Liquid-Vapor Phase-Change Phenomena: An Introduction to the Thermophysics of Vaporization and Condensation Processes in Heat Transfer Equipment, Third Edition [3 ed.] 149871661X, 9781498716611

Table of contents :
Cover
Half Title
Title Page
Copyright Page
Dedication
Contents
Preface
Nomenclature
Author Biography
Introductory Remarks
PART I: Thermodynamic and Mechanical Aspects of Interfacial Phenomena and Phase Transitions
Chapter 1: The Liquid-Vapor Interfacial Region: A Nanoscale Perspective
1.1. A Molecular Perspective on Liquid-Vapor Transitions
1.2. The Interfacial Region - Molecular Theories of Capillarity
1.3. Nanoscale Features of the Interfacial Region
1.4. Molecular Dynamics Simulation Studies of Interfacial Region Thermophysics
1.5. Small System Effects
Reference
Problems
Chapter 2: The Liquid-Vapor Interface: A Macroscopic Treatment
2.1. Thermodynamic Analysis of Interfacial Tension Effects
2.2. Determination of Interface Shapes at Equilibrium
2.3. Temperature and Surfactant Effects on Interfacial Tension
2.4. Surface Tension in Mixtures
2.5. Near Critical Point Behavior
2.6. Effects of Interfacial Tension Gradients
References
Problems
Chapter 3: Wetting Phenomena and Contact Angles
3.1. Equilibrium Contact Angles on Smooth Surfaces
3.2. Wettability, Cohesion, and Adhesion
3.3. The Effect of Liquid Surface Tension on Contact Angle
3.4. Adsorption and Spread Thin Films
3.5. Contact Angle Hysteresis
3.6. Other Metrics for Wettability
3.7. A Nanoscale View of Wettability
3.8. Wetting of Microstructured and Nanostructured Surfaces
References
Problems
Chapter 4: Transport Effects and Dynamic Behavior at Interfaces
4.1. Transport Boundary Conditions
4.2. Kelvin-Helmholtz and Rayleigh-Taylor Instabilities
4.3. Interface Stability of Liquid Jets
4.4. Waves on Liquid Films
4.5. Interfacial Resistance in Vaporization and Condensation Processes
4.6. Maximum Flux Limitations
4.7. Non-Equilibrium and Heat Flux Effects on Interface Boundary Conditions
References
Problems
Chapter 5: Phase Stability and Homogeneous Nucleation
5.1. Metastable States and Phase Stability
5.2. Thermodynamic Aspects of Homogeneous Nucleation in Superheated Liquid
5.3. The Kinetic Limit of Superheat
5.4. Comparison of Theoretical and Measured Superheat Limits
5.5. Thermodynamic Aspects of Homogeneous Nucleation in Supercooled Vapor
5.6. The Kinetic Limit of Supersaturation
5.7. Wall Interaction Effects on Homogeneous Nucleation
5.8. Nanobubbles
References
Problems
PART II: Boiling and Condensation Near Immersed Bodies
Chapter 6: Heterogeneous Nucleation and Bubble Growth in Liquids
6.1. Heterogeneous Nucleation at a Smooth Interface
6.2. Nucleation from Entrapped Gas or Vapor in Cavities
6.3. Criteria for the Onset of Nucleate Boiling
6.4. Bubble Growth in an Extensive Liquid Pool
6.5. Bubble Growth Near Heated Surfaces
6.6. Bubble Departure Diameter and the Frequency of Bubble Release
References
Problems
Chapter 7: Pool Boiling
7.1. Regimes of Pool Boiling
7.2. Mechanisms and Models of Transport During Nucleate Boiling
7.3. Correlation of Nucleate Boiling Heat Transfer Data
7.4. Limitations of Nucleate Boiling Processes and the Maximum Heat Flux Transition
7.5. Minimum Heat Flux Conditions
7.6. Film Boiling
7.7. Transition Boiling
References
Problems
Chapter 8: Other Aspects of Boiling and Evaporation in an Extensive Ambient
8.1. Additional Parametric Effects on Pool Boiling
8.2. The Leidenfrost Phenomenon
8.3. Fluid-Wall Interactions and Disjoining Pressure Effects
8.4. Pool Boiling Heat Transfer on Micro and Nano Structured Surfaces
8.5. Fundamentals of Pool Boiling in Binary Mixtures
References
Problems
Chapter 9: External Condensation
9.1. Heterogeneous Nucleation in Vapors
9.2. Dropwise Condensation
9.3. Film Condensation on a Flat, Vertical Surface
9.4. Film Condensation on Cylinders and Axisymmetric Bodies
9.5. Effects of Vapor Motion and Interfacial Waves
9.6. Condensation in the Presence of a Noncondensable Gas
9.7. Enhancement of Condensation Heat Transfer
References
Problems
PART III: Internal Flow Convective Boiling and Condensation
Chapter 10: Introduction to Two-Phase Flow
10.1. Two-Phase Flow Regimes
10.2. Basic Models and Governing Equations for One-Dimensional Two-Phase Flow
10.3. Determination of the Two-Phase Multiplier and Void Fraction
10.4. Analytical Models of Annular Flow
10.5. Effects of Flow Passage Size and Geometry
References
Problems
Chapter 11: Internal Convective Condensation
11.1. Regimes of Convective Condensation in Conventional (Macro) Tubes
11.2. Analytical Modeling of Downflow Internal Convective Condensation
11.3. Correlation Methods for Convective Condensation Heat Transfer
11.4. Convective Condensation in Microchannels, Advanced Modeling, and Special Topics
11.5. Internal Convective Condensation of Binary Mixtures
References
Problems
Chapter 12: Convective Boiling in Tubes and Channels
12.1. Regimes of Convective Boiling in Conventional (Macro) Tubes
12.2. Onset of Boiling in Internal Flows
12.3. Subcooled Flow Boiling
12.4. Saturated Flow Boiling
12.5. Critical Heat Flux Conditions for Internal Flow Boiling
12.6. Post-CHF Internal Flow Boiling
12.7. Internal Flow Boiling in Microchannels and Complex Enhanced Flow Passages
12.8. Internal Flow Boiling of Binary Mixtures
References
Problems
Appendix I: Basic Elements of the Kinetic Theory of Gases.
Appendix II: Saturation Properties of Selected Fluids
Appendix III: Analysis Details for the Molecular Theory of Capillarity
Index

Citation preview

Liquid-Vapor Phase-Change Phenomena

Liquid-Vapor Phase-Change Phenomena An Introduction to the Thermophysics of Vaporization and Condensation Processes in Heat Transfer Equipment Third Edition

Van P. Carey

CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2020 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Printed on acid-free paper International Standard Book Number-13: 978-1-4987-1661-1 (Hardback) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copyright.com (http://www. copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com

Dedication To Judith, Elizabeth and Sean.

Contents Preface...............................................................................................................................................xi Nomenclature.................................................................................................................................. xiii Author Biography.............................................................................................................................xix Introductory Remarks......................................................................................................................xxi

Part I  T  hermodynamic and Mechanical Aspects of Interfacial Phenomena and Phase Transitions Chapter 1 The Liquid-Vapor Interfacial Region: A Nanoscale Perspective..................................3 1.1 1.2 1.3 1.4

A Molecular Perspective on Liquid-Vapor Transitions...................................... 3 The Interfacial Region - Molecular Theories of Capillarity............................ 11 Nanoscale Features of the Interfacial Region.................................................. 17 Molecular Dynamics Simulation Studies of Interfacial Region Thermophysics.................................................................................................. 21 1.5 Small System Effects........................................................................................26 References...................................................................................................................34 Problems...................................................................................................................... 36 Chapter 2 The Liquid-Vapor Interface: A Macroscopic Treatment............................................. 39 2.1 Thermodynamic Analysis of Interfacial Tension Effects................................. 39 2.2 Determination of Interface Shapes at Equilibrium.......................................... 45 2.3 Temperature and Surfactant Effects on Interfacial Tension............................. 51 2.4 Surface Tension in Mixtures............................................................................ 54 2.5 Near Critical Point Behavior............................................................................ 56 2.6 Effects of Interfacial Tension Gradients........................................................... 57 References...................................................................................................................64 Problems...................................................................................................................... 65 Chapter 3 Wetting Phenomena and Contact Angles.................................................................... 69 3.1 Equilibrium Contact Angles on Smooth Surfaces........................................... 69 3.2 Wettability, Cohesion, and Adhesion............................................................... 72 3.3 The Effect of Liquid Surface Tension on Contact Angle................................. 76 3.4 Adsorption and Spread Thin Films.................................................................. 79 3.5 Contact Angle Hysteresis................................................................................. 85 3.6 Other Metrics for Wettability...........................................................................90 3.7 A Nanoscale View of Wettability.....................................................................94 3.8 Wetting of Microstructured and Nanostructured Surfaces..............................96 References................................................................................................................. 103 Problems.................................................................................................................... 105

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Chapter 4 Transport Effects and Dynamic Behavior at Interfaces............................................ 109 4.1 4.2 4.3 4.4 4.5 4.6 4.7

Transport Boundary Conditions..................................................................... 109 Kelvin-Helmholtz and Rayleigh-Taylor Instabilities...................................... 113 Interface Stability of Liquid Jets.................................................................... 121 Waves on Liquid Films................................................................................... 127 Interfacial Resistance in Vaporization and Condensation Processes............. 134 Maximum Flux Limitations........................................................................... 142 Non-Equilibrium and Heat Flux Effects on Interface Boundary Conditions...................................................................................... 146 References................................................................................................................. 149 Problems.................................................................................................................... 151 Chapter 5 Phase Stability and Homogeneous Nucleation.......................................................... 153 5.1 5.2

Metastable States and Phase Stability............................................................ 153 Thermodynamic Aspects of Homogeneous Nucleation in Superheated Liquid.................................................................................... 164 5.3 The Kinetic Limit of Superheat...................................................................... 171 5.4 Comparison of Theoretical and Measured Superheat Limits........................ 176 5.5 Thermodynamic Aspects of Homogeneous Nucleation in Supercooled Vapor..................................................................................... 180 5.6 The Kinetic Limit of Supersaturation............................................................ 184 5.7 Wall Interaction Effects on Homogeneous Nucleation.................................. 189 5.8 Nanobubbles................................................................................................... 192 References................................................................................................................. 197 Problems.................................................................................................................... 198

Part II  Boiling and Condensation Near Immersed Bodies Chapter 6 Heterogeneous Nucleation and Bubble Growth in Liquids....................................... 203 6.1 Heterogeneous Nucleation at a Smooth Interface.......................................... 203 6.2 Nucleation from Entrapped Gas or Vapor in Cavities....................................209 6.3 Criteria for the Onset of Nucleate Boiling..................................................... 217 6.4 Bubble Growth in an Extensive Liquid Pool.................................................. 222 6.5 Bubble Growth Near Heated Surfaces........................................................... 228 6.6 Bubble Departure Diameter and the Frequency of Bubble Release............... 235 References................................................................................................................. 242 Problems.................................................................................................................... 245 Chapter 7 Pool Boiling............................................................................................................... 249 7.1 7.2 7.3 7.4 7.5 7.6

Regimes of Pool Boiling................................................................................. 249 Mechanisms and Models of Transport During Nucleate Boiling.................. 254 Correlation of Nucleate Boiling Heat Transfer Data...................................... 265 Limitations of Nucleate Boiling Processes and the Maximum Heat Flux Transition....................................................................................... 275 Minimum Heat Flux Conditions.................................................................... 291 Film Boiling................................................................................................... 293

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Contents

7.7 Transition Boiling........................................................................................... 316 References................................................................................................................. 322 Problems.................................................................................................................... 327 Chapter 8 Other Aspects of Boiling and Evaporation in an Extensive Ambient....................... 331 8.1 Additional Parametric Effects on Pool Boiling.............................................. 331 8.2 The Leidenfrost Phenomenon........................................................................ 341 8.3 Fluid-Wall Interactions and Disjoining Pressure Effects............................... 351 8.4 Pool Boiling Heat Transfer on Micro and Nano Structured Surfaces............ 362 8.5 Fundamentals of Pool Boiling in Binary Mixtures........................................ 367 References................................................................................................................. 381 Problems.................................................................................................................... 387 Chapter 9 External Condensation.............................................................................................. 391 9.1 Heterogeneous Nucleation in Vapors............................................................. 391 9.2 Dropwise Condensation.................................................................................. 395 9.3 Film Condensation on a Flat, Vertical Surface..............................................404 9.4 Film Condensation on Cylinders and Axisymmetric Bodies......................... 419 9.5 Effects of Vapor Motion and Interfacial Waves............................................. 423 9.6 Condensation in the Presence of a Noncondensable Gas............................... 428 9.7 Enhancement of Condensation Heat Transfer................................................ 438 References................................................................................................................. 442 Problems.................................................................................................................... 447

Part III  Internal Flow Convective Boiling and Condensation Chapter 10 Introduction to Two-Phase Flow............................................................................... 453 10.1 Two-Phase Flow Regimes.............................................................................. 453 10.2 Basic Models and Governing Equations for One-Dimensional Two-Phase Flow............................................................................................. 463 10.3 Determination of the Two-Phase Multiplier and Void Fraction..................... 470 10.4 Analytical Models of Annular Flow.............................................................. 487 10.5 Effects of Flow Passage Size and Geometry.................................................. 498 References.................................................................................................................500 Problems.................................................................................................................... 503 Chapter 11 Internal Convective Condensation............................................................................ 507 11.1 11.2 11.3 11.4

Regimes of Convective Condensation in Conventional (Macro) Tubes......... 507 Analytical Modeling of Downflow Internal Convective Condensation......... 511 Correlation Methods for Convective Condensation Heat Transfer................ 519 Convective Condensation in Microchannels, Advanced Modeling, and Special Topics.......................................................................................... 535 11.5 Internal Convective Condensation of Binary Mixtures................................. 542 References................................................................................................................. 549 Problems.................................................................................................................... 552

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Contents

Chapter 12 Convective Boiling in Tubes and Channels.............................................................. 555 12.1 12.2 12.3 12.4 12.5 12.6 12.7

Regimes of Convective Boiling in Conventional (Macro) Tubes................... 555 Onset of Boiling in Internal Flows................................................................. 561 Subcooled Flow Boiling................................................................................. 567 Saturated Flow Boiling................................................................................... 575 Critical Heat Flux Conditions for Internal Flow Boiling............................... 590 Post-CHF Internal Flow Boiling....................................................................608 Internal Flow Boiling in Microchannels and Complex Enhanced Flow Passages................................................................................ 625 12.8 Internal Flow Boiling of Binary Mixtures..................................................... 639 References................................................................................................................. 650 Problems.................................................................................................................... 661 Appendix I: Basic Elements of the Kinetic Theory of Gases.................................................... 663 Appendix II: Saturation Properties of Selected Fluids............................................................. 673 Appendix III: Analysis Details for the Molecular Theory of Capillarity................................ 681 Index............................................................................................................................................... 687

Preface This book was inspired by the need for instructional material for a graduate-level course on heat transfer with phase change taught in Mechanical Engineering at UC Berkeley on a yearly basis. The initial goal in assembling this book was to provide a coherent presentation of the nonequilibrium thermodynamics and interfacial phenomena associated with vaporization and condensation, as well as the heat transfer and fluid flow mechanisms in such processes. Since the initial publication of this book, this field has grown to encompass microscale and molecular-level perspectives on the physics of liquid-vapor phase transformations. Applications involving vaporization and condensation are often arising in micro- and nanoscale systems, and there is a need to educate graduate students in engineering in a way that includes this perspective. In response to this need, coverage of molecular-level perspectives of interfacial phenomena and phase transformations, and the effects of reducing system size to the microscale and nanoscale level has been expanded in this third edition. The book focuses on the fundamental thermophysics and transport principles that underlie the mechanisms of condensation and vaporization processes. Those who work in the field know that the number of technical reports, papers, and books dealing with boiling and condensation processes is enormous. Coverage of all the work in these areas is clearly impossible within the limited space of a basic text. In the interest of conciseness, the tone of presentation in this book is illustrative rather than exhaustive. In most cases, the basic physical mechanisms associated with a particular phasechange phenomena are describe in detail, followed by a representative sample of the best models applicable to the circumstances of interest. Simple idealized models are sometimes explored to gain insight into how the mechanisms of the process interact, and to provide a more direct understanding of trends and parametric relationships. Throughout the text, the importance of basic phenomena to a wide variety of applications is discussed. The sequence of chapters in this book was chosen to facilitate instruction at the advanced undergraduate or graduate level in mechanical or chemical engineering. The chapters in Part I of the book deal entirely with nonequilibrium thermodynamics and interfacial phenomena. As it is covered first, this material provides a useful foundation on which the later discussion of boiling and condensation phenomena can build. Part II covers boiling and condensation processes on the external surfaces of a body exposed to an extensive ambient. The material on internal flow boiling and condensation in Part III follows that in Part II because many of the concepts that apply to external condensation and boiling apply in a modified form to convective boiling or condensation in tubes. The coverage of these topics focuses on the fundamental physics and methods to analyze and predict performance in applications. Although space limitations do not permit extensive discussion of application-related topics, references are provided throughout the text that will lead an interested reader to more detailed discussion of related special topics and applications. The progressive flow of ideas provided by the book’s structure should make it useful to practicing engineers who wish to gain a further understanding of the thermophysics of vaporization and condensation processes through individual study. The author is indebted to the numerous students at Berkeley and colleagues in the heat transfer and thermophysics community who have provided useful comments, criticisms, and suggestions regarding the content of this text. Changes in this third edition reflect efforts to respond to this input. The changes also have aimed to include results of recent research into micro- and nanoscale aspects of vaporization and condensation processes. The author is particularly grateful to Professor John H. Lienhard IV, Professor Dennis O’Neal, and Professor Ralph Seban for their insightful comments on the earliest versions of this text. More recent input from Professor John Rose, Professor

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Preface

Nenad Miljkovoc, and Professor Jungho Kim has also helped guide revisions to update this text. An expression of gratitude is also due to the many investigators who have contributed to this area over the past 70 years. It is only through their combined efforts that a clear overview of the physics of these processes is possible. Van P. Carey Berkeley, California

Nomenclature A surface or cross-sectional area Af fin area Ao tube open area Ap prime surface area b fin height Bo boiling number [= q ′′ /Ghlv ] BoL Bond number [= g(ρl − ρv ) L2 /σ (where the length scale L depends on the circumstances of interest)] boiling number (= q n /Ghlv ) cpl liquid specific heat cpv vapor specific heat Ca capillary number [= µ lV /σ ] Co convection number {= [(1 – x)/x]0.8[ρv /ρl ]0.5} dd bubble departure diameter dt tube diameter dh hydraulic diameter based on wetted perimeter (dP/dz)fr frictional component of two-phase pressure gradient (dP/dz)l pressure gradient for liquid flow alone through tube (dP/dz)le pressure gradient for entire flow as liquid through tube (dP/dz)lo pressure gradient for entire flow as liquid through tube (dP/dz)v pressure gradient for vapor flow alone through tube D tube diameter D1* binary diffusion coefficient for more volatile component D12 binary diffusion coefficient for species 1 and 2 E mass fraction of liquid phase entrained in the core during annular flow E ′′ rate of entrainment in mass of droplets per unit time per unit of wall area f bubble frequency friction factor fl friction factor for liquid flowing alone in tube fv friction factor for vapor flowing alone in tube F Helmholtz function (= U – TS) force Chen correlation parameter FTD Taitel-Dukler flow regime parameter Frle Froude number [= G 2 /(ρl2 gD)] g gravitational acceleration specific Gibbs function G Gibbs function (= H – TS) mass flux through tube or channel h local heat transfer coefficient h mean heat transfer coefficient ˆ specific enthalpy on per unit mass basis h h* mass transfer coefficient hl heat transfer coefficient for the liquid phase flowing alone in the tube hle heat transfer coefficient for entire flow as liquid hlo heat transfer coefficient for entire flow as liquid hlv latent heat of vaporization per unit mass xiii

xiv

Nomenclature

Hf fin height jl volume flux of liquid [= G (1 − x ) /ρl ] jv volume flux of vapor [= Gx /ρv ] J flux of droplet or bubble embryos through size space; rate of generation of embryos per unit volume J* dimensionless droplet flux in size space Ja Jakob number [= c p ∆T /hlv (where the choices of cp and ∆T depend on the circumstances of interest)] k B Boltzmann constant (= 1.3805 × 10 −23 J/K) k d deposition coefficient in model of entrainment and deposition for annular flow kl thermal conductivity of liquid kv thermal conductivity of vapor KTD Taitel-Dukler flow regime parameter L tube length Lb bubble or capillary length scale = [σ /g(ρl − ρv )]1/ 2 Lf fin length m mass of one molecule mass flow rate of condensate in liquid film per unit width of surface m ′ m ′′ mass flux M mass mass of one molecule M molecular weight NA Avogadro’s number (= 6.02 × 1026 molecules/kg mol) Nl number of liquid molecules per unit volume Nn number of embryos of n molecules at equilibrium per unit volume N n′′ number of embryos of n molecules at equilibrium per unit of interface area P pressure Pc critical pressure Pl ambient liquid pressure Ppi(T) saturation pressure of pure component i in mixture at temperature T Pr reduced pressure [= P /Pc ] Pv ambient vapor pressure Prl liquid Prandtl number Prt turbulent Prandtl number (= ε M /ε H ) Prv vapor Prandtl number q ′′ heat flux qcrit critical heat flux ′′ qmax maximum (critical) heat flux ′′ qmin minimum heat flux on pool boiling curve ′′ qmkc maximum heat flux limit dictated by kinetic theory for condensation ′′ qmkv maximum heat flux limit dictated by kinetic theory for vaporization ′′ q total heat transfer rate R ideal gas constant on a per unit mass basis liquid jet radius R universal gas constant (= 8.3144 kJ/(kg mol K)) Re Reynolds number ReF film Reynolds number Rel Reynolds number for liquid phase flowing alone [= G (1 − x ) D /µ l ] Rele Reynolds number for entire flow as liquid [= GD /µ l ] Relo Reynolds number for entire flow as liquid [= GD /µ l ] ReL film Reynolds number [= 4 m ′ /µ l ]

{

}

xv

Nomenclature

Rev Reynolds number for vapor phase flowing alone [= GxD /µ v ] s specific entropy distance between fins in an offset matrix S entropy supersaturation ratio  = ( Pv )SSL /Psat ( Tv ) suppression factor in Chen correlation slip ratio [= uv /ul ] Sc Schmidt number [= ν /D12 ] Spls spreading coefficient [= − (∂ F / ∂ Asl )] St Stanton number (= h /Gc p ) Su subcooling number {= [c pl (Tsat − Tin )/hlv ][(ρl − ρv )/ρv ]} T temperature Tbp bubble point temperature Tc critical temperature Tdp dew point temperature Ti interface temperature Tin fluid temperature at tube inlet Tr reduced temperature [= T /Tc ] Tsat saturation temperature TTD Taitel-Dukler flow regime parameter Tw wall temperature u specific internal energy velocity component in the x direction ul liquid mean downstream velocity in two-phase flow [= G (1 − x )/ρl (1 − α)] uv vapor mean downstream velocity in two-phase flow [= Gx /ρv α] U internal energy ˆ v molar specific volume (volume per kmol) v specific volume (volume per unit mass) velocity component in the y direction vc critical specific volume vr reduced specific volume [= v /vc or vˆ /vˆc ] V volume velocity w velocity component in the z direction w mean distance between fins W wickability Wg mass fraction of noncondensable gas Wv mass fraction of condensable vapor We Weber number [= G 2 D /ρσ ] x coordinate (downstream coordinate for external flows) mass quality xa actual ratio of vapor mass flow rate to total mass flow rate xcrit dry out quality xe equilibrium quality xi mass fraction of species i in liquid binary mixture x1 mass fraction of more volatile component in liquid binary mixture xˆi mole fraction of species i in liquid binary mixture xˆ1 mole fraction of more volatile component in liquid binary mixture X Martinelli parameter = [(dP /dz )l /(dP /dz ) v ] 1/ 2 Xtt Martinelli parameter for turbulent-turbulent flow y coordinate, surface normal coordinate for external flows

{

}

xvi

yi y1 yˆi yˆ1 y+ z α α c α T α Tl α Tv β β f β max γ δ δ + δ f δ t ∆Tvl ε ε H ε M θ θa θr θrot ,m λ c λ D µ µ l µˆ l µ v µˆ v vl vv ˆ ρ ρl ρv ρn ρN ρN ,l ρN ,v

Nomenclature

mass fraction of species i in vapor binary mixture mass fraction of more volatile component in vapor binary mixture mole fraction of species i in vapor binary mixture mole fraction of more volatile component in vapor binary mixture dimensionless y coordinate [= y τ 0 / ρl /νl ] coordinate (downstream coordinate for tube flows) wave number void fraction critical wave number thermal diffusivity [= k /ρc p ] thermal diffusivity of liquid thermal diffusivity of vapor frequency volume fraction of vapor volume fraction of liquid flowing in liquid film on tube wall frequency of most rapidly growing disturbance multiplier in Baroczy correlation film thickness dimensionless film thickness [= δ τ 0 /ρl /νl ] fin thickness thermal boundary-layer thickness temperature difference across liquid-vapor interface emissivity eddy diffusivity eddy diffusivity of heat for turbulent flow eddy diffusivity of momentum for turbulent flow liquid contact angle angular coordinate advancing contact angle receding contact angle mean rotational temperature for polyatomic molecule critical wavelength most dangerous wavelength absolute viscosity chemical potential liquid viscosity liquid chemical potential liquid molar chemical potential vapor viscosity vapor chemical potential vapor molar chemical potential liquid kinematic viscosity vapor kinematic viscosity molar density liquid density vapor density number density number density liquid number density vapor number density

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Nomenclature

σ σ lv ˆ σ σ SB τ τi τ 0 τ w φl φlo φ v ξ ψ ψ S Ω

interfacial tension, or equivalently, the interfacial free energy per unit areas of interface interfacial tension of liquid vapor interface at equilibrium accommodation coefficient Stefan-Boltzmann constant (= 5.67 × 10 –8 W/m2 K4) shear stress shear stress at interface shear stress at wall shear stress at wall two-phase multiplier = [(dP /dz ) fr /(dP /dz )l ] 1/ 2

{ two-phase multiplier {= [(dP /dz ) two-phase multiplier {= [(dP /dz )

fr

/(dP /dz )lo ]

} } }

1/ 2

/(dP /dz ) v ] 1/ 2 number of translational and rotational energy storage modes in molecule volumetric free energy density [= F/V] normalized heat transfer coefficient [= h /hl ] vorticity angle between tube axis and horizontal fr

SUBSCRIPTS

a actual value b bulk bp bubble point c properties evaluated at the critical point dp dew point ex exit conditions f film fin i interface in inlet conditions l liquid corresponding to the liquid phase flowing alone le corresponding to the entire flow as liquid corresponding to liquid flow in equivalent separate cylinder lo corresponding to the entire flow as liquid sat corresponding to saturation conditions SSL supersaturation limit v vapor corresponding to the vapor phase flowing alone ve corresponding to vapor flow in equivalent separate cylinder w wall value ∞ far ambient conditions 0 wall value

Author Biography Van P. Carey, a Professor in the Mechanical Engineering Department, holds the A. Richard Newton Chair in Engineering at the University of California at Berkeley. Carey is widely recognized for his research in the areas of micro- and nanoscale thermophysics, interfacial phenomena, and transport in liquid-vapor phase-change processes. His research interests also include development of new methods for computational modeling and simulation of energy conversion and transport processes. Carey’s research has covered a variety of applications areas, including solar thermal power systems, building and vehicle air conditioning, phase-change thermal energy storage, Rankine cycle power for manned space missions, heat pipes for aerospace applications, high heat flux cooling of electronics, energy efficiency of information processing systems, microgravity boiling, and nanostructured surfaces for enhancing droplet evaporation and boiling processes. Carey is a Fellow of the American Society of Mechanical Engineers (ASME) and the American Association for the Advancement of Science, and he has also served as the Chair of the Heat Transfer Division of ASME. Carey has received the James Harry Potter Gold Medal in 2004 for his eminent achievement in thermodynamics and the Heat Transfer Memorial Award in the Science category (2007) from the ASME. Carey is also a three-time recipient of the Hewlett Packard Research Innovation Award for his research on electronics thermal management and energy efficiency (2008, 2009, and 2010), and Carey received the 2014 Thermophysics Award from the American Institute of Aeronautics and Astronautics.

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Introductory Remarks Liquid-vapor phase-change processes play a vital role in many technological applications. The virtually isothermal heat transfer associated with boiling and condensation processes makes their inclusion in power and refrigeration cycles highly advantageous from a thermodynamic efficiency viewpoint. Liquid-vapor phase-change processes are also encountered in petroleum and chemical processing, liquefaction of nitrogen and other gases at cryogenic temperatures, and during evaporation or precipitation of water in the earth’s atmosphere. In addition, the high heat transfer coefficients associated with boiling and condensation have made the use of these processes increasingly attractive in the thermal control of compact devices having high heat dissipation rates. Applications of this type include the use of boiling heat transfer to cool electronic components in mainframe computers and the use of compact evaporators and condensers for thermal control of aircraft avionics and spacecraft environments. Liquid-vapor phasechange processes are also of critical importance to nuclear power plant design, both because they are important in normal operating circumstances, and because they dominate many of the accident scenarios that are studied in detail as part of the design evaluation. The heat transfer and fluid flow processes associated with liquid-vapor phase-change phenomena are typically among the more complex transport circumstances encountered in engineering applications. These processes may have all the complexity of single-phase convective transport (nonlinear effects, transition to turbulence, and three-dimensional or time-varying behavior) plus additional elements resulting from motion of the interface, nonequilibrium effects, or other complex dynamic interactions between the phases. Due to the highly complex nature of these processes, development of methods to predict the associated heat and mass transfer has often proved to be a formidable task. Nevertheless, the research efforts of numerous scientists over several decades have provided a fairly clear understanding of many aspects of vaporization and condensation processes in power and refrigeration systems. On the other hand, some elements of vaporization and condensation phenomena are not well understood, and research in these areas continues. There is a vast quantity of published information on liquid-vapor phase-change phenomena in textbooks, monographs, and journal articles. Because such processes occur in a wide variety of applications and because the thermodynamic, fluid mechanics, and heat transfer aspects of these processes appeal to different groups of investigators, technical papers on various aspects of vaporization and condensation processes are found in a number of journals, serial publications, and conference proceedings. English-language publications frequently containing information related to liquid-vapor phase-change phenomena include the following: International Journal of Heat and Mass Transfer, published by Elsevier. International Communications in Heat and Mass Transfer, published by Elsevier. Journal of Heat Transfer, published by the American Society of Mechanical Engineers. AIChE Journal, published by the American Institute of Chemical Engineers. Journal of Fluid Mechanics, published by Cambridge University Press. International Journal of Multiphase Flow, published by Elsevier. Heat Transfer Engineering, published by Taylor & Francis. Numerical Heat Transfer (Parts A & B), published by Taylor & Francis. ASHRAE Transactions, published by the American Society of Heating, Refrigerating and Air-Conditioning Engineers. International Journal of Heat and Fluid Flow, published by Elsevier. Advances in Heat Transfer, published by Academic Press—Elsevier. Advances in Chemical Engineering, published by Academic Press—Elsevier. Experimental Heat Transfer, published by Taylor & Francis. Journal of Thermophysics and Heat Transfer, published by the American Institute of Aeronautics and Astronautics. xxi

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Introductory Remarks Experimental Thermal and Fluid Science, published by Elsevier. Proceedings of the International Heat Transfer Conferences, various publishers. Journal of Physical Chemistry, published by the American Chemical Society. Multiphase Science and Technology, published by Taylor & Francis. Nuclear Engineering and Design, published by Elsevier. Nuclear Science and Engineering, published by the American Nuclear Society. Nanoscale and Microscale Thermophysical Engineering, published by Taylor & Francis. Thermal Science and Engineering, published by the Heat Transfer Society of Japan. Heat and Mass Transfer, published by Springer-Verlag. Journal of Enhanced Transfer, published by Begell House Publishing. Journal of Colloid and Interface Science, published by Elsevier. Langmuir, published by the American Chemical Society. Journal of Chemical Physics, American Institute of Physics. Physics of Fluids, published by the American Physical Society. Soft Matter, published by the Royal Society of Chemistry. ACS Nano, published by the American Chemical Society. Applied Physics Letters, American Institute of Physics. Physical Review Letters, American Physical Society. Nanoletters, published by the American Chemical Society. Proceedings of the Royal Society, The Royal Society Publishing.

The above list is intended to provide a starting point for exploration of this area. There are many other special conference proceedings and publications in applications areas such as electronics cooling, automotive and aerospace applications that also have content relating to liquid-vapor phase-change processes. In addition to the sources listed above, useful information on liquid-vapor phase-change processes may also be obtained from government agency reports (from, e.g., NASA, the National Bureau of Standards, or the Nuclear Regulatory Commission) and the reports of research laboratories such as Sandia National Laboratory, Argonne National Laboratory, and the Electric Power Research Institute. The presentation of material in this text assumes that the reader is familiar with the basic elements of classical thermodynamics, fluid dynamics, heat transfer, and interfacial phenomena at the level of typical coverage in undergraduate mechanical engineering or chemical engineering programs. Throughout this book, facts from these fields will be recalled as needed for explanation of phenomena of interest. It may be useful to refer to basic texts in these fields, if the reader is unfamiliar with the concepts discussed. References [0.1–0.18] are useful resources for background information in these areas. A word regarding notation in this text is also warranted. The physical diversity of the mechanisms involved in vaporization and condensation processes also makes selection of a consistent nomenclature a difficult problem. For example, symbols traditionally used for properties in thermodynamic analysis are commonly used to denote other physical quantities in fluid mechanics analysis or in analysis of heat exchanger performance. To avoid confusion, every effort has been made to make the definition of variables clear at the location in the text where they are introduced. In addition, a listing of the nomenclature for the text is provided in the back of the book to provide a quick means of checking variable definitions. In macroscopic treatments of systems in which liquid and vapor phases coexist, the boundary between the bulk phases is usually idealized as a surface at which a discontinuity in properties occurs. Interfaces between phases are invariably the locations where the net conversion of one phase into the other occurs in vaporization and condensation processes. For that reason, the thermophysics of the liquid-vapor interface is of primary importance in many phase-change processes in nature and in important technological applications. We begin our exploration of liquid-vapor phase-change processes in Chapter 1 by exploring the nanoscale features of the near-interface region in detail.

Introductory Remarks

xxiii

REFERENCES 0.1 Van Wylen, G., and Sonntag, R. E., Introduction to Thermodynamics—Classical and Statistical, 3rd ed., Wiley, New York, NY, 1991. 0.2 Bejan, A., Advanced Engineering Thermodynamics, 4th ed., Wiley, New York, NY, 2016. 0.3 Wark, K., Advanced Thermodynamics for Engineers, McGraw-Hill, New York, NY, 1995. 0.4 Carey, V. P., Statistical Thermodynamics and Microscale Thermophysics, Cambridge University Press, New York, NY, 1999. 0.5 Cengel, Y. A., and Boles, M. A., Thermodynamics—An Engineering Approach, 8th ed., McGraw-Hill, New York, NY, 2014. 0.6 Moran, M. J., and Shapiro, H. N., Fundamentals of Engineering Thermodynamics, 7th ed., Wiley, New York, NY, 2010. 0.7 Carslaw, H. S., and Jaeger, J. C., Conduction of Heat in Solids, 2nd ed., Oxford University Press, Oxford, 1959. 0.8 Ozisik, M. N., Heat Conduction, Wiley, New York, NY, 1980. 0.9 Arpaci, V. S., Conduction Heat Transfer, Addison-Wesley, Reading, MA, 1966. 0.10 Bird, R. B., Stewart, W. E., and Lightfoot, E. N., Transport Phenomena, 2nd ed., Wiley, New York, NY, 2001. 0.11 Kays, W. M., Crawford, M. E., and Weigand, B., Convective Heat and Mass Transfer, 4th ed., McGraw-Hill, New York, NY, 2004. 0.12 Howell, J. R., Siegel, R., and Menguc, M. P., Thermal Radiation Heat Transfer, 5th ed., CRC Press, Taylor & Francis, New York, NY, 2011. 0.13 Incropera, F. P., DeWitt, D. P., Bergman, T. L., and Lavine, A., Fundamentals of Heat and Mass Transfer, 6th ed., Wiley, New York, NY, 2006. 0.14 Mills, A. F., and Coimbra, C. F. M., Basic Heat and Mass Transfer, 3rd ed., Temporal Publishing, LLC, San Diego, CA, 2015. 0.15 Baehr, H. D., Park, N. J., and Stephan, K., Heat and Mass Transfer, Springer-Verlag, New York, NY, 2006. 0.16 Miller, C. A., and Neogi, P., Interfacial Phenomena—Equilibrium and Dynamic Effects, 2nd ed., CRC Press, Boca Raton, FL, 2007. 0.17 Israelachvilli, J. N., Intermolecular and Surface Forces, 3rd ed., Academic Press, Waltham, MA, 2011. 0.18 Sadhal, S. S., Ayyaswamy, P. S., and Chung, J. N., Transport Phenomena with Drops and Bubbles, Springer-Verlag, New York, NY, 1997.

Part I Thermodynamic and Mechanical Aspects of Interfacial Phenomena and Phase Transitions

1

The Liquid-Vapor Interfacial Region: A Nanoscale Perspective

1.1  A MOLECULAR PERSPECTIVE ON LIQUID-VAPOR TRANSITIONS Before considering the mechanisms of vaporization and condensation processes, it is useful to examine the nature of the liquid and vapor states, and the transition between them, from a molecular point of view. In macroscopic treatments of systems in which liquid and vapor phases coexist, the boundary between the bulk phases is usually idealized as a surface at which a discontinuity in properties occurs. Interfaces between phases are invariably the locations where the net conversion of one phase into the other occurs during vaporization and condensation processes. For that reason, the thermophysics of the liquid-vapor interface is of primary importance in many phase-change processes in nature and in important technological applications. At the nanoscale level, there is a region between the bulk phases across which the mean molecular density exhibits a transition between bulk vapor and bulk liquid values. As we will see, the nature of this transition strongly affects the thermophysics and transport in this region. In analyzing vaporization and condensation processes, it will frequently be useful to model the boundary between bulk liquid and vapor phases as a surface. However, to fully understand such processes, we must first establish an understanding of the nanoscale thermophysics in this region. In addition, we will find that in liquid-vapor phase-change processes in microscale and nanoscale systems, the impact of nanoscale features of the interfacial region is especially important. Coverage of the fundamentals of liquid-vapor phase-change phenomena begins in this chapter with a molecular-level examination of the bulk liquid and vapor phases and the interfacial region that exists between them. The mechanisms that give rise to interaction forces between molecules vary widely in character. The forces can be attractive or repulsive in nature and one type may or may not act independently of others. At very short range, two molecules generally exert a repulsive force on one another. This repulsion is a consequence of the interference of the electron orbits of one molecule with those of the other. For polyatomic molecules, the force of interaction and associated potential function may vary with orientation as well as separation distance. These effects may be weak in simple polyatomic molecules and, in general, they may tend to average-out when analyzing the mean behavior of large numbers of molecules. For the purposes of this introductory exploration, we will adopt the simplifying assumption that orientation effects are small and, to a first approximation, the potential function is only a function of separation distance. Although the exact nature of short-range force interactions varies with the types of molecules involved, the resulting repulsive force generally increases rapidly as the center-to-center spacing r decreases. Consequently, they are often represented by a repulsive potential of the form

φ R (r ) =

λ0 , rk

9 ≤ k ≤ 15 (1.1)

where λ0 is a constant that varies depending on the type of molecule. Note that by definition, the potential function is the energy that must be input to bring two molecules from infinite distance apart to center-to-center spacing r. For a specified interaction potential ϕ, the force between the molecules at any separation distance r is –dϕ/dr. 3

4

Liquid-Vapor Phase-Change Phenomena

At somewhat larger distances, the forces acting between molecules generally fall into one of the three categories: (1) electrostatic forces, (2) induction forces, or (3) dispersion forces. Electrostatic forces between molecules often arise because the molecules have a finite dipole moment (i.e., opposite sides of the molecule have opposite charges). Common polar molecules (those having non-zero dipole moments) include water and alcohols. Because the potential function for electrostatic interactions rapidly becomes small as r increases, dipole interactions are generally significant only at very short range. Induction forces arise when a permanently charged particle or dipole induces a dipole in a nearby neutral molecule. The strength of induction forces depends directly on how easily the initially neutral molecule is polarized. The potential function associated with the induction force interaction between a dipole molecule and a neutral molecule is inversely proportional to r6. Dispersion forces are a consequence of transient dipoles that can be induced in molecules or atoms. Such transient dipoles are mutually induced in adjacent molecules as a result of instantaneous asymmetries in the electric field due to moving electrons in each molecule. This type of force interaction occurs in molecular species of all types and produces an attractive force between the molecules. The associated potential function for dispersion force interactions is proportional to r–6,

φ DIS = −

λ DIS α 2p (1.2) r6

where αp is the polarizability of the molecules and λDIS is a constant that varies with the type of molecule. Short-range attractive dispersion forces play an important role in the thermophysical behavior of virtually all liquids and vapors near saturation conditions. For real molecules, multiple force interaction mechanisms may come into play. Hence, the overall potential function ϕ(r) is generally assumed to reflect the repulsive force behavior at small spacing and the attractive behavior due to electrostatic, induction and/or dispersion forces at intermediate distances. Several model variations of ϕ(r) have been proposed which more or less conform to this general behavior. Perhaps the most well-known of these models is the Lennard-Jones 6–12 potential

φ LJ (r ) = 4ε[(r0 /r )12 − (r0 /r )6 ] (1.3)

which is plotted in Fig. 1.1. Appropriate values of the parameters ε and r0 vary with the type of molecule. A study of all the implications of the Lennard-Jones potential, and other models like it, is beyond the scope of this book. The interested reader can find more information on this subject in references [1.1–1.5]. It is sufficient for our current purposes to note that this variation of the potential

FIGURE 1.1  The Lennard-Jones 6–12 potential.

5

The Liquid-Vapor Interfacial Region

implies that to bring two molecules that are initially very far apart into closer proximity, we must remove energy. Conversely, if two molecules are close enough to feel attractive forces, but not so close that repulsive forces come into play, then energy must be supplied to increase the spacing of the molecules. Clearly, the energy exchanges associated with these processes are consistent with the input or removal of the latent heat of vaporization during vaporization (moving closely spaced molecules apart) or condensation (moving widely spaced molecules closer) processes. The force interactions among molecules largely dictate the thermophysical properties of the substance. For fluids containing molecules that obey a spherically symmetric potential function ϕ(r), statistical thermodynamic theory can used to generate relations among thermodynamic properties. For a system of volume V containing N molecules with a spherically symmetric interaction function, this type of theoretical model can be used to derive the following relation for the logarithm of the canonical partition function:



3 N   2πMk B T (V − Nbv′ )2/3    (ξ − 5) ln ln Q = N +   + N  2 ln π − ln σ s   2   N 2/3 h 2  (1.4)   +

(ξ − 3) N  T  av′ N 2 ln  + 2  θrot ,m  Vk B T

where ∞



∫

av′ = −2π r 2φ(r )dr (1.5) D

and bv′ is a mean volume occupied per molecule in the system. In Eq. (1.4), kB is the Boltzmann constant, h is Planck’s constant, M is the molecular mass of the molecule, ξ is the number of translational and rotational molecular energy storage modes, σs is the symmetry number for the molecule, and θrot,m is the mean rotational temperature for the molecule if it is a polyatomic species. In Eq. (1.5), D is the effective diameter of the molecule. When the Lennard-Jones potential function is used, D is usually taken to be about equal to r0. A simple model to relate bv′ to D is to treat the molecules as hard spheres of diameter D and make two additional idealizations. The first is to interpret bv′ as the volume per molecule in the densest possible condition. The second is to take the densest condition to be cubic close packing of the molecules. Geometry then requires that bv′ is related to the diameter D as

bv′ = 1.350 ( π / 6 ) D 3 (1.6)

Note that the constant 1.350 in the above relation is the ratio of system volume to volume occupied by molecules for cubic close packing of spheres. A complete discussion of the derivation of the relation (1.4) for ln Q for a van der Waals fluid can be found in Carey [1.1]. With the following relations from statistical thermodynamic theory, the equation for ln Q can be used to generate relations for thermodynamic properties:

 ∂ln Q  P = k BT  (1.7)  ∂V  T , N



 ∂(ln Q)  µ = − k BT  (1.8)  ∂ N  V ,T



 ∂(ln Q)  U = − k BT 2  (1.9)  ∂T  V , N

6

Liquid-Vapor Phase-Change Phenomena

Substituting Eq. (1.4) into the equations above yields the relations





P=

Nk B T a′ N 2 − v 2 (1.10) V − Nbv′ V

µ 3   2πMk B T (V − Nbv′ )2/3   (ξ − 5) Nbv′  − ln π + ln σ s  = −   ln   2/3 2   2 k B T V − Nbv′ N h     2 (ξ − 3)  T  2av′ N − ln  − 2  θrot ,m  Vk B T U=

(1.11)

ξN a′ N 2 k B T − v (1.12) 2 V

Equation (1.10) is the well-known van der Waals equation of state that can be reorganized to the form

P=

RT aˆ − v2 (1.13) vˆ − bˆv vˆ

where vˆ is the molar specific volume, NA is Avogadro’s number, R = NAkB is the universal gas constant, and aˆ v and bˆv are the molar van der Waals constants:

aˆ v = av′ N A2 (1.14a)



bˆv = bv′ N A (1.14b)

The relation for chemical potential can similarly be written in terms of molar specific chemical potential and molar specific volume:



µˆ bˆv 3   2πMk B T ( vˆ − bˆv )2/3   (ξ − 5)  = −   ln  ln π + ln σ s  − RT vˆ − bˆv  2   N A2/3 h 2 2    (ξ − 3)  T  2aˆ v − − ln  ˆ 2  θrot ,m  vRT

(1.15)

Equation (1.13) can be reorganized in terms of the mass specific volume v, leading to

P=

RT a − v (1.16) v − bv v 2

Here M is the molecular mass, R = R / M is the specific gas constant, and av and bv are the massbased van der Waals constants:

av = aˆ v / M 2 (1.17a)



bv = bˆv / M (1.17b)

This model analysis thus leads to property relations for a van der Waals fluid. Although it incorporates many idealizations, the van der Waals model analysis provides a workable framework for predicting properties of dense gases and liquids that relates macroscopic thermodynamic properties to

7

The Liquid-Vapor Interfacial Region

molecular characteristics. In particular, it relates the van der Waals constants to the physical size of the molecules and the intermolecular potential function that characterizes force interactions among molecules. Consistency of the thermodynamic properties with the critical point conditions requires that the van der Waals constants must be related to the critical pressure Pc and temperature Tc as av =



27( R / M )2 Tc2 (1.18a) 64 Pc

bv =



( R / M )Tc (1.18b) 8 Pc

The model analysis for a fluid of interacting molecules described above incorporates several idealizations. While its property predictions do not exactly match the behavior of real fluids, they are qualitatively consistent with the behavior of real fluids. Improved property predictions of this type can be developed by incorporating more accurate non-symmetric potential function models (see the discussions in references [1.2–1.5]). Because the predictions of the van der Waals model are expected to be qualitatively consistent with the behavior of many fluids, and the resulting relations are mathematically simple, we will use it here to explore the main thermodynamic features of liquid-vapor phase change for a pure fluid system. The van der Waals equation of state (1.16) can be used to predict the isotherm shapes on a P–v diagram. Introducing the following reduced properties normalized with their critical point values

Tr = T /Tc (1.19a)



Pr = P /Pc (1.19b)



vr = v /vc (l.19c)

the van der Waals equation of state can be written in the dimensionless form Pr =



8Tr 3 − (1.20) 3vr − 1 vr2

Note that in Eqs. (1.19a)–(1.19c), temperatures are absolute (K) and vc is the critical specific volume. Using Eq. (1.20), the variation of Pr with vr for fixed reduced temperature Tr defines a dimensionless van der Waals isotherm on the plot of Pr versus vr . Reduced isotherms of this type are plotted in Fig. 1.2. Below the critical temperature (Tr < 1), the van der Waals model predicts that isotherms exhibit a local minimum and maximum, as indicated in Fig. 1.2. For coexisting liquid and vapor phases, thermodynamics theory dictates that the necessary conditions for equilibrium are that the temperature, pressure, and chemical potential in the two phases must be equal. Tl = Tv ,



Pl = Pv ,

µˆ l = µˆ v (1.21)

In reduced properties, these requirements become

Tr ,l = Tr ,v ,

Pr ,l = Pr ,v ,

µ r ,l = µˆ l / RTc = µ r ,v = µˆ v / RTc

(1.22)

In Fig. 1.2, this implies that the points corresponding to the reduced specific volume of liquid, vr ,l and vapor, vr ,v must lie on a line of constant pressure (horizontal line) and be on the same isotherm. The additional constraint that the chemical potentials be equal is implemented by using Eq. (1.15) to

8

Liquid-Vapor Phase-Change Phenomena

FIGURE 1.2  Reduced property van der Waals isotherms.

evaluate µˆ l and µˆ v and requiring that Psat = Pl = Pv be chosen so that the computed µˆ l and µˆ v values are equal. This defines the link between the vapor pressure Psat and the specified temperature as well as the temperature dependence of the saturation specific volumes. It should be noted that if a liquid is depressurized isothermally in a quasi-equilibrium process from initial state A in Fig. 1.2, classical equilibrium thermodynamics predicts that the state point of the system follows the isotherm (constant Tr line) down until it reaches state point B. Upon reaching point B, classical theory implies that further reduction in pressure can only occur after a phase change occurs. Heat must be input to convert liquid into vapor as the state point shifts from B to F. Once all liquid is converted to saturated vapor at point F, further reduction in pressure will cause the state point to move along the isotherm toward point G. Note that this implies that at specific volumes between vr ,l and vr ,v , the system contains a mixture of two coexisting phases: saturated liquid at vr ,l and saturated vapor at vr ,v. Classical theory further implies that state points on the equation of state isotherm between B and F are not observed in an equilibrium system. We will see, however, that real systems in applications often depart from equilibrium, and it is sometimes possible for the system to exist in some states on the isotherm between B and F. In fact, these departures from equilibrium will be shown to play a central role in the onset of phase-change processes. The relationship between the molecular behavior and macroscopic characteristics for the vapor phase can, at least qualitatively, be understood from the kinetic theory of gases. Basic elements of the kinetic theory are developed in Appendix I. Readers unfamiliar with at least the basic aspects of the kinetic theory of gases are advised to review this appendix before proceeding further in this section. In Appendix I, the following Maxwell-Boltzmann speed distribution for molecules in a gas is derived from the kinetic theory of gases:

 m  dN c = 4 πN   2πk B T 

3/2

c 2e − mc

2

/2 k B T

dc (1.23)

9

The Liquid-Vapor Interfacial Region

The relation ε = (1/2)mc2 for the kinetic energy of a molecule can be inverted to obtain c = (2ε /m)1/2 (1.24)

Differentiating this relation yields

dc = (1/2m)1/2 ε −1/2 dε (1.25)

Substituting Eqs. (1.24) and (1.25) into Eq. (1.23), the following energy distribution is obtained

dN ε = 2πN (πk B T )−3/2 ε1/2e −ε / k BT dε (1.26)

In the above relation, dNε is interpreted as the number of molecules having kinetic energies between ε and ε + dε. It is often useful to know the fraction of the molecules in the gas that have energies exceeding a specified value ε*. The number of molecules in the gas with energies above ε*, N>ε* , is given by ∞



∫

N >ε* = dN ε (1.27) ε*

The fraction of molecules with energies above ε* is just N >ε* / N . Substituting Eq. (1.26) for the integrand in Eq. (1.27), dividing both sides by N and evaluating the integral yields

N >ε*  4 ε*  =  πk B T  N

1/2

e −ε*/ k BT + erfc

(

)

ε* k B T (1.28)

When the threshold energy ε* is much larger than kBT, the complementary error function term will be very small and can be neglected. For such conditions N>ε*/N is given by

N >ε*  4 ε*  =  πk B T  N

1/2

e −ε*/ k BT

(ε* >> k B T ) (1.29)

Perhaps the most significant aspect of Eq. (1.29) is that it predicts that the fraction of molecules having energies above the threshold value ε* increases rapidly with temperature. This behavior, which is characteristic of the kinetic energy of molecules in liquids as well as in gases, plays an important role in determining chemical reaction rates and the equilibrium conditions in two-phase systems. The equilibrium vapor pressure and its variation with temperature in a system containing saturated liquid and vapor are also consequences of the fact that the energy distribution among the molecules in the liquid and vapor phases is similar to the Maxwell-Boltzmann distribution. Because the system exhibits this energy distribution, even at low temperatures some fraction of the molecules in the liquid will have sufficient energy to escape the cohesive forces of other liquid molecules at the liquid-vapor interface. If the kinetic energy distribution in the liquid is similar to the MaxwellBoltzmann result given by Eq. (1.29), the fraction capable of escaping in this manner will increase rapidly with temperature. The rapid increase of the equilibrium vapor pressure suggested by these arguments is characteristic of most substances. This line of reasoning also suggests that at a given temperature, liquid with a small cohesive energy will have a higher vapor pressure than one with a large cohesive energy. Note that if a Lennard-Jones potential is used to model force interactions among molecules in the fluids, the cohesive energy is expected to vary proportional to the depth of the potential well (ε in Eq. (1.3)). Since the latent heat of vaporization is also a macroscopic indicator of the cohesive energy

10

Liquid-Vapor Phase-Change Phenomena

of the liquid, it follows that at the same temperature, a liquid with a high latent heat of vaporization should have a lower vapor pressure than a liquid with a smaller latent heat. This trend is also observed for most liquids. It is clear from this development that the constant bv in the van der Waals equation is associated with the volume occupied by the molecules themselves and the av /v2 term accounts for attractive forces between the molecules. These attractive forces, sometimes referred to as van der Waals forces, may in general be a combination of the dispersion, electrostatic, and induction forces described above. Because the attractive forces represented by the av /v2 term are the same forces that must be overcome to separate molecules that are initially closely spaced, the constant av is roughly proportional to the latent heat of vaporization of the liquid. Note also that the contribution of the av /v2 term varies strongly with the density (and therefore the spacing) of the molecules. For low-density gases v is large and the contribution of av /v2 is small. For a liquid, the specific volume is very small and the contribution of av /v2 will be much larger than for the vapor phase of the same substance. These arguments can be made more concrete by considering a specific example. It can easily be shown using density and molecular weight data from Appendix II that for saturated liquid water at atmospheric pressure (101 kPa) the mean volume occupied per molecule is about 3.1 × 10 –29 m3. If we envision the molecules as being evenly spaced in a cubic lattice, the mean spacing is approximately the cube root of the mean volume, or 3.1 × 10 –10 m apart. Since the diameter of a water molecule is about 2Å (2.0 × 10 –10 m), the center-to-center spacing of the molecules is only about 1.5 molecular diameters in the saturated liquid. For saturated water vapor at 101 kPa, the same line of reasoning suggests that the volume per molecule is 5.0 × 10 –26 m3 and the molecular spacing is 3.7 × 10 –9 m or about 19 molecular diameters. If r0 in the Lennard-Jones potential shown in Fig. 1.1 is taken as being about 1 molecular diameter, then it is clear from these numbers that attractive forces will be very small in the saturated vapor, but very important in the saturated liquid. This is completely consistent with the magnitude of these forces implied by the av/v2 term in the van der Waals equation. Example 1.1 To model nitrogen molecule force interaction with the Lennard-Jones potential (Eq. (1.3)), a value of 1.31 × 10 –21 J is recommended for ε. Note that this sets the depth of the potential well (Fig. 1.1), and the value of ε is the energy that must be input for one molecule to escape the attractive pull of another. In a two-phase system, escape of molecules from a liquid phase into a vapor phase at the interface is more probable if the translational kinetic energy is larger than ε. For saturated nitrogen at 77 K, estimate the fraction of the molecules that have translational kinetic energies larger than ε = 1.31 × 10 –21 J. Equation (1.28) can be used to predict the fraction of molecules with translational energies greater than ε. In the equation, we set ε* = ε = 1.31 × 10 –21 J and T = 77 K. ε* 1.31× 10 −21 = = 1.23 kBT 1.38 × 10 −23 (77)

Substituting into Eq. (1.28) yields



N>ε *  4ε*  =  πkBT  N N>ε *  4(1.23)  =  π  N

1/ 2

1/ 2

e −ε */ kBT + erfc

e −1.23 + erfc

(

(

ε* / kBT

)

)

1.23 = 0.366 + 0.117 = 0.483

Thus, the Boltzmann distribution predicts that almost half the molecules have translation energy values greater than ε = 1.31 × 10 –21 J.

