Coupled Heat and Mass Transfer in Binary Mixtures at Supercritical Pressures (Springer Theses) 9789811678059, 9789811678066, 9811678057

Table of contents :
Supervisor’s Foreword
Preface
Acknowledgements
Contents
Acronyms
Roman Symbols
Greek Symbols
Superscripts
Subscripts
Abbreviations
1 Introduction to Binary Mixtures at Supercritical Pressures and Coupled Heat and Mass Transfer
1.1 Supercritical Pressure Fluids and Near-Critical Fluids
1.1.1 Critical Phenomena
1.1.2 Anomalies in Physical Properties
1.1.3 Engineering Applications of Supercritical Pressure Fluids
1.2 Coupled Heat and Mass Transfer in Binary Mixtures at Supercritical Pressures
1.3 Literature Review
1.3.1 Introduction to Three Fundamental Problems
1.3.2 Piston Effect
1.3.3 Rayleigh-Bénard Instability
1.3.4 Buoyancy-Driven Flows
1.4 Motivation
1.5 Thesis Outline
References
2 Basic Equations and Physical Properties of a Reference Binary Mixture
2.1 Basic Equations
2.2 Physical Properties
2.3 Phase Diagram and Behavior of Physical Properties of C2H6-CO2
2.4 Summary
References
Part I Coupling Through Cross-Diffusion Effects: Relaxation and Diffusion Problems
3 Coupled Transfer in a Relaxation Process: Mass Piston Effect
3.1 Problem Statement
3.2 Traveling-Wave Theory
3.2.1 Wave Generation
3.2.2 Wave Propagation
3.3 Numerical Validations
3.3.1 Numerical Method
3.3.2 Comparisons and Discussions
3.4 Energy Balance Analysis
3.5 Conclusions
References
4 Coupled Transfer in a Diffusion Problem: Concentration Gradient in the Coexisting Liquid-Like and Gas-Like States
4.1 Problem Statement
4.1.1 Physical Model
4.1.2 Governing Equations for the Motionless Steady State
4.1.3 Summary of Cases
4.2 Numerical Method
4.3 Characteristics for the Coexistence of LL and GL States
4.3.1 State Variables
4.3.2 Physical Properties
4.3.3 Physical Explanation for the Pressure Drop
4.3.4 Heat Transfer Characteristics
4.4 Assessments on the Influences of Concentration Gradient
4.4.1 Influences of Concentration Gradient on the PB
4.4.2 Influences of Concentration Gradient on Pressure Drop and Heat Flux
4.5 Conclusions
References
Part II Coupling Through Cross-Diffusion Effects: Instability and Bifurcation
5 Interactions Between Coupled Transfer and Gravity: Rayleigh-Bénard Instability
5.1 Problem Statement and Approximation
5.1.1 Physical Model
5.1.2 Mathematical Model
5.2 Linear Stability Analysis
5.2.1 Formulation of Stability Problem
5.2.2 Analytical Solution under Ideal Boundary Conditions
5.2.3 Stability Threshold under Various Conditions
5.3 Further Discussions
5.3.1 Origin of the Oscillatory Instability
5.3.2 Relative Importance of Gravity-Related Effects
5.4 Conclusions
References
6 Interactions Between Coupled Transfer and Gravity: Nonlinear Rayleigh-Bénard Convection
6.1 Problem Statement
6.1.1 Physical and Mathematical Models
6.1.2 Response Parameters
6.1.3 Numerical Method
6.2 Preliminary Numerical Simulations
6.3 Weakly Nonlinear Theory
6.3.1 Conditions for Backward Bifurcation at R=Rmonocrit
6.3.2 Relative Magnitude Between Rf.a. and Rosccrit
6.4 Theory-guided Numerical Simulations and Discussions
6.4.1 Results under Partially Satisfied Necessary Conditions
6.4.2 Results under Fully Satisfied Necessary Conditions
6.4.3 Physical Explanation for the Conditions of FA Instability below Rcrit
6.5 Conclusions
References
Part III Coupling Through Boundary Reactions: Buoyancy-Driven Flows
7 Coupled Transfer Through Boundary Reactions: An Application-Oriented Cavity Flow Problem
7.1 Problem Description
7.1.1 Physical Model
7.1.2 Mathematical Description
7.1.3 Initial and Boundary Conditions
7.1.4 Governing Parameters
7.2 Formulations for CO2-Naphthalene System
7.2.1 Solubility
7.2.2 Adsorption and Desorption Reactions
7.2.3 Thermodynamic and Transport Properties
7.3 Theory
7.3.1 Thermodynamic Optimization
7.3.2 Hydrodynamic Optimization
7.4 Numerical Simulations
7.4.1 Numerical Method
7.4.2 Cooperative Regime
7.4.3 The Effects of Cavity Height
7.5 Conclusions
References
8 Summary and Perspectives
8.1 Summary
8.2 Perspectives
Appendix A Numerical Methods for the Linear Stability Analysis of Rayleigh-Bénard Instability under Realistic Boundary Conditions
A.1 Discretization of the Generalized Eigenvalue Problem
A.2 Algorithm for the Optimization Problem
A.3 Numerical Codes for the Linear Stability Analysis
Appendix B The Solubility of Naphthalene in Supercritical CO2
Appendix C The Isobaric Specific Heat of Supercritical CO2-C2H6 System

Citation preview

Springer Theses Recognizing Outstanding Ph.D. Research

Zhan-Chao Hu

Coupled Heat and Mass Transfer in Binary Mixtures at Supercritical Pressures

Springer Theses Recognizing Outstanding Ph.D. Research

Aims and Scope The series “Springer Theses” brings together a selection of the very best Ph.D. theses from around the world and across the physical sciences. Nominated and endorsed by two recognized specialists, each published volume has been selected for its scientific excellence and the high impact of its contents for the pertinent field of research. For greater accessibility to non-specialists, the published versions include an extended introduction, as well as a foreword by the student’s supervisor explaining the special relevance of the work for the field. As a whole, the series will provide a valuable resource both for newcomers to the research fields described, and for other scientists seeking detailed background information on special questions. Finally, it provides an accredited documentation of the valuable contributions made by today’s younger generation of scientists.

Theses may be nominated for publication in this series by heads of department at internationally leading universities or institutes and should fulfill all of the following criteria • They must be written in good English. • The topic should fall within the confines of Chemistry, Physics, Earth Sciences, Engineering and related interdisciplinary fields such as Materials, Nanoscience, Chemical Engineering, Complex Systems and Biophysics. • The work reported in the thesis must represent a significant scientific advance. • If the thesis includes previously published material, permission to reproduce this must be gained from the respective copyright holder (a maximum 30% of the thesis should be a verbatim reproduction from the author’s previous publications). • They must have been examined and passed during the 12 months prior to nomination. • Each thesis should include a foreword by the supervisor outlining the significance of its content. • The theses should have a clearly defined structure including an introduction accessible to new PhD students and scientists not expert in the relevant field. Indexed by zbMATH.

More information about this series at https://link.springer.com/bookseries/8790

Zhan-Chao Hu

Coupled Heat and Mass Transfer in Binary Mixtures at Supercritical Pressures Doctoral Thesis accepted by Peking University, Beijing, P.R. China

Author Dr. Zhan-Chao Hu School of Aeronautics and Astronautics Sun Yat-sen University Shenzhen, Guangdong, P.R. China

Supervisor Prof. Xin-Rong Zhang College of Engineering Peking University Beijing, P.R. China

ISSN 2190-5053 ISSN 2190-5061 (electronic) Springer Theses ISBN 978-981-16-7805-9 ISBN 978-981-16-7806-6 (eBook) https://doi.org/10.1007/978-981-16-7806-6 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 This work is subject to copyright. All rights are solely and exclusively licensed 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, expressed 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. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

I would like to dedicate this book to my family members, for their endless love and support.

Supervisor’s Foreword

During the last 15 years, my lab at Peking University has focused on the in-depth exploration and utilization of supercritical and near-critical fluids. It is widely known that the liquid-vapor critical point is unique in the phase diagram. When there is a variation in temperature and the pressure is higher but close to the critical value, a fluid undergoes a continuous transition from a high-density liquid-like state to a lowdensity gas-like one. Along with the drastic change in density, thermophysical properties exhibit abnormal behavior, distinguishing supercritical pressure fluids from normal incompressible ones in terms of hydrodynamic and transport phenomena. Previously, studies regarding heat and mass transfer at supercritical pressures are performed from an engineering viewpoint, and pure-fluid-related works are mainstream. During the past 6 years, Dr. Hu’s research involves binary mixtures at supercritical pressures. He discovered that the coupled heat and mass transfer is a core mechanism, which has not been systematically investigated. He endeavored to reveal novel processes, hydrodynamic instabilities, and potential applications. His research outcomes include new phenomena, new mechanisms, and a new application design, providing innovative insights to this vivid field. Dr. Hu, initially as a fourth-year undergraduate student, began working with me in July 2014. Dr. Hu is an active member of the research group, providing guidance for new graduates and organizing group activities. He is engaging, pleasant, humorous, and reliable. Throughout my interactions with Dr. Hu, I have been continually impressed with both his enthusiasm and integrity. As his mentor, I have had many discussions with him. Dr. Hu always impressed me with his ability to think innovatively and overcome theoretical and computational difficulties. He has achieved so much through his unremitting efforts and unceasing curiosity. There is no doubt that he has opened new possibilities in supercritical fluids in recent years. Now, he is working at Sun Yat-sen University. I believe he can be an excellent researcher, lead new research directions in frontier areas, and achieve fruitful outcomes in a harmonious and cordial team atmosphere.

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Supervisor’s Foreword

This book summarizes Dr. Hu’s last 6 years of research. It is my great pleasure to recommend this book to all readers, especially those interested in supercritical fluids, applied mathematics, and general physics and mechanics. I believe the reader will find useful information and be attracted to this field of study. Beijing, P.R. China August 2021

Xin-Rong Zhang

Preface

The supercritical pressure state is a thermodynamic concept, which refers to the state above the pressure of liquid-vapor critical point. Indeed, under a slightly supercritical pressure, upon increasing temperature, a fluid can undergo a continuous transition from a high-density liquid-like state to a low-density gas-like one. In this process, a drastic change of density takes place in a narrow temperature interval, and thermophysical properties exhibit abnormal behavior, such as the divergence of isobaric specific heat and the reduction of thermal diffusivity. These singular properties are collectively termed as critical anomalies. Besides, under each supercritical pressure, the separating point for the liquid-like state and the gas-like one is usually the temperature where the maximal isobaric specific heat achieves. States under slightly supercritical pressure and around the separating temperature are termed as near-critical ones, whose physical properties are greatly influenced by critical anomalies, distinguishing near-critical fluids from normal incompressible fluids in terms of hydrodynamics and transport phenomena. In this book, supercritical pressure states are general concerns, with special attention paid to near-critical ones. Previous works usually consider pure fluids (i.e., one-component fluids). The hydrodynamics and transport phenomena of mixtures at supercritical pressures have not been systematically investigated. For this, binary mixtures at supercritical pressures are the research objects of this book, which are two-component miscible (singlephase) fluids,1 and contain coupled heat and mass transfer. The coupled transfer comes from two resources. First, cross-diffusion effects, which include the Soret effect and Dufour effect, representing the mass flux driven by temperature gradient and heat flux by concentration gradient, respectively. In fact, under near-critical states, the relevant physical property grows dramatically, enhancing the cross-diffusion effects. Second, temperature-dependent boundary reactions. These reactions exist in chemical engineering, where supercritical fluids are widely used as solvents.

1

Generally speaking, for a binary mixture, a supercritical pressure does not guarantee a single phase. The reader can refer to Chap. 2 for more details, especially the concept of cricondenbar. ix

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In this book, through theoretical modelings, analyses, and numerical calculations, fundamental insights are given into the coupled heat and mass transfer in binary mixtures at supercritical pressures. Due to the feasibility of accurate models or data of physical properties, the two kinds of coupled heat and mass transfer are treated separately, with an emphasis placed on the cross-diffusion effects. The structure of this book is introduced as follows. Chapter 1 introduces the general background of this book. Chapter 2 elaborates the basic equations and physical properties of a reference binary mixture, paving the way for studies in subsequent chapters. The new findings are divided into three parts. Part I focuses on relaxation and diffusion problems. In Chap. 3, the relaxation of a binary mixture at a supercritical pressure after subjecting to a cross-boundary mass diffusion flux is studied. The concept of the mass piston effect as an efficient thermalization mechanism is proposed and its physics is analyzed. In Chap. 4, temperature differences with the separating temperature in between are applied to the same fluid layer. It proposes the idea that when liquid-like and gas-like states coexist, there is a concentration gradient, as an analogy to the situations at subcritical pressures. It confirms that under near-critical conditions, the concentration gradient greatly alters buoyancy force, laying down a basis for Part II of this thesis. Part II includes Chaps. 5 and 6, where the Rayleigh-Bénard instability and bifurcation are investigated. On the one hand, this work is motivated by the conclusion regarding buoyancy in Chap. 4. On the other hand, it is also motivated by the curiosity about the fluid dynamics of a bounded fluid layer after adding gravity. To be specific, Chap. 5 investigates the linear stability of small perturbations and obtains the convection onset criteria and their underlying physical mechanisms. Chapter 6 investigates the bifurcation around the onset threshold, with special attention paid to the conditions of finite amplitude instability. In Part I and Part II, the coupled heat and mass transfer is achieved through crossdiffusion effects. While in Part III (Chap. 7), the coupling through boundary reactions is investigated through an application-oriented cavity flow problem. A conceptual design of coupled extraction and crystal growth apparatus is proposed therein, and the resulting double-diffusive convection is analyzed and simulated. The main conclusions of the above chapters, as well as prospects for future works, are summarized in Chap. 8. This book, combining critical anomalies and coupled heat and mass transfer, reports a series of new phenomena, novel mechanisms, and an innovative engineering design in hydrodynamics and transport phenomena of binary mixtures at supercritical pressures. Supercritical pressure fluids have been widely used in many engineering fields, and also get involved in many emerging techniques. Therefore, apart from the theoretical significance, these findings give new fundamental insights into the supercritical hydrodynamics and transport phenomena in various engineering fields, promote the development of process intensification, and inspire new designs and applications. After graduating from Peking University and during this year working as an Assistant Professor at Sun Yat-sen University, the content of this book has been modified to be more comprehensive and systematic, making it a qualified research

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monograph. This book is a summary of the current stage. I believe as researches move on, more new findings will be revealed. This book can serve as a good reference for graduate students, engineers, and researchers working with supercritical fluids in various fields such as applied mathematics, thermal engineering, and chemical engineering. This work has been supported by the National Natural Science Foundation of China (Grant No. 51476001, 51776002) and the China Scholarship Council (Grant No. 201706010268). Shenzhen, P.R. China August 2021

Zhan-Chao Hu

Acknowledgements

I am grateful for the care and help of many people along the whole way. I would like to thank Prof. Xin-Rong Zhang for his continuous guidance. He introduced the concept of supercritical fluid and piston effect to me, which gives birth to the subsequent six-year exploration. Prof. Zhang always emphasized the importance of innovation and gave me sufficient freedom and opportunities to practice in all aspects. I would like to express my gratitude to Prof. Stephen H. Davis, who was my supervisor during my third year of Ph.D. study at Northwestern University. Prof. Davis systematically guided me to conduct research on linear stability and weakly nonlinear stability, which enabled me to master the methods of stability analysis. He taught me the methods of modeling and simplifying problems from the experimenter’s point of view, and also helped me establish the good habit of writing research records. I worked with Prof. Davis when he was 80 years old, but he was still active in research, rewriting the paper’s abstracts, deriving formulas, and explaining theories to me at our weekly meeting. These details are still fresh in my mind. I admire him for his humorous and easy-going character, as well as his simple and pure love for research. I would like to thank my friends and labmates for their company over the past 5 years. Whenever I was in a glum mood or felt anxious about the future, they were always willing to give me hands and help me get back to the right track of life. I would like to thank Shengjia Cai, Qiao Liu, and Xingyu Shang for their companionship and care during my Ph.D. study. I would like to thank Menghe Sun, Chao Wang, Jiawei Li, Jiaqi Sun, Huang Zhuang, Yating Xiao, Guanbang Wang, Yisai Gao, Junming Yin, Zhaorui Peng, Yudong Zhu, Bin Fang, and Xuegang Lu, for many memorable academic communications and brainstorming opportunities in the past 5 years. I would like to thank my roommates Peng Ao and Yong Zhao; my labmates Qian Zhang, Katarzyna Kowal, and Hamid Karani; Mohan Liu and Chen Chen from LowPass Band; all my friends from the Beike Drama Club, Ian Hammond and Hanna Hammond from the International Exchange Student Office; and all other friends. I would like to thank my family for their hard work and dedication during my doctoral study, and for creating a comfortable environment enabling me to concentrate on my research. I would like to express deep gratitude to my grandfather, who xiii

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loved me the most. He went through the draft of this thesis and expressed his approval. Unfortunately, he was unable to see my graduation. I think the only way to repay his kindness is to work hard to be better. Most of the calculations in this thesis were performed on the high-performance computing platform of Peking University and the QUEST computing platform at Northwestern University. I am grateful to the National Natural Science Foundation of China (NSFC) for programs No. 51476001, 51776002, and the Chinese Scholarship Council for No. 201706010268. Shenzhen, P.R. China August 2021

Zhan-Chao Hu

Contents

1 Introduction to Binary Mixtures at Supercritical Pressures and Coupled Heat and Mass Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Supercritical Pressure Fluids and Near-Critical Fluids . . . . . . . . . . . . 1.1.1 Critical Phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.2 Anomalies in Physical Properties . . . . . . . . . . . . . . . . . . . . . . . 1.1.3 Engineering Applications of Supercritical Pressure Fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Coupled Heat and Mass Transfer in Binary Mixtures at Supercritical Pressures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Introduction to Three Fundamental Problems . . . . . . . . . . . . 1.3.2 Piston Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.3 Rayleigh-Bénard Instability . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.4 Buoyancy-Driven Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Basic Equations and Physical Properties of a Reference Binary Mixture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Basic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Physical Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Phase Diagram and Behavior of Physical Properties of C2 H6 − CO2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 1 1 3 5 6 6 7 8 10 12 12 13 21 21 22 25 29 29

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Part I

Contents

Coupling Through Cross-Diffusion Effects: Relaxation and Diffusion Problems

3 Coupled Transfer in a Relaxation Process: Mass Piston Effect . . . . . . 3.1 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Traveling-Wave Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Wave Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Wave Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Numerical Validations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Numerical Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Comparisons and Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Energy Balance Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Coupled Transfer in a Diffusion Problem: Concentration Gradient in the Coexisting Liquid-Like and Gas-Like States . . . . . . . 4.1 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Physical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2 Governing Equations for the Motionless Steady State . . . . . 4.1.3 Summary of Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Numerical Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Characteristics for the Coexistence of LL and GL States . . . . . . . . . 4.3.1 State Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Physical Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.3 Physical Explanation for the Pressure Drop . . . . . . . . . . . . . . 4.3.4 Heat Transfer Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Assessments on the Influences of Concentration Gradient . . . . . . . . 4.4.1 Influences of Concentration Gradient on the PB . . . . . . . . . . 4.4.2 Influences of Concentration Gradient on Pressure Drop and Heat Flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Part II

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Coupling Through Cross-Diffusion Effects: Instability and Bifurcation

5 Interactions Between Coupled Transfer and Gravity: Rayleigh-Bénard Instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Problem Statement and Approximation . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 Physical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.2 Mathematical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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5.2 Linear Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Formulation of Stability Problem . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 Analytical Solution under Ideal Boundary Conditions . . . . . 5.2.3 Stability Threshold under Various Conditions . . . . . . . . . . . . 5.3 Further Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Origin of the Oscillatory Instability . . . . . . . . . . . . . . . . . . . . . 5.3.2 Relative Importance of Gravity-Related Effects . . . . . . . . . . . 5.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Interactions Between Coupled Transfer and Gravity: Nonlinear Rayleigh-Bénard Convection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.1 Physical and Mathematical Models . . . . . . . . . . . . . . . . . . . . . 6.1.2 Response Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.3 Numerical Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Preliminary Numerical Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Weakly Nonlinear Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 Conditions for Backward Bifurcation at R = Rcrit mono . . . . . . . . . . . ............ 6.3.2 Relative Magnitude Between Rf.a. and Rcrit osc 6.4 Theory-guided Numerical Simulations and Discussions . . . . . . . . . . 6.4.1 Results under Partially Satisfied Necessary Conditions . . . . 6.4.2 Results under Fully Satisfied Necessary Conditions . . . . . . . 6.4.3 Physical Explanation for the Conditions of FA Instability below Rcrit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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75 75 78 80 81 81 85 86 87 89 90 90 91 91 92 94 97 98 101 101 106 109 112 113

Part III Coupling Through Boundary Reactions: Buoyancy-Driven Flows 7 Coupled Transfer Through Boundary Reactions: An Application-Oriented Cavity Flow Problem . . . . . . . . . . . . . . . . . . . . 7.1 Problem Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.1 Physical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.2 Mathematical Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.3 Initial and Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . 7.1.4 Governing Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Formulations for CO2 −Naphthalene System . . . . . . . . . . . . . . . . . . . 7.2.1 Solubility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.2 Adsorption and Desorption Reactions . . . . . . . . . . . . . . . . . . . 7.2.3 Thermodynamic and Transport Properties . . . . . . . . . . . . . . .

117 118 118 119 121 122 123 124 125 127

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Contents

7.3 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.1 Thermodynamic Optimization . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.2 Hydrodynamic Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Numerical Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.1 Numerical Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.2 Cooperative Regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.3 The Effects of Cavity Height . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

127 128 129 135 135 135 138 139 140

8 Summary and Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 8.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 8.2 Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 Appendix A: Numerical Methods for the Linear Stability Analysis of Rayleigh-Bénard Instability under Realistic Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 Appendix B: The Solubility of Naphthalene in Supercritical CO2 . . . . . . . 155 Appendix C: The Isobaric Specific Heat of Supercritical CO2 − C2 H6 System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

Acronyms

Roman Symbols t ta x z T TPB p pd pth ps c u u w va cp cv cs kT kp d dBL D DT H¯ M M1 M2

Time [s] or [-] Acoustic timescale [s] Horizontal coordinate [m] or [-] Vertical coordinate [m] or [-] Temperature [K] or [-] Temperature at maximal cp for a given supercritical pressure [K] Pressure [Pa or MPa] Dynamic pressure [Pa] or [-] Thermodynamic pressure [Pa] Steady-state or hydrostatic pressure [Pa or MPa] Concentration (mass fraction of C2 H6 or naphthalene) [-] Velocity vector [m/s] Velocity in x direction [m/s] or [-] Velocity in z direction [m/s] or [-] Sound speed [m/s] Isobaric specific heat [J/(kg · K)] Isochoric specific heat [J/(kg · K)] Concentration susceptibility [kg/J] Thermodiffusion factor [-] Barodiffusion factor [-] Thickness or height of the fluid layer [m] Boundary layer thickness [m] Diffusion coefficient [m2 /s] Thermal diffusivity [m2 /s] Partial enthalpy of the mixture [J/kg] Molar mass [kg/mol] Molar mass of the first component (or the solvent) [kg/mol] Molar mass of the second component (or the solute) [kg/mol]

xix

xx

q q Q i i g g g0 k N E˙ in E˙ bulk T˙ bulk r rT R RS Rad Pr L Q Nu Sh Re m ˙ VT Vc VS K s ys

Acronyms

Heat flux vector [W/m2 ] One-dimensional heat flux [W/m2 ] Steady-state heat flux [W/m2 ] Mass flux vector [kg/(m2 · s)] One-dimensional mass flux [kg/(m2 · s)] Gravitational acceleration vector [m/s2 ] Gravitational acceleration [m/s2 ] Negative gradient of c at x = 0 [m−1 ] Wave number [-] Number of grid points [-] Energy injection rate [W/(m2 )] Energy reception rate of bulk fluid [W/(m2 )] Temperature change rate of bulk fluid [K/s] Energy ratio [-] Temperature gradient ratio [-] Rayleigh number [-] Solutal Rayleigh number [-] Adiabatic Rayleigh number [-] Prandtl number [-] Lewis number [-] Dufour number [-] Nusselt number [-] Sherwood number [-] Reynolds number [-] Cross-boundary flow rate of the solute [kg/(m2 · s · K)] Total volume [m3 ] Growth rate of the crystal [m/s] Molar volume of the solute [m3 /mol] Growth rate constant [mol/(m2 · s)] Solubility in mass fraction (maximum mass fraction of the solute) [-] Solubility in mole fraction (maximum mole fraction of the solute) [-]

Greek Symbols α β κ ρ ρs ρS T t λ

Isothermal compressibility [Pa−1 ] Thermal expansion coefficient [K−1 ] Concentration contraction coefficient [-] Density [kg/m3 ] Stratified initial density [kg/m3 ] Density of solute [kg/m3 ] Temperature difference [K] Time step [s] Thermal conductivity [W/(m · K)]

Acronyms

η ν μ μ1 μ2  θ p ρ c ∇ad ∇c ∇T ξE ξT 

ξp ξc ϕ ψ (·) ζ σ σr ω γ  ς χ

Dynamic viscosity [Pa · s] Kinematic viscosity [m2 /s] Chemical potential of the mixture [J/kg] Chemical potential of the first component [J/mol] Chemical potential of the second component [J/mol] Viscous dissipation [W/m3 ] Stream function [m2 /s] or indicator for the regime of DDC [degree] Pressure drop = pref − ps [Pa or MPa] Overall density difference [kg/m3 ] Overall concentration difference [-] Adiabatic temperature gradient [K/m] Typical concentration gradient [m−1 ] Typical relative temperature gradient [Km−1 ] Energy efficiency [-] Temperature efficiency [-] or partial derivative of the solubility with respect to temperature [K−1 ] Partial derivative of the solubility with respect to pressure [Pa−1 ] Partial derivative of the solubility with respect to concentration [-] The amplitude of the acoustic wave in response to unit boundary velocity [-] Separation ratio [-] Auxiliary operator [-] Auxiliary variable [-] Complex growth rate [-] or supersaturation [-] Real part of complex growth rate [-] Angular frequency [-] Specific heat ratio [-] Auxiliary quantity [-] or solubility efficiency [-] Auxiliary variable [-] Gravitational compressibility factor [-]

Superscripts  _ ∼ avg crit ideal mod max osc f.a. o.f.

xxi

Amplitude of the acoustic wave or dimensionless variable Base state or initial state Small perturbation Spatially average Neural stability Ideal Modified Maximum value Oscillatory instability Finite-amplitude instability Oscillatory instability and finite-amplitude instability

xxii

Acronyms

Subscripts C mod max mono osc ref f.a. S LCEP UCEP

Critical parameter Modified Maximum value Monotonic instability Oscillatory instability Reference state Finite-amplitude instability Solute Lower critical end point Upper critical end point

Abbreviations AARD ATG BC BL CP1 CP2 CEP COO DDC EOS FA FIN GL LCEP LL MPE OSC PB PE RB SLV STA TVD UCEP

Average absolute relative deviation Adiabatic temperature gradient Boundary condition Boundary layer Critical point of the solvent Critical point of the solute Critical end point Cooperative regime Double-diffusive convection Equation of state Finite amplitude Fingering regime Gas-like Lower critical end point Liquid-like Mass piston effect Oscillatory regime Pseudo-boiling Piston effect Rayleigh-Bénard Solid-liquid-vapor equilibrium Stable regime Total variation diminishing Upper critical end point

Chapter 1

Introduction to Binary Mixtures at Supercritical Pressures and Coupled Heat and Mass Transfer

1.1 Supercritical Pressure Fluids and Near-Critical Fluids 1.1.1 Critical Phenomena Liquid-vapor phase transition is a common phenomenon in natural and engineering processes. Figure 1.1 (a) shows a typical phase diagram of a pure (i.e., onecomponent) fluid on the temperature-pressure plane. The transition takes place on the saturation line, which is also the boundary for liquid and vapor. However, as pressure grows, the saturation line has an end point, termed as the critical point, designating a critical temperature TC and a critical pressure pC . Above the critical pressure (i.e., p > pC ), there is no liquid and vapor phase transition, and fluids at this region are called supercritical pressure fluids. Besides, along the liquid-vapor saturation line, a transition from subcritical state to supercritical one can be experimentally observed. Under subcritical conditions (Fig. 1.1b), gas and liquid coexist and an obvious interface can be observed. At the critical point (Fig. 1.1c), the critical opalescence caused by molecular scattering can be observed. Under supercritical conditions (Fig. 1.1d), only a single-phase fluid exists with no separation of gas and liquid.

1.1.2 Anomalies in Physical Properties For a supercritical pressure fluid, its properties are not a simple function of pressure p and temperature T . In a specific narrow temperature interval, physical properties undergo abnormal variations. For example, density ρ drops rapidly, thermal expansion coefficient β and isobaric specific heat c p develop a peak, while thermal diffusivity DT shows a local minimum. These behaviors are termed as critical © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 Z.-C. Hu, Coupled Heat and Mass Transfer in Binary Mixtures at Supercritical Pressures, Springer Theses, https://doi.org/10.1007/978-981-16-7806-6_1

1

2

1 Introduction to Binary Mixtures at Supercritical Pressures ...

Pressure

High Density (liquid-like) Low Density (gas-like)

Supercrical Pressure Fluid Liquid Crical point

Vapor

a

b

c

d

Temperature Fig. 1.1 Phase diagram of a pure fluid and the observation along going-up the saturation line [1]

(a)

(b)

Fig. 1.2 Several representative physical properties of CO2 at a supercritical pressure p = 7.5 MPa. a Solid line: density ρ and dashed line: thermal expansion coefficient β. b Solid line: isobaric specific heat c p and dashed line: thermal diffusivity DT . Each physical property has its own vertical axis. The data are acquired from NIST database [2]

anomalies. Figure 1.2 demonstrates these facts using CO2 at p = 7.5 MPa as a reference1 . Since the rapid drop of density in a narrow temperature interval mimics the boiling process under a subcritical pressure, the supercritical pressure region can be divided into high-density liquid-like (LL) state and low-density gas-like (GL) one. There are generally two types of separating boundaries for them. One type is termed as the Widom line [3, 4], defined as the maximum of thermodynamic response function and usually depicted based on the local maxima of a thermodynamic property. Apart from the Widom line with a pure thermodynamic definition, there is also the Frenkel line defined in a dynamic way associated with different diffusion mechanisms [5, 6]. Even though identifying the separating line between LL and GL states is a developing subject [7], the most popular one is the Widom line based on the isobaric specific heat [8, 9], which is also adopted in this thesis. Now, for a given supercritical pressure, there is a temperature that marks the Widom line, often denoted by TPB . Therefore, TPB is the temperature at which the isobaric specific heat reaches 1

The critical parameters of CO2 are pC = 7.3773 MPa and TC = 304.13 K.

1.1 Supercritical Pressure Fluids and Near-Critical Fluids

3

its maximal value. Note that the subscript PB means pseudo-boiling, reflecting the similarity between the fast density drop and boiling [8]. After defining the Widom line, the other important concept—“near-critical” should be introduced. In this thesis, only the supercritically near-critical states are concerned with. Even though one may find different definitions for the term “nearcritical”, the underlying idea is the same. That is, to highlight the states where fluids are greatly influenced by critical anomalies. Therefore, near-critical states are used to denote the region around the Widom line with p close to pC . In this thesis, states at supercritical pressures are general concerns, with special attention paid to near-critical ones.

1.1.3 Engineering Applications of Supercritical Pressure Fluids During the last 50 years, due to the pursuit of high parameters and high efficiencies, also because of their excellent properties, supercritical pressure fluids have been widely used in many engineering fields, and also involved in many emerging techniques. In energy engineering, Brayton cycle using supercritical CO2 as the working fluid has been developed for applications in fourth-generation nuclear reactors [14] and concentrated solar power plants [15] (see Fig. 1.3a,b). Supercritical pressure fluids have been proposed for high-capacity thermal energy storage due to the critical enhancement of isobaric specific heat [12, 16] (see Fig. 1.3c). In addition, because supercritical CO2 has chemical stability, high natural convection intensity, and the importance of its geological storage in combating global warming, it has also been proposed to be applied to geothermal utilization [13, 17] (see Fig. 1.3d). Besides, supercritical pressure fluids have also been applied to refrigeration [18], thermal management of fuel cells, and waste heat utilization [19]. In aeronautics and astronautics, in the regenerative cooling of rocket engines and hypersonic vehicles, the fuel first flows through the combustion chamber walls to absorb heat and then enters the combustion chamber [21, 23], during which the fuel undergoes a complex heat transfer process at supercritical pressure (see Fig. 1.4a). Meanwhile, the fuel injection process is also a complex hydrodynamic process from supercritical pressure to supercritical or subcritical pressure [22, 24] (see Fig. 1.4b). In chemical engineering [25], supercritical pressure fluids have been widely used as solvents in extraction and separation processes of natural substances, such as vitamins, natural pigments and essential oils, because the state of supercritical pressure fluids can be easily regulated by changing the temperature or pressure. In polymer processing, supercritical pressure fluids are used as solvents or antisolvents to achieve the preparation of particles of different sizes, reducing energy consumption and pollution. As alternatives to organic solvents, supercritical pressure fluids also serve as

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1 Introduction to Binary Mixtures at Supercritical Pressures ...

Fig. 1.3 Examples for the applications of supercritical pressure fluids in energy engineering. a Lead-cooled fast reactor based on supercritical CO2 Brayton cycle [10]. b Molten salt heat storage solar power station based on supercritical CO2 Brayton cycle [11]. c Supercritical sensible heat storage [12]. d Supercritical CO2 enhanced geothermal system [13]

(a)

(b)

Fig. 1.4 Examples for the applications of supercritical pressure fluids in aeronautics and astronautics. a Active thermal protection in hypersonic vehicles [20, 21]. b Fuel injection process in the main combustion chamber of a liquid rocket engine [22]

1.1 Supercritical Pressure Fluids and Near-Critical Fluids

5

media for chemical and biochemical reactions. Besides, supercritical pressure fluids are also used as solvents in applications such as drying and cleaning. From the above examples, it can be further inferred that supercritical mixtures are more widely used than pure fluids. In energy engineering, the application of CO2 -based blends as the working fluids can achieve higher efficiency of Brayton cycle [26–30], through a good matching with ambient temperature [31, 32] and reducing compression work. In supercritical pressure fluid heat storage, fluids with different critical points are obtained by mixing to meet the demand for thermal energy storage at different temperature levels. In enhanced geothermal systems, a mixture system forms by dissolution [13]. In aeronautics and astronautics, the fuels are natural mixtures. In chemical engineering, supercritical pressure fluids are used as solvents or reaction media, and the resulting systems are also mixtures. However, most studies regarding supercritical fluid dynamics and heat transfer usually consider pure fluids. Mixtures have been rarely investigated. Therefore, this thesis takes the simplest case of a mixture, a binary mixture, as the object to investigate its fluid dynamics and transport phenomena, which can provide details of physical mechanisms for relevant engineering applications.

1.2 Coupled Heat and Mass Transfer in Binary Mixtures at Supercritical Pressures Compared with pure fluids, the prominent nature of a binary mixture at supercritical pressures in transport phenomena is coupled heat and mass transfer. In fact, the coupling is achieved in two ways: 1. Coupling through cross-diffusion effects. Cross-diffusion effects include the Soret effect (mass diffusion driven by temperature gradient) and the Dufour effect (thermal diffusion induced by concentration gradient) [33]. The critical anomalies enhance the cross-diffusion effects [34] at near-critical states. 2. Coupling through temperature-dependent boundary reactions. This is closely related to practical applications in chemical engineering. For example, temperature tends to influence solubility [35], crystallization rate, adsorption and desorption rates [36], etc, which in turn changes the concentration at the fluid-solid interface, leading to coupled transfer through boundary conditions. Coupled heat and mass transfer has deep effects on the fluid dynamics and transport phenomena of binary mixtures at supercritical pressures. This thesis is devoted to developing fundamental insights into this topic. Before introducing the outline of this thesis, some fundamental problems in the fluid dynamics and transport phenomena of supercritical pressure fluids, especially near-critical ones, are overviewed in the following section.

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1.3 Literature Review 1.3.1 Introduction to Three Fundamental Problems The understanding of critical anomalies in pure and binary mixtures is attributed to a series of theoretical works in the 1960s and 1970s [37–39], especially the renormalized group theory [39]. The explorations on fluid dynamics and transport phenomena that are related to critical anomalies in supercritical pressure fluids start from the 1990s. During the last 30 years, these researches have revealed a series of new phenomena and mechanisms. After reviewing available research papers, review papers [40–44] and monographs [45–47], three fundamental problems in this field can be summarized as follows. 1. Piston effect. The piston effect (PE) is a rapid thermalization phenomenon, also known as adiabatic heating [48], thermomechanical effect [47, 49], thermoacoustic heating [50, 51] and thermally driven acoustic wave [52, 53]. The PE can be understood as follows. Heating an enclosed near-critical fluid cavity, a thermal boundary layer forms. Because of large thermal expansion coefficient in a near-critical fluid, the thermal boundary layer will expand and squeeze the other fluid (bulk fluid), resulting in a uniform temperature rise. On the acoustic timescale, the squeezing is done by the propagation and accumulation of acoustic waves (compression waves), so the PE can be approximated as an isentropic effect. Viewed from the pressure field, the PE is actually a relaxation of the pressure, and the thermalization is a side effect. The accelerating effect of the PE on the thermal equilibrium exactly compensates the diminishing thermal diffusivity at near-critical states. This fact is termed as critical speeding-up [47]. 2. Hydrodynamic instability. Hydrodynamic instability concerns the conditions of flow occurrence and transition, which is a basic problem in fluid dynamics. In near-critical fluids, Rayleigh-Bénard (RB) instability is the most widely studied problem [54, 55], i.e., the onset of convection in a horizontal fluid layer heated from below. Other instabilities include thermo-vibrational instability [47, 56], Rayleigh-Taylor instability [47, 57], Kelvin-Helmholtz instability [45, 58], dissolved RB instability [59], the double-diffusive instability [60, 61], and Poiseuille-RB instability [62]. 3. Buoyancy-driven flows. Buoyancy-driven flows have been extensively studied because the density of a near-critical fluid is sensitive to temperature variations. Therefore, a small temperature difference can lead to strong thermal convection. The square cavity and the cylindrical container have received much attention due to their simplicity and representation. The literature reviews given below are organized according to the above classification, where the hydrodynamic instability part only focuses on RB instability.

