 2835 Heat-mass transfer analogy, 365, 367 Heat of oxidation, 2735 Heat of the reaction, 2735 Heat pipes analysis and numerical simulation, 2188–2196 applications, 2196–2203 compatibility tests, 2174

3022 Heat pipes (cont.) heat exchangers, 2201 maximum capillary head, 2175 temperature range, 2172–2174 thermal network modeling, 2182–2188 thermal resistance, 2187–2188 transport limitations, 2176–2182 types, 2168–2172 wick properties, 2176 wick structure, 2174, 2176, 2178 working fluids, 2172, 2173 Heat pumps, 1311–1313 Heat recovery boiler, 1601 Heatric, 1563, 1585 Heat sink, 353, 863, 1508, 1975 Heat sources, 182 Heat transfer, 684–687, 813, 1296, 1505, 1511, 2281, 2283, 2289, 2290, 2846 anode, 2535 cathode, 2529 circular, 550 computational fluid mechanics, 663–664 correlations, 2105 critical point and bifurcation theory, 681 enclosed rotating disks, 676 enhancement, 1508, 1606 enhancement techniques, 1984–1986, 1991, 2022 equations and theoretical methods, 658–663 experimental investigations, 655–658 flow configurations, 673–675 gas flow vs. cylindrical channel wall, 2543 horizontally adjacent, 540 inside the kerf, 2772–2781 in liquid deficient region, 1927 mechanisms, 1298, 1476 mechanisms, in nucleate boiling, 1671–1676 in metal, 2711 numerical investigations, 671–672, 677–678 parallel stream of air, rotating disk, 679–681 perpendicular jets on rotating disks, 678–679 Prandtl number, effect of, 668 problems, 182 regimes, 1913 rotating spheres, 672 similarity numbers, 654 square, 550 surface, 1296, 1298, 1299 surface area density, 1502 vertically separated, 541 vertically spaced, 544

Index Heat transfer, in vivo Bian stone, 2337 bradymetabolism, 2335 classical (Fourier) continuum models, 2346–2349 cryomedicine, 2340 cryopreservation, 2340 cryosurgery, 2340 cylindrical tissue region freezing, 2367–2368 destruction of biological tissues, 2342–2343 ectothermy, 2335 endothermy, 2335 forensic medicine, 2343 fractal models, 2351–2353 heat transfer in finite tissue region, 2369–2370 high temperature effect, on living tissues, 2339 homeothermic, 2335 low temperature effect, on living tissues, 2339–2343 mathematical modelling, 2337–2338 non-Fourier continuum models, 2349–2350 one-dimensional multiregion bioheat equation, 2362–2365 Porous media models, 2350 respiratory system models, 2345 sinusoidal heat flux, on skin surface, 2365–2367 skin and deep tissues, 2345–2346 temperature fluctuations in living tissues, 2356–2357 thermal properties of tissues, 2357–2361 vascular models, 2353–2356 whole-body models, 2343–2345 Heat transfer, micro/nanoscale, see Microscale Heat transfer coefficient (HTC), 1479, 1481–1484, 1487, 1490, 1491, 1496, 1497, 1759, 1770, 1777, 1779, 1785, 1787, 1809, 1993 dense hole array, 441–442 line-averaged, 443 spatially-averaged, 439, 441 Helical fins, 455 Helical swirl flow, 462 HELIXCHANGER ®, 1305 Helmholtz equation, 70 Hemispherical total emissivity, 1207 Henyey-Greenstein phase function, 1212, 1213 Hermite approximation, 40 Hertz–Knudsen–Langmuir equation, 2696 Heterocharge layers, 501

Index Heterogeneous nucleation, 307, 1754–1760, 2238 Hexagonal boron nitride (hBN), 1025, 1053 HIDiC, see Heat exchange distillation (HIDic) Hierarchical, 319 High-conductivity, 351, 352, 354, 356 Higher power computing systems, 1975 High-pressure arc discharges, 2525 High-Resolution TRANsmission (HITRAN) molecular absorption database, 1091–1098 High temperature exhausts, 1596 High temperature furnaces, 2203 High Voltage Circuit Breaker (HVCB), 2630, 2635, 2646, 2650 Histologic analyses, 2476 Hollow optical fiber drawing, 241 Homeothermic, 2335 Homogeneous flow model, 1878 Homogeneous model, 2019 Homogeneous nucleation, 307 Homogeneous void fraction, 1881 Homogenous nucleation, 2238 Hopf bifurcation, 708, 737 Horizontal heated surfaces, 528 pairs of, 528 Horizontal two-sided circular, 556, 557 Horizontal wavy surfaces, 578, 580 “Hot’ cathodes, 2530 Hot disk method, 202, 2229 Hot fluid, 1316, 1326, 1335, 1337 Hot probes method, 180 Hotspot location, 2003 Hotspot region, 2001 Hot spots, 336, 345, 351, 353 Hot-wire method, 181, 2228 H-shaped cavity, 347–348 Hsu’s criterion, 1653 Human biological systems, thermal properties, see Thermal properties Human body temperature regulation, 2202 Human core temperature measurement, 2336 Human eye, see Eye Human thermal comfort, 911 Humidifying chamber, 1299 Humidity ratio, 1524, 1525, 1527 Hydrated salt, 862 Hydration, 2224 Hydraulic conductivity, 2424, 2432 Hydraulic diameter, 376, 1809 Hydrocarbons, 2843 Hydrodynamic entry length, 484, 1467 Hydrogen, 2843

3023 Hydrophilicity, 310, 1839 Hydrophilic surfaces, 1771, 1775 Hydrophobicity, 1839 Hydrophobic membrane cover, 1967 Hydrophobic surface, 1775 Hyperbolic metamaterials, 1024 Hyperbolic phonon polaritons (HPPs), 1055 Hyperosmotic stress, 2422 Hypertension, 819 Hyperthermia, 818, 2456 Hypertonic solution, 2423 Hypo-osmotic solution, 2422 Hypotonic solution, 2423 Hysteresis, 684, 1701, 1733 I Icephobic, 312 Ideal dropwise condensation, 2055 Ideal porous solid, 836 IIF, see Intracellular ice formation (IIF) Ill-posed problem, 184, 1245 discrete, 1248 Imperfection, 333 Impermeable solute, 2426, 2427 Implant, 2505, 2507 Improved heat transfer, 1373 IMSL library, 84 Inactive velocities, 395 Inanimate, 330, 332 Inception superheat, 1652–1655 Incident radiation, 943 Incineration, 2833–2834 Incineration plant, 2834 Inclusions, 2241 Indirect contact type, 1476, 1478, 1480 Indirect evaporative coolers, 1528, 1529 Indirect evaporative cooling, 1525, 1528 Induced emission, 2607, 2621 Induction heating, theory of, 2574, 2576, 2577 Induction pumping, 496, 508 Inductively Coupled Plasmas (ICP), 2645, 2648 Industrial furnace, 1192 Industrial processes, 920 Industrial-scale, 1302 Industrial waste heat, 1596 Infrared camera, 210 Infrared (IR), 2602 Infrared thermography, 656 emissivity estimation and correction, 798 heat transfer augmentation, 798–801 Inhibitors, 1638

3024 Inhomogeneous model, 2019 Initial conditions, 9, 2495, 2498, 2505 Injection, 450, 2889 Injection pumping, 493 Injury, 2280 Inlet orifice, 2004 Inlet restrictors, 1781–1782 Innovative flow-boiler, 2082 Innovative flow-condenser, 2082 Inorganic components, 2845 Inorganic materials, 862 Instabilities, 1781, 1782 Insulating materials properties, 1400 Integral balance procedure, 111 Integral equations of first-kind (IFKs), 1246 Integral technique, 1712, 1720 Integral transform pair, 75 Integrated autocorrelation time, 200 Integrated circuit (IC), 299 Integro-differential equation, 928 Intensification, 448 Intensification, small scale turbulence, 404 Intensity, 2625 Interaction, 394, 2910 Interband transitions, 1044 Interface temperature drop, 2061–2062 Interfacial area concentration transport equation, 1934 Interfacial area transport equation (IATE), 1871 Interfacial dynamics, 1825 Interfacial dynamics closures, 1931 Interfacial force model, 1933 Interfacial mass and heat transfer, 1934 Interfacial properties of liquids, 1829 Interfacial tension, 1825, 1841 relaxation, 1830–1836, 1842 Interferometer, 766 Intermittency levels, 395, 400 Internal combustion gas engines, 2871 Internal elastic lamina (IEL), 819 Internal flows, 1467–1471 in channel, 381–384 characterization, 375 in circular pipe, 377–378 temperature solution, 378–381 Internal heat generation, 614 Inter-segmental ventilation, 907, 911 Interstices, 333, 355 Intima, 819 Intraband transitions, 1046 Intracellular ice formation (IIF), 2428, 2429, 2431 Intraocular tumors, 2386, 2387