The Liquid-Vapor Interfacial Region

11

For molecules in the center of the large body of liquid, the attractive forces from surrounding molecules are nominally spherically symmetric and hence they balance out to zero. Near a liquid-vapor interface, however, things are quite different. Within a few molecular diameters of the interface, molecules in the liquid must redistribute themselves to accommodate the lack of spherical symmetry in the molecular force interactions. Liquid-vapor phase-change processes necessarily involve the coexistence of a liquid and vapor phase, and the conversion of one phase to the other typically occurs at the boundary between the bulk phases. This boundary region is usually of central importance in such processes. To gain a better understanding of the thermophysics of this region, in the following sections we will examine its characteristics in detail. In the remaining sections of this chapter we will consider it from a molecular perspective. In Chapter 2, we will examine how the effects of this region can be treated from a macroscopic point of view.

1.2  THE INTERFACIAL REGION - MOLECULAR THEORIES OF CAPILLARITY Macroscopic thermodynamic and fluid mechanics treatments of the boundary between two phases invariably assume a sharp discontinuity in density and/or composition across the boundary or interface. In considering the details of physical processes at the interface, however, it is important to recognize that at a molecular level, changes in fluid or material properties actually occur over a thin interfacial region that separates the bulk liquid and vapor phases. Consideration of molecular interactions in this interfacial region has led to a better understanding of several aspects of interfacial phenomena. Although it will often be more convenient to treat the boundary between the phases as if it were a two-dimensional surface, in this and the next section we will examine the nanoscale features of the interfacial region. Anyone who has watched small bubbles rise in a carbonated beverage or a pot of boiling water has undoubtedly noted that the bubbles are almost perfectly spherical, as if an elastic membrane were present at the interface to pull the vapor into a spherical shape. From a thermodynamic point of view, this apparent interfacial tension may be interpreted in terms of energy stored in the molecules near the interfacial region separating the bulk phases. Figure 1.3 schematically shows the density distribution near a liquid-vapor interface. On the liquid side of the interfacial region, the density is lower than that in the bulk liquid phase. As described in the previous section, because the molecules attract one another, energy must be supplied to move them apart. Hence, the energy per molecule is greater in the interfacial region than in the bulk liquid. The system thus has an additional free energy per unit area of interface due to the presence of the interface. There is also the more obvious mechanical interpretation of the interfacial tension as a force per unit length parallel to the interface (and perpendicular to the density gradient). This interpretation can also be viewed as being a consequence of attractive and repulsive interactions among molecules that result from the molecular forces described in the previous section. In the bulk liquid, molecules are subject to forces of repulsion from their close neighbors and forces of attraction from all others. The repulsive forces are generally stronger, but since both types of forces act symmetrically in all directions, on the average, the resultant on each molecule is zero. As indicated in Fig. 1.3, the mean spacing of the molecules in the liquid near the interface is greater than in the bulk liquid. Note that the potential functions discussed in the previous section imply that the close-range repulsive force (force being –dϕ/dr) varies more rapidly with spacing than the longer-range attractive forces. Hence, for a given molecule, this slightly increased spacing would significantly weaken the repulsive force it feels from its immediate neighbors, but would likely produce only a small change in the attractive force between it and more distant surrounding molecules. In the direction normal to the interface, this combination of effects would produce a force imbalance that would draw the molecules toward the bulk liquid. The mean spacing of the molecules in the direction normal to the interface could decrease slightly to establish a balance between longrange attractive forces and repulsion from close neighboring molecules.

12

Liquid-Vapor Phase-Change Phenomena

FIGURE 1.3  Variation of the molecular density across the interfacial region.

In the directions parallel to the interface, however, the decrease in repulsion force between molecules produced by the increased spacing does not create a force imbalance because of the radial symmetry of the force interactions. Consequently, there is no impetus to decrease the mean spacing in this direction. The decrease in repulsive forces between immediate neighbors, with little change in the longer-range attractive forces may thus produce a net tension force among molecules in the interface region acting equally in all directions parallel to the interface. Although crude, these arguments clearly imply that the existence of a net tension force on the molecules is a direct consequence of the increased mean molecular spacing and the density gradient in the interface region. As discussed in the previous section, the intermolecular attractions that give rise to interfacial tension may result from several different types of molecular forces. These may include forces that are specific to particular types of molecules, such as the metallic bond or the hydrogen bond, as well as dispersion forces, which exist in all types of matter and always give an attractive force between adjacent atoms or molecules no matter how dissimilar they are chemically. The dispersion forces vary with the electrical properties of the substances involved and the distance between interacting elements, but they are independent of temperature. While the magnitude of the attractive force between molecules may vary, virtually all molecules exhibit long-range attractive force interactions and short-range repulsive interactions that can give rise to the increased interfacial free energy described above. The characteristics of the interfacial region can be explored in a more quantitative way by extending classical thermodynamic analysis to the interfacial region with the idealization that properties vary continuously across the region and that local thermodynamic equilibrium applies in a time averaged sense within small control volumes within the interfacial region. This approach was pioneered by van der Waals [1.6] and is usually referred to as the van der Waals theory of capillarity or the molecular theory of capillarity. It is also sometimes referred to as a mean field theory because it is based on the idealization that the behavior of each molecule in a localized region is dictated by the mean field associated with the surrounding molecules.

13

The Liquid-Vapor Interfacial Region

FIGURE 1.4  Variation of mean molar density and volumetric free energy ψ = F/V across the interfacial region.

The classical van der Waals mean field theory of capillarity is based on the postulate that the mean properties vary continuously across the transition region between the bulk phases (see the discussions in Rowlinson [1.6] and Rowlinson and Widom [1.7]). Specifically, the van der Waals model postulates a continuous variation of mean local density (number density ρn or molar density ρˆ = ρn/NA) across the interfacial region, as indicated in Fig. 1.4. The van der Waals analysis leads to the conclusion that the interfacial tension is equivalent to the excess interfacial free energy per unit area σ, which is the free energy per unit area above that for step changes in ρˆ and the free energy per unit volume at z = 0 (Fig. 1.4). The model postulates that, in accordance with mass conservation and the second law of thermodynamics, the density distribution spontaneously adjusts to minimize the total excess free energy (per unit area) in the interfacial region. In the van der Waals theory of capillarity, the z = 0 location in the interfacial region is chosen so that 0

∫



−∞

(ρˆ − ρˆ v ) dz +

∞

∫ (ρˆ − ρˆ ) dz = 0 (1.30) l

0

Equation (1.30) ensures that the mass in the interfacial region with a distributed density profile is the same as would exist in the region with a discontinuous density step change at z = 0. The free energy per unit volume is defined as ψ = F /V (1.31)



Here F is the Helmholtz free energy and V represents the volume of a local system within the interfacial region. The relation below stipulates that σ is the free energy per unit area of interface in excess of that for step changes in ρˆ and ψ at the interface at z = 0 (Fig. 1.4). 0



σ=

∫

−∞

 ψ − ψ ( ρˆ v )  dz +

∞

∫ ψ − ψ (ρˆ ) dz (1.32) l

0

The van der Waals model of capillarity is postulated to apply to a system held at constant temperature with a volume that encompasses the interfacial region over a unit area of the interface. The

14

Liquid-Vapor Phase-Change Phenomena

second law of thermodynamics requires that for such a system (with fixed V and T), equilibrium corresponds to a minimum in the Helmholtz free energy. Since volume is fixed, this also corresponds to a minimum in volumetric free energy ψ = F/V. This equilibrium free energy σlv is the property commonly referred to as interfacial tension or surface tension. Determination of the equilibrium interfacial tension using van der Waals theory thus requires solution of a constrained minimization problem. We must determine the mean density variation ρˆ ( z ) that minimizes the right side of Eq. (1.32). The interfacial tension or interfacial free energy σlv is the value of the integral on the right side of Eq. (1.32) for ρˆ ( z ) that satisfies Eq. (1.30) and minimizes the integral. To execute this scheme requires a means of predicting ψ ( ρˆ , z ) in the interfacial region that accounts for the effects of the density gradient there. The classical van der Waals theory of capillarity extends the statistical thermodynamics model for properties of a van der Waals fluid described in the previous section to evaluate ψ and other properties in the interfacial region. Once property relations are generated, one of the two approaches can be used: 1. An empirical relation with an adjustable constant can be postulated for the density profile and the integrals on the right side of Eq. (1.32) can be numerically evaluated for chosen values of the adjustable constant to determine the value that minimizes σ. 2. The calculus of variations [1.8] can be applied to derive an integral relation for the minimum σ and the density variation across the interface. Numerical evaluation of the integral relation predicts the interfacial tension σlv and the density variation across the interfacial region. A more detailed discussion of the two methods described above can be found in Appendix III. Using one of these methods, it is therefore possible to compute predictions of the surface tension and density profile in the interfacial region for a van der Waals fluid using Eqs. (1.30) and (1.32), together with the Lennard-Jones model constants and other molecular parameters for the fluid. The classic van der Waals mean field theory of capillarity described above has generally been viewed as an approximate theory. Its predictions are qualitatively similar to real fluid behavior, but its numerical predictions are inaccurate. The predicted property values and the trends in the variations of properties with temperature do not agree with those observed for real fluids. Near the critical point, it is well known that the corresponding states correlation for surface tension that best matches data for real fluids typically varies proportional to (1 − T/Tc) to an exponential power of about 1.22 (see the discussion in Poling et al. [1.9]). The van der Waals mean field model predicts that surface tension varies proportional to (1 − T/Tc)3/2. The van der Waals model also predicts that the interfacial region thickness varies proportional to (1 − T/Tc) –1/2, whereas recent measurements (by, for example, Beysens and Robert [1.10]) indicate that the interfacial region thickness varies about proportional to (1 − T/Tc) –0.62. Despite its shortcomings, the van der Waals theory of capillarity provides useful insight into the connection between molecular properties and interfacial region thermophysics. In particular, it clearly indicates that interfacial free energy is a consequence of the attractive force interactions among molecules and the density gradient that exists in the interfacial region. However, the crudeness of the van der Waals model limits the usefulness of its quantitative predictions. In a recent investigation, Carey [1.11] has developed an alternative version of the molecular theory of capillarity that incorporates a Redlich-Kwong model of fluid properties. The RedlichKwong fluid model was chosen because it is a simple model and its predictions generally agree better with real fluid saturation property data than predictions of the van der Waals model. The formulation of Carey [1.11] used the calculus of variations method described in Appendix III with properties determined using the Redlich-Kwong property model extended to the conditions in the interfacial region. Using this model analysis, Carey [1.11] obtained integral relations for

15

The Liquid-Vapor Interfacial Region

the interfacial tension and the density variation across the interfacial region that can be cast in the following dimensionless forms:



σ lv 1.929 = 1/4 0.17 Pc Li Tr (1 − Tr ) –



ρr ,l

  Pr ,sat  ρr ,v (1 − ρr br )  1  − ρr Tr ln   − ρr Tr  1 + br ρr )  3 ρr (1 − ρr ,v br )  (    ρr , v 

∫

 1 + ρr br  br Tr ρr ,v ρr ar ρr aρ ln – + r r 3br Tr1/2  1 + ρr ,v br  1 − ρr ,v br 3br Tr1/2

z − zv 0.1639 ar = 1/4 0.17 Li Tr (1 − Tr )

1/2

 ρr ,v br     1 + ρ b    r ,v r   

(1.33)

dρr

ρr

  P  ρr ,v (1 − ρr br )  (1 + br ρr )  r ,sat − ρr Tr ln     ρr (1 − ρr ,v br )   3 ρr , v +δρr 

∫

 1 + ρr br  br Tr ρr ,v ρr a ρr a ρr  ρr ,v br    −ρr Tr − r 1/2 ln  − + r 1/2   3br Tr  1 + ρr ,v br  1 − ρr ,v br 3br Tr  1 + ρr ,v br   

(1.34)

−1/2

dρr

In these relations, the earlier definitions of Pr and Tr apply and

ρr = ρˆ / ρˆ c (1.35)



ar = 3.84732 (1.36)



br = 0.25992 (1.37)



Li = [ k B Tc /Pc ]1/3 (1.38)

In the above formulation, Li is a characteristic length associated with the size of the interfacial region. Note that to compute the dimensionless interfacial tension and density profile for a specified Tr , the values of the dimensionless saturation properties must first be determined. The dimensionless saturation pressure can be computed using the following theoretically based curve-fit to the saturation pressure predictions for a Redlich-Kwong fluid [1.11]:

{

(

)}

Pr ,sat = Tr exp −2.9327 Tr−3/2 − 1 (1.39)

The above relation predicts values of the saturation pressure that agree closely with full calculations of Pr,sat for a Redlich-Kwong fluid. If the predicted Pr,sat value from Eq. (1.39) is used, the reduced saturation density for the liquid and vapor are determined by solving the Redlich-Kwong equation of state

Pr =

3Tr a − 1/2 r (1.40) vr − br Tr vr ( vr + br )

for three density values that satisfy the equation for the specified Tr and computed Pr,sat. The highest and lowest density solutions are the liquid and vapor saturation densities. The saturation properties can be determined to higher accuracy if Pr,sat and the subsequent density calculations are iterated to find the values that, when substituted in the chemical potential relations, more accurately satisfy the equilibrium requirement that µˆ v / RTc = µˆ l / RTc (see reference [1.11] for details).

16

Liquid-Vapor Phase-Change Phenomena

Carey [1.11] used Redlich-Kwong thermodynamic property relations to determine the saturation properties, and then determined the dimensionless density profile and interfacial tension by numerically integrating the dimensionless Eqs. (1.33) and (1.34). Asymptotic analysis of the RedlichKwong property relations (1.33) and (1.34) indicates the following power-law dependence of the surface tension at small 1 − Tr:

σ lv

Pc ( k B Tc Pc )

∼ (1 − Tr )

1.33

1/3

In fact, the relation

σ lv

Pc ( k B Tc Pc )

1/3

= 14.65 (1 − Tr )

1.33

(1.41)

closely matches the theoretical predictions computed by Carey [1.11] for Tr between 0.6 and 0.98 shown as the solid line in Fig. 1.5. Also shown in Fig. 1.5, are recommended surface tension values for a variety of fluids from the ASHRAE Fundamentals Handbook [1.12]. It can be seen that the recommended dimensionless surface tension values exhibit a power-law dependence proportional to about (1 − Tr)1.25. In contrast, the van der Waals model predicts that surface tension varies proportional to (1 − Tr)1.5. Although the agreement is not perfect, the modified Redlich-Kwong model agrees fairly well with the recommended values and is a better fit than the classical van der Waals model.

FIGURE 1.5  Theory prediction of interfacial tension for a Redlich-Kwong fluid computed using Eqs. (1.33) and (1.34). Also shown are recommended values from the ASHRAE Fundamentals Handbook [1.12].

17

The Liquid-Vapor Interfacial Region

In addition to providing a prediction of the interfacial tension, molecular theories of capillarity also provide insight into the structure of the interfacial region. This is explored further in the next section.

1.3  NANOSCALE FEATURES OF THE INTERFACIAL REGION As noted in the previous section, Eq. (1.34) can be integrated to determine the correspondence between density and z location in the interfacial region. The density profiles predicted in this manner using Carey’s [1.11] model are plotted in Fig. 1.6. A dimensionless thickness of the interfacial region δzi/Li can be defined as

(ρr ,l − ρr ,v ) δzi = (1.42) Li Li ( dρr dz )z = 0



where the derivative is evaluated at the z = 0 location determined from the integral computation of the density profile. Differentiating Eq. (1.34) to evaluate the derivative, this equation can be written in the form



 P  ρr ,v (1 − ρr br )  0.1639 ( ρr ,l − ρr ,v ) ar  δzi (1 + br ρr )  r ,sat − ρr Tr ln  =  0.17 1/4 Li Li Tr (1 − Tr )   ρr (1 − ρr ,v br )   3 −ρr T −

ar ρr 3br Tr1/2

 1 + ρr br  br Tr ρr ,v ρr a ρr  ρr ,v br    − + r 1/2 ln     1 + ρr ,v br  1 − ρr ,v br 3br Tr  1 + ρr ,v br   

−1/2

(1.43)

z=0

It should be noted that the subscript z = 0 on the right side of Eq. (1.43) implies that the expression in the square brackets is to be evaluated at the ρr value at z = 0. Full application of this relation thus requires that the density profile be determined first to establish the ρr value at z = 0. Alternatively, a good approximate value for δzi/Li can be computed using Eq. (1.43) if the ρr value at z = 0 is taken to be (ρr,v + ρr,l)/2. This is a good approximation since the density at z = 0 is generally close to the average for the two phases (Fig. 1.6).

FIGURE 1.6  Reduced density profiles across the interfacial region predicted for a Redlich-Kwong fluid at various reduced temperatures.

18

Liquid-Vapor Phase-Change Phenomena

FIGURE 1.7  Variation of interfacial region thickness with reduced temperature.

The variation of the interfacial region thickness with reduced temperature determined using Eq. (1.43) with the ρr value at z = 0 determined from the density profile is plotted in Fig. 1.7. The length scale Li used to normalize z in Figs. 1.6 and 1.7 is on the order of one nanometer for many common fluids (Table 1.1). Consequently, the thickness of the interfacial region for many fluids is predicted to be on the order of a few nanometers for temperatures corresponding to 0.6 ≤ Tr ≤ 0.98. As indicated in Fig. 1.7, the variation of interfacial region thickness predicted by the theory is well represented by the power-law relation δzi −0.67 = 0.683 (1 − Tr ) (1.44) Li



This relation provides a means of estimating the interfacial region thickness for coexisting vapor and liquid phases of a pure substance at a specified saturation temperature. While not exact, molecular theories of capillarity provide a theoretical understanding of the variations of surface tension and interfacial region thickness with temperature that are observed in real fluids. Note that the molecular capillarity theories described above indicate that interfacial tension

TABLE 1.1 Values of Li for Various Fluids N2 CH4 Ar O2 H2O NH3 C3H8 (propane) SF6 R-134a Hg

Tc(K)

Pc (MPa)

Li (nm)

126.2 190.6 150.7 154.5 647.3 405.6 369.9 318.7 374.3 1763.2

3.400 4.599 4.865 5.043 22.129 11.290 4.248 3.760 4.059 151.0

0.800 0.830 0.753 0.751 0.739 0.793 1.063 1.054 1.084 0.544

19

The Liquid-Vapor Interfacial Region

and interfacial region thickness are only a function of temperature for a pure fluid. They indicate that the surface tension vanishes at the critical point and varies about proportional to (1 − Tr)n where n is about 1.2–1.5. They similarly indicate that the thickness of the interfacial region varies about proportional to (1 − Tr) –m where m is about 0.5–0.6. The theories predict that the thickness of the interfacial region grows without bound as T → Tc. These theories further imply that the variation of these properties conforms to thermodynamic similitude. This suggests that relations among the properties for different fluids can be reduced to a single relation in terms of properties normalized with critical point parameters. This insight is useful in developing relations that can be used to predict surface tension values for different fluids systems. Molecular theories of capillarity also predict the physical size of the interfacial region. From Figs. 1.6 and 1.7 and Table 1.1, it can be surmised that the thickness of the interfacial region is on the order of a few nanometers for a wide variety of fluids under commonly encountered conditions. This suggests that for systems in which characteristic length scales are on the order of a few nanometers, the properties associated with the interfacial region, including surface tension, will be altered from those observed in an interface between two extensive bulk phases. This may be particularly important in interfacial phenomena that occur in micro- and nanoscale system applications. Carey [1.11] also noted that the molecular theory of capillarity predicts that the center of the interfacial region lacks intrinsic stability, suggesting that the interior of the interfacial region may exhibit high levels of property fluctuations. As discussed in the next section, the interfacial region property statistics for molecular dynamic simulations appear to be consistent with this observation. Example 1.2 For water at atmospheric pressure, use the Redlich-Kwong capillarity theory to estimate the surface tension and the interfacial region thickness. Compare the surface tension value to the tabulated value in Appendix II. For water, from Table 1.1, Tc = 647.3 K, Pc = 22.1 MPa and Li = 0.739 nm. It follows that

Tr = T /Tc = (100 + 273.2) / 647.3 = 0.577

Solving Eq. (1.41) for σlv and substituting σ lv = 14.65Pc (kBTc /Pc )1/ 3 (1− Tr )

1.33



= 14.65 (22.1× 106 )[1.38 × 10 −23 (647.3) / 22.1× 106 ]1/ 3 (1− 0.577)1.33 = 0.076246 N/m

Solving Eq. (1.44) for δzi and substituting

δzi = 0.683Li (1− Tr )−0.67 = 0.683(0.739)(1− 0.577)−0.67 = 0.906 nm

The computed surface tension is within about 29% of the tabulated value of 0.0589 N/m in Appendix II for water at these conditions. Also, the effective diameter of a water molecule is estimated to be about 0.28 nm, which implies that the thickness of the interfacial region is about 3 molecular diameters.

Another noteworthy outcome from the molecular theory of capillarity is that it predicts specific interrelationships among bulk fluid properties and interfacial region properties. Based on the results of molecular capillarity theory for a Redlich-Kwong fluid, Carey and Wemhoff [1.13] derived the

20

Liquid-Vapor Phase-Change Phenomena

FIGURE 1.8  Variation of interfacial region thickness with reduced temperature predicted using recommended property values from the ASHRAE Fundamentals Handbook [1.12].

following relation relating interfacial region thickness to temperature, interfacial tension and the changes in molar density and internal energy between the two bulk phases: δzi = 0.071L2i (1 − T / Tc )

−0.34



 ρˆ l − ρˆ v   σ  uˆlv (1.45) lv

This relation can be used to predict interfacial region thickness from the other properties that can be more directly measured. Figure 1.8 shows a plot of the interfacial region thickness predicted using Eq. (1.45) for a variety of fluids. The data plotted in this figure were computed using saturation properties at several temperatures for each fluid. The saturation property data were taken from recommended values in the ASHRAE Fundamentals Handbook [1.12]. The computed δzi values plotted in Fig. 1.8 indicate that for a variety of common fluids under commonly encountered saturation conditions, the theory described here predicts that the interfacial region thickness is in the range of 1–10 nm. This result, which is consistent with the observed trends in a similar model analysis devised by Hey and Wood [1.14], is remarkable, given that it seems to apply to molecular species with widely different molecular structures and interaction potentials. Example 1.3 Use the property relation (1.45) to predict the interfacial region thickness for saturated nitrogen at atmospheric pressure using property data from Appendix II. For nitrogen, from Table 1.1, Tc = 126.2 K, Li = 0.800 nm and Tr

= Tsat /Tc = 77.4 / 126.2 = 0.613

ρˆ l − ρˆ v = (807.1− 4.621) / 28.0 = 28.66 kmol / m3



uˆ lv =  hlv − Psat (1/ ρv − 1/ ρl )  M = 197.6 − 101.3 (1/ 4.621− 1/ 807.1)  28.0 = 4923 kJ/kmol σ lv = 0.00885 N/m

21

The Liquid-Vapor Interfacial Region Substituting into Eq. (1.45) yields δzi = 0.071L2i (1− T / Tc )

(

−0.34

= 0.071 0.800 × 10 −9

)

2

 ρˆ l − ρˆ v   σ  uˆ lv lv

(1− 0.613)−0.34 

28.66   4,923,000 0.00885 

= 1.00 × 10 −9 m = 1.00 nm This relation predicts an interfacial region thickness of only about 1 nm for saturated nitrogen at atmospheric pressure. Since the effective diameter of a nitrogen molecule is estimated to be about 0.31 nm, the thickness of the interfacial region is about 3 molecular diameters.

In more recent investigations, Wemhoff [1.15, 1.16] extended the capillarity theory model developed by Carey [1.11] for a Redlich-Kwong fluid to Soave-Redlich-Kwong, and Peng-Robinson fluid models, which include an acentric factor as an added parameter. The resulting Soave-Redlich-Kwong, and Peng-Robinson equations of state predict vapor pressure values that agree better with experimental data. However, Wemhoff [1.15, 1.16] found that the surface tension predicted by the Redlich-Kwong capillarity model agreed better with experimental data than surface tension predictions of the SoaveRedlich-Kwong, and Peng-Robinson models. Wemhoff [1.15] also proposed modified forms of the Soave-Redlich-Kwong, and Peng-Robinson fluid models that yielded surface tension predictions that agree with measured data better than the Redlich-Kwong model for a broad spectrum of fluids.

1.4 MOLECULAR DYNAMICS SIMULATION STUDIES OF INTERFACIAL REGION THERMOPHYSICS The notion that the behavior of a system of many molecules can be modeled by mathematically simulating the dynamics of their individual motions can be traced back to the development of kinetic theory in the late 19th century. It was not until the 1950s, with the appearance of the first modern computing machines, that serious efforts were made to computationally attempt such simulations. Early work by Metropolis et al. [1.17] and Alder and Wainwright [1.18] were among the first efforts to compute the time evolution of a system of molecules by solving Newton’s equations of motion for the molecules subject to the intermolecular forces between the molecules and the overall system constraints. In deterministic molecular dynamics simulations, different boundary conditions can be imposed to simulate different system physical constraints. Many simulation studies have considered molecules in a system with fixed total energy E, fixed number of molecules N, and fixed volume V. These are sometime referred to as NVE simulations. Alternate strategies have also been developed to model systems held at constant pressure, constant temperature, and/or constant chemical potential (free mass exchange with a surrounding reservoir). Simulations with all three of these constraints are sometimes referred to as μPT simulations. Subject to the system constraints, molecules in the simulation are allowed to move in accordance with Newton’s laws of motion, as dictated by the influence of intermolecular forces. Positions, velocities, orientations, and angular velocities for each molecule are tracked over the duration of the simulation. As computational power has increased, molecular dynamic (MD) simulation has become an increasingly valuable tool for exploring the physics of two-phase liquid-vapor systems at the nanoscale level. It is useful to consider the basic features of MD simulations for such systems because doing so illuminates the essential physics and the challenges associated with constructing an accurate simulation. MD simulations typically included the following components: Initialization. Each molecule in the simulation must be given an initial position, translational velocity components, and rotational energies. This can be assigned arbitrarily or values can be sampled from Boltzmann distribution.

22

Liquid-Vapor Phase-Change Phenomena

Once the initial microstate is defined, the simulation program evolves the system microstate over time by iteratively executing the following sequence of calculations at each time step: i. Determine forces from interaction potentials for the current positions and orientations of the molecules. This is accomplished by considering each pair of molecules in the simulation that are within a reasonable cutoff distance, forces are determined as the derivative of the potential with respect to separation distance. ii. Advance the simulation in time by applying Newton’s laws governing the motion response due to the computed intermolecular forces. This is often accomplished numerically using the well-known Verlet [1.19] algorithm. Note that this generates new translational and rotational energy values for each molecule, and changes the potential energy storage due to forces between molecules. iii. Handle molecular motion at computational domain boundaries. The specifics of this feature depend on the constraints on the system and its boundary conditions. Commonly used constraints include systems with constant number of molecules N, volume V, and internal energy U, or constant N, V, and temperature T, or constant N, pressure P, and T. Sometimes schemes are included to intermittently add or subtract small amounts of energy to maintain constant U or T. Symmetry conditions may dictate reinjection of leaving molecules at a new location on the domain boundary surface. Alternatively, if the domain boundary surface borders and interacts with a large equilibrium system, molecules may be removed from the domain and molecules with energies and velocity components randomly sampled from Boltzmann distributions may be injected at random points on the surface. iv. Sample data from local cells. Molecule energy and momenta data in localized cells are collected and stored to facilitate determination of instantaneous and average properties at locations throughout the computational domain. Long time averages of these data are used to compute steady state or equilibrium properties for the system. The data can also be used to compute statistical parameters such as standard deviation (fluctuation) levels of properties in the system. The sequence of operations (i)–(iv) are typically executed for each time step to march the simulation forward in time. Once the system moves beyond the initialization transient, it is typically run for tens of thousands of time steps to collect statistics for determination of local macroscopic properties (T, density ρ, pressure, free energy, etc.) Surface tension associated with interfacial regions in the simulation can be computed from mechanical analysis by first determining pressure tensor, or by statistical thermodynamic analysis of free energy and other properties in the model domain. Note that the accuracy of the simulation depends strongly on the accuracy of the intermolecular force potential used in the simulation. Running realistic MD simulations is computationally challenging for three reasons. First, simulations for systems containing bubble or droplets with sizes of interest in applications require a large number of molecules. Second, obtaining good statistics for steady state system behavior requires running the simulation for long time intervals, and third, modeling transient behavior can require running multiple simulations to define and ensemble of transient system behaviors that can be averaged to define the macroscopic transient system response. Equilibrium macroscopic properties are calculated by averaging the appropriate function of molecular positions and velocities over time. The interested reader can find more detailed information on fundamental aspects of molecular dynamic simulation methods in the references by Hoover et al. [1.20], Hoover [1.21], Haile [1.22], and Frenkel and Smit [1.23]. In this section, we will consider specific studies that have used this type of methodology to simulate microscale thermophysics of the interfacial region between coexisting liquid and vapor phases. Many molecular dynamics (MD) simulation studies of interfacial region thermophysics fall into one of the two categories: those that model a free liquid film, and those that model a droplet or molecular cluster. Studies of a free liquid film usually model the type of system shown in Fig. 1.9.

The Liquid-Vapor Interfacial Region

23

FIGURE 1.9  System model used in the simulation of a periodic free liquid film.

As shown in Fig. 1.9, the model system for the free liquid film is a liquid layer bounded in the z direction by vapor regions. The system is periodic in the x, y, and z directions. The vapor region is generally made large enough so the force interactions between molecules in the simulation liquid layer and those in mirror image liquid layers are small. The periodicity requires that any molecule that leaves the simulation region on one bounding surface re-enters across the opposite face. To model the interfacial region thermophysics, the liquid layer must be thick enough so that a region of bulk liquid exists in the interior of the layer. A key component of these simulations is the potential function that models the force interactions among the molecules. Earlier simulations often used the simple radially symmetric Lennard-Jones interaction potential (1.3) discussed in Section 1.1. More recent simulation studies have explored the use of more complex interaction potentials that better model specific molecules. The free liquid film MD simulation has been used to explore the interfacial region properties for a variety of fluids, including Lennard-Jones fluids [1.24], argon [1.24–1.27], nitrogen [1.27], water [1.28, 1.29], water and alcohol mixtures [1.30, 1.31], hydrocarbons [1.32], and fluorocarbons [1.33]. We will not review these investigations in detail here. Instead we will briefly examine some of the more noteworthy results of these MD simulation studies. The results of all the studies cited above confirm the existence of the interfacial region predicted by molecular capillarity theory. Mean density profiles defined by statistics from the MD simulations are generally consistent with the density profiles predicted by capillarity theory (Fig. 1.6) for comparable conditions. The accuracy of the predictions is, of course, limited by the accuracy of the intermolecular potential model used and the appropriateness of the simulation parameters. Computations of the surface tension using MD simulation data from the studies with the best interaction potential models have generally produced values that are in fairly good agreement with recommended values based on experimental data. Figure 1.10 shows the surface tension values for water predicted by the MD simulations of Alejandre et al. [1.28]. Also shown are tabulated values recommended by the ASHRAE Fundamentals Handbook [1.12]. The trend in the simulations predictions agrees well with that in the recommended values. The good agreement is particularly noteworthy since the interaction among polar water molecules is difficult to model in a way that is both computationally simple and physically accurate. This study indicates that the extended simple point charge (SPC/E) interaction potential for water molecules can yield good results in water MD simulations. Other successful efforts to predict surface tension from MD simulation data for other polyatomic species (i.e., nitrogen [1.27], hydrocarbons [1.32], and fluorocarbons [1.33]) also demonstrate that MD simulation can be a useful predictor of interfacial region thermophysics if it incorporates a physically realistic interaction potential model.

24

Liquid-Vapor Phase-Change Phenomena

FIGURE 1.10  Comparison of MD simulation predictions of surface tension for water with recommended values from the ASHRAE Fundamentals Handbook [1.12].

MD simulations can also provide insight into temporal variation of properties in the interfacial region. In this regard, it is interesting to note that molecular capillarity theory implies that the core of the interfacial region corresponds to conditions that do not satisfy (∂ Pr / ∂ vr )Tr < 0, which is a necessary condition for thermodynamic phase stability. As an example, Fig. 1.11 shows the density variation across the interfacial region and the range of conditions that do not satisfy the necessary condition for stability predicted for a Redlich-Kwong fluid at Tr = 0.7. The predicted lack of intrinsic stability in the core region of the interfacial layer suggests that this region may be subject to high levels of property fluctuations. The results of MD simulations appear to corroborate this prediction. In the MD simulation study of Wemhoff and Carey [1.26],

FIGURE 1.11  Density profile and interfacial region structure for a Redlich-Kwong fluid predicted by the model of Carey [1.11].

The Liquid-Vapor Interfacial Region

25

FIGURE 1.12  Variation of the interfacial free energy per unit area with time in the simulation of Tarek et al. [1.34] for a water and ethanol mixture with a mean ethanol mole fraction of 0.1.

for example, the density profile and the variation of the standard deviation in local density across the interfacial region were determined from the simulation statistics for argon. In their simulation results, Wemhoff and Carey [1.26] found that the density standard deviation exhibits a local maximum in the center of the interfacial region. The local peak in the standard deviation of density is consistent with the expectation that the lack of stability in this region results in high levels of property fluctuations. Another result indicating fluctuations in the interfacial region is found among the MD simulations of an ethanol and water free liquid film reported by Tarek et al. [1.34]. Tarek et al. [1.34] determined the time variation of the interfacial tension (interfacial free energy per unit area) in their simulations. An approximate fit to the time variation of the interfacial free energy during one of their simulations is shown in Fig. 1.12. The simulation indicates that over a time scale of a few picoseconds and surface areas with length scales of a few molecular diameters, the interfacial free energy per unit area may fluctuate substantially. Taken as a group, the results of these recent MD simulation studies appear to confirm the tendency for substantial property fluctuations at small length and time scales in the core of the interfacial region between bulk liquid and bulk vapor phases. As noted above, a second category of MD simulation study of interest here is the subset that models the interfacial region thermophysics of a droplet or molecular cluster. Typical of this type of simulation modeling are the studies of Rusanov and Brodskaya [1.35] and Thompson et al. [1.36] who used a molecular dynamics simulation method to study a droplet at equilibrium. In the simulations of Thompson et al. [1.36], the molecules interacted with a truncated Lennard-Jones intermolecular potential. The system molecules were contained by a spherical wall that exerted a repulsive force on molecules in the region near the wall. At any given instant during the simulation, the configuration of the molecules in the system is non-symmetric and non-uniform, as indicated schematically in Fig. 1.13. However, statistically, over time, the position of the density discontinuity that defines the droplet has a spherical mean shape and the surrounding vapor has uniform average density. Thompson et al. [1.36] performed simulation calculations for systems containing 256–2048 molecules. Density profiles, pressure profiles, and the surface tension were determined from the simulation runs. The effects of surface curvature and temperature on these properties were also explored. More recent molecular dynamics simulations of droplets at equilibrium by Nijmeijer et al. [1.37] and Maruyama et al. [1.38] have examined the behavior of molecules at the surface of the droplet in detail. While MD simulations of droplets provide valuable insight into the thermophysics of the interfacial region of the droplets, perhaps the most useful insight provided by studies of this type is their

26

Liquid-Vapor Phase-Change Phenomena

FIGURE 1.13  Droplet simulation model considered by Thompson et al. [1.36].

predictions regarding the effects of droplet radius of curvature on surface tension. These predictions are discussed further in the next section.

1.5  SMALL SYSTEM EFFECTS The molecular capillarity models described above are formulated for a flat interface. These results are expected to be applicable when the thickness of the interfacial region is very small compared to the radius of curvature of the surface defining the outer boundary of the interfacial region, or if the interfacial region is very thin compared to the physical extent of the bulk liquid or vapor regions adjacent to the interfacial region. The case of ultra-small droplets that may exist just after nucleation or just before they evaporate completely, is of particular interest in phase-change applications. Since the interfacial region thickness is frequently on the order of a few nanometers, the interface must have a very small radius of curvature to deviate from the flat-interface theory. One circumstance in which the principal radii of curvature are very small is the interfacial region of very small liquid droplets. Under some common conditions, droplets formed by nucleation in a vapor may form with an initial radius smaller than 100 nm (see, for example, the investigation by Carey [1.39]). For a spherical droplet, the two principal radii of curvature of the outer surface of the interfacial region surface are the same and are equal to the droplet radius. This is depicted in Fig. 1.14. Molecular theories of capillarity, in combination with thermodynamic analysis, predict that the interfacial tension will vary with radius of curvature when the radius of curvature becomes comparable to the thickness of the interfacial region. Perhaps the most well-known analysis of this type is that due to Tolman [1.40] (see also the reference of Rowlinson and Widom [1.7]). A simple model analysis of the effect of radius on surface tension for a nanodroplet can be developed by extending the capillarity theory analysis of the previous section to the interfacial region shown in Fig. 1.14. As discussed in Appendix III, in classic molecular capillarity theory, application

FIGURE 1.14  Interfacial region for a nanodroplet surrounded by vapor.

27

The Liquid-Vapor Interfacial Region

of the calculus of variations leads to the following relation which links the surface tension to the integral of twice the excess free energy per unit volume ψe over the interfacial region. ∞

∫ 2ψ (ρˆ ,T ) dz (1.46)

σ lv =



e

−∞

The excess volumetric free energy ψe is given by

ψ e ( ρˆ , T ) = ψ 0 ( ρˆ , T ) − ψ 0 ( ρˆ v , T ) + µˆ v ( ρˆ − ρˆ v ) (1.47)

where ψ 0 ( ρˆ , T ) is F/V for uniform system density, evaluated at the local density ρˆ and system temperature T, and ψ 0 ( ρˆ v , T ) is F/V for uniform system density, evaluated at the saturated vapor density and temperature T. In the last term on the right side of Eq. (1.47), µˆ v and ρˆ v are the chemical potential and density, respectively, for the saturated vapor at temperature T. Hence the integrand in Eq. (1.46) is a combination of the uniform-density thermodynamic properties evaluated at the system temperature and either the local density in the interfacial region, or the saturated vapor density. Note that even though the limits of integration in Eq. (1.46) are ±∞, the integration is essentially over the interfacial region because thermodynamic relations dictate that the integrand is zero outside it. Also, we can interpret this as a volume integration because the integral with respect to z is a volume integral for a unit area of the outer surface of the interfacial region. Since the integrand is finite and well-behaved, we define a mean value of the volumetric excess free energy ψe as

ψe =

1 δzi

δzi /2

∫

ψ e dz =

−δzi /2

1 Ai δzi

Ai δzi

∫ ψ dV (1.48) e

0

In terms of ψ e, the relation (1.46) can be written in the form

σ lv = 2ψ eVir′′ (1.49)

where Vir′′ is the volume of the interfacial region per unit area of interface. This implies that we can interpret the surface tension as twice the product of the mean excess free energy density and the volume of the interfacial region per unit area. We now extend this interpretation to the interfacial region of the droplet shown in Fig. 1.14. If the droplet radius is large compared to the interfacial region thickness (rd » δzi), the interface volume to area ratio is 4 πrd2 (δzi ) / 4 πrd2 = δzi and the mean volumetric free energy corresponds to that for a flat interface (infinite radius of curvature) ψ e ,∞ Substituting these into Eq. (1.49) leads to

σ lv ,∞ = 2ψ e ,∞ δzi (1.50)

For the spherical droplet in Fig. 1.14, the mean volumetric free energy is expected to deviate from that for the flat-interface case as δzi /rd increases. We therefore define

η = δzi /rd (1.51)

and consider an expansion of ψ e in a Taylor series about ψ e ,∞ for small η

 ∂ψ e  η +  (1.52) ψ e = ψ e ,∞ +   ∂η  η= 0

28

Liquid-Vapor Phase-Change Phenomena

Using the definition of η and defining

α0 =

1  ∂ψ e  (1.53) ψ e ,∞  ∂( δzi / rd )  η= 0

the expansion can be reorganized to the form

   δz  ψ e = ψ e ,∞ 1 + α 0  i  +  (1.54)  rd   

For the spherical droplet, the interfacial region volume to area ratio is

Vir′′ =

3 ( 4π / 3) rd3 − ( rd − δzi ) 

4 πrd2

(1.55)

Expanding the numerator and simplifying yields

 δz δz 2  Vir′′ = δzi 1 − i + i2  (1.56) rd 3rd  

Using the relations (1.54) and (1.56) to evaluate ψ e and Vir′′ in Eq. (1.49), the surface tension for the droplet is given by

   δz δz 2   δz  σ lv = 2ψ e ,∞ 1 + α 0  i  +  δzi 1 − i + i2  (1.57)  rd  rd 3rd    

Since we are interested in the onset of curvature effects on surface tension, we consider δzi /rd small but significant compared to 1. Multiplying the terms in square brackets in Eq. (1.57) and neglecting terms of order (δzi /rd )2 and higher, we obtain

   δz  σ lv = 2ψ e ,∞ δzi 1 − (1 − α 0 )  i  +  (1.58)  rd   

Combining Eqs. (1.50) and (1.58) to eliminate ψ e , the resulting relation is

 (1 − α 0 ) δzi  σ lv = σ lv ,∞ 1 −  (1.59) rd  

The numerator in the fraction inside the square brackets is a length scale that is on the order of the thickness of the interfacial region. In a pioneering study, Tolman [1.16] used thermodynamic analysis to obtain the following relation for the surface tension of a droplet of radius rd:

σ lv =

σ lv ,∞ (1.60) [1 + 2δT / rd ]

In this equation, δT is a characteristic length referred to as the Tolman length. Note that for |δT  /rd| < 1, the right side can be expanded in terms of δT /rd to obtain

σ lv  2δ  = 1 − T +  (1.61) σ lv ,∞  rd 

The Liquid-Vapor Interfacial Region

29

If Eqs. (1.59) and (1.61) are both valid, the length scale (1 − α0)δzi/2 in the analysis developed above must be equivalent to the Tolman length scale. This suggests that the Tolman length scales with the thickness of the interfacial region. These results imply that curvature effects become important when rd becomes comparable to δT. Investigations of ways to predict the Tolman length have included theoretical models [1.41] and molecular dynamic simulations of nanodroplets [1.37, 1.42]. Both types of studies suggest that away from the critical point, the Tolman length is on the order of the effective diameter of the molecule. Numerous investigators have explored computational and theoretical methods to predict the dependence of surface tension on the size of small droplets and the Tolman length at which size effects first become important. Methods used in such studies have included theoretical models [1.41], MD simulations [1.37, 1.42–1.45], Monte Carlo simulations [1.46, 1.47], density functional calculations [1.48–1.51] and semi-empirical thermodynamic models [1.45, 1.52]. These studies suggest that away from the critical point, the Tolman length is on the order of the effective diameter of the molecule. The predictions of such models also generally agree that the droplet surface tension approaches the flat-interface value as the droplet radius rd becomes large, and the surface tension decreases below the flat-interface value as rd becomes extremely small. However, the predicted behavior at intermediate rd has varied somewhat. Some MD models and density functional models predict that the surface tension first increases slightly and then decreases as rd becomes small. This is depicted by the upper curve A in Fig. 1.15. Note that if surface tension initially increases when curvature effects turn-on, this would correspond to a negative value of the Tolman length in Eq. (1.61). On the other hand, a large fraction of the MD simulation model studies have indicated that surface tension decreases continuously when curvature effects turn-on, which corresponds to a positive Tolman length. This is depicted by the lower curve B in Fig. 1.15. Some researchers in this area have suggested that the negative δT values are a consequence of approximations or idealizations in the models that result in non-physical behavior. The alternative possibility is that either type of behavior may result as a consequence of variations of the intermolecular interactions in different systems. Some MD simulation results indicate that the Tolman length sensitively depends on the interaction potential. At this point, it is fair to say that the discrepancy in sign and its dependence on the interaction potential is not fully understood.

FIGURE 1.15  Qualitative variation of surface tension with radius predicted for liquid droplets of small radius with Tolman length δ T < 0 (A) and δ T > 0 (B).

30

Liquid-Vapor Phase-Change Phenomena

Studies that exemplify the trends indicated above include the work of Thompson et al. [1.36] who used data from their simulations to determine the Tolman length δT for a Lennard-Jones fluid at several dimensionless temperatures. The δT values were consistently positive, implying that surface tension decreases as the radius of curvature increases. The resulting values of δT for a fixed temperature varied substantially with droplet size, apparently as a consequence of the small droplet size in the simulation. The scatter suggests a high level of uncertainty in the δT values determined from these simulations. The theoretical model developed by Kalikmanov [1.41] indicates that δT/D is on the order of 0.2, where D is a molecular diameter. It also indicates that as T → Tc, δT becomes negative and diverges. Molecular dynamics simulations of droplets, such as those by Nijmeijer et al. [1.37] and Haye and Bruin [1.42] indicate that the magnitude of δT is slightly smaller than the molecular diameter D. Nijmeijer et al. [1.37] concluded that |δT/D| < 0.7, but they were unable to definitively determine whether δT was positive or negative. Haye and Bruin [1.42] developed molecular dynamic simulations for a Lennard-Jones 12-6 fluid, and used a relation proposed by Blokhuis and Bedeaux [1.53] to determine δT . Haye and Bruin [1.42] found that δT /D was close to 0.2 for 0.696 ≤ 6 T/Tc ≤ 0.835. At T/Tc = 0.881 and above, they observed large fluctuations in instantaneous property values that made accurate prediction of δT difficult. Based on the results of the investigations cited above, a value between 0.2 D and 0.7 D is often used to estimate the effect of radius of curvature on surface tension using Tolman’s relation (1.60) or (1.61). For water in particular, Pruppacher and Klett [1.54] recommended a value of 0.157 nm for δT. Since water has an effective diameter of about 0.28 nm, this value of δT is about 0.56 D. Also of note is that Bartell [1.55], Blokhuis and Kuipers [1.56], and Xioa-Song and Ru-Zheng [1.52] have used thermodynamic analysis to derive simple relations relating the Tolman length, flatinterface surface tension and the liquid isothermal compressibility κ l ≡ ρl−1 ( ∂ρl / ∂ P )T at two-phase coexistence for conditions far from the critical point. The most recent of these relations, derived by Xiao-Song and Ru-Zheng [1.52], is δT =



2πκ l σ lv ,∞ (1.62) 5 AT

where AT is a constant that can be evaluated using MD simulation results. Using results of an earlier study by Lekner and Henderson [1.57], Xioa-Song and Ru-Zheng [1.52] estimated AT to be π/30. This type of relation makes it possible to predict δT from other bulk fluid properties. Note that this relation predicts that δT is strictly positive, whereas a similar relation obtained by Blockhuis and Kuipers [1.57] predicts a negative value for δT. In later sections, we will see that the reduction in the surface tension due to interface curvature can have a significant effect on the equilibrium properties for a droplet-vapor system, and may affect the onset of nucleation. Example 1.4 Just before it evaporates completely, a water droplet at 100°C surrounded by saturated vapor has a diameter of 1.4 nm. Estimate the deviation of the surface tension from the flat-interface value for these conditions. Based on the recommendation of Pruppacher and Klett [1.54], we estimate the Tolman length δT to be 0.157 nm for water. Substituting into Eq. (1.61) yields

σ lv  2δ = 1− T σ lv ,∞  rd

 = 

 2(0.157)  1− 1.4 / 2  = 0.551

This result indicates that the surface tension for the droplet interface is about 55% of the value for a flat interface at the same temperature.