1.3 Literature Review

7

1.3.2 Piston Effect As mentioned earlier, the large compressibility and diminishing thermal diffusivity at near-critical states are responsible for a rapid thermalization process known as the PE [63–66]. Due to buoyant convection on the earth, the PE had long been ignored and was first observed in a microgravity experiment [67] measuring the specific heat capacity of SF6 [67]. Early studies focus on the physical explanation, theoretical analysis and experimental validation of the PE. In 1990, three independent research teams proposed explanations for the PE based on different approaches [63, 65, 66]. Onuki et al. [48, 65] proposed a thermodynamic theory. Boukari et al. [63] and Zappoli et al. [66] performed numerical simulations to intuitively show the process of PE. Subsequently, Zappoli and Carlès conducted a series of matched asymptotic expansion analyses, revealing the decoupling in relaxations of temperature and density [68], and obtaining the analytical solution for the PE in the one-dimensional heating model [69]. Then, they predicted the existence of acoustic saturation (i.e., heat transfer is completed at the speed of sound) [70], viscous regime [71] (i.e., viscosity divergence causes the growth in relaxation time) on the acoustic timescale. They also gave a description of the density relaxation process on the diffusion timescale. Zappoli et al. [72] pointed out that the cooling of near-critical fluids also leads to the cooling PE, which was shown to be highly dependent on the boundary wall material. Zappoli et al. [72] thus pointed out the importance of material selection in experiment design. In terms of experimental studies, early experiments measured temperature or density by thermistors or interferometry to confirm the PE, and the decoupling in relaxations of temperature and density [73–76]. Another research topic with regard to PE is the measurement of acoustic waves. In 2006, Miura et al. [77] achieved the measurement of acoustic waves through an ultra-sensitive interferometer, which sets the benchmark for later numerical studies in terms of code verification. Onuki [51] presented a complete theoretical model for thermoacoustic waves. As for numerical studies, Nakano and Shiraishi [78] carried out numerical calculations of the PE using nitrogen as the working fluid. Shen and Zhang [79, 80] conducted a series of numerical simulations to investigate the influences of initial and boundary conditions on the generation, propagation and reflection of acoustic waves. Hasan et al. [50, 81] carried out experiments using a high-sensitivity pressure measurement device to elucidate the intensities of the acoustic field under different heating rates and initial states. For the active application of the PE, Beysens et al. [82] experimentally investigated the feasibility of long-distance heat transport. Their experimental results show that although the efficiency is in the range of about 10%–30% and decreases further with increasing length, the response of the device is very fast. Therefore, it can aid conventional heat pipes under rapidly changing heating conditions [82]. The above works are concerned with pure fluids. In fact, there are some studies that surveyed the PE in supercritical mixtures. Nakano and Shiraishi [83] experimentally investigated the PE in supercritical air inside a top-heating vessel. Experimental

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1 Introduction to Binary Mixtures at Supercritical Pressures ...

observations exhibit typical temperature variations induced by the PE, and evidences for the Soret effect were also acquired. In a subsequent numerical work [84], the Soret effect was further confirmed by incorporating it into the governing equations, leading to better agreements with experimental results. In addition, Raspo et al. [35] performed numerical simulations for a dissolving problem with temperaturedependent solubility on the boundary, in which the response of concentration field under boundary heating was investigated. The Soret effect was not considered in their study, and the coupled heat and mass transfer was achieved through a boundary reaction. Recently, Long et al. [49] numerically studied the PE, Soret effect and Dufour effect in supercritical binary mixtures. Their work shows that, unlike the PE, the Soret effect and the Dufour effect act mainly on the diffusion timescale [49], and both promote the heat transfer. In summary, previous studies mainly focused on pure fluids, so the PE in binary mixtures has not been sufficiently understood. Specifically, thermal disturbance is always the inducing factor for the PE. How about mass disturbance? Can it cause the PE? What is the role of cross-diffusion effects? Answering these questions is one of the tasks of this thesis.

1.3.3 Rayleigh-Bénard Instability RB convection is a canonical system of buoyancy-driven flow, whose configuration is a horizontal fluid layer heated from below and cooled from above, indicated by the temperature difference T . Indeed, the existence of convection depends on the value of T . As T increases from zero, the diffusive base state undergoes transitions from stable to neutrally stable, and finally to unstable, giving birth to convection. The onset criterion, i.e., T of the neutral stable state, is the most basic question in RB convection, which is also termed as RB instability. Indeed, for a normal incompressible fluid, the criterion for the convection onset describes the interactions among buoyancy, heat conduction, viscosity and boundary condition [85], which is measured by the critical Rayleigh number, denoted by R crit . For no-slip boundary, R crit = 1707.8, while for stress-free one R crit = 657.5. In nearcritical pure fluids, due to the strong compressibility, the thermal effect of hydrostatic pressure gradient matters, leading to the so-called Rayleigh-Schwarzschild criterion [55]. For the sake of brevity, Table 1.1 lists representative previous works of RB instability in near-critical pure fluids. In Table 1.1, Gitterman and Steinberg [86] performed the earliest study for the RB instability in near-critical pure fluids. As pointed by Carlès and Ugurtas [55], the criterion is too complex without taking reasonable simplifications into consideration. Carlès and Ugurtas derived the concise Rayleigh-Schwarzschild criterion using matched asymptotic expansion: R mod =

ρgβ(∇T − ∇ad )d 4 ≥ R crit , ηDT

(1.1)

1.3 Literature Review

9

Table 1.1 Representative previous works of RB instability in near-critical pure fluids Author Year Description Gitterman and Shteinberg [86] 1970 Carlès and Ugurtas [55]

1999

Kogan et al. [87–89]

1999, 2000

Chiwata and Onuki [90, 91]

2001,2002

Amiroudine et al. [54]

2001

Amiroudine and Zappoli [92]

2003

Meakawa et al. [93, 94]

2002, 2004

Accary et al. [95–97]

2004, 2005

Accary et al. [98]

2009

The earliest research on the onset criterion Concise Rayleigh-Schwarzschild criterion Experimental evidence of Schwarzschild criterion Concise governing equations and transient evolution Two thermal boundary layers emerge due to PE PE-induced temperature oscillations in RB convection PE alters the stability of the boundary layer Return to stability due to Schwarzschild criterion Transient development after convection onset

where d is the thickness of the fluid layer, g = 9.81 m/s2 is gravitational acceleration, ∇T = T /d is the temperature gradient, T is the temperature difference, ∇ad = Tβg/c p is the adiabatic temperature gradient (ATG, the thermal effect of hydrostatic pressure gradient) and R mod is the modified Rayleigh number. If ∇ad = 0, the original Rayleigh criterion recovers. In general, ATG is usually considered in large-scale geophysical systems. Kogan et al. [87–89] performed a series of experiments to verify the role of ATG in convection onset, and reported temporal oscillations. The groups led by Onuki [90, 91] and Zappoli [92] conducted numerical simulations to explain the previous experiments, and attributed the temporal oscillations to the PE. The above studies focus on the stability of the whole fluid layer. Since Amiroudine et al. [54] pointed out the PE-induced double-boundary-layer structure in the transient evolution of RB convection, the transient stability of thermal boundary layers became a research hotspot. Maekawa et al. [93, 94] performed linear stability analysis to obtain the criterion for the stability of boundary layer, and the scaling law for convection triggering time. Accary et al. [95–97] revealed the possibility of return to stability due to Schwarzschild criterion, and their subsequent 3D numerical study [98] discussed the nonlinear evolution of convection from aspects such as vortex structure. As for near-critical binary mixtures, the earliest work concerning RB instability was also done by Steinberg [99], in which the ideal boundary condition with fixed-concentration was considered, and approximate criteria were obtained through Bubnov-Galerkin numerical method. Due to the preliminary understanding of critical

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1 Introduction to Binary Mixtures at Supercritical Pressures ...

anomalies in binary mixtures at that time, together with the ideal boundary condition, the criterion is not precise and the form is too complicated. Besides, with the help of the approximate approach in [55], Das and Bhattacharjee [100] performed linear stability analysis with cross-diffusion effects and ATG involved. Their criterion is based on proper boundary condition and more compact than [99]. However, the main drawback of previous works is the neglect of gravity-induced concentration gradient, i.e., the other gravity-related effect apart from ATG, caused by the migration of the heavier component under gravity [101–104]. Since it is enhanced under near-critical conditions, its influences on RB instability are anticipated to be profound, which will be confirmed by this work. Besides, the stability under finite-amplitude perturbations has not been reported yet. All in all, the RB instability in binary mixtures with cross-diffusion effects, ATG and gravity-induced concentration gradient has not been well-understood. As the most fundamental question in RB convection, theoretical analysis and calculations are urgently needed.

1.3.4 Buoyancy-Driven Flows The interests in buoyancy-driven flows of near-critical fluids are originated from the explorations of the PE under gravity. The two-dimensional square cavity model has received much attention. Depending on the arrangement of heating source, there are side heating, central heating, and bottom heating. Zappoli et al. [105] carried out a numerical study for a side-heated cavity flow in a near-critical fluid. The results show that the temperature field can reach an approximate equilibrium rapidly because of PE, and afterwards, a small temperature difference remains. However, the large thermal expansion coefficient still results in considerable density inhomogeneity, giving birth to convection. Due to the feature of temperature, such convection is termed as “quasi-isothermal convection”. Numerical calculations by Hasan and Farouk [106] included the effects of variable physical properties, initial states and bulk viscosity on cavity flows. A correlation equation for the heat transfer coefficient was also proposed. Soboleva [107] revealed the influences of variable physical properties on the side-heated cavity flow. In center-heated cavity flow, Zappoli et al. [108] revealed an interesting interaction between the thermal plume and the cooling PE. Soboleva [109] pointed out that the PE has an inhibiting effect on the convection, while on the contrary, the flow does not have much effect on the PE. As for the bottom-heated cavity flow (RB convection), Shen and Zhang [110, 111] performed a series of 2D and 3D numerical simulations to study the transient and steady-state flow and heat transfer characteristics. Soboleva and Nikitin [112] reported the benchmark data for steady-state RB convection through accurate numerical calculations. Numerical simulations by Wei et al. [113] confirmed the specific heat ratio is an effective parameter for measuring the strength of the PE, suggesting that the PE does not alter the steady state of convection.

1.3 Literature Review

11

Apart from those works directly motivated by the PE, the high compressibility nature of near-critical fluids also makes them good experiment materials for turbulent RB convection, allowing a large Rayleigh number (∼ 1011 ) in a centimeter-scale cell. In fact, a large Rayleigh number in experiments is favored since the mid-nineties due to the quest for “ultimate regime” in RB convection. Chavanne et al. [114] performed experiments of turbulent RB convection using supercritical 3 He as the working fluid, and a 0.41 scaling law between the Nusselt number and Rayleigh number was discovered, providing evidence for the ultimate 1/2 scaling law [115]. Later similar experiments of Niemela et al. [116] confirmed that the scaling exponent tends to 0.31 at high Rayleigh numbers, and no evidence for a transition to the 1/2 regime was reported. Ashkenazi and Steinberg [117, 118] conducted experiments at very high Rayleigh number (up to 1014 ) using near-critical SF6 as the working fluid, and successfully verified the generalized scaling laws for the heat transport and the large scale circulation velocity predicted theoretically by Shraiman and Siggia [119]. Later, researchers realized turbulent RB convection in near-critical fluids contains dramatic non-Boussinesq effects like variable physical properties. Therefore, to some extent, near-critical fluids are not suitable to be employed to verify the theory for Boussinesq fluids. However, the increasing attention regarding nonBoussinesq effects in turbulent RB convection again makes near-critical fluids good experimental materials. Ahlers et al. [120, 121] confirmed the main non-Boussinesq effect in near-critical ethane is the spatial variation of thermal expansion coefficient, and showed the boundary layer under non-Boussinesq effect can be described well by an extended Prandtl-Blasius boundary-layer theory. Burnishev et al. [122, 123] put forward the concept of symmetrical non-Boussinesq effect, and investigated its influences on statistical and scaling properties of temperature field in turbulent RB convection by experiments using near-critical SF6 as working fluid. Recently, Valori et al. [124, 125] and Yik et al. [126] performed measurements of heat transfer in turbulent RB convection of supercritical SF6 with strongly variable properties. They extended the GL theory [127] for variable fluid properties and obtained good predictions [126]. Besides, they also realized the particle image velocimetry measurement of velocity field [128] by using the background-oriented Schlieren technique. Assenheimer and Steinberg [129–132], Roy and Steinberg [133], and Ahlers et al. [134] also studied the influences of non-Boussinesq effects of supercritical pressure fluids on the pattern formation in RB convection. A common feature of the two types of studies mentioned above is that they are mainly concerned with pure fluids. In recent years, there are a small number of studies considering binary mixtures and the coupled transfer in them. Long et al. [135] numerically investigated the natural convection in a supercritical nitrogen/argon binary mixture induced by bottom heating through numerical simulations. It was found that the Soret effect and Dufour effect accelerate the development of natural convection, while their specific roles are difficult to discern in the fully developed stage. They also extended their study to experiments [136]. Wannassi and Raspo [36] numerically studied flows in a side-heated cavity with temperature-dependent adsorption rate at a boundary. The evolution of the concentration field was studied to understand the adsorption process.

12

1 Introduction to Binary Mixtures at Supercritical Pressures ...

In summary, buoyancy-driven flows in near-critical fluids have been extensively studied to understand the turbulent thermal convection, and the interactions of PE with gravity. In recent years, researches have been extended to binary mixtures, in which the coupled heat and mass transfer has gained increasing attention. However, in this specific topic, there are actually only two published papers [36, 136]. Given its rich application background, there are great potentials for further research.

1.4 Motivation The wide applications of supercritical pressure fluids and the frequent occurrences of their mixtures make it essential to have a better understanding of the heat and mass transfer, in which the coupled transfer is a key characteristic. For this, binary mixtures at supercritical pressures are chosen as research objects. A series of theoretical analyses and numerical simulations are reported in this thesis, under the configuration of the three fundamental problems outlined in Sect. 1.3.1. Coupled heat and mass transfer is included, where relevant physical properties are carefully modeled to correctly reflect the critical anomalies at near-critical states. All these efforts are devoted to providing fundamental insights into the coupled heat and mass transfer effects and their influences on fluid dynamics and transport phenomena in binary mixtures at supercritical pressures. From a practical point of view, these insights may inspire new applications and help to realize better designs of relevant apparatus in energy engineering, aeronautics and astronautics, and chemical engineering.

1.5 Thesis Outline For the sake of clarity, the structure of this thesis is summarized in Fig. 1.5. The thesis has been divided into the following parts and chapters. Chapter 2 elaborates the basic equations and physical properties of a reference binary mixture, paving the way for studies in subsequent chapters. Part I includes Chaps. 3 and 4, where relaxation and diffusion problems are the main concerns and gravity is ignored. In Chap. 3, the relaxation of a supercritical binary mixture after subjecting to a cross-boundary mass diffusion flux is studied. The concept of the mass piston effect is proposed and its mechanisms are analyzed. In Chap. 4, temperature differences with TPB in between are applied to the same fluid layer. It proposes the idea that when LL and GL states coexist, there is a concentration gradient, as an analogy to the situations at subcritical pressures. It confirms that at near-critical states, the concentration gradient greatly alters buoyancy force, laying down a basis for Part II. Part II includes Chaps. 5 and 6, where the RB instability and bifurcation are investigated. On the one hand, this work is motivated by the conclusion regarding

1.5 Thesis Outline

13 Binary Mixtures at Supercritical Pressures Coupled Heat and Mass Transfer Cross-diffusion Effects

Boundary Reactions

Part I Relaxation and Diffusion Problems

Part II Instability and Bifurcation

Part III Buoyancy-driven Flows

Mass Piston Concentration Gradient in a Diffusive Steady State Effect

RB Instability and Bifurcation

Cavity Flow

Chapters 5&6

Chapter 7

Chapter 3

Chapter 4

Fig. 1.5 Outline of this thesis

buoyancy in Chap. 4. On the other hand, it is also motivated by curiosity about the fluid dynamics of the bounded fluid layer after adding gravity. To be specific, Chap. 5 investigates the linear stability of small perturbations and obtains the convection onset criterion and the underlying physical mechanisms. Chapter 6 investigates the bifurcation around the onset threshold, with special attention paid to the conditions of finite amplitude instability. In Part I and Part II, the coupled heat and mass transfer is achieved through crossdiffusion effects. While in Part III (Chap. 7), the coupling through boundary reactions is investigated through an application-oriented cavity flow problem. A conceptual design of coupled extraction and crystal growth apparatus is proposed therein, and the resulting double-diffusive convection is analyzed and simulated. The main conclusions and innovations of the above chapters, as well as prospects for future works, are summarized in Chap. 8.

References 1. Wikipedia (2019) Wikipedia contributors: citical point (thermodynamics)—Wikipedia, the free encyclopedia. URL https://en.wikipedia.org/w/index.php?title=Critical_point_ (thermodynamics). Accessed 13 Dec 2019 2. Lemmon EW, Bell I, Huber ML, McLinden MO (2018) NIST standard reference database 23: reference fluid thermodynamic and transport properties-REFPROP, version 10.0. National Institute of Standards and Technology, Gaithersburg 3. Sciortino F, Poole PH, Essmann U, Stanley HE (1997) Line of compressibility maxima in the phase diagram of supercooled water. Phys Rev E 55:727–737 4. Simeoni G, Bryk T, Gorelli F, Krisch M, Ruocco G, Santoro M, Scopigno T (2010) The Widom line as the crossover between liquid-like and gas-like behaviour in supercritical fluids. Nat Phys 6(7):503–507

14

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5. Fomin YD, Ryzhov VN, Tsiok EN, Brazhkin VV (2015) Thermodynamic properties of supercritical carbon dioxide: Widom and Frenkel lines. Phys Rev E 91:022111 6. Yang C, Brazhkin VV, Dove MT, Trachenko K (2015) Frenkel line and solubility maximum in supercritical fluids. Phys Rev E 91(1):012112 7. Yoon TJ, Lee YW (2018) Current theoretical opinions and perspectives on the fundamental description of supercritical fluids. J Supercritical Fluids 134:21–27 8. Banuti D (2015) Crossing the Widom-line—supercritical pseudo-boiling. J Supercritical Fluids 98:12–16 9. Maxim F, Contescu C, Boillat P, Niceno B, Karalis K, Testino A, Ludwig C (2019) Visualization of supercritical water pseudo-boiling at Widom line crossover. Nat Commun 10(1):1–11 10. Mileti´c M, Fukaˇc R, Pioro I, Dragunov A (2014) Development of gas cooled reactors and experimental setup of high temperature helium loop for in-pile operation. Nucl Eng Des 276:87–97 11. Sharan P, Neises T, McTigue JD, Turchi C (2019) Cogeneration using multi-effect distillation and a solar-powered supercritical carbon dioxide Brayton cycle. Desalination 459:20–33 12. Hobold GM, da Silva AK (2017) Critical phenomena and their effect on thermal energy storage in supercritical fluids. Appl Energy 205:1447–1458 13. Avanthi Isaka B, Ranjith P, Rathnaweera T (2019) The use of super critical carbon dioxide as the working fluid in enhanced geothermal systems (EGSs): a review study. Sustainable Energy Technol Assess 36:100547 14. Moisseytsev A, Sienicki JJ (2009) Investigation of alternative layouts for the supercritical carbon dioxide Brayton cycle for a sodium-cooled fast reactor. Nucl Eng Des 239(7):1362– 1371 15. Dunham MT, Iverson BD (2014) High-efficiency thermodynamic power cycles for concentrated solar power systems. Renew Sustain Energy Rev 30:758–770 16. Ganapathi GB, Wirz R (2012) High density thermal energy storage with supercritical fluids. In: ASME 2012 6th international conference on energy sustainability collocated with the ASME 2012 10th international conference on fuel cell science, engineering and technology. American Society of Mechanical Engineers Digital Collection, New York, pp 699–707 17. Higgins BS, Oldenburg CM, Muir MP, Pan L, Eastman AD (2016) Process modeling of a closed-loop sCO2 geothermal power cycle. In: The 5th supercritical CO2 power cycles symposium, pp 1–12 18. Kim Y, Kim C, Favrat D (2012) Transcritical or supercritical CO2 cycles using both low- and high-temperature heat sources. Energy 43(1):402–415 19. Li MJ, Zhu HH, Guo JQ, Wang K, Tao WQ (2017) The development technology and applications of supercritical CO2 power cycle in nuclear energy, solar energy and other energy industries. Appl Therm Eng 126:255–275 20. Goyne C, Hall C, O’Brien W, Schetz J (2006) The Hy-V scramjet flight experiment. In: 14th AIAA/AHI space planes and hypersonic systems and technologies conference, p 7901 21. Zhu Y, Peng W, Xu R, Jiang P (2018) Review on active thermal protection and its heat transfer for airbreathing hypersonic vehicles. Chin J Aeronaut 31(10):1929–1953 22. Wang X, Wang Y, Yang V (2019) Three-dimensional flow dynamics and mixing in a gascentered liquid-swirl coaxial injector at supercritical pressure. Phys Fluids 31(6):065109 23. Denies L (2015) Regenerative cooling analysis of oxygen/methane rocket engines. Master’s thesis, Delft University of Technology 24. Banuti D (2015) Thermodynamic analysis and numerical modeling of supercritical injection. Ph.D. thesis, University of Stuttgart 25. Knez Ž, Markoˇciˇc E, Leitgeb M, Primožiˇc M, Hrnˇciˇc MK, Škerget M (2014) Industrial applications of supercritical fluids: a review. Energy 77:235–243 26. Guo JQ, Li MJ, Xu JL, Yan JJ, Wang K (2019) Thermodynamic performance analysis of different supercritical Brayton cycles using CO2 -based binary mixtures in the molten salt solar power tower systems. Energy 173:785–798 27. Hu L, Chen D, Huang Y, Li L, Cao Y, Yuan D, Wang J, Pan L (2015) Investigation on the performance of the supercritical Brayton cycle with CO2 -based binary mixture as working fluid for an energy transportation system of a nuclear reactor. Energy 89:874–886

References

15

28. Jeong WS, Jeong YH (2013) Performance of supercritical Brayton cycle using CO2 -based binary mixture at varying critical points for SFR applications. Nucl Eng Des 262:12–20 29. Jeong WS, Lee JI, Jeong YH (2011) Potential improvements of supercritical recompression CO2 Brayton cycle by mixing other gases for power conversion system of a SFR. Nucl Eng Des 241(6):2128–2137 30. Lewis TG, Conboy TM, Wright SA (2011) Supercritical CO2 mixture behavior for advanced power cycles and applications. Technical report, Sandia National Laboratory 31. Binotti M, Invernizzi CM, Iora P, Manzolini G (2019) Dinitrogen tetroxide and carbon dioxide mixtures as working fluids in solar tower plants. Sol Energy 181:203–213 32. Manzolini G, Binotti M, Bonalumi D, Invernizzi C, Iora P (2019) CO2 mixtures as innovative working fluid in power cycles applied to solar plants: Techno-economic assessment. Solar Energy 181:530–544 33. Bird R, Stewart W, Lightfoot E (2006) Transport phenomena. Wiley international edition. Wiley, New York 34. Luettmer-Strathmann J (2002) Thermodiffusion in the critical region. Springer, Berlin, pp 24–37 35. Raspo I, Meradji S, Zappoli B (2007) Heterogeneous reaction induced by the piston effect in supercritical binary mixtures. Chem Eng Sci 62(16):4182–4192 36. Wannassi M, Raspo I (2016) Numerical study of non-isothermal adsorption of naphthalene in supercritical CO2 : Behavior near critical point. J Supercritical Fluids 117:203–218 37. Kadanoff LP (1966) Spin-spin correlations in the two-dimensional Ising model. Il Nuovo Cimento B (1965–1970) 44(2):276–305 38. Widom B (1965) Equation of state in the neighborhood of the critical point. J Chem Phys 43(11):3898–3905 39. Wilson KG (1971) Renormalization group and critical phenomena. I. Renormalization group and the Kadanoff scaling picture. Phys Rev B 4:3174–3183 40. Barmatz M, Hahn I, Lipa JA, Duncan RV (2007) Critical phenomena in microgravity: past, present, and future. Rev Mod Phys 79:1–52 41. Beysens D (2014) Critical point in space: a quest for universality. Microgravity Sci Technol 26(4):201–218 42. Beysens DA (2005) Near-critical point hydrodynamics and microgravity. In: Gutkowski W, Kowalewski TA (eds) Mechanics of the 21st century. Springer, Dordrecht, Netherlands, pp 117–130 43. Shen B, Zhang P (2013) An overview of heat transfer near the liquid-gas critical point under the influence of the piston effect: Phenomena and theory. Int J Therm Sci 71:1–19 44. Zappoli B (2003) Near-critical fluid hydrodynamics. Comptes Rendus Mécanique 331(10):713–726 45. Chen L (2016) Microchannel flow dynamics and heat transfer of near-critical fluid. Springer, Berlin 46. Chen L, Iwamoto Y (2017) Advanced applications of supercritical fluids in energy systems. IGI Global, Hershey 47. Zappoli B, Beysens D, Garrabos Y et al (2015) Heat transfers and related effects in supercritical fluids. Springer, Berlin 48. Onuki A, Ferrell RA (1990) Adiabatic heating effect near the gas-liquid critical point. Phys A 164(2):245–264 49. Long Z, Zhang P, Shen B (2016) Thermomechanical effects in supercritical binary fluids. Int J Heat Mass Transf 99:470–484 50. Hasan N, Farouk B (2012) Thermoacoustic transport in supercritical fluids at near-critical and near-pseudo-critical states. J Supercritical Fluids 68:13–24 51. Onuki A (2007) Thermoacoustic effects in supercritical fluids near the critical point: resonance, piston effect, and acoustic emission and reflection. Phys Rev E 76:061126 52. Lei Z, Farouk B (2007) Generation and propagation of thermally induced acoustic waves in supercritical carbon dioxide. In: ASME international mechanical engineering congress and exposition. Vol 8: heat transfer, fluid flows, and thermal systems, Parts A and B. ASME, New York

16

1 Introduction to Binary Mixtures at Supercritical Pressures ...

53. Farouk B, Lin Y, Lei Z (2010) Acoustic wave induced flows and heat transfer in gases and supercritical fluids. In: Cho YI, Greene GA (eds) Advances in heat transfer, vol 42. Elsevier, Amsterdam, Netherlands, pp 1–136 54. Amiroudine S, Bontoux P, Larroudé P, Gilly B, Zappoli B (2001) Direct numerical simulation of instabilities in a two-dimensional near-critical fluid layer heated from below. J Fluid Mech 442:119–140 55. Carlès P, Ugurtas B (1999) The onset of free convection near the liquid-vapour critical point Part I: Stationary initial state. Physica D 126(1–2):69–82 56. Gandikota G, Amiroudine S, Chatain D, Lyubimova T, Beysens D (2013) Rayleigh and parametric thermo-vibrational instabilities in supercritical fluids under weightlessness. Phys Fluids 25(6):064103 57. Amiroudine S, Boutrouft K, Zappoli B (2005) The stability analysis of two layers in a supercritical pure fluid: Rayleigh-Taylor-like instabilities. Phys Fluids 17(5):054102 58. Chen L, Zhang XR, Okajima J, Maruyama S (2014) Abnormal microchannel convective fluid flow near the gas-liquid critical point. Phys A 398:10–24 59. Hu ZC, Zhang XR (2018) Onset of convection in a near-critical binary fluid mixture driven by concentration gradient. J Fluid Mech 848:1098–1126 60. Hu ZC, Davis SH, Zhang XR (2019) Onset of double-diffusive convection in near-critical gas mixtures. Phys Rev E 99:033112 61. Hu ZC, Lv W, Zhang XR (2019) Detour induced by the piston effect in the oscillatory doublediffusive convection of a near-critical fluid. Phys Fluids 31(7):074107 62. Ameur D, Raspo I (2013) Numerical simulation of the Poiseuille-Rayleigh-Bénard instability for a supercritical fluid in a mini-channel. Comput Thermal Sci Int J 5(2):107–118 63. Boukari H, Shaumeyer JN, Briggs ME, Gammon RW (1990) Critical speeding up in pure fluids. Phys Rev A 41:2260–2263 64. Hu ZC, Zhang XR (2016) Numerical simulations of the piston effect for near-critical fluids in spherical cells under small thermal disturbance. Int J Therm Sci 107:131–140 65. Onuki A, Hao H, Ferrell RA (1990) Fast adiabatic equilibration in a single-component fluid near the liquid-vapor critical point. Phys Rev A 41:2256–2259 66. Zappoli B, Bailly D, Garrabos Y, Le Neindre B, Guenoun P, Beysens D (1990) Anomalous heat transport by the piston effect in supercritical fluids under zero gravity. Phys Rev A 41:2264–2267 67. Nitsche K, Straub J (1987) Critical ’HUMP’ of cv under microgravity results from the D1spacelab experiment ’WAERMEKAPAZITAET’. European Space Agency, Paris, pp 109–116 68. Zappoli B (1992) The response of a nearly supercritical pure fluid to a thermal disturbance. Phys Fluids A: Fluid Dyn (1989–1993) 4(5):1040–1048 69. Zappoli B, Carlès P (1995) The thermo-acoustic nature of the critical speeding up. Eur J Mech-B/Fluids 14(1):41–65 70. Zappoli B, Carlès P (1996) Acoustic saturation of the critical speeding up. Physica D 89(3– 4):381–394 71. Carlès P (1998) The effect of bulk viscosity on temperature relaxation near the critical point. Phys Fluids 10(9):2164–2178 72. Jounet A, Zappoli B, Mojtabi A (2000) Rapid thermal relaxation in near-critical fluids and critical speeding up: Discrepancies caused by boundary effects. Phys Rev Lett 84:3224–3227 73. Garrabos Y, Bonetti M, Beysens D, Perrot F, Fröhlich T, Carlès P, Zappoli B (1998) Relaxation of a supercritical fluid after a heat pulse in the absence of gravity effects: Theory and experiments. Phys Rev E 57:5665–5681 74. Guenoun P, Khalil B, Beysens D, Garrabos Y, Kammoun F, Le Neindre B, Zappoli B (1993) Thermal cycle around the critical point of carbon dioxide under reduced gravity. Phys Rev E 47:1531–1540 75. Straub J, Eicher L, Haupt A (1995) Dynamic temperature propagation in a pure fluid near its critical point observed under microgravity during the German Spacelab mission D-2. Phys Rev E 51:5556–5563

References

17

76. Zhong F, Meyer H (1995) Density equilibration near the liquid-vapor critical point of a pure fluid: Single phase T >Tc . Phys Rev E 51:3223–3241 77. Miura Y, Yoshihara S, Ohnishi M, Honda K, Matsumoto M, Kawai J, Ishikawa M, Kobayashi H, Onuki A (2006) High-speed observation of the piston effect near the gas-liquid critical point. Phys Rev E 74:010101 78. Nakano A, Shiraishi M (2005) Piston effect in supercritical nitrogen around the pseudo-critical line. Int Commun Heat Mass Transfer 32(9):1152–1164 79. Shen B, Zhang P (2010) On the transition from thermoacoustic convection to diffusion in a near-critical fluid. Int J Heat Mass Transf 53(21–22):4832–4843 80. Shen B, Zhang P (2011) Thermoacoustic waves along the critical isochore. Phys Rev E 83(1):011115 81. Hasan N, Farouk B (2013) Fast heating induced thermoacoustic waves in supercritical fluids: experimental and numerical studies. J Heat Transf-Trans ASME 135(8):081701 82. Beysens D, Chatain D, Nikolayev VS, Ouazzani J, Garrabos Y (2010) Possibility of longdistance heat transport in weightlessness using supercritical fluids. Phys Rev E 82:061126 83. Nakano A, Shiraishi M (2005) Visualization for heat and mass transport phenomena in supercritical artificial air. Cryogenics 45(8):557–565 84. Nakano A (2007) Studies on piston and Soret effects in a binary mixture supercritical fluid. Int J Heat Mass Transf 50(23):4678–4687 85. Chandrasekhar S (2013) Hydrodynamic and hydromagnetic stability. Courier Corporation, Chelmsford 86. Giterman MS, Shteinberg VA (1970) Criteria for commencement of convection in a liquid close to the critical point. High Temp 8(4):799–805 87. Kogan AB, Meyer H (2001) Heat transfer and convection onset in a compressible fluid: 3 He near the critical point. Phys Rev E 63:056–310 88. Kogan AB, Murphy D, Meyer H (1999) Rayleigh-Bénard convection onset in a compressible fluid: 3 He near Tc . Phys Rev Lett 82(23):4635–4638 89. Kogan AB, Murphy D, Meyer H (2000) Heat transfer in a pure near-critical fluid: Diffusive and convective regimes in 3 He. Physica B 284–288:208–209 90. Chiwata Y, Onuki A (2001) Thermal plumes and convection in highly compressible fluids. Phys Rev Lett 87(14):144301 91. Furukawa A, Onuki A (2002) Convective heat transport in compressible fluids. Phys Rev E 66:016302 92. Amiroudine S, Zappoli B (2003) Piston-effect-induced thermal oscillations at the RayleighBénard threshold in supercritical 3 He. Phys Rev Lett 90:105303 93. Maekawa T, Ishii K, Ohnishi M, Yoshihara S (2002) Convective instabilities induced in a critical fluid. Adv Space Res 29(4):589–598 94. Maekawa T, Ishii K, Shiroishi Y, Azuma H (2004) Onset of buoyancy convection in a horizontal layer of a supercritical fluid heated from below. J Phys A: Math Gen 37(32):7955–7969 95. Accary G, Raspo I, Bontoux P, Zappoli B (2004) Three-dimensional Rayleigh-Bénard instability in a supercritical fluid. Comptes Rendus Mécanique 332(3):209–216 96. Accary G, Raspo I, Bontoux P, Zappoli B (2005) Reverse transition to hydrodynamic stability through the Schwarzschild line in a supercritical fluid layer. Phys Rev E 72(3):035301 97. Accary G, Raspo I, Bontoux P, Zappoli B (2005) Stability of a supercritical fluid diffusing layer with mixed boundary conditions. Phys Fluids 17(10):104105 98. Accary G, Bontoux P, Zappoli B (2009) Turbulent Rayleigh-Bénard convection in a nearcritical fluid by three-dimensional direct numerical simulation. J Fluid Mech 619:127–145 99. Shteinberg V (1971) Convective instability of a binary mixture, particularly in the neighborhood of the critical point. J Appl Math Mech 35(2):335–345 100. Das KS, Bhattacharjee JK (2000) Onset of convection in a binary mixture near the plait point. Phys Rev E 61:5191–5194 101. Giglio M, Vendramini A (1975) Optical measurements of gravitationally induced concentration gradients near a liquid-liquid critical point. Phys Rev Lett 35(3):168–170

18

1 Introduction to Binary Mixtures at Supercritical Pressures ...

102. Greer SC, Block TE, Knobler CM (1975) Concentration gradients in nitroethane+ 3methylpentane near the liquid-liquid critical solution point. Phys Rev Lett 34(5):250–253 103. Maisano G, Migliardo P, Wanderlingh F (1976) Experimental investigation of gravity-induced concentration gradients in critical mixtures. J Phys A: Math Gen 9(12):2149–2158 104. Maisano G, Migliardo P, Wanderlingh F (1976) Gravity-induced gradients in critical mixtures. Optics Commun 19(1):155–158 105. Zappoli B, Amiroudine S, Carlès P, Ouazzani J (1996) Thermoacoustic and buoyancy-driven transport in a square side-heated cavity filled with a near-critical fluid. J Fluid Mech 316:53–72 106. Hasan N, Farouk B (2012) Buoyancy driven convection in near-critical and supercritical fluids. Int J Heat Mass Transf 55(15):4207–4216 107. Soboleva E (2013) Thermal gravitational convection of a side-heated supercritical fluid with variable physical properties. Fluid Dyn 48(5):648–657 108. Zappoli B, Jounet A, Amiroudine S, Mojtabi A (1999) Thermoacoustic heating and cooling in near-critical fluids in the presence of a thermal plume. J Fluid Mech 388:389–409 109. Soboleva EB (2003) Adiabatic heating and convection caused by a fixed-heat-flux source in a near-critical fluid. Phys Rev E 68:042201 110. Shen B (2013) Study of heat and mass transport mechanisms of supercritical fluids under the influence of the piston effect across different timescales. Ph.D. thesis, Shanghai Jiaotong University 111. Shen B, Zhang P (2012) Rayleigh-Bénard convection in a supercritical fluid along its critical isochore in a shallow cavity. Int J Heat Mass Transf 55(23):7151–7165 112. Soboleva E, Nikitin S (2014) Benchmark data on laminar Rayleigh-Bénard convection in a stratified supercritical fluid: A case of two-dimensional flow in a square cell. Int J Heat Mass Transf 69:6–16 113. Wei Y, Hu ZC, Chong B, Xu J (2018) The Rayleigh-Bénard convection in near-critical fluids: Influences of the specific heat ratio. Numer Heat Transf Part A: Appl 74(1):931–947 114. Chavanne X, Chillà F, Castaing B, Hébral B, Chabaud B, Chaussy J (1997) Observation of the ultimate regime in Rayleigh-Bénard convection. Phys Rev Lett 79(19):3648–3651 115. Kraichnan RH (1962) Turbulent thermal convection at arbitrary Prandtl number. Phys Fluids 5(11):1374–1389 116. Niemela J, Skrbek L, Sreenivasan K, Donnelly R (2000) Turbulent convection at very high Rayleigh numbers. Nature 404(6780):837–840 117. Ashkenazi S, Steinberg V (1999) High Rayleigh number turbulent convection in a gas near the gas-liquid critical point. Phys Rev Lett 83:3641–3644 118. Ashkenazi S, Steinberg V (1999) Spectra and statistics of velocity and temperature fluctuations in turbulent convection. Phys Rev Lett 83:4760–4763 119. Shraiman BI, Siggia ED (1990) Heat transport in high-Rayleigh-number convection. Phys Rev A 42:3650–3653 120. Ahlers G, Araujo FF, Funfschilling D, Grossmann S, Lohse D (2007) Non-OberbeckBoussinesq effects in gaseous Rayleigh-Bénard convection. Phys Rev Lett 98:054501 121. Ahlers G, Calzavarini E, Araujo FF, Funfschilling D, Grossmann S, Lohse D, Sugiyama K (2008) Non-Oberbeck-Boussinesq effects in turbulent thermal convection in ethane close to the critical point. Phys Rev E 77:046302 122. Burnishev Y, Segre E, Steinberg V (2010) Strong symmetrical non-Oberbeck-Boussinesq turbulent convection and the role of compressibility. Phys Fluids 22(3):035108 123. Burnishev Y, Steinberg V (2012) Statistics and scaling properties of temperature field in symmetrical Non-Oberbeck-Boussinesq turbulent convection. Phys Fluids 24(4):045102 124. Valori V (2018) Rayleigh-Bénard convection of a supercritical fluid: PIV and heat transfer study. Ph.D. thesis, Delft University of Technology 125. Valori V, Elsinga G, Rohde M, Tummers M, Westerweel J, van der Hagen T (2017) Experimental velocity study of non-Boussinesq Rayleigh-Bénard convection. Phys Rev E 95(5):053113 126. Yik H, Valori V, Weiss S (2020) Turbulent Rayleigh-Bénard convection under strong nonOberbeck-Boussinesq conditions. Phys Rev Fluids 5:103502

References

19

127. Grossmann S, Lohse D (2004) Fluctuations in turbulent Rayleigh-Bènard convection: The role of plumes. Phys Fluids 16(12):4462–4472 128. Valori V, Elsinga GE, Rohde M, Westerweel J, van Der Hagen TH (2019) Particle image velocimetry measurements of a thermally convective supercritical fluid. Exp Fluids 60(9):1– 14 129. Assenheimer M, Steinberg V (1993) Rayleigh-Bénard convection near the gas-liquid critical point. Phys Rev Lett 70(25):3888 130. Assenheimer M, Steinberg V (1994) Transition between spiral and target states in RayleighBénard convection. Nature 367(6461):345–347 131. Assenheimer M, Steinberg V (1996) Critical phenomena in hydrodynamics. Europhysics news 27(4):143–147 132. Assenheimer M, Steinberg V (1996) Observation of coexisting upflow and downflow hexagons in Boussinesq Rayleigh-Bénard convection. Phys Rev Lett 76:756–759 133. Roy A, Steinberg V (2002) Reentrant hexagons in non-Boussinesq Rayleigh-Bénard convection: effect of compressibility. Phys Rev Lett 88:244503 134. Ahlers G, Dressel B, Oh J, Pesch W (2010) Strong non-Boussinesq effects near the onset of convection in a fluid near its critical point. J Fluid Mech 642:15–48 135. Long Z (2014) Study of heat and mass transfer mechanisms and application of supercritical binary fluid. Ph.D. thesis, Shanghai Jiaotong University 136. Long Z, Zhang P, Shen B, Li T (2015) Experimental investigation of natural convection in a supercritical binary fluid. Int J Heat Mass Transf 90:922–930

Chapter 2

Basic Equations and Physical Properties of a Reference Binary Mixture

As a basis of this thesis, this chapter describes the fluid dynamic and thermophysical modelings of a reference binary mixture. The aim of this chapter is twofold: paving the way for later chapters and help readers understand the features of the reference binary mixture.