Index Intratumoral delivery, 2467 Intratumoral injections, 2477 Intravitreal drug delivery, 2403 Intravitreous and transscleral routes, 2383 Inverse ALBDF function, 1116 Inverse design criteria, 928 Inverse heat transfer problems, 182 Inverse parameter estimation problems, 929 Inverse problems, 183, 248–251, 841, 928 Bayesian framework, 184–186 Bayesian inference, 1261–1263 definition, 1244 linear regularization technique, 1258–1261 Markov chain Monte Carlo methods, 188 maximum a posteriori objective function, 187 metaheuristic methods, 1258 nonlinear programming, 1255 objective, 1245 parameter estimation, 1276–1287 property, 1245 radiant enclosure, 1265–1276 types, 1247–1251 Inverted annular flow, 1875 Inward melting, 2235 Ion drag pumping, 493, 507–508, 511–512 Ionization, 2846 Ionization equilibrium, 2767 IR diode laser, 2386, 2387 IRE, see Irreversible electroporation (IRE) Irregular domains, 14, 90 Irreversibility, 330, 333, 355 Irreversible electroporation (IRE), 2491, 2492, 2494, 2496, 2499, 2500, 2502, 2503, 2505, 2508 IR spectrum, 2605 I-shaped, 551 Isochoric cooling, 2444 Isofrequency surface, 1053 Isothermal freezing, 316 Isotherms, 814 Isotonic solution, 2422 Isotropic, 8 Iterative convergence, 162, 166, 169 Iterative methods, 161, 162 J Jet array, 1986, 1987 Jet impinging, 679 Joule heating, 498, 2491, 2495, 2499, 2500, 2502, 2512, 2514

Index Jumping droplet, 311 Junction temperature, 1999 K Kelvin-Helmhotz instability, 1716 Kelvin-Helmholtz Rayleigh-Taylor (KHRT) breakup model, 2952 K-epsilon model (turbulence model), 402, 415, 1932 Kiefer-Wolfowitz algorithm, 1271 Kinetic equilibrium, 2587 Kirchhoff’s law, 1031, 2606, 2607, 2609, 2611, 2615, 2623, 2625, 2644 Knowledge-based design, 257–259 Knudsen, 2893 Knudsen effect, 2951 Koch heat transfer company, 1547 Kolmogorov length scale, 415 Kolmogorov time scale, 416 Kuei-Lai-Mow (KLM) biphasic model, 847 L Lagrange multiplier method, 264–265 Lagrangian, 2895 Lagrangian–Eulerian formulation, 811 Lagrangian stream operator, 939 Lamellae, 2968 Laminar flame, 1175–1178 Laminar flows, 376, 551, 1469 heat transfer augmentation, 404, 414 Laminar layer heat transfer influence of turbulent scale, 405–413 stagnation region, 404–405 turbulent augmentation, 413–417 Laminar mixed convection heat transfer, 810 Landau-de Gennes model, 684 Landau model, 680, 681, 684 Landscape, 331, 356 Large-eddy-simulation (LES), 652, 664, 671, 677, 680, 1175, 2935 Large scales, 404, 407, 412, 418 Laser capsulotomy, 2386 Laser energy, 2462 Laser eye therapies chemical reactions, 2389 corneal laser treatment, 2385 denaturation of protein, 2389 excimer laser, 2388 femtosecond lasers, 2388 lens laser treatment, 2386 microbubble formation, 2389

3025 photo-thermal interactions, 2390 photo vaporization and mechanical damage, 2390 retinal laser treatment (see Retinal laser treatment) TTT, 2387 vitreous humor, 2387 Laser in situ keratomileusis (LASIK), 2386 Laser irradiation, 2396 Laser photothermal therapy, 2457 Laser propagation, 2457 Laser radiance, 2467 Laser retinal surgery, 2386, 2396 Laser schlieren, 763 Laser spot size, 2467 Laser surgery, 2391, 2392, 2396 Laser vitreolysis, 2386, 2387 LASIK, see Laser in situ keratomileusis (LASIK) Latent heat, 861, 1302, 2218, 2360, 2513 Latent heat of vaporization, 1474, 1522 Latent heat thermal energy storage system (LHTESS), 2220 Lateral-axial strain ratio, 846 Lattice Boltzmann method, 610 Lattice structures, 2238 LC circuit model, 1035 Leading edge heat pipe, 2172 Leakage power, 1976 Least squares, 183 Least-squares scheme (LSFEM), 965 Ledinegg instability, 1958 Leidenfrost point, 1725 Leidenfrost temperature, 1725, 1729, 1914 LES, see Large-eddy-simulation (LES) Level of gravity, maximum heat flux, 1691 Levenberg-Marquardt method, 846, 1258 Lewis Number, 368, 609 Lid-driven enclosure, 611 Lift force, 1866 Likelihood function, 183, 186, 193 Linear regularization techniques, 1252–1255 Line broadenings, 2616 Line-by-line (LBL) model, 1096–1097, 1182, 1233 Line heat source, 2772 Lines, 2608, 2615 Line shift, 2618 Line to point, 355 Liquid alkali metals basic properties, 1427, 1433 density (liquid) vs. temperature, 1432 density (vapor) vs. temperature, 1432

3026 Liquid alkali metals (cont.) dynamic viscosity (liquid) vs. temperature, 1436 dynamic viscosity (vapor) vs. temperature, 1436 kinematic viscosity (liquid) vs. temperature, 1437 latent heat of evaporation vs. temperature, 1439 Prandtl number (liquid) vs. temperature, 1438 Prandtl number (vapor) vs. temperature, 1438 specific heat (liquid) vs. temperature, 1435 specific heat (vapor) vs. temperature, 1435 surface tension vs. temperature, 1439 thermal conductivity (liquid) vs. temperature, 1434 thermal conductivity (vapor) vs. temperature, 1434 thermal diffusivity (liquid) vs. temperature, 1437 vapor pressure vs. temperature, 1431 Liquid and gaseous heat transfer agents, 1299 Liquid breakup, 2936 Liquid coolants, 1976 Liquid cooling, 1976, 2022 Liquid crystal thermography, 782 heat transfer, 791–796 surface mounted rib turbulator, 783–789 synthetic jet, 789–791 Liquid film dryout, 1913 Liquid fraction, 867 Liquid–gas phase change, 862 Liquid jet in cross flow, 2937–2940 Liquid metals, 1369 Liquid mixture density, 1803 Liquid–solid contact angles, 2973 Liquid–solid interface, 867, 1825, 1826, 1831 wetting, 1837 Liquid subcooling, 1681–1682, 1690–1691 Liquid-surface wetting, 1828 Liquid–vapor interface, 1806, 1825, 1844 Liquid–vapor phase-change, 1824 Liver, 2494, 2497, 2499, 2501, 2503, 2511, 2513 Π-Loading, 836 Local thermal non-equilibrium (LTNE) model, 862, 863 Local thermodynamic equilibrium (LTE), 2601, 2606, 2615, 2620, 2626, 2631, 2661, 2664, 2686, 2932 Local ventilation, 904

Index Lockhart–Martinelli parameter, 1814 Log-mean temperature difference (LMTD), 1320, 1456–1461, 1481, 1483–1485 correction factor, 1323, 1334, 1336 Lonza, 1560, 1568–1571 Looped-wire mesh insert, 461 Loop heat pipe, 2171, 2195 Lorentz-Lorenz formula, 765 Lorentz profile, 1087 Lorenz-Mie theory (LMT), 1166–1167 extensions, 1168 geometrical optics approximation, 1168 Rayleigh approximation, 1167–1168 Loss of available superheat, 1801 Louvered fins, 459, 460 Low-grade and high-grade waste heat, 1604 Low-temperature cycles, 1602 Lubricant, 1804, 1818 excess layer, 1811 excess surface density, 1811 Luminous flame, 1183 Lumped capacity method, 2952 Lumped formulation, 36 improved, 40 Lumped thermal capacity method, 581 Lymphedema, 849 M Mach-Zehnder interferometer, 762, 763 Macro-scale, 2262 Macular degeneration, 2401 Magnetic field, 2571 Magnetic pinch force, 2663, 2701, 2715 Magnetic polaritons (MPs), 1024, 1032 Magnetic resonance, 1035 Magnetic resonance poroelastography, 850–853 Magnus, 2896 Maisotsenko cycle (M-cycle), 1522, 1526, 1527, 1529 Mal-distribution, 1351 Mangler coordinate transformation, 374 Manifold, 1507 Manifold microchannel (MMC), 297–298 Manual metal-arc (MMA) welding, 2660 Manufacturing, 222, 223, 249, 262 Mapping methods, 1109 Marangoni effect, 2715 Marco encapsulation, 2244 Markov chain Monte Carlo (MCMC) methods, 188–189 Markov chains, 189–191, 1202 Markov random field, 196

Index Mass balance, 2857 Mass burn, 2833 Mass conservation equation, 2682 Mass continuity equation, 2662 Mass flux, 30, 31 Mass fractions, 2850 Mass transfer processes, eye aqueous humor, 2400 choroidal blood flow, 2401 drug delivery (see Drug delivery) tear dynamics, 2399–2400 vitreous humor, 2401 Material decomposition, 2846 Mathematica system, 66 Mating surface, 1507 Maximum a posteriori objective function, 187–188 Maximum drop size, 2069 Maximum heat flux, 1648, 1669, 1679, 1685–1687 on cylinders, 1689, 1690 Maximum internal pressure, 1366 Maximum likelihood approximation, 1237 Maxwell’s equations, 920, 981–988, 990, 1009, 1026 Mean absorption coefficient (MAC), 2635, 2637, 2640, 2642 notion of, 2637 Planck, 2637 Mean free path, 1237 Measurement errors, 182 Measurement techniques, 2227 Measures of error, 166 Mechanical aids, 450, 469 Mechanical cleaning, 1638 Mechanical coupling, 772 Mechanical integrity, 1508 Mechanical interlocking, 2968 Mechanistic model, for nucleate boiling heat transfer, 1650 Media, 819 Medical imaging, 832 Melting, 2846 Melting front, 867 Melting point, 2512, 2514 Melting temperature, 867 Membrane damage, 2427, 2429 Membrane lipid, 2428 Membrane permeability, 2422, 2424, 2426, 2429, 2436, 2437, 2442–2443 Membrane pore, 2422 Membrane transport, 2424, 2432, 2441 Memory storage, 2266