The Liquid-Vapor Interfacial Region

31

FIGURE 1.16  Schematics of the MD simulation of a liquid film on a solid surface used by Carey and Wemhoff [1.58]: (a) computational domain, (b) fluid regimes.

There are two other noteworthy circumstances in which small system effects can alter the interfacial tension from that for a flat interface between two bulk fluids. One is the instance of an ultrathin liquid film on a smooth solid surface. Carey and Wemhoff [1.58–1.60] explored the key nanoscale features of these circumstances using MD simulations for monatomic argon, diatomic nitrogen, and water. Their simulations modeled a layer of liquid phase molecules between a solid metallic surface and a vapor phase. Figure 1.16a depicts this type of simulation [1.58] for monatomic argon liquid and vapor molecules adjacent to a crystalline metal surface. As indicated in Figs. 1.16b and 1.17, at equilibrium, the mean density profile predicted by the MD simulations adjacent to the solid reflects regions of surface affected liquid (in which molecules exhibit some ordering), a region of bulk liquid, an interfacial region, and a vapor region. The noteworthy point here is that as overall film thickness is reduced (say due to evaporation) eventually the bulk liquid region disappears and the interfacial region is affected by the solid-surfaces forces that dictate the structuring in the structured layer immediately adjacent to the solid atoms. For such conditions, where the overall film thickness is comparable to the interfacial region thickness, the interfacial free energy is expected to deviate from the value for an interface between bulk liquid and vapor phases. As will be discussed later in Chapters 3 and 8, the behavior of the liquid film in these circumstances is often parameterized in terms of the disjoining pressure for the film. The third circumstance of interest in which small system effects impact surface tension is for extremely thin free liquid films that can be created, for example, when two bubbles are in close proximity just prior to merging. As will be described later, bubble merging can be an important mechanism in boiling and condensation processes. Figure 1.18 depicts two bubbles during boiling at a heated surface. Growth of the bubbles can thin the liquid film between them until the film thickness at the

32

Liquid-Vapor Phase-Change Phenomena

FIGURE 1.17  Argon liquid film mean mass density profile determined from the MD simulation data of Carey and Wemhoff [1.58] for Tr = 0.57 .

thinnest point is only nanometers thick. Gan and Carey [1.61, 1.62] explored the variation of surface tension with film thickness and the film stability for such circumstances using a hybrid analysis that included an extended capillarity theory model for the entire film (both interfaces and the core region) together with MD simulations. The same SPC/E intermolecular potential model was used in both the capillarity theory analysis and the MD simulation so results for water were directly comparable. The Gan and Carey [1.61] molecular capillarity theory mean density profile predictions for water at reduced temperature Tr = 0.6 for two different film thicknesses are shown in Fig. 1.19. Note that when the film is thick (Fig. 1.18a) the core region has an essentially constant mean density at the bulk liquid value. Reducing the number of molecules in the simulation thins the film, with the result that eventually the core region disappears, leaving two interfacial regions to interact with each other. Further reduction in film thickness results in the two intraspinodal layers merging to form a single core region that lacks intrinsic stability. Gan and Carey [1.61] found that MD simulation run for these conditions did not form a stable film, implying that the film would rupture below this thickness. Gan and Carey [1.61] also found that for pure water, as the film thickness decreased, the interfacial tension decreased below the value for an interface separating bulk phases at the same temperature. Their MD simulation prediction of the surface tension variation with film thickness for pure water at Tr   = 0.6 is shown in Fig. 1.20. Note that this implies that the variation in film thickness between two adjacent bubbles (Fig. 1.18) will result in a variation of the surface tension along the bubble interfaces when the film separating them becomes progressively smaller. The impact of this effect on bubble merging will be discussed in later chapters.

FIGURE 1.18  Thin free liquid film between adjacent bubbles during nucleate boiling.

The Liquid-Vapor Interfacial Region

33

FIGURE 1.19  Density profiles predicted by the film capillarity theory model of Gan and Carey [1.61] for water at Tr = 0.6 : (a) δ f = 4.03 nm resulting in a film with a stable core, (b) δ f = 1.33 resulting in incipient loss of film core stability (rupture).

FIGURE 1.20  Variation of surface tension with film thickness predicted by the MD simulations of Gan and Carey [1.61] for pure water at Tr = 0.6.

34

Liquid-Vapor Phase-Change Phenomena

REFERENCES 1.1 Carey, V. P., Statistical Thermodynamics and Microscale Thermophysics, Cambridge University Press, New York, NY, 1999. 1.2 Hirschfeider, J. O., Curtis, C. F., and Bird, R. B., Molecular Theory of Gases and Liquids, Wiley, New York, NY, 1954. 1.3 Tien, C. L., and Lienhard, J. H., Statistical Thermodynamics, Hemisphere, New York, NY, 1979. 1.4 McQuarrie, D. A., Statistical Thermodynamics, University Science Books, New York, NY, 1997. 1.5 Prausnitz, J. M., Lichtenthaler, R. N., and de Azevedo, E. G., Molecular Thermodynamics of FluidPhase Equilibria, 3rd ed., Prentice Hall, Englewood Cliffs, NJ, 1998. 1.6 Rowlinson, J. S., Translation of van der Waals, J. D., The thermodynamics theory of capillarity under the hypothesis of a continuous variation of density, J. Stat. Phys., vol. 20, pp. 197–245, 1979. 1.7 Rowlinson, J. S., and Widom, B., Molecular Theory of Capillarity, Oxford University Press, Oxford, 1989. 1.8 Sokolnikoff, I. S., and Redheffer, R. M., Mathematics of Physics and Modern Engineering, 2nd ed., Ch. 5, McGraw-Hill, New York, NY, 1966. 1.9 Poling, B. E., Prausnitz, J. M., and O’Connell, J. P., The Properties of Gases and Liquids, 5th ed., McGraw Hill, New York, NY, 2000. 1.10 Beysens, D., and Robert, M., Thickness of fluid interfaces near the critical point from optical reflectivity measurements, J. Chem. Phys., vol. 87, pp. 3056–3061, 1987. 1.11 Carey, V. P., Thermodynamic properties and structure of the liquid-vapor interface: A neoclassical Redlich-Kwong model, J. Chem. Phys., vol. 118, pp. 5053–5064, 2003. 1.12 ASHRAE Fundamentals Handbook, American Society of Heating, Refrigeration and Air-conditioning Engineers, Atlanta, GA, 2001. 1.13 Carey, V. P., and Wemhoff, A. P., Relationships among liquid-vapor interfacial region properties: Predictions of a thermodynamic model, Int. J. Thermophys., vol. 25, pp. 129–161, 2003. 1.14 Hey, M. J., and Wood, D. W., Estimation of liquid/vapor interfacial thicknesses from surface energies, J. Colloid Interf. Sci., vol. 90, pp. 277–279, 1982. 1.15 Wemhoff, Aaron P., Dependence of the equation of state in surface tension prediction by the theory of capillarity, paper IMECE2009-10221, Proc. ASME 2009 Int. Mechanical Engineering Congress Exposition IMECE 2009, November 13–19, Lake Buena Vista, FL. 1.16 Wemhoff, Aaron P., Extension of the neoclassical theory of capillarity to advanced cubic equations of state, Int. J. Thermophysics, vol. 31, pp. 253–275, 2010. 1.17 Metropolis, N. A., Rosenbluth, A. W., Rosenbluth, M. N., Teller, A. H., and Teller, E., Equation of state calculations by fast computing machines, J. Chem. Phys., vol. 21, pp. 1087–1092, 1953. 1.18 Alder, B. J., and Wainwright, T. E., Studies in molecular dynamics. I. General method, J. Chem. Phys., vol. 31, pp. 459–466, 1959. 1.19 Verlet, L., Computer “Experiments” on classical fluids. I. Thermodynamical properties of LennardJones molecules, Phys. Rev., vol. 159, pp. 98–103, 1967. 1.20 Hoover, W. G., Ladd, A. J. C., and Hoover, V. N., Historical development and recent applications of molecular dynamics, ACS Adv. Chem. Ser., vol. 204, pp. 29–46, 1983. 1.21 Hoover, W. G., Molecular Dynamics, Springer Verlag, Berlin, 1986. 1.22 Haile, J. M., Molecular Dynamics Simulation, Elementary Methods, John Wiley & Sons, Inc, New York, NY, 1992. 1.23 Frenkel, D., and Smit, B., Understanding Molecular Simulation, From Algorithms to Applications, 2nd ed., Academic Press, San Diego, CA, 2001. 1.24 Holcomb, C. D., Clancy, P., and Zollweg, J. A., A critical study of the simulation of the liquid-vapor interface of a Lennard-Jones fluid, Mol. Phys., vol. 78, pp. 437–459, 1993. 1.25 Chen, M., Guo, Z.-Y., and Liang, X.-G., Molecular simulation of some thermophysical properties and phenomena, Microscale Thermophys. Eng., vol. 5, pp. 1–16, 2001. 1.26 Wemhoff, A. P., and Carey, V. P., Molecular dynamics exploration of properties in the liquid-vapor interfacial region. Paper HT2003–47158, Proc. ASME Summer Heat Transfer Conf., Las Vegas, NV, 2003. 1.27 Wemhoff, A. P., and Carey, V. P., Surface tension evaluation via thermodynamic analysis of statistical data from molecular dynamic simulations, Paper HT-FED04–56690, Proc. 2004 ASME Heat Transfer/ Fluids Engineering Summer Conf., Charlotte, NC, 2004. 1.28 Alejandre, J., Tildesley, D. J., and Chapela, G. A., Molecular dynamics simulation of the orthobaric densities and surface tension of water, J. Chem. Phys., vol. 102, pp. 4574–4583, 1995.

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1.29 Dang, L. X., and Chang, T.-M., Molecular dynamics study of water clusters, liquid, and liquid-vapor interface of water with many-body potentials, J. Chem. Phys., vol. 106, pp. 8149–8159, 1997. 1.30 Matusmoto, M., Takaoka, Y., and Kataoka, Y., Liquid-vapor interface of water-methanol mixture. I. Computer simulation, J. Chem. Phys., vol. 98, pp. 1464–1472, 1993. 1.31 Daiguji, H., Molecular dynamics study of n-alcohols adsorbed on an aqueous electrolyte solution, J. Chem. Phys., vol. 115, pp. 1538–1549, 2001. 1.32 Alejandre, J., Tildesley, D. J., and Chapela, G. A., Fluid phase equilibria using molecular dynamics: The surface tension of chlorine and hexane, Mol. Phys., vol. 85, pp. 651–663, 1995. 1.33 Hariharan, A., and Harris, J. G., Structure and thermodynamics of the liquid-vapor interface of fluorocarbons and semifluorinated alkane diblocks: A molecular dynamics study, J. Chem. Phys., vol. 101, pp. 4156–4165, 1994. 1.34 Tarek, M., Tobias, D. J., and Klein, M. L., Molecular dynamics investigation of an ethanol-water solution, Physica A, vol. 231, pp. 117–122, 1996. 1.35 Rusanov, A. I., and Brodskaya, E. N., The molecular dynamics simulation of a small drop, J. Colloid Set., vol. 62, pp. 542–555, 1977. 1.36 Thompson, S. M., Gubbins, K. E., Walton, J. P. R. B., Chantry, R. A. R., and Rowlinson, J. S., A molecular dynamics study of liquid drops, J. Chem. Phys., vol. 81, pp. 530–542, 1984. 1.37 Nijmeijer, M. J. P., Bruin, C., van Woerkom, A. B., Bakker, A. F., and van Leeuwen, J. M. J., Molecular dynamics of the surface tension of a drop, J. Chem. Phys., vol. 96, pp. 565–576, 1992. 1.38 Maruyama, S., Matsumoto, S., and Ogita, A., Surface phenomena of molecular clusters by molecular dynamics method, Therm. Sci. Eng., vol. 2, pp. 7–84, 1994. 1.39 Carey, V. P., DSMC modeling of interface curvature effects on near-interface transport, Microscale Thermophys. Eng., vol. 6, pp. 55–74, 2002. 1.40 Tolman, R. C., Effect of droplet size on surface tension, J. Chem. Phys., vol. 17, pp. 333–343, 1949. 1.41 Kalikmanov, V. I., Semiphenomenological theory of the Tolman length, Phys. Rev. E, vol. 55, pp. 3068–3071, 1997. 1.42 Haye, M. J., and Bruin, C., Molecular dynamics study of the curvature correction to the surface tension, J Chem. Phys., vol. 100, pp. 556–559, 1989. 1.43 Vrabec, J., Kedia, G. K., Fuchs, G., and Hasse, H., Comprehensive study of the vapour–liquid coexistence of the truncated and shifted Lennard–Jones fluid including planar and spherical interface properties, Mol. Phys., vol. 104, pp. 1509–1527, 2006. 1.44 Horsch, M., Vrabec, J., and Hasse, H., Modification of the classical nucleation theory based on molecular simulation data for surface tension, critical nucleus size, and nucleation rate, Phys Rev E, vol. 78, 011603-1–011603-8, 2008. 1.45 Julin, J., Napari, I., Merikanto, J., and Vehkamäki, H., A thermodynamically consistent determination of surface tension of small Lennard-Jones clusters from simulation and theory, J. Chem. Phys., vol. 133, 044704-1–044704-6, 2010. 1.46 Ten Wolde, P. R., and Frenkel, D., Computer simulation study of gas–liquid nucleation in a LennardJones system, J. Chem. Phys., vol. 109, pp. 9901–9918, 1998. 1.47 Schrader, M., Virnau, P., and Binder, K., Simulation of vapor-liquid coexistence in finite volumes: A method to compute the surface free energy of droplets Phys Rev E, vol. 79, 061104-1–061104-12, 2009. 1.48 Koga, K., Zeng, X. C., and Shchekin, K. A., Validity of Tolman’s equation: How large should a droplet be?, J. Chem. Phys., vol. 109, p. 4063–4070, 1998. 1.49 Bykov, T. V., and Shchekin, A. K., Thermodynamic characteristics of a small droplet in terms of the density functional methods, Colloid J., vol. 61, pp 144–151, 1999. 1.50 Barrett, J. C., Some estimates of the surface tension of curved surfaces using density functional theory, J. Chem. Phys., vol. 124, 144705-1–144705-6, 2006. 1.51 Napari, I., and Laaksonen, A., Surface tension and scaling of critical nuclei in diatomic and triatomic fluids, J. Chem. Phys., vol. 126, pp. 134503–134503-7, 2007 (doi: 10.1063/1.2714950). 1.52 Xiao-Song, W., and Ru-Zeng, Z., Relation between Tolman length and isothermal compressibility for simple liquids, Chin. Phys. B., vol. 22, pp. 036801-1–036801-3, 2013. 1.53 Blokhuis, E. M., and Bedeaux, D., Derivation of microscopic expressions for the rigidity constants of a simple liquid-vapor interface, Physica A, vol. 184, pp. 42–70, 1992. 1.54 Pruppacher, H. R., and Klett, J. D., Microphysics of Clouds and Precipitation, Reidel, Dordrecht, The Netherlands, 1980. 1.55 Bartell, L. S., Tolman’s δ, Surface curvature, compressibility effects, and the free energy of drops, J. Phys. Chem. B, vol. 105, pp 11615–11618, 2001.

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1.56 Blokhuis, E. M., and Kuipers, J., Thermodynamic expressions for the Tolman length, J. Chemical Physics, vol. 124, pp. 074701-1–074701-8, 2006. 1.57 Lekner, J., and Henderson, J. R., Surface tension and energy of a classical liquid-vapour interface, Mol. Phys., vol. 34, pp. 333–359, 1977. 1.58 Carey, V. P., and Wemhoff, A. P., Disjoining pressure effects in ultra-thin liquid films in micropassages—comparison of thermodynamic theory with predictions of molecular dynamics simulations, J Heat Transfer, vol. 128, pp. 1276–1284, 2006. 1.59 Wemhoff, A. P., and Carey, V. P., Molecular dynamics exploration of thin liquid films on solid surfaces. 1. Monatomic fluid films, Microscale Thermophys. Eng., vol. 9, pp. 331–349, 2005. 1.60 Wemhoff, A. P., and Carey, V. P., Molecular dynamics exploration of thin liquid films on solid surfaces. 2. Polyatomic nonpolar fluid and water films Microscale Thermophys. Eng., vol. 9, pp. 351–363, 2005. 1.61 Gan, Y., and Carey, V. P., Hybrid modeling of interfacial region thermophysics and intrinsic stability of thin free liquid films, Int. J. Heat Mass Transfer, vol. 53, pp. 2169–2182, 2010. 1.62 Carey, V. P., Molecular level modeling of interfacial phenomena in boiling processes, Exp. Heat Transfer, vol. 26, pp. 296–327, 2013.

PROBLEMS 1.1 The van der Waals constants are quoted as being av = 1690 Pa m6/kg2, bv = 0.00175 m3/kg for water, and av = 175 Pa m6/kg2, bv = 0.00140 m3/kg for nitrogen. How do these values compare to those computed using Eqs. (1.18a) and (1.18b)? (see Appendix II for values of Tc and Pc.) Based on the above values of the van der Waals constants, which of these two fluids would you expect to have the higher latent heat of vaporization? Do property data in Appendix II support your conclusion? Briefly explain your answer. 1.2 Effective diameters of water and nitrogen molecules are 0.28 nm and 0.31 nm, respectively. Recommended values of the van der Waals constant bv for water and nitrogen are 0.00175 m3/kg and 0.00140 m3/kg, respectively. Compare these values of bv to values computed from the respective molecular diameter value using Eqs. (1.6), (1.14b), and (1.17b). 1.3 (a) Use values of Tc and Pc from Appendix II and Eqs. (1.18) to determine the van der Waals constants for oxygen. (b) Use Eqs. (1.5) and (1.6) to determine the Lennard-Jones constants that correspond to the computed van der Waals constants. (c) For saturated oxygen vapor at 90 K, use these results to estimate the fraction of oxygen molecules that have translational energies greater than the Lennard-Jones potential well depth ε. 1.4 (a) Use values of Tc and Pc from Appendix II and Eqs. (1.18) to determine the van der Waals constants for mercury. (b) Use Eqs. (1.5) and (1.6) to determine the Lennard-Jones constants that correspond to the computed van der Waals constants. (c) For saturated mercury vapor at 1000 K use these results to estimate the fraction of oxygen molecules that have translational energies greater than the LennardJones potential well depth ε. 1.5 (a) Use Eqs. (1.18) and values of Tc and Pc from Appendix II to determine the van der Waals constants for R-134a. (b) Use Eqs. (1.5) and (1.6) to determine the Lennard-Jones constants that correspond to the computed van der Waals constants. (c) For saturated R-134a vapor at 338 K, use these results to estimate the fraction of R-134a molecules that have translational energies greater than the Lennard-Jones potential well depth ε. 1.6 For rmin = r0 and rmax = ∞, show using Eq. (III.9) in Appendix III that for a system of molecules that interact according to the Lennard-Jones potential, κ v = (9/7)r02 . 1.7 For nitrogen, plot saturation vapor pressure versus temperature using the Redlich-Kwong equation (1.39) for temperatures between 77 K and the critical point. On the same plot, show the recommended saturation pressure values listed in Appendix II for nitrogen. Assess the accuracy of Eq. (1.39) for nitrogen. 1.8 For water, plot saturation vapor pressure versus temperature using the Redlich-Kwong equation (1.39) for temperatures between 373 K and the critical point. Also plot the recommended saturation pressure values listed in Appendix II for water. Assess the accuracy of Eq. (1.39) for water. 1.9 For mercury, plot saturation vapor pressure versus temperature using the Redlich-Kwong equation (1.39) for temperatures between 630 K and the critical point. On the same plot, show the recommended saturation pressure values listed in Appendix II for mercury. Assess the accuracy of Eq. (1.39) for mercury. 1.10 Plot the R-134a vapor pressure data from Appendix II between 247 K and the critical point on a Psat versus 1/T plot with a linear vertical axis and a logarithmic horizontal axis (semi-log plot). In doing so, note that T is absolute temperature (K). Also plot the variation predicted by the Redlich-Kwong equation (1.39) for this fluid over the same range. How do they compare?

The Liquid-Vapor Interfacial Region

37

1.11 Plot the mercury vapor pressure data from Appendix II between 630 K and the critical point on a Psat versus 1/T plot with a linear vertical axis and a logarithmic horizontal axis (semi-log plot). In doing so, note that T is absolute temperature (K). Also plot the variation predicted by the Redlich-Kwong equation (1.39) for this fluid over the same range. How good is the agreement? 1.12 For oxygen, use the critical constants in Appendix II to plot surface tension versus temperature using the Redlich-Kwong equation (1.41) for temperatures between 90 K and the critical point. On the same plot, show the recommended surface tension values listed in Appendix II for oxygen. Assess the accuracy of Eq. (1.41) for oxygen. 1.13 For mercury, use the critical constants in Appendix II to plot surface tension versus temperature using the Redlich-Kwong equation (1.41) for temperatures between 630 K and the critical point. On the same plot, show the recommended surface tension values listed in Appendix II for mercury. Assess the accuracy of Eq. (1.41) for mercury. 1.14 For water, use the critical constants in Appendix II to plot surface tension versus temperature using the Redlich-Kwong equation (1.41) for temperatures between 373 K and the critical point. On the same plot, show the recommended surface tension values listed in Appendix II for water. Assess the accuracy of Eq. (1.41) for water. 1.15 For ethanol at atmospheric pressure, use the Redlich-Kwong capillarity theory to estimate the surface tension and the interfacial region thickness. Compare the surface tension value to the tabulated value in Appendix II. 1.16 For mercury at atmospheric pressure, use the Redlich-Kwong capillarity theory to estimate the surface tension and the interfacial region thickness. Compare the surface tension value to the tabulated value in Appendix II. 1.17 Equation (1.44), derived from the molecular theory of capillarity, indicates how the thickness of the interfacial region increases as the temperature approaches the critical temperature Tc. Consider a carbon nanotube, with an inside diameter dt of 4 nm, filled with saturated ethanol liquid and vapor. If the temperature is slowly raised, estimate how close the temperature would have to be to Tc for the interfacial region thickness δzi to become comparable to the tube diameter (i.e., δzi ∼ dt). 1.18 In Appendix II of the text, the saturation table for nitrogen lists saturation properties for 126 K which is very close to the critical temperature of 126.25 K. Using the data at 126 K, use Eq. (1.45) to predict the thickness of the interfacial region for these conditions. (Note that you will have to convert hlv to uˆlv using the definition of enthalpy.) How does the result compare to the capillarity theory prediction (1.44)? 1.19 Beysens and Robert [1.10] used an optical technique to measure the interfacial region thickness for SF6 at conditions close to the critical point. At a temperature of 317.1 K their data indicate an interfacial region thickness of 30.6 nm. Compare this measurement with the interfacial region thickness predicted by Eq. (1.44) derived from the molecular theory of capillarity for a Redlich-Kwong fluid. 1.20 Immediately following nucleation, a droplet of liquid nitrogen at 77 K surrounded by saturated vapor has a diameter of 1.2 nm. Estimate the surface tension at the droplet interface, including the deviation from the flat-interface value at this temperature due to the small radius of curvature. Take the effective molecular diameter D for nitrogen to be 3.1 Å and take the Tolman length to be 0.2 D. 1.21 In the latter stages of two-phase flow in an evaporator, the flow may consist of a continuous vapor phase carrying entrained droplets that slowly evaporate. Eventually the evaporating droplets become extremely small before evaporating completely. For a droplet of liquid oxygen at 90 K surrounded by saturated vapor, plot the variation of surface tension for the droplet as its diameter decreases from 2 μm to 1 nm. Take the effective molecular diameter D for oxygen to be 3.6 Å and take the Tolman length to be 0.2D. 1.22 A rigid vessel with an internal volume of 0.014 m3 contains 10 kg of water (vapor plus liquid) initially at 20°C. The vessel is then slowly heated. As a result of the heating process, will the liquid-vapor interface move upward toward the top of the vessel or downward toward the bottom? What if the vessel contains 1 kg instead of 10 kg? What would you see if you could view the contents of the vessel during the heating process and the vessel contained a mass equal to 0.014/vc where vc is the critical specific volume? Would you classify this as a condensation or vaporization process? 1.23 Mercury is distinct from most common fluids in that the liquid-vapor interface at saturation exhibits a very high surface tension away from the critical point. A prediction of capillarity theory is that away from the critical point, at Tr = 0.6 for example, the interfacial region thickness is on the order of a few nanometers. Use the results of capillarity theory to estimate the interfacial tension and thickness of the interfacial region for mercury at 1000 K (Tr = 0.57). Assess how well the theory predicts the surface tension, and assess whether mercury, a high surface tension liquid metal, also conforms to the general trend that the interfacial region is a few nanometers thick for these conditions away from the critical point (Tr = 0.57).

2

The Liquid-Vapor Interface: A Macroscopic Treatment

2.1 THERMODYNAMIC ANALYSIS OF INTERFACIAL TENSION EFFECTS In the previous chapter, we examined the liquid-vapor interfacial region from a molecular viewpoint. The interfacial tension associated with a liquid-vapor interface can also be modeled and analyzed from a macroscopic point of view. As discussed in the previous chapter, the intermolecular attractions that give rise to interfacial tension may result from several different types of molecular forces. For a macroscopic assessment of their importance, it is useful to connect these forces to the types of molecules for which they are important. Dispersion forces, also known as London dispersion forces or van der Waals forces, exist in all types of matter and always give an attractive force between adjacent atoms or molecules no matter how dissimilar they are chemically. The London dispersion forces vary with the electrical properties of the substances involved and the distance between interacting elements, but they are independent of temperature. Metals exhibit metallic bond forces and in some fluids, such as water and some hydrocarbons, molecules interact through hydrogen bonding forces. Fowkes [2.1] has shown that the surface tension can be considered as consisting of the sum of two parts: the part due to dispersion forces, σ d , and the part due to specific forces, such as metallic or hydrogen bonding, σ s

σ = σ s + σ d (2.1)

In these terms, it is easy to understand why surface tensions of liquid metals are higher than those of hydrogen-bonded liquids such as water, which in turn are higher than those of non-polar liquids such as pure hydrocarbons. Surface tension in non-polar liquids is due entirely to dispersion forces. In hydrogen-bonded liquids there are contributions due to both dispersion forces and hydrogen bonding, generally resulting in slightly larger values of σ. For liquid metals, a large σs contribution results from the metallic bond attraction which when added to the dispersion force contribution, results in even higher surface tension values. The trends described above can be seen in the surface tension data shown in Table 2.1. The values are listed in the SI system units of milliNewtons per meter (mN/m). Note that 1 mN/m is equal to 1 dyne/cm. Having deduced some of the ways that the nature of the intermolecular forces affects the magnitude of the interfacial tension, we will next use a macroscopic thermodynamic analysis to explore how surface tension and interface curvature affect properties in the adjacent bulk vapor and bulk liquid phases. The interfacial region is presumed to be bounded by two surfaces SI* and SII* which are parallel to the interface surface S * but are located just within the corresponding bulk phases. As discussed in Chapter 1, the molecular theory of capillarity is based on the premise that properties vary continuously across the interfacial region. Consistent with this premise, we assume that the density and internal energy vary continuously across the interfacial region, as depicted in Fig. 2.1. Note that because the transition between the bulk properties occurs over a finite thickness, the actual mass and internal energy of this region may differ from the values which would be calculated by assuming that the phases I and II extended 39

40

Liquid-Vapor Phase-Change Phenomena

TABLE 2.1 Values of Surface Tension for Various Liquids in Contact with Air or its Own Vapor at Saturation Liquid Silver (Ag) Mercury (Hg) Hydrazine (N2H4) Water (H2O) Ethylene glycol (C2H6O2) Ammonia (NH4) Carbon tetrachloride (CC14) n-Butanol (C4H10O) Acetone (CH3COCH3) Ethanol (C2H6O) Methanol (CH4O) R-113 (CC12FCC1F2) R-11 (CC13F) R-12 (CC12F2) R-134a (CF3CH2F) Helium II (HeII) Helium III (HeIII)

Temperature (°C)

Surface Tension (mN/m)

1100 20 25 20 20 –40 20 20 20 20 20 26.7 26.7 17 18 −271 −271

878 484 91.5 72.8 48.4 35.4 27.0 24.6 24.0 22.8 22.6 19.0 18.0 9.4 9.0 0.32 0.069

FIGURE 2.1  Variations of the mean internal energy and density in the interfacial region.

41

The Liquid-Vapor Interface

all the way to S *. This difference is called the surface excess of the property. The surface excess mass Γ eS* is defined as yII



Γ eS* =

∫ ρ dy − ρ y − ρ y (2.2) I

I

II II

− yI

and the surface excess internal energy UeS* is given by yII



UeS* =

∫ ρu dy − ρ u y −  ρ u y (2.3) I I I

II II II

− yI

Surface excess values of other thermodynamic properties (and mass concentrations in multicomponent systems) can be similarly defined. Note that these quantities have been given a superscript S * because their values depend on the location of the surface S *. A specific location may make either Γ eS* or UeS* zero, but in general they may be positive or negative. This concept facilitates a link between the idealized concept of a two-dimensional interface surface, and the actual threedimensional character of the interfacial region. To examine the relationships between properties in the interfacial region and corresponding ones in the surrounding bulk phases at equilibrium, we can proceed using a thermodynamic analysis. The entire system consisting of the interface region (between SI* and SII* ) and bulk fluids I and II in Fig. 2.2 is assumed to be completely isolated from the surroundings. If the shape of the interfacial region (and hence its volume) is assumed to be fixed, its internal energy may be taken to be a function only of its entropy and the number of molecules present, U = U(S, N). It follows that

dU = TdS + µdN (2.4)

where

dU   ∂U  T =  , µ= (2.5)  ds  N ,shape  ∂ N  S ,shape

FIGURE 2.2  Model system for thermodynamic analysis of interfacial region.

42

Liquid-Vapor Phase-Change Phenomena

We now consider possible internal perturbations that leave bulk fluid I unchanged while permitting exchange of energy or mass between the interfacial region and bulk fluid II. Since the overall system is isolated, we must have

dN total = dN + dN II = 0 (2.6)



dStotal = dS + dSII = 0 (2.7)



dUtotal = TdS + µ   dN + TII dSII + µ II dN II = 0 (2.8)

Combining Eqs. (2.6)–(2.8) yields

( T − TII ) dS + ( µ − µ II ) dN = 0 (2.9)

Since Eq. (2.9) must be satisfied for all possible perturbations of S and N, we must have T = TII and μ = μII. An identical set of arguments can be presented for bulk fluid I. The net result is that, at equilibrium, the temperature and chemical potential must be the same in all three regions, that is,

T = TI = TII , µ = µ I = µ II (2.10)

Equation (2.4) applies to the interface region if the interface shape is fixed, regardless of the variation of thermodynamic properties within the region at equilibrium. We can, therefore, take Eq. (2.9) for the actual property variations and subtract from it the corresponding equation which would apply if the properties equaled those of the respective bulk phases all the way to the interface surface S *. By definition, the resulting equation can be written in terms of the excess properties UeS*, SeS* and N eS* as

dUeS* = TdSeS* + µ   dN eS*  (fixed shape) (2.11)

If we now relax the fixed-shape restriction and allow deformation of the interface, both its area and curvature can change. However, if the radii of curvature are much greater than the interfacial thickness, we might expect curvature effects on the internal energy, entropy, and the density in the interface region to be small. We do, however, want to account for the effect of any increase or decrease in the interface area resulting from deformation. Equation (2.11) is therefore modified as

dUeS* = TdSeS* + µ   dN eS* + σdA S* (2.12)

where we have defined the interfacial tension σ as

 ∂U S *  σ =  eS *  (2.13)  ∂ A  SeS * , NeS *

to account for the effect of changing interface area on the energy balance. For the interfacial region, the Helmholtz free energy F is given by

F = U − TS (2.14)

Subtracting from Eq. (2.14) the corresponding equation which would apply if the two parts of the interfacial region were occupied by bulk fluids I and II (up to an interface with a density discontinuity) yields an analogous relation for the surface excess free energy

FeS* = UeS* − TSeS* (2.15)

43

The Liquid-Vapor Interface

Differentiating Eq. (2.15) and substituting Eq. (2.12) for dUeS* yields

dFeS* = − SeS*   dT + µ   dN eS* + σ   dA  i (2.16)

However, if we consider FeS* = FeS* (T , N eS* , A  i ) from a purely mathematical point of view, we can write

 ∂ F S*   ∂ F S*   ∂ F S*  dFeS* =  e  dT +  eS*  dN eS* +  e  dA  i (2.17)  ∂T  NeS * , A  i  ∂ N e  T , A  i  ∂ A i  NeS * ,T

Comparison of Eqs. (2.16) and (2.17) clearly indicates that the interfacial tension is related to the surface excess free energy as

 ∂ F S*  σ= e  (2.18)  ∂ Ai  NeS * ,T

Thus, σ is equal to the change in surface excess free energy produced by a unit increase in interface area, Ai. The definition of σ in terms of excess free energy is consistent with the capillarity theory formulations discussed in Chapter 1. Note that this quantity is usually referred to as simply the surface free energy (per unit surface area) or surface tension even though it is technically an excess property. We can extract an additional useful relation by considering a shift in the reference surface S * uniformly toward region II by some small amount dy. The position of S * is somewhat arbitrary because of the continuous property variations in the interfacial region. However, if our analysis is to yield meaningful results, the values of properties in the three regions in Fig. 2.2 should be independent of where S * is located. In allowing S * to shift by dy, we therefore expect that the overall free energy F for the system as a whole is unchanged since no change occurs in its physical state. From basic thermodynamic relations, it can be shown that for the bulk fluids

dF = − SdT + µ   dN − PdV (2.19)

Summing together the differential changes in F for the three regions in Fig. 2.2, using Eqs. (2.16) and (2.19), and requiring that the overall dF = 0 yields



dF = − SeS* dT + µ   dN eS* + σdAi − SI dT + µ   dN I − PI dVI (2.20) − SII dT + µ   dN II − PII dVII = 0

Since the total number of moles and total volume do not change, and there is no change of temperature,

dV = dVI + dVII = 0 (2.21)



dN = dN I + dN II + dN eS* = 0 (2.22)



dT = 0 (2.23)

and substitution of Eqs. (2.21) through (2.23) reduces Eq. (2.20) to

 dA  PI − PII = σ  i  (2.24)  dVI 

44

Liquid-Vapor Phase-Change Phenomena

FIGURE 2.3  Geometry of the interfacial region between two phases.

If the surface S * can be characterized by two radii of curvature measured in two perpendicular planes containing the local normal to S *, as shown in Fig. 2.3, then the changes in Ai and Vi for a shift in S * of dy are

dAi = s2 ds1 + s1ds2 (2.25)



dVI = s1s2 dy (2.26)

Using simple geometric relations between sides of similar triangles, it can be shown that

ds1 = s1

dy dy ,  ds2 = s2 (2.27) r1 r2

Combining Eqs. (2.25)–(2.27) yields

dAi 1 1 = + (2.28) dVI r1 r2

which can be substituted into Eq. (2.24) to obtain

 1 1 PI − PII = σ  +  (2.29)  r1 r2 

Equation (2.29) is usually called the Laplace or the Young-Laplace equation. It relates the pressure difference across the interface to the interfacial tension and the geometry of the interface at equilibrium. As will be seen in later sections, this relation is a critical element in the thermodynamic and mechanical analysis of interfacial phenomena. In this section, analysis of the interface as a region of finite thickness has demonstrated that the interfacial tension is related to the surface excess free energy by Eq. (2.18). The Young-Laplace equation, which is often derived from force-balance considerations, was also derived from thermodynamic analysis of the interface region. Finally, it should be noted that the analysis presented in this section has consistently considered an interface between two fluid phases of a pure substance, i.e., its liquid and vapor phase. This analysis can be extended to interfaces in multicomponent systems with only slight modifications. Further discussion of interfacial phenomena in multicomponent systems can be found in references [2.2] and [2.3].

45

The Liquid-Vapor Interface

Example 2.1 Liquid droplets that condense to form fog in the atmosphere can be as small as 2 µm in diameter just after they first form. Determine the pressure inside a droplet of this size at 20°C. For a spherical droplet, the Young-Laplace equation (2.30) requires that

Pinside = Poutside +

2σ r

where r is the droplet radius. For water in contact with air at 20°C, σ = 0.0728 N/m and

Pinside = 101+

2(0.0728) (10 −3 ) = 243 kpa 1.0 × 10 −6

Hence the pressure inside the droplet is about 2.5 atm for these conditions.

2.2  DETERMINATION OF INTERFACE SHAPES AT EQUILIBRIUM As might be expected, interfacial tension plays a major role in the determination of the shape of a liquid-vapor interface at equilibrium. The Young-Laplace equation (2.29) derived in the previous section is the principal mathematical tool used to predict the shape of the interface. Because this equation relates the interfacial tension, interface geometry and pressure difference between the fluids at each point along the interface, it can be used with the equations of hydrostatics to compute the shape of a static interface. Alternatively, if the shape of the interface can be determined experimentally, the Young-Laplace equation can be used to infer the interfacial tension. Evaluation of the radii of curvature in Eq. (2.29) for an arbitrary surface shape leads to a differential equation, which can be quite difficult to solve. However, in many instances, the interface shape is sufficiently symmetric that the geometry can be handled without too much difficulty. The simplest case is the spherical drop or bubble in which both r1 and r2 are equal to its radius r. Equation (2.29) then reduces to

PI − PII = Pinside − Poutside =

2σ (2.30) r

A sign convention is adopted here such that the radii of curvature are always taken as positive with the center of curvature on the concave side of the interface. Thus, in Eq. (2.30) above the pressure inside the drop or bubble PI exceeds the pressure outside PII by an amount equal 2σ/r. Another simple geometric configuration of interest is the sessile drop on a flat solid surface, as shown in Fig. 2.4. Determination of the interface shape requires an analysis which combines the

FIGURE 2.4  A sessile drop on a flat solid surface.

46

Liquid-Vapor Phase-Change Phenomena

equations of hydrostatics and the Young-Laplace equation with appropriate boundary conditions. Due to the hydrostatic heads, the pressure distributions in fluids I and II are given by

PI =   PI0 + ρI gz (2.31)



PII = PII0 + ρII gz (2.32)

Note that in both fluids the reference pressure has been chosen as that at the drop apex (point 0) in Fig. 2.4. Cylindrical coordinates are used with the origin at point 0. The drop is assumed to be axisymmetric. Consequently, the radius of curvature r0 at 0 is the same in all vertical planes passing through point 0 and from Eq. (2.30) we have

PI0 − PII0 =

2σ (2.33) r0

Combining Eq. (2.29) with Eqs. (2.31)–(2.33), the Young-Laplace equation for a general point E on the interface can be written as

 1 1  2σ σ +  = + (ρI − ρII ) gz (2.34)  r1 r2  r0

From simple geometric considerations, the principal radius of curvature r1 in the vertical plane is given by

1 d 2 z / dr 2 (2.35) = r1 1 + (dz / dr )2 3/2  

The relation for r2 is somewhat more difficult to derive, but it can be obtained with some manipulation. The radius of curvature r2 is measured in a plane normal to the vertical plane of Fig. 2.4 but containing the local normal to the interface at E. The center of curvature for r2 is at point C2 in Fig. 2.4 where the normal meets the drop axis. From Fig. 2.4 it can be seen that

r sin Ω   =   (2.36) r2

The trigonometric identity

1 + cot 2 Ω = csc 2Ω (2.37)

can be rearranged to obtain

sin Ω =

tan Ω (2.38) (1 + tan 2 Ω)1/2

Substituting Eq. (2.38) into Eq. (2.36) yields

r tan Ω = (2.39) r2 (1 + tan 2 Ω)1/2

47

The Liquid-Vapor Interface

As seen in Fig. 2.4, the normal to the interface at E makes an angle Ω with the horizontal. As a result, we can replace tan Ω with dz/dr in Eq. (2.39), to obtain dz / dr 1 (2.40) = r2 r 1 + (dz / dr )2 1/2  



Substituting Eqs. (2.35) and (2.40) into Eq. (2.34) and writing the resulting equation in terms of the dimensionless variables ξ and λ , defined as ξ=



z r , λ = (2.41) r0 r0

we obtain

ξ ′′ 2 3/2

1 + (ξ ′) 

+

ξ′ 1/2

λ 1 + (ξ ′)2 

= 2 + Bo ξ (2.42)

where Bo is the Bond number defined as

Bo = 

g(ρ1 − ρII )r02 (2.43) σ

and the primes denote derivatives with respect to λ. For the second-order ordinary differential equation (2.42), two boundary conditions are required. As a consequence of the choice of coordinate systems and the symmetry of the sessile drop, it is required that

ξ = 0 and  ξ ′ = 0 at  λ = 0 (2.44)

With the boundary conditions specified by Eq. (2.44), the system is closed and Eq. (2.42) can be solved to determine ξ(λ) for specified Bo. In general, solving Eq. (2.42) analytically is virtually impossible and a numerical approach has been used. Numerical solutions were first obtained by Bashforth and Adams [2.4] in the late 1800s. Complete and highly accurate tables giving ξ as a function of λ for specified Bo values are now available from the computer calculations of, for example, Padday [2.5]. The results obtained for Bo = 20 are plotted in Fig. 2.5 as a function of the angle Ω. It should be noted in Fig. 2.4 that θ is the contact angle of the interface with the solid surface. In liquid-vapor or liquid-gas systems, the convention is that the contact angle is the angle between the interface and the solid surface as measured through the liquid phase. In this chapter, the solid surface in contact with liquid and vapor phases will be modeled as ideal: a smooth surface with a fixed contact angle θ at all locations on the surface. In general, the value of θ will depend on the combination of fluids and solid surface material in the system. Deviations from this ideal surface model, including the effects of surface roughness, will be discussed in Chapter 3. In Fig. 2.4 it can be seen that Ω starts at zero at the apex and increases as z increases until the interface contacts the surface. At the point where the interface contacts the surface, Ω = θ. Hence in the computed results, the maximum value of ξ (where the interface stops) corresponds to Ω = θ. Another circumstance of interest is the shape of a free liquid surface meeting a plane vertical wall. As shown in Fig. 2.6, if the liquid wets the wall (has a contact angle θ < 90°), the liquid level will rise as the wall is approached, meeting the wall at its contact angle θ. For the two-dimensional configuration shown in Fig. 2.6, the principal radii of curvature are such that

1 d 2 z / dy 2 1 = ,  = 0 (2.45) r1 1 + (dz / dy)2 3/2 r2  

48

Liquid-Vapor Phase-Change Phenomena

FIGURE 2.5  Variation of the dimensionless drop radius λ = r/r0 and interface height ξ = z/r0 as a function of angular position for a Bond number of 20.

Substituting these results into the Young-Laplace equation (2.29) yields

PI − PII =

σ (d 2 z / dy 2 ) 1 + (dz / dy)2 

3/2

(2.46)

The hydrostatic pressure variations in the two fluids are given by

PI = Pν = P0 − ρν gz (2.47a)



PII = Pl = P0 − ρl gz (2.47b)

FIGURE 2.6  Rise height of a wetting liquid at the contact line with a solid vertical wall.

49

The Liquid-Vapor Interface

where P0 is the pressure at the interface far from the vertical wall. Substituting these equations into Eq. (2.46) and rearranging, we obtain −3/2 (ρl − ρv ) gz z ′′ = 0 (2.48) − 1 + ( z ′)2  σ



where the primes denote differentiation with respect to y. Multiplying Eq. (2.48) by z′ and integrating gives −1/2 (ρl − ρv ) gz 2 + 1 + ( z ′)2  = CI (2.49) 2σ



Since z and z′ → 0 as y → ∞, C1 = 1. At y = 0, z′ is given by the contact angle as z′(0) = − cot θ (2.50)



Equation (2.49) with C1 = 1 and Eq. (2.50) can be solved for z(0). The resulting relation is 1/2

 2σ (1 − sin θ)  z0 = z (0) =   (2.51)  (ρl − ρν ) g 



Note that z0 is the height to which the liquid climbs at the vertical wall. Using Eq. (2.51) as a boundary condition, integration of Eq. (2.49) yields the following relation for the shape of the interface  2L   y z2   2L  = cosh −1  c  − cosh −1  c  +  4 + 02   z   z0   Lc Lc 



1/2

1/2

 z2  −  4 + 2  (2.52) Lc  

where 1/2

  σ Lc =   (2.53)  (ρl − ρv ) g 



Example 2.2 A cylindrical container is filled with saturated liquid R-134a and its vapor at 32°C. Determine the height to which the liquid will climb the vertical walls of the container if the contact angle with the walls is 5°. At 32°C, the interfacial tension for saturated liquid R-134a liquid in contact with its vapor is 0.0072 N/m and

ρv = 39.8 kg/m3 , ρl = 1,180 kg/m3

The height to which the liquid climbs the vertical wall is given by Eq. (2.51) 1/ 2



 2σ(1− sin θ)  z0 =    (ρl − ρv )g   2(0.0072)[1− sin(5°)]  z0 =    (1180 − 39.8)(9.8) 

1/ 2



= 0.0011 m = 1.1 mm

50

Liquid-Vapor Phase-Change Phenomena

FIGURE 2.7  The rise of a wetting liquid in a capillary tube.

In a similar manner, the free surface of a liquid in small tubes and porous media will rise or fall to satisfy the Young-Laplace equation. This phenomenon is known as capillarity. Consider the small tube of radius ri shown in Fig. 2.7. The tube contains liquid with a free surface and is in contact with an extensive pool of liquid. The liquid is assumed to meet the wall at the contact angle θ < 90° (i.e., the liquid wets the wall). When ri 0, then (∂σ/∂xA)T is negative and the presence of the surface-active material decreases the surface tension. This is the situation when a typical soap is added to water. Because water is a highly polar material, polar molecules are readily accepted into its structure, whereas non-polar entities, such as hydrocarbon chains, are not. Consequently, pure hydrocarbons are rather insoluble in water, while polar materials have considerable solubility in water. With these observations, we can at least qualitatively interpret the effect of soap molecules on the surface tension of water. Typical soap molecules have a hydrocarbon chain and a polar group, as indicated schematically in Fig. 2.9. Given water’s affinity for the polar group and aversion to the

FIGURE 2.9  Orientation of surfactant molecules at the interface between water and another non-polar fluid.

54

Liquid-Vapor Phase-Change Phenomena

non-polar end of the soap molecule, the system clearly would prefer a configuration in which the polar group remains in the water and the hydrocarbon end is removed from the water (this is a low free energy state for the system). As indicated schematically in Fig. 2.9, this preferred configuration can be achieved if the soap molecules take up positions in a monomolecular layer or monolayer at the interface between the water and the vapor or other fluid. Because this configuration is preferred, the soap molecules generally tend to concentrate at the interface, in which case Eq. (2.61) implies that the interfacial tension will be reduced. Many different types of polar groups exist, and they may be combined with many different hydrocarbon groups to form the type of surfactant molecule described here. Some of the more common simple polar groups are amine (NH3+), alcohol (OH), carboxylic acid (COOH) and sulfate (SO4 –). A wide variety of straight or branched hydrocarbon chains may be linked to the polar group, making it possible to construct an almost infinite number of different surfactant molecules. Because so many substances can act as surfactants in water, and because it takes only a minute amount of them to form a monolayer at the interface, it is particularly easy for the surface tension of water to be altered by trace contaminants. The surface tension can, in fact, be a useful indicator of the purity of water. The need for high purity to avoid surface contamination and a reduction in surface tension should be considered when planning or evaluating experiments involving interfacial transport phenomena. Further information regarding methods to predict values of interfacial tension for pure substances and mixtures can be found in reference [2.11].

2.4  SURFACE TENSION IN MIXTURES In the previous section, we noted that when the presence of a species in the interfacial region decreases the surface tension, the concentration of that species in the interfacial region will be higher than in the bulk liquid. Because of this effect, in a mixture, the composition of the interfacial region will often be different than that in the bulk liquid. Usually the component with the lowest pure fluid surface tension will have a higher concentration at the interface, with the result that the surface tension of the mixture will not be equal to a bulk mole fraction average of the pure component surface tensions. In most cases, the concentration of components with lower pure-fluid surface tension at the interface results in the mixture surface tension being less than the bulk mole fraction average of the pure component surface tensions. In binary mixture systems, this results in a very non-linear variation of mixture surface tension with mole fraction of the component with the lower pure fluid surface tension. A number of different computational methods to predict the surface tension of mixtures have been proposed. References [2.11–2.21] provide a sampling of proposed models for predicting surface tension in multicomponent fluid mixtures. Even for a limited category of mixtures, such as water and alcohol mixtures, a single surface tension model often does not accurately predict the mixture surface tension for all mixture combinations. For 2-propanol/water mixtures, in which 2-propanol concentrates at the interface, Meissner and Michaels [2.12] developed the following equation that can be used to predict the mixture surface tension

xp     σ m = σ w 1 − 0.411log  1 +  (2.62)  0.0026   

where xp is the mole fraction of 2-propanol. This equation is good for small liquid mole fractions of 2-propanol (xp < 0.01). McGillis [2.13] developed a similar relation that accurately predicts the mixture surface tension for all mole fractions of 2-propanol/water mixtures. For a water and ethylene glycol mixture, a volume fraction weighted average of the pure fluid surface tensions works well. The mixture surface tension is then given by

σ m = vl ,eg σ eg + vl ,w σ w (2.63)

55

The Liquid-Vapor Interface

Here, vl ,eg and vl ,w are the liquid volume fractions of ethylene glycol and water, respectively. These are defined as

vl ,eg =

xeg vˆl ,eg xeg vˆl ,eg + (1 − xeg ) vˆl ,w



vl ,w =

(1 − xeg ) vˆl ,w ˆ xe ,g vl ,eg + (1 − xeg ) vˆl ,w

where vˆl ,eg and vˆl ,w are the pure liquid molar specific volumes for ethylene glycol and water, respectively. McGillis [2.13] showed that Eq. (2.63) agrees well with recommended surface tension data for water and ethylene glycol solutions. The mixtures above are examples of two categories of aqueous mixtures. Some mixtures, such as water and ethylene glycol, conform to a simple weighted average model. Alcohol-water mixtures, on the other hand, may strongly deviate from a simple weighted average. Figure 2.10 shows the predicted variation of surface tension with concentration for water/ 2-propanol mixtures and water/ethylene glycol mixtures at 25°C. It can be seen that the water ethylene glycol mixture surface tension varies gradually between the two pure fluid values. The variation is monotonic and varies somewhat from a linear variation. The water/2-propanol mixture surface tension drops sharply when small amounts of 2-propanol are added to pure water. This is a consequence of the surface-active nature of 2-propanol in water. At low concentration, the surface tension quickly drops as the concentration is increased, leveling-off at the value for pure 2-propanol. This type of variation is characteristic of mixtures in which one component is a surface-active agent. Generalized methods for predicting the liquid-vapor surface tension of mixtures include those of Suarez et al. [2.14], Zuo and Stenby [2.15], and Escobedo and Mansoori [2.18]. A number of models have been developed for binary mixtures [2.12, 2.13, 2.16, 2.20, 2.21]. A comprehensive review of methods for predicting surface tension in mixtures is beyond the scope of this introductory summary. The interested reader can find an in-depth discussion of different methods for prediction mixture surface tensions in Poling et al. [2.11]. Models developed very recently are summarized in references [2.19–2.21].