2.1 Basic Equations The full governing equations written for a general single phase binary mixture are [1] Dρ Dt Du ρ Dt DT ρc p Dt Dc ρ Dt ρ

= − ρ∇ · u, 2 = − ∇ p + ∇ · (η∇u) + ∇ · [η(∇u)T ] − ∇(η∇ · u) + ρg, 3 D p = − ∇ · q + H¯ ∇ · i + Tβ + , Dt

(2.1) (2.2) (2.3)

= − ∇ · i,

(2.4)

= f (T, p, c),

(2.5)

where ρ the density, T is the temperature, p the pressure, c the concentration (mass fraction [1, 2]), D/Dt is the material derivative, t the time, u the velocity vector, η the dynamic viscosity, c p the isobaric specific heat, β = −1/ρ × (∂ρ/∂ T ) p,c the ther  mal expansion coefficient,  = η ∇u + (∇u)T : ∇u − 2η(∇ · u)2 /3 the viscous

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 Z.-C. Hu, Coupled Heat and Mass Transfer in Binary Mixtures at Supercritical Pressures, Springer Theses, https://doi.org/10.1007/978-981-16-7806-6_2

21

22

2 Basic Equations and Physical Properties of a Reference Binary Mixture

dissipation, and H¯ is the partial enthalpy of the mixture1 . Note that gravity has been omitted. Equation (2.5) is the equation of state (EOS), where (T, p, c) are taken as the independent variables. The heat and mass fluxes are given by kT i + H¯ i − λ∇T, cs   kp kT i = −ρ D ∇c + ∇T + ∇ p , T p

q=

(2.6) (2.7)

respectively. D is the diffusion coefficient, k T is the thermodiffusion (i.e. Soret effect) factor, and k p is the barodiffusion factor satisfying kp κcs =− , p ρ

(2.8)

where the concentration contraction coefficient κ and concentration susceptibility cs are defined as     ∂c 1 ∂ρ , cs = , (2.9) κ= ρ ∂c p,T ∂μ p,T with μ the chemical potential of the mixture2 . The three terms on the right-hand side of Eq. (2.6) represent Dufour effect, inter-diffusion and Fourier’s law, while in Eq. (2.7) the right-hand-side terms are Fick’s law, Soret effect and barodiffusion, respectively.

2.2 Physical Properties The modeling of thermophysical properties is described in this section. To fully describe the fluid dynamics of a binary mixture at supercritical pressures, the required thermodynamic properties include ρ, β, κ, c p , H¯ and cs . Besides, the isothermal compressibility, defined as α = 1/ρ × (∂ρ/∂ p)T,c , is also a general thermodynamic property, which is also counted in. The transport properties are η, λ, k T and D. All of these properties should be modeled as functions of (T, p, c) for a selected reference binary mixture. Due to the importance of physical properties in natural gas industries, the modeling of thermodynamic properties of binary mixtures with natural gas components has The partial enthalpy of the mixture is defined as H¯ = μ − T (∂μ/∂ T ) p,c , with μ the chemical potential as introduced below. 2 According to [1], the chemical potential of the mixture is defined as μ = μ /M − μ /M , where 1 1 2 2 μ1 is the chemical potential of the first component, and μ2 is the chemical potential of the second component. M1 and M2 are their molar masses. 1

2.2 Physical Properties

23

been explored for several decades. Now this task can be accurately achieved by the GERG-2008 EOS [3]. GERG-2008 EOS is based on 21 natural gas components, including CO2 , C2 H6 , N2 , etc. It covers not only the gas and liquid phases, but also the supercritical region and vapor-liquid equilibrium states. It is based on an excess Helmholtz energy approach using pure fluid equations of state and a mixture model that specifies the excess contribution. Therefore, a binary mixture of natural gas components is considered in the present work by virtue of the feasibility and accuracy of predictions by GERG-2008 EOS. Besides, the GERG-2008 EOS has been implemented by many databases, such as NIST REFPROP [4] and CoolProp [5]. The latter is adopted in this thesis. Available models for transport properties, especially k T and D, are much fewer than thermodynamic properties. In order to describe the transport properties around the Widom line, one must include the crossover between the critical behavior and the regular behavior far away from the critical point [6–8]. Fortunately, the crossover models of transport properties have also been extensively reported for binary mixtures of natural gas components. After a literature survey for the available models of transport properties, C2 H6 − CO2 is chosen as the reference pair in the current study. One reason is it has been taken as the reference system in many previous works [6, 7, 9, 10]. On the other hand, CO2 is the most widely used supercritical fluid, its binary systems get involved in various engineering sectors [11, 12]. After selecting the reference fluid, C2 H6 is designated as the first component, while CO2 the second component. c represents the mass fraction of C2 H6 . In details, the transport properties are calculated by the practical representation proposed by Kiselev and Huber [13], which incorporates the crossover from critical contributions to regular background parts. However, because a different EOS is employed here, model parameters for the C2 H6 − CO2 binary mixture reported in [13] should be refitted to produce accurate predictions. First introduce two modifications to the equations. In [13], the definitions of parameters α˜ 0 and β˜0 are given by Eqs. (36) and (37) in [13], namely α˜ 0 =

2 ρ D0 Mmi x x(1 − x), RT 

β˜0 = R α˜ 0 β1 + xβ2 − ln

x 1−x

(2.10)

 .

(2.11)

Now, the expressions for α˜ 0 and β˜0 are modified into         2 ρ D0 Mmi ρ 3 ρ 5 ρ 7 x x(1 − x) κ1 + κ2 α˜ 0 = + κ3 + κ4 , (2.12) RT ρcx ρcx ρcx   R x ˜ , (2.13) α˜ 0 β1 + xβ2 − ln β0 = Mmi x 1−x

24

2 Basic Equations and Physical Properties of a Reference Binary Mixture

respectively. From Eq. (2.10) to Eq. (2.12), a dimensionless modification to α˜ 0 is introduced, which is actually an adjustment to the density dependence of α˜ b in Eq. (29) of [13]. Such a modification is shown to be effective by trial and error. As for Eq. (2.13), a correction is made to guarantee the dimension consistency. Follow the instructions of [13], the background kinetic coefficients αk and βk used in Eqs. (31) and (32) of [13] are refitted, along with the newly imported parameters κk in Eq. (2.12). Table 2.1 lists the fitting results for later reference. Furthermore, critical parameters are also required in the modeling of transport properties. Note that the critical point of a binary mixture is defined as the state where gas and liquid mixtures become identical in concentration and density. Moreover, critical parameters are functions of concentration. For the sake of consistency, the critical parameters are obtained from GERG-2008 EOS through CoolProp [5]. For precise critical parameters of C2 H6 − CO2 binary mixture, one can refer to [15]. Figure 2.1 plots the validations for the thermophysical modeling. In Fig. 2.1a, the calculated λ is compared with the experimental results of Mostert and Sengers [16], Table 2.1 The background coefficients αk , and βk used in Eqs. (31) and (32) of [13] for C2 H6 − CO2 binary mixture, along with the newly imported parameters κk in Eq. (2.12). These parameters are refitted to be compatible with the GERG-2008 EOS αk βk κk α6 α7 α12 α13 α18 α19

(a)

1.19792 × 10−5 1.12169 × 10−5 −4.53763 × 10−6 −3.29428 × 10−6 5.64431 × 10−7 1.16814 × 10−7

β1 β3 β6 β12 β18

−1.93675 −3.29859 × 10−6 2.50317 × 10−6 −2.91075 × 10−7 1.32107 × 10−8

κ1 κ2 κ3 κ4

0.50437 −0.33440 0.12379 −0.01309

(b)

Fig. 2.1 Comparisons between the calculated transport properties and experimental results. a The thermal conductivity λ versus the reduced temperature ε = (T − TC )/TC for two sets of data with different densities (the molar fraction of C2 H6 equaling 0.26). ρC = 382.39 kg/m3 is calculated by GERG-2008 EOS. b The thermodiffusion factor k T versus density ρ for one set of data with T = 305.15 K (the molar fraction of C2 H6 equaling 0.40). All curves represent calculated results, while symbols are experimental measurements (cf. [16] for λ and [17] for k T )

2.2 Physical Properties

25

while in Fig. 2.1b the comparisons are presented for k T with experimental data from [17]. Good agreements between experimental data and calculated values are clearly observed, which validate the calculations of transport properties. Besides, a few words are given below to explain Fig. 2.1b, where obvious oscillations appear in k T . The largest peak is because k T increases in the critical region and reaches a maximum at a density close to the critical density. For other oscillations, it is hard to determine whether they are physical or not (experimental data are very limited), but their amplitudes are not significant. Here it is assumed that these smallamplitude oscillations are acceptable since they also exist in the original papers of Kiselev et al. One can refer to the Fig. 6 in [14] and the Fig. 7 in [13]. For the convenience of latter studies, an open-source code is provided to calculate the physical properties of C2 H6 − CO2 binary mixture, which is available at https://github.com/huzhch7/CO2-ethane/. The code for generating Fig. 2.5 (see below) is provided as an example.

2.3 Phase Diagram and Behavior of Physical Properties of C2 H6 − CO2 For a binary mixture, the phase diagram and the critical behavior of the physical properties are more complicated than a pure fluid, which has been studied extensively since the 1970s. According to the classification of van Konynenburg and Scott [18], there are six basic types of phase behavior. In actual applications such as chemical extractions, the phase behavior is further complicated by the presence of solid phases [19]. The reference mixture C2 H6 − CO2 exhibits type I phase behavior, where the locus of the liquid-gas critical points (i.e., critical locus) is a continuous function of c. Figure 2.2 shows the phase diagram of C2 H6 − CO2 in the T − p plane, where the critical points of pure species and that at a reference concentration c = 0.6721 Fig. 2.2 Phase diagram of C2 H6 − CO2 in the T − p plane. C1 is the critical point of C2 H6 , C2 is the critical point of CO2 , and C3 is the critical point at c = 0.6721. The solid thin lines are saturation lines of pure species. The dotted curve is the critical locus. The bold solid (bubble curve) and dashed (dew curve) curves jointly form the phase envelope at c = 0.6721

C2 Critical locus

C3 Bubble curve Dew curve

C1

26

2 Basic Equations and Physical Properties of a Reference Binary Mixture

(corresponding to the molar fraction of C2 H6 equaling 0.75) are marked. As a signature of type I phase behavior, the critical locus is a continuous curve connecting C1 and C2 . As c increases (from C1 to C2 ), the critical pressure increases monotonically, while the critical temperature drops first and then increases. For a pure fluid, the liquid-gas coexisting states form a single curve in the phase diagram. However, for a binary mixture, it becomes a region bounded by the phase envelope, consisting of the bubble and dew curves joining at the critical point (i.e., C3 ). Inside the phase envelope, the mixture is in a liquid-vapor equilibrium state. Outside it, above the bubble curve is in a single-phase liquid state, and below the dew curve is in a single-phase gas state. One important fact is the mixture critical point can be located anywhere along the phase envelope [20]. Therefore, the critical point is not the top and right limits of the phase envelope. The maximum pressure of the phase envelope is termed as cricondenbar (see point A in Fig. 2.3), whereas the maximum temperature is called cricondentherm (see point B in Fig. 2.3). Therefore, it is possible to obtain twophase fluid under a supercritical pressure. In this thesis, c = 0.6721 is always considered as a reference concentration once C2 H6 − CO2 is employed as the reference fluid. As predicted by GERG-2008 EOS, the cricondenbar for C2 H6 − CO2 at c = 0.6721 is 5.4414 ≈ 5.441 MPa, which is very close to the critical pressure pC = 5.4405 ≈ 5.441 MPa. Hence, it is safe to believe that there is only a single-phase fluid under supercritical pressure in this thesis. Figure 2.4 shows the phase diagram at supercritical pressures. The two-phase region inside the phase envelope for p > pC is very small and is omitted. Since the concentration is an independent variable, the Widom line is no longer a

Fig. 2.3 Phase envelope of C2 H6 − CO2 at c = 0.6721. A and B are top and right limits of the phase envelope. C3 is the critical point. The dotted curve is the Widom line, where isobaric specific heat reaches its maximal value

Widom line

AC3 B Bubble curve

Dew curve

2.3 Phase Diagram and Behavior of Physical Properties of C2 H6 − CO2

27

Fig. 2.4 The supercritical-pressure region ( p − pC > 0) of C2 H6 − CO2 binary mixture. The semi-transparent surface is composed of Widom lines under different concentrations, which is termed as Widom surface

single curve but a continuous surface, which is consequently termed as Widom surface. To exhibit the behavior of physical properties, Fig. 2.5 presents the physical properties of C2 H6 − CO2 under four representative pressures (5.6 MPa, 6.0 MPa, 6.5 MPa and 7.0 MPa) and a reference concentration c = 0.6721. Figure 2.5a–g are thermodynamic properties and Fig. 2.5h–k are transport properties. Figure 2.5a presents the variations of ρ with T . For p = 5.6 MPa and 6.0 MPa, the rapid continuous variations in ρ within a narrow range of T is shown. As p further increases, the variation becomes gradual. For physical properties, it is demonstrated in Fig. 2.5b–k that they depend on the critical enhancement and density. For c p , α, β, κ, cs , H¯ and k T , they have strong critical enhancements. Consequently, if p is close to pC , they rapidly reach local extrema in narrow temperature intervals. As p increases, critical enhancements are weakened. For λ, η and D, they are mainly influenced by density. As density decreases, λ and η drop, while D increases, behaving like normal liquid and gas.

28

2 Basic Equations and Physical Properties of a Reference Binary Mixture (a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)

Fig. 2.5 Variations of properties of C2 H6 − CO2 with temperature at concentration c = 0.6721. a Density ρ. b Isobaric specific heat c p . c Isothermal compressibility α. d Thermal expansion coefficient β. e Concentration contraction coefficient κ. f Concentration susceptibility cs . g Partial enthalpy H¯ . h Thermodiffusion factor k T . i Thermal conductivity λ. j Dynamic viscosity η. k Diffusion coefficient D. Four representative pressures are chosen: 5.6 MPa (solid line), 6.0 MPa (dashed line), 6.5 MPa (dash-dotted line) and 7.0 MPa (dotted line). Note that pC = 5.441 MPa for c = 0.6721, and the red dots denote the values at TPB , i.e., the temperature where maximal c p achieves

2.4 Summary

29

2.4 Summary This chapter offers details about the governing equations of a single phase binary mixture, and the modeling of physical properties of C2 H6 − CO2 . These preparations are important for Chaps. 3–6, where C2 H6 − CO2 is employed as a reference to demonstrate phenomena or show the behavior of relevant parameters. In Chap. 7, since a different form of coupling is involved, the specific details regarding phase behavior and physical properties are described therein.

References 1. Landau L, Lifshitz E (1987) Fluid mechanics, 2nd edn. Pergamon Press, New York 2. Bird R, Stewart W, Lightfoot E (2006) Transport phenomena. Wiley international edition. Wiley, New York 3. Kunz O, Wagner W (2012) The GERG-2008 wide-range equation of state for natural gases and other mixtures: an expansion of GERG-2004. J Chem Eng Data 57(11):3032–3091 4. Lemmon EW, Bell IH, Huber ML, McLinden MO (2018) NIST standard reference database 23: reference fluid thermodynamic and transport properties-REFPROP, version 10.0. National Institute of Standards and Technology, Gaithersburg 5. Bell IH, Wronski J, Quoilin S, Lemort V (2014) Pure and pseudo-pure fluid thermophysical property evaluation and the open-source thermophysical property library CoolProp. Ind Eng Chem Res 53(6):2498–2508 6. Jin GX, Tang S, Sengers JV (1993) Global thermodynamic behavior of fluid mixtures in the critical region. Phys Rev E 47:388–402 7. Luettmer-Strathmann J (2002) Thermodiffusion in the critical region. Springer, Berlin, pp 24–37 8. Luettmer-Strathmann J, Sengers JV (1996) The transport properties of fluid mixtures near the vapor-liquid critical line. J Chem Phys 104(8):3026–3047 9. Hu ZC, Zhang XR (2019) Piston effect induced by cross-boundary mass diffusion in a binary fluid mixture near its liquid-vapor critical point. Int J Heat Mass Transf 140:691–704 10. Sengers JV, Sengers JMHL (1986) Thermodynamic behavior of fluids near the critical point. Annu Rev Phys Chem 37(1):189–222 11. Knez Ž, Markoˇciˇc E, Leitgeb M, Primožiˇc M, Hrnˇciˇc MK, Škerget M (2014) Industrial applications of supercritical fluids: A review. Energy 77:235–243 12. White MT, Bianchi G, Chai L, Tassou SA, Sayma AI (2021) Review of supercritical CO2 technologies and systems for power generation. Appl Therm Eng 185:116447 13. Kiselev S, Huber M (1998) Transport properties of carbon dioxide+ethane and methane+ethane mixtures in the extended critical region. Fluid Phase Equilib 142(1):253–280 14. Kiselev S, Kulikov V (1997) Thermodynamic and transport properties of fluids and fluid mixtures in the extended critical region. Int J Thermophys 18(5):1143–1182 15. Sengers JV, Jin GX (2007) A note on the critical Locus of mixtures of carbon dioxide and ethane. Int J Thermophys 28(4):1181–1187 16. Mostert R, Sengers J (2008) Thermal conductivity of mixtures of carbon dioxide and ethane in the critical region. Int J Thermophys 29(4):1205–1221 17. Walther JE (1957) Thermal diffusion in non-ideal gases. Ph.D. thesis, University of Illinois 18. van Konynenburg PH, Scott RL, Rowlinson JS (1980) Critical lines and phase equilibria in binary van der Waals mixtures. Philos Trans R Soc London Ser A, Math Phys Sci 298(1442):495–540

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19. de Loos TW (1994) Understanding phase diagrams. Springer, Dordrecht, Netherlands, pp 65–89 20. Mokhatab S, Poe WA, Mak JY (2019) Chapter 2: phase behavior of natural gas systems. In: Mokhatab S, Poe WA, Mak JY (eds) Handbook of natural gas transmission and processing, 4th edn. Gulf Professional Publishing, Oxford, pp 37–101

Part I

Coupling Through Cross-Diffusion Effects: Relaxation and Diffusion Problems

Chapter 3

Coupled Transfer in a Relaxation Process: Mass Piston Effect

This chapter enters the first part of this thesis, where a relaxation problem is studied and coupled heat and mass transfer exists through cross-diffusion effects. For near-critical states, the large compressibility and the diminishing thermal diffusivity are responsible for a rapid thermalization process known as the piston effect (PE) [1–4]. As introduced in Chap. 1, in previous studies, the PE is triggered by boundary heating or cooling, so the energy flux is always heat flux. However, what will happen if the energy flux is given in a different form? Inspired by this idea and the chemical extraction process (one of the most important engineering applications of supercritical fluids), a binary mixture at supercritical pressures subjected to boundary concentration perturbations is investigated in this chapter. In this situation, the boundary concentration perturbation mimics the variation in solubility, which is responsible for a cross-boundary mass flux of the solute. This chapter confirms the PE can take place in the current configuration. Such a PE induced by cross-boundary mass flux is termed as mass piston effect (MPE). This chapter is dedicated to reporting these interesting findings. This chapter is organized as follows. In Sect. 3.1, details about the physical and mathematical descriptions of the problem are presented. In Sect. 3.2, the travelingwave theory is proposed to describe the MPE on the acoustic timescale. The theory is validated through numerical simulations in Sect. 3.3. In Sect. 3.4, an energy balance analysis is provided. The chapter is concluded in Sect. 3.5.

3.1 Problem Statement The physical model is the C2 H6 − CO2 at a supercritical pressure confined between two infinite solid plates. The thickness of the fluid layer is d = 10 mm (see Fig. 3.1), where x is the horizontal coordinate with its origin placed at the left boundary. The one-dimensional model is justified because gravity is ignored (no buoyant convection © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 Z.-C. Hu, Coupled Heat and Mass Transfer in Binary Mixtures at Supercritical Pressures, Springer Theses, https://doi.org/10.1007/978-981-16-7806-6_3

33

34

3 Coupled Transfer in a Relaxation Process: Mass Piston Effect

Fig. 3.1 One-dimensional physical model

Diffusion of C2H6

Binary mixture C2H6-CO2

x=0

Adiabatic Impermeable

x = d= 10 mm

and density stratification). The fluid is initially motionless and at thermodynamic equilibrium. The fluid is subjected to a small concentration perturbation at the left side resulting in a cross-boundary mass flux of C2 H6 , while CO2 is rejected by the left boundary. The other side is impermeable. Besides, there is no temperature gradient at the two boundaries. The boundary condition (BC) employed in this study mimics chemical extraction processes, where a solute dissolves into a supercritical solvent from the solid substrate (the left boundary). Since concentration perturbations are small, constant physical properties are assumed. As a result, the initial state is employed as a reference state, denoted by a subscript ref. Given temperature T , pressure p and concentration c (mass fraction of C2 H6 ), the reference state is (Tref , pref , cref ). According to Eqs. (2.6) and (2.7), the one-dimensional energy flux q and diffusion flux i are given by [5] ∂T kT , i + H¯ i − λ cs ∂x ρk T D ∂ T ∂c − , i = −ρ D ∂x T ∂x

q=

(3.1) (3.2)

respectively. Employing the above expressions for q and i, and assuming constant physical properties, the governing equations (2.1)–(2.5) are simplified into ∂ρ ∂ρu + = ∂t ∂x ∂ρu ∂ρu 2 + = ∂t ∂x   ∂ρT ∂ρuT cp + = ∂t ∂x ∂ρc ∂ρuc + = ∂t ∂x ρ − ρref =

0,

(3.3)

∂ p 4η ∂ 2 u + , (3.4) ∂x 3 ∂x2     ∂2T ∂p k T ∂i 4η ∂u 2 ∂p λ 2 + Tβ +u − + , (3.5) ∂x ∂t ∂x cs ∂ x 3 ∂x ∂i − , (3.6) ∂x ρref [α( p − pref ) − β(T − Tref ) + κ(c − cref )]. (3.7) −

Note that in order to get concise equations, q has been fully substituted, while i is retained. Besides, the complete equation of state (EOS) has been linearized around the reference state, leading to Eq. (3.7). Mathematically, the BCs are given by

3.1 Problem Statement

∂c ∂T Dg0 = 0, = −g0 , u = u 0 = , ∂x ∂x 1 − cref ∂T ∂c x = d, = = u = 0, ∂x ∂x x = 0,

35

(3.8) (3.9)

where u 0 denotes the left boundary velocity, and the BC for c is a Neumann type with an adjustable quantity g0 . Note that the u 0 = 0 is caused by the partial permeability of the left boundary [6]:  1 D ∂c  Dg0 i0 = − u0 = ≈ , ρ(1 − c) 1 − c ∂ x x=0 1 − cref

(3.10)

where the approximation is justified since the concentration perturbations considered in this study are small.

3.2 Traveling-Wave Theory In this section, a traveling-wave theory is developed to analytically model the MPE. Figure 3.2 shows a heuristic explanation for the behavior of the boundary layer (BL), where u 0 is the boundary velocity caused by the partial permeability of the left wall (see Eq. 3.10), and u  is the velocity amplitude of the acoustic wave. Without loss of generality, the case of u 0 > 0 (a result of a positive g0 ), k T > 0 and κ < 0 is plotted as an example. In this situation, the cross-boundary mass flux of C2 H6 induces u 0 . Then a concentration BL forms, expanding in the velocity u c owing to κ < 0. Also, an accompanying heat flux due to the Dufour effect enters the fluid (k T > 0), leading to a thermal BL expanding in the speed of u D . Thickness of the BL, denoted by d B L , is defined as the very first location measured from x = 0 where both T and c remain unperturbed. Define the acoustic timescale as ta = d/va , where va is the speed of sound. Because on the acoustic timescale, d B L is very small and va is usually a large value, it is assumed that the perturbations of different origins reach x = d B L at the same time, leading to (3.11) u = u0 + u D + uc. On the one hand, the superposed perturbation is continuously produced at x = d B L in such a manner. On the other hand, the perturbation propagates as a traveling wave. When g0 is fixed, u  is a constant value, corresponding to a step-like wavefront [7]. When the acoustic wave reaches the other boundary, it reflects with −u  [7], and overlaps with the later generated perturbation. In fact, since the signs of k T and κ differ in different fluid systems and initial conditions, u D and u c may be positive or negative. However, it does not influence the relationship given in Eq. (3.11) and the above arguments about the propagation and reflection of the acoustic wave.

36

3 Coupled Transfer in a Relaxation Process: Mass Piston Effect

u0 uD

u0

uc u'

x

x

0

d BL

Fig. 3.2 A schematic diagram for the behavior of the boundary layer (BL), where d B L is the BL’s thickness and u  is the amplitude of the velocity. The situation when u 0 > 0, k T > 0 and κ < 0 is shown, where u  is decomposed into three parts: u 0 the boundary velocity, u D the Dufour-effectinduced velocity, and u c the velocity caused by concentration variation

3.2.1 Wave Generation Assume the expansion of the BL is isobaric [3] and neglect the viscous heating and the velocity in the BL (since their effects are second-order). Equations (3.5) and (3.6) are simplified into λ ∂2T ∂i kT ∂T = , − ∂t ρref c p ∂ x 2 ρref c p cs ∂ x ∂c 1 ∂i =− , ∂t ρref ∂ x

(3.12) (3.13)

respectively. Integrating the two equations over the whole BL leads to ∂

 dB L

T dx

0

∂t

∂ kT = i0 , ρref c p cs

 dB L 0

∂t

c dx

=

1 i0 . ρref

According to the linearized EOS and the relationship between i 0 and u 0 given in Eq. (3.10), Eq. (3.11) can be transformed into 

u =β where

∂

 dB L 0

∂t

T dx

−κ

∂

 dB L 0

∂t

c dx

+ u 0 = ϕu u 0 ,

  u kT β ϕu ≡ = 1 + (1 − cref ) −κ u0 c p cs

(3.14)

(3.15)

is a dimensionless ratio of the total perturbation velocity to the boundary one. ϕu is an important thermodynamic indicator of the MPE. The sign of ϕu reflects the nature of the acoustic wave. If the initial propagation direction of the perturbation is

3.2 Traveling-Wave Theory

37

taken as the positive direction, u  = ϕu u 0 > 0 corresponds to a compression wave or heating MPE, while u  = ϕu u 0 < 0 results in an expansion wave or cooling MPE. In a limiting case of cref = 1, the problem degenerates into an injection (or extraction) of pure C2 H6 , so ϕu = 1. The MPE is thus solely driven by the mechanical perturbation of u 0 . Given u  , the amplitude of density ρ  , pressure p  and temperature T  can be obtained directly from mass conservation and the isentropic nature of the acoustic wave [8]. The expressions for the amplitudes u  , ρ  , p  and T  can be summarized and written in a compact form: ⎡ ⎤ ⎡ ⎤ ϕu u ⎢ ρ  ⎥ ⎢ ϕρ ⎥ ⎢  ⎥ = ⎢ ⎥ u0, (3.16) ⎣ p ⎦ ⎣ ϕp ⎦ T ϕT where ϕρ =

ρref Tref βva ϕu , ϕ p = ρref va ϕu , ϕT = ϕu , va cp

(3.17)

are thermodynamic indicators, standing for the amplitudes of the denoted properties (see their subscripts) induced by unit u 0 . The signs of ϕρ , ϕ p and ϕT are identical to ϕu . Therefore, a compression wave has positive ρ  , p  and T  (namely the heating MPE), while for a expansion wave, they are negative (namely the cooling MPE).

3.2.2 Wave Propagation The following equations are proposed to describe the propagation of the acoustic wave in the bulk fluid: ⎧ ⎨ (−1)n−1 f  [t − τ (x, n)] H [t − τ (x, n)], f ∈ {u}, n δ f (x, t) =  (3.18) f  [t − τ (x, n)] H [t − τ (x, n)], f ∈ {ρ, p, T }, ⎩ n

where

f  [s] = ϕ f u 0 (s),

(namely Eq. 3.16), H[s] is the Heaviside step function satisfying  1, s ≥ 0, H[s] = 0, s < 0, and τ (x, n) = [n − mod(n, 2)]ta + (−1)n−1 x/va , with n being an integer obeying 1 ≤ n < t/ta + 1.

(3.19)

38

3 Coupled Transfer in a Relaxation Process: Mass Piston Effect

The above equations are based on traveling-wave physics. The integer n indicates the minimum upper bound for how many times the perturbations are being superposed in the whole domain (the upper bound of how many acoustic times have been consumed). For a specific x and t, the array τ gives the time taken by these perturbations to reach the location x, and the array (t − τ ) calculates the moments when they were created. The transient amplitude is a superposition of all the perturbations that are reaching x, which is accomplished by Eq. (3.18). For u  , the reflection results in a change in its sign, so its expression is different from those of ρ  , p  and T  since the reflection does not alter the nature of the acoustic wave (compression wave or expansion wave). In this section, a theoretical model is developed for the MPE, including the expressions of various amplitudes and the theoretical representation of the traveling wave. In the subsequent section, the traveling-wave theory will be validated by comparing with numerical simulations of the full equations.

3.3 Numerical Validations 3.3.1 Numerical Method To simulate the MPE on acoustic timescale, Eqs. (3.2)–(3.7) with initial and BCs were solved by SIMPLE algorithm after finite volume discretization implemented based on OpenFOAM [9], an open source C++ library for computational fluid dynamics. Convective terms are discretized using a TVD (total variation diminishing) scheme with OpenFOAM’s limitedLinear limiter. Transient terms are discretized with a first-order Euler scheme. In 2006, Miura et al. [8] observed the acoustic waves experimentally using an ultra-sensitive interferometer. Continuous heating of 1.83 kW/m2 is applied during 0.2 ms to a cell filled with near-critical CO2 . They measured density changes on a timescale of 1 μs. To validate our code, a simulation has been conducted in a 1D configuration and with the same initial and BCs as in the experiments. The comparisons are presented in Fig. 3.3, where generally fair agreements are noticed, with overestimation in the late stage of propagation. Such an overestimation is also observed in the calculation of Amiroudine et al. [10], where the same model but a totally different method was used. It thus suggests that the overestimation is a common issue of the 1D model, where the lateral walls in the experimental apparatus are omitted. However, the no-slip nature of the lateral walls imposes damping effects on the acoustic waves. The neglecting of these effects is believed to result in the overestimation of the 1D calculation in the late stage of propagation. The numerical validations were carried out for the four reference states denoted by red dots in Fig. 2.5. They locate on the Widom line (T = TPB ) with different pressures. The physical properties at these states are summarized in Table 3.1. For the sake of comparison, g0 is designed to generate a constant u 0 = 10−5 m/s according to Eq.

3.3 Numerical Validations Fig. 3.3 The normalized density change at the cell center versus time. The set-up of the numerical model mimics the experiment of Miura et al. [8], serving as the validation of our numerical code

39 10 -7 8 6 4 2 0 0

0.1

0.2

0.3

0.4

(3.10). The mesh, including 1080 points, was refined near the boundaries so as to accurately represent thin BLs. A time step t = 0.01 μs was chosen to assure proper numerical convergence of the solutions.

3.3.2 Comparisons and Discussions Presented in Fig. 3.4 is the temperature profiles at different acoustic times for various cases, which are obtained from the theory (see Sect. 3.2.2) and numerical simulations. The corresponding velocity profiles are shown in Fig. 3.5. In each figure, a step-like compression wave traveling in the fluid is observed, which reflects back at the right boundary. As a result, the velocity changes its sign after reflection and offsets its later part. A traveling increase or decrease in the bulk temperature is identified. The profiles of δρ and δp are not shown here, because their shapes are identical to those of δT with different magnitudes. Comparing between the left and right columns of Figs. 3.4 and 3.5, the theoretical predications are in excellent agreements with the results from numerical simulations, except for the shapes of the wavefronts. In the theory, the wavefront is described by the Heaviside function. Hence, the gradient of the wavefront is infinite. However, in real situations, the gradient is finite and decreases with time due to diffusion. This is the main imperfection of the current theoretical representation. However, since the diffusion is usually very slow and its influence is relatively local, the theory is acceptable especially when the spatial resolution is not the main concern. Because the growth rate of average bulk temperature is usually the main concern avg for a PE, the spatially average bulk temperature, denoted by δTbulk , is additionally calculated. Next, calculate the absolute relative deviation (ARD) of the theoretical results by    Theortical predictation − Simulation  .  ARD (%) =   Simulation

Table 3.1 Details of the simulated cases No. cref pref /(MPa) 1 0.6721 5.6 2 0.6721 6.0 0.6721 6.5 3 4 0.6721 7.0 κ cs /(kg/J) η/(Pa · s) 9.20 1.11 × 10−5 2.39 × 10−5 −6 4.60 5.80 × 10 2.45 × 10−5 −6 4.81 × 10 2.54 × 10−5 2.72 −6 1.88 4.43 × 10 2.62 × 10−5 Tref /(K) 298.79 302.12 305.96 309.55 λ/[W/(m · K)] 0.0694 0.0548 0.0500 0.0477

ρref /(kg/m3 ) 248.77 253.14 260.43 266.81 D/(m2 /s) 3.02 × 10−8 5.45 × 10−8 6.22 × 10−8 6.44 × 10−8

c p /[J/(kg · K−1 )] 3.61 × 104 1.94 × 104 1.26 × 104 9.54 × 103 kT 11.95 3.34 1.76 1.21

α/(Pa−1 ) 2.16 × 10−6 1.06 × 10−6 6.10 × 10−7 4.14 × 10−7 va /(m/s) 188.69 202.63 217.24 230.07

β/(K−1 ) 0.248 0.125 0.0751 0.0531 g0 /(m−1 ) 108.45 60.15 52.68 50.92

40 3 Coupled Transfer in a Relaxation Process: Mass Piston Effect

3.3 Numerical Validations

41

(b) →

3.4

2.2 →

2.8 → →

→

1.9

→

1.5

1.1

→

0.2 →

3.4 2.8 →

1.9

→

→

2.2 →

3.7

→

→

3.7

→

(a)

1.5

1.1

0.2 →

0.8 →

0.8 →

(d) →

3.4

→

→

→

1.5

1.1

→

0.2 →

2.8 →

1.9

→

→ 1.9

3.4

2.2 →

2.8 →

2.2 →

3.7

→

→

3.7

→

(c)

1.5

1.1

0.2 →

0.8 →

(e)

0.8 →

(f) →

3.4

→

→

→

1.1

→

0.2 →

1.9

1.5

1.1

0.2 →

0.8 →

(g)

2.8 → →

→

1.5

3.4

2.2 →

2.8 →

1.9

3.7

→

→

2.2 →

→

3.7

0.8 →

(h)

2.8 →

→

0.8 →

→

→

→

0.2 →

1.1

2.2 →

1.9

1.5

0.2 →

3.4

2.8 →

→

→

1.5

3.7

→

→

1.9

→

2.2 →

→

3.7

3.4

1.1

0.8 →

Fig. 3.4 Temperature profiles at several acoustic times for cases 1 to 4 in Table 3.1 subjected to a constant u 0 = 10−5 m/s. All of the four cases show compression waves (heating MPEs). The figures in each row correspond to cases 1, 2, 3, 4, respectively, from top to bottom. The figures in the right column are obtained from numerical simulations, and the left ones are predicted by the theory elaborated in Sect. 3.2.2. The numbers denote the ratio t/ta , with ta the acoustic timescale, and arrows point out propagation direction. The profiles of δp and δρ are similar to those of δT in shapes but with different magnitudes

42

3 Coupled Transfer in a Relaxation Process: Mass Piston Effect

(b)

3.2

2.2→

0.4→

2.2→

0.4→

→

→

1.4

→

0.4→

→

2.2→

2.2→

0.4→

1.4

→

(a)

3.2

(d)

→

0.4→

1.4

→

2.2→

3.2

(e)

1.4

→

(c)

3.2

(f)

→

→

1.4

3.2

(g)

→

0.4→

1.4

→

2.2→

3.2

(h)

→

→

1.4

3.2

2.2→

0.4→

→

0.4→

1.4

→

2.2→

3.2

Fig. 3.5 Velocity profiles at several acoustic times for cases 1–4 in Table 3.1 subjected to a constant u 0 = 10−5 m/s. See the caption of Fig. 3.4 for more details

3.3 Numerical Validations

43

Fig. 3.6 The absolute avg relative deviation of δTbulk of the theoretical predictions, plotted as functions of time for cases 1–4

The results are plotted in Fig. 3.6. All the curves suggest fast growths in ARDs after the initial emissions of the acoustic waves, followed by stable stages with slight oscillations. The theory is satisfactory since the maximum ARD is less than 2%. So far, the theoretical representations have been compared with numerical simulations and excellent agreements have been shown. The main drawback of the current traveling-wave theory lies in the omitting of diffusion, so its flattening effect on the wavefront is not included. It is also identified that this approximation does not influence the increase rate of average bulk temperature, which is usually the main concern for a PE.