3027 Meniscus, 1752, 1770 Meshless methods, 129 Metabolism, 2472 Metaheuristic methods, 1258 Metal active-gas (MAG) welding, 2660 Metal/dielectric/metal structures, 1024 Metal foam, 862, 1506 Metal inert-gas (MIG) welding, 2660 Metallic, 2914 Metallurgy, 1300 Metals basic properties at normal conditions, 1369, 1391 electrical resistivity vs. temperature, 1375 specific heat vs. temperature, 1371 thermal conductivity vs. temperature, 1370 Metal vapour, 2679, 2696, 2706 Metamaterial absorbers, 1037 Methane, 2873 Method of false transients, 162 Method of Moments, 1162 Method of weighted residuals (MWR), 130, 131 Metropolis-Hastings algorithm, 191–193 MHP, see Micro heat pipe (MHP) Micellization, 1832 Microbubble, 2389–2390 Micro/nanostructured surfaces, 929, 1025 Micro/nanostructures, 1024, 1769, 1771, 1787 Micro/nanostructuring, 1760, 1781 Microchannel, 1506, 1954, 1980, 1992, 1997, 2003, 2004, 2006, 2008, 2013, 2016, 2019 critical heat flux, 1963–1965 definition, 1954–1955 pin fins and nanowires, 1966 radial, 1968 Microchannel flow/convective boiling annular flow, 1781 artificial nucleation sites, 1783 bubbly flow, 1780 COP, 1781 inlet restrictors, 1781 slug flow, 1780 surface structuring, 1784 Microchannel heat exchangers, 1350–1351, 1476, 1538–1545 applications, 1493–1495 thermal design of, 1495–1497 Microchannel heat sink, 1493 Microclimate ventilation rate, 903 Micro-convection, 1826, 1844 Micro-electro mechanical systems (MEMS), 2265

3028 Micro encapsulated PCMs, 2245 Microfin tubes, 1808 Microgap, 2013 Microgravity, 1684, 1691 Micro heat pipe (MHP), 301–305, 2172 Micropillars, 1785–1786 Microreactor, 1560, 1563, 1571, 1572 Microscale, 279, 2262 heat conduction (see Heat conduction) single phase convection, 295–300 two phase convection, 300–318 Microscale boiling, 305–309 Microscale convection condensation, micro/nanotextured surfaces for, 309–312 electronics cooling, 296 micro heat pipe, 301 microscale boiling, 305 nanoengineered icing, 312 Microscale laser induced fluorescence (μLIF), 300 Micro-scale transport mechanisms, 1825 Microstructure, 2971 Mie scattering, 1212, 2462 Mie theory, see Lorenz-Mie theory (LMT) Mikic-Rohsenow model, 1762–1763 Millimeter-scale annular flow-boilers’ design, 2122 Millimeter-scale annular flow-condensers’ design, 2133 Millimeter-scale boilers, 2078 Miniature heat pipe, 2172 Miniaturization, 350 Minichannels, 1954 Minimum film boiling (MFB), 1700, 1703 heat flux and temperature, 1725 and transition boiling, 1738 Mirror-like reflecting surfaces, 1215 Mixed convection, 364, 384, 902 Mixture model, 1915 Mixture thermal conductivity, 868 MMC, see Manifold microchannel (MMC) Model, 2493, 2496, 2498, 2500, 2502, 2505, 2508, 2512 Model-based elastography, 846 Modeling, 2078, 2889 Modeling coating formation, 3001–3005 Mode of experiments, 1689–1690 Modified/enhanced surfaces, 1778 Modified force-balance model, 1771–1772 Modified Gauss-Newton method, 846 Modified Peclet number, 1555, 1559, 1561 Modified SORTE (MSORTE), 950

Index Modulated DSC techniques (MDSC), 2230 Molar fractions, 2850 Molecular adsorption, 1831, 1837 Molecular agglomeration, 1837 Molecular bands, 2619, 2620, 2623, 2635 Molecular diffusion, 889 Molecular dynamics, 1828–1831, 2260 Molecular dynamic simulation, 2038 Molecular gases, 1231 Molecular kinetics, 1830 Molecular-scale adsorption dynamics, 1826 Molecular vibrational-rotational energy transitions, 1077 Momentum boundary layer, 365 Momentum conservation equation, 2663, 2664 Momentum equation, 867 Momentum force-balance model, 1765–1767 Monopolar, 2494, 2497, 2499, 2501, 2503, 2504, 2507, 2508, 2510, 2512, 2514 Monopolar electrodes, 2494, 2497, 2499, 2501, 2503, 2504, 2507, 2508, 2510, 2512 Monte Carlo method, 1182, 1184, 1187, 1189, 1190, 1202, 1219 Monte-Carlo simulations, 3002–3005 Monte Carlo techniques, 928 Moody diagram, 378 Mouse model, 2467 Moving boundary, 2232 Moving boundary problems, 2231 Moving heat source (MHS), 2772, 2775 MR imaging, 850, 2478 MRS-criterion, 662 Multiangle elastic light scattering (MAELS), 1283 Multi-component droplet, 2955 Multigrid methods, 161 Multilayer periodic micro/nanostructures, 1026 Multi-louver fins, 1517 Multi-objective optimization, 269–272 Multipass heat exchangers, 1453 Multiple heat sources, 2193 Multiple scales, 241–245 Multiplication approach, 1128 Multi-scale analysis, 229 Multi-spot, 2396 Municipal solid waste (MSW), 2832 Municipal waste incinerator, 1192 Mushy zone, 867, 2718 N Nanocapsules, 2412 Nanocoating, 1617, 2245

Index Nanocomposite, 1617 Nanoengineered surfaces, 313 Nanofilms, 206–214 Nanofins, 2244 Nanofluids, 533, 754, 864, 1799 Nano lithography, 2263, 2266 Nanolubricants, 1799, 1802 Nanomaterials, 2888 Nano/microscale heat transport, see Microscale Nanoparticle, 594, 613, 862, 1708, 1728, 1799, 1815, 2240 additives, 1545–1549 delivery, 2464 surface density, 1816 thermal plasma (see Thermal plasma, nanoparticles) Nano-PCM, 864 Nanopowder, 2792, 2793, 2816–2817, 2820 Nanorods, 2461 Nano scale encapsulation, 2245 Nanosecond pulsed electric fields (nsPEFs), 2515 Nanoshells, 2461 Nanostructured surfaces, 2055 Nanostructure enhanced phase change material (NEPCM), 2230 Nanostructures magnetic polaritons (MP), 1032 surface plasmon polaritons (SPP), 1032 Nanotextured, 312 Nano-warming, 2443 Nanowires, 285, 286, 1784–1785 Nanowires within microchannels, 1966 Naphthalene sublimation technique, 655, 656 Narrow bands, 1101 model spectrum, 1097 Narrow plates, 528 Narrow vertical plates, 533 Nasolacrimal, 2404 Natural convection, 364, 384–387, 529, 693, 696–699, 702–709, 723, 809, 900, 1385, 2257, 2401 porous media (see Porous media) Taylor-Proudman column, 701 Natural convective heat transfer, 529 Navier–Stokes equation, 482, 651, 658, 660, 662, 673, 682, 2388 Nd-YAG laser, 2386 Near-critical point, 1361 Near-field, 981, 985, 986, 988–1003, 1007 Near-field radiation transfer (NFRT), 920, 929 Near infrared (NIR) range, 2462 Necrosis, 2508

3029 Neovascularization, 2390 Net emission coefficient (NEC), 2625, 2631, 2632, 2638, 2640, 2665, 2668, 2676, 2680, 2682 advantages, 2633 for air plasma, 2629 definition, 2626 metal vapor influence, 2628 vs. temperature, 2627 at very low temperature, 2630 Newtonian fluid, 811, 2927 Newton’s Cooling Law, 10 Newton’s method, 1257 Nix correlation, 407 Nodes, 682 NO emission, 1178, 1192 NO formation, 1177, 1182, 1186 Non-annular flow-regimes, 2085 Non-condensing gas, 2043–2046 Non-continuum effect, 2951 Non-Darcian effects, 809, 810 Non-Darcy model, 745, 751, 752 Non-equilibrium effects, 2572 Non-Fourier continuum models, 2349 Nonhomogeneous heat conduction problem, 80 Nonlinear boundary conditions, 115 Nonlinear eigenfunction expansions, 121 Nonlinear problem, 83 Nonlinear programming (NLP), 1255–1258 Non-luminous flame, 1174, 1183 Non-participating medium, 926 Non-spherical nanoparticles, 1804 Non-thermionic cathode, 2694 Non-transferred arc plasma systems, 2846 Non-transferred arcs, 2647 Non-transferred DC arcs, 2548 Non-uniform heating/cooling, 2104 Normal heat transfer, 1373 Normal integral scale, 394 Normalization integral, 67 Normalized eigenfunction, 84 nsPEFs, see Nanosecond pulsed electric fields (nsPEFs) ε-NTU, see Number of Transfer Units (NTU) method Nuclear fuel properties, 1402 Nuclear fuel rod, 47 Nuclear-reactor coolants basic reference parameters, 1430, 1440 density, 1441 dynamic viscosity, 1442 specific heat, 1442 thermal conductivity, 1441