FIGURE 2.10  Predictions of surface tension variation with concentration for water/2-propanol mixtures [Eq. (2.62)] and water/ethylene glycol mixtures [Eq. (2.63)] at 25°C.

56

Liquid-Vapor Phase-Change Phenomena

2.5  NEAR CRITICAL POINT BEHAVIOR Equilibrium thermodynamics predicts that as the pressure is increased toward the critical point, the liquid and vapor saturation properties become progressively closer. The molecular theory of capillarity discussed in Section 1.2 implies that interfacial tension is a consequence of the variation of mean density across the interfacial region. Clearly, if the liquid and vapor bulk densities approach the same value, in the limit, the density transition will vanish, implying that interfacial tension must also approach zero in this limit. This trend is reflected in recommended surface tension values listed in tables in Appendix II and in references such as the ASHRAE Handbook of Fundamentals [2.22]. The surface tension values in such tables reflect the expectation that surface tension vanishes at the critical point, even though very few measurements of surface tension have been made for conditions very near the critical point. In fact, the extrapolation of surface tension data into the near critical point region is largely based on model analysis of the thermophysics in the near-critical region. The molecular theories of capillarity described in Chapter 1 are a primary means of predicting interfacial region properties and surface tension near the critical point. As noted in Section 1.2, the classic van der Waals theory of capillarity predicts that the surface tension varies proportional to (1 − T/Tc)3/2 and the interfacial region thickness varies proportional to (1 − T/Tc) –1/2. The predictions of van der Waals theory are qualitatively similar to real fluid behavior, but its numerical predictions are inaccurate, both in terms of specific property values and the trends in the variations of properties with temperature. Near the critical point, for example, it is well known that the corresponding states correlation for surface tension that best matches data for real fluids typically varies proportional to (1 − T/Tc) to an exponential power of about 1.22 (see the discussion in Poling et al. [2.11]). The van der Waals model also predicts that the interfacial region thickness varies proportional to (1 − T/Tc) –1/2, whereas recent measurements (by, for example, Beysens and Robert [2.23]) indicate that the interfacial region thickness varies about proportional to (1 − T/Tc) –0.62. As noted in Sections 1.2 and 1.3, the Redlich-Kwong molecular theory of capillarity developed by Carey [2.24] predicts that surface tension varies proportional to (1 − T/Tc)1.33 and the interfacial region thickness varies proportional to (1 − T/Tc) –0.67 near the critical point. These predictions agree better with data for real fluids than the predictions of van der Waals theory. Beysens and Robert [2.23] used measurements of optical reflectivity to determine values of interfacial region thickness for sulfur hexafluoride at near-critical conditions. Some of the interfacial region thickness data obtained by Beysens and Robert [2.23] are shown in Fig. 2.11. Also shown is the variation of interfacial region thickness predicted by the near-critical model proposed by Beysens and Robert [2.23]. It can be seen in Fig. 2.11 that the near critical point variation of interfacial region thickness with temperature predicted by the Redlich-Kwong capillarity model [2.24] is consistent with the measured interfacial region thickness data obtained by Beysens and Robert [2.23]. This model predicts values of the interface thickness that are close to the corresponding data in magnitude, and the slope indicated by the theory is about the same as that of the data. The near-critical model proposed by Beysens and Robert [2.23] is also close to the data and the RedlichKwong theory prediction. Note also that the predicted and measured variations of interfacial region thickness with temperature indicate that the interfacial region thickness can become macroscopic only when T is very close to Tc. Carey [2.24] also noted that the Redlich-Kwong capillarity theory predicts that a central region within the interfacial region lacked intrinsic stability and, as discussed in Section 1.4, molecular dynamics simulations indicate that this region exhibits a high level of property fluctuations. Redlich-Kwong capillarity theory further suggests that the thickness of this unstable region increases in proportion to the thickness of the interfacial region as the critical point is approached. This implies that experimental determination of interfacial tension and other saturation properties may be increasingly difficult very near the critical point because, as the critical point is approached, a subregion within the interfacial region that exhibits large property

The Liquid-Vapor Interface

57

FIGURE 2.11  Comparison of the variation of interfacial region thickness predicted by the theory of Carey [2.24] to measured data and the prediction of the near-critical model of Beysens and Robert [2.23].

fluctuation becomes increasingly large. It also suggests that capillarity theory models based on the assumption of local thermodynamic equilibrium with minimal property fluctuations are suspect close to the critical point. Despite the suspect nature of molecular capillarity theory near the critical point, it nevertheless provides useful insight into the expected variation of the surface tension and interfacial region thickness near the critical point, and the predicted trends agree fairly well with available property data.

2.6  EFFECTS OF INTERFACIAL TENSION GRADIENTS As noted in the previous sections of this chapter, the surface tension at the interface between a liquid and a vapor phase varies with the temperature and species concentrations in the liquid. Consequently, if the temperature or composition varies over the interface, the interfacial tension will also be non-uniform, with the result that liquid near the interface in regions of low surface tension will be pulled toward regions of higher surface tension. If the temperature or concentration non-uniformities are maintained somehow, a steady flow pattern may be established in which the pulling action at the interface resulting from the surface tension gradient is balanced locally be viscous shear stress in the liquid flow. Motion of the liquid due to surface tension gradients at the interface is sometimes called the Marangoni effect. Perhaps the most well-known example of a liquid flow driven by interfacial tension gradients is the formation of Bénard circulation cells in a thin (0.5–1 mm) pool of liquid heated from below. Although Bénard’s experimental observation of these cells inspired Rayleigh’s subsequent analysis of the onset of buoyancy-driven motion in a fluid layer heated from below, the flows observed by Bénard were actually driven by surface tension gradients. Figure 2.12a schematically illustrates the steady cellular flow driven by the Marangoni effect. Once steady state is achieved, warm liquid flows upward to the interface at point A where it turns and flows toward either point B or B′. As the warm fluid flows from point A to point B, it cools due to heat transfer by convection to the cooler surrounding gas. Because the temperature at B and B′ is less than at A, a surface tension gradient is maintained that drives the flow along the interface from A to B and from A to B′. The symmetric convergence of flow at points B and B′ causes the fluid to turn downward toward the wall, whereupon it absorbs heat from the wall and repeats the cycle.

58

Liquid-Vapor Phase-Change Phenomena

FIGURE 2.12  Cellular convection driven by surface tension gradients.

In some cases the cellular flow may be accompanied by significant deflection of the interface, as shown in Fig. 2.12b. Note that moving the interface at location A toward the surface may increase the temperature at A while moving the interface at B away from the surface may reduce the temperature at B. Deflection of the interface in this manner thus has the effect of increasing the surface tension gradient between points A and B. In some systems, the increased surface tension gradient will move fluid rapidly enough from A to B to maintain the deflected interface configuration. For a thin liquid layer heated from below, the conditions for the onset of cellular motion can be predicted using linear stability analysis. The postulated initial condition of the liquid layer is indicated schematically in Fig. 2.13. Initially the liquid is motionless and heat transfer from the wall to the interface is by conduction alone. The temperature profile in the liquid is linear with a constant slope dT/dz equal to ξ. Heat is assumed to be transferred from the interface to the surrounding bulk vapor by convection in such a manner that the heat transfer coefficient hi = q ′′ / (Ti − Tg ) is constant and uniform over the interface. Following the usual practice for linear stability analysis, the local temperature is assumed to be equal to the sum of the base flow temperature T given by the initial linear profile, plus a small sinusoidal fluctuation T ′ representing a Fourier component of random disturbances in the system

T = T + T ′ = (Tw + ξz ) + T ′ (2.64a)

FIGURE 2.13  Model system used in the stability analysis of a thin liquid film.

59

The Liquid-Vapor Interface

Associated with the temperature perturbation are small perturbation velocities w′ and u′. Since all base flow velocities for the initial state w and u are zero,

w = w + w′ = w ′,   u = u + u ′ = u ′ (2.64b)

The linear stability analysis proceeds by substituting Eqs. (2.64a) and (2.64b) into the governing continuity, momentum and thermal energy transport equations, subtracting the corresponding base flow equations, and neglecting products of perturbation quantities. The resulting equations for the perturbation quantities T′, w′ and u′ are solved with appropriate boundary conditions assuming that T′ and w′ are of the form

T ′ = θ( z )eiαx +βt (2.65a)



w ′ = W ( z )eiαx +βt (2.65b)

In the above relations α is the wavenumber of the perturbation and the real and imaginary parts of β represent its amplification factor and temporal frequency. The boundary conditions at the interface insure that the usual force, mass, momentum and energy balances are enforced and the additional condition

 ∂u ′   dσ   ∂T ′  (τ xz ) z =δ = µ l  = − (2.66)  dT   ∂ x  z =δ  ∂ z  z =δ

provides the key link between the liquid flow field and the surface tension variation along the interface. The linear stability analysis of convective instability due to surface tension gradients in thin liquid films is described in detail by Pearson [2.25], Scriven and Sternling [2.26], and Miller and Neogi [2.3]. Although fundamentally the same, each of these three analytical treatments is slightly different. Pearson [2.25] assumed that the interface does not deform normal to itself and that the gas phase is inviscid. Scriven and Sternling [2.26] considered the interface to be deformable and the effects of capillary waves and surface viscosity were included. The treatment in Miller and Neogi [2.3], which is a somewhat simplified version of the more general analysis of Smith [2.27], allows for interface deformation and the presence of capillary and gravity waves, but ignores surface viscosity effects and assumes the gas to be inviscid. In all of the previous studies noted above, both the real and imaginary parts of β were taken to be zero. The resulting stationary wave solutions, which do not amplify or damp with time, are therefore interpreted as being neutrally stable perturbations. The conditions for which these solutions are obtained are assumed to correspond to the onset of instability, which will ultimately lead to the transition to cellular convection. Because the details of the linear stability analysis are described in the references noted above [2.3, 2.25–2.27], a full description of the analysis is not presented here. Some of the results of these analyses are noteworthy, however. If the gas is assumed to be inviscid and surface viscosity effects are ignored, this type of linear stability analysis indicates that the neutral stability conditions can be ˆ the Marangoni number Ma, the Biot number specified in terms of a dimensionless wavenumber α, Bi, the Bond number Bo and the “crispation number” Cr defined as αˆ = αδ (2.67)



Ma = 

ξ(dσ / dT )δ 2 (2.68) α Tl µ l

60

Liquid-Vapor Phase-Change Phenomena



Bi = 



Bo = 



hi δ (2.69) kl

(ρl − ρv ) gδ 2 (2.70) σ

Cr = 

µ l α Tl (2.71) σδ

In the above expressions, kl, μl, and αTl are the thermal conductivity, viscosity, and the thermal diffusivity of the liquid, respectively. Note that the temperature gradient ξ is negative, so ξ(dσ/dT) is positive. The linear stability analysis (see reference [2.3]) predicts that neutral (or marginal) stability corresponds to

Ma =

8αˆ (αˆ coshαˆ  + Bi sinhαˆ )(αˆ − sinhαˆ  coshαˆ ) (2.72) αˆ 3 coshαˆ − sinh 3αˆ − (8 Cr αˆ 5 coshαˆ )/(Bo + αˆ 2 )

The neutral stability curves predicted by Eq. (2.72) for various combinations of Bi, Cr, and Bo are shown in Fig. 2.14. For combinations of αˆ and Ma below the neutral curve, the system is stable; whereas above the curve the system is unstable. As illustrated in Fig. 2.14, an increase in the Biot number (sometimes referred to as a Nusselt number by earlier investigators) shifts the neutral curve upward, implying that the system is more stable. For thin liquid layers in real systems, the range of Bond numbers is small, and its variation typically has only a small effect on the location of the neutral curve. Generally, the position of the neutral curve at small wavenumbers is very sensitive to the value of Cr, while at higher wavenumbers it is very insensitive to changes in Cr. The limit Cr → 0 corresponds to thick layers or high surface tension. For such conditions the interface deflection is small, vanishing in the limit Cr → 0. The early analysis of Pearson [2.25] considered conditions that correspond to the limit Cr → 0.

FIGURE 2.14  Stability plane for the onset of cellular motion as predicted using linear stability analysis.

61

The Liquid-Vapor Interface

For real systems, however, the value of Cr is usually very small, but greater than zero. For example, if we consider a 0.5-mm thick layer of water on a plate held at 30°C and surrounded by air at 25°C and atmospheric pressure, the corresponding values of Cr and Bo are Cr = 3.3 × 10 −6 ,  Bo = 0.034



and if the heat transfer coefficient from the interface to the ambient air is 20 W/m2K (a typical value for natural convection), the value of the Biot number is Bi = 0.0162



The neutral stability curve predicted by Eq. (2.72) for these values of Cr, Bo, and Bi is shown in Fig. 2.14. Even though the value of Cr is small, this system is much less stable to long-wavelength perturbations than would be the case if Cr = 0. The reason for this is that surface deflection is permitted for non-zero values of Cr. Long-wavelength deflection of the interface tends to make the system less stable by depressing high-temperature regions and elevating colder regions, as shown in Fig. 2.12b. For very long wavelengths, gravity has a stabilizing influence that causes the neutral stability curve to level off as αˆ →  0. Because it is generally assumed that random disturbances contain Fourier components of all wavelengths, the system is expected to be unstable if any wavelength is unstable. Hence, the minimum value of Ma on the neutral stability curve is the critical value Mac, above which the system is likely to undergo a transition to cellular convection. The curves in Fig. 2.14 indicate that unless Cr exceeds about 10 –4, the critical Marangoni number is about 80 and the associated unstable dimensionless wavenumber αˆ c is about 2, which corresponds to a wavelength λc of λc  = 



2π 2π  =    =   πδ 2/δ αc

For the 0.5-mm thick layer of water considered, this implies that the layer is unstable for temperature differences across the layer greater than about 12°C. Alternatively, it may be stated that the layer is unstable at heat flux levels above 145 W/m2. In real systems, the stability mechanism and cellular flow resulting from the Marangoni effect can be significantly altered by the presence of small amounts of surface-active contaminants. Flow initially induced by interfacial temperature gradients will carry the contaminants from the high temperature regions to the cooler regions. The resulting concentration of the surfactant at the cooler locations on the interface will reduce the surface tension there. This will tend to counteract the surface tension gradient resulting from temperature differences, which may weaken or suppress the cellular motion. Hence, the presence of surfactant contaminants tends to enhance the stability of the system. Experimentally, instability may not be observed until the Marangoni number has greatly exceeded the critical value Mac predicted by the stability analysis for a pure system. Example 2.4 A thin layer of water covers a surface held at 100°C and is exposed to air at a bulk temperature of 20°C. The heat transfer coefficient between the interface and the bulk air is 8.0 W/m2°C. Estimate the range of layer thicknesses for which Marangoni effects may become important. Initially making the assumption that the Bond and Biot numbers are very small and the crispation number is small but finite, the critical Marangoni number is about 80. Taking the temperature gradient ξ across the layer to be equal to –(Tw – Ti )/δ, the critical layer thickness δc is given by

Ma c = 80 = −

Tw − Ti δc

2  dσ  δ c   dT αTl µ l

62

Liquid-Vapor Phase-Change Phenomena Applying the resistance analogy for steady transport of heat from the surface to the bulk air, it can be shown that

Tw − Ti =

δ / kl (Tw − Tair ) (1/ hc ) + (δ / kl )

Substituting this relation into that above for the critical Marangoni number yields

Tw − Tair    dσ  δ c2 Ma c = 80 = −        (1/ hc ) + (δ c / kl )   dT  αTl µ l kl

For water at 60°C, αTl = 0.158 × 10 –6 m2/s, kl, = 0.653 W/m°C, μl, = 4.67 × 10 –6 Ns/m3, σ = 0.0644 N/m, dσ/dT = –1.79 × 10 –4 N/m°C. Substituting these property values and solving the above equation iteratively, the layer thickness corresponding to the critical Marangoni number is determined to be

δ c = 0.00026 m  = 0.26 mm

Thus the system is unstable and Marangoni effects may become important for δ > 0.26 mm. Using Eqs. (2.69)–(2.71), it can be shown that Cr = 4.41 × 10 –7, Bi = 3.19 × 10 –3, and Bo = 2.67 × 10 –6. These values are consistent with the initial assumption regarding these parameters.

Besides being a driving mechanism for cellular convection, interfacial tension gradients are also responsible for other interfacial flow phenomena. In multicomponent systems, when one or more species is transferred across a liquid-vapor interface, an irregular, quasi-cellular flow pattern can result that is often referred to as interfacial turbulence. Although somewhat periodic in nature, flows of this type are generally much less regular than the Bénard cells previously discussed. This irregular flow is driven by interfacial tension gradients resulting from interfacial concentration gradients. Interfacial tension is typically a strong function of concentration. For evaporation from the interface of a multicomponent liquid, the variation of surface tension with concentration usually dominates over the variation with temperature, and, as a result, it usually plays a dominant role in the development of interfacial turbulence. Because heat and mass transfer at the interface may be greatly enhanced by the presence of interfacial turbulence, this phenomenon may be particularly important to the accurate prediction of heat and mass transfer in equipment where vaporization or condensation of multicomponent mixtures occurs. Further discussion of this phenomenon may be found in reference [2.28]. Marangoni flow can also arise during evaporation from the surface of liquid jets and at the interface of liquid drops or cylinders formed by mixing or agitation in an evaporating two-phase flow. Again, because surface tension usually varies strongly with concentration, the effects of Marangoni flow on the breakup of liquid jets and droplet breakoff from liquid films in two-phase flow may be particularly significant in multicomponent systems. The Marangoni effect may act to stabilize or destabilize the interface, depending on the nature of the multicomponent mixture. The role of the Marangoni effect in these circumstances is discussed in detail in reference [2.29]. The Marangoni effect also causes vapor bubbles in a liquid with an imposed temperature gradient to move toward the high-temperature region. Motion in this direction is thermodynamically favored because it reduces the interfacial free energy of the bubble. The liquid flow that causes the bubble motion toward the high temperature region is driven by interfacial tension gradients. Evaporation of a falling film of a binary liquid mixture on a heated vertical surface may also be affected by Marangoni flow if the more volatile component of the mixture has a higher surface

The Liquid-Vapor Interface

63

FIGURE 2.15  Lateral wavy perturbation that may lead to breakup of a wavy film of a binary liquid mixture.

tension. A transverse wavy perturbation of the film interface, like that shown in Fig. 2.15, would be expected to result in more rapid evaporation of the more volatile component in the thin portion of the film compared to that in the thick portion. This would produce a concentration difference between the thick and thin regions, resulting in an interfacial tension gradient that draws liquid from the thin regions into the thick regions. Thus the resulting Marangoni flow promotes breakup of the film into rivulets. Conversely, the Marangoni effect opposes rivulet formation if the more volatile component has a lower surface tension. When it occurs, the transition from film flow to rivulets often produces a drastic drop in the heat transfer performance for these circumstances. Interfacial tension gradients may also affect the spreading of an evaporating liquid droplet over a warm surface. We consider specifically a binary liquid mixture in which the more volatile component has a lower surface tension. As shown in Fig. 2.16, a thinner advancing layer of liquid forms at the perimeter of the droplet. Because it is thinner, evaporation of the more volatile component is more rapid than in the center of the droplet. The resulting concentration difference produces a surface tension gradient that enhances the motion of liquid from the bulk droplet to the advancing layer at its perimeter. The overall effect is to increase the rate of spreading of the liquid over the solid. The formation of “wine tears” is also a consequence of the effect of Marangoni flow on the spreading of a binary liquid. Wine is, to a first approximation, a mixture of water and ethanol, with ethanol being the more volatile and have the lower surface tension of the two components. The rate of liquid spreading up the walls of the wine glass is enhanced by Marangoni flow until enough liquid accumulates to form droplets or “tears.” Spreading of liquids on solid surfaces is discussed in more detail in Chapter 3.

FIGURE 2.16  Schematic illustrating the possible effects of interfacial tension gradients on spreading of a liquid droplet on a solid surface.

64

Liquid-Vapor Phase-Change Phenomena

REFERENCES 2.1 Fowkes, F. M., Attractive forces at interfaces, Industrial & Engineering Chemistry, vol. 56, pp. 40–52, 1964. 2.2 Adamson, A. W., and Gast, A. P., Physical Chemistry of Surfaces, 6th ed., John Wiley and Sons, New York, NY, 1997. 2.3 Miller, C. A., and Neogi, P., Interfacial Phenomena, Marcel Dekker, Inc., New York, NY, 1985. 2.4 Bashforth, F., and Adams, J. C., An Attempt to Test the Theory of Capillary Action, Cambridge University Press, Cambridge, 1893. 2.5 Padday, J. F., Theory of surface tension, in Surface and Colloid Science, E. Matijevic (editor), vol. 1, John Wiley and Sons, New York, NY, pp. 39–151, 1969. 2.6 National Bureau of Standards, Release of Surface Tension of Water Substance, the International Association for the Properties of Steam, December, 1976, available from the Executive Secretary, IAPS, Office of Standard Reference Data, National Bureau of Standards, Washington, D.C. 2.7 Jasper, J. J., The surface tension of pure liquid compounds, J. Phys. Chem. Ref. Data, vol. 1, pp. 841–1010, 1972. 2.8 Brock, J. R., and Bird, R. B., Surface tension and the principle of corresponding states, AIChE J., vol. 1, pp. 174–184, 1955. 2.9 Reidel, L., Eine neue universelle dampfdruckformel, Chem. Ing. Tech., vol. 26, pp. 83–89, 1954. 2.10 Miller, D. G., On the reduced Frost-Kalkwarf vapor pressure equation, Ind. Eng. Chem. Fundam., vol. 2, pp. 78–79, 1963. 2.11 Poling, B. E., Prausnitz, J. M., and O’Connell, J. P., The Properties of Gases and Liquids, 5th ed., McGraw-Hill, New York, NY, 2001. 2.12 Meissner, H. P., and Michaels, A. S., Surface tensions of pure liquids and liquid mixtures, Ind. Eng. Chem., vol, 41, pp. 2782–2787, 1949. 2.13 McGillis, W. R. Boiling from localized heat sources in pure and binary fluid systems. PhD. Dissertation, Mechanical Engineering Department, University of California at Berkeley, 1993. 2.14 Suarez, J. T., Torres-Marchal, C., and Rasmussen, P., Prediction of surface tension of nonelectrolyte solutions, Chem. Eng. Sci., vol. 44, pp. 782–785, 1989. 2.15 Zuo, Y.-X., and Stenby, E. H., Corresponding-states parachor models for the calculation of interfacial tensions, Can. J. Chem. Eng., vol. 75, pp. 1130–1137, 1997. 2.16 Carey, B. S., Scriven, L. E., and Davis, H. T., Semiempirical theory of surface tension of binary systems, AIChE J., vol. 26, pp.705–711, 1980. 2.17 Rice, P., and Teja, A. S., The prediction of surface tension in mixtures of cryogenic liquids Cryogenics, vol. 22, pp. 588–590, 1982. 2.18 Escobedo, J., and Mansoori, G. A., Surface-tension prediction for liquid mixtures, AIChE J., vol. 44, pp. 2324–2332, 1998. 2.19 Li, Z., and Lu, B. C.-Y., On the prediction of surface tension for multicomponent mixtures, Can. J. Chem. Eng., vol. 79, pp. 402–411, 2001. 2.20 Amooey, A. A., A novel model for predicting the surface tension of binary solutions, J. Eng. Phys. Thermophys., vol. 87, pp. 533–540, 2014. 2.21 Khosharay, S. Mazraeno, M. S., Varaminian, F., and Bagheri, A., A proposed combination model for predicting surface tension and surface properties of binary refrigerant mixtures, Int. J. Refrig., vol. 40, pp. 347–361, 2014. 2.22 ASHRAE Fundamentals Handbook, American Society of Heating, Refrigerating and Airconditioning Engineers, Atlanta, GA, 2012. 2.23 Beysens, D., and Robert, M., Thickness of fluid interfaces near the critical point from optical reflectivity measurements, J. Chem. Phys., vol. 87, pp. 3056–3061, 1987. 2.24 Carey, V. P., Thermodynamic properties and structure of the liquid-vapor interface: A neoclassical Redlich-Kwong model, J. Chem. Phys., vol. 118, pp. 5053–5064, 2003. 2.25 Pearson, J. R. A., On convection cells induced by surface tension, J. Fluid Mech., vol. 4, pp. 489–500, 1958. 2.26 Scriven, L. E., and Sternling, C. V., On cellular convection driven by surface-tension gradients: Effects of mean surface tension and surface viscosity, J. Fluid Mech., vol. 19, pp. 321–340, 1964. 2.27 Smith, K. A., On convective instability induced by surface-tension gradients, J. Fluid Mech., vol. 24, pp. 401–414, 1960. 2.28 Steming, C. V., and Scriven, L. E., Interfacial turbulence: Hydrodynamic instability and the Marangoni effect, AIChE J., vol. 5, pp. 514–523, 1959. 2.29 Bainbridge, G. S., and Sawistowski, H., Surface tension effects in sieve plate distillation columns, Chem. Eng. Sci., vol. 19, pp. 992–993, 1964.

The Liquid-Vapor Interface

65

PROBLEMS 2.1 A small quantity of liquid occupies the space near the contact point between two solid cylindrical rods of radius R as shown in Fig. P2.1. The liquid interface has a concave radius of curvature ri in the plane of the page which becomes smaller as the volume of liquid present becomes smaller. The contact angle of the interface is zero. (a) For small values of the angle ϕ, derive a relation between the volume of liquid present and the force acting to hold the cylindrical rods against each other. (Hint: Use the fact that ϕ is small to simplify geometry relations.) (b) Estimate the magnitude of the force per unit length if the cylinder radius is 1 mm, the interface radius is 0.15 mm and the liquid is water (surrounded by water vapor) at 20°C.

FIGURE P2.1

2.2 A small drop of liquid is squeezed between two flat plates to form a thin disc of height H and diameter D. The contact angle of the liquid interface at the solid surfaces is zero, so that the perimeter of the liquid disc has a concave radius of curvature in the plane normal to the disc, as shown in Fig. P2.2. (a) What are the principle radii of curvature of the interface at the edge of the liquid disk between the solid plates? (b) If D is much larger than H, derive a relation for the force acting to hold the plates together. (c) Determine the magnitude of the force if the liquid is water at 20°C for D = 1.0 cm an H values of 1, 0.1, and 0.01 mm. (Note that this is approximately the diameter and fluid conditions for the liquid film under a contact lens.)

FIGURE P2.2 2.3 One end of a capillary tube with an inside diameter of 0.2 mm is immersed in a pool of water at 20°C. The liquid water rises to a height of 3.2 cm in the tube above the pool interface. Assume the interface at the rise height is a segment of spherical surface with radius ri. The contact line where the spherical interface meets the tube wall is a circle whose radius is the inside radius of the tube rt . (a) Draw a sketch of the interface geometry showing ri and rt. (b) Determine the radius of curvature for the interface ri. (c) The planar surface inside the contact line circle, together with the interface spherical surface define a spherical cap. For a spherical cap with radius ri and liquid contact angle θ, the spherical cap geometry dictates that the radius of the circular base area for the cap is rt = ri / cos θ. Using this information, determine the contact angle for the liquid at the inside tube wall.

66

Liquid-Vapor Phase-Change Phenomena

2.4 Determine the capillary depression of liquid mercury at 20°C in a capillary tube with an inside diameter of 0.5 mm. The contact angle of the mercury-air interface with the wall of the capillary tube is 100°. 2.5 Determine the capillary rise height of water at 20°C in a round tube microchannel with an inside diameter of 20 µm. The contact angle of the interface with the wall of the channel is 10°. 2.6 For water at 20°C in contact with air, use the results shown in Fig. 2.5, to determine and plot the variation of the sessile drop height with liquid contact angle for Bo = 20 and liquid contact angles between 5° and 90°. 2.7 In Fig. 2.5, picking Ω to match the contact angle θ, the curves indicate the values of rfp / r0 and z h / r0 for Bo = 20, where rfp is the radius of the circle that corresponds to the footprint of the droplet on the solid surface, z h is the height of the droplet at the centerline, and r0 is the radius of curvature at the uppermost point of the droplet interface. If the droplet is modeled as a spherical cap, with radius of curvature r0 and contact angle θ, geometry dictates that rfp / r0 = sin θ and z h / r0 = 1 − cos θ. The spherical cap model might be a reasonable model if gravity deformation of the droplet is small. Determine the curve predictions for rfp / r0 and z h / r0 in Fig. 2.5 and the spherical cap model predictions for these ratios at a contact angle θ of 15°. Based on your results, assess the accuracy of the spherical cap model. 2.8 Show that for a liquid having a contact angle θ > 90°, the depression of the liquid-vapor interface at the side wall of the container is given by 1/ 2



 2σ (1 − sin θ)  z0 =    (ρl − ρv ) g 

where the negative square root is taken as the physically realistic solution. 2.9 A cylindrical vessel contains liquid (molten) silver at 1100°C. The contact angle of the silver-air interface with the vessel walls is 105°. Determine the depression of the silver-air interface at the vertical wall of the vessel. Note that for depression of the interface, Eq. (2.51) predicts the depression of the interface if the negative square root is taken as the physically realistic solution (see Problem 2.8). 2.10 A vertical cylindrical vessel contains half saturated liquid water and half water vapor by volume at 100°C under normal gravity. The contact angle of the water interface with the vessel stainless steel walls is 25°. (a) Determine the rise of the liquid-vapor interface at the vertical wall of the vessel. (b) As discussed later in Chapter 3, adding a nanostructured to a flat metal substrate can achieve a much lower contact angle. Determine how much the rise height of the liquid-vapor interface will change, if the vertical wall of the cylindrical vessel has a nanostructured surface with an apparent contact angle of 3°. 2.11 A capillary tube with an inside diameter of 0.2 mm is immersed in a pool of an unknown hydrocarbon, which rises to a height of 2.9 cm in the tube above the pool interface. The contact angle of the liquid at the inside of the tube is virtually zero, and the density of the liquid is measured to be 789 kg/m3. Estimate the surface tension of the liquid for these conditions. 2.12 A capillary tube with an inside diameter of 0.2 mm is heated so that a linear temperature gradient is maintained in its walls and in the fluid in the tube. The temperature is 20°C at the bottom of the tube and the temperature rises linearly with height to 80°C at the height of 60 cm. The bottom of the tube is immersed in a pool of liquid water at 20°C and the top is open to the atmosphere, as is the surface of the pool. Using the fact that the variation of surface tension with temperature is given by Eq. (2.57) (with appropriate values of C1 and C2), find the rise height of the liquid in the capillary tube for these conditions. 2.13 For a 1.5-mm thick layer of methanol, liquid nitrogen and liquid mercury at atmospheric pressure, use the results shown in Fig. 2.13 to estimate the temperature gradient that would have to be imposed to induce Marangoni instability. Use Table 2.2 and Appendix II to evaluate the physical properties in each case. The Biot number Bi may be taken as equal to zero. 2.14 A recommended means of accounting for the variation of surface tension with temperature is to use the following interpolation relation 1.2



 T −T  σ = σ0  c   Tc − T0 

Taking T0 = 20°C and σ0 = 0.0728 N/m, plot and compare the variation of σ with T predicted by this relation to that predicted by Eq. (2.56). Over the range 0°C < T < 374°C, where is the largest difference?

The Liquid-Vapor Interface

67

2.15 Plot the variation of surface tension for water between 0°C and 374°C predicted by the relation obtained from the molecular theory of capillarity in Chapter 1: Eq. (1.41) and Eq. (2.56). Assess how well the predictions of these two relations agree over this range of temperatures. 2.16 For liquid mercury in contact with saturated mercury vapor, determine the surface tension values predicted by Eq. (2.57) and Eq. (1.41) from Chapter 1 at temperatures of 630 K, 800 K and 950 K. Assess the agreement of these values with those in the saturation table for mercury in Appendix II. 2.17 (a) Determine the capillary rise height of pure water at 25°C in a tube with an inside diameter of 0.3 mm. (b) Using Eq. (2.62) to compute the surface tension for the mixture, determine how high we would have to raise the mole fraction of 2-propanol in a water/2-propanol mixture to drop the capillary rise to half that for pure water. 2.18 Determine the capillary rise height of pure water at 20°C in a tube with an inside diameter of 0.5 mm. By how much does the rise height change if the temperature is increased to 90°C? 2.19 One end of capillary tube with an inside diameter of 0.5 mm is immersed in a pool of saturated liquid water at 100°C. The surface of the pool and the free end of the tube are in contact with saturated vapor at atmospheric pressure. The liquid perfectly wets the tube, so the contact angle may be take to be zero. (a) Determine the rise height of liquid in the tube for these conditions. (b) Determine the rise height for saturated liquid water and water vapor at three other pressures between 1 atmosphere and the critical pressure for water. Plot the variation of rise height with pressure. What do you conclude will happen to the rise height as the critical point is approached? 2.20 Saturation property tables for methanol, ammonia and carbon tetrachloride can be found in Appendix II. For each of these fluids, select surface tension data for three temperatures from the table and compute the corresponding values of σlv/(PcLl) and 1 – T/Tc, where Li = (kBTc/Pc)1/3. Then plot all the points on a log-log plot of σlv/(PcLi) versus 1 – T/Tc. On this graph, also plot the variation predicted by Eq. (1.41) from Chapter 1. How well do the saturation table data agree with Eq. (1.41), which is obtained from molecular capillarity theory? 2.21 Saturated liquid oxygen and vapor at 90 K coexist in a simple cylindrical storage tank for a rocket propulsion system. The contact angle is 5°. (a) Calculate the rise height of liquid oxygen at the location where the liquid-vapor interface meets the vertical wall of the storage tank while the vehicle is sitting on the launch pad. (b) As the vehicle launches, the fluid experiences a vertical acceleration equal to 8 times normal earth gravity. Determine the rise height at the wall under these conditions. (c) What happens to the rise height as the vehicle enters orbit and acceleration forces drop toward zero? What do you expect to happen to the fluid in the tank? 2.22 Example 2.5 examines the Marangoni stability of a thin layer of water on a surface at 100°C exposed to air at 20°C with a convective heat transfer coefficient between the water and air of 8.0 W/m2°C. This example indicates that the system is unstable and Marangoni flow is likely for film thicknesses greater than 0.26 mm. Suppose we have a film with thickness 0.3 mm under these conditions. Can we stabilize the system and suppress Marangoni effects by changing the convective heat transfer coefficient? Briefly explain your answer and if it is possible, indicate how much we would have to change the heat transfer coefficient to do so.

3

Wetting Phenomena and Contact Angles

3.1  EQUILIBRIUM CONTACT ANGLES ON SMOOTH SURFACES In most technological applications, a liquid-vapor phase change is accomplished by transferring energy through the walls of a container or channel into or out of a two-phase system. The vaporization or condensation process ultimately takes place at the liquid-vapor interface. However, in these circumstances the manner in which the liquid and vapor contact the solid walls through which the energy is transferred will strongly affect the resulting heat and mass transfer in the system. Consequently, the performance of heat transfer equipment in which vaporization or condensation occurs may depend strongly on the way that the two phases contact the solid walls. In everyday circumstances, it can be observed that the behavior of liquids in contact with solids may vary from one surface to another, and with the type of liquid. If a small amount of liquid acetone is placed on a clean flat aluminum surface, the liquid spreads out to form a thin film. If a small quantity of liquid water is placed on the same surface, a discrete drop is observed. Generally, liquids with weak affinities for a solid wall will collect themselves into beads while those with high affinities for the solid will form films to maximize the liquid-solid contact area. The affinity of liquids for solids is referred to as the wettability of the fluid. The general circumstance of a smooth axisymmetric bead of liquid in contact with a solid and surrounded by vapor is shown in Fig. 3.1. The solid surface is taken to be locally flat and is idealized as perfectly smooth. We will reconsider this idealization later in this chapter, but for the moment adopt it as a plausible first approximation of a real surface. The wettability of the liquid is quantified by the contact angle θ defined as the angle between the liquid-vapor interface and the solid surface, measured through the liquid at the point O in Fig. 3.1 where all three phases meet. Note that for the real axisymmetric drop there is actually a line common to all three phases (the basal circle of the drop) called the contact line. A fixed quantity of liquid will spread more over the surface as θ decreases. In the limit of θ → 0, the liquid spreads over the entire available surface, forming a thin film. The system shown in Fig. 3.1 has three interfaces: one between the vapor and liquid, another between the liquid and solid, and a third between the solid and vapor. For the liquid-vapor interface, σlv is the interfacial tension defined in the previous chapter. There are also interfacial tensions σsl and σsv associated with the solid-liquid and solid-vapor interfaces, respectively. At equilibrium, a force balance at point O in the horizontal direction requires that

σ sv = σ sl + σ lv cos θ (3.1)

The vertical force σlv sin θ at point O must be balanced by a vertical reaction force in the solid. However, this force is usually so small, and the modulus of elasticity of the solid is usually so high, that there is no significant deformation of the solid surface. Equation (3.1) can be rearranged to the form

σ lv cos θ = σ sv − σ sl (3.2)

Although sometimes referred to as Neumann’s formula, this relation is more often called Young’s equation or the Young-Dupré equation. 69

70

Liquid-Vapor Phase-Change Phenomena

FIGURE 3.1  Interfacial tensions acting on a contact line.

We can gain a different perspective on this relation (3.2) by returning to the notion that the interfaces are actually interfacial regions, as shown in Fig. 3.2, and considering the thermodynamics of the system. We specifically consider a perturbation which results in a differential change in the positions of the interfacial surfaces Slv* , Ssv* and Ssl* while the temperature and the volume of the individual phases in the system are constant. The resulting change in the total Helmholtz free energy is equal to the sum of the changes in the bulk phases and the interfacial regions

dF = dFv + dFl + dFs + dFelv + dFesl + dFesv (3.3)

Here, subscripts denote the bulk phase for the bulk phase properties and superscripts denote the interface for properties of the interfacial regions. In Chapter 2, it was shown that for the interfacial region (see Eq. (2.16))

dFeS* = − SeS* dT + µ dN eS* + σdAiS* (3.4)

and for the bulk phases (see Eq. (2.19))

dF = − SdT + µ   dN − PdV (3.5)

Although Eq. (3.4) was originally developed specifically for a fluid-fluid interface, the arguments used in its development are equally valid for a solid-fluid interface. We will therefore use Eq. (3.4) to evaluate the dF terms in Eq. (3.3) for the solid-vapor and solid-liquid interfaces as well as for the liquid-vapor interface. If, in addition, we use Eq. (3.5) for the dF terms for the bulk phases, and simplify using the fact that dT = 0 and dV = 0 (since the temperature throughout the system is fixed) Eq. (3.3) becomes dF = µ dN v + µ dN l + µ dN s

+ µ dN lv + µ dN sl + µ dN sv

(3.6)

+σ lv dAlv + σ sl dAsl + σ sv dAsv

FIGURE 3.2  Interfacial regions associated with a liquid droplet on a solid surface.

71

Wetting Phenomena and Contact Angles

In deriving this result, we have also made use of the fact that, at equilibrium, μ must be the same in all the bulk and interfacial regions, as shown in Chapter 2. Because the total number of molecules in the system is unchanged, we must have

dN = dN v + dN l + dN s + dN lv + dN sl + dN sv = 0 (3.7)

Substituting into Eq. (3.6) yields

dF = σ lv dAlv + σ sl dAsl + σ sv dAsv (3.8)

For a droplet like that in Fig. 3.2, a shift in the interface surfaces that increases the area of the solidliquid interface would result in an equal reduction in the solid-vapor interface area. This implies that

dAsv = − dAsl (3.9)

If we further assume that the liquid-vapor interface is a spherical cap (which will be true if the effects of gravity body forces are small), then it can be shown from geometric considerations that

dAlv = dAsl cos θ (3.10)

Substituting Eqs. (3.9) and (3.10) into Eq. (3.8) and rearranging yields

∂F = σ lv cos θ + σ sl − σ sv (3.11) ∂ Asl

Note that this relation gives the rate of change of the total free energy of the system with changing Asl for a constant drop volume and constant temperature. For a system held at constant volume and temperature, thermodynamic equilibrium corresponds to a minimum in F. Thus, at equilibrium, F must be a minimum with respect to Asl, that is, ∂F/∂Asl = 0. Hence, the right side of Eq. (3.11) must equal zero, which, when rearranged, yields Young’s equation (3.2). The above development demonstrates that Young’s equation can be derived from a thermodynamic analysis of the interface regions and bulk phases in which the minimization of the free energy of system at equilibrium is imposed. In deriving Young’s equation (3.2), we have ignored some aspects of the physical system. In particular, our treatment of the contact line region omits some important features of the problem. In some systems, for example, there can be an absorbed liquid film on the surface we have treated as the solid-vapor interface. In such cases, the presence of the adsorbed film affects the force balance and liquid-vapor interface shape at the contact line. Young’s equation is valid when there is no absorbed liquid film on the solid surface, but when one is present, Young’s equation is not a completely accurate representation of the physics of the system. In addition, it is assumed in the development presented above that the interfacial tensions are constant everywhere along their respective interfaces. It is known, however, that the interfacial tension very near to the contact line may be different from the value far from the contact line. Fortunately, these differences are usually small, and the measured contact angle (for systems with no absorbed film on the solid surface) usually agrees with that predicted by Young’s equation. Because the interfacial tensions are equilibrium properties, the contact angle defined by Young’s equation is also an equilibrium property, sometimes called the equilibrium contact angle. Young’s equation provides a useful framework for relating the observed shape of the liquid-vapor interface near its contact line with a solid surface to thermodynamic requirements for equilibrium. However, it is difficult to do much in the way of practical calculations with Young’s equation, because the solid-liquid and solid-vapor interfacial tensions cannot ordinarily be measured. Thus, there are generally no available σsl and σsv data to predict the contact angle. More rigorous

72

Liquid-Vapor Phase-Change Phenomena

derivations of Young’s equation and discussions of its limitations have been presented by Johnson [3.1] and Buff and Saltzburg [3.2]. Finally, it is noteworthy that the analysis in this section and the next three sections of this chapter focuses on ideal, smooth solid surfaces of one homogenous material. The effects of surface roughness and surface non-homogeniety will be considered in Sections 3.5 and 3.8.

3.2  WETTABILITY, COHESION, AND ADHESION It can be seen in Fig. 3.1 that as the contact angle θ approaches 180°, if gravity body forces are small, the liquid drop becomes spherical, with only one point of contact with the solid. At the other extreme, as θ approaches zero, a drop with a fixed volume approaches a thin film configuration. A liquid for which θ = 0 is said to completely wet the solid surface and the surface would be described as superhydrophilic. By convention, a liquid with a value of θ between 0° and 90° is termed a wetting liquid or hydrophilic. For 90° < θ < 180° the liquid is said to be non-wetting or hydrophobic, and for θ = 180° it is completely non-wetting and the surface would be described as superhydrophobic. Thus the contact angle is a direct index of the wettability of the liquid. Photographs illustrating the very different wetting characteristics of water on polytetrafluoroethylene (Teflon) and clean smooth copper are shown in Fig. 3.3. Note that in Young’s equation (3.2), |cos θ| cannot exceed 1. Hence, for an equilibrium contact angle to be established, it is necessary that |(σsv − σsl)/σlv| < 1. Otherwise the interfacial tensions will be unable to achieve a force balance. If (σsv − σsl)/σlv is less than −1, σsl overcomes the combined horizontal components due to σsv and σlv The contact line will be pulled toward the center of the drop (θ → 180°) until the contact disappears and the liquid will be completely non-wetting. This virtually never happens for a droplet of liquid surrounded by vapor, but it could happen for a droplet of one liquid surrounded by another immiscible liquid. Conversely, if (σsv − σsl)/σlv is greater than 1, σsv overcomes the combined horizontal components due to σsl and σlv The contact line will be pulled away from the center of the drop, spreading the liquid thinner and thinner until the surface of the solid is covered, or until the film becomes so thin that molecular interactions come into play. For a wetting liquid to spontaneously spread on a surface in this manner, the spreading process must result in a decrease in the free energy of the system.

FIGURE 3.3  Profile shapes of liquid droplets on a solid polytetrafluoroethylene (Teflon) surface: (a) water, (b) trichlorotrifluoroethane (R-113). (Archive photo, Multiphase Transport Laboratory, UC Berkeley.)

73

Wetting Phenomena and Contact Angles

For the sessile drop shown in Fig. 3.1, it was shown in Section 3.1 that

∂F = σ sl − σ sv + σ lv cos θ (3.12) ∂ Asl

If we define a spreading coefficient Spls as

Spls = σ sv − σ lv − σ sl (3.13)

then Eq. (3.12) can be written as

∂F = −  Spls + σ lv (1 − cos θ )  (3.14) ∂ Asl

For a fully wetting liquid, θ = 0 and Eq. (3.14) becomes

∂F = −Spls (3.15) ∂ Asl

Equation (3.15) indicates the justification for the definition of the spreading coefficient given by Eq. (3.13). If the spreading coefficient Spls is positive, Eq. (3.15) indicates that the free energy decreases with increasing Asl, as is necessary for spontaneous spreading of the liquid. If the liquid does not spread and instead establishes an equilibrium contact angle, then ∂F/∂Asl = 0. Substituting this result into Eq. (3.14) yields

Spls = −σ lv (1 − cos θ ) (3.16)

which indicates that Spls must be negative. Thus, the spreading coefficient indicates the tendency of the liquid to wet and spread into a thin liquid film. A positive value of Spls indicates that the liquid will wet the solid and spontaneously spread into a thin film. A negative value of Spls indicates that the liquid will partially wet the solid and establish an equilibrium contact angle. Although these results are theoretically satisfying, from a practical point of view, it is generally difficult to evaluate Spls from Eq. (3.13) for most materials because of the lack of data for the interfacial tensions σsv and σsl. Consider now the cylindrical column formed by the liquid phase (l), in contact with an ideal (flat, homogeneous) solid phase(s), which is surrounded by a low-density gaseous phase (g), as shown in Fig. 3.4a.

FIGURE 3.4  Separation of a solid-liquid column surrounded by a gas phase.

74

Liquid-Vapor Phase-Change Phenomena

If the column is torn apart at the l-s interface, as indicated in Fig. 3.4b, then the net reversible work required per unit of interface area is

wsl = σ lg + σ sg − σ sl (3.17)

Note that the terms on the right side are a direct result of the fact that in tearing the column in this way, we have created new liquid-vapor and solid-vapor interfaces and eliminated a solid-liquid interface. Assuming that we must do work on the system to create an interface and can retrieve work reversibly from destroyed interfaces, the right side of Eq. (3.17) follows immediately from the definitions of the interfacial tensions. The work wsl defined by Eq. (3.17) is called the work of adhesion, since it is the minimum reversible work required to tear the liquid off the solid surface. Instead of the lower portion being a different solid phase, suppose that the column torn in half was entirely the same liquid phase. Because we have created two new liquid-vapor interfaces without destroying an existing interface, the same arguments made above imply that the work required to break the column is

wll = 2σ lg (3.18)

Because wll is the work required to break internal bonds in the material, it is called the work of cohesion. We can use a similar line of reasoning to analyze the spreading of a liquid on a solid. The spreading process can be idealized as indicated schematically in Fig. 3.5. In this idealized process the liquid is sliced along the dotted lines in Fig. 3.5a and the cut-off portions are brought into contact with the solid to obtain the final configuration shown in Fig. 3.5b. In this manner, the spreading process can be viewed as a sequence of creating and destroying interfaces. The process shown in Fig. 3.5 results in the creation of new liquid-gas and liquid-solid interface areas and the elimination of sections of solid-gas interface. The total work interaction for the processes (assuming δ σ sg ,d cos θ =   σ lg  (3.33)  for σ lg ≤ σ sg ,d  1



The variation of θ with σlg indicated by Eq. (3.33) is, in fact, observed in experimental data. Figure 3.6 shows, as an example, measured combinations of cos θ and surface tension for various hydrocarbon liquids on polytetrafluoroethylene (Teflon) which were obtained by Fox and Zisman [3.8]. It can be seen that the points distribute themselves in a fairly narrow band. The intersection of this band with the horizontal line at cos θ = 1 corresponds to σlg = σsg,d. This value of σlg = σsg,d is called the critical surface tension of the solid. For values of σlg > σsg,d but close to σsg,d, the relation given by Eq. (3.33) may be expanded in a Taylor series about σlg = σsg,d. Retaining only the first term of the series yields cos θ = 2 −



σ lg σ sg ,d

for

σ lg > σ sg ,d ,

σ lg − σ sg ,d σsg,d linearly on this type of plot to determine the critical surface tension. Example 3.2 Polytetrafluoroethylene (Teflon) is a low-energy surface having a critical surface tension of 0.018 N/m (Fig. 3.6). Determine the degree to which the following fluids wet this material: liquid helium II at –271°C and saturated liquid methanol at 20°C and 160°C.

78

Liquid-Vapor Phase-Change Phenomena

FIGURE 3.6  Plot of the observed contact angle θ versus liquid-vapor interfacial tension for various liquids on polytetrafluoroethylene at 20°C. The solid curve is a best fit to the data.

From Table 2.1. Helium II at –271°C has a surface tension of 0.00032 N/m, which is far below the critical surface tension of polytetrafluoroethylene. This implies, from Eq. (3.33) that cos θ = 1, from which it follows that θ = 0 and the liquid fully wets the surface. For methanol at 20°C, the interfacial tension is 0.0226 N/m and from Eq. (3.33)

{

θ = cos −1 2( 0.018 / 0.0226)

1/ 2

}

− 1 = 38.3°

For methanol at 160°C, the interfacial tension is 0.0069 N/m, which is less than the critical surface tension for the polytetrafluoroethylene. Hence, by the same logic as for the HeII, θ = 0 and the liquid fully wets the surface. This demonstrates the often-observed fact that the wetting characteristics of the fluid vary with temperature. Generally, the liquid becomes more wetting as the temperature increases.