3.4 Energy Balance Analysis Another important aspect of the MPE is the energy balance. To look into it, neglecting the viscous heating, the equivalent form of Eq. (3.5) written for the specific heat at constant volume cv is given by ∂ρcv T u ∂2T ∂ρcv T =− +λ 2 + ∂t ∂x ∂x



kT Tβκ − ρα cs



Tβ ∂u ∂i − . ∂x α ∂x

(3.20)

To facilitate theoretical analysis, this equation is linearized as ∂ρref cv Tref u ∂2T ∂ρref cv T =− +λ 2 + ∂t ∂x ∂x



kT Tref βκ − ρref α cs



∂i Tref β ∂u − , (3.21) ∂x α ∂x

since the perturbations are small. Integrating the above equation over the whole domain and applying the BCs yield

44

3 Coupled Transfer in a Relaxation Process: Mass Piston Effect

∂ E˙ in (t) = ∂t



d 0

  Tref β u0 ρref cv T dx = ρref cv Tref + α    I

kT Tref βκ u0, + ρref (1 − cref ) u 0 −ρref (1 − cref ) cs ρref α      II

(3.22)

III

where E˙ in is energy injection rate. The right-hand side of Eq. (3.22) can be divided into three parts: The term I stands for the energy carried by the boundary velocity, where the second quantity in the parentheses is a correction for this non-isochoric process. The term II represents the thermal effect of diffusion, namely the Dufour effect. The term III comes from Tβ(∂ p/∂c)T,ρ (Dc/Dt), representing the concentrationrelated pressure work, which vanishes for an isobaric process. The energy reception rate of the bulk fluid is actually the rate of the BL’s mechanical work, which can be expressed as (neglecting small pressure variations) E˙ bulk (t) = pref u  .

(3.23)

In previous studies about thermal PE, the energy efficiency is defined as ξ E ≡ E˙ bulk / E˙ in [11], reflecting the energy transfer capability of the PE. After a calculation, this traditional definition leads to ξ E = r u 0 ξ Eu 0 + r D ξ ED + r c ξ Ec , where r u0 =

I II III , rD = , rc = , I + II + III I + II + III I + II + III

(3.24)

(3.25)

are the ratios of the individual terms to their sum, and  ξ Eu 0 =

ρref cv Tref 1 + c pref ξE

−1

, ξ ED =

pref β pref α , , ξc = ρref c p E Tref β

(3.26)

are the energy efficiencies of boundary velocity, Dufour effect and concentration variation, respectively. These efficiencies are pure thermodynamic properties. Note that because the expansion due to Dufour effect is essentially a thermal effect, ξ ED is the same as the energy efficiency of the thermal PE [8]. In Eq. (3.24), ξ E is expressed as a weighted average of the three sub-efficiencies in Eq. (3.26). Notice that when both k T and −κ are positive (i.e., II, III > 0), Eq. (3.24) is physical. However, in other situations, Eq. (3.24) no longer reflects the average effect of three aspects and one may get puzzling results such as ξ E < 0. Hence, to fix this issue, the modified ξ Emod is proposed as u0 D c ξ Eu 0 + rmod ξ ED + rmod ξ Ec , ξ Emod = rmod

(3.27)

3.4 Energy Balance Analysis

45

where u0 rmod =

|II| |III| I D c = = , rmod , rmod , I+ |II| + |III| I+ |II| + |III| I+ |II| + |III|

(3.28)

and the definitions of ξ Eu 0 , ξ ED and ξ Ec remain unchanged. The other important aspect of the MPE is the efficiency of thermalization, which is measured by the temperature efficiency ξT , defined as the ratio of the bulk temperature ideal , the expected change rate T˙bulk induced by the acoustic wave, to the ideal one T˙bulk bulk temperature change rate if the energy is uniformly distributed in the fluid. According to the feature of the acoustic wave and Eq. (3.22), they are given by va E˙ in (t) ideal . (t) = T˙bulk (t) = T  , T˙bulk d ρref cv d

(3.29)

The traditional definition of ξT contains the same issue as in ξ E . Therefore, the modified temperature efficiency ξTmod is proposed, and its expression is obtained as u0 D c ξTu 0 + rmod ξTD + rmod ξTc , ξTmod = rmod

where

 ξTu 0 =

ρref cv α +1 β

−1

, ξTD =

Tref β 2 , ξ c → 1, ρref c p α T

(3.30)

(3.31)

are the temperature efficiencies of boundary velocity, Dufour effect and concentration variation, respectively. In fact, the calculation gives ξTc = 1, which is somehow unphysical since ξ Ec < 1. This puzzling value is caused by the linearization applied throughout the analysis, which results in the disregard of high-order terms. Hence in real situations, ξTc should tend to but be smaller than 1. So ξTc → 1 is written in Eq. (3.31). Figure 3.7a shows ξ Emod and ξTmod for C2 H6 − CO2 at cref = 0.6721. It is obvious that ξTmod is much higher than ξ Emod . Figure 3.7b presents ξTmod for four representative supercritical pressures, which shows strong critical enhancement. As p approaches pC = 5.441MPa, the peak value of ξTmod tends to 1. Therefore, MPE is more efficient in terms of thermalization than energy transfer, especially at near-critical states. Besides, the gas-like (GL) state (T > TPB ) generally has a smaller temperature efficiency than the liquid-like (LL) state (T < TPB ) under the same pressure. Figure 3.7c plots ξ Emod for four representative supercritical pressures. Indeed, ξ Emod shows strong negative density-dependence. As a result, the GL (T > TPB ) state generally has a larger energy efficiency than the LL (T < TPB ) state under the same pressure. Besides, ξ Emod gets weakly enhanced around TPB .

46

3 Coupled Transfer in a Relaxation Process: Mass Piston Effect

(b)

(a)

(c)

Fig. 3.7 a The modified total energy efficiency ξ Emod defined in Eq. (3.27) and modified total temperature efficiency ξTmod defined in Eq. (3.30) for C2 H6 − CO2 at cref = 0.6721. b ξTmod for four selected supercritical pressures. c ξ Emod for four selected supercritical pressures. In Figs. (b) and (c), the red dots denote the values at TPB , i.e., the temperature where maximal c p achieves

3.5 Conclusions This chapter is devoted to reporting the MPE induced by the cross-boundary mass flux in a binary mixture under supercritical pressures. MPE is driven by three cooperative or competing mechanisms: boundary velocity, Dufour effect, and concentration variation. A traveling-wave theory is developed to represent the amplitudes of the acoustic wave and the wave’s propagation. Through comparing with numerical simulations, the theoretical predictions are quite satisfactory. Furthermore, in the energy balance analysis, the modified energy and temperature efficiencies measuring the capabilities of the MPE in terms of energy transfer and thermalization are derived. It shows that MPE is more efficient in terms of thermalization than energy transfer, especially at near-critical states. This chapter enriches the concept of the PE by introducing the energy flux carried by a cross-boundary mass diffusion, which is a common occurrence in chemical extraction processes. In the future, a promising direction is to explore other forms of cross-boundary energy flux to discover more new members of the PE’s family.

References

47

References 1. Boukari H, Shaumeyer JN, Briggs ME, Gammon RW (1990) Critical speeding up in pure fluids. Phys Rev A 41:2260–2263 2. Hu ZC, Zhang XR (2016) Numerical simulations of the piston effect for near-critical fluids in spherical cells under small thermal disturbance. Int J Therm Sci 107:131–140 3. Onuki A, Hao H, Ferrell RA (1990) Fast adiabatic equilibration in a single-component fluid near the liquid-vapor critical point. Phys Rev A 41:2256–2259 4. Zappoli B, Bailly D, Garrabos Y, Le Neindre B, Guenoun P, Beysens D (1990) Anomalous heat transport by the piston effect in supercritical fluids under zero gravity. Phys Rev A 41:2264– 2267 5. Landau L, Lifshitz E (1987) Fluid mechanics, 2nd edn. Pergamon Press, London 6. Westphal G, Rosenberger F (1978) On diffusive-advective interfacial mass transfer. J Cryst Growth 43(6):687–693 7. Shen B, Zhang P (2010) On the transition from thermoacoustic convection to diffusion in a near-critical fluid. Int J Heat Mass Transf 53(21–22):4832–4843 8. Miura Y, Yoshihara S, Ohnishi M, Honda K, Matsumoto M, Kawai J, Ishikawa M, Kobayashi H, Onuki A (2006) High-speed observation of the piston effect near the gas-liquid critical point. Phys Rev E 74:010101 9. https://www.openfoam.org 10. Amiroudine S, Caltagirone JP, Erriguible A (2014) A Lagrangian-Eulerian compressible model for the trans-critical path of near-critical fluids. Int J Multiph Flow 59:15–23 11. Garrabos Y, Bonetti M, Beysens D, Perrot F, Fröhlich T, Carlès P, Zappoli B (1998) Relaxation of a supercritical fluid after a heat pulse in the absence of gravity effects: Theory and experiments. Phys Rev E 57:5665–5681

Chapter 4

Coupled Transfer in a Diffusion Problem: Concentration Gradient in the Coexisting Liquid-Like and Gas-Like States

This chapter also belongs to the first part of this thesis, where a diffusion problem is studied and coupled heat and mass transfer exists through cross-diffusion effects. As mentioned in Chap. 1, upon crossing the Widom line marked by TPB , a supercritical fluid undergoes a sharp change in density ρ, similar to a subcritical boiling. Therefore, the pseudo-boiling (PB) [1] has been introduced to denote this process. In recent years, PB has gained increasing attention in the scientific community [1–6]. However, the PB in binary mixtures has not received much attention, especially its analogy to subcritical boiling. It is well understood that for a zeotropic binary mixture, the subcritical boiling occurs in a temperature range instead of a fixed boiling point (the so-called temperature glide), and the concentrations in liquid and gas are different. So, is there concentration difference between liquid-like (LL) and gas-like (GL) states? To survey this problem, the coexistence of LL and GL states should be achieved first. Under subcritical pressure, the coexistence of liquid and gas can be achieved as long as T is maintained between the bubble and dew points, as liquid and gas separate naturally. However, if p is elevated above pC , the coexistence of LL and GL states requires an externally imposed temperature difference, within which TPB should be embraced. In this chapter, such a temperature difference is termed as a transcritical temperature difference. Then, there is naturally concentration difference due to the cross-diffusion effects. To give more insights into the behavior and influences of concentration difference, the steady-state responses of C2 H6 − CO2 mixture at supercritical pressures subjected to transcritical temperature differences are explored in this chapter. Under this configuration, the concentration difference arises as a steady-state concentration gradient. This chapter is arranged as follows. In Sect. 4.1, the details about the physical and mathematical descriptions of the problem are provided. In Sect. 4.2, the numerical method is proposed. In Sects. 4.3 and 4.4, the characteristics for the coexistence © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 Z.-C. Hu, Coupled Heat and Mass Transfer in Binary Mixtures at Supercritical Pressures, Springer Theses, https://doi.org/10.1007/978-981-16-7806-6_4

49

50

4 Coupled Transfer in a Diffusion Problem …

of LL and GL states, and the influences of concentration gradient are described, respectively. The chapter is concluded in Section 4.5.

4.1 Problem Statement 4.1.1 Physical Model The physical model is shown in Fig. 4.1. An enclosed binary mixture layer (C2 H6 − CO2 ) subjected to a transcritical temperature difference under a supercritical pressure is studied. A concept of reference state is introduced, which is the thermodynamic equilibrium state before applying the temperature difference. The reference state is denoted by a subscript ref. The state variables, i.e., temperature, pressure, concentration (mass fraction of C2 H6 ) and density, are denoted as T , p, c and ρ, respectively, so the reference state is (Tref , pref , cref , ρref ). In addition, to fulfill the coexistence conditions, Tref is set as TPB , and pref is larger than the critical pressure pC . Note that the critical parameters are functions of c (see the dotted curve in Fig. 2.2). As for TPB , it is a function of p and c (see the surface in Fig. 2.4). The temperatures at the left and right walls are kept constant as (Tref + ΔT /2) and (Tref − ΔT /2), respectively, with ΔT a positive temperature difference. d is the thickness of the fluid layer. In this problem, since the steady state is governed by a thermal gradient, the local equilibrium assumption is employed to guarantee the binary mixture can be described using properties obtained from thermodynamic equilibrium states. Besides, gravity has been omitted in this study, thus no natural convection, supporting the one-dimensional configuration with the coordinate x.

4.1.2 Governing Equations for the Motionless Steady State In Section 2.1, the complete governing equations are introduced. However, in this study, the purely diffusive steady state is the main concern, thus bringing about huge

Tref

pref

T 2

d

pC

Tref

T 2

x Fig. 4.1 Sketch of the physical model: A binary mixture layer (C2 H6 − CO2 ) subjected to a temperature difference under supercritical pressure and zero gravity. Temperatures at the left and right boundaries are kept constant, where Tref is reference temperature equaling TPB , pref is the reference pressure larger than the critical pressure pC , and ΔT is a positive value representing the temperature difference. x is the coordinate. d is the thickness of the fluid layer

4.1 Problem Statement

51

simplifications into the governing equations. After setting u = 0 and ∂/∂t = 0, Eqs. (2.1)–(2.5) are simplified into (one-dimensional) ∂q = 0, ∂x ∂i = 0, ∂x ρ = f (T, p, c),

(4.1) (4.2) (4.3)

where q is still given by Eq. (2.6) but i by  i = −ρ D

∂c kT ∂ T + ∂x T ∂x

 ,

(4.4)

where the barodiffusion effect in Eq. (2.7) has been neglected, since there is no pressure gradient in the current motionless steady state. Besides, since a one-dimensional problem is considered in this study, x is the only spatial coordinate. The boundary conditions are impermeable (i.e., i = 0) walls with fixed temperatures, which can be mathematically expressed by ΔT ∂c kT , =− 2 ∂x T kT ΔT ∂c , =− − 2 ∂x T

x = 0, T = Tref + x = d, T = Tref

∂T , ∂x ∂T . ∂x

(4.5) (4.6)

4.1.3 Summary of Cases To investigate the steady-state responses under coexistence conditions, the following conditions are applied in this chapter Tref = TPB = f ( pref ), cref = 0.6721, d = 10 mm.

(4.7)

That is to set Tref as TPB (temperature of maximal c p ) at pref , and fix cref and d as reference values. Now, the independent variables are ΔT and pref . Given pC = 5.441 MPa, the following seven pairs of ( pref , Tref , ρref ) are configured ( pref , Tref , ρref ) ∈ {(6.0 MPa, 302.04 K, 255.69 kg/m3 ), (6.5 MPa, 305.95 K, 260.72 kg/m3 ), (7.0 MPa, 309.63 K, 265.72 kg/m3 ), (7.5 MPa, 313.11 K, 270.45 kg/m3 ), (8.0 MPa, 316.39 K, 275.11 kg/m3 ),

52

4 Coupled Transfer in a Diffusion Problem …

Fig. 4.2 Widom line and the locations of seven reference states: C2 H6 − CO2 at c = 0.6721

(8.5 MPa, 319.48 K, 279.70 kg/m3 ), (9.0 MPa, 322.39 K, 284.23 kg/m3 )}.

(4.8)

By setting Tref = TPB , the above seven reference states are located on the Widom line, as shown in Fig. 4.2. As for ΔT , eleven values are configured ΔT = 0.1, 0.5, 1, 2, 4, 6, 8, 10, 12, 16, 20 K.

(4.9)

Therefore, the above setup leads to 77 cases of simulations. Note that even though pC is a function of c and the steady-state pressure is different from pref , all steady states own supercritical pressures. The steady-state pressures are also higher than the cricondenbar, which is 1 kPa higher than pC for C2 H6 − CO2 at cref = 0.6721. Next, the numerical method will be introduced first in Sect. 4.2, followed by the results and discussions in Sect. 4.3.

4.2 Numerical Method In this study, the one-dimensional physical domain is discretized into (N − 1) elements of equal size. To obtain a steady-state solution, one can solve the transient equations and proceed until a steady state is reached, or solve the steady-state equations iteratively. In this paper, the former approach is adopted. For this, the transient version of Eqs. (4.1) and (4.2) is employed in the numerical algorithm: ∂q ∂i ∂T =− + H¯ , ∂t ∂x ∂x ∂c ∂i ρ =− . ∂t ∂x

ρc p

(4.10) (4.11)

Note that compared with Eq. (2.3), u = 0 and D p/Dt = 0 are adopted here, so the piston effect (PE) has been neglected. As a consequence, the transient process is

4.2 Numerical Method

53

not real. Since only steady states are concerned in this chapter, this treatment is acceptable. The two equations are discretized by the finite difference method. The spatial derivatives are discretized using the second-order central difference scheme, while the temporal terms by the explicit Euler scheme. To treat variable physical properties, three-dimensional lookup tables are created in which (T, p, c) are independent variables. As mentioned earlier, local equilibrium assumption is employed to guarantee the binary mixture can be described using properties obtained from thermodynamic equilibrium states. During each time step, Eqs. (4.10) and (4.11) are solved first to update (T, c) from (T n , cn ) to (T n+1 , cn+1 ), where the superscript (n + 1) denotes the current time step, while n the previous one. Then, ρ and other variable properties are interpolated from three-dimensional lookup tables according to the the given (T n+1 , p n , cn+1 ): n+1 } = f (T n+1 , p n , cn+1 ). {ρ n+1 , k Tn+1 , H¯ n+1 , csn+1 , λn+1 , D n+1 , cn+1 p ,α

(4.12)

Note that p is a value, while T and c are functions of x. Finally, p n is updated to guarantee the overall mass conservation: p

n+1

1 =p + 2 d n

  ρref d − 0

d

 ρ

n+1

dx 0

d

1 dx. ρ n+1 α n+1

(4.13)

In each case, the calculation always proceeds from a initial solution with c = cref , p = pref and T = Tref − ΔT (x/d − 1/2), until a steady state is reached. Owing to the explicit temporal discretization, the time step Δt should be small enough to guarantee numerical stability. For the 77 cases considered in this study, Δt = 0.001 s works well. As for the number of grid nodes N , comparing between the results of N = 1000 and N = 10000 for p = 6 MPa and ΔT = 20 K, the profiles of various properties coincide, suggesting N = 1000 is adequate.

4.3 Characteristics for the Coexistence of LL and GL States This section is dedicated to describing the phenomenon from various aspects, with an emphasis on the characteristics in the profiles of the state variables and physical properties, along with the behavior of several response parameters.

4.3.1 State Variables The state variables T , c, ρ and p are of significant concern. To give an overview, Fig. 4.3 shows the initial state and final steady state for the case of pref = 6.5 MPa and ΔT = 20 K. The initial state (i.e., reference state) is denoted by a red pentagram

54

4 Coupled Transfer in a Diffusion Problem …

gas-like liquid-like

Fig. 4.3 The initial state (red pentagram) and finial steady state (blue solid line) plotted in the (T, p − pC , c) parametric space for the case of pref = 6.5 MPa and ΔT = 20 K. The green surface (Widom surface) is composed of Widom lines at different concentrations, served as the boundary for liquid-like and gas-like states

(a)

(b)

Fig. 4.4 The profiles of temperature T under various (a) temperature differences ΔT and (b) reference pressures pref . In Fig. (a), pref = 6.0 MPa is fixed, while in Fig. (b), ΔT = 20 K is chosen as a representative value. Besides, T has been normalized as (T − Tref )/ΔT for the sake of comparison

on the Widom surface, while the final steady state by a solid curve penetrating the Widom surface. A concentration gradient forms and a pressure drop relative to the initial state is observed. Compared with the initial state, even though pC varies with c, the final steady state is featured by supercritical pressure and coexistence of LL and GL states. Next, discuss these profiles in detail. First, look into the profile of T . As can be inferred from the governing equations and boundary conditions, a steady state is featured by a constant heat flux and zero mass flux, making T as the solution of

4.3 Characteristics for the Coexistence of LL and GL States

55

(b)

(a)

(c)

Fig. 4.5 The profiles of concentration c under various (a) temperature differences ΔT and (b) reference pressures pref , and (c) the contour plot of Δc with respect to ΔT and pref , where Δc = max(c) − min(c) is an indicator for the strength of concentration gradient. In Fig. (a), pref = 6.0 MPa is fixed, while in Fig. (b), ΔT = 20 K is chosen as a representative value

∂ ∂x

  ∂T λ = 0. ∂x

(4.14)

Therefore, the profile of T is continuous. Note that λ now depends on T and c, leading to noticeable nonlinearity in T . The profiles of T are plotted in Fig. 4.4. For the sake of comparison, T has been non-dimensionalized as (T − Tref )/ΔT . As suggested by Fig. 4.4a, when pref is fixed (currently pref = 6.0 MPa), the nonlinearity in T grows with ΔT . An opposite trend is identified in Fig. 4.4 (b), namely the nonlinearity vanishing gradually with increasing pref under a fixed ΔT (currently ΔT = 20 K). According to Eq. (4.14), these two distinct trends can be understood through referring to the spatial variations in λ, which naturally grow with ΔT , but reduce with pref (see Fig. 2.5 i). Second, survey the profile of c. As has been mentioned earlier, ΔT induces concentration gradient, making c a spatial variable. From the steady-state condition, i.e., i = 0, the profile of c is actually governed by

56

4 Coupled Transfer in a Diffusion Problem …

(a)

(b)

(c)

Fig. 4.6 The profiles of density ρ under various (a) temperature differences ΔT and (b) reference pressures pref , and (c) the contour plot of Δρ with respect to ΔT and pref , where Δρ = max(ρ) − min(ρ) is an indicator for the strength of PB. In Fig. (a), pref = 6.0 MPa is fixed, while in Fig. (b), ΔT = 20 K is chosen as a representative value

kT ∂ T ∂c =− . ∂x T ∂x

(4.15)

Therefore, the slope of c is greatly coupled with that of T , but more complex due to the additional variations in k T . As demonstrated in Fig. 4.5a,b, the nonlinearity in the profile of c is also positively correlated to ΔT , but negatively to pref . Moreover, it is also revealed that the nonlinearity is strongly related to the strength of the concentration gradient, which can be measured by the overall concentration difference, namely (4.16) Δc = max(c) − min(c) = c|x=d − c|x=0 . Therefore, one could expect Δc is an increasing function of ΔT , but a decreasing one of pref . Such trends are clearly presented in Fig. 4.5c. Next, the profiles of ρ are presented in Fig. 4.6. Under the coexistence conditions, ρ changes dramatically if ΔT is not very small, reminiscent of the subcritical boiling. To measure the strength of PB, the overall density difference is introduced:

4.3 Characteristics for the Coexistence of LL and GL States

57

Fig. 4.7 Contour plot of Δp with respect to ΔT and pref , where Δp is the pressure drop with respect to the reference pressure pref

Δρ = max(ρ) − min(ρ) = ρ|x=d − ρ|x=0 .

(4.17)

Comparing between Figs. 4.6 and 4.5, one can immediately notice the similarities between the profiles of c and ρ. Hence, previous conclusions concerning c also accommodate ρ, and those for Δc suit Δρ. That is to say, upon increasing ΔT or decreasing pref , the strength of PB (measured by Δρ) and the nonlinearity in ρ increase. Finally, in all of the simulated cases, pressure drops are observed between the initial pressure (i.e., the reference pressure pref ) and steady-state one, denoted as ps . The pressure drop is measured by Δp = pref − ps .

(4.18)

The contour plot of Δp versus ΔT and pref is shown in Fig. 4.7. Δp becomes increasingly significant as approaching the lower-right corner. Therefore, as suggested by Figs. 4.5c, 4.6c and 4.7, the behavior of Δp at different ΔT and pref is the same as the aforementioned ones for Δc and Δρ. The above descriptions reveal the similarities among Δρ, Δc and Δp. Since they are extensive properties, there is no surprise that they are positively correlated to ΔT . Besides, because critical anomalies in physical properties play important roles in them, they are naturally get enhanced as pref approaches pC .

4.3.2 Physical Properties Figure 4.8 presents the profiles of various properties at the steady state for the case of pref = 6.0 MPa and ΔT = 20 K. Because T and c are nonlinear functions of x, it is not strictly correct to compare Fig. 4.8 with Fig. 2.5, where c is constant and T varies linearly. However, it is easy to notice their qualitative similarities. Therefore,

58

(a)

4 Coupled Transfer in a Diffusion Problem …

(b)

Fig. 4.8 Profiles of various properties at the steady state for the case of pref = 6.0 MPa and ΔT = 20 K. Every property has been adjusted into a proper range by multiplying a factor. Note that in Fig. (a), k T and D are represented by solid curves of different thicknesses, so as to α and η in Fig. (b)

in a steady state under coexistence conditions, the profiles of λ, D and η are governed by density, while other properties are dominated by the critical enhancement. In this study, the profiles of λ and k T are of great interest. Equation (4.14) suggests the steady-state slope of T solely depends on λ. Figure 4.8 shows λ is generally an increasing function of x, which is because the LL state is more conductive than GL state, just like normal liquid and gas. Consequently, T is a convex function of x, making the mean temperature lower than Tref . Such results are clearly observed in Fig. 4.4. As for k T , since the variations in ∂ T /∂ x are relatively mild, Eq. (4.15) implies the slope of c should mainly be determined by k T . Being a critical-enhancementdominated property, k T develops a peak around x/d = 0.5, corresponding to the rapid variations of c around x/d = 0.5. It explains the profiles of c presented in Fig. 4.5.

4.3.3 Physical Explanation for the Pressure Drop From the discussions in Sect. 4.3.1, a binary mixture layer under coexistence conditions is strongly nonlinear in state variables and physical properties, featured by concentration gradient and pressure drop. Section 4.3.2 confirms the nonlinearities are closely related to the variable physical properties. Besides, it is already known that the concentration gradient is a result of the heat and mass coupling caused by Soret effect. However, the mechanism of pressure drop remains unknown.

4.3 Characteristics for the Coexistence of LL and GL States

59

From Eq. (4.13), it is reasonable to argue that the behavior of pressure depends on the influences of T and c on ρ. If an expansion effect is exerted, the pressure increases to guarantee the overall mass conservation. Otherwise, the pressure decreases. Therefore, the mechanism can be explained from two aspects: • Redistribution of temperature. As has been discussed in the previous section, the mean temperature of a steady state is lower than Tref . In other words, redistribution of temperature yields a contraction effect, leading to a pressure drop. From a physical point of view, the mean temperature reflects the intensity of the mean kinetic energy of the molecules. Therefore, as the mean temperature drops, the thermal motions of the molecules are weakened, corresponding to the pressure drop. • Redistribution of concentration. As shown in Fig. 4.5, it is not straightforward to infer the change of average concentration from its profiles. (Note that the average value of ρc is a conserved quantity, not the average value of c.) A survey on the mean value of c for all simulated cases confirms it is also a decreased value. That is to say, redistribution of concentration exerts an expansion effect (because κ is positive), leading to a pressure increase. Since eventually a pressure drop is observed, it is concluded that the influences of temperature win those of concentration. Indeed, the pressure drop indicates the possibility for a subcritical ps . Therefore, to maintain a supercritical pressure, ΔT cannot be too large, as has been guaranteed in the current work. Because the behavior of λ is similar, the above arguments also suggest the pressure drop is a common phenomenon in the present problem.

4.3.4 Heat Transfer Characteristics Another important aspect for such a fluid layer under coexistence conditions is the steady-state heat flux, denoted by a scalar Q, expressed by Q = −λ

∂T . ∂x

(4.19)

Contour plot of Q versus ΔT and pref are presented in Fig. 4.9. As can be inferred from Fig. 4.9, like Δc, Δρ and Δp, Q is an increasing function of ΔT , but a decreasing one of pref . However, the difference is, the dependence of Q on pref is rather weak. In fact, the strong dependence of Q on ΔT is natural since it determines the magnitude of the temperature gradient. The weak dependence on pref should be attributed to the weak dependence of λ on the critical enhancement. In summary, a binary mixture layer under coexistence conditions is strongly nonlinear in state variables and physical properties. Moreover, the heat and mass coupling due to Soret effect induces noticeable concentration gradient, and the low

60

4 Coupled Transfer in a Diffusion Problem …

Fig. 4.9 Contour plot of steady-state heat flux Q with respect to ΔT and pref

heat-conducting nature of GL state compared to LL state gives rise to the pressure drop. Reducing pref or enlarging ΔT , the profiles of T , c and ρ own growing nonlinearities. Meanwhile, the concentration gradient, the strength of PB, the pressure drop, and the heat flux get enhanced, reflected by the increased Δc, Δρ, Δp and Q, respectively.

4.4 Assessments on the Influences of Concentration Gradient Characteristics for the coexistence of LL and GL states have been discussed in the previous section. In general, the most striking feature is the concentration gradient. However, the accurate prediction of concentration gradient strongly relies on the accurate database of k T , but such databases are very limited. Therefore, a common practice is to introduce the pseudo-pure fluid approximation, in which k T = 0 and the binary mixture is treated as an equivalent pure one, namely no concentration gradient. Therefore, the question that arises naturally is the applicability of the pseudo-pure fluid approximation for the current configuration (closed-volume system). To answer this question, the influences of concentration gradient will be assessed in this section. To this end, additional 77 cases were supplemented, in which concentration gradients have been switched off through setting k T = 0. In the following, comparisons will be made to elaborate the influences of concentration gradient on Δρ, Δp, and Q.

4.4 Assessments on the Influences of Concentration Gradient

(a)

61

(b)

10 %

5%

Fig. 4.10 The underestimations of overall density difference Δρ after switching off the concentration gradient. a (Δρ − ΔρΔc=0 ) scaled by the reference density ρref . b (Δρ − ΔρΔc=0 ) scaled by Δρ with isolines indicating 5 % and 10 % Fig. 4.11 The profiles of density ρ for pref = 6.0 MPa and ΔT = 20 K with and without concentration gradient

4.4.1 Influences of Concentration Gradient on the PB First look into the influences of concentration gradient on Δρ. In the current problem, Δc has a positive effect on Δρ. Therefore, switching off Δc results in underestimations of Δρ, as presented in Fig. 4.10. In Fig. 4.10a, (Δρ − ΔρΔc=0 ) is scaled by ρref to measure the influences relative to the absolute density. It is demonstrated that when pref or ΔT is large enough, or when ΔT is substantially small, the underestimations of Δρ are negligible. Indeed, the decreasing of (Δρ − ΔρΔc=0 ) as ΔT approaching zero is natural, because there is no concentration migration without thermal disturbances. In fact, the effect of pref is also straightforward since it controls the strength of concentration gradient as presented in Fig. 4.5 (c). However, when ΔT is sufficiently large, the diminishing of (Δρ − ΔρΔc=0 ) is somehow counter-intuitive. Compare Figs. 4.10a and 4.5c, the counter-intuitive point is when concentration gradient is large, the resulting underestimation of Δρ is unusually negligible! To give more insights into this point, Fig. 4.11 shows the profiles of ρ for pref = 6.0 MPa and ΔT = 20 K with and without concentration gradient. Indeed, Δc = 0 leads to an underestimation in the slope of ρ. But the influences on Δρ are very small.

62

(a)

4 Coupled Transfer in a Diffusion Problem …

(b)

10 % 5 %

Fig. 4.12 The influences of the concentration gradient on the pressure drop Δp. a (Δp Δc=0 − Δp) scaled by the steady-state pressure ps . b (Δp Δc=0 − Δp) scaled by Δp with isolines indicating 5 % and 10 %

Since ρ is determined by the equation of state, this can be explained by the weak dependence of ρ on c when ΔT is large enough, since both (T = Tref + ΔT /2) and (T = Tref − ΔT /2) are far from Tref = TPB . No matter how the trend is, Fig. 4.10a shows (Δρ − ΔρΔc=0 )/ρref is less than 6.5%. Therefore, as far as the absolute value of ρ is concerned, the deviations are not vital. In Fig. 4.10b, (Δρ − ΔρΔc=0 )/Δρ is plotted, together with two isolines indicating 5% and 10%. In the left-bottom region, the relative differences are larger than 10 %, in which the largest value is 27.2%, so Δρ has been obviously underestimated. Therefore, in situations where Δρ works as the key mechanism, the neglect of concentration gradient is unacceptable when ΔT and ( pref − pC ) are small. In reality, Δρ plays an important role in buoyancy flows. So this conclusion implies that in the natural convection of a binary mixture at a supercritical pressure, concentration gradient is crucial when ΔT is small and pref is close to pC (i.e., nearcritical states). For example, if the fluid layer is placed under a Rayleigh-Bénard configuration, i.e., heated from below under gravity, the onset of convection would be different. This point will be confirmed by the next chapter.

4.4.2 Influences of Concentration Gradient on Pressure Drop and Heat Flux Figure 4.12 presents the influences of concentration gradient on Δp, where (ΔpΔc=0 − Δp) is considered as the indicator. In Fig. 4.12a, (ΔpΔc=0 − Δp) has been scaled by ps to examine the influences on absolute pressure, where (ΔpΔc=0 − Δp)/ ps is less than 0.2 %. Therefore, as far as ps is concerned, omitting concentration gradient is fairly acceptable. To explore the influences on relative pressure change, (ΔpΔc=0 − Δp)/Δp is plotted in Fig. 4.12b. In the left-bottom region of the figure, the relative difference is larger than 10 %, with a maximum value of 452 %, so Δp has been severely

4.4 Assessments on the Influences of Concentration Gradient

63

Fig. 4.13 The influences of the concentration gradient on the heat flux Q

amplified. Therefore, if Δp is the main concern, it is unwise to apply the pseudopure fluid approximation, especially when ΔT is small and pref is close to pC . In reality, Δp is the key factor of the PE. Note that the PE is a fast thermalization phenomenon due to the rapid expansion or contraction of the thermal boundary layer in a near-critical fluid confined within a fixed volume. It is caused by the heating source term TβD p/Dt on the right-hand side of Eq. (2.3). Indeed, in a classical thermodynamic treatment of PE [7], velocity field can be omitted, and TβD p/Dt degenerates into Tβ∂ p/∂t. Since Δp is the time integral of ∂ p/∂t, it can be inferred that once Δp is altered after switching off concentration gradient, the strength of the PE would be different, leading to a different route to the final steady state. In this study, the PE has not been taken into consideration since only steady states are involved. Therefore, once the PE is the main concern, it is unacceptable to apply the pseudo-pure fluid approximation, especially when ΔT is small and pref is close to pC . Finally, Fig. 4.13 presents the quantitative influences of concentration gradient on Q, where the relative deviation (Q Δc=0 − Q)/Q has been taken as the indicator. For all of the calculated cases, the relative deviations are smaller than 4 %. Therefore, the pseudo-pure fluid approximation is acceptable as far as Q is concerned. According to Eq. (4.19), it is because ∂ T /∂ x is mainly determined by the boundary conditions, and λ is not sensitive to c. The above discussions suggest in the current configuration (closed-volume system), the pseudo-pure fluid approximation is inapplicable in problems that are featured by a binary mixture close to the Widom line, subjected to small thermal disturbances and with a pressure close to pC . Besides, it also requires that Δρ or Δp is the main concern, such as buoyancy flows or the PE, respectively. Indeed, this conclusion is consistent with the previous work of Long et al. [8]. They performed experiments regarding the natural convection of a supercritical N2 − Ar binary mixture in an enclosed cylinder domain, where the applicability of pseudo-pure fluid approximation was identified. This work shows that such applicability is caused by the large temperature differences applied in their experiments.

64

4 Coupled Transfer in a Diffusion Problem …

4.5 Conclusions In this chapter, the steady-state responses of an enclosed binary mixture subjected to transcritical temperature differences under supercritical pressures are numerically simulated to study the coexistence of LL and GL states and the influences of the concentration gradient. It is found that the steady state is strongly nonlinear in state variables and physical properties. The Soret effect induces the concentration gradient, and the low heatconducting nature of GL state compared to LL state gives rise to the pressure drop. As the critical pressure is approached and the temperature difference is enlarged, the profiles of T , c and ρ own growing nonlinearities. Meanwhile, the concentration gradient, the strength of PB, the pressure drop, and the heat flux are also enhanced. Besides, when ΔT is small and pref is close to pC , the concentration gradient has deep effects on the relative variations of density and pressure. Therefore, it plays a vital role in buoyancy flows and transient phenomena like the PE. This work gives new insights into the similarities of binary mixtures under suband supercritical pressures, also extends the analogy between subcritical boiling and PB. It inspires a unified view of miscible zeotropic binary mixtures under sub- and supercritical pressures. Therefore, previous phenomena related to the concentration difference in liquid-gas coexistence also possibly exist under supercritical pressures, such as the concentration shift observed in the organic Rankine cycle using zeotropic mixtures as working fluids [9, 10]. Besides, comprehensive studies on the fluid dynamics and heat transfer when the concentration gradient exists are also promising future works.

References 1. Banuti D (2015) Crossing the Widom-line—supercritical pseudo-boiling. The Journal of Supercritical Fluids 98:12–16 2. Kim K, Hickey JP, Scalo C (2019) Pseudophase change effects in turbulent channel flow under transcritical temperature conditions. J Fluid Mech 871:52–91 3. Lapenna PE (2018) Characterization of pseudo-boiling in a transcritical nitrogen jet. Phys Fluids 30(7):077106 4. Maxim F, Contescu C, Boillat P, Niceno B, Karalis K, Testino A, Ludwig C (2019) Visualization of supercritical water pseudo-boiling at Widom line crossover. Nat Commun 10(1):1–11 5. Maxim F, Karalis K, Boillat P, Banuti DT, Marquez Damian JI, Niceno B, Ludwig C (2021) Thermodynamics and dynamics of supercritical water pseudo-boiling. Advanced Science 8(3):2002312 6. Migliorino MT, Scalo C (2020) Real-fluid effects on standing-wave thermoacoustic instability. J Fluid Mech 883:A23. https://doi.org/10.1017/jfm.2019.856 7. Boukari H, Shaumeyer JN, Briggs ME, Gammon RW (1990) Critical speeding up in pure fluids. Phys Rev A 41:2260–2263 8. Long Z, Zhang P, Shen B (2015) Natural convection heat transfer of supercritical binary fluid in a long closed vertical cylinder. Int J Heat Mass Transf 80:551–561

References

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9. Zhao L, Bao J (2014) The influence of composition shift on organic Rankine cycle (ORC) with zeotropic mixtures. Energy Convers Manage 83:203–211 10. Zhou Y, Zhang F, Yu L (2017) The discussion of composition shift in organic Rankine cycle using zeotropic mixtures. Energy Convers Manage 140:324–333

Part II

Coupling Through Cross-Diffusion Effects: Instability and Bifurcation

Chapter 5

Interactions Between Coupled Transfer and Gravity: Rayleigh-Bénard Instability

This chapter enters the second part of this thesis, where a hydrodynamic instability problem is studied and coupled heat and mass transfer exists through cross-diffusion effects. In Chap. 4, it is confirmed that when temperature difference ΔT is small and reference pressure pref is close to pC , the concentration gradient greatly alters the strength of buoyancy. Therefore, it is reasonable to infer that the concentration gradient may have profound effects on the onset threshold of Rayleigh-Bénard (RB) convection in binary mixtures at near-critical states. The onset threshold, i.e., ΔT of the neutral stable state, or the critical Rayleigh number R crit , is the most basic question in RB convection, which is also termed as RB instability. Besides, the RB instability in such a binary mixture with cross-diffusion effects, adiabatic temperature gradient (ATG) and gravitational diffusion has not been wellunderstood. The task of this chapter is to derive the expressions of R crit , and to reveal the interactions of buoyancy, viscosity, heat conduction, cross-diffusion effects, ATG and gravitational diffusion. This chapter is arranged as follows. Section 5.1 presents the mathematical modeling and simplifications. Section 5.2 is the main body of linear stability analysis, including the derivation and solution of the generalized eigenvalue problem. Section 5.3 provides results and discussions regarding the criteria and underlying physics, where the origin of oscillatory instability is elaborated. This chapter is summarized in Sect. 5.4.