3030 Nucleate boiling, 1825, 1828, 2106 curve, 1827 on flat plate heaters, 1686 heat transfer, 1674, 1679–1685 heat transfer in polymeric solutions, 1842–1844 heat transfer in reagent solutions, 1839–1842 on inclined surfaces, 1680 Nucleate boiling curve, 1827 Nucleate boiling suppression, 1496, 1497 Nucleate pool boiling, 1386 Nucleation, 2009, 2237, 2428, 2429, 2431, 2437, 2441, 2443, 2794, 2795, 2798–2800, 2803, 2804, 2808, 2809, 2815, 2817, 2820, 2822, 2914 Nucleation promoters, 2238 Nucleation sites, 1844, 2060 Nucleation site density, 1937 Number density of active nucleation sites, 1650, 1656–1658, 1663, 1675 Number of Transfer Units (NTU) method, 1325–1328 Numerical diffusion, 970 Numerical methods, conduction-type phenomena, see Conduction-type phenomena Numerical simulations, 1660, 1661, 1672, 1675, 1683, 2397 Numerical stability, 946 Numerical strategies, 2234 Nusselt, 2898 Nusselt approximations, 2039 Nusselt number (Nu), 365, 380, 382, 654, 658, 665, 670, 672, 676, 679, 681, 683, 685, 729, 732, 754, 788, 790–791, 801, 1540, 1808, 1810, 2001, 2953 O Oberbeck-Boussinesq approximation, 607 Ocular tumors, 2393 Offset-strip fins, 459 Oil compatibility, 1633 On demand roughness, 453 1D anisotropic gratings, 1027 One-dimensional Energy Balance, 2101 One-dimensional heat transfer model, 793, 797 One-dimensional modeling, 2101 One-dimensional multiregion bioheat equation, 2362 One-sided surfaces, 548 Onsager field enhanced dissociation, 500

Index Onset of nucleate boiling, 1956–1957 Opaque surface, 1214 Open cavities H-shaped cavity, 347 isothermal elemental, 334–339 T-shaped cavity, 339 X-shaped cavity, 348 Y-shaped cavity, 345 OpenFOAM, 1555 Open microchannels with manifold (OMM), 1967, 1968 Optical and radiative properties of graphene, 1044–1046 Optical fiber drawing, 223, 229, 231, 236, 238, 241, 245, 253, 266 Optically smooth surface, 1215 Optically thick media, 1236 Optically thin, 2910 Optically thin approximation (OTA), 1176 Optically thin fluctuation approximation (OTFA), 1179 Optical methods, 2762 Optical phonon, 1053 Optical properties, 2457 Optical properties of surface, 929 Optical theorem, 1153 Optical window, 2457 Optimization, thermal systems basic aspects, 262–264 Lagrange multiplier method, 264 multi-objective optimization, 269 response surfaces, 267 search methods, 265 Optimization strategies, see Optimization, thermal systems Order of accuracy, 164 Order-of-magnitude cost estimation, 2876 Order parameter, 683, 684 Ordinary differential equations, 84 Organic material, 862 Organics, 2512 Organic substances, 2840 Orthogonality property, 67 Orthotropic materials, 202–206 Oscillator strength, 2617 Oseen, 2890 Osmolality, 2423, 2426 Osmotic injury, 2421–2423, 2428, 2433 Osmotic process, 2428 Osmotic stress, 2423 Ovens, 1308, 2202 Overall heat transfer coefficient, 1454, 1481, 1511, 1517

Index Overall surface efficiency, 1456 Overheated vapor, 1361 Oxidation agents, 2848 Oxy-fuel, 1190 Oxytactic microorganisms, 620 P P1, 1182, 1183 Pacific Northwest National Laboratory (PNNL), 1539 Pan retinal photocoagulation (PRP), 2390 Papermaking, 1300 Paraffin, 862, 2221 Parallel channel instability, 1958–1959 Parallel flow exchanger, 1515 Parallel flow heat exchangers, 1451, 1464 Parallelization, 1238 Parallel processing, 151, 1238 Parameter estimation, 1245, 1276 Partial characteristics, 2642 Partial integral transformation scheme, 85 Partial oxidation, 2848 Partial pressure of water vapor, 1524 Participating media, 1231 Participating medium, 926 Particles, EM wave interaction, 920 Particle size, 2460 Particle swarm optimization (PSO), 1260 Particulate drug delivery, 2412 Particulates, 2888 Passive technique, 448, 450 Pathlength based algorithm, 1227, 1236 Pattern-preserving grid-refinement, 164, 168 PCM, see Phase change materials (PCMs) Pdf transport method, 1187 Peak heat flux, 1648 Peclet number, 365, 698, 715, 717, 2973 PEFs, see Pulsed electric fields (PEFs) Penetration depth, 997, 1004 of electric field, 1036 Penetration height, 2936 Pennes bioheat equation, 2498 Pennes Heat Transfer Equation, 2394 Pennes’s bioheat equation, 819 Performance evaluation criteria, 470–473 Periocular administration, 2407 Periodic metamaterial applications, 1039 Periodic nano/microstructure applications, 1039 Periodic nanostructures, 1026 Permafrost stabilization, 2203

3031 Permeability, 608, 2422, 2423, 2427, 2429, 2435, 2436, 2442 of vacuum, 1030 poroelastogram, 848 Permeability index, 891 Permeable solute, 2426 Permittivity, of vacuum, 1030 Permittivity tensor, 1029 Petrochemical, glass, 1300 Phase, 2492, 2512, 2515 Phase change, 300, 1305, 2285, 2287, 2290, 2291 Phase change heat transfer, 1672, 1831 Phase change materials (PCMs), 861, 2216, 2492, 2512–2514 analytical solutions, 2230–2236 chemical kinetics considerations, 2226 chemical properties, 2226 classes of, 2221 inorganic, 2222–2225 macro-encapsulated, 2244–2245 material property measurement techniques, 2226–2230 micro/nano encapsulated, 2245–2248 organic, 2220–2222 physical properties, 2225–2226 thermal properties, 2226 thermodynamic properties, 2225 types, 2220 Phase-change process, 1825 Phase Change Slurries (PCS), 2254 Phase equilibrium diagram, 1800 Phase function, see Scattering Phase segregation, 2226, 2237 Phase transformations, 862 Phase transition temperature, 2227 Phenomenological model DNB prediction, 1938–1941 dryout prediction, 1938 Phonons, 280, 282, 286, 291, 293 Photocoagulation, 2385, 2387, 2390, 2393, 2413 Photocoagulation temperatures, 2390 Photodissociation, 2613 Photodynamic therapy (PDT), 2385, 2393 Photoionization, 2613 Photon bundles, 1203, 1206, 1227, 1239, 1240 Photonic crystals, 1024 Photons, 2469 Photoreceptors, 2384, 2389, 2395, 2401 Photo-thermal interactions, 2390 Physical characteristics, 2236 Physisorption, 1830, 1837

3032 Piecewise constant angular quadrature (PCA), 967 Piercing, 2733 Pilot arc (PA), 2731, 2733, 2735, 2786 Pin fin array, 1979, 1992, 2015 Pin fins within microchannels, 1966 Planck blackbody emissive power, 1074 Planck constant, 922 Planck function, 2616 Planck Law, 922, 2605 Planck mean, 2641 Planck Mean Absorption Coefficient, 2637 Planck’s law, 2603, 2604 Plane of incidence, 1033 Plane wave, 1033 Plasma, 2889 Plasma arc gasification, 2836–2840 Plasma arc welding (PAW), 2661 Plasma coatings coating build up, 2999–3005 droplet impacts, 2971–2984 impacts, mathematical model of, 2984–2999 initial conditions, 2988 Plasma column (plasma jet), 2648 experimental observation ionization equilibrium in, 2767 modeling, 2764 Townsend breakdown, 2771 turbulent and laminar flow, 2766 Schlieren method, 2764 spectroscopy, 2762 Plasma composition, 2664 Plasma enthalpy, 2858 Plasma fields, 2586 Plasma gases, 2846 Plasma generating gases, 2565 Plasma jet, 2561 See also Plasma column (plasma jet) Plasma jet fluctuations, 2537 Plasma jet modeling, 2929–2936 Plasma modeling, 2764–2767 Plasma-particle, 2910 Plasma pyrolysis, 2847 Plasma swirl and erosion rate, 2754–2755 Plasma torch, 2846 Plasma waste-to-energy (WTE) process, 2870 biotechnology, 2873 cogeneration and CHP, 2872 electricity production, 2871 energy products, 2871 financial viability, 2876–2879 Fischer-Tropsch process, 2872