For non-polar solids, σsg,d = σsg, and determination of the critical surface tension also indicates the interfacial tension of the solid. As noted in Section 3.2, this interfacial tension indicates the surface excess free energy per unit of surface area of the solid. Two categories of solid surfaces can be considered. There are high-energy surfaces having a chemical binding energy on the order of 1 eV. Surfaces of this type are ioninc, covalent, or metallic, with interfacial tensions of 500 mN/m or greater. Glasses and metals are typically in this high-energy category of materials. Because of the high surface energy of these solids, many liquids spread on and wet this type of surface. The second category is low-energy surfaces with chemical binding energy on the order of k B T . Note that this implies that thermal motion can often overcome the binding energy. Materials in this category typically have interfacial tensions in the range from about 15 to 50 mN/m, and

Wetting Phenomena and Contact Angles

79

are poorly wetted by most common liquids. Hydrocarbons, polymers, and plastics are generally low-energy materials. The surface tensions of non-metallic liquids are typically between 15 and 75 mN/m. For highenergy surfaces, this means that σlg is invariably below the critical surface tension, leading to the conclusion that clean high-energy surfaces are almost always wettable. For low-energy surfaces, the critical surface tension may be above or below that of the liquid. High surface tension liquids such as water (Table 2.1) may poorly wet surfaces of this type, while low surface tension liquids like R-134a (Table 2.1) may fully or almost fully wet the surface. Fluorocarbon surfaces in general and polytetrafluoroethylene in particular, have very low critical surface tensions. In Fig. 3.6, it can be seen that the critical surface tension for polytetrafluoroethylene is about 18 mN/m. As a result, very few liquids fully wet polytetrafluoroethylene. Polyethylene, on the other hand, has a critical surface tension of about 31 mN/m and, consequently, it is wet by many liquids. Water, having a particularly high surface tension, has an equilibrium contact angle of about 108° on polytetrafluoroethylene. Although higher contact angles are theoretically possible, in practice, liquid-vapor intrinsic contact angles above 108° are almost never found on flat, homogeneous, clean surfaces. As discussed later in Section 3.8, higher contact angles can be achieved on some microstructured and nanostructured surfaces. The above results suggest that the high-energy metal surfaces of heat transfer equipment in which liquid and vapor phases are present will almost always be fully wetted by the liquid. This would be true if the fluids were pure and the surfaces were perfectly clean. However, these circumstances are almost never achieved in practice. A contact angle of about 20° is more typical of observed values for water on metal surfaces of heat transfer equipment. Because the metals are high-energy surfaces, they typically are also wetted by contaminants in the system, which may spread to form a thin layer over all or part of the surface. The working fluid then may not wet portions of the surface covered by a thin absorbed film of the contaminant. The characteristics of such thin films are discussed further in the next section.

3.4  ADSORPTION AND SPREAD THIN FILMS Adsorption As noted in Section 2.1, the variation of properties in the interfacial region between two phases is actually continuous, and the values of the properties in this region are generally different from those in the bulk phases. For a pure vapor in contact with a solid, attraction between solid and vapor molecules can thus increase the density of vapor molecules in the interface region near the solid surface. The excess vapor density above the bulk value is the amount said to be adsorbed onto the surface. In a similar manner, if a mixture of gases is in contact with a solid, attraction between solid and gas molecules may increase the concentration of one or more of the component gases in the interfacial region above that in the bulk mixture. In this way, gases that have a particularly high affinity for molecules in the solid may be preferentially adsorbed onto the solid surface. In general, adsorption is the retention at the interface of solid, liquid, or gas molecules, atoms or ions by a solid or a liquid. Adsorption may affect the wetting of a liquid on a solid in at least two ways. First, we have already seen in Chapter 2 that concentration of surfactant molecules at a liquid-vapor interface can reduce the interfacial tension. We have also seen in the Section 3.3 that the liquid-vapor interfacial tension directly affects the contact angle. Hence, substances adsorbed at the liquid-vapor interface can directly affect the wetting characteristics of the liquid. In addition, adsorption of a substance onto the solid surface can alter the interfacial tension of the solid-vapor interface. The difference between the interfacial tension without and with

80

Liquid-Vapor Phase-Change Phenomena

the adsorbed species present is termed the surface pressure of the adsorbed material on the solid surface, πS:

π S = σ sv − σ sv ,a (3.35)

In this equation, σsv,a is the interfacial tension with the adsorbed substance present. It can be seen from Young’s equation (3.2) or Eq. (3.33) that changing σsv will alter the contact angle. In fact, we can generalize Young’s equation to include this effect as

σ lv cos θ = ( σ sv − π S ) − σ sl (3.36)

In this equation, the πS term is usually unimportant for non-wetting liquids. For these circumstances σsv is small compared to σsl anyway (note that cos θ < 0 for 90° < θ < 108°) and the presence of adsorbed material usually just decreases it a bit more. On the other hand, high-energy surfaces like metals or glass have large values of σsv which may be reduced appreciably by the presence of an adsorbed species. Equations (3.33) and (3.36) both indicate that this can lead to significant changes in the wetting angle for such cases. Strictly speaking, the validity of Young’s equation is suspect when additional adsorbed materials are present at one or more interfaces, because such effects are not considered in its derivation. Equation (3.36) is plausible if one considers the problem from a macroscopic force-balance viewpoint. However, this relation assumes that the sole effect of the adsorbed layer is to uniformly modify σsv, which may not be true in all cases. The above discussion indicates that adsorbed materials on a solid metal surface may cause its wetting characteristics to deviate significantly from those for a perfectly clean surface. Hence, for a system whose operation is sensitive to liquid wetting characteristics, exposure of the system to substances that may adsorb on its surfaces should be avoided if at all possible. The consequences of heterogeneous surface wetting that can result from local adsorption of contaminant materials are explored later in Sections 3.5 and 3.8.

Spread Thin Films As discussed in Section 3.3 spreading of liquid when the spreading coefficient, Spls, is greater than zero can also cover a solid surface with a thin liquid film. This type of film differs from an adsorbed film in that it is created by spreading of a liquid phase over the surface, as opposed to a vapor phase deposition process that is typical of adsorbed films. From Eq. (3.32) it can be seen that Spls will be positive if

σ sg ,d > σ lg (3.37)

Hence a spread thin film is more likely to be observed for low surface tension liquids on high surface energy solids, such as glass or metals. The most well-known examples of spread films do, in fact, correspond to these conditions. Perhaps the most spectacular example of a spread film is the spreading of liquid helium. The surface tension of liquid helium varies with temperature, but is usually near or below 1 mN/m (Table 2.1). A glass Dewar flask containing the liquid may have a surface free energy (σsg) on the order of several hundred millinewtons per meter. Thus, the work of adhesion is much greater than the work of cohesion, and spreading will occur. The consequences of this strong tendency for spreading are shown in Fig. 3.7. The liquid helium spreads up the walls of the Dewar against gravity, over the rim of the opening at the top, and forms a thin film on the outside that either evaporates or collects and drips off the bottom of the flask. Low viscosity silicone oils (e.g., Dow Corning 200 series) also spread spontaneously on most metal surfaces. The surface tension of these fluids is typically near 18 mN/m, and consequently the

Wetting Phenomena and Contact Angles

81

FIGURE 3.7  Schematic illustration of the spontaneous spreading of liquid helium over the walls of a Dewar flask.

work of cohesion wll = 2σlg is small compared to the work of adhesion on high-energy solids. Hence, liquids of this type readily spread to form a thin film on such materials. It should be noted, however, that adsorption of substances on the solid surface may sometimes inhibit the spreading of liquids over the surface. An adsorbed monolayer of a cationic surfactant can change a high-energy surface into a low-energy one, which can reduce the work of adhesion to the point that Splg < 0 and the liquid cannot spread. Surfactants of this type may be adsorbed selectively at specific locations resulting in localized barriers to the spread of the otherwise wetting liquid. To analyze more fully the spread thin film that results for these circumstances, we must first develop a new concept. Consider the system shown in Fig. 3.8 in which liquid is in contact with and fully wets the horizontal solid surface. A hemispherical housing with a trapped bubble of gas inside is brought into close proximity to the solid surface so that a thin film of liquid of thickness δ exists between the vapor and solid interfaces. The liquid does not wet the inside of the housing and a tube connects the bubble inside the housing to a system that holds the pressure constant.

FIGURE 3.8  Model system considered in analysis of disjoining pressure effects.

82

Liquid-Vapor Phase-Change Phenomena

If the pressure inside the bubble is Pb and the liquid-gas interface is flat, then at equilibrium, Pb must balance the liquid pressure across the interface in the liquid. If δ is large, then the pressure across the interface, P f , will just equal the local ambient pressure in the liquid Pl = Patm + ρl gz h (note that z h is distance measured downward from the liquid-air interface). However, if the housing is brought very close to the solid surface, the pressure inside the bubble must not only balance the ambient liquid pressure Pl, but it must also counteract the attractive forces between the liquid molecules and the solid surface, which otherwise would maintain a thicker film of liquid on the surface. When the film is very thin, these attractive forces act to pull liquid into the layer as if the pressure in the layer were reduced below the ambient pressure Pl by an amount Pd, which is known as the disjoining pressure. By convention, if the affinity of the liquid for the solid draws liquid into the film, Pd is taken to be negative. For the circumstances shown in Fig. 3.8, the local liquid pressure Pl and the disjoining pressure effect (due to solid-liquid attraction) act in tandem to thicken the film. To maintain a thin liquid film, the pressure force in the bubble must balance both effects, which implies that at equilibrium

Pb = Pl +   Pd = Pl − Pd = ρl gzh + Patm − Pd (3.38)

Note in Eq. (3.38) above that since Pd is negative by convention, –Pd is positive. Since the attractive forces between liquid molecules and those of the solid are expected to be stronger for molecules closer to the surface, the required disjoining pressure difference –Pd to overcome them is expected to increase as δ gets smaller, as suggested in Fig. 3.9. If we consider the process of slowly bringing the housing progressively closer to the solid wall while adjusting Pb, to maintain equilibrium, the pressure required to thin the layer becomes continually larger until finally the last monolayer of the liquid phase is removed. The molecules of the vapor are then in contact with those of the solid to within an interfacial separation δ0, which is less than the thickness of the liquid monolayer just removed. The work required to remove a unit area of the film in this manner wf is given by δ=∞



wf =

∫ − P (δ ) dδ (3.39) d

δ=δ 0

In forcing the spread layer of liquid away so a unit area is no longer wetted, we have just reversed the spreading process shown in Fig. 3.5 and described in Section 3.2. From this point of view, the work

FIGURE 3.9  Expected variation of disjoining pressure with film thickness δ.

83

Wetting Phenomena and Contact Angles

required to remove a unit area of the film wf must be equal to the negative of the work of spreading, wsp, given by Eq. (3.20):

w f = − wsp = σ sg − σ lg − σ sl = Spsl (3.40)

Hence, the disjoining pressure is related to the spreading coefficient as δ=∞



Spsl =

∫ − P (δ ) dδ (3.41) d

δ=δ o

In Section 2.3, we considered the shape of a free liquid surface meeting a vertical wall when the liquid only partially wets the surface, forming a finite contact angle. When the liquid fully wets and spontaneously spreads on the surface, the interface near the vertical wall differs in that there is a thin liquid film above the intrinsic meniscus, as indicted in Fig. 3.10. Here the intrinsic meniscus is defined as that portion of the meniscus profile that is dictated by the Young-Laplace equation (2.29) with the effects of disjoining pressure being negligible. The portion of the meniscus above the intrinsic meniscus will be referred to simply as the thin film. The thin film and intrinsic meniscus together will be termed the extended meniscus. Here we will limit our attention to a two-dimensional extended meniscus system that is in equilibrium with no evaporation occurring. (In Section 8.3, characteristics of an evaporating extended meniscus will be examined.) In the intrinsic region, the interface profile is determined by solving Eq. (2.48) as described in Section 2.3. In solving Eq. (2.48), the requirement that z → 0 as y → ∞ is one boundary condition, and matching the profile to the thin film at the wall provides the other required boundary condition. In the thin film, the interface profile is determined largely by the variation of the disjoining pressure. The hydrostatic pressure gradient in the liquid film is given by

dPl = −ρl g (3.42) dz

FIGURE 3.10  The extended meniscus formed where the interface of a highly wetting liquid contacts a solid wall.

84

Liquid-Vapor Phase-Change Phenomena

At the lower limit of the thin film (z = z0), the film is relatively thick and the pressure in the film equals the pressure in the surrounding vapor. Integrating Eq. (3.42) from this point to an arbitrary location z, we obtain Pl − Pl 0 = Pd = −ρl ( z − z0 ) g (3.43)



Hence, the change in the hydrostatic pressure along the film is equal to the variation of the disjoining pressure. Differentiating (3.43) yields dPd = −ρl g (3.44) dz



The disjoining pressure is primarily a function of the film thickness and the nature of the liquid and solid surface. Because, for a given solid-liquid system, the disjoining pressure is only a function of film thickness δ, Eq. (3.44) can be written dPd  dδ    = −ρl g (3.45) dδ  dz 



Following Potash and Wayner [3.9], we further assume a power-law dependence of Pd on δ Pd = − Aδ − B (3.46)



This functional form was found to agree well with measurements of disjoining pressure and film thickness obtained by Deryagin and Zorin [3.10]. Differentiating Eq. (3.46) with respect to δ and substituting into Eq. (3.45) yields  dδ  ABδ − B −1   = −ρl g (3.47)  dz 



Rearranging Eq. (3.47) and integrating from the lower limit of the thin film z0 to an arbitrary z location δ

∫

z

∫

AB δ − B −1 dδ = −ρl g dz (3.48)



δ0

z0

Completion of the integration and solving for δ yields ρ g ( z − z0 )   δ = δ −0 B + l  A  



−1/ B

(3.49)

Thus the disjoining pressure difference supports a thin liquid film against gravity above the intrinsic meniscus. The diminishing film thickness results in a gradient in the disjoining pressure that matches the hydrostatic pressure gradient. Additional thermodynamic aspects of the extended meniscus, and transport during evaporation of an extended meniscus will be described in Section 8.3. Example 3.3 Estimate the disjoining pressure for a layer thickness of 0.1 and 0.01 mm using the values of A = 1.782 Pa mB and B = 0.6 given for carbon tetrachloride (CC14) and glass in reference [3.9].

85

Wetting Phenomena and Contact Angles How thick would the film be at a distance of 10 cm above a liquid pool of CC14 on the vertical wall of the container? The disjoining pressure is determined from Eq. (3.46) as Pd = − Aδ − B



For δ = 0.10 mm: Pd = –1.782(0.0001) –0.6 = –448 Pa For δ = 0.01 mm: Pd = –1.782(0.00001) –0.6 = –1782 Pa For CC14, ρl = 1590 kg/m3. Using Eq. (3.49) and taking δ0 = ∞ for z0 = 0 at the level of the free surface of the liquid-vapor interface, it follows that



 ρ gz  δ =  l   A 

−1/ B

 1590 ( 9.8)( 0.1)  =  1.782  

−1/ 0.6

= 1.25 × 10 −5m = 12.5 µm

3.5  CONTACT ANGLE HYSTERESIS Equilibrium contact angles can usually be determined by simply photographing the contact location and measuring the angle on enlarged photographs. Data obtained in this manner generally indicate that the contact angle of the liquid varies depending on the motion history of the contact line. The nature of this phenomenon is illustrated schematically in Fig. 3.11. Consider the process of partially immersing a thin slab of the solid into the liquid, moving it vertically downward very slowly and stopping when the contact line reaches point z1. The partially wetting liquid forms an equilibrium contact angle at point z1, which we will designate as θa. Because this contact angle was established after advancing the contact line over dry solid, it is referred to as the advancing contact angle. We now move the slab vertically upward, causing the contact line to move down the surface, stopping when the contact line reaches point z2. After stopping, the equilibrium contact angle formed at z2 is called the receding contact angle θr because it was formed after the contact line receded over portions of the surface initially covered with liquid. After repeated dipping and removal of the slab, θa and θr may reach steady values, but for most systems they will not be equal. This difference between advancing and receding contact angles is usually referred to as contact angle hysteresis. In our discussion of wetting to this point, we have implicitly assumed that the surface is smooth and homogeneous. The equilibrium contact angle θE defined for such circumstances is an intrinsic property for a perfectly flat surface of a homogeneous solid with no contamination brought into

FIGURE 3.11  The immersion and removal sequence illustrating advancing and receding contact angles.

86

Liquid-Vapor Phase-Change Phenomena

contact with a specific liquid and gas pair. For an idealized solid surface that is perfectly smooth, clean and homogeneous in composition there would appear to be no reason for θa and θr to be different. However, such an idealized surface does not exist. Real solid surfaces are never perfectly smooth, their composition may vary slightly with location, and molecules, atoms or ions of other substances may be adsorbed on the surface. Metals of the type used in heat transfer equipment (copper, brass, steel, aluminum) are particularly susceptible to these imperfections. Manufacturing processes always leave some degree of roughness on the surface. Alloys of these metals invariably have some distinct grain structure resulting from processing the material. Consequently, the surface is often a patchwork of two or more different grain types, which means that on a microscopic scale the surface is intrinsically heterogeneous. In addition, as noted in Section 3.4, clean metals are high-energy surfaces that easily adsorb thin films of many substances. As a result, metal surfaces are easily contaminated by substances in the environment, even if their concentrations are very low. Contact angle hysteresis is generally acknowledged to be a consequence of three factors: (1) surface inhomogeneity, (2) surface roughness, and (3) impurities on the surface. The effects of surface inhomogeneity and roughness can be more clearly understood by considering Fig. 3.12. Figure 3.12a schematically shows the behavior of advancing and receding two-dimensional liquid fronts on an idealized heterogeneous surface having alternating bands of a strongly wetted and poorly wetted material. For an advancing liquid front, interface 1 in Fig. 3.12a establishes a contact angle appropriate to the poorly wetted material. However, when the contact line reaches the boundary between the bands of different material, the interface changes to accommodate the smaller contact angle for

FIGURE 3.12  Effects of surface inhomogeneity and roughness on the apparent contact angle for advancing and receding liquid fronts.

Wetting Phenomena and Contact Angles

87

the strongly wetted material. This produces a curvature in the interface (see interface 2), which, because of the interfacial tension, lowers the pressure in the liquid near the solid surface. This lower pressure causes liquid to flow toward the contact line region, allowing the contact line to move rapidly across the strongly wetted material. At the next band of poorly wetted material, the contact line motion is slowed as the larger contact angle is re-established. The tendency for the contact line to move rapidly across the strongly wetted surface means that if the front motion is stopped, the larger contact angle of the poorly wetted surface is more likely to be established as θa. For the receding liquid front, interface 1 has established the low contact angle for the strongly wetted material that persists until the contact line reaches the boundary between the different materials. At this point the larger contact angle of the poorly wetted material is established which produces curvature in the interface near the contact line for receding interface 2 in Fig. 3.12a. Due to the interfacial tension, this curvature produces a rise in pressure in the liquid, which causes liquid to flow away from the contact line region, allowing the contact line to move rapidly across the poorly wetted material. At the next band of strongly wetted material, the motion of the contact line is slowed while the lower contact angle is established. In this case, the rapid motion of the contact line across the poorly wetted material means that if the front motion is stopped, the lower contact angle of the strongly wetted material is most likely to be established as θr . Figure 3.12b schematically shows the behavior of advancing and receding liquid fronts on an idealized rough but homogeneous surface. The actual contact angle for liquid in contact with a flat section of the material θ is acute, indicating that the liquid actually moderately wets the surface. However, advancing interface 1 contacts a downward slope such that when a contact angle of θ is established with the slope, the angle of the interface to the horizontal, which will be referred to as the apparent contact angle, is somewhat larger than θ. As the front advances, the contact line moves down the slope until it reaches the bottom of the groove. At this point the interface must curve as shown for interface 2 to preserve the actual contact angle θ. Because of the interfacial tension, this curvature reduces the pressure in the liquid near the contact line, causing liquid to flow into this region. This, in turn, allows the contact line to rapidly move up the slope to the apex of the groove. At the apex, the contact line motion is slowed while the interface adjusts to establish the contact angle θ with the next downward slope. As a consequence of the more rapid motion of the contact line along the upslope, if the front is stopped, the observed contact angle of the interface is most like to be the apparent contact angle for the downslope θa. Similar arguments may be applied to the receding fronts in Fig. 3.12b. Now, on the downslope, the apparent contact angle will be lower than θ if the angle θ is preserved on the sloping surface. When the interface 2 contacts the opposite wall of the groove, it does so at the apex, and the contact line simply transfers to the next downslope. As a result, if the front is stopped, the observed contact angle is most likely to be the apparent contact angle for the downslope θr , which is smaller than θ and smaller than θa. Real surfaces are, of course, different from the idealized surfaces shown in Fig. 3.12. In general, the surface may be rough and have a non-systematic variation of solid material properties over the surface. It is noteworthy, however, that consideration of the idealized surfaces in Fig. 3.12 indicates that both roughness and surface inhomogeneity may cause the advancing contact angle to be greater than the receding one. We may expect, then, that when both conditions are present, the same trend will be observed. Adsorption of contaminants onto the surface may further contribute to the non-uniformity of its wetting characteristics, with the net effect being the same as the non-homogeneous surface considered above. Hence, it can be argued that all three of the mechanisms of contact angle hysteresis mentioned earlier in this section can cause advancing contact angles to be larger than receding ones. This trend is, in fact, observed in contact angle measurements. Another insightful observation can be extracted from the sequences of interface motion in Fig. 3.12. When the contact line for an advancing front encounters a surface defect in the form of a high θ surface region or a downslope surface deformation, adjustment to the new local surface

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contact angle condition at the boundary of the surface defect can produce interface curvature and an associated capillary pressure difference that drives liquid away from the contact line region. When a liquid droplet is spreading over a surface, the liquid front momentum may not be great enough to overcome this flow of liquid away from the contact line region when a surface defect is encountered. If this happens, the forward motion of the front may stall, bringing the advancing of the contact line to a halt. If liquid were then extracted from the droplet with a small syringe, the contact line would tend to move backward across the defect boundary (with the interface profile changing from interface 3 to interface 2 in the Advancing diagrams in Fig. 3.12). However, doing so would reverse the curvature near the contact line, drawing liquid toward the contact line, which would counteract the motion of the interface away from the boundary of the defect. The net effect of these tendencies is that the contact line is likely to stop at the edge of surface defects, and once stopped, it tends to resist moving away from the defect boundary. This tendency is referred to as pinning the interface contact line at defect edges. The scenario described here considers idealized surface variations, but the implication that pinning of a droplet contact line is most likely to occur at the edge of a surface defect is consistent with observed behavior of droplet or liquid front contact lines on real surfaces. The presence of a kinked surface morphology or the presence of structures with acute angles can cause the contact line of a droplet to be trapped in such a pinned state. Although this discussion has considered the defect areas to be naturally occurring, similar effects can also give rise to pinning of droplet contact lines at the boundaries of fabricated microstructures or nanostructures on surfaces. Contact angle hysteresis plays an important role in the behavior of liquid droplets on vertical or inclined surfaces and in small diameter tubes. If a drop of liquid is placed on a horizontal solid surface, the resulting forward motion of the liquid front as the droplet contacts the surface and achieves an equilibrium shape leaves the droplet with the advancing contact angle all around its perimeter. If the surface is then rotated through 90°, the vertical components of the forces exerted on the liquidvapor interface at the upper and lower contact lines effectively cancel, leaving the gravitational body force on the droplet initially unbalanced. However, as indicated in Fig. 3.13a, as the droplet begins to move downward, the advancing contact angle is maintained near the bottom of the droplet, while the smaller receding contact angle is established near the top. This difference between the top and bottom contact angles makes it

FIGURE 3.13  (a) Illustration of how contact angle hysteresis can allow a liquid droplet to resist downward motion when a horizontal surface is rotated to a vertical orientation. (b) Illustration of how contact angle hysteresis can allow a liquid slug in a small vertical tube to resist downward motion.

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possible for the droplet to adopt a shape that may support the weight of the liquid against gravity. Beginning at the top of the droplet and moving downward, the hydrostatic pressure in the liquid inside the droplet increases more rapidly than that in the surrounding gas because of the density difference between the phases. Note in Fig. 3.13a that the interface radius of curvature is greater over the upper portion of the droplet and smallest near the bottom. As a result, the capillary pressure difference across the interface is largest at the bottom and smallest at the top of the droplet. The resulting variation of the surface tension forces over the interface may thus serve to balance the hydrostatic pressure difference across the interface, allowing the droplet to hold its position on the vertical wall against the force of gravity. It should also be noted that a bubble on a vertical solid surface surrounded by liquid can similarly resist the upward buoyancy force on the vapor and remain fixed on the surface. For the bubble the mechanisms are virtually identical to those described above for the droplet surrounded by gas. In addition, differences in the contact angle and associated variations of the interface curvature can similarly allow the bubble or droplet to adopt an interface profile that resists drag forces resulting from motion of the surrounding fluid. A similar phenomenon is exhibited by a droplet or slug of liquid in a small-diameter tube such as a soda straw. If a slug of liquid enters at the top of a vertical tube (open at both ends), gravity naturally tends to pull the liquid downward. However, as downward motion is initiated, the contact angles at the top and bottom of the slug must be the receding and advancing values, respectively, as indicated in Fig. 3.13b. As a result, a smaller mean radius of curvature exists at the top of the slug than at the bottom end. If the interface at each end of the liquid slug in Fig. 3.13b is idealized as a portion of a sphere, then the Young-Laplace equation requires that

P1 − P2 =

2σ (3.50) rt



P4 − P3 =

2σ (3.51) rb

where rt and rb are the radii of curvature at the top and bottom, respectively. Because the tube is open at both ends, the hydrostatic pressure variations are such that

P4 − P1 = ρg gL (3.52)



P3 − P2 = ρl gL (3.53)

If all four Eqs. (3.50)–(3.53) are satisfied, a force balance is achieved and the slug will remain fixed at its vertical location, despite the downward force of gravity on the slug. Combining these equations, it can be shown that a necessary condition for this to be true is

1 1 2σ  −  = ( ρl − ρg ) gL (3.54)  rt rb 

This result implies that the length of a slug that can be supported against gravity depends directly on the difference between the radii of curvature at the upper and lower interfaces of the slug. Although the curvature of the interfaces most directly affects the force balance, contact angle hysteresis is instrumental in establishing the different radii of curvature necessary to achieve equilibrium. Using that fact that the interface radius of curvature in the tube is essentially equal to the tube radius divided by cos θ (i.e., ri = rt/cos θ) a similar analysis of capillary rise in a tube can be used to show that fluid rises for a contact angles less than 90° and is depressed for contact angles greater than 90° (see Fig. 3.14).

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Liquid-Vapor Phase-Change Phenomena

FIGURE 3.14  Contact angle effect on capillary rise.

In preceding sections, we have treated the contact angle as a well-defined constant property of a system. In most circumstances of interest we may still do so and obtain useful results. However, the results of this section indicate that when evaluating the contact angle, we must be cognizant of its dependence on the prior motion of the contact line. Further discussion of wetting and contact angles may be found in references [3.11–3.13].

3.6  OTHER METRICS FOR WETTABILITY The static equilibrium contact angle described in previous sections of this chapter is the most commonly used metric for wettability. There are at least two measures of wettability that are related to the interfacial tensions and equilibrium contact angle. One is the spreading coefficient Spls defined by Eq. (3.13). The definition of Spls together with Young’s equation (3.2) indicates that

Spls = σ sv − σ lv − σ sl = −σ lv (1 − cos θ ) (3.55)

As discussed in Section 3.2, Spls > 0 implies that the liquid spontaneously spreads into a thin film, and Spls < 0 implies that liquid partially wets the surface and establishes an equilibrium contact angle. Hence, this parameter also quantifies the degree to which the liquid wets the solid surface. Another way to quantify wettability is in terms of the work of adhesion defined in Eq. (3.17). This parameter indicates the minimum (reversible) work required to tear liquid off a unit area of surface:

wsl = σ lv + σ sv − σ sl = σ lv (1 − cos θ ) (3.56)

Because it is difficult to measure solid-vapor and solid-liquid interfacial tensions, it is difficult to directly measure Spls and wsl. The easiest way to determine these parameters is by measuring contact angle and surface tension and using the relations above. These parameters provide an alternate metric of wettability, but they contain basically the same information as the contact angle, and they are subject to the same limitations. While the interpretation of static contact angle is straightforward, it has the disadvantage that in many systems, it depends on system history. Specifically, as noted in the previous section, it can depend on whether the liquid has recently advanced or receded over the surface. This shortcoming can be avoided by specifying both the advancing and receding contact angles as a specification

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of wettability. In some cases, differences between static and dynamic contact angles have been observed. The latter case corresponds to advancing or receding contact angles observed during motion of the contact line. Thus, dynamic or static pairs of advancing and receding contact angles could be used a metric for wetting. However, even if advancing and receding parameters are employed together as a metric of wettability, their separate variation with temperature and surface conditions can make determination of changes in wettability a challenging task. Figure 3.15 shows the measured variation of these different contact angle definitions with temperature for water on a single crystal sapphire plate. These data are from a study by Nagai et al. [3.14]. The simultaneous variations of advancing and receding contact angle indicated by these data make it difficult to assess how much the wettability changes with a given change in temperature. Another possible parameter that could be used to quantify wettability is the wickability W, which can be defined as

W = σ sv − σ sl (3.57)

Note that using Young’s equation (3.2), the wickability can be related to the equilibrium contact angle θ E as

W = σ lv cos θ E (3.58)

which indicates that as the contact angle approaches zero, the wickability approaches the magnitude of the liquid-vapor interfacial tension. The definition adopted above links the wickability to other wetting metrics. Equation (3.58) above links this parameter to the contact angle, and combining Eq. (3.57) with Eq. (3.16) provides the following relation linking the wickability and the spreading coefficient:

Spl ,s = W − σ lv (3.59)

An interesting interpretation of the wickability can be established by reconsidering the rise or depression in a capillary tube for the circumstances depicted in Figure 3.14. In Chapter 2,

FIGURE 3.15  Contact angle data measured by Nagai et al. [3.14].

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Liquid-Vapor Phase-Change Phenomena

the following relation was derived for the rise height of a wetting liquid (θ E < 90°) in a capillary tube.

σ lv cos θ E (3.60) (ρl − ρv ) gri

zi =

Although this relation was derived for a wetting liquid that rises in the capillary tube, it correctly predicts the depression (negative zi) that results for liquid with contact angles greater that 90°. Using the definition of W in Eq. (3.58), this equation for the rise height can be written in the form.

W (3.61) (ρl − ρv ) gri

zi =

Thus, the wickability W is a direct indicator of the tendency for liquid to rise in a capillary passage. The more positive W is, the higher the capillary rise, and if W is negative, the liquid exhibits a negative rise height, which effectively inhibits capillary entry of liquid into the capillary tube. Because porous media are effectively a matrix of capillary passageways, this interpretation can be extrapolated to interpret W as an indicator of the degree to which liquid will wet and permeate a porous media against gravity. This suggest interpretation of W as a wetting metric for capillary passages and, at least in some cases, porous media. Shoji and Zhang [3.15] proposed to quantify wettability in terms of two parameters derived from thermodynamic considerations. The first is the product r Φ where r is the ratio of actual liquid-solid contact area Als,act to projected liquid-solid contact area Als,proj

r = Als ,act / Als , proj (3.62)

and Φ is defined as

Φ=

σ sv − σ sl (3.63) σ lv

Note that Φ is the wickability W normalized with the liquid-vapor interfacial tension. The second parameter is the product rf, where f is the friction coefficient associated with the force required to move the contact line. This parameter is defined as

    dG dG f = − = (3.64)   d (rAls , proj )  θ=θa  d (rAls , proj )  θ=θr

Note in the above relation that the derivatives represent the rate of change of free energy (G) with actual contact area. This parameter can be viewed as an increment or reduction of the interfacial tension due to resistance to contact line motion. Shoji and Zhang [3.15] showed that these parameters are related to the advancing and receding contacting angles as cos θr + cos θa (3.65) 2



rΦ =



rf = σ lv

cos θr − cos θa (3.66) 2

The analysis of Shoji and Zhang [3.15] indicates that specifying r Φ and rf is an effective way of quantifying the wetting characteristics of a fluid-solid system. These parameters more directly

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Wetting Phenomena and Contact Angles

quantify the interfacial energy features that dictate wettability. These investigators also indicated that these parameters more directly tie into the effects of system temperature and roughness on surface wettability. However, for a given system, it is difficult to determine these parameters from their definition relations. The easiest way to determine them is to measure contact angles and use Eqs. (3.60) and (3.61) above. Kunkle and Carey [3.16] proposed quantifying surface wetting with a wetting number NW defined as the ratio of the droplet wetted footprint area ( Asl ) to the footprint area of a droplet of equal volume with a contact angle of 90°. NW =



Asl (3.67) Asl ,90°

If the deposited droplet is modeled as a spherical cap, the geometry dictates that NW is related to the apparent contact angle θapp as

NW =

22/3 (1 − cos2 θapp ) (3.68) (2 − 3 cos θapp + cos3 θapp )2/3

This metric has the advantage that the footprint area of a test droplet can be accurately measured using a photo or video frame and image processing software, even when the contact angle is very small. NW can then be computed using the spherical cap model to calculate Asl ,90°. This is an especially sensitive wetting metric for highly wetted (low contact angle) surfaces. A completely satisfactory wettability metric would be easily measurable, have a clear physical interpretation and would have the capacity to account for the effects of temperature, surface roughness, and surface contamination. While each of the wettability metrics described above offers a slightly different perspective, none provides a completely satisfactory means of quantifying wettability for systems in which phase-change processes occur in applications. There is clearly a need for further exploration of the mechanisms of wetting and ways of quantifying the wetting characteristics that more directly reflect the physics of wetting. One aspect of wetting that makes experimentation and theoretical modeling difficult is that experiments have demonstrated that wettability can vary with time due to slow oxidation and other physical or chemical changes at the surface of the solid. The data shown in Fig. 3.16 are an example of the time variation of wetting characteristics. These equilibrium contact angle data were obtained by Nagai et al. [3.14] for a water droplet on a single crystal sapphire surface in air. The data indicate

FIGURE 3.16  Time variation of contact angle data for water on a sapphire crystal (data from Nagai et al. [3.14]).

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a substantial variation in contact angle with time, due to contaminant adsorption onto the surface over a number of hours. These results demonstrate the difficulty of controlling surface wettability in experiments. It also indicates that even if satisfactory metrics of wettability are available, surface contamination and/or roughness in the real system of interest may vary over time, making prediction of wettability at any one instant a difficult task.

3.7  A NANOSCALE VIEW OF WETTABILITY The wettability of a surface is directly dependent on the attractive force interactions between fluid molecules and the attractive force interactions between fluid molecules and atoms or molecules in the solid. The macroscopic view taken in the previous sections considers the role of interfacial tension force interactions at the contact line in determining the contact angle. Useful insight can also be obtained by considering wettability from a molecular point of view. It should be noted, however, that a nanoscale perspective on wettability relates most directly to the intrinsic contact angle associated with an ideal flat homogeneous surface because at the nanoscale, the liquid-vapor interface interacts with a portion of the solid surface that is small compared to the length scales of surface roughness and/or material inhomogeneity on typical surfaces in applications. Advancement of accessible computing power has made it possible to explore wetting from a molecular viewpoint using molecular dynamics (MD) simulations. Recent investigations of this type have provided valuable insight into how molecular interactions dictate the wetting characteristics of the system. In the mid-1990s, Matsumoto et al. [3.17] and Maruyama et al. [3.18] examined the wetting characteristics of a Lennard-Jones fluid that interacts with wall atoms or molecules via a Lennard-Jones type potential. For fluid-fluid molecule interactions, the conventional Lennard-Jones 6–12 potential function was used

12 6 φll = 4 εl ( σ l / r ) − ( σ l / r ) 

whereas for fluid-solid molecule interactions, these investigators used a correlation of similar form but with different constants:

12 6 φsl = 4 ε sl ( σ sl / r ) − ( σ sl / r ) 

A snapshot of the system during this type of simulation is shown in Fig. 3.17. The droplet in these simulations generally contains a very small number of molecules, which in this case are argon atoms. A particularly noteworthy result of these studies was that the simulation indicated that the cosine of the contact angle θ for the droplets was a function of the ratio εsl/εl. As εsl/εl increased, cos θ also increased, implying that θ decreases and the wettability increases. This is consistent with the physical argument that as fluid-solid attractive forces become stronger relative to forces between fluid molecules, the liquid will have a greater tendency to spread over the surface. Tang and Harris [3.19] constructed an MD simulation of a Lennard-Jones fluid confined between two identical solid surfaces under conditions that result in formation of both a liquid and vapor region. They modeled a smooth surface as three layers of solid particles in a face-centered cubic lattice. They also modeled molecular-level roughness by adding to the smooth surface a fourth layer containing less than a full monolayer of solid particles. For smooth surfaces, the results of these simulations indicate that the contact angle decreases with increasing depth of the potential well in the fluid-solid molecule interaction potential. An interesting result of this investigation is that the simulations predict that molecular level roughness tends to increase the contact angle. This is contrary to the results of experiments that indicate that macroscopic roughness tends to reduce the apparent contact angle, enhancing wettability of the surface. The fluid density profile near a molecularly rough surface was observed to be different from that near a smooth surface, resulting in differences in the interfacial tensions and different contact angles.

Wetting Phenomena and Contact Angles

95

FIGURE 3.17  A snapshot of a liquid droplet of argon on a solid surface at a system temperature of 95 K (Adapted by permission from Matsumoto et al. [3.17].).

These early MD simulation studies considered the simple Lennard-Jones potential model of molecular interactions. This model is most suitable to monatomic species. Accurate simulation of water, which is polar, and more complex polyatomic molecules requires a more sophisticated treatment. More recent studies by Kimura and Maruyama [3.20] and Kandlikar et al. [3.21] have constructed MD simulations that used the SPC/E [3.22] model for interactions between water molecules, and models for interactions between water molecules and platinum clusters and surfaces developed by Spohr and Heinzinger [3.23] and Zhu and Philpott [3.24]. These simulations indicated that a water droplet with a finite contact angle tended to form on top of a monolayer of water molecules on the platinum surface. They also concluded that the contact angle is determined by the surface energy between the monolayer water film and bulk liquid water in the droplet. They further noted that using different platinum crystal arrangements altered the droplet contact angle, apparently because the crystal structure affects the monolayer on which the droplet attaches. De Ruijter, et al. [3.25], Blake et al. [3.26], Heine et al. [3.27], and Sedighi et al. [3.28] also used MD simulations to model droplet spreading at the molecular level. MD simulation has also been used to model dynamics of dewetting in investigations by Berttrand et al. [3.29] and Koplik et al. [3.30]. MD simulations have also been constructed to model wetting in more complex systems such as liquids spreading on heterogeneous [3.31], chemically patterned [3.32], or porous [3.33] surfaces. A full description of MD simulation studies that explore surface wetting physics is beyond the scope of this introductory summary. However, several general observations about the work in this area are noteworthy. Increasing computing capabilities have made it possible to model increasingly complex systems with larger numbers of liquid molecules. Simulations using 106 or more liquid molecules are now common. Although this is a small number of molecules compared to real systems, data from such simulations have provided useful insight into the molecular level features of wetting phenomena. Overall MD simulation models of wetting physics have confirmed that the general wetting behavior of the simulated systems and key macroscopic variables, such as the dynamic contact angle, are consistent with the behavior observed in real physical experiments. MD simulation studies have also made it possible to test existing

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Liquid-Vapor Phase-Change Phenomena

theoretical descriptions and in some cases to verify these at the molecular level. A more complete description of work in this area can be found in the review articles of Bertrand et al. [3.29], Maruyama et al. [3.34], and De Coninck et al. [3.35]. The molecular-kinetic theory (MKT) of dynamic wetting has also been used in some recent molecular-level explorations of the dynamics of contact line motion associated with wetting physics. In this treatment, the random surface diffusion of the molecules at the contact line is modeled, taking into account the energy barrier and the unbalanced Young force associated with interfacial tensions at the contact line. The initial formulation of the MKT model by Blake et al. [3.36] was subsequently extended to include fluid viscosity effects [3.37] and account for liquid adhesion forces [3.38]. The model is noteworthy because it predicts the dependence of the dynamic contact angle on the speed of a moving meniscus. De Ruijter et al. [3.25], for example, have used the MKT in large scale molecular dynamic simulations of spreading droplets to predict the dynamic contact angle relaxation in the three phase zone. The interested reader can find further description of use of the MKT for modeling wetting dynamics in references [3.38–3.41].

3.8  WETTING OF MICROSTRUCTURED AND NANOSTRUCTURED SURFACES Wetting on a Rough Surface Previous sections of this chapter have considered wetting on an idealized single-material surface, and the impact of roughness and surface inhomogeneity on contact angle hysteresis. In this section, we will explore the effects of surface roughness, surface micro- and nanostructuring, and local variations of surface material properties on apparent contact angle in a more quantitative way. This will encompass effects of highly controlled nano and micro structuring of surfaces with specialized fabrication processes, as well as more randomized structuring associated with less controlled surface nanoscale coating processes and roughness resulting from common surface manufacturing processes like machining, grinding and sanding. For an ideal flat solid surface of one material, the equilibrium contact angle dictated by the horizontal force balance θ E is often referred to as Young’s angle cos θ E =



σ sv − σ sl (3.69) σ lv

The definition implies that θ E is a function of the solid, liquid and gas/vapor materials in the system of interest. Note here that if the surface is rough, microstructured, nanostructured, or nonhomogeneous (chemically heterogeneous), we expect that if the contact line is at a specified location, θ E evaluated with the local surface free energies is the local contact angle established there. For a liquid that completely wets a rough solid surface, displacement of the contact line a distance dz, as depicted in Fig. 3.18, results is a differential change in total free energy dG which equals

dG = r (σ sl − σ sv )dz + σ lv dz cos θapp (3.70)

We designate the energy change as free energy because it accounts for reversible mechanical energy changes at constant temperature and pressure, which corresponds to a change in Gibbs free energy. Also, in the above equation, r is the ratio of rough surface area to plane (or footprint) area

r=

actual surface area (3.71) apparent (footprint) area

and θapp is the apparent contact angle relative to plane corresponding to the mean surface position.

Wetting Phenomena and Contact Angles

97

FIGURE 3.18  Displacement of the contact line on a rough solid surface.

Since equilibrium at constant T and P corresponds to a minimum of free energy, the contact line will find an equilibrium where dG = 0. Setting dG = 0, solving for cos θapp and using Eq. (3.69) to replace (σ sv − σ sl ) / σ lv with cos θ E yields

cos θapp = r cos θ E (3.72)

This analysis of wetting on a rough surface is essentially Wenzel’s model [3.41]. It should be noted that this model suggests that there are two wetting regimes for a surface of this type, corresponding to two possible ranges for θ E: 1. θ E < 90°, for which θapp < θ E (since r ≥ 1). Note that for this range of θ E, increasing roughness (increasing r) makes the surface more wetting. 2. θ E > 90°, for which θapp > θ E (for r ≥ 1). For this range of θ E, increasing roughness (increasing r) makes the surface less wetting. And, for θ E exactly equal to 90°, changing r produces no change in wetting (θapp = θ E ) . These predictions are based on a thermodynamic model (that minimizes free energy at equilibrium), with the idealizations that θ E is fixed locally everywhere on the surface, and the liquid penetrates and wets the rough surface completely in regions of the surface covered by liquid. The latter condition is sometimes referred to as a Wenzel state for a liquid droplet on a surface. Note that this model would apply to manufactured surfaces with a regular pattern of small-scale roughness elements, as depicted in Fig. 3.19, if the interstitial spaces between the elements are flooded with liquid in regions where the surface is covered with liquid.

FIGURE 3.19  Droplet on a surface with a regular pattern of small-scale roughness elements.

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Liquid-Vapor Phase-Change Phenomena

Wetting on a Smooth Heterogeneous Surface An alternate wetting analysis can be constructed for a planar, but chemically heterogeneous surface. In this case envision a surface that is a mixture of planar regions of two different materials, 1 and 2, with different intrinsic equilibrium contact angles θ E ,1 and θ E ,2. This surface model is depicted in Fig. 3.20. Here f1 and f2 are the fractions of the surface corresponding to type 1 and type 2 materials, respectively. Similar to the Wenzel model described above, here we construct a relation for the change in free energy associated with motion of the contact line through a distance dz across the composite surface

dG = f1 (σ sl ,1 − σ sv ,1 )dz + f2 (σ sl ,2 − σ sv ,2 )dz + σ lv dz cos θapp (3.73)

Setting dG = 0 for equilibrium and dividing by σ lv dz yields

f1

σ sl ,1 − σ sv ,1 σ − σ sv ,2 + f2 sl ,2 + cos θapp = 0 (3.74) σ lv σ lv

For each material, Young’s equation (3.69) relates the interfacial tensions to the equilibrium contact angle

cos θ E ,1 =

σ sv ,1 − σ sl ,1 σ − σ sl ,2 , cos θ E ,2 = sv ,2 (3.75) σ lv σ lv

Using these relations to replace the terms involving interfacial tensions, and solving for cos θapp converts Eq. (3.74) to

cos θapp = f1cos θ E ,1 + f2 cos θ E ,2 (3.76)

This model thus predicts that the cosine of the apparent contact angle is the weighted average of the cosines of the equilibrium contact angles for each of the pure materials. This is essentially the model proposed by Cassie and Baxter [3.43] for a composite surface made up of planar regions with different pure material wetting characteristics.

Rough and Non-homogeneous Surfaces Real surfaces may be both rough and non-homogeneous chemically. Here we will consider the two categories of rough and composite surfaces indicated below: Rough and Hydrophilic Surfaces We first consider a rough or porous hydrophilic surface with θ E < 90°, with liquid filling the interstitial space (pores) between roughness elements. As depicted in Fig. 3.21, this circumstance is

FIGURE 3.20  Displacement of the contact line on a flat composite surface.

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FIGURE 3.21  A droplet on a rough or porous hydrophilic surface with θ E < 90°, with liquid filling the interstitial space (pores) between roughness elements.

similar to the Wenzel system considered in Fig. 3.19, but here we consider the system from a different viewpoint. The body of liquid in the droplet above the dotted line in Fig. 3.21 can be considered as a droplet in contact with a composite surface composed of alternating planar regions of a solid having an intrinsic surface equilibrium contact angle θ E, and planar regions of a liquid surface that perfectly wets with an equilibrium contact angle of 0°. For this model, we further define φ S as the fraction of the overall surface that is planar solid with contact angle θ E. Note that the fraction of surface that is liquid is 1 − φ S . Since the viewpoint of this model is a droplet on a composite surface of planar regions with different contact angles, we can apply the Cassie-Baxter relation (3.76), setting f1 = φ S , f2 = 1 − φ S , θ E ,1 = θ E and θ E ,2 = 0 . Doing so converts Eq. (3.76) to

cos θapp = φ S cos θ E + (1 − φ S ) (3.77)

Note that this model predicts that complete wetting (θapp = 0, cos θapp = 1) cannot be achieved for finite solid surface contact angle θ E, regardless of the roughness or fraction of solid surface φ S . This rough hydrophilic surface wetting model is based on the idealization that liquid establishes a penetrating liquid film that fills the interstitial space between protruding elements of the rough surface (Fig. 3.21). Formation of this type of penetrating film might be expected for conditions between total wetting (spreading coefficient Sp > 0, θ E = 0 ) and hydrophilic with finite θ E < 90°. Criteria for the formation of such a penetrating film can be explored from a thermodynamic viewpoint. We consider the rough surface depicted in Fig. 3.22 onto which a penetrating liquid film can spread that fills the interstitial space between roughness elements except for outer horizontal planar portions of the surface that remain dry. Here we designate r as the ratio of rough surface area to footprint area of the surface in the plane corresponding to the mean surface position, and we use φ S to designate the ratio of dry surface area

FIGURE 3.22  A penetrating liquid film that spreads and fills the interstitial space between roughness elements.

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to footprint area. The change in free energy per unit length of film associated with motion of the penetrating liquid film advancing through a distance dz across the composite porous surface is

dG = (r − φ S )(σ sl − σ sv )dz + (1 − φ S )σ lv dz (3.78)

The first term on the left side of the above equation is the energy to convert solid-vapor interface to solid-liquid interface. The second term is the energy to create the liquid-vapor free surface at the top of the film between locations where dry solid surface existed. Spontaneous spreading is thermodynamically favored for

dG < 0.

Substitution of the right side of Eq. (3.78) into this inequality and dividing both sizes by dz yields the following necessary condition for spontaneous spreading of the penetrating film:

(r − φ S )(σ sl − σ sv ) + (1 − φ S )σ lv < 0 (3.79)

Rearranging, this requirement can be written in the form

σ sl − σ sv 1 − φ S > (3.80) σ lv r − φS

Note that from Young’s relation (3.69), the left side of the above relation is equal to cos θ E . Because r ≥ 1, the right side of the above equation is always between 0 and 1. Defining a critical angle θc such that

cos θc =

1 − φS (3.81) r − φS

the necessary condition for spontaneous spreading of the penetrating film can be stated in the form

cos θ E > cos θc (3.82)

or equivalently as

θ E < θc (3.83)

Thus when the material intrinsic angle θ E defined by Young’s relation (3.69) is below the critical value θc , the liquid will penetrate the rough surface matrix in the manner that can give rise to the rough hydrophilic model behavior described above. Higher roughness r makes cos θc smaller, and θc larger, thereby making permitting penetration spreading for a broader range of θ E. The penetration of liquid into the interstitial spaces in a porous surface matrix ahead of the macroscopic contact line of a droplet has been referred to as hemi spreading [3.44] or a Cassie penetrating (impregnating) state [3.45]. As discussed above, for such circumstances, liquid penetrates into interstitial spaces in the porous surface structure and a drop deposited on the surface finds itself on a substrate which is essentially a patchwork of solid and liquid (solid “islands” ahead of the drop as it spreads are dry, as shown in Fig. 3.21). Theoretical models of the conditions leading to hemi spreading behavior have been developed [3.46–3.48] that are consistent with the analysis presented above. Experimental studies have also demonstrated hemi spreading at the perimeter of droplets on rough hydrophilic surfaces [3.49, 3.50]. The observed morphology for a droplet in these circumstances has been described as looking “like an egg fried without [being] flipped over

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(sunny-side up) with a well-defined radius of the egg yolk” [3.48]. Experiments have demonstrated that the outer perimeter of the penetrating film can have a non-circular shape while the droplet macroscopic contact line perimeter is circular [3.50]. This non-circular shape appears to be a consequence of the pattern geometry in the porous surface morphology [3.50]. The rate of droplet spreading on porous surfaces that may produce hemi spreading, and the resulting effects on droplet evaporation have been explored by Joung and Buie [3.51], Kim et al. [3.52], and Wemp and Carey [3.53]. Rough and Hydrophobic Surfaces We next consider a rough or porous hydrophobic surface with θ E > 90°. In this model, the interstitial space (pores) between roughness elements under the droplet are presumed to be occupied by trapped vapor, as depicted in Fig. 3.23. The body of liquid in the droplet in Fig. 3.23 can be considered as a droplet in contact with a composite surface composed of alternating planar regions of a solid having an intrinsic surface equilibrium contact angle θ E, and planar regions of a vapor surface that is perfectly non-wetting, with an equilibrium contact angle of 180°. For this model, we again adopt the definition of φ S as the fraction of the overall surface that is planar solid with contact angle θ E. It follows that the fraction of surface that is vapor is 1 − φ S . Since the viewpoint of this model is also a droplet on a composite surface of planar regions with different contact angles, we can again apply the Cassie-Baxter relation (3.76). Here we set f1 = φ S , f2 = 1 − φ S , θ E ,1 = θ E and θ E ,2 = 180°, which converts Eq. (3.76) to

cos θapp = φ S − 1 + φ S cos θ E (3.84)

Note that this model predicts that at θ E = 90°, trapping of air decreases φ S below one and drops cos θapp to the negative value 1 − φ S . This has the effect of increasing θapp above θ E = 90°. For fixed φ S , as θ E is increased above 90°, the φ S cos θ E term becomes increasingly negative, which has the effect of increasing θapp toward 180°. Achieving θapp = 180° is only possible if φ S → 0 or θ E → 180°, neither of which is attainable for real surfaces. Hence, the model relation (3.84) indicates that surfaces with large apparent contact angles are possible if φ S is small and/or θ E is large, but a perfectly non-wetting surface with θapp = 180° is not achievable. This is, of course, a model, and real system behavior could deviate from its predictions. It is based on the premise that gas or vapor is trapped in the interstitial spaces between roughness elements under the droplet, meeting the liquid at flat interfaces. It is possible that real surfaces may have surface roughness geometries that deviate from this postulate morphology in the model, resulting in somewhat different apparent contact angle behavior. In general, it is expected that very rough surfaces will likely entrap gas or vapor under the droplet described in the above model, and achieve an apparent contact angle that agrees reasonably well with the prediction of Eq. (3.84). On the other hand, less rough surfaces will less likely trap vapor effectively, and will be more likely to flood the interstitial spaces under the droplet.