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 Z.-C. Hu, Coupled Heat and Mass Transfer in Binary Mixtures at Supercritical Pressures, Springer Theses, https://doi.org/10.1007/978-981-16-7806-6_5

69

70

5 Interactions Between Coupled Transfer and Gravity: Rayleigh-Bénard Instability

5.1 Problem Statement and Approximation 5.1.1 Physical Model Figure 5.1 presents the physical model of RB instability, namely a 2D binary mixture layer at a supercritical pressure bounded by two walls, heated from below and cooled from above. The thickness is denoted by d. C2 H6 − CO2 is taken as a reference fluid pair, and c represents the mass fraction of C2 H6 . The horizontal and vertical directions are denoted by x and z, respectively. g = (0, −g) = (0, −9.81 m/s2 ) is the gravitational acceleration in the vector form. The two walls have fixed temperatures, with a temperature difference denoted by ΔT . Besides, ideal boundary conditions (BCs) (stress-free walls with fixed concentrations) are employed to facilitate theoretical analyses. The values of c at the walls are determined by the impermeable condition. For a binary mixture, three independent variables should be employed to specify its state, and the ones considered in this study are temperature T , pressure p and concentration c. Besides, the thermodynamic equilibrium state when ΔT = 0 and g = 0 is termed as the reference state, denoted by the subscript ref.

5.1.2 Mathematical Model 5.1.2.1

Complete Governing Equations

The complete governing equations have been introduced in Sect. 2.1, see Eqs. (2.1)– (2.5). Previous studies suggest at near-critical states, ATG has profound effects on

z

T

Tref

d

T 2

g

Binary mixture Reference state: (Tref , pref , cref )

T

Tref

T 2

x

Fig. 5.1 The physical model of RB instability. T is temperature, p is pressure, ρ is density, c is concentration, the subscript ref represents the reference state, g is the gravitational acceleration, d is the thickness of the fluid layer, ΔT is temperature difference, x and z are the horizontal and vertical coordinates. The two walls have fixed temperatures. In this study, ideal BCs (stress-free walls with fixed concentrations) are employed

5.1 Problem Statement and Approximation

71

the onset of RB convection [1]. In order to explicitly show the role of ATG, equations are processed as follows. Denote physical quantities at the purely conductive steady state by overlines, so u¯ = 0. Then, Eq. (2.2) leads to ∇ p¯ = ρg. ¯

(5.1)

¯ and subtracting Eq. (5.1) from Eq. Introducing the dynamic pressure as pd = p − p, (2.2) result in ρ

Du 2 = −∇ pd + ∇ · (η∇u) + ∇ · [η(∇u)T ] − ∇(η∇ · u) + (ρ − ρ)g. ¯ Dt 3

(5.2)

Then substituting p = p¯ + pd into Eqs. (2.3) and (2.7), using Eq. (5.1), one obtains ρc p

DT = − ∇ · q + H¯ ∇ · i − ρc p ∇ad w − Dt  kp kT i = − ρ D ∇c + ∇T + ρg ¯ + T p

T ρ



∂ρ ∂T



D pd + , p,c Dt

 kp ∇ pd , p

(5.3) (5.4)

where ATG has been denoted by ∇ad , with its expression given by ∇ad = −

  ∂ρ ρgT ¯ , ρ 2 c p ∂ T p,c

(5.5)

normally being positive. Equations (5.3) and (5.4) contain two special consequences of gravity. On the one hand, the third term on the right-hand side of Eq. (5.3) implies an upward motion (w > 0) induces a cooling effect. Therefore, fluid at a lower position has a lower temperature than the measured value, corresponding to an implicit positive temperature gradient, which will hinder the onset of convection. This is the implication of ATG, which originates from the hydrostatic pressure gradient. On the other hand, the third term on the right-hand side of Eq. (5.4) accounts for the gravitational diffusion, corresponding to the barodiffusion effect of hydrostatic pressure gradient.

5.1.2.2

Approximations of the Governing Equations

The above governing equations contain abundant nonlinear effects: Besides advection terms, there are compressibility, variable density and physical properties, making the theoretical analyses extremely difficult. Therefore, the following approximations are considered in this study: (1) Applying the linearized equation of state with respect to the reference state. ρ − ρref = ρref [α( p − pref ) − β(T − Tref ) + κ(c − cref )] ,

(5.6)

72

5 Interactions Between Coupled Transfer and Gravity: Rayleigh-Bénard Instability

where isothermal compressibility α, thermal expansion coefficient β and concentration contraction coefficient κ are respectively defined as 1 α= ρref



∂ρ ∂p



1 ,β = − ρ ref T,c



∂ρ ∂T



1 ,κ = ρ ref p,c



∂ρ ∂c

 .

(5.7)

T, p

(2) Assuming constant physical properties given by the reference state. (3) Ignoring the variable density except for the buoyancy term (Boussinesq approximation). (4) Neglecting the viscous heating term:  = 0. (5) Neglecting the barodiffusion of dynamic pressure gradient: ∇ pd  ρref g. ¯ (6) Neglecting the thermal effect of changing dynamic pressure: D pd /Dt  D p/Dt. (7) Neglecting the dependence of density on the dynamic pressure: α = 0. There are some comments for the above approximations. The applicability of approximations (1), (2) and (3) are closely related to the thickness of the fluid layer [2, 3]: The higher d is, the stronger the density stratification, ATG and gravitational diffusion become. Since they are inhibiting factors for convection onset, ΔT should be increased to trigger the convection, so the variations in physical properties and density are significant. In spite of these limitations on the precision, approximations of (1), (2) and (3) actually make the stability problem concentrate on the reference state. The conclusions obtained in this study can be extended to another situation as long as a representative state is input. Regarding approximation (4), the viscous heating effect is important in strong shear flows with high viscosity, which is usually neglected in buoyancy-driven flows of supercritical pressure fluids [4]. At last, the approximations (5), (6) and (7) are related to the hydrodynamic pressure, namely ignoring its mass diffusion effect, thermal effect and compressibility effect. This is justified because the current study focuses on the convection near the onset threshold, the hydrodynamic pressure gradient is small. Note that after these approximations, the piston effect has been ignored. This is acceptable because the piston effect is a transient mechanism, it does not change the nature of the dynamic system, i.e., the existence and stability of steady states [5, 6], which is the main concern of the current work. After approximations (1)-(7), the governing equations are simplified into ∇ · u =0,   pd Du =−∇ + ν∇ 2 u + −β(T − T¯ ) + κ(c − c) ¯ g, Dt ρref   k T2 D kT D 2 DT ∇2T + = DT + ∇ c − ∇ad w, Dt Tref c p cs c p cs Dc kT D 2 =D∇ 2 c + ∇ T, Dt Tref

(5.8) (5.9) (5.10) (5.11)

5.1 Problem Statement and Approximation

73

where ν = η/ρref is kinematic viscosity, DT = λ/(ρref c p ) is thermal diffusivity and ∇ad = Tref βg/c p is ATG.

5.1.2.3

Initial and Boundary Conditions

The realistic no-slip, fixed-temperature and impermeable (i = 0) BCs can be mathematically written as u = w = 0, T = Tref + u = w = 0, T = Tref

⎫ kT ∂ T ΔT ∂c ⎪ , =− − κcs g, at z = 0 ⎪ ⎪ ⎬ 2 ∂z Tref ∂z

⎪ ⎪ kT ∂ T ΔT ∂c ⎭ , =− − κcs g, at z = d ⎪ − 2 ∂z Tref ∂z

.

(5.12)

To facilitate theoretical analysis, the corresponding ideal BCs (stress-free walls with fixed concentrations) are actually employed in the linear stability analysis ⎫

 ∂u ΔT ΔT k T d ⎪ = 0, T = Tref + , c = cref − , at z = 0 ⎪ − κcs g ⎪ ⎪ ⎬ ∂z 2 d Tref 2 .

 ⎪ ⎪ ∂u ΔT ΔT k T d ⎪ w= = 0, T = Tref − , c = cref + , at z = d ⎪ − κcs g ⎭ ∂z 2 d Tref 2 (5.13) The motionless base state (initial condition) is w=

ΔT u¯ = w¯ = p¯ d = 0, T¯ = Tref − d

 z−

d 2



, c¯ = cref +

ΔT k T − κcs g d Tref

 z−

d 2

 ,

(5.14)

accounting for linear distributions of T¯ and c. ¯

5.1.2.4

Nondimensionalization

Nondimensionalization is the preliminary procedure for stability analysis. Introduce the following dimensionless quantities (denoted by the prime symbol, and also note that the prime symbol has been used to represent wave amplitude in Chap. 3) [7, 8] x z , ,  d d w u (u  , w  ) = , , DT /d DT /d t t = 2 , d /DT (x  , z  ) =

(5.15) (5.16) (5.17)

74

5 Interactions Between Coupled Transfer and Gravity: Rayleigh-Bénard Instability

gβd 3 , ν DT gκd 3 c =(c − cref ) , ν DT pd pd = . ρref DT2 /d 2 T  =(T − Tref )

(5.18) (5.19) (5.20)

Substituting them into the governing equations yields ∇ · u Du Dt  DT  Dt  Dc Dt 

= 0,

(5.21)

  = −∇ pd + Pr ∇ 2 u + Pr (T  − T¯  ) − (c − c¯ ) k,

(5.22)

= (1 + L Qψ 2 )∇ 2 T  + L Qψ∇ 2 c − Rad w  ,

(5.23)

= L(∇ 2 c + ψ∇ 2 T  ).

(5.24)

Substituting the dimensionless quantities into Eq. (5.14), the base state reads     1 1 , c¯ = (Rψ − Q −1 Rad ) z  − . u¯  = w¯  = p¯ d = 0, T¯  = −R z  − 2 2 (5.25) Similarly, the ideal BCs are ⎫ ∂u  R  Rψ − Q −1 Rad ⎪   ⎬ , c , at z = 0, T = = − = 0 ∂z  2 2 .  −1 ∂u R Rψ − Q Rad ⎪ , at z  = 1 ⎭ w  =  = 0, T  = − , c = ∂z 2 2 w =

(5.26)

The above equations contain six dimensionless parameters: R=

gβΔT d 3 gβ∇ad d 4 ν D κk T Tref β 2 , Rad = , Pr = ,L= ,ψ= , Q= . ν DT ν DT DT DT βTref c p κ 2 cs

(5.27) R is the Rayleigh number, showing up in the expressions of base state and BCs, measuring the competitions between the driving factor (buoyancy) and the inhibiting factors (viscosity and heat conduction). The larger R is, the stronger the convection becomes. Because ΔT only appears in the definition of R, R is usually taken as an experimentally controllable parameter. Rad is the adiabatic Rayleigh number, namely the Rayleigh number written for ∇ad , evaluating the inhibiting effect of ∇ad . Pr is the Prandtl number, defined as the ratio of momentum diffusivity to thermal diffusivity. L is the Lewis number, defined as the ratio of concentration diffusivity to thermal diffusivity. ψ is the separation ratio, standing for the cross-diffusion effects. When ψ > 0, temperature gradient and its resulted concentration gradient work

5.1 Problem Statement and Approximation

75

cooperatively to drive the convection. While when ψ < 0, the concentration gradient is an inhibiting factor. Q is termed as Dufour number [9], which is positive due to cs > 0. The influences of Q are twofold. On the one hand, it couples with L and ψ, constituting the Dufour effect (see Eq. 5.23) and the enhancement in thermal diffusivity. On the other hand, it couples with Rad , constituting the gravitational diffusion (see Eq. 5.25). The dimensionless governing equations (5.21)–(5.24), together with base state (Eq. 5.25) and BCs (Eq. 5.26), constitute the full mathematical model of RB instability in a binary mixture at supercritical pressures. The linear stability analysis is to find the neutrally stable state where the growth rate equaling zero. The neutrally stable state corresponds to a critical Rayleigh number, denoted by R crit . Therefore, the problem under study can be expressed by R crit = f (Rad , Pr, L , ψ, Q).

(5.28)

Figure 5.2 presents the behavior of Pr , L, ψ, Q and Rad for four representative pressures and cref = 0.6721. All of the five properties are greatly influenced by the critical effects. To be specific, Pr , L, ψ and Rad develop peaks around TPB , while Q reaches a local minimum around TPB . Therefore, one can expect from Eq. (5.28) that R crit should also vary singularly around TPB , which will be discussed in Sect. 5.2.3. This section discusses the physical and mathematical models of RB instability in binary mixtures at supercritical pressures, paving the way for the linear stability analysis. It should be highlighted that, only ψ > 0 is considered in this study after setting cref = 0.6721 (see Fig. 5.2 c), so the concentration gradient will always enhance the buoyancy. However, the RB instability in a binary mixture has been extensively studied mainly for ψ < 0 [10–12]. Under such a situation, concentration gradient has a negative effect on buoyancy, whose competition with temperature gradient gives rise to intriguing dynamic phenomena. The problem studied here is featured by ψ > 0 and gravity-related effects (ATG and gravitational diffusion), which is distinguished greatly from previous works.

5.2 Linear Stability Analysis 5.2.1 Formulation of Stability Problem This section is the main body of linear stability analysis, arriving at the generalized eigenvalue problem. For the sake of brevity, hereinafter the prime symbols accompanying dimensionless variables are omitted. The first step in linear stability analysis is to obtain the governing equations of small perturbations. For this, subject the base state to a small perturbation (denoted by tilde symbol):

76

5 Interactions Between Coupled Transfer and Gravity: Rayleigh-Bénard Instability

(a)

(b)

(c)

(d)

(e)

Fig. 5.2 The dimensionless numbers (a) Pr , (b) L, (c) ψ, (d) Q and (e) Rad plotted against T for C2 H6 − CO2 at four representative pressures and cref = 0.6721. d = 0.05 m is employed in the calculation of Rad . The red dots denote the properties at TPB , i.e., the temperature where maximal c p achieves

˜ pd = p¯ d + p˜ d = p˜ d , T = T¯ + T˜ , c = c¯ + c. u = u¯ + u˜ = u, ˜ Experimentally, small perturbations arise naturally via thermal or mechanical background noise. Substituting the above perturbed forms into the governing equations (5.21)–(5.24), neglecting high-order terms, and introducing an auxiliary variable [7] ζ˜ = c˜ + ψ T˜ , one arrives at a linearized equation set governing the dynamics of perturbations:

5.2 Linear Stability Analysis

Pr −1

 ∂∇ 2 w˜ ∂  =∇ 4 w˜ + 2 (1 + ψ)T˜ − ζ˜ , ∂t ∂x ˜ ∂T =(R − Rad )w˜ + ∇ 2 T˜ + L Qψ∇ 2 ζ˜ , ∂t ∂ ζ˜ =(Q −1 − ψ)Rad w˜ + ψ∇ 2 T˜ + L(1 + Qψ 2 )∇ 2 ζ˜ . ∂t

77

(5.29) (5.30) (5.31)

In order to get Eq. (5.29), the continuity and moment equations are processed yielding an expression of ∇ 2 pd , which is then used to eliminate the pressure term in the moment equation of w. ˜ The ideal BCs for perturbations are w˜ =

∂ 2 w˜ = 0, T˜ = 0, ζ˜ = 0 at z = 0 and z = 1. ∂z 2

(5.32)

The auxiliary variable ζ˜ makes the above expressions compact. The second step for linear stability analysis is the normal mode expansion. The form of Eqs. (5.29)–(5.31) allows one to seek separable normal mode solutions: ⎤ ⎤ ⎡ w(z) ˆ exp(ikx + σ t) w˜ ⎣ T˜ ⎦ = ⎣ Tˆ (z) exp(ikx + σ t) ⎦ , ζ˜ ζˆ (z) exp(ikx + σ t) ⎡

(5.33)

where k is the wave number and σ = σr + iω is the growth rate (ω the angular frequency). Physically speaking, this equation means subjecting the base state to a series of perturbations that is periodic in x direction and BC-compatible in z direction. Then judge the behavior of each perturbation, i.e., growth (σr > 0) or decay (σr < 0). If (∀k, σr < 0), the system is linearly stable, while if (∃k, σr > 0), the system is linearly unstable. The difficulty in linear stability analysis lies in the determination ˆ Tˆ , c] ˆ T has the same BCs as [w, ˜ T˜ , c] ˜ T , see Eq. of [w(z), ˆ Tˆ (z), ζˆ (z)]T . Note that [w, (5.32). Substituting Eq. (5.33) into Eq. (5.30) yields   (Pr −1 σ − D 2 + k 2 )(D 2 − k 2 )wˆ = −k 2 (1 + ψ)Tˆ − ζˆ , (σ − D 2 + k 2 )Tˆ = (R − Rad )wˆ + L Qψ(D 2 − k 2 )ζˆ , (5.34)   σ − L(1 + Qψ 2 )(D 2 − k 2 ) ζˆ = (Q −1 − ψ)Rad wˆ + ψ(D 2 − k 2 )Tˆ , where D ≡ d/dz represents the derivative with respect to z. The above equations are equivalent to ⎡ ⎤ ⎡ ⎤ wˆ wˆ (5.35) A ⎣ Tˆ ⎦ = σ B ⎣ Tˆ ⎦ , ζˆ ζˆ where the matrices are

78

5 Interactions Between Coupled Transfer and Gravity: Rayleigh-Bénard Instability

 A=

2

(D 2 − k 2 ) −k 2 (1 + ψ) k2 2 2 R − Rad D −k L Qψ(D 2 − k 2 ) (Q −1 − ψ)Rad ψ(D 2 − k 2 ) L(1 + Qψ 2 )(D 2 − k 2 )



 ,B=



Pr −1 (D 2 − k 2 )

.

1 1

Equation (5.35) is the final generalized eigenvalue problem, where σ is the eigenvalue, and [w, ˆ Tˆ , ζˆ ]T is the eigenvector. This problem will be solved analytically in the next section.

5.2.2 Analytical Solution under Ideal Boundary Conditions The ideal BCs wˆ =

∂ 2 wˆ = 0, Tˆ = 0, ζˆ = 0 at z = 0 and z = 1, ∂z 2

(5.36)

and Eq. (5.35) allow the solutions in the following form [7] ⎤ ⎡ ⎤ w(z) ˆ a1 sin nπ z ⎣ Tˆ (z) ⎦ = ⎣ a2 sin nπ z ⎦ , n = 1, 2, 3... a3 sin nπ z ζˆ (z) ⎡

(5.37)

where n is a positive integer. Substituting the above equations into Eq. (5.35) yields ⎡

⎤⎡ ⎤ ⎡ ⎤ −Pr kn2 a1 a1 Pr (1 + ψ)k 2 /kn2 −Pr k 2 /kn2 ⎣ ⎦ ⎣ a2 ⎦ = σ ⎣ a2 ⎦ , R − Rad −kn2 −L Qψkn2 (Q −1 − ψ)Rad −ψkn2 −L(1 + Qψ 2 )kn2 a3 a3 (5.38) where kn2 = n 2 π 2 + k 2 . Equation    −Pr k 2 − σ Pr (1 + ψ)k 2 /k 2  −Pr k 2 /kn2 n n   2 2  =0 R − Rad −kn − σ −L Qψkn  −1  2 2 2  (Q − ψ)Rad −ψkn −L(1 + Qψ )kn − σ 

(5.39)

provides the relationships between dimensionless numbers, k and σ , called dispersion relation. Under ideal BCs, the task of linear stability analysis is to find R crit so that if and only if R > R crit , there are k and n making σr > 0. The most unstable mode is defined by k crit and n crit . Therefore, when R = R crit , k = k crit and n = n crit , σr = 0. The above discussion confirms (R crit , k crit , n crit ) can be determined from σr = 0. Substituting σ = 0 + iω into Eq. (5.39), the two sets of solutions for unknowns (R, ω) are obtained as R=

1 kn6 (1+ψ)2 + L −1 Q −1 + Rad , ω = 0, k 2 (1 + ψ)(1 + Qψ 2 ) + ψ L −1 (1 + ψ)(1 + Qψ 2 ) + ψ L −1

(5.40)

5.2 Linear Stability Analysis

79

  ⎧ kn6 L(1 + Qψ 2 ) 2 + Pr + L Pr −1 + L(1 + Qψ 2 ) + 1 + Pr + L Pr −1 ⎪ ⎪ ⎪ + R = ⎪ ⎪ k2 1 + Pr (1 + ψ) ⎪ ⎪ 2 ⎪ ⎪ L Q(ψ − Q −1 ) + Pr (Q −1 +1)+1 ⎪ ⎪ ⎨ Rad ,  1 + Pr (1 + ψ)2    ⎪ L 1 + Pr (1+Qψ ) ψ + L(1 + ψ)(1 + Qψ 2 ) +Pr ψ [1 + L Qψ(1 + ψ)] ⎪ 2 = −k 4 ⎪ ω ⎪ n ⎪ 1 + Pr (1+ψ) ⎪   ⎪ ⎪ 2 1 + L Q(1+ψ)(ψ − Q −1 ) (ψ − Q −1 )Pr R ⎪ k ad ⎪ ⎪ ⎩ − 2 . kn 1 + Pr (1+ψ)

(5.41) The former set has ω = 0, which means the initial growth of small perturbations is monotonic, called monotonic instability. The latter set owns ω = 0 so that small perturbations will initially grow in an oscillatory manner, called oscillatory instability. The two sets of equations provide the neutral stability conditions for two kinds of instabilities. (k crit , n crit ) can be determined by minimizing R. That is to minimize R0 = kn6 /k 2 = (n 2 π 2 + k 2 )3 /k 2 ,

(5.42)

√ which immediately leads to n crit = 1 and k crit = π/ 2. Therefore, the neutral stability conditions are finally obtained as crit = Rmono

L L(ψ + 1)2 + Q −1 R0 + Rad , (ψ + 1)Ξ1 − 1 (ψ + 1)Ξ1 − 1

(5.43)

Ξ1 + Pr + L Pr −1 1 + Pr (1 + Q −1 ) + L Q(Q −1 − ψ) Ξ1 R 0 + Rad , 1 + Pr (1 + ψ) 1 + Pr (1 + ψ) (5.44)     9π 4 L 1 + Pr (1+Qψ 2 ) ψ + L(1 + ψ)(1 + Qψ 2 ) ωcrit2 = − 4 1 + Pr (1+ψ) 4 9π Pr ψ [1 + L Qψ(1 + ψ)] − 4 1 + Pr (1+ψ)   1 1 + L Q(1+ψ)(ψ − Q −1 ) (ψ − Q −1 )Pr Rad , (5.45) − 3 1 + Pr (1+ψ) 2

crit Rosc =

where R0 = 27π 4 /4 = 657.5, and an operator Ξ(·) = L Qψ 2 +1+L(·)

(5.46)

crit crit < Rosc , the threshold of monotonic is introduced to simplify the expressions. If Rmono instability will first be overshot, which is also termed as exchange of stability [13]. Otherwise, the oscillatory instability takes place, which is also called overstability [14].

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5 Interactions Between Coupled Transfer and Gravity: Rayleigh-Bénard Instability

Fig. 5.3 Stability threshold ΔT crit calculated for C2 H6 − CO2 binary mixture at cref = 0.6721 and pref = 5.6 MPa under different temperatures and thicknesses. Note that TPB = 298.79 K

5.2.3 Stability Threshold under Various Conditions Given the reference state and the thickness d, one can obtain the physical properties and all dimensionless numbers. With the help of Eqs. (5.43) and (5.44), R crit can be calculated by crit crit , Rosc ), (5.47) R crit = min(Rmono and additionally ΔT crit (the stability threshold of ΔT ) by ΔT crit =

ν DT crit R , gβd 3

(5.48)

according to the definition of Rayleigh number. Using the above methodology, ΔT crit is calculated for pref = 5.6 MPa and 7.0 MPa under different temperatures and thicknesses. The results are shown in Fig. 5.3. The behavior of ΔT crit is greatly influenced by d. In small-thickness regime, ΔT crit has a local minimum value around TPB , so ΔT crit grows rapidly once T leaves TPB . However, an obvious transition occurs as d increases. On the one hand, in the bulk region away from TPB , ΔT crit drops drastically as d increases. On the other hand, small peaks of ΔT crit emerge around TPB , instead of previous local minimums. The same calculations are additionally carried out for pref = 7.0 MPa and presented in Fig. 5.4. As pref increases, the above-mentioned singularities around TPB still exist, but become weaker. The shift in the behavior of ΔT crit implies the physics governing ΔT crit alters as d increases. Indeed, d is a key factor governing the strength of gravity-related effects, including ATG and gravitational diffusion. Hence, it is reasonable to anticipate that the transition is a result of gaining relative superiority of gravity-related effects. This topic will be discussed in Sect. 5.3.2.

5.3 Further Discussions

81

Fig. 5.4 Stability threshold ΔT crit calculated for C2 H6 − CO2 binary mixture at cref = 0.6721 and pref = 7.0 MPa under different temperatures and thicknesses. Note that TPB = 309.55 K

5.3 Further Discussions 5.3.1 Origin of the Oscillatory Instability Oscillatory instability is an important feature of RB instability in a binary mixture, which has been reported for fluids with ψ < 0 [10–12]. That is because the oscillatory instability is a result of interactions between destabilizing temperature gradient and stabilizing concentration gradient, where the latter is originated from the Soret effect [10–12] of ψ < 0. However, this study confirms there is oscillatory instability even for ψ > 0. What is the origin of the oscillatory instability? This is a puzzle because the Soret-effect-induced concentration gradient is destabilizing for ψ > 0. This section is dedicated to answering this question based on Eqs. (5.43) and (5.44). Oscillatory instability requires crit crit > Rosc . Rmono

(5.49)

Substituting Eqs. (5.43) and (5.44) into the above equation leads to • If (Q −1 − ψ)Ξ Qψ−ψ−1 > 0: osc Rad > Rad =

[(ψ + 1)Ξ1 − 1] (Ξ1 2 − 2L) − L [(ψ + 1)Ξ1 − 1] (Pr −1 Ξ1 + 1)L R + R0 , 0 (Q −1 − ψ)Ξ1 Ξ Qψ−ψ−1 (Q −1 − ψ)Ξ1 Ξ Qψ−ψ−1

• If (Q −1 − ψ)Ξ Qψ−ψ−1 < 0: osc Rad < Rad =

(5.50)

[(ψ + 1)Ξ1 − 1] (Ξ1 2 − 2L) − L [(ψ + 1)Ξ1 − 1] (Pr −1 Ξ1 + 1)L R0 + R0 . −1 (Q − ψ)Ξ1 Ξ Qψ−ψ−1 (Q −1 − ψ)Ξ1 Ξ Qψ−ψ−1

(5.51) From the following two relations (given ψ > 0):

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5 Interactions Between Coupled Transfer and Gravity: Rayleigh-Bénard Instability

[(ψ + 1)Ξ1 − 1] (Ξ1 2 − 2L) − L = L 3 (ψ + 1)(Qψ 2 + 1)3 + L 2 ψ(Qψ 2 + 1)2 + 2Qψ 2 L 2 (ψ + 1)(Qψ 2 + 1)+ Lψ(3Qψ 2 + Qψ + 1) + ψ >0, and

[(ψ + 1)Ξ1 − 1] (Pr −1 Ξ1 + 1)L > 0,

osc is identical to that of (Q −1 − ψ)Ξ Qψ−ψ−1 (i.e., one can find that the sign of Rad the denominator). Therefore, Eq. (5.51) is unphysical since it gives Rad < 0, so only Eq. (5.50) is the condition of oscillatory instability. The premise of Eq. (5.50), (Q −1 − ψ)Ξ Qψ−ψ−1 > 0, is the necessary condition of oscillatory instability, which can be decomposed into

−

1 < ψ − Q −1 < 0. L Q(ψ + 1)

(5.52)

The above equation is equivalent to ψ 0 is attributed to the gravitational diffusion, serving as the new source for the stabilizing concentration gradient. Since oscillatory instability arouses rich dynamic phenomena in normal binary mixtures with ψ < 0 [1–5], given the present new oscillatory instability under ψ > 0, it is natural to be curious about the nonlinear convection after it occurs. Motivated by curiosity and its theoretical importance, this chapter is devoted to exploring the nonlinear dynamics of RB convection near the stability threshold by means of theoretical analyses and numerical simulations. Note that the realistic boundary conditions (BCs) (no-slip and impermeable walls) are the main concern of the present work, while the ideal BCs (stress-free walls with fixed concentrations) will be employed to facilitate theoretical analyses, serving as guides for numerical explorations. Because the expression of R crit is only known for ideal BCs, a numerical approach is developed to obtain R crit under realistic BCs. This chapter is arranged as follows. Section 6.1 describes the details about the current problem, including physical model, mathematical model, response parameters, and numerical methods. Section 6.2 is dedicated to presenting preliminary numerical

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 Z.-C. Hu, Coupled Heat and Mass Transfer in Binary Mixtures at Supercritical Pressures, Springer Theses, https://doi.org/10.1007/978-981-16-7806-6_6

89

90

6 Interactions Between Coupled Transfer and Gravity ...

results regarding the nonlinear RB convection near the threshold of oscillatory instability, where the existence of FA instability is reported. Section 6.3 elaborates the weakly nonlinear analysis under ideal BCs, confirming the dominance of FA instability in the near-threshold dynamics, and the connection between the oscillatory and FA instabilities. More importantly, the conditions for FA instability are conjectured from the weakly nonlinear theory, serving as guides for a series of numerical simulations in Section 6.4 under realistic BCs. After thorough analyses of the numerical results, the conditions for FA instability are identified, and their underlying physical mechanisms are interpreted therein. This chapter is concluded in Section 6.5.

6.1 Problem Statement 6.1.1 Physical and Mathematical Models Figure 6.1 sketches the physical model of the two-dimensional RB problem, which is a C2 H6 − CO2 mixture layer at a supercritical pressure bounded by two rigid walls heated from below and cooled from above. The horizontal and vertical directions are denoted by x and z, respectively. g = (0, −g) = (0, −9.81 m/s2 ) is the gravitational acceleration in the vector form. The no-slip and impermeable BCs are employed for the bottom and top walls. The dimensionless width of the model is 2, and the lateral walls are periodic to realize the horizontal infinity. Besides, the two walls have fixed temperatures, with a dimensionless temperature difference denoted by R.

T

g

1

z

R 2

no-slip & impermeable

Binary mixture Reference state: (Tref , pref , cref ) Periodic

2

x T

R 2

no-slip & impermeable

Fig. 6.1 The physical model employed in this study: a binary mixture layer at a supercritical pressure heated from below and cooled from above. The subscript ref is used to mark the reference state, namely (Tref , pref , cref ) for reference temperature, pressure, and concentration. g is the vector of gravitational acceleration. x and z are the coordinates of horizontal and vertical directions, respectively. Lateral walls are periodic to represent the horizontally infinite fluid layer, with the width-to-height ratio equaling 2. The bottom and top walls are no-slip and impermeable

6.1 Problem Statement

91

The dimensionless governing equations are given by Eqs. (5.21) to (5.24). The prime symbols accompanying dimensionless variables are omitted in this chapter. All later results and discussions will be presented from a dimensionless viewpoint. The realistic BCs are u = w = 0, T =

R ∂c , = Rψ − Q −1 Rad , 2 ∂z

⎫ ⎪ at z = 0 ⎪ ⎪ ⎬

⎪ ⎪ R ∂c ⎭ = Rψ − Q −1 Rad , at z = 1 ⎪ u = w = 0, T = − , 2 ∂z

.

(6.2)

6.1.2 Response Parameters In order to measure the intensity of natural convection, the response parameters, i.e., Nusselt number N u and Reynolds number Re, are introduced as 1 Nu = − R



  ∂ T  u rms , , Re =  ∂z b,avg Pr

(6.3)

where ... represents the temporal average in a (statistically) steady state, and the subscript (b, avg) means the spatial average value over one boundary. Note that when a (statistically) steady state is reached, the values of N u calculated from the two boundaries are identical. N u serves as an indicator for the heat transfer enhancement of convective motions at the (statistically) steady state. In the definition of Re, u rms is the dimensionless root-mean-square velocity, calculated by u rms =

1 2 0

0

(u 2 + w 2 )dxdz . 2

(6.4)

Here, Re works as an indicator for the strength of transient convective motions. The problem studied in this chapter can be briefly written as [N u, Re(t)] = f (R, Rad , Pr, L , ψ, Q).

(6.5)

6.1.3 Numerical Method This study involves the numerical simulations of Eqs. (5.21)–(5.24), along with initial conditions Eq. (5.25) and BCs Eq. (6.2). This system was solved using the opensource CFD toolbox OpenFOAM, which is based on finite-volume discretization. The solver has a second-order accuracy in space and first-order accuracy in time. The pressure-velocity coupling is treated using the block-coupled algorithm [6, 7].

92

6 Interactions Between Coupled Transfer and Gravity ...

The advection terms are discretized by the scheme limitedLinear, which is a second-order TVD scheme [8]. For more details about the numerical method, one can refer to [9]. The physical domain is discretized into square elements that are equally sized in x direction and locally refined in the vicinity of the bottom and top walls in z direction to resolve the physical boundary layers. Before massive computations for different parameters, the grid-independent study was performed for a typical case. Three different sizes of meshes were considered: 247 × 132, 388 × 204 and 483 × 253. Comparisons among the results for different meshes reveal the mesh of 247 × 132 is adequate. The near-wall refinement makes the largest width-to-height ratio of square elements being 450. In each simulation, variable time step is realized through controlling the maximum Courant number as 0.5. The calculation proceeds until a (statistically) steady state is reached through checking the agreement between the Nusselt numbers calculated at two boundaries. Furthermore, to study the RB convection near the stability threshold, R crit should be predicted to guide the settings of R. In this work, R crit and its accompanying angular frequency ωcrit is obtained by numerically solving the generalized eigenvalue problem derived from the linear stability analysis for Eqs. (5.21)–(5.24). The details are provided in the Appendix A.

6.2 Preliminary Numerical Simulations The initial intention of the present work is to explore the RB convection near the threshold of oscillatory instability. This section presents preliminary numerical simulations regarding this question. Such an arrangement is necessary because later explorations in Sects. 6.3 and 6.4 are actually motivated by the interesting findings reported here. The parameters considered in this section are: Rad = 4.921 × 108 , Pr = 9.483, L = 1.272, ψ = 13.647 and Q = 0.0716. The numerically predicted critical Rayleigh number is R crit = 5.028432 × 108 , with an angular frequency ωcrit > 0, corresponding to an oscillatory instability. In this section, numerical simulations were carried for (R − R crit ) ∈ {−43200, 0, 200, 26800, 56800, 356800}.

(6.6)

The values of N u for the six cases were obtained in post-processing and plotted in Fig. 6.2. In the first two cases, N u = 1 means fluid is stable with respect to small perturbations, thus no convection. In the third case, natural convection occurs. However, an jump in N u from 1 to about 1.6 is observed. Note that one usually expects that as R exceeds R crit , N u increases gradually from 1. Therefore, here the jump in N u is somehow unexpected. Then, as R increases further, the variation in N u becomes rather gradual.

6.2 Preliminary Numerical Simulations

93

Fig. 6.2 The calculated Nusselt numbers N u (denoted by circles) for the Rayleigh numbers R given in Eq. (6.6), where R crit is the numerically predicted critical Rayleigh number. The arrows indicate the direction of increasing R

Fig. 6.3 Log-log plot of the temporal evolution of Reynolds number Re for the case of (R − R crit ) = 200 to demonstrate the oscillatory growth of small perturbations. The horizontal axis is the dimensionless time t. A partially enlarged view for t > 1.85 is provided to clearly visualize the oscillations and rapid transition

To look into the temporal evolution after oscillatory instability sets in, Fig. 6.3 shows Re versus t for the case of (R − R crit ) = 200. The oscillatory growth of small perturbations is clearly seen, which lasts for a long period until t = 1.88. Then, the nonlinear effects dominate and drive the convection away from the previous linear regime, resulting in a rapid transition into an aperiodic convection state. During the rapid transition starting from t = 1.91, Re increases more than tenfold in a short period. The behavior presented in Figs. 6.2 and 6.3 suggests the nonlinear dynamics of RB convection near the stability threshold is more complex than that predicted by linear theory. Both the jump in N u and the rapid transition of Re imply the possibility of FA instability below R crit [10, 11]. That is to say, when R < R crit , the fluid is possibly unstable with respect to FA perturbations, i.e., large perturbations. To confirm this point, four additional cases were supplemented: (R − R crit ) ∈ {−2843200, −4843200, −5043200, −5053200},

(6.7)

where the fields in the fully-developed stage of (R − R crit ) = 200 were used as the initial conditions, serving as the FA perturbations. Their results are plotted in Fig. 6.4.

94

6 Interactions Between Coupled Transfer and Gravity ...

Fig. 6.4 The calculated Nusselt numbers N u (denoted by squares) for the Rayleigh numbers R given in Eq. (6.7), where R crit is the numerically predicted critical Rayleigh number. The case of (R − R crit ) = 200 is represented by the single circle on the top-right corner, and the arrows indicate the direction of decreasing R

It indicates that the convective solution branch indeed extends leftward until (R − R crit ) = −5053200, where the FA perturbations decay and a pure diffusive solution recovers. In other words, in the parametric range of (R − R crit ) ∈ (−5053200, 0), the fluid is stable with respect to small perturbations, while unstable to FA ones. The above interesting findings pose the necessity to explore several questions regarding FA instability below R crit : 1. What is the condition for the FA instability? 2. What is the physical mechanism for the FA instability? 3. What is the relationship between oscillatory instability and FA instability? Answering these questions is the motivation of Sects. 6.3 and 6.4. To be specific, later explorations start analytically in Sect. 6.3 by virtue of ideal BCs. The conclusions in Sect. 6.3 serve as guides for numerical explorations in Sect. 6.4 under realistic BCs.

6.3 Weakly Nonlinear Theory Surveying the existence of FA instability below R crit needs exploring the bifurcation of governing equations at R crit , which falls into the nonlinear dynamics of partial differential equations. From a mathematical point of view, the existence of FA instability below R crit is a consequence of the so-called backward bifurcation (or subcritical bifurcation) [11, 12]. The previous chapter has revealed that R crit is equal to the smaller one between crit crit . In this section, the ideal BCs will be employed to facilitate theoretical Rmono and Rosc crit will be studied by means of truncated Fourier analysis. The bifurcation at Rmono expansion, with particular attention paid on the conditions of backward bifurcation. crit crit on Rosc will be analyzed. Then, the influences of the backward bifurcation at Rmono As will been seen later in Sect. 6.4, the conclusions drawn from this section are indeed adequate to guide the numerical simulations to fully resolve the three questions raised at the end of Sect. 6.2 for realistic BCs.