Index methane and alcohol production, 2873 process layout, 2873 waste type, 2874–2876 Plasmon resonant absorption, 2457 Plate-fin, 458 Plate-fin heat exchanger, 1492 Plate heat exchangers, 1490–1491, 1623–1624 Plate-type heat exchangers, 1453, 1489–1491 P1 method, 2636, 2637 PN approximation, 955 PNNL, see Pacific Northwest National Laboratory (PNNL) Poiseuille flow, 377, 381, 900 Poisson equation, 494 Poisson’s ratio, 834 time constant elastogram, 848 Polaritons, 929 Polarization, 492 Polarization angle, 1038 Polarization dependence, of radiative properties, 1037 Polished surfaces, 1215 Pollutant emissions, 1174 Polyatomic molecules, 2037 Poly-ethylene terephthalate (PET), 2229 Polymeric solutions, 1833 Polymeric surfactants, 1831 Pool boiling, 306–308, 1677, 1678, 1683, 1758–1760, 1807, 1811 boiling curve hysteresis, 1701 critical heat flux, 1763–1767 definition, 1698 film boiling (see Film boiling) isolated bubble regime, 1760–1763 MFB point (see Minimum film boiling (MFB)) micro/nanostructures, boiling surface, 1758–1760 Nukiyama’s experiment, 1698 parametric effects, 1707 surface thermal conductivity variation, 1763–1765 surfactant adsorption, 1767 transition boiling, 1700, 1732 wettability of surface, 1774 Porcine, thermal properties, see Thermal properties Pore density, 862 Pore pressure, 834 Pore size, 863 Poroelastic response, 833 Poroelastography, 833 Porosity, 836, 862, 889, 2971, 3000–3001

Index Porous, 301, 307 Porous media, 693, 695, 748–753, 808, 2017 anisotropic effects, 753–754 binary alloys, 754 centrifugal body forces, 728–740 convective flows, classification of, 722 Coriolis effect, 740 flow, modeling of, 709–713 heat transfer, modeling of, 713–716 heterogeneous, 720–722 homogeneous, 716–720 nanofluids, 754 Taylor-Proudman columns and geostrophic flow, 724 Porous media models, 2350, 2410 Posterior probability density, 185, 186 Post processing, 1220 Power balance, 2859 Power dissipation, 1996 Power dissipation density, 1031 Power machinery, 1300 Poynting vector, 984, 987, 992, 998, 1055 definition, 1151 time averaged, 1151 Prandtl, 2899 Prandtl number, 365, 367, 371, 606, 654, 658, 668–672, 676, 678, 684, 699, 706, 708, 715, 717, 720, 737, 745, 746, 751, 753, 1712, 1741, 1809 Predictive error model, 213 Preliminary cost estimation, 2877 Pre-processing step, 1217 Pre-sheath, 2686 Pressure difference, 889 Pressure drop, 1345–1350, 1382, 1479, 1481, 1485–1486, 1490, 1558–1559, 2021 determination, 1466–1471 during flow boiling, 1959–1962 Pressure effects, 2616 Pressure number, 711, 717 Primary droplet breakup liquid jet in cross flow, 2937 numerical modeling, 2943–2945 two-fluid atomizers, 2940 Primary drops, 2060 Printed circuit, 1505 Prior density, 186 Probability-based methods, 129 Probability density function (PDF), 1207 Process function, 1478 Process heating, 1596 Processing, 2888 Process intensification

3033 bubble injection, 1552 classification of, 1538 electrostatic fields, 1553 flow rotation, 1550 gas phase reaction with catalyst wall, 1555–1559 liquid-phase reactions, 1559–1562 microchannel heat exchangers, 1538 nanoparticle additives, 1545 suction, 1554 surface modifications, 1549 swirl flow enhancements, 1545 ultrasonic enhancement, 1551 Producer gas, 2871 Production tools, 2202 Promoter, 2053 Propagating modes, 988, 989, 1004 Propagating waves, 995 Properties effective, 1148 electromagnetic, 1148 optical, 1147–1148 particle shape, 1146 particle size, 1144–1145 radiative, 1146, 1150–1156 determination methodology, 1157–1158 inferring from experiments, 1168 predicting, 1158–1160 Proposal distribution, 192 Propylene glycol, 2427 Protein crystal growth, 777, 778 Protein denaturation, 2281, 2286, 2297 Protrusions, 1984, 1985 Pseudo-boiling, 1374 Pseudocritical line, 1361 Pseudocritical point, 1361 Pseudo-film boiling, 1374 Pseudo-random numbers, 1216 PSI-Cell method, 2709 Psychrometric chart, 1522, 1524, 1526, 1530 Psychrometrics, 1523–1525 Pulsatile, 2410 Pulsatile blood flow, 819 Pulsatile flow realizations, 2150 Pulsating heat pipe, 2172, 2196 Pulsation frequency, 819 Pulsed electric fields (PEFs), 2490, 2491 Pulse parameters, 2491, 2502, 2505, 2508, 2514 Pumping power, 1781, 1787 Pyrolysis, 2834, 2843

3034 Q Quadratic model, 1255, 1258 Quality of cut, 2781–2786 Quantum confinement, 284, 286 Quasi-Newton method, 1257 Quasi steady state, 2234 Quenching, 1700, 1707, 1721, 1728, 1732, 1738, 1740 R Radial microchannels, 1968 Radiant enclosure analysis, 1265 Radiation, 10, 45, 47, 891, 1710, 1711, 1723, 2675, 2676, 2853, 2899 Radiation emission, 2628, 2641 from volume, 1207 Radiation energy density, 944 Radiation heat transfer, 1389 Radiation intensity, 922, 2603, 2604, 2621, 2623 Radiation losses, 2648, 2649 Radiation power, 2648 Radiation transfer, 920 Radiative and optical properties of graphene, 1044 Radiative attachment, 2611 Radiative energy balance equation, 1204 Radiative flux, 982, 984, 986, 988, 992, 998, 1004, 1008, 2625, 2651 Radiative flux profile, 921 Radiative heat flux, 2651 Radiative heat flux vector, 943 Radiative losses, 2648 Radiative properties, of selected materials, 1401 Radiative properties of surface, 929 Radiative recombination, 2611 Radiative transfer, 1071, 2623, 2626, 2627, 2636 bases of, 2621 Radiative transfer equation (RTE), 921, 922, 936, 1175, 1176, 1234 classical, 936 in gradient index medium (GRTE), 952 Lagrangian form, 938 refractive media, 950 Radiators, 1307–1310 Radio frequency, 2570 Radiofrequency ablation, 2491 Radio frequency plasma, 2846 Rainbow schlieren, 767–769 Random number, 1214, 1216 Random number generators, 1216

Index Random sampling, 1206 Rank-deficient, 1248 Rapid distortion theory, 404 Rarefied vapor flow, 2194 Rating problem, 1332–1334, 1510, 1511 Ray effects, 972 Ray equation, 953 Rayleigh approximation, 1164 for spheres, 1167 Rayleigh-Benard convection, 771 Rayleigh-Debye-Gans for fractal aggregates (RDG-FA) approximation, 1165 Rayleigh-Debye-Gans fractal aggregate (RDF-FA) theory, 1286 Rayleigh distribution, 194 Rayleigh-Gans approximation, 1164–1165 Rayleigh number, 369, 385, 699, 706, 707, 719, 720, 733, 735, 737, 746, 747, 751, 753, 754, 809 Rayleigh scattering, 1212, 2462 Rayleigh–Taylor instability, 1715 Ray path coordinate, 936 Ray tracing, 1219, 1221 RDG-FA approximation, see Rayleigh-DebyeGans for fractal aggregates (RDG-FA) approximation Reactive distillation, 1579–1580 Recalescent, 316 Recess depth, 565 Recessed heated horizontal circular surface, 562, 565 Reciprocity, 1205 Recrystallization, 2421, 2443 Rectangular enclosure, 2257 Rectangular-finned surface, 1817 Rectangular waves, 573 Recuperation, 1599 Recuperative heat exchangers, 1450, 1476, 1486–1487 Recuperative type heat exchanger, 1449 Recuperators, 1298 Reduced gravity, 1684 Reentrant cavities, 1783–1784, 1787, 1817 REFPROP, 1801 Refractive index, 2602, 2603 Refractive index techniques color schlieren, 768 data analysis, 765–767 experimentally recorded images, 771–781 optical configurations, 762–764 tomography, 771 Refractory cathodes, 2530 Refrigerants, 1522–1523, 1979

Index Refrigerant/lubricant/additive mixture interface, 1818 Refrigerant/lubricant mixtures, 1811, 1814 Refrigerant/nanolubricant mixture, 1816 Refrigerators, 1306 Regeneration, 1599 Regenerative heat exchangers, 1298, 1476, 1487 Regularization parameter, 1253 Relative humidity, 1524 Relaxation, 2890 Reliability, 1997 Reliability-based design, 261–262 Renewable energy, 2832 Reordering criterion, 72 Reordering scheme, 72 Representative elementary volume (REV), 709, 711, 713, 752 Resistance, 332, 336, 340, 347, 349, 352, 353 Resistive losses, in electrode, 2720 Resonance effect, 2616 Respiratory system models, 2345 Response surfaces, 267–269 Reststrahlen band, 1053 Retention of condensates, 2047 Retinal detachment, 2386, 2390, 2392, 2407 Retinal laser treatment, 2387 age-related macular degeneration, 2392 central serous chorioretinopathy, 2393 diabetic retinopathy, 2391 geometry and properties of the eye, 2395–2396 heat and bioheat transfer equations, 2393 ocular histoplasmosis, 2392 ocular tumors, 2393 retinal breaks and detachment, 2392 retinal vein occlusions, 2391–2392 typical 3-D modeling, 2396 Retinal pigmented epithelium (RPE), 2394 Retinal surgery, 2396–2398 Retinopathy, 2391, 2399 REV, see Representative elementary volume (REV) Reverse Monte Carlo method, 1230, 1236 Reversible electroporation, 2491 Reynolds, 2899 Reynolds analogy, 365, 368 Reynolds-averaged Navier–Stokes (RANS), 663, 671, 677, 680, 1175, 1182, 1183, 2935 Reynolds number, 365, 613, 698, 713, 717, 823–824, 1468, 1809 Rhodamine B, 2404