FIGURE 3.23  Droplet on a surface with the interstitial spaces (pores) between roughness elements under the droplet occupied by trapped vapor.

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Less rough surfaces are therefore expected to behave in accordance to the Wenzel model prediction embodied in Eq. (3.72).

An Overall Viewpoint Considering the collection of models described above, the following overall picture emerges. The intrinsic contact angle (Young’s angle) for a flat surface of specified material θ E is, at least theoretically, between 0° and 180°. For an intrinsically wetting fluid with θ E between 0° and 90° (1 > cos θ E > 0), two possible regimes are indicated. When cos θc   < cos θ E   < 1 and 0 < θ E < θc , the penetrating liquid film fills the rough surface interstitial space on the surface beyond the contact line of the droplet. The droplet then behaves as if it is on a composite surface of solid and liquid surface regions, and its apparent contact angle behavior is predicted by Eq. (3.77) for the rough hydrophilic surface model discussed above. When θc < θ E < 90° (0 < cos θ E < cos θc ), conditions do not thermodynamically favor penetration of the liquid into the rough surface interstitial spaces. Liquid then only fills the pores of the rough surface under the droplet. This is the configuration specified in the Wenzel model, and the apparent contact angle is therefore predicted by Eq. (3.72) developed for that model. The variations of cos θapp with cos θ E indicated by Eqs. (3.72) and (3.77) are depicted in Fig. 3.24. Note that the transition between these regimes (where liquid film penetration turns on or off) occurs at θ E = θc , which is also the location where the θapp predictions of Eqs. (3.72) and (3.77) cross. Thus, as θ E increases, the apparent contact angle is expected to follow Eq. (3.70) for 0 < θ E < θc , and at θ E = θc , it transitions to follow Eq. (3.72) for θc < θ E < 90°. The overall behavior for θ E between 0° and 90° is the composite dark line in the upper right quadrant of Fig. 3.24. For a rough hydrophobic surface with θ E between 90° and 180° (1 < cos θ E < 0), it is expected that vapor and/or gas is trapped in interstitial spaces under the droplet. The appropriate model for these circumstances is a liquid droplet sitting on a composite surface having regions of solid and vapor surface. As noted above, this model predicts that at θ E = 90°, trapping of air decreases φ S below one and drops cos θapp to the negative value φ S − 1. For fixed φ S , as θ E is increased above 90°, θapp increases toward 180°. In Fig. 3.24, this trend is reflected in the model prediction curve for Eq. (3.84) shown in the lower left quadrant. This curve, together with the two curve segments in the upper right quadrant represent the predicted variation of cos θapp with cos θ E for θ E values between 0° and 180°. Data for nanostructured and microstructured surfaces with experimentally determined values of r, φ S , θ E, and θapp have been found to agree reasonably well with the trends indicated in Fig. 3.24 [3.54].

FIGURE 3.24  Regimes of apparent contact angle θapp variation with intrinsic contact angle θ E .

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Example 3.4 A nanostructured surface can be fabricated that is an array of pillars, looking in cross section like the surface depicted Fig. 3.23. The fabricated surface morphology is such that the top area of pillars in contact with liquid is 40% of the nominal surface area, and the roughness ratio r defined by Eq. (3.71) is 7.5. At 20°C, the intrinsic contact angle (Young’s angle θ E) for liquid water on a flat surface of the material used to fabricate the nanostructure is 95°. Estimate the apparent contact angle achievable for water on this nanostructured surface. The operating condition for the proposed surface is on the left, hydrophobic side of Fig. 3.24, and Eq. (3.84) predicts the apparent contact angle:

cos θapp = φ S − 1+ φ S cos θE

Here φ S = 0.4 , θE = 95° and r does not affect the contact angle. Substituting yields

cos θapp = 0.4 − 1+ 0.4cos(95π / 180) = −0.635

and

θapp = cos −1(−0.635) = 129.4°

So the nanostructured surface makes the surface strongly hydrophobic, with an estimated apparent contact angle of 129°.

Two final observations should be noted here. One is that the regimes in terms of cos θapp verses cos θ E noted above correspond to states in thermodynamic equilibrium. It is possible for droplets on real surfaces to exist in metastable states that deviate from these model predictions. As a result, it is possible for a droplet in a Wenzel state and a droplet in a Cassie state to coexist, at least for some limited time, if one or the other is in a metastable state. A transition from one state to another can then return the overall systems state to that expected under thermodynamic equilibrium. A consequence of this is that the state of a droplet at any point in time, can depend on the history of drop deposition or droplet nucleation and growth. These issues are important considerations for dropwise condensation which is considered in more detail in Chapter 9.

REFERENCES 3.1 Johnson, R. E. Jr., Conflicts between Gibbsian thermodynamics and recent treatments of interfacial energies in solid-liquid-vapor systems, J. Phys. Chem, vol. 63, pp. 1655–1658, 1959. 3.2 Buff, F. P., and Saltzburg, H., Curved fluid interfaces. II. The generalized Neumann formula, J. Chem. Phys., vol. 26, pp. 23–31, 1957. 3.3 Fowkes, F. M., Additivity of intermolecular forces at interfaces. I. Determination of the contribution to surface and interfacial tensions of dispersed forces in various liquids, J. Phys. Chem., vol. 67, pp. 2538–2541, 1963. 3.4 Fowkes, F. M. Attractive forces at interfaces, in Chemistry and Physics of Interfaces, American Chemical Society, 1965. 3.5 Girifalco, L. A., and Good, R. J., A theory for estimation of the surface and interfacial energies. I. Derivation and application to interfacial tension, J. Phys. Chem., vol. 61, pp. 904–909, 1957. 3.6 Good, R. J., Girifalco, L. A., and Krause, G., A theory for estimation of the surface and interfacial energies. II. Application to surface thermodynamics of teflon and graphite, J. Phys. Chem., vol. 62, pp. 1418–1421, 1958. 3.7 Good, R. J., and Girifalco, L. A., A theory for estimation of the surface and interfacial energies. III. Estimation of surface energies of solids from contact angle data, J. Phys. Chem., vol. 64, pp. 561–565, 1960.

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3.8 Fox, H. W., and Zisman, A., The spreading of liquids on low energy surfaces. I. Polytetrafluoroethylene, J. Colloid Sci., vol. 5, pp. 514–531, 1950. 3.9 Potash, M. Jr., and Wayner, P. C. Jr., Evaporation from a two-dimensional extended meniscus, Int. J. Heat Mass Transf., vol. 15, pp. 1851–1863, 1972. 3.10 Deryagin, B. V, and Zorin, A. M., Optical study of the adsorption and surface condensation of vapors in the vicinity of saturation on a smooth surface, Proc. 2nd Int. Congress on Surface Activity (London), vol. 2, pp. 145–152, 1957. 3.11 Chappuis, J., Contact angles, Multiphase Science and Technology, McGraw Hill, New York, NY, vol. 1, pp. 387–505, 1982. 3.12 Padday, J. F., Adhesion in a low-gravity environment, Adhesion, vol. 6, pp. 1–18, 1982. 3.13 Miller, C. A., and Neogi, P., Interfacial Phenomena, Marcel Dekker, Inc., New York, NY, 1985. 3.14 Nagai, N., Takeuchi, M., Kimura, T., and Kanashima, T., Attempts for measuring contact angles on superheated walls, Proc. 11th Int. Heat Transfer Conf., vol. 4, pp. 137–141, 1998. 3.15 Shoji, M., and Zhang, X. Y., Study of contact angle hysteresis (In relation to boiling surface wettability), Trans. JSME(B), vol. 58, pp. 1853–1859, 1992 (in Japanese). 3.16 Kunkle, C.M. and Carey, V.P., paper HT2016-7203, Proc. ASME 2016 Heat Transfer Summer Conf. HT2016, July 10–14, 2016, Washington, DC 3.17 Matsumoto, S., Maruyama, S., and Saruwatari, H., A molecular dynamics simulation of a liquid droplet on a solid surface, Proc. ASME/JSME Thermal Eng. Conf. Maui, Hawaii, vol. 2, pp. 557–562, 1995. 3.18 Maruyama, S., Kurashige, T., Matsumoto, S., Yamaguchi, Y., and Kimura, T., Liquid droplet in contact with a solid surface, Microscale Thermophys. Eng., vol. 2, pp. 49–62, 1998. 3.19 Tang, J. Z., and Harris, J. G., Fluid wetting on molecularly rough surfaces, J. Chem. Phys., vol. 103, pp. 8201–8208, 1995. 3.20 Kimura, T., and Maruyama, S., Molecular dynamics simulation of a water droplet in contact with a platinum surface, Proc. 12th Int. Heat Transfer Conf., pp. 537–542, 2002. 3.21 Kandlikar, S. G., Maruyama, S., Steinke, M. E., and Kimura, T., Measurement and molecular dynamics simulation of contact angle of water droplet on a platinum surface, Proc. ASME Heat Transfer Division 2001, vol. HTD-369, pp. 343–348, 2001. 3.22 Berendsen, H. J. C., Grigera, J. R., and Straatsma, T. P., The missing term in effective pair potentials, J. Phys. Chem., vol. 91, pp. 6269–6271, 1987. 3.23 Spohr, E., and Heinzinger, K., A molecular dynamics study on the water/metal interfacial potential, Ber. Bunsenges. Phys. Chem., vol. 92, pp. 1358–1363, 1988. 3.24 Zhu, S.-B., and Philpott, M. R., Interaction of water with metal surfaces, J Chem. Phys., vol. 100, pp. 6961–6968. 1994. 3.25 De Ruijter, M. J., Blake, T. D., and De Coninck, J., Dynamic wetting studied by molecular modeling simulations of droplet spreading, Langmuir, vol. 15, pp. 7836–7847, 1999. 3.26 Blake, T. D., Decamps, C., De Coninck, J., de Ruijter, M., and Voué, M., The dynamics of spreading at the microscopic scale, Colloids Surf., vol. 154, pp. 5–11, 1999. 3.27 Heine, D. R., Grest, G. S., and Webb, E. B. III, Spreading dynamics of polymer nanodroplets, Phys. Rev. E, vol. 68, p. 061603, 2003. 3.28 Sedighi, N., Murad, S., and Aggarwal, S. K., Molecular dynamics simulations of nanodroplet spreading on solid surfaces, effect of droplet size, Fluid Dyn. Res., vol. 42, p. 035501, 2010. 3.29 Berttrand, E., Blake, T. D., Ledauphin, G., Ogonowski, G., and De Coninck, J., Dynamics of dewetting at the nanoscale using molecular dynamics, Langmuir, vol. 23, pp. 3774–3785, 2007. 3.30 Koplik, J., and Banavar, J. R., Molecular simulations of wetting. Phys. Rev. Lett., vol. 84, pp. 4401–4404, 2000. 3.31 Adao, M. H., de Ruijter, M., Voué M, and De Coninck, J., Droplet spreading on heterogeneous substrates using molecular dynamics, J. Phys. Rev. E, vol. 59, pp. 746–750, 1999. 3.32 Grest, G., Heine, D. R., and Webb, E. B. III, Liquid nanodroplets spreading on chemically patterned surfaces, Langmuir, vol. 22, pp. 4745–4749, 2006. 3.33 Seveno, D., Ledauphin, V., Martic, G., Voué, M., and De Coninck, J., Spreading drop dynamics on porous surfaces, Langmuir, vol. 18, pp. 7496–7502, 2002. 3.34 Maruyama, S., Kimura, T., and Lu, M.-C., Molecular aspects of liquid contact on a solid surface, Therm, Sci. Eng., vol. 10, pp. 23–30, 2002. 3.35 De Coninck, J., and Blake, T. D., Wetting and molecular dynamics simulations of simple liquids, Annu. Rev. Mater. Res., vol. 38, pp. 1–22, 2008. 3.36 Blake, T. D., and Haynes, J. M., Kinetics of liquid/liquid displacement. J. Colloid Interface Sci., vol. 30, pp. 421–423, 1969.

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3.37 Blake, T. D., Dynamic contact angles and wetting kinetics, in Wettability, J. C. Berg (editor), Dekker, New York, NY, pp. 251–309, 1993. 3.38 Blake, T. D., and De Coninck, J., The influence of solid-liquid interactions on dynamic wetting. Adv. Colloid Interface Sci., vol. 96, pp. 21–36, 2002. 3.39 Sedev, R., The molecular-kinetic approach to wetting dynamics: Achievements and limitations, Adv. Colloid Interface Sci., vol. 222, pp. 661–669, 2015. 3.40 Li, H., Sedev, R., and Ralston, J., Dynamic wetting of a fluoropolymer surface by ionic liquids, Phys. Chem. Chem. Phys., vol. 13, pp. 3952–3959, 2011. 3.41 Ramiasa, M., Ralston, J., Fetzer, R., and Sedev, R., The influence of topography on dynamic wetting, Adv. Colloid Interface Sci., vol. 206, pp. 275–293, 2014. 3.42 Wenzel, R. N., Resistance of solid surfaces to wetting by water, Ind. Eng. Chem, vol. 28, pp. 988–994, 1936. 3.43 Cassie, A. D. B., and Baxter, S., Wettabiity of porous surfaces, Trans. Faraday Soc., Vol. 40, p. 546, 1944. 3.44 Quéré, D., Wetting and roughness, Annu. Rev. Mater. Res. vol. 38, pp. 71–99, 2008. 3.45 Bormashenko, E., Progress in understanding wetting transitions on rough surfaces, Adv. Colloid Interface Sci., vol. 222, pp. 92–103, 2015. 3.46 Bico, J., Thiele, U., and Quéré, D., Wetting of textured surfaces. Colloids Surf A., vol. 206, pp.41–46, 2002. 3.47 Ishino C, Okumura K., and Quéré D., Wetting transitions on rough surfaces, Europhys Lett., vol. 68, pp. 419–425, 2004 3.48 Ishino, C., and Okumura, K., Wetting transitions on textured hydrophilic surfaces, Eur Phys. J. E., vol. 25, pp. 415–24, 2008. 3.49 Ishino, C., Reyssat, M., Reyssat, E., Okumura, K., and Quéré, D., Wicking within forests of micropillars, Europhys. Lett., vol. 79, p. 56005, 2007. 3.50 Courbin, L., Denieul, E., Dressaire, E., Ajdari, A., Roper, M., Ajdari, A., and Stone, H. A., Imbibition by polygonal spreading on microdecorated surfaces. Nat. Mater., vol. 6, pp. 661–64, 2007. 3.51 Joung, Y. S., and Buie, C. R. Scaling laws for drop impingement on porous films and papers, Phys. Rev. E., vol. 89, p. 013015, 2014. 3.52 Kim, B., Lee, H., Shin, S., Choi, G., and Cho, H. Interfacial wicking dynamics and its impact on critical heat flux of boiling heat transfer, Appl. Phys. Lett., vol. 105, p. 191601, 2014. 3.53 Wemp, C. K., and Carey, V. P., Water wicking and droplet spreading on randomly structured thin nanoporous layers, Langmuir, vol. 33, pp. 14513−14525, 2017. 3.54 Shibuichi, S., Onda, T., Satoh, N., and Tsujii, K., Super water-repellent surfaces resulting from fractal structure, J. Phys. Chem., vol. 100, pp. 19512–19517, 1996.

PROBLEMS 3.1 (a) Using Young’s equation (3.2) and a suitable predictive equation for the surface tension from Chapter 2, calculate σsv – σsl for liquid water and water vapor on a gold surface at 20°C if the equilibrium contact angle θ is 20°. (b) Assuming that σsv – σsl does not change significantly with temperature, use an appropriate relation from Chapter 2 for a σlv to determine and plot the variation of contact angle θ with temperature from 20°C to 100°C. How significantly does this analysis predict that the surface wettability will change over this temperature range? 3.2 Use Eq. (2.57) from Chapter 2 to predict the surface tension for an air-octane and an air-heptane interface at 20°C. Then use Eq. (3.25) to determine the interfacial tension for a water-octane and a waterheptane interface and compare the results with the following experimental data reported by Girifalco and Good [3.5]: σwater-octane = 0.0508 N/m, σwater-heptane = 0.0502 N/m. 3.3 For octane: Pc = 2490 kPa, Tc = 569 K and the molecular mass M is 114.2 kg/kmol. Assuming that the surface tension variation with temperature for pure water and pure octane are in good agreement with the reduced-property relation (1.41) for the surface tension, use Eq. (3.25) to derive a relation that predicts how the interfacial tension between liquid water and liquid octane varies with temperature. Calculate the interfacial tension value with this relation for a temperature of 20°C and compare the result to the following experimental value reported by Girifalco and Good [3.5]: σwater-octane = 0.0508 N/m. 3.4 At 20°C, to what degree will the following liquids wet polytetrafluoroethylene (Teflon): liquid mercury, ethylene glycol, acetone, R-134a, and methanol? Briefly explain your answer.

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3.5 The photo in Fig. 3.3a shows a water droplet at room temperature, which is about 25°C. The photo indicates that the equilibrium contact angle for water on polytetrafluoroethylene (Teflon) at this temperature is approximately 100°. Use an appropriate relation from Chapter 2 to evaluate the surface tension for water at 25°C. Show the resulting data as a plotted point on a plot of cos θ versus surface tension σlv like the plot shown in Fig. 3.6. Also show in your plot the best-fit curve shown in Fig. 3.6. Does this point agree with the best-fit curve from Fig. 3.6? 3.6 (a) For liquid carbon tetrachloride, use the liquid density quoted in Example 3.4 to determine the mean volume occupied by one molecule in the liquid, (b) Assuming that the mean volume occupied by a molecule in the liquid is a cube of side Lm, compute Lm. (c) Taking Lm as an estimate of the thickness of a monolayer film, for the circumstances in Example 3.4, use the relation for δ(z) to estimate the height at which the thickness of the film just equals Lm (i.e., the film is virtually a monolayer). (The molecular weight of carbon tetrachloride can be obtained in Appendix II.) 3.7 In Fig. 3.12, if the sawtooth pattern on the solid surface has an apex angle of 80° and the actual contact angle θ is 60°, what would be the advancing and receding contact angles? 3.8 At 20°C, the surface tension for water in contact with air is 0.0728 N/m. As indicated in Fig. 3.6, water is hydrophobic on polytetrafluoroethylene (Teflon) at this temperature. (a) Estimate how the wettability will change if the water is at 60°C. (b) At what temperature (if any) will liquid water fully wet polytetrafluoroethylene? 3.9 Consider a capillary tube of radius rt containing a liquid slug like that is shown in Fig. 3.14. Determine a relation for the pressure drop required to initiate downward motion of the plug in terms of the liquid and vapor densities, the length of the slug L, and the advancing and receding contact angles θa and θr, respectively. You may assume that the gas-liquid interfaces are both portions of a spherical surface. 3.10 (a) For water at 20°C and atmospheric pressure, if the advancing contact angle is 75° and the receding contact angle is 15°, estimate the length of a liquid slug that can be supported in air against gravity in a vertical tube with an inside diameter of 0.1 mm. Assume that the interface surfaces are portions of spherical surfaces, and note that for a spherical interface with radius ri and liquid contact angle θ, the spherical surface geometry dictates that the relationship between the radius of the interface ri and the radius of the tube rt is ri = rt / cos θ. (b) How does the supported length change if the temperature increases to 100°C, but the contact angles stay the same? 3.11 A slug of liquid water is supported against gravity inside a vertical tube with an inside diameter of 2 mm. For the slug, which is surrounded by water vapor at the same temperature, the advancing contact angle is 60° and the receding contact angle is 15°. Assuming that the contact angles do not change, determine and plot the variation of the length of the slug that can be supported with temperature for saturation temperatures between 100°C and the critical point. 3.12 A lens-shaped inclusion of fluid b can, under the right conditions, be trapped at the interface between two other fluids a and c, as shown in Fig. P3.1. Assuming the lens is radially symmetric, use force balance requirements at the location where the three fluids meet to derive equations for the angles γ1 and γ2. Also derive two inequality constraints on the three interfacial tension values that must be satisfied if an equilibrium configuration of this type is to exist.

FIGURE P3.1 3.13 Determine the capillary rise height on the polytetrafluoroethylene (Teflon) side wall of a container for the following liquids in air at 20°C: (a) water, (b) carbon tetrachloride, (c) R-134a, and (d) ethanol. Use Eq. (2.51) and contact angle values determined using Eq. (3.33) with information from Fig. 3.6 for your calculations. What would these rise heights be in a slowly spinning space station where the effective gravity is only 2% of earth-normal gravity?

Wetting Phenomena and Contact Angles

107

3.14 (a) Using Young’s equation (3.2) and a suitable predictive equation for the surface tension from Chapter 2, calculate σsv – σsl for liquid water and water vapor on a copper surface at 20°C if the equilibrium contact angle θ is 20°. (b) Estimate how much the contact angle will change if 2-propanol is added to the water so that the resulting solution has a 2-propanol mole fraction of 0.005. In doing so, assume that σsv – σsl does not change significantly with temperature or concentration, and use Eq. (2.62) from Chapter 2 to determine the mixture surface tension. 3.15 For liquid water at 20°C on a flat, smooth surface of a specific solid material, the intrinsic contact angle (Young’s angle) θ E is 45°. Use the Wenzel model to estimate the apparent contact angle if a droplet of water at 20°C is placed on a rough surface of this material with roughness ratio r defined by Eq. (3.72) equal to 5.0. 3.16 The intrinsic contact angle (Young’s angle) θ E for liquid water at 20°C on a flat, smooth surface of a specific solid material is 98°. Use the Wenzel model to estimate the apparent contact angle θ A if a droplet of water at 20°C is placed on a rough surface of this material with roughness ratio r (defined by Eq. (3.72)) equal to 5.0. 3.17 A nanostructured surface fabricated as an array of pillars, looks in cross section like the surface depicted in Figs. 3.21 and 3.23. The fraction of the nominal surface area corresponding to the top of the pillar, φ S , is 0.25. The intrinsic contact angle (Young’s angle) θ E for liquid water at 20°C on a flat, smooth surface of the solid material in the nanostructure is 20°. Estimate the achievable apparent contact angle θ A for a liquid water droplet at 20°C on this nanostructured surface. 3.18 A nanostructured surface fabricated as an array of pillars. In cross section the surface looks like that depicted in Figs. 3.21 and 3.23. The fraction of the nominal surface area corresponding to the top of the pillar, φ S , is 0.2. The intrinsic contact angle (Young’s angle) θ E for liquid water at 20°C on a flat, smooth surface of the solid material in the nanostructure is 70°. Estimate the achievable apparent contact angle θ A for a liquid water droplet at 20°C on this nanostructured surface. 3.19 Nanostructured surfaces can be fabricated as an array of pillars, looking in cross section like the surface depicted in Figs. 3.21 and 3.23. Two alternative designs for the fabricated surface morphology are such that the fraction of the nominal surface area corresponding to the top of the pillar, φ S , is either 0.2 or 0.35. In either case the roughness ratio r defined by Eq. (3.72) is 7.5. At 20°C, the intrinsic contact angle (Young’s angle θ E) for liquid water on a flat surface of the material used to fabricate the nanostructure is 95°. Which of the two φ S values will result in the most hydrophobic apparent contact angle on this nanostructured surface? 3.20 A nanostructured surface can be fabricated that is an array of pillars, looking in cross section like the surface depicted in Figs. 3.21 and 3.23. The fabricated surface morphology is such that the top area of pillars in contact with liquid is 25% of the nominal surface area, and the roughness ratio r defined by Eq. (3.72) is 1.9. At 20°C, the intrinsic contact angle (Young’s angle θ E) for liquid water on a flat surface of the material used to fabricate the nanostructure is 65°. Estimate the apparent contact angle achievable for water on this nanostructured surface.

4

Transport Effects and Dynamic Behavior at Interfaces

4.1  TRANSPORT BOUNDARY CONDITIONS In the preceding chapters, we have treated the interface between the two fluid phases to be a region over which the mean fluid properties vary continuously. Several important concepts and fundamental relations emerge from this view of the interface. At this juncture, we will depart from this viewpoint. Taking a more macroscopic perspective, we will now consider the interface to be a surface separating a liquid and its vapor. To facilitate solution of the governing equations for heat, mass, and momentum transfer in the two fluids on either side of the interface, we must specify appropriate boundary conditions at the liquid-vapor interface. As a prelude to considering specific phase-change phenomena in later chapters, here we will formulate, in general terms, the interface conditions that will serve as boundary conditions for the transport equations in the adjacent phases. At the interface, the system must satisfy the principles of conservation of mass, momentum, and energy. The transport of mass at the interface is schematically indicated in Fig. 4.1. In the liquid region, mass moves toward the interface with a velocity wl,n with respect to a stationary observer. However, the interface is also moving with a velocity dZi/dt, so that the rate of liquid mass flow toward the control volume moving with the interface is ρl(wl,n – dZi/dt). In a similar manner, it can be argued that the rate of vapor mass flow rate out of the control volume moving with the interface is ρv(wv,n – dZi/dt). Note that we have arbitrarily adopted a sign convention whereby coordinates and velocities are positive to the right in Fig. 4.1. The control volume in Fig. 4.1 is assumed to be so thin that there can be negligible accumulation of mass within it. Conservation of mass for this control volume then requires that



dZ  dZ   ρl  wl ,n − i  = ρv  wv ,n − i     dt dt 

(4.1)

which can be rearranged to obtain

ρl wl ,n − ρv wv ,n = (ρl − ρv )

dZi dt

(4.2)

Note in Eqs. (4.1) and (4.2) that the velocities wl,n, wv,n, and dZi/dt are all in the direction of the unit vector normal to the interface, n. The transport of momentum normal and tangential to the interface is depicted in Figs. 4.2a and 4.2b, respectively. Because of the motion of the interface, momentum in the direction of the unit normal vector n is convected into the control volume moving with the interface at the relative velocity of the fluid with respect to the interface. Including the effects of pressure and surface tension forces, the force and momentum balance normal to the interface requires dZ  dZ   1 1   Pl − Pv = σ  +  + ρv  wv ,n − i  wv ,n − ρl  wl ,n − i  wl ,n     r1 r2  dt dt 

(4.3) where r1 and r2, are the principal radii of curvature of the interface surface. Consistent with our sign convention, r1 and r2, are positive if measured on the left (liquid) side of the interface and negative if measured on the right (vapor) side. 109

110

Liquid-Vapor Phase-Change Phenomena

FIGURE 4.1  Mass fluxes across a liquid-vapor interface.

For some phase-change processes, the motion of the interface is limited by the heat transfer to or from it. We shall later see that, for these conditions, the interface motion is usually relatively slow, and the liquid and vapor momentum terms on the right side of Eq. (4.3) are very small. Equation (4.3) then reduces to



 1 1 pl − pv = σ  +   r1 r2 

(4.4)

which is just the Young-Laplace equation (2.29) derived in Chapter 2. Thus, even if the liquid and vapor are not motionless, the Young-Laplace equation may still hold if the fluid momentum terms in Eq. (4.3) are small. Note also that the pressure difference across the interface for these circumstances is often referred to as the capillary pressure difference. As indicated in Fig. 4.2b, momentum in the direction of the unit vector s (tangent to the interface surface) is also convected into the interface control volume at the velocity of the fluid relative to the interface. Including the effects of shear stresses at the interface and the variation of the surface tension along the interface, the force and momentum balance in the direction of the unit tangent vector s requires that



dZ  dZ   ∂σ   = ρl  wl ,n − i  wl ,s − ρv  wv ,n − i  wv ,s τ l ,s − τ v ,s −      ∂s  dt dt 

(4.5)

If, in addition, we assume that the velocity fields vary continuously in each fluid, and we impose a no-slip condition on the tangential velocity components, it follows that, at the interface,

wl ,s = wv ,s



(4.6)

Substituting Eq. (4.6) together with the conservation of mass relation (4.1) into Eq. (4.5), and assuming constant surface tension ( ∂σ / ∂s = 0 ), Eq. (4.5) reduces to

τ l ,s = τ v ,s



(4.7)

Transport Effects and Dynamic Behavior

FIGURE 4.2  Normal (a) and tangential (b) force-momentum interactions at a liquid-vapor interface.

111

112

Liquid-Vapor Phase-Change Phenomena

Thus, for constant surface tension, the tangential momentum balance at the interface reduces to simple equality of the shear stress in the two fluids on either side of the interface. If both the liquid and the vapor are Newtonian fluids, Eq. (4.7) may be written as



∂ wv , s  ∂w   ∂w  ∂w = µ v  v ,n + µ l  l ,n + l ,s      ∂s ∂s ∂ z  z = Zi ∂ z z = Zi

(4.8)

Note that, in general, Eqs. (4.5)–(4.8) are applicable to the momentum balance and no-slip condition in either of two orthogonal directions in the plane tangent to the interfacial surface at the point of interest. The mechanisms included in the balance of energy at the interface are indicated in Fig. 4.3. As in the transport of mass and momentum, energy is convected into the control volume moving with the interface at the velocity of the fluid relative to the interface. The convected energy content includes fluid enthalpy and kinetic energy associated with the velocity field in the fluid. Note also that cˆcv ,v and cˆcv ,l are the speed of the vapor and liquid, respectively, relative to the control volume. 2



dZ  cˆcv ,l =  wl ,n − i  + wl2,s1 + wl2,s2  dt 



dZ  cˆcv ,v =  wv ,n − i  + wv2,s1 + wv2,s2  dt 

(4.9a)

2

(4.9b)

The terms containing these speeds represent fluid kinetic energy contributions to the energy balance. Thermal energy can also be transported to or from the interface by Fourier conduction or, in the vapor, by radiative transport. The heat flux terms ql′′ and qv′′ include the transport due to these mechanisms. Conservation of energy thus requires



dZ   dZi   ˆ 1 2  1 2    qv′′ − ql′′= ρl  wl ,n − i   hˆl + cˆcv ,l  − ρ v  w v , n −   hv + cˆcv ,v       dt dt   2 2

FIGURE 4.3  Energy transport across a liquid-vapor interface.

(4.10)

113

Transport Effects and Dynamic Behavior

Substituting Eq. (4.1), Eq. (4.9) can be rearranged to obtain



dZ   cˆ 2 − cˆ 2   qv′′ − ql′′= ρl  wl ,n − i  hlv 1 + cv ,l cv ,v   dt   2hlv 

(4.11)

Note that, in obtaining Eq. (4.11), it has been assumed that local thermodynamic equilibrium exists at the interface so that ĥl – ĥv = hlv. For many systems of interest, the contribution of fluid kinetic energy to the energy balance is small, and the factor in square brackets in Eq. (4.11) is close to one. For the vapor and liquid phases of a pure substance, the assumption of thermodynamic equilibrium at the interface also implies a unique relation between the temperature and vapor pressure when momentum and interface curvature effects are sufficiently small that Pl = Pv, that is,

Tsat = Tsat ( Pv )

(4.12)

When curvature effects are small, Eq. (4.12) is a necessary boundary condition for thermal transport in the vapor and liquid regions. In many cases of practical interest, radiation effects in the vapor are small and the heat flux terms in Eq. (4.11) are due to Fourier conduction alone. Equation (4.11) may then be rewritten as



 cˆ 2 − cˆ 2  dZ  dT   dT  − kl  = ρl  wl ,n − i  hlv 1 + cv ,l cv ,v  kv      dz  z = Zi  dz  z = Zi dt   2hlv 

(4.13)

which can often be used as a thermal boundary condition at the interface.

4.2  KELVIN-HELMHOLTZ AND RAYLEIGH-TAYLOR INSTABILITIES Instability associated with liquid-vapor interfaces can have a strong impact on the heat and mass transfer at the interface during phase-change processes. Often these instabilities cause a change in the morphology of the two-phase system at a particular set of transition conditions. Altering the interface morphology invariably results in dramatic changes in heat and mass transport at the interface. In this and subsequent sections of this chapter, we will examine several different instability mechanisms that (we will later see) play important roles in some commonly encountered vaporization and condensation processes. We begin here by considering the circumstances shown in Fig. 4.4. A vapor or gas phase is presumed to overlay a heavier liquid in a gravitational field that exerts a downward body force on the fluids.

FIGURE 4.4  Perturbed interface considered in linear analysis of interface stability.

114

Liquid-Vapor Phase-Change Phenomena

Initially, the interface between the two phases is assumed to be a flat horizontal plane at z = 0. The liquid and vapor phases are moving with free-stream velocities ul and uv, respectively, in the x direction parallel to the undisturbed interface. The object of our analysis here is to determine the range of conditions for which the interface is stable with respect to an arbitrary perturbation of the interface. In general, determination of the time-dependent variation of the interface position in response to an arbitrary initial perturbation requires solution of a set of complex nonlinear equations with the associated boundary conditions. We can, however, simplify the analysis, while retaining most of the important physics, by adopting some appropriate idealizations. As a first means of simplification, we will restrict the analysis to arbitrary disturbances δ(x,t) that vary in the x direction but not in the y direction. More general disturbances that vary in both the x and y directions can be considered, at the expense of a more complicated analysis. However, because the method of analysis and the physical conclusions differ only slightly, we will consider simpler disturbances that vary only with x and t. In addition, we will also limit the analysis to circumstances in which viscous effects are negligible (i.e., the inviscid flow case). With these assumptions, the governing equations for the subsequent fluid motion in either of the two fluids are the following two-dimensional forms of the laminar transport equations, simplified for zero viscosity:



∂u ∂ w + =0 ∂ x ∂z

(4.14)



∂u ∂u  ∂P  ∂u ρ + u +w =− ∂x ∂x ∂z   ∂t

(4.15)



∂w ∂w  ∂P  ∂w ρ +u +w =− − ρg  ∂z ∂ t ∂ x ∂ z  

(4.16)

The velocities u and w and the pressure P are decomposed into base flow and perturbed components:

u = u + u ′,

w = w + w ′,

P = P + P′



(4.17)

Because the governing Eqs. (4.14)–(4.16) must hold for the unperturbed flow, the base flow quantities u, w, and P must satisfy these equations. Following the usual practice of linear stability analysis, Eqs. (4.17) for u, w, and P are substituted into Eqs. (4.14)–(4.16), products of perturbation (primed) quantities are neglected, and the corresponding equations for the base flow are subtracted. The resulting equations are further simplified using the facts that ∂u / ∂ x = ∂u / ∂ z = w = 0 to obtain the following equations for the perturbation quantities:



∂u ′ ∂ w ′ + =0 ∂x ∂z

(4.18)



∂ p′ ∂u ′   ∂u ′ ρ =− +u  ∂x ∂x   ∂t

(4.19)



∂ p′ ∂w′   ∂w′ ρ =− +u  ∂z ∂x   ∂t

(4.20)

115

Transport Effects and Dynamic Behavior

Differentiating Eq. (4.19) with respect to x and Eq. (4.20) with respect to z, adding the resulting equations together, and substituting the continuity equation yields the Laplace equation for the perturbation pressure field:



∂2 P ′ ∂2 P ′ + =0 ∂x 2 ∂z 2

(4.21)

Before the perturbation is applied, the base flow pressure variation in the two fluids is hydrostatic:

Pv = P0 − ρv gz

(4.22a)



Pl = P0 − ρl gz

(4.22b)



where P0 is the pressure at the undisturbed (flat) interface. Any arbitrary two-dimensional initial disturbance can be represented as a sum of Fourier series components with contributions over a broad range of wavelengths. Because a wide variety of initial perturbations δ(x,0) may occur, all possible Fourier wavelengths are expected to be present, and the stability of the interface will depend on whether the amplitude of one or more of the Fourier waves will grow and cause the system to become unstable. The above observations imply that the question of interface stability for these circumstances can be answered by considering the Fourier component waves themselves as initial perturbations,

δ( x ,0) = Aeiαx

(4.23)

where the disturbance wavelength is equal to 2π/α, and it is understood that only the real part of this relation is physically significant. For the system considered here, it is plausible to expect that the spatial variations of δ, u′, w′, and P′ with x will be similar to that for the original perturbation δ(x, 0), and that their temporal variation will be oscillatory. We therefore postulate the following functional forms for the subsequent position of the interface δ and the perturbation quantities w′ and P′:

δ( x , t ) = Aeiαx +βt



(4.24)



w ′( x , z , t ) = wˆ ( z )eiαx +βt



P ′( x , z , t ) = Pˆ ( z )eiαx +βt (4.26)

(4.25)

Substituting Eq. (4.26) for P′ into Eq. (4.21) yields the following equation for Pˆ :



d 2 Pˆ = α 2 Pˆ dz 2

(4.27)

Solutions for Pˆ in the liquid and vapor regions, which satisfy Eq. (4.27) and the condition that P′ → 0 far from the interface, are

Pˆv = av e −αz

(4.28)



Pˆl = al e −αz

(4.29)

116

Liquid-Vapor Phase-Change Phenomena

Substituting Eqs. (4.25) and (4.26) into Eq. (4.20), we obtain



ˆ −1  dP  wˆ ( z ) = − [ ρ(β + iαu ) ]    dz 

(4.30)

Using Eqs. (4.28) and (4.29) to evaluate dPˆ / dz in this equation yields the following relations for wˆ in the liquid and vapor regions

wˆ v ( z ) = α   av [ ρv (β + iαuv ) ] e −αz



(4.31)



wˆ l ( z ) = −α   al [ ρl (β + iαul ) ] eαz



(4.32)

−1

−1

Substituting Eq. (4.25) into the continuity Equation (4.18), the resulting differential equation for u′ can be integrated. Using the condition that u′ → 0 far from the interface, the following relation for u′ is obtained:



i  dwˆ  iαx +βt e α  dz 

u′ =

(4.33)

Direct substitution of the above relation for u′ with Eqs. (4.25) and (4.26) for w′ and P′ verifies that Eq. (4.19) is identically satisfied. Having established solutions for u′, w′, and P′ that satisfy the governing equations, we now apply appropriate boundary conditions at the interface. Note first that at a given x location, Eq. (4.24) indicates that the interface position oscillates sinusoidally with time. This produces a contribution to w′ near the interface equal to ∂δ/∂t. In addition, for the fluid immediately adjacent to the interface to be convected in the x direction with a velocity u and yet follow the interface contour, at any instant of time w′ must have a component equal to u ∂δ / ∂ x. To satisfy both of these requirements, w′ at the interface must be equal to



wz′→ 0 =

∂δ ∂δ +u ∂t ∂x

(4.34)

Imposing this condition on both the vapor and liquid sides of the interface, substitution of Eqs. (4.24) and (4.25) with either Eq. (4.31) or (4.32) yields the following two relations:

α   av [ ρv (β + iαuv ) ] = βA + iαuv A



(4.35)



−α   al [ ρl (β + iαul ) ] = βA + iαul A



(4.36)

−1

−1

These relations can be solved for av and al in terms of A and the other parameters:



av =

ρv (β + iαuv )2 A α

(4.37)

ρl (β + iαul )2 A α

(4.38)

al = −

117

Transport Effects and Dynamic Behavior

For inviscid flow, the momentum balance tangential to the interface and continuity of the tangential velocity component at the interface cannot be imposed. However, the force and momentum balance in the direction normal to the interface must be enforced even for inviscid flow. Because the perturbation velocities normal to the interface are small, their contributions to the force and momentum balance normal to the interface are negligible. As described in Section 4.1, for such circumstances the force balance normal to the interface simply becomes the Young-Laplace equation:  1 1 Pl − Pv = σ  +   r1 r2 



(4.39)

For the system considered here,

(

)

∂2 δ / ∂ x 2 1 =− 3/ 2 , r1 1 + (∂δ / ∂ x )2 



1 =0 r2

(4.40)

and Pl = Pl + Pl ′,      Pv = Pv + Pv′



(4.41)

The analysis may be continued by substituting the relations (4.40) and (4.41) into Eq. (4.39), using Eqs. (4.22a) and (4.22b) with z = δ to evaluate Pl and Pv, and using Eqs. (4.24), (4.26), (4.28), and (4.29) to evaluate Pl ′, Pv′, and δ. Neglecting terms of order δ2 compared to l, we obtain the following relation from Eq. (4.39): al − av = (ρl − ρv ) g + σα 2  A



(4.42)

Substituting Eqs. (4.37) and (4.38) for al and av into Eq. (4.42), A cancels out of the equation. After doing so, this equation can then be rearranged to get the following relation for β:

{α ρ ρ (u − u ) β=± 2



l

v

l

v

2

}

−  σα 3 + ( ρl − ρv ) gα  ( ρl + ρv ) ρl + ρv

1/ 2

−

iα ( ρl ul + ρv uv ) ρl + ρv

(4.43)

The relation (4.43) for β above actually provides a great deal of information about the type of system considered here. The fact that β always has an imaginary part as long as ul or uv is greater than zero implies that any horizontal velocity component in either fluid will produce waves at the interface. The amplitude of the perturbation will grow with time only if β has a positive real part. This can happen only if the sum of the two terms inside the braces is greater than zero. The second term inside the braces, being negative, diminishes the sum. Thus surface tension and gravity tend to stabilize the interface. Setting the term inside the braces to zero and rearranging, the condition for an unstable interface becomes 1/ 2



  σα + ( ρl − ρv ) g / α  ( ρl + ρv )   ul − uv >      ρ ρ l v  

(4.44)

118

Liquid-Vapor Phase-Change Phenomena

The term on the right side of relation Eq. (4.44) above varies with the wave number of the disturbance, α. It can be shown that the right side of this inequality has a minimum value at a critical wavenumber αc equal to 1/ 2



(ρ − ρv ) g  α c =  l  σ 

(4.45)



As a result, if ul − uv is greater than the right side of inequality (4.44) evaluated at α = αc, there will be some range of disturbances that are unstable. Thus, if disturbances of all wavelengths are present in the system, the interface will be unstable when

ul − uv > uc

(4.46)

where uc is the critical velocity equal to the right side of relation (4.44) evaluated at αc: 1/ 2



 2(ρ − ρv )  uc =  l  ρl  

1/ 4

 σ (ρl − ρv ) g    ρ2v  

(4.47)

This type of interface instability is referred to as Kelvin-Helmholtz instability or simply Helmholtz instability. If a specific disturbance wavelength λ is imposed on the system, the inequality (4.44) indicates that the interface will be unstable for 1/ 2



 [2πσ / λ + (ρl − ρv ) gλ / 2π](ρl + ρv )  ul − uv >   ρl ρv  

(4.48)

Note also that when the interface is unstable, the oscillatory component of the disturbance is solely a result of the second term of Eq. (4.43), and the wave propagates in the x direction with a speed equal to ( ρl ul + ρv uv ) / ( ρl + ρv ) which is between the fluid velocities ul and uv. In addition to the system shown in Fig. 4.4, Eq. (4.43) also provides information on the stability of other similar systems. For example, if we change the sign on g, we get the dispersion relation for waves on an interface between a lower vapor region and an upper liquid region that have some relative velocity parallel to the interface. Gravity then acts to further destabilize the interface, with the interface being unstable for  [σα − ( ρl − ρv ) g / α]( ρl + ρv )  ul − uv >   ρl ρv   1/ 2



(4.49)

Note that the right side of the above inequality now does not exhibit a minimum value as α varies. Hence if disturbances of all wavelengths are present, there will always be some long-wavelength (small-α) disturbances that will amplify. If g is set to zero in Eq. (4.43), we get the dispersion relation for a vertical interface, which implies that it is unstable for 1/ 2



 σα(ρl + ρv )  ul − uv >   ρl ρv  

(4.50)

This would also apply to an interface in a spacecraft at zero gravity. Again, it is clear that if disturbances of all wavelengths are present, there will be some disturbances at small α and long wavelength that will amplify and cause the interface to be unstable.

119

Transport Effects and Dynamic Behavior

If we set ul = uv = 0 and change the sign of g in Eq. (4.44), we obtain the following condition for interface instability of a motionless liquid overlaying a motionless vapor region:  ( ρl − ρv ) g  α < αc =   σ  

1/ 2



(4.51)

This condition is referred to as Rayleigh-Taylor instability. The critical wavelength λ c = 2π / α c corresponding to the critical wavenumber is 1/ 2



  σ λ c = 2π   g ρ − ρ v)  ( l

(4.52)

For a steam-water system at 100°C, λc is about 1.6 cm. Only perturbations having wavelengths greater than λc will grow, leading to instability of the interface. The stability of the interface can be assured by limiting the lateral extent of the interface. If the lateral dimensions of the interface are smaller than λc, no disturbance of wavelength greater than λc can arise, and the interface will be stable. The value of β for Rayleigh-Taylor instability is obtained by setting ul = uv = 0 and changing the sign of g in Eq. (4.43). This yields the relation 1/ 2



 (ρl − ρv ) gα − σα 3   β = ±  ρl + ρv  

(4.53a)

If ρv is neglected compared to ρl, this relation can be written approximately as 1/ 2



 (ρ − ρv ) gα σα 3  β = ± l −  ρl ρl  

(4.53b)

From the form of the relation for the disturbance (see Eq. (4.24)) it is clear that the positive β value is equal to the fractional growth rate of the disturbance with time (dδ/dt)/δ. The variation of β with α given by Eq. (4.53b) indicates that there is a specific value of α where β is a maximum. Differentiating Eq. (4.53b) with respect to α and setting dβ/dα = 0, it can be shown that the maximum value of β and the corresponding α are given by



α max =

{

(ρl − ρv ) g 3σ

}

1/ 2

(4.54)

1/ 4



 4(ρ − ρv )3 g 3  β max =  l  2  27σρl 

(4.55)

The disturbance wavelength corresponding to αmax is often referred to as the most dangerous wavelength λD, given by 1/ 2



  3σ λ D = 2π    (ρl − ρv ) g 

= 3λ c

(4.56)

120

Liquid-Vapor Phase-Change Phenomena

Because they grow most rapidly, disturbances with wavelengths near λD are expected to dominate the early stages of the instability where the linear theory applies. In real systems, the dominant disturbance wavelength observed experimentally is often close to λD, even though the linear analysis is not applicable to the larger interface deformations required to detect the instability. Example 4.1 Determine the critical velocity for a stratified horizontal flow of saturated steam above liquid water at atmospheric pressure. What happens if the pressure is increased to 8.59 MPa? For saturated steam and water at atmospheric pressure, ρl = 958 kg/m3, ρv = 0.598 kg/m3, σ = 0.05878 N/m. Using Eq. (4.47), 1/ 2



 2(ρ − ρv )  uc =  l  ρl   1/ 2



 2(958 − 0.598)  uc =   958 

1/ 4

 σ(ρl − ρv )g    ρv2  

1/ 4

 0.05878(958 − 0.598)9.8    (0.598)2  

= 7.793 m/s

At 8.59 MPa, ρl = 712.5 kg/m3, ρv = 46.2 kg/m3, σ = 0.01439 N/m, and 1/ 2



 2(712.5 − 46.2)  uc =   712.5 

1/ 4

 0.01439(712.5 − 46.2)9.8    (46.2)2  

= 0.627 m/s

The density difference and surface tension that act to restore a perturbed interface to its initial flat configuration are weaker at high pressure, and the system becomes unstable at a lower relative velocity.

As noted at the beginning of this section, the linear stability analysis for two-dimensional waves presented here can be extended to three-dimensional waveforms. The procedure used to solve the three-dimensional problem is basically the same as for the two-dimensional one. Although the mathematical details are a bit more complex, solutions can be obtained for more general threedimensional waveforms. In a very early study, however, Squire [4.1] showed for Rayleigh-Taylor instability that three-dimensional waves are generally more stable than two-dimensional waves for these circumstances. Hence two-dimensional waves are more likely to be the mechanisms to initiate instability in systems of this type. In addition, the results of a more recent study by Sernas [4.2] indicate that the most dangerous three-dimensional Rayleigh-Taylor waveform has peaks and valleys in a square grid pattern with the side of the square unit cell equal to the two-dimensional wavelength λD. Thus the two-dimensional wave analysis presented above provides a physically realistic indication of the stability criteria and the wavelength of the disturbance that is most rapidly amplified. Strictly speaking, the analysis presented above applies only when the liquid and vapor depths are infinite. However, in later sections we will be interested in film boiling processes in which a vapor layer of finite depth exists under a liquid pool of virtually infinite depth. In a discussion of the paper by Hsieh [4.3], Dhir and Lienhard showed that for Rayleigh-Taylor instability, there was no effect of finite vapor depth on the most-dangerous wavelength and its effect on the corresponding amplification rate was small. Hence the results for infinite layer depths may be used for finite vapor layers with little loss in accuracy.

121

Transport Effects and Dynamic Behavior

Although the analysis presented here ignores viscous effects, the conclusions based on the results of the analysis are usually valid for most liquid-vapor systems of interest in engineering systems. A further discussion of Kelvin-Helmholtz and Rayleigh-Taylor instabilities may be found in reference [4.4]. As we shall see, these types of instability also play a role in some of the vaporization and condensation processes we shall consider in later sections. Example 4.2 Determine the most dangerous wavelength for saturated liquid water in contact with its vapor at (a) 20°C and (b) 300°C, and (c) saturated liquid R-134a in contact with its vapor at 345 K. a. For saturated liquid water at 20°C, ρl = 998 kg/m3, ρv = 0.0173 kg/m3, σ = 0.0728 N/m. It follows, using Eq. (4.56), that 1/ 2



  3σ λ D = 2π    (ρl − ρv )g 

1/ 2

3(0.0728)   = 2π    (998 − 0.0173)9.8 

= 0.0297 m

b. Similarly, for water at 300°C, ρl = 712 kg/m3, ρv = 46.2 kg/m3, σ = 0.0144 N/m, and 1/ 2



 3(0.0144)  λ D = 2π    (712 − 46.2)9.8 

= 0.0162 m

c. For saturated R-134a at 345 K, ρl = 983.8 kg/m3, ρv = 122.4 kg/m3, σ = 0.0024 N/m, and 1/ 2



3(0.0024)   λ D = 2π    (983.8 − 122.4)9.8 

= 0.0058 m

Note that at low to moderate saturation pressures (away from the critical point), ρl – ρv, changes slowly as the pressure and temperature decrease, while σ increases about linearly with decreasing temperature (see Chapter 2), having the stronger effect on λD.