6.3 Weakly Nonlinear Theory

95

Introduce the stream function θ as u=

∂θ ∂θ ,w = − . ∂z ∂x

(6.8)

Then using the Jacobian determinant J (A, B) =

∂A ∂B ∂A ∂B ∂(A, B) = − , ∂(x, z) ∂ x ∂z ∂z ∂ x

(6.9)

and the auxiliary variable ζ = c + ψ T , the governing Eqs. (5.21)–(5.24) are finally transformed into 1 ∂∇ 2 θ 1 ∂ζ ∂T = J (θ, ∇ 2 θ ) + ∇ 4 θ − (1 + ψ) + , Pr ∂t Pr ∂x ∂x ∂T ∂θ =J (θ, T ) + Rad + ∇ 2 T + L Qψ∇ 2 ζ, ∂t ∂x ∂ζ ∂θ =J (θ, ζ ) + ψ Rad + ψ∇ 2 T + L(1 + Qψ 2 )∇ 2 ζ, ∂t ∂x

(6.10) (6.11) (6.12)

where the continuity equation has been employed to eliminate pressure terms in moment equations. To facilitate theoretical analysis, ideal stress-free walls with fixed temperature and concentration are considered here: ⎫ Q −1 Rad ∂ 2θ R ⎪ , at z = 0 ⎪ θ = 2 = 0, T = , ζ = ⎪ ⎬ ∂z 2 2 . (6.13) ⎪ ⎪ ∂ 2θ R Q −1 Rad ⎪ θ = 2 = 0, T = − , ζ = − , at z = 1 ⎭ ∂z 2 2 The motionless base state is

  1 1 , ζ¯ = −Q −1 Rad z − . θ¯ = 0, T¯ = −R z − 2 2

(6.14)

For this system, the previous chapter yields the expressions for two critical Rayleigh numbers crit Rmono =

L L(ψ + 1)2 + Q −1 R0 + Rad , (ψ + 1)Ξ1 − 1 (ψ + 1)Ξ1 − 1

(6.15)

Ξ1 + Pr + L Pr −1 1 + Pr (1 + Q −1 ) + L Q(Q −1 − ψ) Ξ1 R 0 + Rad , 1 + Pr (1 + ψ) 1 + Pr (1 + ψ) (6.16) 2

crit Rosc =

with R0 = (n 2 π 2 + k 2 )3 /k 2 ,

(6.17)

96

6 Interactions Between Coupled Transfer and Gravity ...

Ξ(·) = L Qψ 2 +1+L(·),

(6.18)

where k is the horizontal wave number, n is a positive integer. Consider the following truncated Fourier representation [12–16]: θ (x, z, t) =θ¯ + θ1 (t) sin kx sin π z, T (x, z, t) =T¯ (z) + T1 (t) cos kx sin π z + T2 (t) sin 2π z,

(6.19)

ζ (x, z, t) =ζ¯ (z) + ζ1 (t) cos kx sin π z + ζ2 (t) sin 2π z. Physically speaking, the above expansions decompose the solutions into three parts: the base state (denoted by overlines), columnar vortices, and advection-induced distortions of mean fields [12]. Note that the third part is only considered for T and ζ . The main advantage of this truncated representation is that it allows an analytical treatment of the equations, which in turn provides a guide for further numerical explorations. Substituting Eq. (6.19) into Eqs. (6.10)–(6.12) and using Galerkin projection, a set of ordinary differential equations of amplitude coefficients (θ1 , T1 , T2 , ζ1 , ζ2 ) is obtained: ∂θ1 ∂t ∂ T1 ∂t ∂ T2 ∂t ∂ζ1 ∂t ∂ζ2 ∂t

= − Pr (k 2 + π 2 )θ1 − Pr

k [(1 + ψ)T1 − ζ1 ] , k2 + π 2

(6.20)

= − (k 2 + π 2 )T1 − (k 2 + π 2 )L Qψζ1 − k(R − Rad )θ1 − kπ θ1 T2 ,

(6.21)

πk θ1 T1 , 2

(6.22)

= − 4π 2 T2 − 4π 2 L Qψζ2 +

= − (k 2 + π 2 )L(Qψ 2 + 1)ζ1 − (k 2 + π 2 )ψ T1 − k(Q −1 − ψ)Rad θ1 − kπ θ1 ζ2 , (6.23) = − 4π 2 L(Qψ 2 + 1)ζ2 − 4π 2 ψ T2 +

πk θ1 ζ1 . 2

(6.24)

At the steady state (∂θ1 /∂t = ∂ T1 /∂t = ∂ T2 /∂t = ∂ζ1 /∂t = ∂ζ2 /∂t = 0), the above ordinary differential equations degenerate into five algebraic equations, which can be further combined leading to a quadratic equation of θ12 :  R0

k 2 θ12 π 2 + k2 8

2



− R − (Ξ1 2 − 2L)R0 − Ξ Q −1 −2ψ Rad



k 2 θ12 π 2 + k2 8



crit ) = 0. −L [(ψ + 1)Ξ1 − 1] (R − Rmono (6.25) crit It can been seen from Eq. (6.25) that when R = Rmono , θ12 = 0 is a solution. Therecrit , fore, Eq. (6.25) actually represents the branch of solution emanated from Rmono which can be consequently employed to investigate the nature of bifurcation at R = crit crit . The conditions for backward bifurcation at R = Rmono will be investigated Rmono next.

6.3 Weakly Nonlinear Theory

97

crit 6.3.1 Conditions for Backward Bifurcation at R = Rmono

The solution of Eq. (6.25) can be represented by k 2 R0 θ12 = A(R) ± π 2 + k2 4

 A(R)2 + B(R),

(6.26)

where A(R) =R − Ξ Q −1 −2ψ Rad + (2L − Ξ1 2 )R0 ,

(6.27)

B(R) =4R0 L [(ψ + 1)Ξ1 − 1] (R −

(6.28)

crit Rmono ).

crit , there is real positive solution for Backward bifurcation requires when R < Rmono crit . Eq. (6.26). Given ψ > 0 and Ξ1 > 1, Eq. (6.28) implies B(R) < 0 when R < Rmono Therefore, A(R) must be positive to allow a real positive solution of Eq. (6.26):

A(R) > 0 ⇒ R > Ξ Q −1 −2ψ Rad + (2L − Ξ1 2 )R0 .

(6.29)

crit , one arrives at the conditions for backward bifurcation Combining with R < Rmono crit > (Ξ1 2 − 2L)R0 + Ξ Q −1 −2ψ Rad . Rmono

(6.30)

Using Eq. (6.15), Eq. (6.30) can be further decomposed into ψ Rad shown in diagram 4 to 5. As Rad further increases, Eqs. (6.40) and (6.41) suggest, o.f. crit will first tends to Rf.a. , coincide with Rf.a. at Rad = Rad , and then separates. Rosc This process denotes the transition from diagram 5 to 7. For the sake of clarity, the details of each bifurcation diagram are shown in Table 6.1. 2. When at least one of Eqs. (6.31) and (6.32) is failed to be met, there is no oscillatory instability and FA instability below R crit . The bifurcation diagram is always like diagram 1 in Fig. 6.5. Table 6.1 Summary of the bifurcation diagrams presented in Fig. 6.5 No. Range of Rad Small perturbations FA perturbations below R crit 1 2 3 4 5 6 7

f.a. Rad < Rad f.a. Rad = Rad f.a. osc Rad < Rad < Rad osc Rad = Rad o.f. osc < R Rad ad < Rad o.f. Rad = Rad o.f. Rad > Rad

Monotonic instability Monotonic instability Monotonic instability Codimension-2 point Oscillatory instability Oscillatory instability Oscillatory instability

Stable Stable Unstable unstable Unstable Stable Unstable

6.3 Weakly Nonlinear Theory

101

It should be emphasized that the weakly nonlinear analysis performed in this section is based on ideal BCs and severely truncated representation. Besides, only crit is studied. Due to these limitations, the bifurcation branch emanated from R = Rmono one can hardly expect a real system under realistic BCs can behave exactly as Fig. 6.5. However, the current theory still provides valuable information to guide numerical simulations to answer the questions raised at the end of Sect. 6.2. The subsequent section will exploit the results here as a guide to depict various bifurcation diagrams under different parametric conditions via numerical simulations. Meanwhile, the conditions for FA instability will also be elaborated.

6.4 Theory-guided Numerical Simulations and Discussions The above weakly nonlinear analysis reveals that the backward bifurcation at crit plays an important role in the dynamics of convection near threshold. When Rmono crit crit , the backward bifurcation at Rmono is conditional (Eqs. 6.31–6.33 R crit = Rmono crit (because Eqs. 6.31–6.33 should be satisfied), while it is definite when R crit = Rosc crit are automatically satisfied). Besides, as long as the backward bifurcation at Rmono occurs, there is always FA instability in the parametric range of Rf.a. ≤ R < R crit . Note that these conclusions are obtained from a severely truncated representation and ideal BCs. In this section, they provide a guide for numerical exploration of the near-threshold dynamics of RB convection under realistic BCs. To be specific, Eq. (6.33) can be met as long as Rad is large enough (corresponding to increase the thickness of the fluid layer), and in Fig. 6.5, the local bifurcation around R crit has been divided into two scenarios depending on whether the necessary conditions, i.e., Eqs. (6.31) and (6.32), are fully satisfied or not. Therefore, the numerical simulations presented in this study are also divided into two parts depending on whether the necessary conditions are partially or fully satisfied, where Rad is set as some typical values. Inspired by Sect. 6.2, the bifurcation branch is depicted using a two-step method: forward detection and backward detection. In forward detection, numerical simulations start from motionless base state, i.e. Eq. (5.25). Small random perturbations are added to the temperature field. The forward detection is performed for a series of Rayleigh numbers around the numerically predicted R crit . In backward detection, the stability of FA perturbations is surveyed. Backward detection starts from the minimum R observed in forward detection with N u > 1, whose fully-developed flow field is employed as the initial conditions. Then a series of smaller R is employed to perform numerical simulations.

6.4.1 Results under Partially Satisfied Necessary Conditions Table 6.2 lists the parameters considered in this section. The dimensionless parameters in the fourth group (No. IV) are selected to fulfill Eq. (6.32) but not Eq. (6.31).

102

6 Interactions Between Coupled Transfer and Gravity ...

Table 6.2 Summary of parameters of groups I to IV No. Rad Pr L I II III IV

Rad Rad Rad Rad

4.991 × 105

= = 7.985 × 106 = 4.043 × 107 = 3.119 × 108

12.036 12.036 12.036 12.036

6.000 6.000 6.000 13.536

(a)

ψ

Q

0.803 0.803 0.803 0.803

1.000 1.000 1.000 1.292

(b) C3

C2

C1

C3 C2 C1

R crit Fig. 6.6 The bifurcation diagram for group I in Table 6.2. a Bifurcation diagram: the horizontal axis is Rayleigh number R, and the vertical one is Nusselt number N u. There are only monotonic instabilities for small perturbations, indicated by ↑, and no FA instability below R crit . R crit is the numerically predicted stability threshold. b Log-log plot for the temporal evolution of convection: the horizontal axis is time t, and the vertical axis is the Reynolds number Re. Ci denotes the ith typical case, see Table 6.3 for details

On the contrary, parameters in other three groups (No. I, II and III) are modified directly from group IV to satisfy Eq. (6.31) but not Eq. (6.32). The modifications include changing L from 13.536 to 6.0, and Q from 1.292 to 1.0. Besides, a series of gradually increased Rad is adopted for groups I–III. Figures 6.6, 6.8-6.10 present the bifurcation diagrams obtained from the two-step depiction procedure. These diagrams will be explained in the ensuing paragraphs. For the sake of later reference, Table 6.3 summarizes the details of typical cases in Figs. 6.6, 6.8-6.10. Figure 6.6 presents the bifurcation diagram for group I, in which Fig. 6.6a is the bifurcation diagram, and Fig. 6.6b shows the temporal evolution of convection for three typical cases. In forward detection procedure, small perturbations decay for R < R crit , and increase monotonically for R > R crit , finally evolving into fully-developed convective states. For a Rayleigh number between cases C1 and C2 , convection will finally enter a steady state, as shown in Fig. 6.6b. On the right side of C2 , the convection undergoes a transition from steady state, periodic state and finally the aperiodic state, as demonstrated by Fig. 6.7. As shown in Fig. 6.6a, N u increases crit . There from 1 in forward detection, suggesting a forward bifurcation at R crit = Rmono crit is no FA instability below R . Above all, the results of group I agree well with the

6.4 Theory-guided Numerical Simulations and Discussions

103

Table 6.3 Summary of typical cases in Figs. 6.6, 6.8-6.10. N u is the Nusselt number, and Re is the average Reynolds number at (statistically) steady state. In the third column, two arrows are used to indicate the stability with respect to small perturbations (left) and FA perturbations (right), with (↑) for unstable and (↓) for stable. The subscript mono means monotonic instability. Ci denotes the ith typical case. For small perturbations, there is no oscillatory instability observed in the current four groups No. R Stability Nu Re Reference C1 C2 C3 C4 C5 C6 C7 C8 C9 C10 C11 C12 C13 C14 C15 C16

553000 590000 640000 8810000 8814000 8815000 8810000 8805000 44584000 44589000 44590000 43800000 43560000 306640000 307200000 309000000

Fig. 6.7 The variation of fully-developed convection states along with Rayleigh number R. The three cases R = 55500000, 62000000 and 6400000 are extracted from the bifurcation diagram shown in Fig. 6.6, corresponding to steady state, periodic state and aperiodic state

(↑mono ) (↑) (↑mono ) (↑) (↑mono ) (↑) (↑mono ) (↑) (↑mono ) (↑) (↑mono ) (↑) (↓) (↑) (↓) (↓) (↑mono ) (↑) (↑mono ) (↑) (↑mono ) (↑) (↓) (↑) (↓) (↓) (↑mono ) (↑) (↑mono ) (↑) (↑mono ) (↑)

1.004 1.323 1.776 1.0002 1.0039 1.2846 1.2473 1 1.0000 1.0006 1.8584 1.5479 1 1.0014 1.0052 1.0201

0.4740 4.1294 7.8601 0.2156 0.97604 10.1472 9.2593 0 0.1397 0.4645 42.8177 28.0122 0 2.6731 5.8332 13.7139

Fig. 6.6 Fig. 6.6 Fig. 6.6 Fig. 6.8 Fig. 6.8 Fig. 6.8 Fig. 6.8 Fig. 6.8 Fig. 6.9 Fig. 6.9 Fig. 6.9 Fig. 6.9 Fig. 6.9 Fig. 6.10 Fig. 6.10 Fig. 6.10

R = 6400000

R = 6200000 R = 5550000

weakly nonlinear theory: When at least one of Eqs. (6.31) and (6.32) is failed to be met, the bifurcation diagram is like diagram 1 in Fig. 6.5. The parameters for group II are obtained after increasing Rad on the basis of group I. Figure 6.8 presents the depicted bifurcation diagrams. Like the forward detection for group I, small perturbations are found to grow monotonically as long as R > crit . As shown in Fig. 6.8 (a), there is a solution branch confined between R crit = Rmono 4 5 C and C , along which N u increases gradually from 1, obviously corresponding to

104

6 Interactions Between Coupled Transfer and Gravity ...

(a)

(b) C5

C5

C4 C4

R crit (c)

(d) C7

C6 C7

C6 C8

C4

C8

C5

R crit



Fig. 6.8 The bifurcation diagram for group II in Table 6.2. a, c The bifurcation diagrams: the horizontal axis is the Rayleigh number R, and the vertical axis is Nusselt number N u. For small perturbations, there are only monotonic growths, denoted by ↑. Besides, there is FA instability below R crit , where the evolution of FA perturbations are denoted by . R crit is the stability threshold for small perturbations. b, d Log-log plots for the temporal evolution of convection: the horizontal axis is time t, and the vertical axis is the Reynolds number Re. Ci denotes the ith typical case, see Table 6.3 for details

a forward bifurcation emanated from R crit , similar to Fig. 6.6a. However, on the right side of C5 , a sudden increase in the amplitude of convection is observed, resulting in a jump in N u, see C6 in Fig. 6.8c for instance. On the right side of C6 , N u increases gradually with R. Therefore, an additional branch of solutions is actually depicted in the forward detection, which should emanate from a Rayleigh number beyond R crit , corresponding to a backward bifurcation. Besides, the solutions on this new branch own higher flow intensities, as reflected by the comparisons between Fig. 6.8b and d. In backward detection, the fully-developed flow fields of C6 are employed as the initial conditions. As shown in Fig. 6.8c, the upper branch continues extending leftward until returning to motionless state at C8 . Above all, two bifurcation branches are depicted for group II. The first is the result crit , and the second one is caused by the backward of forward bifurcation at R crit = Rmono

6.4 Theory-guided Numerical Simulations and Discussions

(a)

105

(b) C10

C10

C9

C9

R crit (c)

(d)

C12

C11 C11

C12 C13 C13

C9,10

R crit

Fig. 6.9 The bifurcation diagram for group III in Table 6.2. See the caption of Fig. 6.8 for more details

bifurcation of a Rayleigh number beyond R crit . Due to the second bifurcation branch, there is FA instability below R crit (between C8 and C4 in Fig. 6.8c). The parameters of group III are obtained as Rad is further increased on the basis of group II, whose bifurcation diagrams are shown in Fig. 6.9. Compared with Fig. 6.8, the two groups have identical bifurcation structure near R crit , consisting of a forward crit (Fig. 6.9a), and a backward bifurcation emanating bifurcation originated from Rmono crit . Besides, there is FA instability between C13 from a Rayleigh number beyond Rmono and C9 . The difference is the range and intensity of FA instability are enhanced due to increased Rad . Figure 6.10 exhibits the bifurcation diagram for group IV. Results suggest when Eq. (6.31) holds but Eq. (6.32) doesn’t, there is only one solution branch as a forward crit . There is no FA instability below R crit . Therefore, the results bifurcation from Rmono of group IV agree well with the weakly nonlinear theory: The bifurcation diagram is like diagram 1 in Fig. 6.5. Table 6.4 summarizes the main results of current four groups of numerical simulations. Consistent with previous linear theory, only monotonic instabilities of small crit . In general, when Eq. (6.31) holds perturbations are observed, i.e., R crit = Rmono

106

6 Interactions Between Coupled Transfer and Gravity ...

(a)

(b) C16 C15 C14

C16

C14

C15

R crit Fig. 6.10 The bifurcation diagram for group IV in Table 6.2. See the caption of Fig. 6.6 for more details Table 6.4 Summary of numerical results for groups I to IV No. ψ < Q −1 L + Lψ < Rad 1 + L Qψ + L Qψ 2 Small Medium Large I II III IV

   ×

× × × 

   

Theoretical

Numerical

Fig. 6.5 -(1) Fig. 6.5 -(1) Fig. 6.5 -(1) Fig. 6.5 -(1)

Fig. 6.6 (a) Fig. 6.8 (c) Fig. 6.9 (c) Fig. 6.10 (a)

and Eq. (6.32) does not hold, there is always a forward bifurcation emanating from R crit , consistent with weakly nonlinear analysis. However, in reality, when Rad is large enough, a subcritical bifurcation branch will emerge from a Rayleigh number beyond R crit , extend leftward and result in FA instability below R crit . Because only the bifurcation at R crit is analyzed in the current theory, the second upper bifurcation branch has not been discussed in Fig. 6.5, which leads to the failure in correct prediction of FA instability below R crit . In other words, the existence of FA instability below R crit relies on Eq. (6.31) and a sufficiently large Rad . That explains the results for group IV: When Eq. (6.31) does not hold, there is no FA instability below R crit even though Rad is sufficiently large.

6.4.2 Results under Fully Satisfied Necessary Conditions This section presents and discusses the numerical results when both Eqs. (6.31) and (6.32) hold. For this end, three groups of numerical simulations were performed, whose parameters referred to as groups V, VI and VII are summarized in Table 6.5. From group V to VII, Rad increases gradually. Figures 6.11–6.13 show the results of

6.4 Theory-guided Numerical Simulations and Discussions Table 6.5 Summary of parameters of groups V to VII No. Rad Pr L V VI VII

7.874 × 105

Rad = Rad = 1.260 × 107 Rad = 6.378 × 107

9.483 9.483 9.483

1.272 1.272 1.272

107

ψ

Q

13.647 13.647 13.647

0.0716 0.0716 0.0716

(b)

(a)

C19

C17

R

C19 C18

C18

C17

crit

Fig. 6.11 The bifurcation diagram for group V in Table 6.5. Ci denotes the ith typical case, see Table 6.6 for details. See the caption of Fig. 6.6 for more details

(a)

(b) C21

C21 C C22

20

C20

C22

R crit



Fig. 6.12 The bifurcation diagram for group VI in Table 6.5. a Bifurcation diagram: the horizontal axis is Rayleigh number R, and the vertical one is Nusselt number N u. There are only monotonic instabilities for small perturbations, indicated by ↑. Besides, there is FA instability below R crit , where the evolution of FA perturbations are denoted by . R crit is the numerically predicted stability threshold. b Log-log plot for the temporal evolution of convection: the horizontal axis is time t, and the vertical axis is the Reynolds number Re. Ci denotes the ith typical case, see Table 6.6 for details

numerical bifurcation detections. For the sake of clarity, eleven typical cases denoted by C17 − C27 are summarized in Table 6.6. Figure 6.11 presents the results of group V, where Fig. 6.11a is the bifurcation diagram, and Fig. 6.11b presents the evolution of flow intensity for selected cases

108

6 Interactions Between Coupled Transfer and Gravity ...

(a)

(b)

C26

C25

C25 C23,24 C26 C27

C24 C23 C27

R crit



Fig. 6.13 The bifurcation diagram for group VII in Table 6.5. a Bifurcation diagram: the horizontal axis is Rayleigh number R, and the vertical one is Nusselt number N u. There are monotonic and oscillatory instabilities for small perturbations, indicated by ↑ and , respectively. Besides, there is FA instability below R crit , where the evolution of FA perturbations are denoted by . R crit is the numerically predicted stability threshold. b Log-log plot for the temporal evolution of convection: the horizontal axis is time t, and the vertical axis is the Reynolds number Re. C i denotes the i th typical case, see Table 6.6 for details 

Table 6.6 Summary of typical cases in Figs. 6.11–6.13. N u is the Nusselt number, and Re is the average Reynolds number at (statistically) steady state. In the third column, two arrows are used to indicate the stability with respect to small perturbations (left) and FA perturbations (right), with (↑) for unstable and (↓) for stable. The subscript mono means monotonic instability, and osc for oscillatory instability. Ci denotes the ith typical case No. R Stability Nu Re Reference C17 C18 C19 C20 C21 C22 C23 C24 C25 C26 C27

804833 809000 830000 12873987 12872000 12870000 65171873 65173250 65700000 65000000 64898000

(↑mono ) (↑) (↑mono ) (↑) (↑mono ) (↑) (↑mono ) (↑) (↓) (↑) (↓) (↓) (↑osc ) (↑) (↑osc ) (↑) (↑mono ) (↑) (↓) (↑) (↓) (↓)

1.0006 1.0277 1.2142 1.0295 1.0174 1 1.1976 1.2084 1.4862 1.0967 1

0.2887 1.4270 4.5410 6.3095 5.2854 0 33.4890 35.6803 57.0514 25.7133 0

Fig. 6.11 Fig. 6.11 Fig. 6.11 Fig. 6.12 Fig. 6.12 Fig. 6.12 Fig. 6.13 Fig. 6.13 Fig. 6.13 Fig. 6.13 Fig. 6.13

along with time. Only monotonic instabilities of small perturbations are observed, crit . Based on the experiences from Fig. 6.6, one can directly tell that so R crit = Rmono the depicted branch is a result of forward bifurcation originated from R crit , along which N u increases gradually from 1. From C17 to C19 , influenced by the increasing nonlinearity, final flow states undergo a transition from steady state to aperiodic state. The bifurcation behavior differs as Rad increases. Figure 6.12 shows the bifurcation diagram for group VI. Only monotonic growths of small perturbation are found

6.4 Theory-guided Numerical Simulations and Discussions Table 6.7 Summary of numerical results for groups V to VII No. ψ < Q −1 L + Lψ < Rad 1 + L Qψ + L Qψ 2 Small Medium Large V VI VII

  

  

  

109

Theoretical

Numerical

Fig. 6.5 -(1) Fig. 6.11 Fig. 6.5 -(3) Fig. 6.12 Fig. 6.5 -(5), (7) Fig. 6.13

crit in forward detection, so R crit = Rmono . Compared with Fig. 6.11, there is a sudden crit increase in N u at R (see Fig. 6.12a). In backward detection, this solution branch continues to extend until C22 . These features confirm a backward bifurcation occurs at R crit , leading to FA instability below R crit . Moreover, only aperiodic convection is observed over the whole branch. Indeed, the transition from Fig. 6.11 to Fig. 6.12 is consistent with the shift from diagram 1 to diagram 3 in Fig. 6.5. Figure 6.13 shows the bifurcation diagram for group VII. The bifurcation behavior differs again as Rad is further increased. The biggest difference is at R crit , oscillatory growth of small perturbations is identified, see C23 in Fig. 6.13b. However, as the oscillatory growth develops to some extent, a huge jump in flow intensity occurs. The flow rapidly enters a new aperiodic state, similar to Fig. 6.3. As R increases, the time for oscillatory growth is substantially reduced, as indicated by the shift from C23 to C24 in Fig. 6.13b. For all the cases on the right side of C24 , small perturbations tend to grow monotonically, like C25 in Fig. 6.13b. Besides, there is FA instability below R crit , confined by C27 and C23 in Fig. 6.13a. Similar to Fig. 6.12a, the solution branch depicted in Fig. 6.13a only consists of aperiodic flow states. Table 6.7 briefly summarizes the main results for groups V, VI and VII. Indeed, these results suggest when Eqs. (6.31) and (6.32) hold, as Rad increases, the bifurcacrit changes from a forward type to a backward one. Besides, the backward tion at Rmono crit occurs prior to oscillatory instability. Moreover, when oscillabifurcation at Rmono crit crit extends below Rosc , resulting in tory instability occurs, the solution branch of Rmono crit crit FA instability below R = Rosc . These conclusions agree well with the theoretical predictions of Fig. 6.5. Therefore, the numerical simulations for groups V, VI and VII validate the main conclusions obtained from weakly nonlinear theory.

6.4.3 Physical Explanation for the Conditions of FA Instability below Rcrit The above seven groups of numerical simulations suggest when Eq. (6.31) holds and Rad is large enough, there is FA instability below R crit . Mathematically, there are two situations. (i) When Eq. (6.32) does not hold, FA instability is caused by the backward crit . (ii) When Eq. (6.32) holds, bifurcation of a Rayleigh number large than R crit = Rmono crit , no matter R crit = FA instability is induced by the backward bifurcation at Rmono crit crit crit Rmono or R = Rosc . This section discusses the underlying physical mechanisms.

110

6 Interactions Between Coupled Transfer and Gravity ...

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

Fig. 6.14 Horizontally and temporally averaged profiles (dashed curves) of (a,c,e,g) temperature and (b,d,f,h) concentration for (a,b) C7 , (c,d) C12 , (e,f) C21 and (g,h) C26 , along with their initial linear profiles (solid curves). The vertical axis is the coordinate z. Horizontal axis for the four figures in the left column is the temperature after compensating ATG, i.e., T + (z − 1/2)Rad . The horizontal axis for the four figures in the right column is concentration. The maximum and minimum values in the dashed profiles are denoted by max and min, respectively. In all of these cases, temperature is the destabilizing factor, and concentration is the stabilizing one

6.4 Theory-guided Numerical Simulations and Discussions

111

Let us start from several typical cases of FA instability below R crit . Figure 6.14 plots the horizontally and temporally averaged profiles (only functions of z, dashed curves) of temperature and concentration for C7 , C12 , C21 and C26 , along with their initial linear profiles (solid curves). Note that temperature profiles are plotted after compensating ATG, i.e., T + (z − 1/2)Rad . According to the linear profiles, the common characteristic of these typical cases is that the temperature is the destabilizing factor, and the concentration is the stabilizing one. Moreover, in a fully-developed convective state, the overall temperature difference (equaling maximum value minus minimum value) remains unchanged or slightly increased, but the overall concentration difference is apparently reduced. In other words, convective motions almost do not influence the strength of destabilizing effect but reduce the stabilizing one remarkably, corresponding to a buoyancy-release mechanism. This finding is very helpful to understand two typical situations encountered in numerical simulations. (i) Rapid transition to an aperiodic state, e.g., C6 , C11 , C20 and C23 , observed in forward detection. Note that in forward detection, a calculation is initialized by the diffusive base state, where the buoyancy is just enough to trigger convection. As the amplitude of convection increases, the buoyancy stored in concentration gradient is gradually released, causing the rapid transition into the high-intensity aperiodic state. (ii) FA instability below R crit , e.g., C7 , C12 , C21 and C26 , observed in backward detection. Note that in backward detection, a calculation is initialized by the fully-developed convective state of a higher Rayleigh number, where the buoyancy in concentration gradient has already been effectively released. That is why a convective state can maintain. It is concluded from above discussions that FA instability below R crit requires a large enough stabilizing concentration gradient in the base state when R = R crit . Now, let us discuss why this requirement is satisfied when Eq. (6.31) holds and Rad is large enough. Recall that the concentration gradient in the motionless base state when R = R crit is ψ R crit − Q −1 Rad (see Eq. 5.25). If Eq. (6.31) holds, there is R crit > Rad due to (ψ − Q −1 )Rad < 0. If Rad is small, R crit /Rad  1, leading to R crit Q −1 > 1,  Rad ψ

(6.42)

namely ψ R crit − Q −1 Rad > 0. So concentration indeed yields a destabilizing effect when R = R crit . Only when Rad is large enough, there is Q −1 R crit > > 1, ψ Rad

(6.43)

or equivalently ψ R crit − Q −1 Rad < 0, there can be stabilizing effect of concentration when R = R crit , giving birth to the rapid transition to aperiodic state and FA instability below R crit . In summary, FA instability below R crit requires a large enough stabilizing concentration gradient at R crit , which is equivalent to Eq. (6.31) along with a sufficiently large Rad . Besides, the underlying mechanism of FA instability is attributed to the buoyancy-release mechanism of convective motions.

112

6 Interactions Between Coupled Transfer and Gravity ...

6.5 Conclusions Through weakly nonlinear analysis and a series of numerical simulations, the nonlinear dynamics of RB convection near the stability threshold is studied for a binary mixture at supercritical pressures, which is featured by cross-diffusion effects (Soret and Dufour effects under positive separation ratio) and gravity-related effects (ATG and gravitational diffusion), allowing an oscillatory instability. This work confirms the near-threshold RB convection is deeply influenced by a FA instability mechanism. Once it happens, a rapid transition in flow intensity takes place when R is slightly larger than R crit . Then reducing R below R crit , the convective state can remain stable. In fact, the FA instability requires ψ < Q −1 and a sufficiently large Rad , which imply a sufficiently large stabilizing concentration gradient. crit , i.e., oscillatory instability. Such conditions are naturally satisfied when R crit = Rosc Therefore, when oscillatory instability sets in, small perturbations undergo initial oscillations and then rapidly evolve into an intense aperiodic flow state. However, crit , i.e., monotonic instability, the FA instability is conditional. when R crit = Rmono Physically speaking, FA instability is attributed to the reduction effect of convective motions on the stabilizing concentration gradient, leading to a release in buoyancy. Indeed, ψ < Q −1 is a prerequisite for the existence of a stabilizing concentration gradient, while Rad controls its strength. A stronger stabilizing concentration gradient means a higher potential in buoyancy release, thus being more susceptible to FA instability. Mathematically, this chapter also provides the bifurcation diagrams around R crit . crit . Solutions around R crit are determined by the First, when ψ > Q −1 , R crit = Rmono crit forward bifurcation at Rmono , as shown in Fig. 6.10a). Second, when ψ < Q −1 and crit . As Rad increases, solutions around (L + Lψ > 1 + L Qψ + L Qψ 2 ), R crit = Rmono crit crit , and then additionR are first solely determined by the forward bifurcation at Rmono crit , ally determined by the backward bifurcation at a Rayleigh number higher than Rmono whose bifurcation diagram exhibits a transition from Figs. 6.6a to 6.9c. Finally, when ψ < Q −1 and (L + Lψ < 1 + L Qψ + L Qψ 2 ), as Rad increases, R crit undergoes crit crit crit to Rosc . Besides, the bifurcation at Rmono changes from a a transition from Rmono forward type to a backward one, and always determines the solution around R crit , crit crit or R crit = Rosc . Correspondingly, the bifurcation diagram no matter R crit = Rmono evolves from Figs. 6.11a to 6.12a, and finally to Fig. 6.13a. As an extension of Chap. 5, this chapter preliminary addresses the intriguing nonlinear dynamics around R crit . Since numerical simulation is the main methodology in the present work, only stable parts of the solution branches were obtained and depicted in the bifurcation diagrams. Future works could investigate this problem using the continuation method to depict full bifurcation diagrams and give more insights into the nonlinear dynamics. The other promising future direction is investigating turbulent RB convection in binary mixtures at supercritical pressures with strong variable properties, which contributes to understanding the roles of crossdiffusion effects in supercritical heat transfer.

References

113

References 1. Batiste O, Knobloch E (2005) Simulations of localized states of stationary convection in He3 − He4 mixtures. Phys Rev Lett 95(24):244501 2. Knobloch E (2015) Spatial localization in dissipative systems. Ann Rev Condensed Matter Phys 6(1):325–359 3. Kolodner P (1991) Drifting pulses of traveling-wave convection. Phys Rev Lett 66:1165–1168 4. Kolodner P, Surko CM (1988) Weakly nonlinear traveling-wave convection. Phys Rev Lett 61:842–845 5. Mercader I, Batiste O, Alonso A, Knobloch E (2013) Travelling convectons in binary fluid convection. J Fluid Mech 722:240–266 6. Darwish M, Sraj I, Moukalled F (2009) A coupled finite volume solver for the solution of incompressible flows on unstructured grids. J Comput Phys 228(1):180–201 7. Jareteg K (2012) Block coupled calculations in OpenFOAM. CFD with OpenSource software. Chalmers University of Technology, Goteborg 8. Sweby PK (1984) High resolution schemes using flux limiters for hyperbolic conservation laws. SIAM J Numer Anal 21(5):995–1011 9. Hu ZC, Zhang XR (2017) An improved decoupling algorithm for low Mach number nearcritical fluids. Comput Fluids 145:8–20 10. Hu ZC, Lv W, Zhang XR (2019) Detour induced by the piston effect in the oscillatory doublediffusive convection of a near-critical fluid. Phys Fluids 31(7):074107 11. Huppert HE, Turner JS (1981) Double-diffusive convection. J Fluid Mech 106:299–329 12. Veronis G (1965) On finite amplitude instability in thermohaline convection. J Mar Res 23(1):1– 17 13. Malashetty M, Biradar BS (2012) Linear and nonlinear double-diffusive convection in a fluidsaturated porous layer with cross-diffusion effects. Transp Porous Media 91(2):649–675 14. Mamou M, Vasseur P, Hasnaoui M (2001) On numerical stability analysis of double-diffusive convection in confined enclosures. J Fluid Mech 433:209 15. Platten J, Chavepeyer G (1975) Finite amplitude instability in the two-component Bénard problem. Adv Chem Phys 32:281–322 16. Pritchard D, Richardson CN (2007) The effect of temperature-dependent solubility on the onset of thermosolutal convection in a horizontal porous layer. J Fluid Mech 571:59–95

Part III

Coupling Through Boundary Reactions: Buoyancy-Driven Flows

Chapter 7

Coupled Transfer Through Boundary Reactions: An Application-Oriented Cavity Flow Problem

This chapter enters the third part of this thesis, where a cavity flow problem is studied and coupled heat and mass transfer exists through temperature-dependent boundary reactions. Supercritical fluids have been widely used in chemical engineering. For example, extraction of natural compounds, such as caffeine, vitamins and active pharmaceutical ingredients, from materials is one of the main applications of supercritical fluids [1]. There are also various techniques for the precipitation or crystallization of compounds by supercritical fluids, where supercritical fluids can work either as solvents or antisolvents, including crystallization from supercritical solution, rapid expansion of a supercritical solution, gas antisolvent crystallization and supercritical antisolvent precipitation [2]. There are two types of supercritical solvent-solute phase diagrams [3], among which type II is more relevant than type I to extraction or crystallization using supercritical fluids. The emphasis of this chapter is placed on the type II phase diagram, as shown in Fig. 7.1. The solid curve at low temperature represents the vapor pressure curve of the pure solvent, which ends at the critical point denoted by CP1 . In this vicinity, there is a dotted-dashed curve standing for the solid-liquid-vapor (SLV) equilibrium of the mixture due to the solute dissolving into the solvent. This SLV curve ends at the lower critical end point (LCEP), where the liquid and vapor phases become identical in the presence of the solid phase [4]. The solid curves at high temperature are the sublimation curve, the vapor pressure curve (ending at CP2 ) and the melting curve of the pure solute [4], which intersect at the triple point. The other SLV curve appears at high pressure due to the solvent dissolving into the liquid solute [4], which emanates from the triple point and terminates at the upper critical end point (UCEP). The LCEP and UCEP are connected to CP1 and CP2 , respectively, by the dashed vapor-liquid critical curves of the mixture. The shaded region between the two CEPs, where an equilibrium is achieved between the solid and single-phase fluid, is the region where extraction and crystal growth would take © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 Z.-C. Hu, Coupled Heat and Mass Transfer in Binary Mixtures at Supercritical Pressures, Springer Theses, https://doi.org/10.1007/978-981-16-7806-6_7

117

118

7 Coupled Transfer Through Boundary Reactions …

Fig. 7.1 Type II phase diagram of the supercritical solvent-solute system in the temperature-pressure plane. CP1 : the critical point of the solvent. CP2 : the critical point of the solute. LCEP: lower critical end point. UCEP: upper critical end point. SLV: solid-liquid-vapor equilibrium

place. More importantly, the solubility of the solvent changes dramatically with small variations in temperature and/or pressure [5, 6] near the two CEPs. The question that naturally arises is whether one can combine the two processes in one apparatus in order to take advantage of the critical behavior of the solubility. On the one hand, it is particularly suitable for substances that require both processes in industrial production. On the other hand, such an apparatus may also operate in a single-purpose mode. This chapter presents preliminary theoretical explorations regarding this idea based on a conceptual model. This chapter is arranged as follows. Section 7.1 is devoted to the presentation of the model and its general mathematical descriptions, followed, in Sect. 7.2, by the specific formulations for the CO2 -naphthalene system. In Sect. 7.3, detailed theoretical analyses are presented to optimize the model from both thermodynamic and hydrodynamic points of view. In Sect. 7.4, the performance of the optimized model is evaluated by means of numerical simulations. The chapter is summarized in Sect. 7.5.