3035 Ribs and indentations, 451 Richardson–Dushman equation, 2692, 2695 Rigid blood vessel, 819 Rigorous coupled-wave analysis (RCWA), 1027 Robust, 353 Rohsenow correlation, 1761–1762 Rossby number, 655, 698 Rosseland approximation, 957 Rosseland diffusion approximation, 1237 Rosseland mean, 2641 Rosseland Mean Absorption Coefficient, 2641 Rotary kiln, 1192 Rotating boundary layer, 652 Rotating heat pipe, 2171 Rotating systems, heat transfer, see Heat transfer Rotational population, 1079 Rotational Reynolds number, 651, 654, 665, 669, 673, 677, 679, 680, 684, 685 Rotation Froude number, 698 Rotor-stator systems, 652, 661, 672, 677 Roughness, 1708, 1725, 1733, 1752, 1770, 1772, 1776 Rough surfaces, 449, 451–454 RTE, see Radiative transfer equation (RTE) S Saddle points, 682 Saffman, 2896 Salt hydrates, 2223, 2512 Sandia National Laboratory, 1551 SARA, see Saturates, Aromatics, Resins, and Asphaltenes (SARA) Saturated liquid, 1361 Saturated-liquid line, 1361 Saturated liquid–vapor mixture, 1361 Saturated liquid–vapor mixture region, 1361 Saturated vapor, 1361 Saturated-vapor line, 1361 Saturates, Aromatics, Resins and Asphaltenes (SARA), 1633 Saturation pressure, 1361 Saturation temperature, 1361, 1481, 1494 Scaling, 1614 Scattering, 280 albedo, 1213 angles, 1150 boundary, 284, 285, 293 coefficient, 938, 2470 cross-section, 1153, 1154 matrix, 1148–1150

3036 Scattering (cont.) mechanisms, 921 model, 1286 particle, 280 path length, 1211 phase function, 927, 938, 1150, 1154, 1211 plane, 1149 process, 1146 scattered power, 1150–1153 approximate, 1156 Scattering regime dependent, 1146 independent, 1146 for radiative transfer, 1155–1156 Scattering techniques infrared thermography, 797–801 liquid crystal thermography (see Liquid crystal thermography) Schlieren method, 2764 Schlieren system, 766 color, 768 crystal growth, 778 optical configurations, 764 Schmidt number, 365, 655 Schottky correction, 2741 Schrödinger equation, 1077 Sclera, 2383, 2394, 2395, 2399, 2403, 2406–2408, 2410 Scleral permeability, 2408 Screw extruder, 229, 231, 232, 244, 251 Search methods, 265–267 Secondary droplet breakup, 2945–2948 Second construct, 347–348 Second law, 2257 Second order RTE (SORTE), 949 boundary condition, 949 Security, thermal systems, see Thermal systems Sedimentation, 1613 Self-adjoint eigenvalue problem, 112 Self-similarity variable, 651 Sensible enthalpy, 867 Sensible heat loss, 903 Sensitivity, 236, 253 Sensitivity analysis, 253–254 Sensitivity coefficients, 205 Separation of variables, 64–69 Sequential heating, 2397 Sequential iterative variable adjustment (SIVA) procedure, 160, 161 SF6 plasma, 2611 SF6 radiation, 2641 Shadowgraph, 763, 764, 767 +
shaped, 551

Index Shape factor analysis, 926 Shape functions, 964 Shear modulus, 854 Shear stress, 1616 Sheath, 2686 Sheet conductivity, 1044 Shell-and-tube heat exchangers (STHX), 1305–1306, 1451, 1465, 1487–1489, 1503 Sherwood number, 366, 655, 2954 Short isothermal cylinder, 590 Short vertical cylinders, 585, 587 Similarity approach, 651, 652, 665, 673, 678 Similarity numbers, 654–655 Simulated annealing, 1258 Simulation, 222, 223, 2397 and concurrent experimentation, 260 and modeling, 225–230 simulation results, 230–247 Single domain formulation, 89 Single-insertion, 2508 Single-pass crossflow exchanger, 1515 Single-pass heat exchangers, 1453, 1466 Single phase convection, 1807 Single-phase cooling, 2002 Single-phase model, 594, 864 Single scattering albedo, 1213 Singular value decomposition, 1247 Sinusoidal heat flux, on skin surface, 2366 Sinusoidal waves, 573, 574 SIVA procedure, see Sequential iterative variable adjustment (SIVA) procedure Size parameter, 1144 Sizing problems, 1320, 1331–1332 Skin friction, 368 Skin friction coefficient, 366 Skin temperature, 904 Slip factor model, 1928 Slip flow, 2894 Slug flow, 1780 SLW reference approach, 1123 Small scales, 404, 412 Smith and Kuethe’s parameter, 404, 406 Smoothed particle hydrodynamics, 2995 SN-approximation, 959 Snell’s law, 990 Solar energy systems, 2250–2251 Solar power, 2251 Solid angle, 922, 924 Solidification front, 2989 Solidification parameter, 2983 Solidified layer thickness, 2973

Index Solid–liquid interface, 1830, 1836 Solid–liquid phase change, 861 Solid-solid phase change, 862 Sonic limit, 2180 Soot, 1176, 1178, 1183 Soret effect, 34, 35 SORTE, see Second order RTE (SORTE) South Africa, 2831 Spacings, 350 Sparse hole array vs. dense hole array adiabatic film effectiveness, 442 heat transfer coefficient, 442 Sparse matrix, 151, 157 Specific absorption rate, 2471 Specific heat capacity, 2227, 2281, 2282, 2284, 2285, 2287, 2289, 2297, 2300 Speckle tracking algorithms, 844 Spectra, normal component of turbulence, 394 Spectral absorption coefficient, 1233 Spectral group model, 1114–1115 Spectral line-based weighted-sum-of-greygases (SLW) models, 1182 Spectral line shape, 1086 Spectral line weighted-sum-of-gray-gases (SLW) method, 1115, 1232 Spectral methods, 129 Spectral properties, 1232 Spectral radiative heat flux, divergence of, 1204 Spectral radiative properties, of different gases, 927 Spectroscopic investigations, 2762 Specularly reflecting surface, 1215, 1229 Specular reflection, 1215 Speed-up, 1238, 1239 Spheres, 1707, 1708, 1721 Spheres and vertical surfaces, 1688 Spherical coordinate system, 942 Spherical harmonics method, 955, 2636 Spheroidisation, 2888 Spinodal temperature, 1728 Spiral fins, 458 Spiral plate exchangers, 1624 Spiral plate heat exchanger, 1491 Spiral tube heat exchanger, 1489 Splashing, 2976 Splats, 2968, 2981 Sponge balls, 1637 Spontaneous emission, 2615 Spray cooling, 1298 Spray trajectory, 2940 Spread factor, 2973 Squareness of the cut (bevel angle), 2784–2786 Stable film boiling, 1927

3037 Stagnation region, heat transfer augmentation, 404, 407, 408, 414 Standard deviation, 1217 Stand-off, 2733, 2739, 2740, 2786 Stanton number, 365, 368 Stark effect, 2617, 2618 Start erosion, 2758, 2760–2761 Statistical narrow band, 1176, 1179 Statistical narrow band correlated-k, 1176 Statistical narrow band (SNB) modeling, 1101 fictitious gases and mapping methods, 1109–1110 with Malkmus’ distribution of line strengths, 1105–1107 in non-uniform gaseous media, 1107–1108 principle, 1102–1105 Statistical uncertainty, 1216 Steady drag force, 1864 Steady-in-the-mean flows, 2087 Steady operations, 2085 Steady state, 2192, 2388, 2403 Steam–air mixtures, 2046 Steam generators, 1302, 1475, 1487 Stefan–Boltzmann constant, 1207 Stefan–Boltzmann law, 922, 2605 Stefan number (Ste), 755, 2231, 2973 Stefan number problems, 2234 Stefan problem, 2232 Stem, 345 Stents, 2505 Stephan Boltzmann constant, 10 Step scheme, 961 Sterling engines, 2871 Stewartson flow, 661, 674 Stiffness, 815 Stochastic, 2895 Stochastic method, 1179, 1181 Stokes, 2891 parameters, 1150 vector, 1150 Stokes number, 2958 Storage capacity, 2232, 2267 St0 parameter, 398, 402 Streamlines, 814 Stream operator, 941 Striations, 2781, 2783–2784 Structured roughness, 451, 453, 470 Sturm-Liouville problem, 66 Subconjunctival injections, 2406 Subcool, 1753, 1757, 1762 Subcooled boiling, 2022 Subcooling, 2218, 2226 Subcritical bifurcation, 683, 684, 686