4.3  INTERFACE STABILITY OF LIQUID JETS Careful observation of a jet of water issuing from a faucet or water hose reveals that although the air-water interface is cylindrical near the nozzle, farther downstream the jet invariably breaks up into droplets. The photograph in Fig. 4.5 illustrates this phenomenon. The tendency of the liquid jet to break up into droplets is actually advantageous in some applications. When a jet of liquid fuel is injected into a combustion engine, the breakup of the jet increases the interfacial area, which, in turn, leads to more rapid vaporization of the fuel. A similar enhancement of vaporization results in the case of cooling ponds, where water sprayed into the air is cooled by partial evaporation. On the other hand, the breakup phenomenon is a disadvantage for sprinkler systems, because loss of water by evaporation and the drift of small droplets is undesirable. In addition to these direct applications, the instability of liquid jets also provides a basis for understanding the breakdown of inverted annular flow into dispersed flow during convective film boiling in tubes. In this section, the stability of a thin liquid jet will be analyzed to try to determine the conditions that lead to breakup of the jet. We wish to consider a cylindrical jet of radius R0 issuing from

122

Liquid-Vapor Phase-Change Phenomena

FIGURE 4.5  Breakup of a round water jet in air. The smallest divisions on the scale shown are 1 mm (From D. F. Rutland and G. J. Jameson, J. Fluid Mech., vol. 40, pp. 267–271, 1971, reproduced with permission, copyright © 1971, Cambridge University Press).

a nozzle at a uniform velocity ul, as shown in Fig. 4.6a. To simplify the stability analysis, inviscid flow is assumed, gravity body forces are neglected, and the analysis will employ a coordinate system moving with the liquid at uniform velocity ul. Hence in this coordinate system the unperturbed liquid velocity is zero. With the idealizations noted above, the problem becomes that of analyzing the stability of the motionless liquid cylinder shown schematically in Fig. 4.6b. We specifically wish to know whether an arbitrarily small perturbation of the interface will tend to grow in amplitude. As in the stability analyses considered in the previous sections, we will consider sinusoidal disturbances of the interface because random disturbances in real systems are expected to be composed of Fourier components having wavelengths over virtually the entire spectrum.

FIGURE 4.6  Model system considered in analysis of the stability of a liquid jet.

123

Transport Effects and Dynamic Behavior

For inviscid incompressible flow in the liquid jet, the governing continuity and momentum equations in cylindrical coordinates are



l  ∂(rur )  ∂uz 1  ∂uθ  +  +  =0 r  ∂r  ∂ z r  ∂θ 

(4.57)

(4.58)



2  ∂u ∂P  ∂u  u   ∂u  u  ∂u  ρl  r + ur  r  + θ  r  + uz  r  − θ  = −       ∂ ∂θ ∂r t r r z r ∂ ∂  





 ∂u  ∂u  u ρl  θ + ur  θ  + θ  ∂r  r  ∂t

1  ∂P   ∂uθ  ur uθ   ∂uθ  =−   +  + uz     ∂z ∂θ r  r  ∂θ 

(4.59)

 ∂u ∂P  ∂u    ∂u  u  ∂u  ρl  z + ur  z  + θ  z  + uz  z   = −       ∂z  r ∂θ ∂r ∂z  ∂t

(4.60)

Following the usual procedure for linear stability analysis, we postulate that the pressure and velocities are the sum of steady plus perturbed quantities



ur = ur + ur′ ,

uθ = uθ + uθ′ ,

uz = uz + uz′ ,

P = P + P′



(4.61)

Substituting these relations into Eqs. (4.57)–(4.60), ignoring products of primed quantities, and using the fact that the base flow velocities are zero (ur = uθ = uz = 0), the following equations for the perturbation quantities are obtained:



1  ∂(rur′ )  ∂uz′ 1  ∂uθ′  + +   =0 r  ∂r  ∂ z r  ∂θ 

ρl

∂ur′ ∂P ′ =− ∂t ∂r

(4.63)

∂uθ′ 1  ∂P ′  =−   r  ∂θ  ∂t

(4.64)

∂uz′ ∂P ′ =− ∂t ∂z

(4.65)

ρl



(4.62)

ρl

Applying the operators (1/r)∂[r( )]/∂r, (1/r)∂/∂θ, and ∂/∂z to Eqs. (4.63), (4.64), and (4.65), respectively, adding the resulting equations, and using the continuity equation (4.62) yields



1  ∂(r ∂ P ′ / ∂r )  1  ∂2 P ′  ∂2 P ′  + r 2  ∂θ2  + ∂ z 2 = 0 ∂r r 

(4.66)

where the left side of Eq. (4.66) is the expansion of ∇ 2 P ′ in cylindrical coordinates. For an initial interface perturbation of the form

R = R0 + δ 0 ei ( mθ+αz )

(4.67)

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Liquid-Vapor Phase-Change Phenomena

we postulate that the subsequent motion of the interface will result in variations of P′ and ur′ of the form

P ′(r , z , θ, t ) = Pˆ (r ) A(t )ei ( mθ+αz )

(4.68)



ur′ (r , z , θ, t ) = uˆr (r ) A(t )ei ( mθ+αz )

(4.69)

In these equations, m can be any integer to allow for perturbations that vary sinusoidally around the circumference of the jet. Note that for m = 0 the disturbance perturbation is radially symmetric, with the radius of the jet varying sinusoidally with z, as schematically shown in Fig. 4.6b. Values of m ≠ 0 correspond to more complicated, nonaxisymmetric perturbations. Substituting the relation (4.68) for P′ into Eq. (4.66) yields the following equation for Pˆ (r ):



r2

d 2 Pˆ dPˆ +r − (m 2 + α 2 r 2 ) Pˆ = 0 2 dr dr

(4.70)

For a specified value of m, the two independent solutions of this second-order linear differential equation are Im(α r) and Km(α r), where Im and Km are the modified Bessel functions (of order m) of the first and second kinds, respectively. In general, the solution may be a linear combination of Im(α r) and Km(α r):

Pˆ = p1 I m (α r ) + p2 K m (α r )

(4.71)



We note, however, that Bessel functions of the second kind are unbounded as r → 0. To ensure that the pressure field is finite throughout the liquid domain, we therefore take p2 = 0.

Pˆ = p1 I m (α   r )

(4.72)

Substituting the relations (4.68) and (4.69) for P′ and ur′ into the r-direction momentum equation (4.63), we obtain



dA / dt (dPˆ / dr ) =− =β A ρl uˆr (r )

(4.73)

Because the left-most term above is only a function of time and the center term is only a function of r, they can be equal for all t and r only if they both equal a constant, which we denote as β. Setting the left-most term equal to β and solving for A yields

A(t ) = a0 eβt

(4.74)

Similarly, using Eqs. (4.72) and (4.73) together with the identity



dI m ( x ) = I m +1 ( x ) + ( m / x ) I m ( x ) dx

(4.75)

(see reference [4.5]) we obtain the following relation for uˆr (r ) :



 p α m uˆr (r ) = −  1   I m +1 (α   r ) + I m (α, r )  α  r   ρl β  

(4.76)

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Transport Effects and Dynamic Behavior

To obtain the dispersion relation between β and α, we now impose two boundary conditions. First, we require that the rate of change of the radius location of the interface must equal the radial velocity at the interface: dR = ur′ ( R0 ) dt



(4.77)

The variation of R with time is postulated to be of the form



δ = δ 0 ei ( mθ+αz ) B(t )

R = R0 + δ,

(4.78)



where B(0) = l. Substituting the relations (4.69), (4.76), and (4.78) into (4.77) and solving for B(t) yields



 uˆ ( R )a  B(t ) =  r 0 0  eβt  βδ 0 

(4.79)

We further note that B(0) = 1 requires that a0 =

βδ 0 uˆr ( R0 )

(4.80)

The second boundary condition imposed here is the force-momentum balance normal to the interface, which reduces to the Young-Laplace equation 1



1

( Pl − Pv )r = R0 = σ  +   r1 r2 

(4.81)

Using the fact that δ/R λ c = 2π   R0

(4.87)

In addition, by differentiating Eq. (4.85), with respect to α for m = 0 and setting the derivative equal to zero, it is possible to obtain an equation that can be solved for the value of α = αmax that results in the maximum value of β = β max. The resulting value of α = αmax is the wavenumber of the disturbance that will be most rapidly amplified. Because of the presence of the Bessel functions in Eq. (4.85), explicit relations for αmax and β max cannot be obtained. However, the equation for αmax can be solved numerically to yield α max =

0.70 R0

(4.88) 1/ 2



 σ  β max = 0.34  3   ρl R0 

(4.89)

These results were first obtained by Rayleigh [4.6]. Equation (4.88) implies that the fastest-growing disturbance has a wavelength λmax equal to 9.0R0. It is interesting to note that λmax depends only on the radius of the jet and not on the surface tension, in spite of the fact that surface tension is the primary cause of the instability. On the other hand, the growth rate of the most rapidly amplified disturbance, which is proportional to eβmax t , increases as σ increases. If the Fourier component of a small perturbation of the interface at the jet nozzle with wavelength λmax has amplitude δ0, the above analysis indicates that this disturbance component will grow most rapidly and may ultimately be responsible for breakup of the jet. The distance downstream of the nozzle where breakup occurs, z B, for a jet injected with uniform velocity ul can be estimated to be

z B = ul t B

(4.90)



where tB is the time required for the disturbance amplitude to grow from δ0 to R0. If the disturbance grows as indicated by Eqs. (4.78)–(4.80) with β = β max, then tB is given by



 1   R0  ln tB =   β max   δ 0 

(4.91)

Because of the logarithmic variation of tB with R0/δ0, tB is not strongly dependent on R0/δ0 at large values of R0/δ0. Hence if we take a reasonable estimate of R0/δ0, we can estimate tB and z B for a given system (see Example 4.3).

127

Transport Effects and Dynamic Behavior

The above analysis specifically considers the temporal amplification of spatial wave distortions of the interface. Keller et al. [4.7] have also examined the spatial instability of liquid jets. Their analysis postulates wave distortions of the interface that oscillate with a fixed real temporal frequency, and it examines the spatial growth rates of waves as they propagate downstream from the nozzle. The analysis of Keller et al. [4.7] indicates that the maximum growth rate obtained from the temporal analysis above is valid for high-velocity jets when ρR ul  l 0   σ 



1/ 2

(4.92)

1

Even when this condition is satisfied, the spatial stability analysis indicates that there are longwavelength modes that amplify more rapidly than the axisymmetric (m = 0) mode that amplifies most rapidly in the temporal analysis. Apparently, however, these long-wavelength modes cannot occur in a short jet, and the axisymmetric m = 0 mode of the temporal analysis dominates. In many real circumstances, the jet velocity and liquid density are high enough and the surface tension is low enough that the condition (4.92) is satisfied, and the results of the temporal analysis are valid. When the jet velocity is so low that ul (ρlR0/σ)1/2 is small compared to 1, the spatial instability analysis of Keller et al. [4.7] predicts results that are very different from those obtained in the temporal stability analysis presented above. Example 4.3 A jet of liquid water at 20°C is injected into air at the same temperature. If the jet diameter is 0.5 mm and the liquid velocity is 3.0 m/s, estimate the distance downstream of the nozzle where the jet is likely to break up into droplets. For water at 20°C, ρl = 998 kg/m3 and σ = 0.0728 N/m. Using Eq. (4.89), we have



 σ  βmax = 0.34   ρl R03 

1/ 2

 0.0728  = 0.34   998(0.0005)3 

1/ 2

= 259.7 s −1

If we take δ0/R0 = 10 –3, Eq. (4.91) implies that



 1  tB =  ln (103 ) = 0.0266 s  259.7 

and

zB = ul tB = (3.0)(0.0266) = 0.080 m = 8.0 cm.

If δ0/R0 = 10 –4, a similar set of calculations yields zB = 10.6 cm. This suggests that the jet will break up somewhere around 9 cm downstream of the nozzle. For these circumstances, ul(ρlR0/σ)1/2 is equal to 7.9, implying that the growth rate obtained from the temporal analysis should be applicable to this system.

4.4  WAVES ON LIQUID FILMS We will see in later sections that engineering applications often involve vaporization or condensation at the interface of a thin liquid film flowing over a solid surface. These films are typically thin compared to their lateral extent, but not so thin that disjoining pressure effects are important. The flow in films of this type is usually dominated by viscous, gravity, and surface tension effects.

128

Liquid-Vapor Phase-Change Phenomena

In many instances, circumstances may give rise to interfacial waves on the thin liquid film. The presence of such waves may strongly affect the rate of vaporization or condensation at the surface of the film, because the waves increase the interfacial area and can enhance convective transport near the interface. In Section 4.2, the Kelvin-Helmholtz stability of a liquid-vapor interface between two semiinfinite liquid and vapor regions was analyzed. It was shown in that analysis that any velocity component parallel to the interface in either fluid will produce waves at the interface. This suggests that any motion of an overlying vapor region parallel to the interface of a thin liquid film on a horizontal solid surface may produce small amplitude waves. The analysis presented in Section 4.2 further suggests that if the horizontal vapor velocity exceeds a critical value, the amplitude of the waves may become very large, with a correspondingly large effect on interfacial heat and mass transfer. Another circumstance of interest is that in which a thin liquid film surrounded by motionless vapor flows down a vertical or inclined surface due to gravity. This situation occurs during film condensation on a solid object surrounded by vapor, and in falling-film evaporators in which liquid is intentionally allowed to flow downward over heated surfaces in the evaporator to facilitate the vaporization process. In film flows of this type, three different flow regimes have been observed experimentally. Generally, these regimes correspond to different ranges of the film Reynolds number, ReF , defined as

Re F = wm δ /vl

(4.93)

where wm is the mean liquid downstream velocity over the cross section of the film and δ is the film thickness. For values of ReF below about 30, the film flow is laminar and, if the inclination of the surface is constant, the film thickness is uniform. At Reynolds numbers between about 30 and 1500, a wavy flow regime is observed in which a wavy motion is superimposed on the forward motion of the film. At a value of ReF of about 1500, the film undergoes a transition from laminar to turbulent flow. To further investigate the existence of interfacial waves in film flows of this type, we will specifically consider the laminar downward-flowing thin liquid film shown schematically in Fig. 4.7. Because the film thickness is small and we wish to consider waves with wavelengths long compared to the film thickness (small wavenumbers), it is expected that velocity derivatives across the film (in the y direction) will be large compared to those in the z direction. Based on the usual boundary-layer

FIGURE 4.7  Model system considered in the analysis of waves on a liquid film flowing down an inclined surface.

129

Transport Effects and Dynamic Behavior

approximations, we therefore assume that the fluid motion in the film is governed by the continuity equation and the boundary-layer form of the w momentum equation:



∂v ∂w + =0 ∂ y ∂z

(4.94)



∂2 w ∂w ∂w ∂w 1  ∂P  + g sin θ + νl +v +w =−   ∂ y2 ∂t ∂y ∂z ρl  ∂ z 

(4.95)

In the absence of any wave motion, steady, downward, fully developed flow exists in the liquid film, implying that the time and z derivatives and v are all zero. Noting that ∂ P / ∂ z is just equal to the vapor hydrostatic gradient, the governing equation for w is then

(ρl − ρv ) g  sin θ + v

l

ρl



∂2 w =0 ∂ y2

(4.96)

Using the boundary conditions

w = 0  at   y = 0,

∂w = 0 at  y = δ ∂y

(4.97)

integration of Eq. (4.96) yields the following expression for the w velocity profile: w=

(ρl − ρv ) g sin θ(2 yδ − y2 ) 2µ l



(4.98)

Defining the mean film velocity wm as δ



1 (ρ − ρv ) gδ 2 sin θ wm = w( y)dy = l 3µ l δ

∫ 0

(4.99)

we can rewrite Eq. (4.98) as w=

3wm  2 y y 2  − 2  δ δ 2 

(4.100)

We now will examine the unsteady flow that occurs when waves are present at the interface. For these circumstances, we retain the full time-dependent boundary-layer Equation (4.95) for the w momentum balance. Assuming that the pressure variation in the liquid and vapor regions varies only hydrostatically, and that the Young-Laplace equation (4.4) holds at the interface, the pressure variation in the liquid film is given by



Pl = Pv + (ρl − ρv ) g  cos θ (δ − y) − σ

∂2 δ ∂z 2

(4.101)

130

Liquid-Vapor Phase-Change Phenomena

Note that we are now allowing for the variation of δ with time and z location. Differentiating this equation with respect to z, we obtain



∂ Pl ∂ Pv ∂δ ∂3 δ = + (ρl − ρv ) g  cos θ −σ 3 ∂z ∂z ∂z ∂z

(4.102)

Using the fact that ∂Pv /∂z is just equal to the hydrostatic gradient in the vapor,



∂ Pv = ρv g sin θ ∂z

(4.103)

Equations (4.102) and (4.103) can be substituted into Eq. (4.95) to obtain



∂2 w ∂w ∂w ∂ w ( ρl − ρv ) g  ∂δ  σ ∂3 δ + vl +v +w =  sin θ − cos θ  + 3 ρl ∂ z ∂ y2 ρl ∂z ∂t ∂y ∂z

(4.104)

The analysis is continued by integrating both sides of this equation across the film from y = 0 to δ and using the continuity relation (4.94), integration by parts, Leibniz’s rule, and the relation



w y→δ =

3wm 2

to evaluate terms in the equation. After doing so, the w momentum relation becomes



2 3  3wm  ∂δ +  3wm  ∂δ =  (ρl − ρv ) g   sin θ − cos θ ∂δ  +  σ  ∂ δ − 3wm vl      3      ρl δ2 ∂z 2δ ∂t  2δ  ∂ z   ρl  ∂ z 

(4.105)

By considering a control volume that extends across the film, it can be shown that conservation of mass requires that



∂δ ∂( wm δ) =− ∂t ∂z

(4.106)

Substituting Eq. (4.99) for wm and carrying out the differentiation, Eq. (4.106) becomes



∂δ ∂δ = −3wm ∂t ∂z

(4.107)

Substituting Eq. (4.107) into Eq. (4.105) reduces that equation to the form



3  3wm2  ∂δ  ( ρl − ρv ) g   ∂δ   σ  ∂ δ 3w v − sin θ   −   3 + m2 l     cos θ  δ  ∂ z =    ρl  ∂ z ρl ∂z δ  

(4.108)

Because wm is a function of δ, we are left with a pair of nonlinear partial differential equations, (4.107) and (4.108), which must be solved together to determine δ(z, t). To avoid the difficulties of solving this nonlinear system, we postulate that the interface location δ is equal to its base flow

131

Transport Effects and Dynamic Behavior

value (with no waves present) δ0 plus a small fluctuating component δ′. In addition, it is assumed that the perturbation of the interface is accompanied by perturbations of the local w velocity field, but that the mean mass flow rate, and hence the mean velocity in the film wm, remain equal to the base flow values with no waves present.

δ = δ0 + δ′

wm = wm 0

(4.109)



We further note that from the definition of wm in Eq. (4.99), the following relation must hold:



 ( ρl − ρv ) g  3wm 0 vl   sin θ   =    δ 20 ρl  

(4.110)

Substituting the relations (4.109) into (4.108), using Eq. (4.110) and the fact that ∂δ0/∂z = ∂wm0/∂z = 0, and neglecting products of primed (fluctuating) quantities, we obtain



 3wm2 0  ∂δ ′  ( ρl − ρv ) g  ∂δ ′  σ  ∂3 δ ′ = θ +    cos   δ  ∂ z  ∂ z  ρl  ∂ z 3 ρl 0  

(4.111)

The fluctuating component δ′ in the linearized equation above is postulated to be a two-dimensional sinusoidal wave of the form

δ ′ = δ* eiα ( z − ct )

(4.112)

where α is the wavenumber and c is the wave velocity. Substituting Eqs. (4.109) and (4.112) into Eq. (4.107), we find that c = 3wm0. Thus the wave velocity is three times the mean base flow velocity in this model analysis. Substitution of Eq. (4.112) into Eq. (4.111) yields the following relation for the wavenumber α:



ρ  (ρ − ρv )2 g 2 δ 30 sin 2 θ (ρl − ρv ) g cos θ  α 2 =    l     l −   σ  3µ l2 ρl 

(4.113)

If the right side of Eq. (4.113) is not greater than zero, α is a pure imaginary number and the solution for δ′ is not a sinusoidal wave. It appears, therefore, that waves exist at the interface only if the right side of Eq. (4.113) is greater than zero. Rearranging this equation, it can be shown that this condition is met, and hence waves exist, for 1/3



  3µ12 cos θ δ0 >   2   (ρl − ρv )ρl g sin θ 

(4.114)

Alternatively, if we combine Eqs. (4.110) and (4.114), we can write this criterion in terms of the film Reynolds number ReF = ρlwm0 δ0/μl simply as

Re F > cot θ



(4.115)

In light of the above results, we interpret the limiting case ReF = cot θ as the boundary between a stable film with no waves (for ReF < cot θ) and conditions for which wavy flow may occur (ReF > cot θ).

132

Liquid-Vapor Phase-Change Phenomena

It is interesting to note that the criterion for the appearance of wavy flow obtained from the approximate analysis above is consistent with the results of Benjamin’s [4.8] more thorough analysis of wave formation in film flow down an inclined plane. The full stability analysis presented by Benjamin [4.8] predicts that very long waves (small α) become unstable (and hence amplify) for 5 Re F >   cot θ  6



(4.116)

In the case of a vertical surface, θ = 90° and both Eqs. (4.115) and (4.116) reduce to ReF > 0. This implies that conditions favor the presence of waves at the interface for any finite Reynolds number, no matter how small. At first glance, this appears to contradict the fact that waves are not observed experimentally at very low film Reynolds numbers. This apparent contradiction is resolved by considering the amplification rate of wave amplitudes. For a vertical surface, Benjamin [4.8] also showed that for long-wavelength surface waves having the form given by Eq. (4.112), there is a specific wavelength that amplifies more rapidly than all others, and that the wavenumber and the imaginary component of the wave speed for this wavelength are: α max



−1/ 2   2/3 1/6  σ  =  1.12δ  vl g     Re5/6 F  ρl    −1 0

−1/ 2   11/6  σ Im(c)max = 0.336 wm 0  vl2/3 g1/6     Re F  ρl   

(4.117)

(4.118)

The amplitude of this wavelength must increase according to δ = exp {α max Im(c)max t } δ t=0



(4.119)

If we consider the amplification over the time interval required for the wave to travel 100 times the film thickness downstream, then t = 100δ0/3wm0 and the amplification factor given by Eq. (4.119) becomes δ

δ t=0

−1     σ  = exp  12.5vl4 /3 g1/3    Re8/3 F   ρl     

(4.120)

Generally, the term in square brackets in Eq. (4.120) is small. For water at 19°C under earth-normal gravity, it is 3.8 × 10 –3. Equation (4.120) is plotted for this value in Fig. 4.8. For these conditions, the amplification of the most rapidly amplified wavelength is extremely small for film Reynolds numbers below 6 and extremely large for ReF > 13. Hence, although wave disturbances are unstable for all ReF > 0, the growth rate is so low for ReF < 6 that waves may not be observed over the finite length of any real falling film for these conditions. Furthermore, the very rapid increase in amplification rate over a narrow range of ReF makes it likely that wavy flow will be observed at Reynolds numbers just above this range. This implies that although there is no critical Reynolds number in the usual sense (dividing wave-amplifying and wavedamping conditions), there is, nevertheless, a quasi-critical value of ReF below which wavy flow is unlikely and above which it is very likely to occur. This is consistent with the experimentally observed behavior of low-Reynolds-number liquid films, as described at the beginning of this section.

133

Transport Effects and Dynamic Behavior

FIGURE 4.8  Wave amplitude growth predicted for a water film flowing down a vertical surface.

Some additional aspects of wave motion on flat liquid-vapor interfaces in general, and thin liquid films in particular, are discussed in reference [4.9]. The effects of interfacial waves on evaporation and condensation processes involving liquid films will be explored in detail in later sections of this text. Example 4.4 Estimate the quasi-critical value of ReF = ρlwm0 δ0/μl for a film flow of saturated liquid water at 300°C. For saturated water at 300°C, ρl = 712.5 kg/m3, vl = 0.127 × 10 –6 m2/s, σ = 0.01439 N/m, and Eq. (4.120) requires that −1    σ  δ   = exp  12.5v l4 / 3 g1/ 3    ReF8/ 3  ρ | δ |t = 0   l    



−1    0.01439   8/ 3  = exp  12.5(0.127 × 10 −6 )4 / 3(9.8)1/ 3   Re   712.5   F     

The resulting variation of the disturbance amplification with ReF is summarized in the following table: ReF

|δ|/|δ|t=0

5 10 20 30 35 40

1.06 1.48 12.1 1570 6.63 × 104 7.66 × 106

Hence ReF = 35 is about the quasi-critical value at which disturbances are expected to be amplified enough to be detectable.

134

Liquid-Vapor Phase-Change Phenomena

4.5 INTERFACIAL RESISTANCE IN VAPORIZATION AND CONDENSATION PROCESSES The extremely high heat transfer coefficients typically associated with vaporization and condensation processes make it possible to transfer thermal energy at high heat flux levels with relatively low driving temperature differences. The ability to handle high heat flux levels is particularly important in applications such as electronics cooling and power system thermal control. A question of central interest in such applications is “What is the highest heat flux possible in a given vaporization or condensation process?” To explore this question, we must consider the liquid-vapor interface at the molecular level. Because motion of vapor molecules in the vicinity of the interface plays a central role in heat flux limitations during vaporization and condensation processes, we will first examine some relevant aspects of kinetic theory of gases. One of the most useful results of the classical development of the kinetic theory of gases is the Maxwell velocity distribution:



dnuvw  m  =  2πk B T  n

3/ 2

e − m (u

2

+ v 2 + w 2 )/ 2 k B T

(4.121)

dudvdw

The left side of this equation is the fraction of the total number of molecules n with Cartesian velocities, u, v, and w in the ranges u to u + du, v to v + dv, and w to w + dw, respectively. In Eq. (4.121), m is the mass of one molecule, T is the absolute temperature, and kB is the Boltzmann constant

kB = R / N A

(4.122)

where NA is Avogadro’s number. Fundamental aspects of kinetic theory of gases and derivation of the Maxwell distribution are described in Appendix I. More detailed treatments of these subjects are also presented in most texts that treat statistical thermodynamics (see, e.g., references [4.10–4.12]). Using the velocity distribution given by Eq. (4.121), it is possible to derive the following relation for the fraction of the molecules with speed in the range c to c + dc regardless of direction (see Appendix I):



dnc  m  = 4π   2πk B T  n

3/ 2

c 2 e − mc

2

/2 k B T

(4.123)

dc

As discussed in Appendix I, relations for the most probable speed cmp, the mean speed 〈c〉, and the root-mean-square speed 〈c2〉1/2 can be obtained from this speed distribution. We now wish to determine the flux of molecules through an arbitrary plane for a gas that has a Maxwell velocity distribution. To do so, we consider the motion of molecules in the box shown in Fig. 4.9. We specifically wish to determine the average number of molecules within the box that strike the shaded surface S x* per unit area and per unit time. A molecule with a given x component velocity u and any v and w must lie within a distance uΔt of the surface at the beginning of the time interval in order to pass through it. The fraction of molecules having an x component of velocity between some value u and u + du and having any v and w is found by integrating Eq. (4.121) over all possible v and w values:



dnu = n

∫∫ v w

dnuvw n

(4.124)

135

Transport Effects and Dynamic Behavior

FIGURE 4.9  Geometry used in analysis of molecular flux through a surface.

Using Eq. (4.121) to evaluate the right side of Eq. (4.124), we obtain



dnu  m  =  2πk B T  n

1/ 2

e − mu

2

/ 2 kBT

(4.125)

du

The above observations indicate that only molecules within a fraction uΔt/L x of the total box volume will pass through the surface S x* in the time interval Δt. If the total number of molecules in the box is n, the number of molecules with an x component of velocity equal to u is dn u, as given by Eq. (4.125). The number of molecules with velocity u in the x direction that will pass through the surface S x* per unit area per unit time dju is therefore equal to



 1  1   u∆t  dju =  dnu     Lx   Lz L y   ∆t 

(4.126)

Substituting Eq. (4.125) for dnu and integrating the resulting expression over all possible values of u (0 to ∞), we obtain the following relation for the total rate at which molecules pass through the surface S x* per unit area, jn:



1  n  8k T  jn =      B   4   V   πm 

1/ 2

 M  =  2πR 

1/ 2

P mT 1/ 2



(4.127)

where M is the molecular weight of the vapor. It is noteworthy that the flux of molecules jn through the surface S x* is independent of the original box dimensions. The flux jn must therefore be the same for any planar surface in the gas. We can now use some of the above results from the kinetic theory of gases to interpret the motion of vapor molecules near a liquid-vapor interface. We now consider a plane surface in the

136

Liquid-Vapor Phase-Change Phenomena

FIGURE 4.10  Mass fluxes at a liquid-vapor interface.

vapor phase but immediately adjacent to the interface, as shown in Fig. 4.10. Even if no (net) vaporization or condensation occurs at the interface, a dynamic equilibrium is established in which molecules from the vapor phase that hit the interface and become part of the liquid phase are balanced (on the average) by an equal number of molecules that escape the liquid into the vapor region. When condensation occurs, the flux of vapor molecules joining the liquid must exceed the flux of liquid molecules escaping into the vapor phase. When vaporization occurs, the opposite must be true. In considering evaporation or condensation from the interface, we will, at least initially, admit the possibility that the pressures and temperatures in the two phases may be different, as indicated in Fig. 4.10. For the system shown in this figure, the net mass flux across surface Si* immediately adjacent to the interface must be equal to the difference between the mass fluxes ml′′ and mv′′ passing through surface Si* in opposite directions just inside the vapor region: mnet ′′ = mv′′ − ml′′



(4.128)

To facilitate the evaluation of ml′′ and mv′′ in Eq. (4.128), it will be assumed here that the flux of molecules ml′′ is characterized by Tl and Pl, whereas mv′′ is characterized by Tv and Pv. With this assumption, it would appear that ml′′ and mv′′ can be obtained by multiplying the relation (4.127) by the mass of one molecule and evaluating the resulting expression at the appropriate T and P values. However, this approach is inadequate in two respects. First, Eq. (4.127) predicts the flux of molecules in a stationary gas, whereas in the system shown in Fig. 4.10, the vapor must have a bulk velocity w = w0 in the z direction as a result of the phase change at the interface. The bulk velocity is toward the interface (w0 > 0) for condensation and away from the interface (w0 < 0) for vaporization. By extending the analysis presented above, it can be shown from kinetic theory (see Schrage [4.13]) that when the gas moves normal to a planar surface at a speed w0, the flux of molecules through the plane in the direction of the motion is jnw +

 M  = Γ (a)  2πRT 

1/ 2

 P   m

(4.129a)



and the flux of molecules in the direction opposite to that of the bulk motion is



 M  jnw − = Γ ( − a )   2πRT 

1/ 2

 P   m



(4.129b)

137

Transport Effects and Dynamic Behavior

where a=

w0

( 2 RT / M )

(4.130)

1/ 2



and the factor Γ, which corrects for the effects of bulk gas motion, is given by one of the following relations:

Γ (a) = exp (a 2 ) + aπ1/ 2 [1 + erf(a)]

(4.131a)



Γ (− a) = exp (a 2 ) − aπ1/ 2 [1 − erf(a)]

(4.131b)

The variations of Γ(a) and Γ(–a) with a are shown in Fig. 4.11. Because the vapor in Fig. 4.10 is expected to be in bulk motion relative to the surface Si* Eq. (4.129a) or (4.129b) must be used to compute mv′′. Following Schrage [4.13], we assume that the surface Si* is an infinitesimally small distance from the phase interface, so that there is no bulk motion effect on molecules emerging from the liquid and passing through the surface Si* . The mass flux ml′′ is therefore calculated using Eq. (4.127). In addition to the effect of bulk vapor motion, it is clear that only a fraction σˆ e of the molecules crossing the surface Si* in the negative z direction is actually due to vaporization. The remaining fraction 1 – σˆ e is due to the “reflection” of vapor molecules that strike the interface but do not condense. The fraction of the molecules crossing the surface Si* in the positive z direction that condense and are not “reflected” is designated as σˆ c. If no phase change occurs at the interface, equilibrium requires that σˆ e = σˆ c = σˆ . Usually σˆ e and σˆ c are assumed to be equal even for the dynamic case when phase change occurs at the interface, although the validity of this assumption is suspect. As a result, σˆ is often referred to as a vaporization, evaporation, condensation, or accommodation coefficient. The term accommodation coefficient has been perhaps most commonly used for σˆ in recent years and will be adopted here. Allowing for the bulk motion of the vapor and the role of the accommodation coefficient as described above, the portion of the mass flux mv′′ that actually enters the liquid phase upon striking the liquid, mvc ′′ , is given by



 mσˆ jnw + mvc ′′ =   mσˆ jnw −

for condensation for vaporization

(4.132)

FIGURE 4.11  The Γ(a) and Γ(–a) functions developed in Schrage’s [4.13] analysis of molecular fluxes at a liquid-vapor interface.

138

Liquid-Vapor Phase-Change Phenomena

and the portion of ml′′ that is due to molecules that actually emerged from the liquid phase, mle′′, is given by

mle′′ = mσˆ jn

(4.133)

The net mass flux to or from the interface as a result of the phase change, ml′′, is just equal to the difference between mvc ′′ and mle′′:

ml′′= mvc ′′ − mle′′

(4.134)

Substituting Eqs. (4.132) and (4.133) into (4.134), evaluating jnw+ or jnw– at Pv and Tv and jn at Pl and Tl, and rearranging yields



 M  ml′′=   2πR 

1/ 2

 Γσˆ Pv σˆ Pl   T 1/ 2 − T 1/ 2  v l

(4.135)

Because the heat flux to the interface qi′′ must equal the net mass flux multiplied by the latent heat, a relation for qi′′ can be obtained directly from Eq. (4.135)



 M  qi′′= σˆ hlv   2πR 

1/ 2

 ΓPv Pl   T 1/ 2 − T 1/ 2  v l

(4.136)

It should be noted that because the rate of phase change is dictated by the heat flux, the resulting bulk velocity of the vapor is given by w0 =

qi′′ ρv hlv

(4.137)

which implies that



q ′′  2 RTv  a= i  ρv hlv  M 

−1/ 2

(4.138)

Because Γ is a function of a, Eq. (4.136) is actually an implicit relation for qi′′ . If we assume that Pv and Pl are the saturation pressures corresponding to Tv and Tl, Eq. (4.136) provides an implicit relation for the dependence of qi′′ on the temperatures Tv and Tl:



 M  qi′′= σˆ hlv   2πR 

1/ 2

 ΓPsat (Tv ) Psat (Tl )   T 1/ 2 − T 1/ 2  v l

(4.139)

For vaporization and condensation processes at high temperatures, a is often small. For example, for saturated pure water at 100°C condensing or vaporizing at a heat flux of 100 kW/m 2 , a is 1.3 × 10 –4. On the other hand, the values of a for the vaporization or condensation of cryogenic liquids at low absolute temperatures may be much higher. In the limit of small a, Eq. (4.131) is well approximated by

Γ = 1 + aπ1/ 2

(4.140)

139

Transport Effects and Dynamic Behavior

Substituting this relation into Eq. (4.136) and using Eq. (4.138) with the ideal gas relation to evaluate ρv, after some manipulation, the following relation is obtained for qi′′



 2σˆ   M  qi′′=    hlv   2 − σˆ   2πR 

1/ 2

 Pv Pl   T 1/ 2 − T 1/ 2  v l

(4.141)

The associated relation for the interfacial mass flux (obtained by dividing the above equation by hlv),



 2σˆ   M  mi′′=     2 − σˆ   2πR 

1/ 2

 Pv Pl   T 1/ 2 − T 1/ 2  v l

(4.142)

was suggested by Silver and Simpson [4.14] in their study of condensing steam. In the Soviet literature, Eq. (4.142) has been referred to as the Kucherov-Rikenglaz equation [4.15]. An alternate form of the relation for qi′′ can be obtained if the relations

∆Pvl = Pv − Pl and ∆Tvl = Tv − Tl

(4.143)

are substituted into Eq. (4.136) to eliminate Pl and Tl:



 M  qi′′= σˆ hlv   2πR 

1/ 2

Pv − ∆Pvl   ΓPv  T 1/ 2 − (T − ∆T )1/ 2  v v vl  

(4.144)

To develop a more useful form of this equation, we expand the second term in square brackets in terms of ΔTvl/Tv, using the assumption that ΔPvl/Pv 0

(5.39) (5.40)



 ∂µ   >0  ∂ N  V ,T



These are both necessary and sufficient conditions for stability of a pure fluid system. If these conditions are met, the system is said to be intrinsically stable. Although we are limiting our considerations to a pure (single component) system, necessary conditions for intrinsic stability of mixtures can also be derived (see Carey [5.1] and Modell and Reid [5.2]). Extension of this type of analysis to mixtures is straightforward, but the mathematics becomes increasingly complicated as the number of components increases.

159

Phase Stability and Homogeneous Nucleation

The following relation from thermodynamics theory [5.1]:



V 2  ∂P   ∂µ    = − 2   ∂ N V ,T N ∂V  N ,T

(5.41)

can be used to modify the second requirement, Eq. (5.40), into a more useful form. Since V and N are positive definite, the condition (5.40) is equivalent to (∂P/∂V)N,T < 0. Note that if we pick N so the system contains a unit mass, then V is equal to the specific volume v and (∂P/∂V)N,T = (∂P/∂v)T. We can therefore state the necessary and sufficient conditions for intrinsic stability of a pure system as



cˆv > 0  ∂P  re, ΔG decreases as r increases and the bubble is most likely to spontaneously grow. Thus, if density fluctuations in a metastable superheated liquid produce an embryo bubble of radius r < re, the bubble is likely to collapse. Conversely, if the embryo bubble has a radius greater than re, it is expected to spontaneously grow, resulting in homogeneous nucleation of the vapor phase in the system. In the above analysis, the assumption of mechanical equilibrium embodied in the use of the Young-Laplace equation is reasonable if the bubble radius readjusts rapidly to mechanical force variations. However, if the mechanical forces change too rapidly for the liquid to readjust, then mechanical equilibrium may not be maintained. For such conditions, a better assumption would be that the values of chemical potential in the two phases are equal. Use of this idealization, along with the others noted above, results in an expansion for ΔG that is identical to Eq. (5.80) except that the term in square brackets is simply equal to 3. Zeldovich [5.3] and Kagen [5.4] have shown that this modified form of the expansion for ΔG is more appropriate for cavitation processes in which a superheated liquid condition is created by a sudden (isothermal) decrease in pressure. Regardless of whether chemical or mechanical equilibrium is assumed, the stability characteristics of the vapor bubble are qualitatively the same, and quantitatively only slightly different.

171

Phase Stability and Homogeneous Nucleation

Example 5.3 Estimate the number of molecules needed to make up a bubble of critical size for superheated water at atmospheric pressure and 250°C. The radius of a bubble of critical size is just that for a bubble in equilibrium, given by Eq. (5.55): re =



2σ Psat (Tl )exp {v l[Pl − Psat (Tl )] / RTl } − Pl

For water at 250°C, Psat = 3978 kPa, vl = 0.001251 m3/kg, σ = 0.0262 N/m, and R = R /M = 8314.4/18 = 462 J/kg. Substituting into the above equation: re =

2(0.0262) 3,978,000 exp {0.001251[101,000 − 3,978,000] / 462(250 + 273)} − 101,000

= 1.38 × 10 −8  m

The pressure inside the bubble is given by



Pve = Pl +

2σ 2(0.0262) = 101,000 + = 3,900,000 Pa re 1.38 × 10 −8

and the specific volume of the gas is found from the ideal gas law:



vv =

RT 462(250 + 273) = = 0.0620 m3 /kg Pv 3,900,000

The number of molecules in the bubble, ne, is found as ne =



VbNA vv M

where NA and M are Avogadro’s number and the molecular mass, respectively, and the bubble volume Vb is equal to 4πre3 / 3 = 1.1× 10 −23m3. Substituting, it is found that



ne =

1.1× 10 −23(6.02 × 10 26 ) = 5940 molecules 0.0620(18)

Thus, to form an embryo bubble of critical size requires about 6000 molecules – a relatively small number.

5.3  THE KINETIC LIMIT OF SUPERHEAT The results of the analysis presented in the preceding section indicate that the likelihood that homogeneous nucleation will occur depends on the kinetics of the vapor embryo formation process. A vapor embryo of n vapor molecules is continuously gaining molecules due to evaporation and losing molecules due to condensation at the bubble interface. The difference between the rates of these two processes dictates whether the embryo will grow or decrease in size. We consider a superheated liquid held at constant Tl and Pl in which heterophase fluctuations give rise to vapor embryos of various sizes. Let us first consider an idealized model of such a system in which an equilibrium distribution of embryo sizes is established. It is plausible to expect that the

172

Liquid-Vapor Phase-Change Phenomena

probability of finding an embryo of a particular size in such a system will vary with the change in free energy associated with the formation of the embryo. We postulate, therefore, that the number distribution of embryos for such an equilibrium system would be of the form



 ∆G (r )  N n = ρN ,l exp −   k B Tl 

(5.82)

where Nn is the number of embryos of n molecules at equilibrium per unit volume, ρN,l is the number of liquid molecules per unit volume, and ΔG(r) is the free energy change for formation of the embryo of radius r discussed in the previous section. Note that r is related to the number of molecules n in the embryo by



4 πr 3 / 3 nM = nm = vv NA

(5.83)

where m is the mass of one molecule, M is the molecular mass (kg/kmol), and NA is Avogadro’s number (6.02 × 1026 molecules/kmol). The number density of liquid molecules is given by ρN ,l = NA / v l M . For an embryo of size n, jne is the number of molecules evaporating from the interface per unit time, per unit area, and jnc is the number of molecules condensing at the interface per unit time, per unit area. An equilibrium number distribution can be established only if the following condition is satisfied:

N n An jne = N n +1 An +1 j( n +1)c

(5.84)

In Eq. (5.84), An and An+1 are the interfacial areas of embryos of size n and n + 1, respectively. This equality enforces the condition that the rate at which embryos of size n are converted to size n + 1 by evaporation is equal to the rate at which embryos of size n + 1 are converted to size n by condensation. As a result, the equilibrium number of embryos in each size category Nn is maintained, and there is no net flux of bubbles through the range of sizes (i.e., there is no steady stream of embryos that grow progressively in size). For a real superheated liquid, the system is not expected to be in equilibrium, and Eq. (5.84) is not necessarily satisfied. Denoting the difference between the right and left sides of Eq. (5.84) as Jn, and the size distribution of embryos as N * for the nonequilibrium circumstances, we can write that

Jn = N n* An jne − N n*+1 An +1 j( n +1)c

(5.85)

Note that Jn is the excess number of embryos of size n that (due to evaporation) pass to size n + 1 over the number of size n + 1 that (due to condensation) pass to size n. Thus Jn is the net flux of the number of embryos in size space from n to n + 1. Using Eq. (5.84) to eliminate An+1, Eq. (5.85) can be rearranged to obtain



 N *   N *   Jn = N n An jne  n  −  n +1     N n   N n +1  

(5.86)

For an unsteady size distribution, the rate of change of the number of embryos in each category ∂ N n* / ∂t is given by



∂ N n* = J n −1 − J n ∂t

(5.87)

173

Phase Stability and Homogeneous Nucleation

To facilitate further analysis of the kinetics of the embryo formation process, we will treat n as a continuous variable, whereupon Eqs. (5.86) and (5.87) can be rewritten as



 ∂[ N * (n) / N (n)]  J (n) = − N (n) A(n) je (n)   ∂n 



∂ N * (n) ∂ J (n) =− ∂t ∂n

(5.88) (5.89)

To simplify the analysis somewhat, we allow a steady flow of embryos through size space, but we insist that the number distribution of embryos is steady, so that ∂ N n* / ∂t = 0. From Eq. (5.89), it follows that



∂ J (n) =0 ∂n

(5.90)

J (n) = constant = J

(5.91)

which implies that

We know from the thermodynamic analysis presented above that embryos larger in size than the critical radius will spontaneously grow and allow the superheated liquid system to undergo a transition to an equilibrium state. Consequently, if we consider such a superheated system prior to such an event, there can be no embryos present with r > re. We therefore assume that

N * → 0 at  n = n0 ≅ ne

(5.92)

where ne is the number of molecules in an embryo of radius r = re for the actual superheated liquid. The qualitative variation of N(n) is indicated in Fig. 5.10. Note that because ΔG(r) has a maximum at r = re, there is a corresponding minimum in N at ne. The expected variation of N * with n is also qualitatively shown in Fig. 5.10. N * goes to zero at n = n0, based on the reasoning described above. The number densities of very small embryos (small n) are so large that departure from equilibrium will likely have little effect on their numbers. We therefore postulate that



N* → 1 as  n → 0 N

as graphically suggested in Fig. 5.10.

FIGURE 5.10  Qualitative variations of N(n) and N *(n).

(5.93)

174

Liquid-Vapor Phase-Change Phenomena

Integrating Eq. (5.88) and using the condition (5.92), the following relation is obtained:



N * (n) =J N (n)

n = n0

∫ [N (n) A(n) j (n)] e

−1

(5.94)

dn

n=n

which can be rearranged to yield



n = n0   N * (n)    −1 J=  [ N (n) A(n) je (n)] dn    N (n)   n = n 

−1

∫

(5.95)

Because N(n) possess a sharp minimum at n = ne, the integrand in Eq. (5.95) is significantly greater than zero only in the vicinity of n = ne. We therefore take je(ne) to be equal to its value at n = ne, and we use Eq. (4.129a), developed using kinetic theory of gases in Chapter 4, to evaluate je(ne): je (n) ≅ je (ne ) =

Pve (2πmk B Tl )1/ 2

(5.96)

Note that in obtaining the above approximation for je(ne), we have taken Γ(a) = 1 and rearranged Eq. (4.129a) slightly, writing it in terms of the Boltzmann constant k B = R / N A. In addition, because the integrand vanishes rapidly as |n – ne| becomes large, we can take the limits of integration to be 0 and ∞ with no significant loss of accuracy. Also, since J is a constant (independent of the value of n), we can elect to evaluate the right side of Eq. (5.95) at small n where N * (n) / N (n) → 1. Evaluating the right side of Eq. (5.95) as described above, we obtain the following simpler relation for J:



∞  Pve     −1 J= [ N ( n ) A ( n )] dn  1/ 2   (2 π mk T ) B l    0 

∫

−1

(5.97)

The integral on the right side of Eq. (5.97) can be converted to an integral with respect to r using Eq. (5.82) for N and the relations A = 4 πr 2

dn =

4 2  Pve   P  2 − l  dr πr   RTl m   3 Pve 

(5.98) (5.99)

where the second relation (5.99) is obtained by differentiating Eq. (5.83) and using the ideal gas law and the Young-Laplace equation to evaluate the dependence of vv on r. After converting the integral in this manner, Eq. (5.97) becomes



 3ρN ,l   k B Tl  J=    2 − Pl / Pve   2πm 

1/ 2

−1

∞  ∆G (r )     exp   dr   k B Tl    0

∫

(5.100)

175

Phase Stability and Homogeneous Nucleation

Because of the maximum in ΔG at r = re, the integrand takes on its greatest value at r = re and rapidly becomes smaller as |r – re| increases. Hence the value of the integral is dictated almost entirely by the variation of the integrand near r = re. The expansion (5.80) for ΔG around r = re is therefore adequate for evaluating the integral. We proceed by substituting the expansion for ΔG into 1/ 2 Eq. (5.100), changing the integration variable to ξ = [4 πσ (2 − Pl / Pve ) / 3k B Tl ] (r − re ), and taking the lower integration limit in terms of |ξ| to be –∞ (because the integrand goes rapidly to zero as |ξ| becomes large). The integral can then be evaluated in closed form. The resulting equation for J is 1/ 2



6σ   J = ρ N ,l   (2 / ) π m − P P l ve  

 −4 πre2 σ  exp   3k B Tl 

(5.101)

Using the Young-Laplace equation to evaluate re and neglecting Pl / Pve compared to 2, the relation for J can be written as 3σ  J = ρN ,l   πm 



1/ 2

  −16πσ 3 exp  2   3k B Tl ( Pve − Pl ) 

(5.102)

From Eq. (5.54), we can write Pve = ηPsat (Tl )

(5.103)

 v [ P − Psat (Tl )]  η = exp  l l  RTl  

(5.104)

where



Substituting Eq. (5.103) into Eq. (5.102), we obtain



3σ  J = ρN ,l   πm 

1/ 2

  −16πσ 3 exp  2  3 [ ( ) ] k T η P T − P sat l l  B l 

(5.105)

The above equation for J is basically the same as that obtained by Kagen [5.4] and by Katz and Blander [5.5]. Note that J represents the rate at which embryo bubbles grow from n to n + 1 molecules per unit volume of liquid. Because in this steady-state solution J is constant for all n, it is also equal to the rate at which bubbles of critical size are generated. Hence, as J increases, the probability that a bubble will exceed critical size and grow spontaneously also increases. Because most of the temperature-dependent quantities appear in the exponential term of Eq. (5.105), a slight change in Tl can have a very strong effect on the rate of embryo formation. For typical circumstances, a change of 1°C can change J by as much as three or four orders of magnitude. Hence, although the embryo bubble formation rate increases continuously with temperature, the rate of nucleation changes so rapidly with temperature that we expect that there will exist a narrow range of temperature below which homogeneous nucleation does not occur, and above which it occurs almost immediately. The median temperature of this range is usually referred to as the kinetic limit of superheat. Equation (5.105) can be written in laboratory units as



 ρ2 σ  J = 1.44 × 10 40  l 3  M 

1/ 2

 −1.213 × 10 24 σ 3  exp  2   Tl [ ηPsat (Tl ) − Pl ] 

(5.106)

176

Liquid-Vapor Phase-Change Phenomena

where J is in m −3s −1, Psat and Pl are in Pa, T l is in K, σ is in N/m, ρl is in kg/m 3, and M is in kg/kmol. For a specific pure substance, if the physical properties are known and a suitable threshold value of J is chosen, then Eq. (5.106) can be solved iteratively to determine the limiting superheat temperature Tl = TSL. The above analysis of the kinetics of the bubble formation process can also be made to apply to a cavitation process. It can be shown that doing so amounts to using the form of the expansion for ΔG that is applicable to such processes (see Section 5.2) and taking the terms in right most round brackets in Eq. (5.99) to be equal to 3. The resulting expression for J is identical to Eq. (5.105), except that the factor of 3 inside the round square root brackets is replaced by 2. Example 5.4 Estimate the kinetic limit of superheat for water at 1000 kPa. At 1000 kPa, Tsat = 180°C + 273 = 453 K. The kinetic limit of superheat is defined by Eqs. (5.104) and (5.106) with the threshold value of J = 1012 m−3s−1. 1/ 2



 ρ2σ  J = 1.44 × 10 40  l 3  M 



[P − Psat (Tl )]  η = exp  l   ρl RTl 

 −1.213 × 10 24 σ 3  exp  2   Tl[ ηPsat (Tl ) − Pl ] 

Note that varying Tl causes ρl , Psat(Tl ) and σ to vary in the above equations. Using values of ρl, Psat(Tl ), and σ from the saturation table for water at various Tl values, the resulting values of log10 J are: Tl (°C)

Psat(Tl) (kPa)

ρl (kg/m3)

σ (N/m)

log10 J

300 310 305

8592 9869 9231

712 691 702

0.0144 0.0121 0.0133

–12.0 17.5 5.24

Interpolating between 305°C and 310°C to determine the value of Tl that corresponds to log10 J = 12 yields TSL = (Tl )SL = 308°C.