7.1 Problem Description 7.1.1 Physical Model The discussions of this chapter are based on a conceptual model that combines extraction and crystal growth by virtue of the critical behavior of the solubility. Figure 7.2 shows the schematic of the proposed model, where a cavity with thermostats mounted at the top and bottom walls is presented, while its lateral walls are adiabatic. The operation of the model can be understood as follows: The solution in the cavity is initially saturated and in thermal equilibrium. Then, the thermostats are adjusted to achieve a temperature difference between the two walls. Since the solubility depends on temperature, as the heat penetrates into the fluid, the regions near the walls are either unsaturated or supersaturated depending on the behavior of the solubility to temperature change. The unsaturated region provides the opportunity for extraction,

7.1 Problem Description

119

Fig. 7.2 Schematic of the proposed model that combines extraction and crystal growth processes. The situation plotted here is when the solubility increases with temperature and the top wall is hotter than the bottom wall or when the solubility decreases with temperature and the bottom wall is hotter than the top wall

Fig. 7.3 Schematic of the physical model for mathematical modeling where x and z are the horizontal and vertical directions, respectively, g is the gravitational acceleration vector, d is the cavity’s height, T1 and T2 are the temperatures at the bottom and top walls, respectively

while the supersaturated region favors crystal growth. To combine extraction and crystal growth, the unsaturated region is filled with solute-enriched nutrients, and a crystal growth plate (with crystal seeds on it) is inserted in the supersaturated region. In the transport region, the solute transfer is accomplished by diffusion or natural convection.

7.1.2 Mathematical Description To study the conceptual model shown in Fig. 7.2, the emphasis of the mathematical modeling is placed on the transport region, where the extraction and crystal regions are regarded as two idealized rigid interfaces subjected to desorption and adsorption reactions, respectively. Figure 7.3 presents the simplified model, a square cavity of height d containing a supercritical binary mixture initially at saturation. The horizontal and vertical direc-

120

7 Coupled Transfer Through Boundary Reactions …

tions are denoted by x and z, respectively. The subscripts 1 and 2 are used to denote a physical property at z = 0 and d, respectively. The mixture is initially motionless, saturated, in thermal equilibrium and stratified under gravity g(0, −g). The reference state is defined as the initially average state, denoted by the subscript ref. The lateral boundaries are adiabatic and impermeable, and all of the four boundaries are no-slip. At t = 0 s, the temperature T at the bottom and the top walls are adjusted from Tref to T1 and T2 , respectively, resulting in a solubility difference. The desorption reaction due to extraction occurs at the horizontal wall of the higher solubility, while the adsorption reaction caused by crystallization takes place at the other horizontal wall. Depending on the value of the temperature difference, the physical process in the cavity can be either pure diffusion or natural convection. The governing equations are given by [7] • conservation of mass:

∂ρ + ∇ · (ρu) = 0, ∂t

(7.1)

• conservation of momentum: ∂ρu 1 + ∇ · (ρuu) = −∇ pd + η∇ 2 u + η∇(∇ · u) + (ρ − ρs )g, ∂t 3

(7.2)

• conservation of energy: ∂ρT ∇ad ∂ pth λ + ∇ · (ρuT ) = ∇ 2 T + − ρs ∇ad w, ∂t cp g ∂t

(7.3)

• conservation of concentration: ∂ρc + ∇ · (ρuc) = D∇ · (ρ∇c), ∂t

(7.4)

• equation of state (EOS): ρ = ρs + ρref [α( pth − pref ) − β(T − Tref ) + κ(c − cref )] ,

(7.5)

where ρ is the density, t is the time, u(u, w) is the velocity vector, η is the viscosity, λ is the thermal conductivity, c is the concentration of the solute (mass fraction), c p is the isobaric specific heat, D is the diffusion coefficient, ∇ad = Tref βg/c p is the adiabatic temperature gradient, and α=

1 ρref



∂ρ ∂p

 , β=− T,ρ

1 ρref



∂ρ ∂T

 , κ= p,c

1 ρref



∂ρ ∂c

 ,

(7.6)

T, p

are the isothermal compressibility, thermal expansion coefficient and concentration contraction coefficient, respectively. The cross-diffusion effects and barodiffusion effect are omitted since the unavailability of relevant physical properties.

7.1 Problem Description

121

In the derivation of the above equations, the total pressure p was decomposed into three parts: (7.7) p = pth + ps + pd , where pth is the thermodynamic pressure representing the average thermodynamic state (a function of t), ps is the hydrostatic pressure due to gravity (a function of z) obeying (7.8) d ps /dz = −ρs g, (ρs is the stratified initial density) and pd is the dynamic pressure working to balance the inertial, viscous and body forces (time- and space-dependent). According to Eq. (7.8), by letting the mean ρs and ps equal ρref and 0, respectively, ρs and ps are obtained as exp (−χ z/d) , 1 − exp(−χ )   1 exp (−χ z/d) − , ps = ρref gd 1 − exp(−χ ) χ

ρs = χ ρref

(7.9) (7.10)

where χ = αρref gd is a dimensionless parameter. pth is calculated according to the overall mass conservation:  pth = pref +



VT [β(T − Tref ) − κ(c − cref )] dV + VT (ρ − ρs )/ρref dV

αVT

,

(7.11)

where VT is the total volume. In addition to the pressure decomposition in Eq. (7.7), through the low-Mach number approximation [8], pd , as a high-order term, was omitted in Eqs. (7.3) and (7.5). In addition, in Eqs. (7.2), (7.3) and (7.5), ps has been substituted by ρs through Eq. (7.8). In this study, a linearized EOS is used, and all physical properties are treated as constant values of those at the reference state. These approximations are perfectly satisfactory when |T1 − T2 | is small. Even though the thermostats are adjusted in such a way that (T1 + T2 )/2 = Tref to guarantee that the physical properties are representative, there will be deviations if |T1 − T2 | is too large. However, they are still employed because they reduce the nonlinearities of the problem and facilitate theoretical analyses and numerical simulations.

7.1.3 Initial and Boundary Conditions According to the description in Sect. 7.1.2, the initial fields, denoted by bars, are governed by

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7 Coupled Transfer Through Boundary Reactions …

⎫ u¯ = w¯ = p¯ d = 0, T¯ = Tref , ⎪ ⎪ ⎪ ⎪ ∇ ρ∇ ¯ c¯ = 0,  ⎬ )/ρ − κ( c ¯ − c )] dV ρ ¯ − ρ [( s ref ref V ,⎪ p¯ th = pref + T ⎪ ⎪ αVT ⎪ ⎭ ρ¯ = ρs + ρref [α( p¯ th − pref ) + κ(c¯ − cref )] .

(7.12)

Note that initially, c¯ = cref because of the nonuniformity of ρ. ¯ Mathematically, the boundary conditions can be expressed as ∂c ∂T = = 0, ∂x ∂x D ∂c z = 0, u = 0, w = − , T = T1 , 1 − c ∂z D ∂c , T = T2 . z = d, u = 0, w = − 1 − c ∂z

x = 0 and d, u = w = 0,

(7.13)

The nonzero w is a result of the partial permeability of the reactive walls (only the solute is allowed to go across) [9]. According to previous discussions, the boundary conditions of c at the two reactive walls can be generally expressed by s1 − s2 ≥ 0, c1 : desorption, c2 : adsorption, s1 − s2 < 0, c1 : adsorption, c2 : desorption,

 (7.14)

where s is the solubility in the unit of mass fraction. The adsorption is caused by crystal growth while desorption is the result of extraction. The exact expressions for these reactions are very complicated and rely strongly on the correlations reported in experimental measurements. In the next section, the modeling of the CO2 -naphthalene system will be presented, together with the formulations of Eq. (7.14).

7.1.4 Governing Parameters Under the current configuration, the natural convection is driven by the gradients of T and c, and they generally have different rates of diffusion [10]. This kind of natural convection is called double-diffusive convection (DDC). The DDC of a compressible binary fluid is governed by seven parameters [7]: thermal Rayleigh number R, solutal Rayleigh number R S , Lewis number L, Prandtl number Pr , temperature gradient ratio r T , gravitational compressibility factor χ and specific heat ratio γ . The definitions of these parameters are summarized in Table 7.1, where ∇ˆ T = (T1 − T2 )/d − ∇ad is the typical relative temperature gradient, ∇c = (c1 − c2 )/d is the typical concentration gradient, DT = λ/(ρref c p ) is the thermal diffusivity and cv = c p − Tref β 2 /(αρref ) is the specific heat at constant volume. Physically, the two Rayleigh numbers measure the competition between the driving force (buoyancy) and the drag force (diffusion and viscosity) in natural con-

7.1 Problem Description

123

vection. In particular, R > 0 indicates that the buoyancy produced by T is positive, while R < 0 means that T depresses natural convection. In contrast, R S < 0 suggests that c promotes convection, while R S > 0 means that the effect of c is negative. L is defined as the ratio of mass diffusivity to thermal diffusivity. r T measures the relative importance of ∇ad to ∇ˆ T , and χ the density stratification caused by gravity. γ indicates the intensity of the piston effect, which mainly influences the route to the fully developed state [11, 12]. Two response parameters are defined to quantitatively describe the convection. The intensity of convective heat transfer is measured by the Nusselt number: Nu =

∂ T1 /∂z or ∂ T2 /∂z convective heat transfer = . conductive heat transfer (T2 − T1 )/d

(7.15)

Similarly, the performance of mass transfer is measured by the Sherwood number: ρ1 D ∂c1 ρ2 D ∂c2 or convective mass transfer 1 − c1 ∂z 1 − c2 ∂z Sh = = . diffusive mass transfer ρref D(c2 − c1 )/d

(7.16)

The emphasis of this study is placed on the fully developed natural convection, which is featured by the statistical consistency between the values of Sh and N u calculated at the two reactive walls.

7.2 Formulations for CO2 −Naphthalene System A reference fluid system should be chosen to conduct detailed analyses. In this chapter, the CO2 -naphthalene system is taken as a reference system with a type II phase diagram (see Fig. 7.1). CO2 , as a nontoxic, nonflammable, environmentally safe and widely available solvent, is the most popular substance in industrial applications. Moreover, the binary system has been studied extensively, and there are

Table 7.1 Definitions and names of the governing parameters Symbols Definitions Names ρref gβ ∇ˆ T d 4 R Thermal Rayleigh number ηDT ρref gκ∇c d 4 Solutal Rayleigh number RS ηDT L D/DT Diffusivity ratio (Lewis number) η Prandtl number Pr ρref DT ∇ad /∇ˆ T Temperature gradient ratio rT αρref gd Gravitational compressibility factor χ γ c p /cv Specific heat ratio

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7 Coupled Transfer Through Boundary Reactions …

sufficient data for the solubility [13–16], transport properties [17–20] and crystal growth kinetics [21, 22]. It is legitimately supposed that due to the critical universality, the conclusions based on one reference system could be extended to other systems with the same type of phase diagram. The task of this section is twofold. On the one hand, the adsorption and desorption reactions at the two reactive walls are formulated to obtain explicit expressions of Eq. (7.14). On the other hand, the thermodynamic and transport properties in the governing equations are calculated. Before dealing with the first task, the modeling of solubility is first introduced.

7.2.1 Solubility The model proposed by Škerget et al. [16] is implemented in this study to estimate the solubility of naphthalene in CO2 . The details are summarized in Appendix B. In their model, the solubility, denoted by ys , is expressed by the mole fraction of the solute in the saturated mixture. s can be calculated from ys by s=

ys M1 , (1 − ys )M1 + ys M2

(7.17)

where M1 and M2 are the molar masses of CO2 and naphthalene (M1 = 0.04401 kg/mol, M2 = 0.1282 kg/mol), respectively. Note that ys and s are different representations of the solubility, and the emphasis of this study is placed on the latter one. For the CO2 -naphthalene system, TLCEP = 307.65 K [23] is close to 304.21 K, the critical temperature of CO2 . McHugh and Paulaitis [13] reported that TUCEP is between 333.55 K and 338.05 K. Solubility data are usually experimentally reported at different temperatures between TLCEP and TUCEP as functions of p. Many data are available for T = 308.15 K, 328.15 K and 333.55 K [13–16] scattered between TLCEP and TUCEP . To estimate the solubility, the methodology of Škerget et al. [16] is employed in this work, detailed in Appendix B. To be specific, Škerget et al. [16] fitted the parameters in Eq. (B.3), i.e., k12 and l12 , at individual temperatures by minimizing the average absolute relative deviation (AARD) between the calculated ys and the experimental data. Their results are [16]: For T = 308.15 K, k12 = −0.0272, l12 = −0.1289; for T = 328.15 K, k12 = −0.0177, l12 = −0.1255; and for T = 333.55 K, k12 = 0.0533, l12 = −0.0166. Symbolic computations are employed to avoid lengthy algebraic calculations. The computations were implemented based on the Symbolic Math Toolbox in MATLAB [24]. The variations in s against p are presented in Fig. 7.4 for later reference. To facilitate theoretical analyses, similar to Eq. (7.5), s is linearized with respect to p, T , and c, leading to

7.2 Formulations for CO2 −Naphthalene System

125

Fig. 7.4 Maximum mass fraction of the solute (naphthalene) in the supercritical solvent (CO2 ), i.e., the solubility s, plotted against pressure p at 308.15 K (solid curve), 328.15 K (dash-dotted curve) and 333.55 K (dotted curve)

/  s = sref + ξ p ( pth − pref ) + ξ p ρref gd



where ξp =

∂s ∂p

1 exp(−χ z/d) − 1 − exp(−χ ) χ



 , ξT = T,c

∂s ∂T

 + ξT (T − Tref ) + ξc (c − cref ),



(7.18) 

, ξc = p,c

∂s ∂c

 ,

(7.19)

T, p

can be determined directly by virtue of the symbolic computations. Note that sref = cref because the reference state is saturated. In the derivation of Eq. (7.18), Eq. (7.7) was employed, where pd was omitted according to the conclusion from low-Mach number approximation and ps has been substituted by Eq. (7.10).

7.2.2 Adsorption and Desorption Reactions As mentioned earlier, the extraction and crystal growth regions in Fig. 7.2 are simplified into two rigid walls in Fig. 7.3 where desorption (extraction) and adsorption (crystal growth) reactions occur. According to Eq. (7.14), the sign of (s1 − s2 ) determines the designation of the types of reactions. After some computations based on Eq. (7.18), (s1 − s2 ) is obtained as s 1 − s2 =

ξT (T1 − T2 ) + ξ p ρref gd , 1 − ξc Γ

where Γ =

c1 − c2 s1 − s2

(7.20)

(7.21)

is the solubility efficiency, defined as the ratio of actual concentration difference to its theoretically maximum value. Physically, Γ indicates that the system is either diffusion-limited (when Γ is close to unity) or reaction-limited (when Γ is close to zero). Furthermore, computations confirm that 0 < ξc < 1 is satisfied at the three

126

7 Coupled Transfer Through Boundary Reactions …

temperatures for p values ranging from 5 MPa to 30 MPa. Therefore, the denominator is always positive, and the sign of (s1 − s2 ) is determined by the numerator. The desorption reaction at the extraction wall depends on the specific circumstance. Modeling a practical situation would lose generality. Thus, the desorption reaction is assumed to be diffusion-limited. That is, local equilibrium is achieved at any time [25], namely, desorption : c = s. (7.22) Now, consider the adsorption reaction due to crystal growth. At the interface, the outflow solute crystallizes into a new layer of the solid solute of density ρS with a building velocity of V c [9]. This mass balance relationship yields [9] ρD 1−c

∂c = ρS V c , ∂z

(7.23)

where V c is the growth rate of the crystal (always positive), ρS = M2 /VS (VS is the molar volume of the solute, and VS = 1.1 × 10−4 m3 /mol for naphthalene). V c is modeled by the experimental correlation proposed by Tai and Cheng [21]:  V c = K VS σ  = 100K VS

 c s − , 1 − c 1 − s

(7.24)

where K is the growth rate constant, σ  is the bulk supersaturation, c the bulk concentration and s the bulk solubility. The bulk quantities reflect the environment for crystal growth in their experiments. In [21], the solubility is expressed in grams of the solute per 100 g of solvent. That is why a unit conversion appears in Eq. (7.24). Combining Eqs. (7.23) and (7.24) yields: ρD adsorption : 1−c

  ∂c c s = 100K M2 − . ∂z 1 − c 1 − s

(7.25)

Finally, using Eqs. (7.20), (7.22) and (7.25), Eq. (7.14) can be written explicitly as s1 − s2 ≥ 0, c1 = s1 , −

ρ2 D ∂c2 = 100K M2 1 − c2 ∂z

s1 − s2 < 0, c2 = s2 ,

ρ1 D ∂c1 = 100K M2 1 − c1 ∂z

 

c2 s2 − 1 − c2 1 − s2 c1 s1 − 1 − c1 1 − s1

 ⎫ ⎪ ,⎪ ⎪ ⎪ ⎬  ⎪ ⎪ ⎪ .⎪ ⎭ (7.26)

Tai and Cheng [21] measured the growth rate of naphthalene in supercritical CO2 at T = 318.15 K under various p values and provided K = 3.6 × 10−3 mol/(m2 · s). Since the real situation is complicated and accurate experimental data at other temperatures are not available, K is treated as a constant throughout this study. In fact, it is valid as long as K does not vary dramatically with T and p, as confirmed by

7.2 Formulations for CO2 −Naphthalene System

127

the experiments of Tai and Cheng [21] and Uchida et al. [22] for the CO2 -naphthalene system.

7.2.3 Thermodynamic and Transport Properties Calculations of the thermodynamic properties, i.e., α, β, κ and c p , are based on the Peng-Robinson EOS introduced in Appendix B. Some additional equations for c p are detailed in Appendix C. In regard to the transport properties, because the experimental data are scarcely available, λ and η of the mixture are approximated as those of pure CO2 [25] at Tref and ρref . The state-of-the-art correlations developed by Laesecke and Muzny [26] and Huber et al. [27] are employed to calculate η and λ, respectively. In these correlations, α is required in the calculations of the critical enhancement contributions. Therefore, α of the mixture is used to ensure the consistencies in the critical behavior of various properties. Such an approximation is highly accurate when the state is close to LCEP, since the solubility is substantially low; see Fig. 7.4. When close to the UCEP, the approximation is still satisfactory. The results from the pure-CO2 approximation are compared to the experimental data of η at T = 328.15 K reported by Lamb et al. [18]; for p ranging from 12 MPa to 100 MPa, the AARD is 6.17 %. D is obtained from the model developed by Vaz et al. [28], where they report the AARD is 9.68 % for the CO2 -naphthalene system by comparing with 114 data points.

7.3 Theory So far, the conceptual model and its mathematical formulation have been elaborated. In this section, theoretical analyses are developed to optimize the model and provide detailed strategies concerning its operation. An objective function is a key component in an optimization problem. Here, it is natural to take the objective function as the cross-boundary flow rate of the solute, given by ρ2 D ∂c2 ρ1 D ∂c1 or , (7.27) m˙ = 1 − c ∂z 1 − c ∂z 1

2

i.e., m˙ can be calculated at both walls. In the sense of the temporal average, m˙ 1 is equal to m˙ 2 in a fully developed convection. Combining Eqs. (7.16), (7.20) and (7.21), m˙ can be further expressed by ξT (T1 − T2 ) + ξ p ρref gd ρref DSh . m˙ = Γ −1 − ξc d

(7.28)

128

7 Coupled Transfer Through Boundary Reactions …

Substituting Eq. (7.26) into Eq. (7.16), using the definition of Γ given by Eq. (7.21) and neglecting high-order terms, one arrives at  Γ =

−1 ρref DSh . +1 100K M2 d

(7.29)

Combining the above two equations leads to the final expression of m˙ m˙ = ξT (T1 − T2 ) + ξ p ρref gd



1 1 − ξc d + 100K M2 Sh ρref D

−1

.

(7.30)

The optimization goal is to maximize m. ˙ The elements of Eq. (7.30) can be divided into six groups: (T1 − T2 ), d, K , Sh, g, and physical properties at the reference state. Among these elements, K and g have been treated as constants in this study. Moreover, in this section, d = 0.01 m is fixed as a reference length. The effects of d on m˙ will be separately discussed later. After these simplifications, the independent factors in the current optimization problem are the reference state and (T1 − T2 ), while Sh is a response parameter of these two variables. Therefore, the task of this section can be briefly interpreted as exploring the best reference state and the strategies of controlling T1 and T2 to maximize m. ˙ Moreover, 0 < ξc < 1 implies m˙ is a monotonic increasing function of Sh. The optimization is performed in two steps. First, the thermodynamic optimization maximizes the sensitivity of m˙ in terms of (T1 − T2 ). Second, the hydrodynamic optimization maximizes Sh for a given (T1 − T2 ).

7.3.1 Thermodynamic Optimization In thermodynamic optimization, the natural convection is omitted. By letting Sh = 1, the thermodynamic contribution of m, ˙ denoted by m˙ th , is obtained as m˙ th ≡ m| ˙ Sh=1 = ξT (T1 − T2 ) + ξ p ρref gd



1 d(1 − ξc ) + 100K M2 ρref D

−1

. (7.31)

The sensitivity of m˙ th with respect to (T1 − T2 ) is measured by T m˙ th =

  dm˙ th 1 d(1 − ξc ) −1 = |ξT | + . d(T1 − T2 ) 100K M2 ρref D

(7.32)

T Figure 7.5 shows the variations in m˙ th with p at various temperatures. In the lowpressure region, influenced by the LCEP, all three curves develop local minimum values denoted by A, B and C. As T departs from TLCEP = 307.65 K, the locations of the minimum values move toward the higher pressure and the slopes decay gradually.

129

/

7.3 Theory

/

/

T , given by Fig. 7.5 Thermodynamic contribution of the cross-boundary flow rate of the solute, m˙ th Eq. (7.32), versus pressure p at 308.15 K (solid curve), 328.15 K (dash-dotted curve) and 333.55 K (dashed curve) for the CO2 -naphthalene system. The local extreme values (denoted by A to F) are marked by circles, two of which are enlarged in the two insets. The pressure at the upper critical end point, pUCEP = 22.6 MPa, is obtained from [29], while that at the lower critical end point, pLCEP = 7.5 MPa, is a rough estimation

Meanwhile, intriguingly, local peak values emerge in the high-pressure region (D, E and F), which increase with T , along with the locations moving toward pUCEP . The trends described above indicate a transition from the LCEP-dominance to the UCEP-dominance. T , m˙ th is sensitive to the variation of (T1 − T2 ). At the local extreme values of m˙ th Therefore, they are preferable reference states. On the basis of the concept of the pseudocritical point, it is convenient to call them pseudo-CEPs. According to Fig. 7.5, the performance of a pseudo-CEP increases as its corresponding CEP is approached. In fact, the pseudo-LCEPs and the pseudo-UCEPs can be represented by two respective continuous curves in the p − T plane. Due to the lack of data, they canT tends to infinity (i.e., |ξT | tends to infinity) not be obtained. However, because m˙ th at both CEPs, it is reasonable to argue that one can always obtain two equivalent T in both near-LCEP and near-UCEP regions. reference states that own the same m˙ th Therefore, the thermodynamic optimization cannot predict the preferable near-CEP region, which should be investigated from a hydrodynamic point of view.

7.3.2 Hydrodynamic Optimization This section is dedicated to comparing the two near-CEP regions from a hydrodynamic point of view. Figure 7.5 suggests there are six favorable reference states,

130

7 Coupled Transfer Through Boundary Reactions …

which can be divided into two groups: the pseudo-LCEPs, including A, B and C, and the pseudo-UCEPs, including D, E and F. Table 7.2 lists the physical properties, and Table 7.3 lists the dimensionless parameters at these points. In the near-LCEP region, i.e. A, B and C, s decreases with T (ξT < 0), density is sensitive to the change in T and c, and the dissolved naphthalene makes the fluid heavier (κ > 0). However, in the near-UCEP region, i.e., D, E and F, these features are just the opposite. The preliminary conclusion is that different CEP regions have distinct hydrodynamic behavior, which further influences the performance.

7.3.2.1

Regime of Convection

As mentioned earlier, the natural convection in the cavity is categorized as DDC, where the two governing components are T and c. In fact, according to the roles of the two components and their diffusion rates, DDC can be divided into four regimes [10]: • Stable Regime (STA): Convection is inhibited by both components. Sh = 1. • Fingering Regime (FIN): Convection is promoted by the slow diffuser and is inhibited by the fast one. Sh ≥ 1. • Oscillatory Regime (OSC): Convection is promoted by the fast diffuser and is inhibited by the slow one. Sh ≥ 1. • Cooperative Regime (COO): Convection is promoted by both components. Sh ≥ 1. It is fundamental to investigate the regime of convection prior to understanding the behavior of Sh. As discussed in Sect. 7.1.4, promotion or inhibition can be determined from the signs of R and R S , while the relative size of diffusion rates can be inferred from L. To avoid lengthy descriptions, through heuristic derivations, the judgment can be elegantly performed by the θ − criterion: θ = arg(R + i R S ) + 180◦

max(R R S , 0) min(L − 1, 0) , R RS L −1

(7.33)

where arg() is a function that returns the argument of a complex number in the range of [0◦ , 360◦ ]. Through this equation, the regimes of DDC are projected onto the phase space of θ . The correspondences between θ and the regimes are FIN : θ = 0◦ − 90◦ , 360◦ − 450◦ , STA : θ = 90◦ − 180◦ , OSC : θ = 180◦ − 270◦ , COO : θ = 270◦ − 360◦ .

(7.34) (7.35) (7.36) (7.37)

Label A B C D E F Label A B C D E F

Tref /(K) 308.15 328.15 333.55 308.15 328.15 333.55 c p /[kJ/(kg · K)] 22.445 4.329 3.862 1.960 1.983 1.870

pref /(MPa) 7.93 11.7 12.7 26.1 24.4 24.3 ξT /(K−1 ) −3.208 × 10−3 −7.323 × 10−4 −2.529 × 10−4 2.248 × 10−3 4.316 × 10−3 7.035 × 10−3

Table 7.2 Reference states and corresponding properties ρref /(kg/m3 ) 455.534 526.066 504.748 935.964 855.178 893.198 ξ p /(MPa−1 ) 1.923 × 10−2 1.027 × 10−2 7.753 × 10−3 8.410 × 10−5 7.839 × 10−4 1.466 × 10−3

sref /(-) 5.221 × 10−3 2.880 × 10−2 2.400 × 10−2 5.160 × 10−2 1.171 × 10−1 2.095 × 10−1 ξc /(-) 0.723 0.526 0.415 0.075 0.284 0.797

α/(MPa−1 ) 9.352 × 10−1 1.097 × 10−1 1.034 × 10−1 7.387 × 10−3 9.437 × 10−3 7.147 × 10−3 η/(Pa · s) 3.121 × 10−5 3.534 × 10−5 3.366 × 10−5 9.700 × 10−5 7.703 × 10−5 8.491 × 10−5

β/(K−1 ) 1.729 × 10−1 2.472 × 10−2 2.171 × 10−2 4.772 × 10−3 5.003 × 10−3 4.058 × 10−3 λ/[W/(m · K)] 0.081 0.061 0.058 0.113 0.098 0.106

κ/(-) 33.790 4.090 3.585 −0.439 −0.260 −0.268 D/(mm2 /s) 2.354 × 10−2 1.945 × 10−2 2.054 × 10−2 8.153 × 10−3 1.036 × 10−2 9.695 × 10−3

7.3 Theory 131

132

7 Coupled Transfer Through Boundary Reactions …

Fig. 7.6 Evolution of the regimes of double-diffusive convection with T = |T1 − T2 | in the phase diagram of the indicator θ defined by Eq. (7.33) for the pseudocritical end points A to F in Table 7.2, where Sh = 1 is fixed. T1 and T2 are the temperatures at the lower and upper walls, respectively. STA: stable regime. FIN: fingering regime. OSC: oscillatory regime. COO: cooperative regime. Solid curves: (T1 > T2 ), dashed curves: (T1 < T2 )

7.3 Theory

133

Table 7.3 Dimensionless parameters at the reference states Label L Pr χ A B C D E F

2.973 0.724 0.682 0.132 0.179 0.153

8.652 2.501 2.213 1.678 1.554 1.499

4.179 × 10−5 5.662 × 10−6 5.122 × 10−6 6.783 × 10−7 7.917 × 10−7 6.262 × 10−7

γ 27.507 5.065 4.537 2.074 2.055 1.852

Moreover, the modeling of the CO2 -naphthalene system allows R S to be expressed as a function R. According to the definitions of R and R S in Table 7.1 and Eqs. (7.20), (7.21) and (7.29), the function is expressed as  −1  ρref DSh ρref gκd 4 + 1 − ξc R S = ξT κβ −1 R + (ξT ∇ad + ξ p ρref g) , ηDT 100K M2 d (7.38) where R is a function of (T1 − T2 ). Based on Eqs. (7.33) and (7.38), the evolutions of the regimes of convection varying with (T1 − T2 ) can be investigated for the reference states A to F in Table 7.2. As indicated by Eq. (7.38), because 1 − ξc > 0, Sh has no effect on the sign of R S . In other words, Sh does not alter the regime of DDC. Without loss of generality, the following discussions begin with Sh = 1. Figure 7.6 shows evolutions of the regimes of convection with T = |T1 − T2 | at the pseudo-CEPs for Sh = 1. It is demonstrated that the evolutions of the regimes are complex when T is small. However, the common feature is, when T is large enough, all cases evolve into the STA if (T1 < T2 ) and the COO if (T1 > T2 ). Because T is anticipated to be higher than several mKs in actual applications, it is concluded that (T1 > T2 ) is preferable for both near-CEP regions to take advantage of the convection instead of being motionless. The influence of Sh > 1 can be understood by considering the limiting case of Sh = ∞, which leads to R S = 0 and θ = 0◦ . Therefore, because Sh > 1, the solid curves in the COO actually bend toward (but never go across) θ = 0◦ as (T1 − T2 ) increases gradually. However, the dashed curves in Fig. 7.6 remain unchanged since Sh = 1 in the STA.

7.3.2.2

Evaluation of the Sherwood Number

Figure 7.6 identifies that COO is the most relevant regime in actual applications. Therefore, in terms of Sh, the two near-CEP regions should be compared in this regime.

134

7 Coupled Transfer Through Boundary Reactions …

Fig. 7.7 Values of [R − (R S /L)] at six pseudocritical end points A to F of the CO2 -naphthalene system (see Fig. 7.5) for T1 − T2 = 10 mK. R is the thermal Rayleigh number, R S is the solutal Rayleigh number, L is the diffusivity ratio, T1 and T2 are the temperatures at the bottom and top walls, respectively

The level of Sh is proportional to the intensity of convection. The estimation is relatively straightforward in the COO since both T and c promote the convection. According to the classical result of linear stability analysis, the criterion for the onset of DDC in the COO is given by [10] R − (R S /L) > threshold.

(7.39)

This equation suggests that the left-hand side, [R − (R S /L)], can work as an indicator for the intensity of DDC. Figure 7.7 presents the comparisons of [R − (R S /L)] at various reference states for T1 − T2 = 10 mK, under which all reference states are in COO. Comparing the values of [R − (R S /L)], for pseudo-LCEPs, the ranking is A B > C, and for pseudo-UCEPs, F > E > D. Therefore, in each CEP region, the intensity of DDC increases as the corresponding CEP is approached. However, comparisons between the two regions suggest A B > C > F > E > D, namely, the pseudoLCEPs are much higher than the pseudo-UCEPs. Therefore, due to the hydrodynamic advantages, the pseudo-LCEPs are preferable to the pseudo-UCEPs. The explanations for the hydrodynamic advantages of the pseudo-LCEPs involve the critical behavior of physical properties. For a pure fluid, it is well known that the variable physical properties in the critical region make it susceptible to natural convection [30]. For a binary fluid mixture, there are also anomalies in physical properties. However, the extents of these anomalies are generally weaker than those of pure fluids [31]. Nevertheless, exceptions occur for dilute mixtures, whose anomalies in physical properties are also strong [31]. Since the solubility at the LCEP is relatively lower than that at the UCEP (see Fig. 7.4), it is concluded that the physical properties exhibit stronger anomalous behavior near the LCEP than near the UCEP. Since the susceptibility in terms of buoyancy-driven convection is proportional to the extent of anomalies, pseudo-LCEPs are hydrodynamically preferable to pseudo-UCEPs. The theoretical optimization performed in this section identifies that the pseudoCEPs are preferable reference states. At these states, the best strategy to control T1 and T2 is (T1 > T2 ) so that the DDC is cooperatively driven by T and c. In

7.3 Theory

135

each near-CEP region, the performance is promoted when the corresponding CEP is approached. In addition, the pseudo-LCEPs are superior to the pseudo-UCEPs due to their hydrodynamic excellence.

7.4 Numerical Simulations The optimization presented in the previous section suggests that the state A is the best reference state among the available pseudo-CEPs in Table 7.2. According to Fig. 7.6a, as (T1 − T2 ) grows from zero, the convection evolves into the STA, FIN, and COO in sequence, in which the COO is the main concern. In this section, by taking state A as the reference state, the actual performance of the system is evaluated through a series of numerical simulations for d = 0.01 m and R up to 1 × 107 in the COO. Then, the influences of d are studied separately.

7.4.1 Numerical Method The governing equations along with the initial and boundary conditions were solved numerically by the finite-volume method. The numerical schemes have second-order accuracy in space and first-order accuracy in time. The pressure-velocity coupling is treated by the block-coupled algorithm [32]. The time step is controlled by limiting the maximum Courant number to be lower than 0.5. The numerical method has been elaborated in [33]. A nonuniform wall-refined grid with 256 × 256 grid points is used in this study. A grid-independence test has been performed by comparing the results of R = 1 × 107 from two refined grids: 388 × 388 and 483 × 483. The values of N u and Sh produced by the three grids agree well, suggesting that the resolution of 256 × 256 is sufficient.

7.4.2 Cooperative Regime According to Eq. (7.20), when (T1 − T2 ) surpasses −ξ p ρref gd/ξT = 0.2678 mK, s1 is smaller than s2 , and correspondingly, the DDC enters COO. The setup of the cases and the main results are summarized in Table 7.4. As (T1 − T2 ) increases, the response parameters, i.e., N u and Sh, increase monotonically (see Table 7.4). Moreover, the form of convection changes progressively from steady state to periodic motion and further to chaotic state, i.e., in the direction of increasing nonlinearity. Figure 7.8 presents Sh versus [R − (R S /L)] in the double logarithmic coordinates. As discussed earlier, [R − (R S /L)] is regarded as a measurement of the intensity of DDC in the COO. The values of Sh for the chaotic cases, i.e., cases 7 to 30 in

136

7 Coupled Transfer Through Boundary Reactions …

Table 7.4 Summary of the selected cases and results in the COO. For chaotic and periodic cases, time-averaged values of R S , Γ , N u, Sh and m˙ are shown here (T1 − T2 )/(mK)

R

RS

1

0.2678

109000

0

Steady

0.943

1.4158

2.6132

0

2

0.2681

110000

−1851

Steady

0.942

1.4242

2.6536

8.624 × 10−12

3

0.2809

150000

−75525

Periodic

0.940

1.6211

2.7321

3.622 × 10−10

4

0.3001

210000

−178040

Periodic

0.927

1.9807

3.3931

1.060 × 10−9

5

0.3161

260000

−260210

Periodic

0.920

2.2689

3.7426

1.710 × 10−9

6

0.3321

310000

−341910

Periodic

0.916

2.3938

3.9476

2.369 × 10−9

7

0.3480

360000

−422720

Chaotic

0.913

2.7288

4.1076

3.048 × 10−9

8

0.3640

410000

−498010

Chaotic

0.909

3.0620

4.2979

3.757 × 10−9

9

0.3800

460000

−571900

Chaotic

0.903

3.4573

4.6482

4.666 × 10−9

10

0.3960

510000

−646930

Chaotic

0.899

3.6304

4.8147

5.468 × 10−9

11

0.4069

544000

−696469

Chaotic

0.898

3.8262

4.9509

6.053 × 10−9

12

0.5329

938000

−1251121

Chaotic

0.878

5.5735

5.9851

1.315 × 10−8

13

0.6589

1332000

−1787886

Chaotic

0.868

6.8768

6.5707

2.062 × 10−8

14

0.7849

1726000

−2294765

Chaotic

0.858

7.9307

7.1379

2.875 × 10−8

15

0.9109

2120000

−2812321

Chaotic

0.852

8.7330

7.4469

3.676 × 10−8

16

1.0369

2514000

−3313633

Chaotic

0.848

9.3988

7.6910

4.474 × 10−8

17

1.2889

3302000

−4273488

Chaotic

0.838

10.5478

8.3070

6.232 × 10−8

18

1.4150

3696000

−4791228

Chaotic

0.836

11.0505

8.4494

7.107 × 10−8

19

1.6670

4484000

−5745444

Chaotic

0.829

11.9622

8.8902

8.967 × 10−8

20

1.9190

5272000

−6664723

Chaotic

0.821

12.7637

9.3512

1.094 × 10−7

21

2.0450

5666000

−7099038

Chaotic

0.818

13.2059

9.5608

1.191 × 10−7

22

2.1710

6060000

−7515324

Chaotic

0.815

13.4426

9.7708

1.289 × 10−7

23

2.2970

6454000

−7997895

Chaotic

0.813

13.7742

9.8779

1.387 × 10−7

24

2.4230

6848000

−8495331

Chaotic

0.813

13.9954

9.9031

1.477 × 10−7

25

2.5490

7242000

−8890377

Chaotic

0.810

14.2959

10.1215

1.580 × 10−7

26

2.6750

7636000

−9255017

Chaotic

0.803

14.8627

10.5388

1.712 × 10−7

27

2.8010

8030000

−9796852

Chaotic

0.806

14.9027

10.3674

1.783 × 10−7

28

3.1791

9212000

−11119753

Chaotic

0.799

15.7025

10.8346

2.115 × 10−7

29

3.3051

9606000

−11467015

Chaotic

0.798

15.8132

10.8907

2.192 × 10−7

30

3.4311

10000000

−11672524

Chaotic

0.795

16.0681

11.1055

2.275 × 10−7

Fig. 7.8 Log-log plot of the Sherwood number Sh versus [R − (R S /L)], where R is the thermal Rayleigh number, R S is the solutal Rayleigh number and L is the diffusivity ratio. Dots: data in Table 7.4 for the CO2 -naphthalene system. Solid line: power law given by Eq. (7.40), which is fitted from cases 7 to 30 in Table 7.4

Final state

Γ

2 · s)] m/[kg/(m ˙

Case No.