3038 Submerged arc welding (SAW), 2660 Substrate thermophysical properties, 1689 Subzero, 2285, 2287, 2294, 2297, 2300 Suction, 450 Sulzer Chemtech Ltd., 1576 Summation rule, 1205 Summation rule of exchange factors, 1237 Supercooled, 312 Supercritical bifurcation, 680, 683, 686 Supercritical fluids critical parameters, 1416, 1417 density vs. temperature, 1417, 1423, 1425, 1429 dynamic viscosity vs. temperature, 1420, 1423, 1424, 1428, 1431 Prandtl number vs. temperature, 1421 specific heat vs. temperature, 1419, 1423, 1424, 1427, 1430 thermal conductivity vs. temperature, 1418, 1423, 1424, 1426, 1430 volumetric vs. temperature, 1422 Supercritical steam, 1361 Supercritical vapor, 1362 Superheat, 1753, 1755, 1758, 1767, 1768, 1776, 1783 Superheated steam, 1362 Superheated vapor, 1362 Superhydrophilic surface, 1751, 1752, 1775, 1776 Superhydrophobic, 307, 1707, 1708 Superhydrophobic surface, 1751, 1752, 1776 Superlattices, 284–286 Suppressed nucleation, 2140 Suppression factor, 1808 Suprazero, 2285, 2295, 2297, 2300 Suprazero temperatures, 2422 Surface-active additives, 1826 Surface age, 1830, 1835 Surface area density, 1502 Surface coatings, 2011 Surface compactness, 1476 Surface contamination, 1682–1683 Surface curvature effect, 2061 Surface energy, 1818 Surface heat transfer coefficients, 433 area-averaged, 436 line-averaged, 436 Surface modifications, 1549–1550 Surface mounted rib turbulator, 783, 784 Surface phonon polaritons (SPhPs), 990, 996, 1006, 1007, 1024 Surface plasmon polaritons (SPPs), 1002, 1024, 1031–1037

Index Surface plasmon resonance (SPR), 2461 Surface polariton, 990, 994, 997, 1002, 1004, 1007 Surface roughness, 1628, 1657, 1678, 1687 Surfaces, EM wave interaction, 920 Surface structuring, 1769, 1784, 1787 Surface sweeping, by falling drops, 2060 Surface tension, 1505, 1751, 1765, 1767, 1778, 1785 Surface-tension devices, 450 Surface treatments, 1617 Surface vibration, 450, 469, 1552 Surface waves, 1024 Surface wettability, 1657, 1678, 1679, 1688, 1751, 1768, 1771, 1786 Surface wetting, 1825, 1836–1839, 1844 Surfactant, 1803 Surfactant adsorption, 1767–1770 Suspension, 2924 properties, 2926–2929 Suspension plasma spray (SPS), 2925 droplet/particle phase modeling, 2948–2957 heat transfer, 2956–2960 plasma jet modeling, 2929 primary droplet breakup (see Primary droplet breakup) secondary droplet breakup, 2945 Sustainable development, 1594 Swirl-flow devices, 449, 462 Swirl flow enhancements, 1545–1547 Swirl-flow length, 463 Swirl parameter, 462 Swirl velocity, 463 Symbolic computation, 85 Syngas, 2843 composition of, 2852 production, 2868 Syngas-fuelled gas engines, 2872 Synthetic jet actuator, 789, 790 System level, 2258 System level performance, 2258 System pressure, 1682, 1691 T Tabu search algorithm, 1260 Tapered gap microchannels, 1967–1968 Tapestry, 356 Tar, 2843 Taylor analogy breakup (TAB) model, 2951 Taylor bubbles, 1854 Taylor flow, 1553, 1563 Taylor Gortler vortices, 420

Index Taylor number, 655, 702, 706, 708, 746, 747, 751, 753, 755 Taylor–Proudman columns, 699–702, 724–728 Taylor–Proudman theorem, 700, 725 Techno-economic needs, 2262 TEMA, see Tubular Exchanger Manufacturers Association (TEMA) Temperature, 2280, 2282, 2285–2290, 2293, 2299, 2300, 2491, 2492, 2494, 2500, 2502, 2513, 2515 Temperature-based method, 864 Temperature difference, 1297, 1300, 1318, 1320–1326, 1328, 1330, 1332 Temperature distribution, 2232 Temperature drop, in promoter layer, 2062 Temperature elevations, 2467 Temperature glide, 1800 Temperature regulation, 818 Temperature swing adsorption (TSA), 1578 Textile, 2264 Thawing, 2422, 2436 Thawing injury, 2443 Therapeutic processes, 2383 Thermal, 2491, 2501, 2507, 2514, 2889 ablation, 2505 application, 1296 behavior, 2260 boundary conditions, 2494, 2505 boundary layer, 484, 1813 comparator method, 2310 conduction, 1364, 2672, 2675 coupling, 772 damage, 2390, 2393, 2466, 2492, 2502, 2504, 2506, 2512, 2514 devices, 1502 diffusivity, 8, 17, 208–215, 367 distribution, 2505 dose, 2503 effects, 2492, 2502, 2514 energy, 2495, 2496, 2498, 2500 exposure, 2502 history, 2280, 2282 images, 208 losses, 1598 mitigation strategies, 2492, 2508 modeling, 2280 properties, 2500 protection, 2492 Thermal conductivity, 8, 281, 283, 287, 289, 291, 293, 297, 303, 311, 316, 862, 2228, 2281, 2282, 2285, 2290, 2297, 2300, 2308, 2495, 2497, 2499, 2501, 2513, 2853

3039 guarded hot plate method, 2308–2310 heated thermocouple method, 2311–2312 of nanoparticle, 1804 tensor, 14 thermal comparator method, 2310–2311 Thermal contact resistance, 2970 Thermal discrete dipole approximation (T-DDA), 981, 1003, 1009–1015 Thermal emission, 980, 982, 984, 986, 988, 990, 991, 994, 998, 1002, 1003, 1009, 1014 near-field, 994 Thermal energy storage (TES), 2219–2220 Thermal energy storage system, 861 Thermal equilibrium, 811, 893 Thermal interface materials (TIMs) carbon nanotube and nanoparticles in polymer matrix, 291–293 graphene based, 293 thermal resistance of, 291 Thermal isoeffective dose, 2476 Thermal lattice Boltzmann method, 864 Thermal management, 281, 296, 301, 307, 318, 1494 electronic systems, 244 Thermal management applications (TMA), 2219 Thermal pinch effect, 2671, 2672, 2676 Thermal plasma, 2661, 2840 Thermal plasma, nanoparticles DC arc plasma technique, 2805–2810 equilibrium size distribution, 2797–2799 feedstock particles, dynamics and evaporation of, 2793 general dynamic equation, 2802–2803 heterogeneous condensation, 2800 homogeneous nucleation, 2799 Kelvin relation and critical diameter of nuclei, 2795–2797 modulated induction thermal plasmas, 2815–2823 particle growth, coagulation, 2802 particle growth, modelling approach for, 2803–2804 particle size and nanoparticle features, 2792–2793 RF inductively coupled thermal plasma, 2810–2815 saturation vapor pressure and supersaturation ratio, 2794–2795 Thermal power generation, 1299

3040 Thermal properties, 2281, 2282, 2284, 2286, 2288, 2293, 2297, 2301 cryopreservation, 2291 cryoprotectants, 2300 protein phase change and water loss, 2286–2287 specific heat capacity measurement, 2284–2285 subzero temperatures I and II, porcine systems at, 2294–2295 suprazero temperatures, porcine systems at, 2295 suprazero temperatures I and II, human systems at, 2296–2298 thermal conductivity measurement, 2282–2284 water phase change and cryoprotectant effects, 2287–2290 Thermal quadrupole formalism, 203 Thermal radiation, 1024, 1043–1058, 1298 Thermal relaxation time, 819 Thermal resistance, 1476, 1481, 1492, 1494, 1976, 2182 Thermal resistance method, 2228 Thermal sciences, 332, 358 Thermal (species) boundary layer, 365 Thermal spray process, 2557 Thermal switch, 2265 Thermal systems acceptable design domain, 254 basic design strategy, 248 boundary conditions, 236–238 chemical vapour deposition, 223, 225 combined mechanisms, 238–241 complex transport phenomena, 241 concurrent experimentation and simulation, 259–261 data centers, 233 definition, 220 feasibility, 251 inverse problem, 248 knowledge-based design, 257 modeling and simulation, 225 multiple scales, 242 optical fiber drawing process, 231 optimization of (see Optimization, thermal systems) screw extruder, 231 sensitivity analysis, 253 uncertainty and reliability-based design, 261 validation, 245 Thermal tagging technique, 394 Thermal therapy, 817, 2280

Index Thermal tolerance, 2475 Thermal transport, micro and nanoscale systems, see Microscale Thermionic, 2741 cathodes, 2691 electron emission, 2529 emission, 2692 Thermochromic liquid crystals (TLC), 656 Thermodynamics, 2889 equilibrium, 2572, 2850 First Law of, 6 properties, 2666, 2668, 2680 Thermoelectricity, 284–291, 318 Thermo-field emission, 2694 Thermo-hydraulic design, 1474, 1480, 1495 Thermophoresis, 34, 2889, 2951 Thermo-physical properties, 180, 1523, 2226, 2664, 2676, 2680 inversion problems (see Inverse problems) nanofilms, 206 orthotropic materials, 202 Thermosolutal convection, 609 Thermosyphons, 2194 Thermosyphon thermal resistance, 2184–2187 Thickening, 2239 Thin fabric model for clothing ventilation, 894–897 Three-dimensional boundary layer flow, 653 Three-dimensional model, 2396 3D numerical simulation, 1987 3D printing, 1505 3D stacked ICs, 1975, 1979, 2001 Tikhonov regularization, 184, 1253 Tilted cut, 2773–2775 Time constants, 854 Time-domain method, 1027 TIMs, see Thermal interface materials (TIMs) Tissue, 2490, 2491, 2493, 2496, 2498, 2500, 2502, 2504–2507, 2509, 2511, 2513, 2514 Tissue electroporation, thermal considerations boundary conditions and initial conditions, 2494–2495 dynamic electrical properties, 2496–2497 dynamic thermal properties, 2500 electric field distribution, 2492–2493 heat diffusion equation, 2496–2499 Joule heating, 2495 numerical modeling, 2503–2505 Pennes bioheat equation, 2498–2500 thermal damage, 2500–2503 thermal mitigation strategies, 2505–2514 TLR parameter, 399–400