Nu

Sh

137

/

7.4 Numerical Simulations

(

(

/

Fig. 7.9 Cross-boundary flow rate of the solute calculated from Eq. (7.30), m, ˙ plotted against (T1 − T2 ) ranging from 3.48 × 10−4 K to 0.5 K. T1 and T2 are the temperatures at the bottom and top walls, respectively. Solid curve: with double diffusive convection, namely, the Sherwood number Sh obeying Eq. (7.40). Dashed curve: without double-diffusive convection, namely, Sh = 1. The physical properties are those of state A in Table 7.2 for the CO2 -naphthalene system

Table 7.4, are identified to obey the following power law (represented by a straight line on a log-log plot): (7.40) Sh = n 1 [R − (R S /L)]n 2 , where n 1 = 0.1123 and n 2 = 0.2793 are fitted coefficients. Note that n 2 is close to 2/7 = 0.2857, an exponent for N u in the Rayleigh-Bénard convection [34]. Note that since the mathematical modeling is based on the linearized EOS and constant properties, Eq. (7.40) is likely to deviate from real situations when obvious variations in physical properties set in (when R is large enough). In fact, numerical simulations incorporating real EOS and variable properties are extremely challenging. Equation (7.40) can serve as a reference for real situations. Due to Eqs. (7.38) and (7.40), m˙ can be calculated by Eq. (7.30) given (T1 − T2 ). The results are plotted in Fig. 7.9 for (T1 − T2 ) up to 0.5 K, along with those obtained by setting Sh = 1 for the purpose of comparison. In both situations, m˙ varies almost linearly with (T1 − T2 ). When T1 − T2 = 0.5 K, m˙ = 5.08 × 10−5 kg/(m2 · s), corresponding to the crystal growth rate of 4.98 × 10−8 m/s. Uchida et al. [22] constructed a crystal growth chamber to measure the crystal growth rate of naphthalene in supercritical CO2 under constant T and p. In their experiments, supersaturation was achieved by cooling the continuously supplied and fresh saturated solution. The cooling was maintained at 0.5 K, 1 K, or 2 K. However, the crystal growth rates were reported to be much lower than the one achieved by the present model for T1 − T2 = 0.5 K. Because there was a magnetic stirrer in their experiments to externally improve the mass transfer, the enhancement in the current model should be attributed to the optimized reference state, which guarantees a large solubility difference to promote the reaction. However, in their experiments,

138

7 Coupled Transfer Through Boundary Reactions …

Fig. 7.10 a Sherwood number Sh plotted against [R − (R S /L)], where R is the thermal Rayleigh number, R S is the solutal Rayleigh number and L is the diffusivity ratio. The arrow indicates the direction of increasing cavity height d. b Cross-boundary flow rate of solute m˙ calculated from Eq. (7.30), plotted against d from 0.01 m to 1 m. In both figures, the physical properties are those of state A in Table 7.2 for the CO2 -naphthalene system and T1 − T2 = 0.5 K is fixed

the reference states were chosen to cover large ranges of p and T , so the solubility differences were not significant. Furthermore, by comparing the two curves, a tenfold increase in m˙ is achieved by the DDC. Although Sh is higher than 10 when T1 − T2 > 3.43 mK (see Table 7.4), a better amplification does not appear due to the limitation of the crystal growth kinetics. Therefore, if somehow K can be increased, the amplification effect of DDC on m˙ is anticipated to be improved. In a limiting case when K → ∞ (when the reaction is absolutely diffusion-limited), the amplification factor is exactly Sh, according to Eq. (7.30).

7.4.3 The Effects of Cavity Height In previous sections, the external factor d is treated as a constant of 0.01 m. Here T1 − T2 = 0.5 K is fixed and the effects of d on the performance are investigated. The range of d considered in this section is from 0.01 m to 1 m. Figure 7.10a plots Sh as a function of [R − (R S /L)] for various cavity heights, where a dramatic increase in the intensity of DDC is clearly seen. As d increases from 0.01 m to 1 m, [R − (R S /L)] amplifies almost six orders of magnitude. As a result, Sh shows an increase of nearly fiftyfold. However, as shown in Fig. 7.10b, the enhanced convection does not bring about a sharp growth in m, ˙ which contrarily decreases gently with d. This counterintuitive phenomenon can be interpreted by Eq. (7.30). In the COO of state A, m˙ can be expressed by

7.4 Numerical Simulations

 m˙ = −ξT (T1 − T2 ) − ξ p ρref gd

139



1 1 − ξc d + 100K M2 ρref D Sh

−1

.

(7.41)

The numerator decreases with d (since ξ p > 0), while d is coupled with Sh in the denominator. According to Eq. (7.40), d/Sh ∝ d 1−3×0.2793 = d 0.1621 is an increasing function. Therefore, m˙ decreases monotonically with d, as shown in Fig. 7.10b. The above arguments suggest that the exponent in Eq. (7.40) is crucial to the behavior of m˙ versus d. If the exponent is larger than 1/3, the denominator also decreases with d. Therefore, the relationship becomes rather complicated and depends on specific circumstances. In this section, the performance of the model operating at reference state A was evaluated based on a series of numerical simulations of DDC in COO. A power-law relation between Sh and [R − (R S /L)] was identified for the chaotic motions. The exponent in this power law influences the dependence of m˙ on d. When the exponent in this power law is smaller than 1/3, as is currently the case, m˙ decreases with d. Otherwise, the relationship depends on specific situations. The DDC achieves an overall tenfold increase in m. ˙ The performance of the conceptual model operating at the point A was confirmed to be better than the previous experimental results of crystal growth rates reported in [22].

7.5 Conclusions In this chapter, a conceptual model is developed to combine extraction and crystal growth in a binary mixture by virtue of the critical behavior of solubility near the CEPs. The CO2 -naphthalene system is employed as a reference system. Theoretical optimizations are performed to maximize the performance, indicated by the cross-boundary flow rate of the solute m. ˙ Thermodynamic optimization identifies that the newly defined pseudo-CEPs are preferable reference states. The temperature at the bottom wall should be hotter than that at the top wall to take advantage of the convection cooperatively driven by T and c. In each near-CEP region, the performance is promoted when the corresponding CEP is approached. Moreover, hydrodynamic optimization reveals that pseudo-LCEPs are preferable to pseudoUCEPs due to their hydrodynamic excellence. Numerical simulations based on the most favorable reference state among all available states are carried out to evaluate the performance in the COO. A power-law relation between Sh and [R − (R S /L)] is identified. If the exponent of the power law is less than 1/3, m˙ decreases with d. Otherwise, the relationship depends on specific situations. The performance of the current model operating at the optimized reference state is confirmed to be better than the previous experimental results in terms of crystal growth rates. The model presented in this study features a simple structure and excellent performance, which provides a promising configuration for coupled extraction and crystal growth apparatuses. Furthermore, the methods and procedures reported here will serve as a reference when applying this model to other systems.

140

7 Coupled Transfer Through Boundary Reactions …

References 1. Knez Ž, Markoˇciˇc E, Leitgeb M, Primožiˇc M, Hrnˇciˇc MK, Škerget M (2014) Industrial applications of supercritical fluids: A review. Energy 77:235–243 2. Bertucco A (1999) Precipitation and crystallization techniques. Wiley, New York, pp 108–123 3. Lu BCY, Zhang D (1989) Solid-supercritical fluid phase equilibria. Pure Appl Chem 61:1065– 1074 4. Xu G, Scurto AM, Castier M, Brennecke JF, Stadtherr MA (2000) Reliable computation of high-pressure solid-fluid equilibrium. Ind Eng Chem Res 39:1624–1636 5. Gitterman M, Procaccia I (1983) Quantitative theory of solubility of supercritical fluids. J Chem Phys 78:2648–2654 6. Lu BY, Zhang D, Sheng W (1990) Solubility enhancement in supercritical solvents. Pure Appl Chem 62:2277–2285 7. Hu ZC, Davis SH, Zhang XR (2019) Onset of double-diffusive convection in near-critical gas mixtures. Phys Rev E 99:033112 8. Paolucci S (1982) Filtering of sound from the Navier-Stokes equations. Technical report (1982) 9. Westphal G, Rosenberger F (1978) On diffusive-advective interfacial mass transfer. J Cryst Growth 43(6):687–693 10. Turner JS (1973) Buoyancy effects in fluids. Cambridge monographs on mechanics. Cambridge University Press, Cambridge 11. Hu ZC, Lv W, Zhang XR (2019) Detour induced by the piston effect in the oscillatory doublediffusive convection of a near-critical fluid. Phys Fluids 31(7):074107 12. Wei Y, Hu ZC, Chong B, Xu J (2018) The Rayleigh-Bénard convection in near-critical fluids: Influences of the specific heat ratio. Numer Heat Transf Part A: Appl 74(1):931–947 13. McHugh M, Paulaitis ME (1980) Solid solubilities of naphthalene and biphenyl in supercritical carbon dioxide. J Chem Eng Data 25:326–329 14. Sauceau M, Fages J, Letourneau JJ, Richon D (2000) A novel apparatus for accurate measurements of solid solubilities in supercritical phases. Ind Eng Chem Res 39:4609–4614 15. Schmitt WJ, Reid RC (1986) Solubility of monofunctional organic solids in chemically diverse supercritical fluids. J Chem Eng Data 31:204–212 16. Škerget M, Novak-Pintariˇc Z, Knez Ž, Kravanja Z (2002) Estimation of solid solubilities in supercritical carbon dioxide: Peng-Robinson adjustable binary parameters in the near critical region. Fluid Phase Equilib 203:111–132 17. Kong CY, Sone K, Sako T, Funazukuri T, Kagei S (2011) Solubility determination of organometallic complexes in supercritical carbon dioxide by chromatographic impulse response method. Fluid Phase Equilib 302:347–353 18. Lamb DM, Adamy ST, Woo KW, Jonas J (1989) Transport and relaxation of naphthalene in supercritical fluids. J Chem Phys 93:5002–5005 19. Lauer HH, McManigill D, Board RD (1983) Mobile-phase transport properties of liquefied gases in near critical and supercritical fluid chromatography. Anal Chem 55:1370–1375 20. Sassiat PR, Mourier P, Caude MH, Rosset RH (1987) Measurement of diffusion coefficients in supercritical carbon dioxide and correlation with the equation of Wilke and Chang. Anal Chem 59:1164–1170 21. Tai CY, Cheng C (1995) Growth of naphthalene crystals from supercritical CO2 solution. AIChE J 41:2227–2236 22. Uchida H, Manaka A, Matsuoka M, Takiyama H (2004) Growth phenomena of single crystals of naphthalene in supercritical carbon dioxide. Crystal Growth Design 4:937–942 23. Nicolas C, Neau E, Meradji S, Raspo I (2005) The Sanchez-Lacombe lattice fluid model for the modeling of solids in supercritical fluids. Fluid Phase Equilib 232:219–229 24. MATLAB (2018) MATLAB symbolic math toolbox, version 8.1. The MathWorks, Natick, MA 25. Raspo I, Meradji S, Zappoli B (2007) Heterogeneous reaction induced by the piston effect in supercritical binary mixtures. Chem Eng Sci 62(16):4182–4192 26. Laesecke A, Muzny CD (2017) Reference correlation for the viscosity of carbon dioxide. J Phys Chem Ref Data 46:013107

References

141

27. Huber ML, Sykioti EA, Assael MJ, Perkins RA (2016) Reference correlation of the thermal conductivity of carbon dioxide from the triple point to 1100 K and up to 200 MPa. J Phys Chem Ref Data 45:013102 28. Vaz RV, Magalhaes AL, Silva CM (2014) Prediction of binary diffusion coefficients in supercritical CO2 with improved behavior near the critical point. J Supercritical Fluids 91:24–36 29. Lamb DM, Barbara TM, Jonas J (1986) NMR study of solid naphthalene solubilities in supercritical carbon dioxide near the upper critical end point. J Chem Phys 90:4210–4215 30. Amiroudine S, Bontoux P, Larroudé P, Gilly B, Zappoli B (2001) Direct numerical simulation of instabilities in a two-dimensional near-critical fluid layer heated from below. J Fluid Mech 442:119–140 31. Sengers JMHL (1994) Critical behavior of fluids: concepts and applications. Springer, Dordrecht, Netherlands, pp 3–38 32. Darwish M, Sraj I, Moukalled F (2009) A coupled finite volume solver for the solution of incompressible flows on unstructured grids. J Comput Phys 228(1):180–201 33. Hu ZC, Zhang XR (2017) An improved decoupling algorithm for low Mach number nearcritical fluids. Comput Fluids 145:8–20 34. Kadanoff LP (2001) Turbulent heat flow: structures and scaling. Phys Today 54:34–39

Chapter 8

Summary and Perspectives

8.1 Summary In this thesis, fundamental transport phenomena and related effects in binary mixtures at supercritical pressures are explored, where coupled heat and mass transfer exists via cross-diffusion effects or temperature-dependent boundary reactions. The main conclusions are summarized as follows. (1) Mass piston effect. Mass piston effect is a newly identified fast thermalization phenomenon driven by three cooperative or competing mechanisms: boundary velocity, Dufour effect, and concentration variation. A traveling-wave theory is developed to represent the amplitudes of the acoustic wave and its propagation, which shows good agreements with numerical simulations. The energy balance analysis suggests that mass piston effect is more efficient in terms of thermalization than energy transfer, especially at near-critical states. Therefore, mass piston effect is an efficient thermalization mechanism at near-critical states. (2) Concentration gradient in a diffusive steady state. Under a transcritical temperature difference, the steady state of a binary mixture at supercritical pressures is strongly nonlinear in state variables and physical properties. The Soret effect induces the concentration gradient. As the critical pressure is approached and the temperature difference is enlarged, the concentration gradient is enhanced. Besides, the concentration gradient has profound effects on the relative variations of density at near-critical states, thus playing a vital role in buoyancy-driven flows. (3) Rayleigh-Bénard (RB) instability and bifurcation. The RB instability in binary mixtures at supercritical pressures is characterized by cross-diffusion effects (Soret and Dufour effects under positive separation ratio) and gravity-related effects (adiabatic temperature gradient and gravitational diffusion). Analytical criteria are derived for ideal stress-free and fixed-concentration boundary conditions, namely Eq. (5.43) for monotonic instability and Eq. (5.44) © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 Z.-C. Hu, Coupled Heat and Mass Transfer in Binary Mixtures at Supercritical Pressures, Springer Theses, https://doi.org/10.1007/978-981-16-7806-6_8

143

144

8 Summary and Perspectives

for oscillatory instability. This study discovers a new oscillatory instability under positive separation ratio, whose conditions are Eqs. (5.53), (5.54) and (5.50). They collectively guarantee that there is a sufficiently large stabilizing concentration gradient and the total diffusivity rate of concentration is less than that of temperature. Through weakly nonlinear analysis and numerical simulations, the nonlinear dynamics of RB convection near the stability threshold is studied. It is confirmed that the near-threshold RB convection is deeply influenced by a finite-amplitude (FA) instability. In fact, the FA instability requires a sufficiently large stabilizing concentration gradient, whose necessary condition is Eq. (5.53). Interestingly, when oscillatory instability occurs, the required concentration gradient is naturally established. That is, small perturbations undergo initial oscillations and then rapidly evolve into an intense aperiodic flow state. Physically speaking, FA instability is attributed to the reduction effect of convective motions on the stabilizing concentration gradient, leading to a release in buoyancy. Mathematically, this work also provides the bifurcation diagrams around the critical Rayleigh number. (4) Application-oriented cavity flow problem. A conceptual cavity flow model is developed to combine extraction and crystal growth in a binary mixture by virtue of the critical behavior of solubility near the critical end points (CEPs). The CO2 -naphthalene system is employed as a reference system. The thermodynamic optimization identifies that the newly defined pseudo-CEPs are preferable reference states. In each near-CEP region, the performance is promoted when the corresponding CEP is approached. The hydrodynamic optimization reveals that pseudo lower CEPs are preferable to pseudo upper ones due to their hydrodynamic excellence. In addition, the temperature at the bottom wall should be hotter than that at the top wall to take advantage of the convection cooperatively driven by temperature and concentration. The performance of the current model operating at the optimized reference state is confirmed to be better than the previous experimental results in terms of crystal growth rates.

8.2 Perspectives It is well illustrated in this thesis that coupled heat and mass transfer in binary mixtures at supercritical pressures brings new phenomena and mechanisms in the fluid dynamics and transport phenomena, as well as possibilities for new engineering designs. Based on the researches in this thesis, in-depth explorations in terms of experiments, theories, and applications are needed in the future. • Experimentally, a series of theories proposed in this thesis requires experimental observations, including the measurement of acoustic waves in the mass piston

8.2 Perspectives

145

effect, as well as the criterion for the onset of RB convection and the existence of FA instability. • Theoretically, future researches can start with considering variable physical properties. In this regard, possible directions include the mass piston effect induced by large diffusion flux, transient turbulent flows of pseudo-boiling, and the RB convection with strong non-Boussinesq effects. • In terms of applications, future attempts can be made to propose new specific applications in several engineering fields involving supercritical pressure fluids from the principles of coupled heat and mass transfer. It should also be noted that even in the basic theory of binary mixtures at supercritical pressures, there are little experimental data on physical properties, so the physical model is still very inadequate. This brings about huge difficulties to the study of fluid dynamics and transfer phenomena. Efforts from various fields are still in need.

Appendix A

Numerical Methods for the Linear Stability Analysis of Rayleigh-Bénard Instability under Realistic Boundary Conditions

This appendix provides the numerical method for predicting critical Rayleigh number R crit , which is related to performing linear stability analysis for the system of governing equations (5.21)–(5.24), base state Eq. (5.25) and boundary conditions Eq. (6.2). For the sake of brevity, the prime symbols accompanying dimensionless variables are omitted.

A.1

Discretization of the Generalized Eigenvalue Problem

The formulation of stability problem has been derived in Sect. 5.2.1. Equation (5.35) is the final generalized eigenvalue problem, where σ is the eigenvalue, and [w, ˆ Tˆ , ζˆ ]T is the eigenvector. For no-slip and impermeable realistic boundary conditions, the boundary conditions for [w, ˆ Tˆ , ζˆ ]T are given by wˆ =

∂ wˆ ∂ ζˆ = 0, Tˆ = 0, = 0, at z = 0 and z = 1. ∂z ∂z

(A.1)

There is no analytical solutions for the generalized eigenvalue problem under the above boundary conditions. Therefore, a numerical approach will be adopted. To solve the generalized eigenvalue problem given by Eqs. (5.35) and (A.1), discretize the physical space z = [0, 1] into (N − 1) elements of equal size. Then, write finite difference approximations to spatial derivatives at grid points using the five-point stencils:  2  −ς j−2 +16ς j−1 − 30ς j + 16ς j+1 − ς j+2 d ς = , D 2ς j = 2 dz j 12(Δz)2

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 Z.-C. Hu, Coupled Heat and Mass Transfer in Binary Mixtures at Supercritical Pressures, Springer Theses, https://doi.org/10.1007/978-981-16-7806-6

(A.2)

147

148

Appendix A: Numerical Methods for the Linear Stability Analysis …



d4 ς D ςj = dz 4

 =

4

j

ς j−2 − 4ς j−1 + 6ς j − 4ς j+1 + ς j+2 , (Δz)4

(A.3)

where Δz = 1/(N − 1) is the distance between two adjacent grid points, and ς ∈ {w, ˆ Tˆ , ζˆ }, j = {4, 5, ..., N − 3} is the index of grid points (from bottom to top). For near-boundary grid points, the above schemes should be modified combining the boundary conditions: 21wˆ 2 − 12wˆ 3 + 3wˆ 4 , 4(Δz)4 −4wˆ 2 + 6wˆ 3 − 4wˆ 4 + wˆ 5 = , (Δz)4 wˆ N −4 − 4wˆ N −3 + 6wˆ N −2 − 4wˆ N −1 = , (Δz)4 3wˆ N −3 − 12wˆ N −2 + 21wˆ N −1 = , 4Δz 4 −2wˆ 2 + wˆ 3 = , (Δz)2 16wˆ 2 − 30wˆ 3 + 16wˆ 4 − wˆ 5 = , 12(Δz)2 −wˆ N −4 + 16wˆ N −3 − 30wˆ N −2 + 16wˆ N −1 = , 12(Δz)2 wˆ N −2 − 2wˆ N −1 = , (Δz)2 −2Tˆ2 + Tˆ3 = , (Δz)2 16Tˆ2 − 30Tˆ3 + 16Tˆ4 − Tˆ5 = , 12(Δz)2 −TˆN −4 + 16TˆN −3 − 30TˆN −2 + 16TˆN −1 = , 12(Δz)2 TˆN −2 − 2TˆN −1 = , (Δz)2 −2ζˆ2 + 2ζˆ3 = , 3(Δz)2 44ζˆ2 − 89ζˆ3 + 48ζˆ4 − 3ζˆ5 = , 36(Δz)2 −3ζˆ N −4 + 48ζˆ N −3 − 89ζˆ N −2 + 44ζˆ N −1 = , 36(Δz)2

(D 4 w) ˆ 2= (D 4 w) ˆ 3 (D 4 w) ˆ N −2 (D 4 w) ˆ N −1 ˆ 2 (D 2 w) (D 2 w) ˆ 3 (D 2 w) ˆ N −2 (D 2 w) ˆ N −1 (D 2 Tˆ )2 (D 2 Tˆ )3 (D 2 Tˆ ) N −2 (D 2 Tˆ ) N −1 (D 2 ζˆ )2 (D 2 ζˆ )3 (D 2 ζˆ ) N −2

Appendix A: Numerical Methods for the Linear Stability Analysis …

(D 2 ζˆ ) N −1 =

149

2ζˆ N −2 − 2ζˆ N −1 . 3(Δz)2

Given R, k, Rad , Pr , L, ψ and Q, using above stencils, Eq. (5.35) can be transformed into a linear system of the unknowns X = [wˆ 2 , wˆ 3 , ..., wˆ N −1 , Tˆ2 , Tˆ3 , ..., TˆN −1 , ζˆ2 , ζˆ3 , ..., ζˆN −1 ]T .

(A.4)

Meanwhile, A and B are square matrices of order (3N − 6). Next, the eig function in MATLAB is employed to obtain the (3N − 6) eigenvalues and their corresponding eigenvectors. Each eigenvector is a solution of X, and the corresponding eigenvalue σ = σr + iω is its growth rate. Finally, the maximum value of σr is the largest growth rate for the given (R, k, Rad , Pr, L , ψ, Q), denoted as σrmax  . Given (Rad , Pr, L , ψ, Q), linear stability analysis is to determine (R crit , k crit ), satisfying that if and only if R > R crit , there is k = k crit making σr > 0. This is essentially a two-degree-of-freedom optimization problem of (R, k). In the following, a numerical method will be proposed based on the nature of the problem.

A.2

Algorithm for the Optimization Problem

For the two-degree-of-freedom optimization problem, a stepwise algorithm is developed in this study. As shown in Fig. A.1, the algorithm has a double-layer structure. 1. The inner layer realizes the optimization of k for a given R, namely finding  the most unstable k (denoted by k crit ) within a preassigned range [0, kmax ]. The crit  is denoted by σrmax  , which is the initial growth rate of corresponding σr of k perturbations for the given R. 2. The outer layer achieves the optimization of R = (Rl + Rr )/2 by successive  shrinking [Rl , Rr ] to make σrmax  → 0. As a result, R and k crit tend to R crit and crit k , respectively. The complex of inner layer for the calculation of σrmax  is caused by two situations encountered in practice. (i) When k = 0, σ = 0 is always an eigenvalue. Figure A.2 plots the process of losing stability as R increases for the case of Rad = 4.6546 × 108 , Pr = 8.9037, L = 1.0462, Q = 0.0619 and ψ = 15.7096. The situation of k = 0, σrmax  = 0 is clearly seen as all curves start from the origin. Besides, the losing stability is accomplished by the hump part in the curve. Therefore, the local extreme value of σrmax  in the hump part should be captured by the algorithm instead of the overall maximum value. (ii) k crit → 0 is possible. Figure A.3 plots the process of losing stability as R increases for the case of Rad = 6.4744 × 108 , Pr = 14.7900, L = 6.8439, Q = 0.3044 and ψ = 3.3074. The hump part originates from k = 0 and bulges out gradually as R increases, leading to the instability.

150

Appendix A: Numerical Methods for the Linear Stability Analysis …

Outer layer

Start

Inner layer

Find Rl and Rr obeying max r

' |R

Rl

R

0

max r

( Rl

Rr ) / 2

Calculate

max

' |R

max r

Calculate r '' for several k uniformly distributed in [0, kmax].

Rr

0

max

'

Is Rl and Rr close enough?

max r

' = Maximum value of

max r

''

Yes No

Does extreme value of rmax '' exist?

Adjust Rl and Rr

No

No

Is maximum r '' found at k = 0?

max r

'=

Yes max r

' = Extreme value of

max r

''

Yes End

Fig. A.1 The algorithm for the two-degree-of-freedom optimization problem Fig. A.2 The process of losing stability as Rayleigh number R increases. The case of Rad = 4.6546 × 108 , Pr = 8.9037, L = 1.0462, Q = 0.0619 and ψ = 15.7096 is considered here. k is the wave number, and σrmax  is the maximum real part of the growth rate for any given (R, k) pair

Based on these two situations, the inner layer takes both the maximum and local extreme values of σrmax  as alternatives of σrmax  . If the maximum value is obtained at k = 0, then turn to find the local extreme value. Once found, set σrmax  as this local extreme value; Once not found, set σrmax  = −∞ to indicate the stable state. If the maximum value is obtained at k > 0, the fluid is unstable, so set σrmax  as this maximum value. Besides, the algorithm proposed in this work also requires kmax > k crit . In practice, one can start from a sufficiently large value, such as kmax = 100, then adjust the range of k and recalculate until sufficient accuracy of k crit is reached.

Appendix A: Numerical Methods for the Linear Stability Analysis …

151

Fig. A.3 The process of losing stability as Rayleigh number R increases. The case of Rad = 6.4744 × 108 , Pr = 14.7900, L = 6.8439, Q = 0.3044 and ψ = 3.3074 is considered here. k is the wave number, and σrmax  is the maximum real part of the growth rate for any given (R, k) pair

A.3

Numerical Codes for the Linear Stability Analysis

This section provides the examples of MATLAB codes for the linear stability analysis of Rayleigh-Bénard instability under realistic boundary conditions. Code A.1 Function calculate_Dmat realizes the calculations of D 4 w, ˆ D 2 w, ˆ D 2 Tˆ and D 2 ζˆ . 1

f u n c t i o n [ Dw4 , Dw2 , DT2 , D z e t a 2 ] = c a l c u l a t e _ D m a t ( N )

2 3 4 5 6 7 8 9 10 11 12 13

dz = 1/( N -1) ; Dw4 = 6* diag ( ones ([1 , N -2]) ) + ... - 4* diag ( ones ([1 , N -3]) ,1) - 4* diag ( ones ([1 , N -3]) , -1) + ... + diag ( ones ([1 , N -4]) ,2) + diag ( ones ([1 , N -4]) , -2) ; Dw4 (1 ,1) = 21/4; Dw4 (1 ,2) = -3; Dw4 (1 ,3) = 3/4; Dw4 ( end ,end ) = 21/4; Dw4 ( end ,end -1) = -3; Dw4 ( end ,end -2) = 3/4; Dw4 = Dw4 /( dz ^4) ;

14 15 16 17 18 19 20 21 22 23 24 25 26

Dw2 = -2 .5 * diag ( ones ([1 , N -2]) ) + ... 4/3* diag ( ones ([1 , N -3]) ,1) + ... -1/12* diag ( ones ([1 , N -4]) ,2) + ... 4/3* diag ( ones ([1 , N -3]) , -1) + ... -1/12* diag ( ones ([1 , N -4]) , -2) ; Dw2 (1 ,1) = -2; Dw2 (1 ,2) = 1; Dw2 (1 ,3) = 0; Dw2 ( end ,end ) = -2; Dw2 ( end ,end -1) = 1; Dw2 ( end ,end -2) = 0; Dw2 = Dw2 /( dz ^2) ;

27 28

DT2 = Dw2 ;

29 30 31 32 33

Dzeta2 = -2 .5 * d i a g ( ones ([1 , N -2]) ) + ... 4/3* diag ( ones ([1 , N -3]) ,1) + ... -1/12* diag ( ones ([1 , N -4]) ,2) + ... 4/3* diag ( ones ([1 , N -3]) , -1) + ...

152 34 35 36 37 38 39 40 41 42 43 44 45

Appendix A: Numerical Methods for the Linear Stability Analysis … -1/12* diag ( ones ([1 , N -4]) , -2) ; D z e t a 2 (1 ,1) = -2/3; D z e t a 2 (1 ,2) = 2/3; D z e t a 2 (1 ,3) = 0; D z e t a 2 (2 ,1) = 1 1 / 9 ; D z e t a 2 (2 ,2) = -89/36; D z e t a 2 ( end ,end ) = -2/3; D z e t a 2 ( end ,end -1) = 2/3; D z e t a 2 ( end ,end -2) = 0; D z e t a 2 ( end -1 ,end ) = 11/9; D z e t a 2 ( end -1 ,end -1) = -89/36; D z e t a 2 = D z e t a 2 / dz ^2;

46 47

end

Code A.2 Function calculate_sigma_max_pp realizes the calculation of σrmax  . 1 f u n c t i o n [ s i g m a _ m a x ] = ... c a l c u l a t e _ s i g m a _ m a x _ p p (N , R , k , PSI ,L ,Q , Pr , Rad ) 2 3

[ Dw4 , Dw2 , DT2 , D z e t a 2 ] = c a l c u l a t e _ D m a t ( N ) ;

4 5 6

M1 = diag ( ones (1 , N -2) ) ; M0 = diag ( zeros (1 , N -2) ) ;

7 8 9 10

M11 = Dw4 - 2* k ^2 . * Dw2 + k ^4* M1 ; M21 = ( R - Rad ) * M1 ; M31 = (1/ Q - PSI ) * Rad * M1 ;

11 12 13 14 15 16 17

M12 M13 M22 M23 M32 M33

= = = = = =

-k ^ 2 * ( 1 + PSI ) * M1 ; k ^2* M1 ; DT2 - k ^2 . * M1 ; L * Q * PSI *( Dzeta2 - k ^2* M1 ) ; PSI *( DT2 - k ^2 . * M1 ) ; L *(1+ Q * PSI ^2) *( Dzeta2 - k ^2* M1 ) ;

18 19 20

A = [[ M11 , M12 , M13 ];[ M21 , M22 , M23 ];[ M31 , M32 , M33 ]]; B = [[1/ Pr *( Dw2 - k ^2* M1 ) , M0 , M0 ];[ M0 , M1 , M0 ];[ M0 , M0 , M1 ]];

21 22 23

[ ¬ , sigma ] = eig (A , B ) ; sigma = diag ( sigma ) ;

24 25 26

[ ¬ , order ] = sort ( real ( sigma ) , ' d e s c e n d ' ) ; s i g m a _ m a x = sigma ( order (1) ) ;

27 28

end

Code A.3 Function calculate_sigma_max_p realizes the calculation of σrmax  . 1 f u n c t i o n [ s i g m a _ m a x _ p , k _ c r i t _ p ] = ... c a l c u l a t e _ s i g m a _ m a x _ p ( kmax , N , R , PSI ,L , Q , Pr , Rad ) 2 3 4

k_range = l i n s p a c e (0 , kmax ,51) ; s i g m a _ m a x _ p p = [];

5 6

for k = k _ r a n g e

Appendix A: Numerical Methods for the Linear Stability Analysis … 7

8

[ s i g m a _ m a x _ p p ( end +1) ] = ... c a l c u l a t e _ s i g m a _ m a x _ p p (N , R , k , PSI ,L ,Q , Pr , Rad ) ; end

9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27

[ ¬ , loc ] = max ( real ( s i g m a _ m a x _ p p ) ) ; if loc == 1 [ val_pks , l o c _ p k s ] = f i n d p e a k s ( real ( s i g m a _ m a x _ p p ) ) ; v a l _ p k s ( l o c _ p k s ==1) = []; l o c _ p k s ( l o c _ p k s ==1) = []; if i s e m p t y ( l o c _ p k s ) k_crit_p = 0; s i g m a _ m a x _ p = - inf ; else [¬ , l o c _ m p ] = max ( v a l _ p k s ) ; loc_pks = loc_pks ( loc_mp ); k_crit_p = k_range ( loc_pks ); sigma_max_p = sigma_max_pp ( loc_pks ); end else k_crit_p = k _ r a n g e ( loc ) ; s i g m a _ m a x _ p = s i g m a _ m a x _ p p ( loc ) ; end

28 29

end

Code A.4 Function calculate_R_crit realizes the calculations of R crit , k crit and crit . 1 f u n c t i o n [ R_crit , k_crit , o m e g a _ c r i t ] = ... calculate_R_crit (

kmax ,N , PSI ,L ,Q , Pr , Rad )

2 3 4 5 6 7 8

9 10 11 12 13 14

% get the range R_l , R_r Rr = 1; s i g m a r = -1; while r e a l ( s i g m a r ) < 0 Rr = Rr *10; [ sigmar , ¬ ] = ... c a l c u l a t e _ s i g m a _ m a x _ p ( kmax , N , Rr , PSI ,L , Q , Pr , Rad ) ; end if Rr == 10 Rl = 0; else Rl = Rr /10; end

15 16 17 18 19 20

21 22 23 24 25 26

% s h r i n k [ Rl , Rr ] k _ c r i t _ p = 1; while ( 2*( Rr - Rl ) /( Rr + Rl ) ≥ 1 e -3 || k _ c r i t _ p == 0 ) R = ( Rl + Rr ) /2; [ s i g m a _ m a x _ p , k _ c r i t _ p ] = ... c a l c u l a t e _ s i g m a _ m a x _ p ( kmax , N , R , PSI ,L , Q , Pr , Rad ) ; if real ( s i g m a _ m a x _ p ) >0 Rr = R ; else Rl = R ; end end

27 28

R_crit = R;

153

154 29 30

Appendix A: Numerical Methods for the Linear Stability Analysis … k_crit = k_crit_p ; o m e g a _ c r i t = imag ( s i g m a _ m a x _ p ) ;

31 32

end

Appendix B

The Solubility of Naphthalene in Supercritical CO2

As proposed in [4], when T and p are known, the solubility is given by ys =

  VS ( p − psub ) psub exp , φp RT

(B.1)

where ys is the mole fraction of the solute in saturated solution (the solubility in mole fraction), psub is the sublimation pressure of the solid, VS is the molar volume of the solid, R = 8.314 J/(mol · K) is the ideal gas constant and φ is the fugacity coefficient of the solute. psub of naphthalene is calculated by two coupled equations [4,3,1], which are combined and simplified into ln psub = 37.7612 −

10890.1411 , T

(B.2)

where the unit of T is K, and the unit of the calculated psub is Pa. VS = 1.1 × 10−4 m3 /mol is approximated as a constant. φ is calculated from the Peng-Robinson equation of state written for a mixture: p=

a RT − 2 , V −b V + 2bV − b2

(B.3)

where V is the molar volume, and a and b are functions of the mole fraction of naphthalene (denoted by y): a = (a1 + a2 − 2a12 )y 2 + 2(a12 − a1 )y + a1 ,

(B.4)

b = (b1 + b2 − 2b12 )y + 2(b12 − b1 )y + b1 ,

(B.5)

2

with

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 Z.-C. Hu, Coupled Heat and Mass Transfer in Binary Mixtures at Supercritical Pressures, Springer Theses, https://doi.org/10.1007/978-981-16-7806-6

155

156

Appendix B: The Solubility of Naphthalene in Supercritical CO2

ai = 0.45724 

R 2 TCi2 × pCi





1 + (0.37464 + 1.54226ωi −

0.26992ωi2 )

1−

T TCi

2 ,

(B.6)

RTCi bi = 0.07780 , pCi √ a12 = a1 a2 (1 − k12 ), b1 + b2 b12 = (1 − l12 ), 2

(B.7) (B.8) (B.9)

where i = 1 and 2 represent CO2 and naphthalene, respectively. The properties required to calculate ai and bi are listed in Table B.1. The two adjustable parameters k12 and l12 are determined by fitting experiment data of solubility [4]. For the Peng-Robinson equation of state, the expression for φ is [2] ln φ =

   pV p(V − b) − 1 − ln + RT RT

⎡ √  ⎤   V + 1− 2 b a aN bN

− ln ⎣ √ √  ⎦, a b 2 2b RT V + 1+ 2 b bN b



(B.10)

where a N = 2y(a2 − a12 ) + 2a12 , b N = (y 2 − 2y)(2b12 − b1 − b2 ) + 2b12 − b1 .

(B.11) (B.12)

Since φ depends on y, iterations among Eqs. (B.1)–(B.10) are required to determine ys .

Table B.1 Critical properties, acentric factors and molar masses of CO2 and naphthalene Name i TC /(K) pC /(MPa) ω/(-) M/(kg/mol) CO2 Naphthalene

1 2

304.21 748.40

7.38 4.051

0.225 0.302

0.04401 0.1282

Appendix B: The Solubility of Naphthalene in Supercritical CO2

157

References 1. Baum E (1997) Chemical property estimation: theory and application. CRC Press, Boca Raton 2. McHugh M, Krukonis V (1994) Supercritical fluid extraction: principles and practice, 2nd ed. Elsevier, Amsterdam 3. Sako S, Ohgaki K, Katayama T (1988) Solubilities of naphthalene and indole in supercritical fluids. J Supercritical Fluids 1:1–6 4. Škerget M, Novak-Pintariˇc Z, Knez Ž, Kravanja Z (2002) Estimation of solid solubilities in supercritical carbon dioxide: Peng-Robinson adjustable binary parameters in the near critical region. Fluid Phase Equilibria 203:111–132

Appendix C

The Isobaric Specific Heat of Supercritical CO2 − C2 H6 System

The isobaric specific heat, denoted by c p , is calculated by [1] c p1 = R(3.259 + 1.356 × 10−3 T + 1.502 × 10−5 T 2 −2.374 × 10−8 T 3 + 1.056 × 10−11 T 4 ); c p2 = R(2.889 + 14.306 × 10−3 T + 15.978 × 10−5 T 2 −23.930 × 10−8 T 3 + 10.173 × 10−11 T 4 ); (1 − y)c p1 + yc p2 + (dH R /dT ) p,ρ cp = , (1 − y)M1 + y M2

(C.1) (C.2) (C.3)

where the unit of T is K, and c p1 and c p2 are in J/(mol · K). In the framework of Peng-Robinson equation of state, the residual enthalpy H R is expressed as 

√ V + (1 + 2)b pV T da/dT − a HR = −1+ ln . √ √ RT RT 2 2b RT V + (1 − 2)b

(C.4)

Reference 1. Poling BE, Prausnitz JM, O’Connell JP (2001) The properties of gases and liquids. 5th ed. McGraw-Hill, New York

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 Z.-C. Hu, Coupled Heat and Mass Transfer in Binary Mixtures at Supercritical Pressures, Springer Theses, https://doi.org/10.1007/978-981-16-7806-6

159