Index T-matrix method, 1162–1163 TMP, see Transmembrane potential (TMP) TNO, 1560, 1562 Tomography, 771 Torches in analytical chemistry, 2581 with inter-electrode inserts (cascaded torch), 2561 stability and electrode wear, 2569 Total clothing ventilation, 904 Total internal reflection, 990, 991, 993 Total transformation scheme, 85 Total variation, 197 Townsend-like breakdown, 2771 Toxicity, 2464 Trail road facility, 2880 Transferred arcs, 2647, 2846 Transient conduction, 1365 Transient design-analytical tools, 2269 Transient heat pipe, 2193–2194 Transient methods, 2228 Transient natural convection, 2388 Transitional flow, 2894 Translational Reynolds number, 654, 678, 680, 684, 685 Transition boiling, 1700, 1704, 1707, 1709, 1732, 1740, 1914 Transition to turbulence, 366 Transition Weber number, 2947 Transmembrane potential (TMP), 2490, 2500, 2514 Transport, 2889 Transportation, 1300 systems and deicing, 2203 Transport coefficients, 2666, 2668, 2680 Transport processes, 2382 Transpupillary thermal therapy (TTT), 2383, 2387–2388 Transscleral drug delivery, 2407 modeling, 2409–2410 sclera, anatomy of, 2407 transscleral administration of drugs, modes of, 2408–2409 Transscleral routes, 2383 Transverse electric (TE) waves, 1032 Transverse magnetic (TM) waves, 1032 Treated surfaces, 449 Tree links, 355 Tree-shaped inserts, 351 Triangular waves, 573 Tributary branches, 345 Triple point, 1362 TRL model, 408

3041 TRL parameter, 406, 408, 409, 412, 413 Truncated Gaussian distribution, 194 Truncated singular value decomposition, 1252 TSA, see Temperature swing adsorption (TSA) T-shaped cavity, 339–345 TSV, 2002 Tube-fin, 458 Tube-fin heat exchanger, 454, 1451, 1452, 1491–1492, 1503 Tubular Exchanger Manufacturers Association (TEMA), 1619 Tumor cells, 817 Tumor shrinkage, 2476 Tumor sizes, 2469 Tungsten inert-gas (TIG) welding, 2659 Turbulence, 607, 1714, 2895, 2914 conditions, 1468 dispersion force, 1867 dissipation scale, 398 energy scale, 398 flame, 1175, 1178–1184 flow, 366 inactive motions, 395 interaction with flat surface, 394 intermittency, 395 large and small scales, 404 models, 1932, 2766 normal integral scale, 394 plasma, 2766 wall normal attenuation, 395 Turbulence-radiation interaction (TRI), 1178 Turbulent Prandtl number, 668, 677 Turbulent time scale, 406, 409 Twist angle, 1809 Twisted-tape inserts, 462, 463 2D grating, 1028 Two-dimensional (2D) materials, 1025 Two-fluid atomizers, 2940–2942 Two-fluid model, 1915 Two modes, 2747, 2749 Two-phase convective heat transfer, 1478 Two-phase cooling, 2008, 2015, 2016 Two-phase flow in horizontal tube, 1854 pattern maps, 1855–1862 in vertical tube, 1854 Two-phase fluid dynamics, 1911 Two-phase heat exchangers applications of, 1475 basic equations for, 1479–1481 basic types of, 1476–1479 extended surface heat exchangers, 1491–1493

3042 Two-phase heat exchangers (cont.) heat exchanger pressure drop, 1485 log mean temperature difference, 1483 microchannel heat exchangers, 1493–1497 overall heat transfer coefficient, 1481 plate-type heat exchangers, 1489 recuperative exchangers, 1486 tubular heat exchangers, 1487–1489 Two-phase model, 595 Two-phase thermosyphon, 2168 Two-phase pressure drop, 1892–1902 Two-temperature chemical non-equilibrium model, 2589 T-Y-shaped cavity, 350 U Ultrasonic enhancement, 1551–1552 Ultrasonic speckle, 843 Ultrasound elastography, 841–846 Ultraviolet (UV), 2602 Under-relaxation techniques, 162 Unfinned heat exchangers, 1452 Uniform distribution, 194 Uniform heat flux, 538 Uniform surface temperature, 535 UNIT code, 85 Unsteady RANS (URANS) methods, 663, 677 Unsteady simulation, 2147 Upstream compressible volume instability, 1958 URANS methods, see Unsteady RANS (URANS) methods Uveal blood flow, 2407 UV spectrum, 2605, 2621 V Vacuum plasma spraying (VPS), 2567 Vacuum Ultra Violet (VUV) spectrum, 2605 Vadasz coefficient, 717 Vadasz number, 717, 753 Validation, 225, 226, 245–247 Validation and verification, 162 Van der Waals effect, 2617 Van Fossen parameter, 407 Vapor, 2911 Vapor chamber, 2171 Vapor entrapment, 1756, 1767 Vapor extraction, 1966–1967 Vaporization, 2911 Vaporized cathode material, re-deposition of, 2566

Index Vapor–liquid interface, 1829 Vapor production, 2014 Vapor superheat effect, 2042 Variable metric method, 1257 Variance, 1216, 1217, 1220 Vascularization, 332 Vascular models, 2353, 2472 Velocys, 1565 Vertical wavy surfaces, 574 Vibration, 634 Virtual mass, 2892 Virtual mass force, 1868 Viscosity, 864, 1803, 2254 Viscous boundary layer, 483 Viscous (vapor pressure) limit, 2181 Visible spectrum, 2605, 2621 Vitreous circulation, 2403 Vitreous hemorrhage, 2386, 2387, 2399 Vitreous humor, 2383, 2387, 2388, 2401, 2409 Vitrification, 2280, 2281, 2285, 2342, 2443, 2840 v0 l model, 415 VOF method, 2019 Void fraction, 1851–1852 distribution, 1879 drift flux model, 1882–1886 empirical correlations, 1886 homogeneous flow model, 1879–1881 one-dimensional model, 1881–1882 stratified flow, 1888–1892 Voigt profile, 1089–1090, 2618, 2619 Volatilization, 2846 Volt equivalent of the heat, 2738, 2749 Volterra integral equation, 1277 Volume, 2495, 2498, 2505, 2512, 2513 Volume fraction of nanoparticles, 1804 Volume integral equation formulation, 1161 methods, 1161–1162 Volume of fluid (VOF) method, 2704, 2705, 2943, 2986 Volume to point, 355 Volumetric heat source, 945 Von-Karman swirling flows, 651 Vortex amplification theory, 405 Vortex generators, 462, 470 Vortex shedding, 319 Vortex stabilized plasma torch, 2556 Vortex stretching, 405 VUV region, 2646 VUV spectrum, 2649

Index W Waiting period between successive nucleations, 1675 Wall boiling closures multi-dimensional, 1936 1-dimensional, 1924–1927 Wall friction, 1923–1924 Wall functions for turbulence model, 1933 Wall heat flux, 365, 2541 Wall lift force, 1867 Wall temperature, 365 Waste destruction methods gasification, 2834–2837 (see also Gasification) incineration, 2833 pyrolysis, 2834 Waste heat, 1596 losses, 1598 Water heaters, 1302 Water loss, 2281, 2286–2287 Water transport, 2434–2437 Wave breakup model, 2951 Wavelength, 1715, 1716, 1719, 1721, 1730 of electromagnetic wave spectrum, 921 Wavy channels, 1984, 2022 Wavy fins, 459, 460 Wavy shape, 573 Wavy surfaces, 528 Wedge flow, 372–375 Weighted residual approach, 965 Weighted-sum-of-grey-gases (WSGG) model, 1112–1114, 1177, 1182, 1233 Welded spiral heat exchanger, 1545 Welding, 2648, 2650 Welding efficiency, 2720

3043 Weld pool, 2698 flow, 2715 surface profile, 2712 Well-posed problem, 184 Westinghouse plasma, 2840 Wet air, 1524 Wet bulb effectiveness, 1528, 1530 Wet bulb temperature, 1524, 1527, 1529, 1531 Wettability, 301, 1701, 1704, 1707, 1708, 1725, 1727, 1730, 1765, 1774–1776 See also Surface wettability Wetting, 1828, 1831, 1841 Wetting and wickability of fabrics, 890 Whole-body models, 2343 Wick, 1785 Wickability, 1772–1774 Wien law, 923 Wind tunnel, 652, 658 Wire-coil inserts, 451 Wood’s anomaly (WA), 1034 Work function, 2741, 2744, 2750 X X-shaped cavity, 348 Y Young’s modulus, 834 Y-shaped cavity, 345–347 Z Zeotropic mixtures, 1799, 1801 Zeta potential, 506, 1839 Zuber hydrodynamic model, 1763–1765