5.4  COMPARISON OF THEORETICAL AND MEASURED SUPERHEAT LIMITS We now have two theoretical means of predicting the limiting superheat for metastable liquid: the thermodynamic or intrinsic limit conditions derived in Section 5.2, and the kinetic limits specified by the analysis in Section 5.3. We will now examine how each of these theoretical limits compares to experimental measurements. The reduced forms of the van der Waals and Redlich-Kwong equations of state described in Chapter 1 can both be represented by the relation Pr =

γTr ar − vr − br1 Trλ vr ( vr + br 2 )

(5.107)

where

Pr = P / Pc ,

Tr = T / Tc ,

vr = vˆ / vˆc



(5.108)

In Eq. (5.107), ( γ ,  λ,  ar ,  br1 ,  br 2 ) = ( 83 , 0, 3, 13 , 0 ) corresponds to the van der Waals equation, and (γ, λ, ar, br1, br2) = (3, 12 , 3.84732, 0.25992, 0.25992) yields the Redlich-Kwong equation.

177

Phase Stability and Homogeneous Nucleation

In terms of the reduced properties, the condition defining the spinodal limit (∂ P / ∂ vˆ)T = 0 can be written as



 ∂ Pr   ∂ v  = 0 r Tr

(5.109)

Differentiating Eq. (5.107) for Pr with respect to vr, and setting the result equal to zero, the following relation is obtained for the spinodal limits: 1/( λ+1)



 a (2 v + b )( v − b )2  Tr ,s =  r r 2 r 2 r 2 r1  γ vr ( vr + br 2 )  

(5.110)

Once values of the constants are specified, Tr can be computed for a given value of vr using Eq. (5.110), and then Eq. (5.107) can be used to compute the corresponding Pr. The variation of Tr as a function of Pr along the spinodal curve determined in this manner is shown in Fig. 5.11 for constants corresponding to the van der Waals and Redlich-Kwong equations, respectively. Also shown in Fig. 5.11 are measured superheat limit data summarized in reference [5.6] for various liquids at atmospheric pressure. The spinodal curves in Fig. 5.11 imply that a liquid can generally be superheated to more than 80% of its critical temperature before the spinodal limit is reached. The thermodynamic limit of superheat computed from the Redlich-Kwong equation is consistently higher than that computed from the van der Waals equation at a given reduced pressure. Experimental observations of the superheat limit temperature have typically been obtained using a test apparatus in which a drop of the liquid to be tested is injected through a capillary tube into the bottom of a vertical column filled with an immiscible liquid having a higher density and boiling point. Because of the density difference, the small drop rises slowly through the fluid in the column. The temperature in the column is controlled with external heaters so that the top is

FIGURE 5.11  Comparison of measured superheat limit data with spinodal limit predictions.

178

Liquid-Vapor Phase-Change Phenomena

FIGURE 5.12  Variation of the rate of critical embryo production with liquid temperature predicted by analysis of the kinetics of vapor embryo formation for several fluids.

hotter than the bottom. The droplet is slowly heated as it rises, until homogeneous nucleation causes it to explode. Thermocouples at a number of locations indicate the temperature variation along the column, which in turn is used to determine the temperature of the droplet at the location where it is observed to explode. The superheat limits determined in this manner by Blander and Katz [5.6] for a number of fluids are plotted in Fig. 5.11. Also shown are superheat limit data obtained by Skripov [5.7]. The experimentally observed superheat limits for a number of fluids exceed the spinodal limit predicted by the van der Waals equation, implying that this equation does not provide an accurate indication of the thermodynamic limit of superheat. The experimental data for a wide variety of fluids agree well with the spinodal limit predicted by the Redlich-Kwong equation. Equation (5.106) can be solved iteratively to determine the kinetic limit of superheat, if a threshold value of J corresponding to the onset of homogeneous nucleation is assumed. Blander and Katz [5.6] have compared the kinetic limit of superheat TSL predicted in this manner for J = 1012 m−3s−1 with experimentally observed superheat limits at atmospheric pressure for a wide variety of hydrocarbon fluids. The intersection of Eq. (5.106) with this threshold value for several common fluids at atmospheric pressure is shown graphically in Fig. 5.12. A comparison of the measured superheat limits with those predicted by Eq. (5.106) with J = 1012m−3s−1 is shown in Fig. 5.13 for some of the fluids considered by Blander and Katz [5.6]. The excellent agreement between the predicted and measured values of TSL seen in Fig. 5.13 was typical of the results for virtually all the organic fluids considered by Blander and Katz [5.6]. For water, however, the agreement is not quite as good. Equation (5.106) with J = 1012 m−3s−1 predicts a superheat limit of about 300°C for water at atmospheric pressure. Attempts to experimentally determine the superheat limit for water at atmospheric pressure by Blander et al. [5.8] and Apfel [5.9] produced measured values of TSL between 250°C and 280°C. While the discussion here has focused on systems with positive values of pressure, the results of these analyses can also be used to predict homogeneous nucleation in liquids subjected to negative pressure (i.e., in tension). Eberhart and Schnyders [5.10] found that the superheat limit data

Phase Stability and Homogeneous Nucleation

179

FIGURE 5.13  Comparison of measured superheat limit data with the superheat limit temperature predicted by analysis of the kinetics of vapor embryo formation.

obtained by Apfel [5.9] at negative pressures fell slightly above the van der Waals spinodal curve and close to the Redlich-Kwong spinodal for Pr < 0. Superheat limit data obtained by Skripov [5.7] and Avedisian [5.11, 5.12] for pure liquids at high positive pressure are also consistent with the theoretical relations derived above. The above analysis of the kinetics of the nucleation process in a superheated liquid makes use of the assumption that the number density of an embryo bubble is proportional to e − ( ∆G / k BT ), as indicated by Eq. (5.82). Note that the exponent of this term is the ratio of the change in free energy (reversible work) associated with the embryo formation to a term proportional to the mean kinetic energy of the molecules. As a result of this hypothesis, which has become conventional in analyses of this type [5.3–5.5], the model analysis predicts that J is proportional to e − ( ∆Ge / k BT ) (see Eqs. (5.74) and (5.101)). Lienhard and Karimi [5.13] have suggested that it may be more appropriate to normalize ΔG with kBTc. They argue that the probability of embryo formation ought to be dependent on the ratio of the reversible work of formation (ΔG) to the fixed value of the energy required to separate one molecule from another, which is denoted here as ε. It has been demonstrated using experimental data that for many substances ε is approximately equal to 43 kBTc (see reference [5.14]). Normalizing ΔG with kBTc for such substances thus accomplishes the objective of normalizing ΔG with a quantity proportional to ε. Using this line of reasoning, Lienhard and Karimi [5.13] developed the following relation for the spinodal limit for liquid water:



  −16πσ 3 10 −5 = exp  2 2   3k B Tc (1 − vl / vv ) [ Psat (Tl ) − Pl ] 

(5.111)

Equation (5.111) was found to closely match the spinodal limits obtained using isotherms fit to superheated liquid data at negative pressures. The spinodal limit for positive pressures generally corresponds to TSL > 0.8Tc (Fig. 5.11). As a result, replacing kBTl with kBTc in Eq. (5.105) only slightly changes the argument of the exponential term. The choice of threshold value of J could be altered to compensate, resulting in a relation that agrees as well with superheat data as Eq. (5.106). However,

180

Liquid-Vapor Phase-Change Phenomena

at negative pressures the superheat limit temperature may be substantially lower than Tc, and the choice of kBTc over kBTl as the quantity used to normalize ΔG may have a greater impact on the predictive capabilities of the relation for the spinodal limit. Treatment of homogeneous nucleation in multicomponent systems is beyond the scope of this text. It is worth noting, however, that the analytical techniques described here for pure liquids can be extended to nucleation processes in multicomponent liquid mixtures. More information on nucleation in multicomponent systems can be obtained in references [5.6, 5.15–5.19]. The practical significance of homogeneous nucleation processes in liquids is usually related to the fact that, when it does occur, vapor is generated at an extremely rapid rate. In some instances, vapor explosions may occur, with disastrous consequences for nearby equipment and/or people. Perhaps the most well-known example is the vapor explosion that can result from a spill of cryogenically liquefied natural gas (LNG) on water. The explosive boiling resulting from homogeneous nucleation can generate shock waves much like a chemical explosion, with the same potentially destructive result. Further discussion of bubble growth after the initiation of nucleation is presented in Chapter 6.

5.5 THERMODYNAMIC ASPECTS OF HOMOGENEOUS NUCLEATION IN SUPERCOOLED VAPOR The vapor phase of a pure substance may be brought to a supercooled or supersaturated state either by transferring heat through the walls of a containing structure or by rapidly changing the pressure of the gas. For either type of process, once a saturated state is reached, further change in the system pressure or further removal of heat may initiate condensation of some vapor to liquid if nuclei of the liquid phase are present in the system. Often there are adsorbed molecules in a near-liquid state on the containment walls (see Sections 3.4 and 3.5) or on dust particles suspended in the vapor that serve as nuclei. However, in the absence of these types of nuclei, the processes mentioned above may alter the pressure or cool the vapor to a state point in the metastable vapor region of Fig. 5.4. Heterophase fluctuations of the density of a metastable supersaturated vapor may result in the formation of an embryo droplet that may initiate the condensation process. Such a homogeneous nucleation process will, as in the case of vaporization, depend on the kinetics of the embryo formation process in the vapor. To explore these matters further, we will specifically consider the system shown in Fig. 5.14 containing a liquid droplet of radius r in equilibrium with a surrounding vapor held at fixed temperature Tv and pressure Pv. At equilibrium, the temperature and chemical potential in the vapor and the droplet must be the same

µ v = µl

(5.112)



The pressures in the two phases are related through the Young-Laplace equation (2.29): Ple = Pv +

2σ re

(5.113)

Using the integrated form of the Gibbs-Duhem equation for a constant temperature process (derived in Section 5.2), P

µ − µ sat =

∫

(5.114)

vdP

Psat ( Tv )



181

Phase Stability and Homogeneous Nucleation

FIGURE 5.14  System model considered in the thermodynamic analysis of the formation of an embryo liquid droplet.

We evaluate the integral on the right side using the ideal gas law (v = RTv/P) for the vapor to obtain



 Pve  µ ve = µ sat , v + RTv ln    Psat ( Tv ) 

(5.115)

For the liquid phase inside the droplet, the chemical potential can again be evaluated using Eq. (5.114). The liquid is taken to be incompressible, with v equal to the value for saturated liquid at Tv. With this assumption, evaluation of the integral in Eq. (5.114) for P = Ple yields

µ le = µ sat ,l + vl [ Ple − Psat (Tv )]

(5.116)

Equating the values of μv and μl given by Eqs. (5.115) and (5.116) to satisfy Eq. (5.112), and using the fact that μsat,v = μsat,l the following relation is obtained:



 vl  Ple − Psat ( Tv )  Pv = Psat ( Tv ) exp    RTv  

(5.117)

Comparison of the above relation with Eq. (5.54) reveals that the relation (5.117) for the equilibrium vapor pressure surrounding a small liquid droplet is identical to that for the pressure inside a small vapor bubble in equilibrium with a surrounding liquid pool. As seen in Fig. 5.15, if the vapor state point is on the metastable supercooled vapor curve at point a, the liquid state corresponding to equal μ must lie on the subcooled liquid line at point b. For a liquid droplet with finite radius, equilibrium can therefore be achieved only if the liquid is subcooled and the vapor is supersaturated, relative to the normal saturation state for a flat interface. Equation (5.117) also indicates that if Pv is greater than Psat(Tv), then Ple must also be greater than Psat(Tv), which is consistent with the state points indicated in Fig. 5.15. Substituting Eq. (5.113) to eliminate Ple, Eq. (5.117) becomes



 vl  Pv − Psat ( Tv ) + 2σ / re   Pv = Psat ( Tv ) exp    RTv  

(5.118)

182

Liquid-Vapor Phase-Change Phenomena

FIGURE 5.15  The liquid and vapor states for a liquid droplet in equilibrium with surrounding vapor.

In most instances, the steep slope of the subcooled vapor line in Fig. 5.15 results in values of Pv that are much closer to Psat(Tv) than to Ple. When this is the case, P v – Psat(Tυ) is small compared to 2σ/re = Ple – Pv, and the relation (5.118) for Pv is well approximated by  2v σ  Pv = Psat ( Tv ) exp  l   re RTv 



(5.119)

Equations (5.118) and (5.119) can also be inverted to solve for the equilibrium droplet radius for a given set of pressure and temperature conditions. re = re =

2σ

(5.120)

( RTv / vl ) ln  Pv / Psat ( Tv ) − Pv + Psat ( Tv )

2σvl RTv ln  Pv / Psat ( Tv )

for Pv − Psat ( Tv ) re, where ΔG decreases as r increases, which favors further growth of the droplet. Hence, if density fluctuations in a metastable vapor result in the formation of an embryo droplet of radius r < re, the droplet is likely to lose molecules and disappear. If the radius is greater than re, it will grow spontaneously, initiating the condensation process in the vapor via homogeneous nucleation. Example 5.6 Determine the critical radius of a water droplet for supersaturated steam at atmospheric pressure and a temperature of 80°C. The critical radius is the equilibrium radius given by Eq. (5.120):



re =

2σ

(RTv / v l ) ln Pv / Psat (Tv )  − Pv + Psat (Tv )

184

Liquid-Vapor Phase-Change Phenomena

For saturated water at Tv = 80°C, Psat = 47,400 Pa, vl = 0.001029 m3/kg, σ = 0.0629 N/m, and R = R /M = 8314.4/18 = 462 J/kg. Substituting into the above equation, re =

2 ( 0.0626)  462(80 + 273) / 0.001029  ln[101,000 / 47,400 ] − 101,000 + 47,400

= 1.04 × 10 −9 m

The number of molecules in the droplet ne is found as ne =



( 4πr

3 e

)

/ 3 NA

vlM

where NA and M are Avogadro’s number and the molecular mass, respectively. Substituting, it is found that



ne =

[4π(1.04 × 10 −9 )3 / 3](6.02 × 10 26 )  = 155 molecules 0.001029(18)

Thus, for these conditions, to form an embryo droplet of critical size requires about 155 molecules – a very small number.

5.6  THE KINETIC LIMIT OF SUPERSATURATION As in the case of bubble formation in a superheated liquid, the likelihood of homogeneous droplet nucleation in a supersaturated vapor depends on the kinetics of the embryo formation process. In analyzing the kinetics, the droplet formation process is typically assumed to be analogous to the bubble formation process considered in Section 5.3. Using arguments virtually identical to those presented in Section 5.3 for bubble formation, the following relation is obtained for the net flux of the number of droplet embryos in size space, J:



n = n0   N ∗ ( n )    −1 J=  [ N ( n ) A ( n ) jc ( n )] dn   N n ( )    n = n 

∫

−1

(5.124)

The above equation is, in fact, identical to Eq. (5.95) except that it incorporates the condensing flux at the droplet interface. N(n) is the equilibrium number distribution of droplet embryos for a supercooled vapor, postulated as being equal to



 ∆G ( r )  N ( n ) = ρN ,v exp −   k B Tv 

(5.125)

where ρN,v is the number of vapor molecules per unit volume, and ΔG(r) is the free energy change for formation of the embryo of radius r discussed previously. The radius of the droplet r is related to the number of molecules n in the embryo by



4 πr 3 / 3 nM = nm = vl NA

(5.126)

where, as before, m is the mass of one molecule, M is the molecular mass, and NA is Avogadro’s number.

185

Phase Stability and Homogeneous Nucleation

N *(n) in Eq. (5.124) is the number distribution of droplet embryos in the supersaturated vapor for the actual nonequilibrium conditions. The surface area of an embryo is A(n) = 4πr 2 , and jc(n) is the number of molecules condensing at the interface of a droplet of size n per unit time, per unit area. As in the analysis of bubble formation, the integrand in Eq. (5.124) rapidly approaches zero as |n – ne| or |r – re| becomes large. We therefore approximate ΔG(r) for the droplet with the expansion about r = re described in Section 5.5. In addition, we take jc(n) to be equal to its values at n = ne and use Eq. (4.129a), developed from the kinetic theory of gases in Chapter 4, to evaluate jc ( ne ). jc ( n ) ≅ jc ( ne ) =



Pv

(5.127)

( 2πmk B Tv )1/ 2

In obtaining the approximate relation for jc ( ne ) we have taken Γ(a) = 1 and substituted R = k B N A in Eq. (4.129a). Use of this relation also implicitly assumes that the accommodation coefficient is unity. For small n, we again assume that N*(n)/N(n) → 1, based on the argument that there will be so many very small embryos that departure from equilibrium has little effect on their numbers. The assumption that J is independent of time again leads to the conclusion that it is independent of n, and we therefore evaluate the right side of Eq. (5.124) at n ≅ 0 so that N */N can be taken equal to 1. Substituting Eqs. (5.125) and (5.127) into Eq. (5.124) and using Eq. (5.126) to convert the integral in terms of n to an integral in terms of r, the following relation for J is obtained:



∞  ρN ,v Pv vl m    ∆G ( r )   J= exp   1/ 2  k T  dr   B l    ( 2πmk B Tv )   0

−1

∫

(5.128)

Substituting the expansion (5.123) for ΔG into Eq. (5.128), changing the integration variable to ξ = (4πσ/kBTv)1/2 (r – re), and taking the lower limit of integration to be –∞ (because the integrand goes rapidly to zero as |ξ| becomes large), the following relation is obtained for J:



 ρ P v  2σm  J =  N ,v v l     k B Tv   π 

1/ 2

 −4 πre2 σ  exp   3k B Tv 

(5.129)

Using Eq. (5.121) to eliminate re, the above relation for J can be rearranged to obtain



 2σM  J=  πN A 

1/ 2

2  −16π ( σ / k T )3 Mv / N 2  ( l A)  B v  Pv   exp v 2  k T  l   B v 3 ln  Pv / Psat ( Tv )

{

}

(5.130)

For a given pure substance with known physical properties, a threshold value of J can be chosen, corresponding to onset of homogeneous nucleation, and Eq. (5.130) can be solved iteratively for the limiting supersaturation temperature at a given vapor pressure. Supersaturation limit conditions determined experimentally have generally been found to be consistent with the theoretical relations developed in this chapter. Figure 5.16 shows a comparison between measured supersaturation limit data obtained by Frank and Hertz [5.20] and the vapor spinodal limits predicted by Eqs. (5.107) and (5.110) for the van der Waals and

186

Liquid-Vapor Phase-Change Phenomena

FIGURE 5.16  Comparison of measured supersaturation limit data with spinodal limit predictions.

Redlich-Kwong equations of state. The data shown in Fig. 5.16 were obtained for methanol and ethanol using a diffusion cloud chamber. It can be seen that the data in Fig. 5.16 are all well above the predicted spinodal limit curve for both the van der Waals and Redlich-Kwong equations of state. This trend is consistent with the interpretation of the spinodal as the ultimate stability limit for metastable vapor. Of the two curves shown in Fig. 5.16, the Redlich-Kwong spinodal is somewhat closer to the actual supersaturation limits observed in the experiments of Frank and Hertz [5.20], but both curves are well below the measured data. In Fig. 5.17, the supersaturation limit data for a variety of fluids are compared with the corresponding limit predicted by Eq. (5.130) with J taken to be 106 m−3s−1. Note that the data are plotted in terms of the supersaturation ratio S = (Pv)SSL/Psat(Tv), where (Pv)SSL is the vapor pressure at the supersaturation limit for T = Tv. As in the case of bubble nucleation, the choice of J = 106 m−3s−1 for droplet nucleation was made to provide a best fit to available experimental data. It is interesting to note that J = 106 m−3s−1 provides a best fit to droplet nucleation data, whereas for bubble nucleation, J = 1012 m−3s−l provides better agreement. The value of (Pv)SSL predicted by Eq. (5.130) varies weakly with J, so uncertainty in J does not strongly affect the accuracy of the predicted supersaturation limit. As seen in Fig. 5.17, in general, the agreement between the experimentally determined S values and the theoretical predictions is fairly good. The fact that a single J value produced (Pv)SSL values that agree well with data for a wide variety of fluids argues strongly for the validity of this classical model analysis of the nucleation kinetics, and the resulting predictive equation (5.130). However, in their experimental investigation, Hung et al. [5.21] found systematic differences between measured nucleation rate data and the predictions of the classical model described above. Efforts have been made to develop an improved model of the nucleation kinetics (see, e.g. Reiss et al. [5.23]). Hung et al. [5.21] found that the agreement between measured data and the model predictions for alternative models was not significantly better than that for the classical model. To determine (Tv)SSL for a given vapor pressure Pv using Eq. (5.130) with J = 106 m−3s−1 requires an iterative calculation because of the implicit temperature dependence of Psat and σ. However, it can be shown that the following relation approximates Eq. (5.130) to a high degree of accuracy: 



1/ 2



  E ∗ ( − ln J ∗ ) ( Pv )SSL     S= = exp  1/ 2 3/ 2 ∗ ∗ Psat ( Tv )   2 ( E ) + ( − ln J )   

1/ 2



(5.131)

187

Phase Stability and Homogeneous Nucleation

FIGURE 5.17  Comparison of measured supersaturation ratios for various fluids with values predicted by Eq. (5.130).

where J∗ =

MJ  πM  N A vl  2σN A  E∗ =



1/ 2

2

 RTv     Psat ( Tv ) 

16πσ 3 vl2 3k B R 2 Tv3

(5.132) (5.133)

Equation (5.131) expresses S = (Pv)SSL/Psat(Tv) as a function of the dimensionless variables J * and E *, which are functions only of Tv. Hence if Tv is specified, the limiting supersaturation ratio S = (Pv)SSL/Psat(Tv) can be computed explicitly using Eqs. (5.131)–(5.133). Values of S predicted by Eq. (5.131) agree with values obtained by iteratively solving Eq. (5.130) to within less than 0.5%. The variation of S with these parameters predicted by Eq. (5.131) is shown in Fig. 5.18 for ranges of J * and E * commonly encountered in real systems. It is worth noting that the supersaturation limit data represented in Fig. 5.17 were obtained in experiments in which the saturation condition was carefully obtained in a diffusion or expansion chamber. Additional supersaturation limit data were also obtained by Yellot [5.24] for steam and Goglia and Van Wylen [5.25] for nitrogen undergoing high-speed expansion in diverging nozzles. It is generally acknowledged that the conditions at which a condensation shock occurs in such flows may differ significantly from those in nonflow expansions. The supersaturation limit data reported by Yellot [5.24], Goglia and Van Wylen [5.25], and others for expanding gas flows generally scatter widely about the kinetic limits indicated by Eq. (5.130) or Eq. (5.131). Experimentally determined values of S obtained by Yellot [5.24] for steam are somewhat higher than the values predicted by Eq. (5.131), whereas for nitrogen, the data of Goglia and Van Wylen [5.25] are significantly lower.

188

Liquid-Vapor Phase-Change Phenomena

FIGURE 5.18  Supersaturation limit conditions plotted as a function of the parameters E * and J   *.

As noted by Mason [5.26], the time scales associated with rapid expansions in supersonic wind tunnels may be comparable to the time required to establish the quasi-equilibrium embryo size distribution postulated to exist in the analysis described above. When this is true, this type of analysis may fail to accurately predict the onset of condensation in the tunnel. The scattering of data for the onset conditions about the limits predicted by Eq. (5.131) apparently is due, at least in part, to this inadequacy of the model. Note that diffusion chamber experiments like those of Frank and Hertz [5.20] are characterized by time scales much longer than the time needed to establish the embryo size distribution postulated in the analysis. Data from experiments of this type are therefore expected to agree well with the predictions of Eq. (5.131). It is clear from the above discussion that it is, in fact, possible for a vapor to be in a highly supersaturated state without condensation occurring. This behavior can play a major role in many circumstances of practical interest, including cloud (fog) formation and precipitation in the atmosphere, expanding vapor flow in nozzles, and vapor-deposition manufacturing processes. As noted in Section 5.5, vapor can be maintained in a metastable supersaturated state only if there is not a liquid film on the solid surface of the system, and only in the absence of dust particles or surface imperfections that can serve as nucleation sites. In most heat transfer equipment, however, it is difficult, if not impossible, to avoid the presence of these types of nucleation sites. As a result, condensation is often initiated by a heterogeneous nucleation process. This type of nucleation event is considered in detail in Chapter 6. Example 5.7 Estimate the kinetic limiting supersaturation vapor pressure for steam at 20°C. For saturated water at Tv = 20°C, Psat = 2.36 kPa, vl = 0.001002 m3/kg, σ = 0.07278 N/m, and R = R / M = 8314.4 / 18 = 462 J/kg. Using Eqs. (5.131)–(5.133), 1/ 2

MJ  πM  NAv l  2σNA 

( )



2

 RTv     Psat (Tv )  1/ 2 2 18 106    462( 20 + 273)  π (18) =  6.02 × 10 26 ( 0.001002)  2 ( 0.07278) 6.02 × 10 26   2360  = 7.87 × 10 −26

J∗ =

189

Phase Stability and Homogeneous Nucleation from which it follows that –ln (J *) = 57.80. Continuing with E∗ =

16π ( 0.07278) ( 0.001002) 16πσ 3v l2 = 2 3 = 87.47 2 3 3kBR Tv 3 1.381× 10 −23 ( 462) ( 20 + 273)

S =

  E ∗ ( − ln J ∗ ) (Pv )SSL   = exp   ∗ 1/ 2 ∗ 3/ 2 Psat (Tv )   2(E ) + ( − ln J ) 

3



(

)







2

1/ 2

     1/ 2

1/ 2 1/ 2    87.47 ( 57.80 )  = exp     = 3.34 1/ 2 3/ 2   2 (87.47) + ( 57.80 )  

The supersaturation limit is therefore predicted to be

(Pv )SSL = 3.34Psat (Tv ) = 3.34(2.36) = 7.88 kPa

5.7  WALL INTERACTION EFFECTS ON HOMOGENEOUS NUCLEATION The previous sections have examined homogeneous nucleation within an extensive bulk phase. In some applications that lead to the onset of bubble formation, the tendency for homogeneous nucleation may be highest – very close to solid surfaces where force interactions between solid and fluid molecules may affect the mechanisms of homogeneous nucleation. This may be the case when a liquid is bounded by solid containment walls with flow of heat through the wall into the system. Heating of the liquid through the containment wall may result in a region of superheated liquid very close to the solid surface. This type of circumstance can arise, for example, during rapid transient heating of a surface in contact with a liquid. In high heat flux rapid pulse heating experiments, Skripov [5.7] and Asai [5.27] observed that the dominant bubble generation mechanism is the spontaneous nucleation due to thermal motion of liquid molecules (homogeneous nucleation). Lin et al. [5.28] also concluded that sudden energizing of polysilicon micro resistors in fluorinert liquids resulted in formation of a bubble by homogeneous nucleation. Experimental studies of transient heating in inkjet printer heads indicate that if the liquid highly wets the surface, homogeneous nucleation may occur in superheated liquid next to the heated surface when the transient heating raises the liquid temperature far into the metastable range [5.29, 5.30]. In these instances, homogeneous nucleation is most likely to occur in the hottest fluid very close to the solid surface where force interactions between solid and fluid molecules may affect the nucleation process. As discussed in Chapter 3, the long-range attractive dispersion force interactions are known to attract fluid molecules to the solid surface. As discussed in Chapter 3, these attractive forces facilitate the formation of adsorbed liquid films on solid surfaces and disjoining pressure effects in ultra-thin liquid films. Recent investigations by Gerweck and Yadigaroglu [5.31] and Carey and Wemhoff [5.32] have theoretically examined how near-wall attractive forces on fluid molecules affect the onset of homogeneous nucleation. Gerweck and Yadigaroglu [5.31] developed an analytical model of the effects of wall-fluid attractive forces on the equation of state of the fluid. They used a thermodynamic analysis to assemble an equation of state for the near-wall region by combining a repulsive force interaction model for a hard sphere fluid with attractive force models for interactions between the fluid molecules and solid surface molecules. Although it is difficult to extract specific predictions from the study of Gerweck and Yadigaroglu [5.31], it does provide qualitative predictions of the effect of attractive forces near the wall. It predicts that for typical conditions, the pressure is very high near the wall and the spinodal temperature is substantially higher than in the bulk fluid farther from the wall. This would imply that fluid immediately adjacent to the wall must be heated to a higher temperature to initiate homogeneous nucleation than bulk fluid farther away.

190

Liquid-Vapor Phase-Change Phenomena

In a more recent investigation, Carey and Wemhoff [5.32] used statistical thermodynamics analysis to derive a modified version of the Redlich-Kwong fluid property model that accounts for attractive forces between the solid surface molecules and liquid molecules in the near-wall region. In this model, the wall-fluid attractive forces were quantified in terms of Hamaker constants. Since values of Hamaker constants and Redlich-Kwong constants are available for a variety of fluid and solid surface combinations, this model can be used to assess the effect of wall-fluid force interactions on the spinodal conditions for a variety of fluid and surface material combinations. The relations that predict the spinodal condition near the surface obtained from their model can be cast in reduced form as Pr =

3ρr Tr ρ2 a A′ρ2  2 Dr6, f  − 1/ 2 r r + ll 3r 1 − 1 − ρr br Tr (1 + ρr br ) 2πzr  15zr6 

3Tr

(1 − ρr br )2



−

ar ρr ( 2 + ρr br )

Tr1/ 2 (1 + ρr br )

2

+

All′ρr πzr3

(5.134)

 2 Dr6, f  1 − 15z 6  = 0 r  

(5.135)

where

Pr = P /Pc

(5.136)



Tr = T /Tc

(5.137)



ρr = ρˆ / ρˆ c

(5.138)



zr = z /[k B Tc /Pc ]1/3 All′ = A ll / k B Tc ,





Als′ = A ls / k B Tc

(5.139)

Dr , f = D f /[k B Tc /Pc ]1/3 , Dr ,m = Dm /[k B Tc /Pc ]1/3



(5.140)



(5.141)



In the above equations, z is the distance from the solid surface, A ll and A ls are the Hamaker constants for liquid-liquid and liquid-solid interfaces, and ar and br are the numerical constants specified below: ar = 3.84732, br = 0.25992



(5.142)

Df and Dm are the closest approach distances for pairs of fluid molecules and solid and fluid molecules, respectively. The local density is determined by setting the local temperature and chemical potential (which depend on Als′ and Dr,m) equal to the bulk fluid values. Equality of the local and bulk temperature and chemical potential requires that the local density ρr satisfy the equation  1 − ρr br  ρr br Tr ,∞ ar a  ρr  −Tr ,∞ ln  + − ln ( l + ρr br ) − 1/r 2   1/ 2 3 3Tr ,∞  1 + ρr br   ρr N Aρˆ c Λ  1 − ρr br 3br Tr ,∞



 2 Dr6, f  Als′ 1 − 15 z 6  − 6πz 3 r r  

 1 − ρr ,∞ br  ρr ,∞ br Tr ,∞  2 Dr6,m  1 − 15 z 6  = −Tr ,∞ ln  ρ N ρˆ Λ 3  + 1 − ρ b r  r ,∞ r   r ,∞ A c 

+

All′ρr 3πzr3

−

ar a  ρr ,∞  ln (1 + ρr ,∞ br ) − 1/r 2  3br Tr1/,∞2 3Tr ,∞  1 + ρr ,∞ br 

(5.143)



Phase Stability and Homogeneous Nucleation

191

where Λ = [h2/2πMkBT]1/2, Tr,∞ is the bulk fluid reduced temperature, and h is Planck’s constant. Equation (5.143) above is solved iteratively to determine the local reduced density ρr . Then Eqs. (5.134) and (5.135) are used to determine the local reduced pressure Pr and the reduced temperature Tr that corresponds to the spinodal limit. Computed results obtained with this model indicate that for a wide variety of fluids, surface forces affect the fluid only within a few nanometers of the solid surface. Two interesting predictions of the Carey and Wemhoff [5.32] model are that as the wall is approached (z → 0) the local pressure and the local spinodal temperature increase rapidly in the region within a few nanometers of the wall. Figure 5.19 shows the variation of the pressure and spinodal temperature with distance from the wall predicted by the model for carbon tetrachloride on a gold metallic surface. The rise in pressure as the wall is approached is consistent with the predictions of the model of near-wall effects developed by Gerweck and Yadigaroglu [5.31]. The increase in pressure near the wall is similar to the rise in pressure caused by gravity forces acting on fluid molecules above a horizontal solid surface. The only difference here is that the attractive forces are negligible beyond a few nanometers from the wall, so the rise in pressure is limited to that region. The pressure variation predicted by the thermodynamic model of Carey and Wemhoff [5.32] was shown to agree well with the near-wall pressure variations determined by a continuum hydrostatic model and a molecular dynamics simulation model. The near-wall force interaction effects suggested by the studies of Gerweck and Yadigaroglu [5.31] and Carey and Wemhoff [5.32] may have important consequences for boiling heat transfer during rapid transient heating processes. These will be explored in more detail in Chapter 8. Other studies have suggested that the proximity of solid surfaces may also affect homogeneous nucleation in other ways. Zhang et al. [5.33] have argued that embryo formation due to density

FIGURE 5.19  Near-wall variations of pressure (a) and spinodal temperature (b) predicted by the model of Carey and Wemhoff [5.32] for CCl4 on gold.

192

Liquid-Vapor Phase-Change Phenomena

fluctuations produces pressure waves that propagate into the adjacent fluid. In microchannels or microchambers, they argued that these waves will be rapidly reflected back to the locations of origin when the wall is close, and the returning pressure fluctuation may serve to inhibit homogeneous nucleation. Another effect of adjacent surrounding walls can arise in liquid filled microchambers that are heated to the onset of homogeneous nucleation. The conventional models used to predict the superheat limit or spinodal condition generally are based on the premise that the system is held at constant pressure. Carey [5.34] has shown that when the chamber walls are elastic, pressure and specific volume in the chamber vary in a manner dictated by the fluid equation of state and the elastic deflection characteristics of the wall. The model analysis presented by Carey [5.34] accounts for these effects and predicts the spinodal conditions when a liquid-filled chamber with elastic walls is heated. The model predictions indicate that the confining effects of the elastic walls may significantly affect the conditions for the onset of homogeneous nucleation in a system of this type. It should be noted that in a supersaturated vapor, wall attractive forces affect nucleation in a much different manner. Unlike the superheated liquid case, in a supersaturated vapor, wall attraction forces do not suppress nucleation. Instead, the attractive forces may result in formation of an adsorbed film on the solid surface on which liquid condensation may occur. Condensation on solid surfaces is considered in more detail in Chapter 9.

5.8 NANOBUBBLES Nanobubbles are of interest in several contexts of technological interest. As discussed in Section 5.3, in pure fluid systems, the population of ultra-small embryo bubbles produced by random density fluctuations increases as liquid is superheated to progressively higher levels, eventually resulting in production of embryos slightly larger than critical size, which triggers macroscopic homogeneous nucleation. Experiments have also indicated that under transient cyclic surface heating of the type that occurs in some inkjet printers, nanobubble vapor remnants from one cycle can affect bubble nucleation and growth in a subsequent cycle [5.35]. In addition, there is currently a strong interest in creating nanostructured surfaces that will enhance nucleate boiling heat transfer (e.g., [5.36, 5.37]). The interstitial spaces in such surfaces are typically nanometer in scale, which means that any bubble nucleation in the interstitial spaces would involve nanoscale bubbles or liquid-vapor interfaces with nanometer scale principle radii of curvature. This suggests that the thermophysics of nanobubbles in superheated liquid may play a central role in the nucleation of bubbles during nucleate boiling on nanostructured heated surfaces. Researchers have also suggested that the presence of sub-critical-radius bubbles produced by density fluctuations can cause nucleation and boiling in superheated liquid in microchannels to differ from the corresponding processes in macroscopic systems [5.33, 5.38]. Absorption of solar radiation at the surface of nanoparticles in liquid to produce vapor initially as nanobubbles has also been proposed as a means of efficiently producing steam from solar energy for power production, or other applications that can use steam heat [5.39, 5.40]. Production of nanobubbles by laser heating of nanoparticles in biological system fluids has also been proposed as a means of delivering destructive energy to kill cells or tissue at precisely defined locations (see, for example the discussion by Wagnera et al. [5.41]). Geometry dictates that nanobubbles have a very high interface surface-to-volume ratio, which facilitates high mass and heat transfer rates at the interface. For this reason, infusing a liquid with nanobubbles containing a second soluble gaseous species will facilitate rapid transport of the second species into the liquid. This can be useful for manufacturing or chemical processes where high mass and/or heat transfer rates enhance the process. An example of applications of this type is the injection of nanobubbles of gas in water purification processes [5.42]. In addition, the presence of gas nanobubbles at a hydrophobic surface (which may be spheroids or sphere segments) may also promote slip to reduce significantly the shear stress and drag [5.43, 5.44].

Phase Stability and Homogeneous Nucleation

193

FIGURE 5.20  A vapor bubble in equilibrium with superheated liquid at 101 kPa and 120°C.

Because of the interest in potential applications like those described above, we will explore some of the thermophysics of nanobubbles that relate to liquid-vapor phase change in this final section of Chapter 5. To do so, consider first the system shown in Fig. 5.20. Water vapor is injected through a channel with a radius of about a micron, so that a bubble forms at the point where the channel enters a chamber containing superheated liquid water at 101 kPa and 120°C. The bubble is a spherical cap with a radius exactly equal to the equilibrium radius re computed using Eq. (5.55) to be 1.2 × 10 −6 m. The bubble is therefore in equilibrium with the surrounding superheated liquid. The discussion in Section 5.3 is based on the thermodynamic argument that the size number distribution of small bubble embryos in superheated liquid is given by



 ∆G (r )  N n = N (r ) = ρN ,l exp −   k B Tl 

(5.144)

To predict the equilibrium bubble size distribution N(r) using Eq. (5.144) above for arbitrary circumstances obviously requires a relation for the free energy of formation ∆G (r ) for a bubble of size r. In Section 5.2, the expansion relation (5.81) was derived that can be used to predict ∆G (r ) near the equilibrium radius re . This relation can be written in the form



2     r  1 ∆G = (4 / 3)πσ i re2 1 −  2 + − 1 +      1 + 2σ i / re Pl   re   

(5.145)

Here, we are interested in bubbles with radius values between the minimum possible radius rmin and the equilibrium radius re . It is reasonable to expect that rmin cannot be smaller than the mean diameter of the fluid molecule, and is likely on the order of the mean free path of vapor molecules for the vapor pressure and temperature inside the bubble. Since formation of a bubble of radius larger than re would lead to a homogeneous nucleation event, we confine our interest to r < re to explore the distribution of bubble sizes in metastable superheated liquid at temperatures below homogenous nucleation limit. For the small bubble radius values of interest here, the term 2σ i / re Pl is large compared to one, and the value of the left factor in round parentheses on the right side or Eq. (5.145) is well approximated as 2. For our purposes here, this relation can be simplified to



2   r  ∆G = (4 / 3)πσ i re2 1 − 2  − 1 +   re   

(5.146)

194

Liquid-Vapor Phase-Change Phenomena

Unfortunately, this relation is not accurate as r becomes much smaller than re , so an alternative relation for ∆G (r ) is needed that will work for the full range rmin < r < rc of interest. Equation (5.75) is a general relation for ∆G (r ) which is applicable to the conditions of interest here.

∆G = Nˆ v ( gˆ v − gˆl ) + 4 πr 2 σ i − (4 / 3)πr 3 ( Pv − Pl )

(5.75)

Using the Young-Laplace equation to replace Pv − Pl with 2σ i / r , and the relation Nˆ v = (4 / 3)πr 3ρˆ v for the number of moles of vapor in the bubble, Eq. (5.75) can be reorganized to

∆G = (4 / 3)πr 3ρˆ v ( gˆ v − gˆl ) + (4 / 3)πr 2 σ i

(5.147)

To evaluate the molar specific free energies in Eq. (5.147), the following idealizations and thermodynamic requirements are invoked here: i. The vapor is treated as an ideal gas at Tv = Tl , and therefore ρˆ v = Pv / RTv , and changes in hˆv , sˆv and gˆ v are computed using relations for an ideal gas with constant specific heats. ii. The liquid is treated as being incompressible, so its enthalpy and entropy and free energy are functions only of temperature, and are therefore equal to the corresponding saturated liquid properties at the specified temperature. Also, changes in hˆl , sˆl , and gˆl are computed using relations for an incompressible liquid with constant specific heats. iii. The liquid and vapor temperatures are equal, and as required thermodynamically, gˆl ,sat = gˆ v ,sat at Tv = Tl .

Using these requirements and idealizations [i]–[iii] with the definition gˆ = hˆ − Tsˆ , the relation (5.147) can be converted to



 P + 2σ i / r  2σ  ∆G = (4 / 3)πr 3  Pl + i  ln  l + (4 / 3)πr 2 σ i  r   Psat (Tl ) 

(5.148)

Substituting the right side of the above equation for ∆G (r ) in Eq. (5.144), the following relation for the number distribution is obtained:



 (4 / 3)πr 3  N A ρl ,sat (Tl )  N (r ) =  exp −   M k B Tl 

 2σ i   Pl + 2σ i / r  σ i   +   ln  Pl + r   Psat (Tl )  r   

(5.149)

Note that since Psat, ρˆ l ,sat , and σ i are functions only of Tl , once Tl and Pl are specified, the relation above predicts the variation of the bubble number density N as a function of bubble radius r. It is also worth noting that the above equation merges into the expansion relation (5.146) as r approaches re , so Eq. (5.149) can be used to predict N(r) for small r and either relation can be used to predict N(r) near r = re. The distribution of bubble sizes at equilibrium predicted by Eq. (5.148) for water at Pl = 101 kPa and Tl = 120°C is shown in Fig. 5.21. Bubbles with a radius smaller than about 1.5 × 10 −10 m are not physically possible because the effective diameter of a water molecules is about 3.0 × 10 −10 m. The free energy increase for formation of an embryo with radius r normalized with k B Tl (∆G / k B Tl) is also plotted in Fig. 5.21. This ratio is positive and bigger than one even at the smallest possible bubble sizes, and ∆G / k B Tl gets larger as the embryo radius increases. This, in turn, results in a very rapid decrease in the number of bubbles with increasing radius. The decrease is so much that there are virtually no density-fluctuation-produced bubbles found with a radius near the equilibrium

Phase Stability and Homogeneous Nucleation

195

FIGURE 5.21  Predicted variation of normalized free energy of formation and number distribution of embryo bubbles with radius for bubbles formed by density fluctuations in superheated liquid at 101 kPa and 120°C.

radius of 1.2 × 10 −6 m at the liquid temperature of 120°C. This is consistent with the observation that for these conditions (101 kPa, 120°C liquid), homogeneous nucleation in the liquid phase is generally not observed experimentally. The distribution does indicate, however, that superheated liquid water at this temperature contains only a very few bubble embryos produced by density fluctuations with diameters of about one nanometer. It predicts, for example, that if you had 1040 m3 of liquid water at 120°C, it likely contains one embryo with a radius of 0.5 nm at any instant of time, and virtually zero embryos with radii above 0.7 nm. Figure 5.22 indicates how the distribution of bubble sizes changes as the liquid temperature is increased, with the pressure held at 101 kPa. At Tl = 180°C, the equilibrium bubble radius decreases to 9.4 × 10 −8 m, and the number density decreases rapidly with increasing diameter, becoming negligible at radius values well below the equilibrium radius. Figure 5.22 also shows the bubble size distribution for 101 kPa and Tl = 310°C. Note that by raising the superheat level, more bubbles exist at larger sizes and the equilibrium bubble radius decreases to 2.6 × 10 −9 m. Superheating water at atmospheric pressure to 310°C is predicted to produce significant number concentrations of bubbles at sizes at or near the critical radius. This is consistent with the fact that the onset of homogeneous nucleation has been observed for water heated to this superheat level at atmospheric pressure. The distribution relation further indicates that significant concentrations of nanobubbles exist in the water at atmospheric pressure as the superheat limit is approached. The thermodynamic model analysis described above predicts the equilibrium distribution of bubble sizes expected in liquid under specified temperature and pressure conditions as a result of heterophase density fluctuations in the liquid. It indicates that to establish a significant sustained

196

Liquid-Vapor Phase-Change Phenomena

FIGURE 5.22  Predicted variation of the number distributions of embryo bubbles with radius for bubbles formed by density fluctuations in superheated liquid at 101 kPa for temperatures of 180°C and 310°C.

(equilibrium) population of vapor nanobubbles in pure water, the temperature of the liquid must be raised to a level close to the heterogeneous nucleation limit. It also indicates that the tendency for nanobubbles to persist in a liquid is directly connected to the bubble’s free energy change of formation (or collapse). Note that in the cyclic transient surface heating processes considered by Cavicchi and Avedisian [5.35], nanobubbles exist periodically in the cyclic process. Nanobubbles are generated as from remnants of microbubbles that collapse in a previous heating cycle. Since they are below critical radius size, they tend to collapse, but the data of Cavicchi and Avedisian [5.35] indicate that they exist long enough to play a role in nucleation of bubbles in the next heating cycle. Likewise, for the processes of nucleate boiling on a nanostructured surface considered by Wang et al. [5.36] and Lu et al. [5.37], transient formation of nanobubbles can play a role in bubble growth and release at the surface, which dictates the heat transfer there. Wang et al. [5.36] concluded that it appears that the ability of their copper surface with nanorods to generate stable nucleation of bubbles at low superheated temperatures results from a synergistic coupling effect between the nanoscale gas cavities (or nanobubbles) formed within the nanorod interstices and micrometer-scale defects (voids) that form on the film surface during nanorod deposition. The nanobubbles in the interstitial nanoscale spaces may transiently form from remnants of vapor bubbles departing the surface or vapor entrapped in nanoscale crevices in the surface. Again because the nanobubble radius of curvature is less than the critical radius, the nanobubble may be thermodynamically favored to collapse, but the cyclic process of vaporization-driven bubble growth and release may drive the periodic formation of nanobubbles that participate in the bubble growth and release mechanisms for the nucleate boiling process on the nanostructured surface. Formation of nanobubbles that surround a nanoparticle as a means of generating steam using a radiant energy input provides a means of steady creation of nanobubbles. The free energy of formation for bubbles surround a nanoparticle may differ from that for a free nanobubble, which may affect bubble stability. In general, it appears that nanobubbles generated in this way in water would still tend to collapse. In biological system applicators, where the objected is to allow the bubble to

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collapse to release its energy, this is an advantage. If the objective is steam generation from solar energy, the system conditions must be chosen so that the nanobubbles coalesce to allow collection of the resulting vapor for a useful purpose before the vapor bubbles collapse. The stability of nanobubbles of a second gas species in a liquid, such as air bubbles in water, generally depends on a collection of system-specific factors. Effects of gas/liquid interface charge distribution, surface tension variation with gas concentration, and self-organization of nanobubble clusters may alter the free energy of formation in a way that enhances the stability of nanobubbles in liquids. In some systems, at least some of these factors stabilize gas nanobubbles sufficiently that they persist long enough to affect processes in a technologically useful way. This can, for example, make it possible to use nanobubbles or truncated sphere nanobubbles at a surface to reduce surface drag in a liquid flow, as described in the references mentioned above. A comprehensive exploration of the physics of gas nanobubbles and their applications is beyond the scope of the introductory summary provided here. The interested reader can find further discussions of the physics of nanobubble stability in liquids in references [5.40, 5.42–5.46].

REFERENCES 5.1 Carey, V. P., Statistical Thermodynamics and Microscale Thermophysics, Cambridge University Press, New York, NY, 1999. 5.2 Modell, M., and Reid, R. C., Thermodynamics and Its Applications, 2nd ed., Prentice-Hall, Englewood Cliffs, NJ, 1983. 5.3 Zeldovich, Y. B., On the theory of new phase formation: Cavitation, Acta Physiochem. URSS, vol. 18, pp. 1–22, 1943. 5.4 Kagen, Y., The kinetics of boiling of a pure liquid, Russ. J. Phys. Chem., vol. 34, pp. 42–46, 1960. 5.5 Katz, J. L., and Blander, M., Condensation and boiling: Corrections to homogeneous nucleation theory for nonideal gases, J. Colloid Interface Sci., vol. 42, pp. 496–502, 1973. 5.6 Blander, M., and Katz, J. L., Bubble nucleation in liquids, AIChE J., vol. 21, pp. 833–848, 1975. 5.7 Skripov, V. P., Metastable Liquids, Wiley, New York, NY, 1974. 5.8 Blander, M., Hengstenberg, D., and Katz, J. L., Bubble nucleation in n-pentane, n-hexane, n-pentane + hexadecane mixtures, and water, J. Phys. Chem., vol. 75, p. 3613, 1971. 5.9 Apfel, R. E., Vapor nucleation at a liquid-liquid interface, J. Chem. Phys., vol. 54, p. 62, 1971. 5.10 Eberhart, J. G., and Schnyders, H. C., Applications of the mechanical stability condition to the prediction of the limit of superheat for normal alkanes, ether and water, J. Phys. Chem., vol. 77, pp, 2730–2736, 1973. 5.11 Avedisian, C. T., The homogeneous nucleation limits of liquids, J. Phys. Chem. Ref. Data, vol. 14, pp. 695–729, 1985. 5.12 Avedisian, C. T., Effect of pressure on bubble growth within liquid droplets at the superheat limit, J. Heat Transf., vol. 104, pp. 750–757, 1982. 5.13 Lienhard, J. H., and Karimi, A., Homogeneous nucleation and the spinodal line, J. Heat Transf., vol. 103, pp. 61–64, 1981. 5.14 McQuarrie, D. A., Statistical Thermodynamics, Harper & Row, New York, NY, 1973. 5.15 Holden, B., and Katz, J. L., The homogeneous nucleation of bubbles in superheated binary mixtures, AIChE J., vol. 24, pp. 260–267, 1978. 5.16 Ward, C. A., Balakrishnan, A., and Hooper, F. C., On the thermodynamics of nucleation in weak gas-liquid solutions, J. Basic Eng., vol. 85, pp. 695–704, 1970. 5.17 Forest, T. W., and Ward, C. A., Effect of dissolved gas on the homogeneous nucleation pressure of a liquid, J. Chem. Phys., vol. 66, pp. 2322–2330, 1977. 5.18 van Stralen, S., and Cole, R., Boiling Phenomena, vol. 1, Chap. 3, Hemisphere, New York, NY, 1979. 5.19 Avedisian, C. T., and Glassman, I., High pressure homogeneous nucleation of bubbles within superheated binary liquid mixtures, J. Heat Transf., vol. 103, pp. 272–280, 1981. 5.20 Frank, J. P., and Hertz, H. G., Messung der kritischen Übersättigung von Dampfen mit der Diffusionsnebelkammer, Z. Phys., vol. 143, pp. 559–590, 1956. 5.21 Hung, C.-H., Krasnopler, M. J., and Katz, J. L., Condensation of a supersaturated vapor. VIII. The homogeneous nucleation of n-nonane, J. Chem. Phys., vol. 90, pp. 1856–1865, 1989. 5.22 Volmer, M., and Flood, H., Tröpfchenbildung in Dampfen, Z. Phys. Chem., vol. 170, pp. 273–285, 1934.

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5.23 Reiss, H., Katz, J. L., and Cohen, E. R., Translation–rotation paradox in the theory of nucleation, J. Chem. Phys., vol. 48, pp. 5553–5560, 1968. 5.24 Yellot, J. I., Supersaturated steam, Trans. ASME, vol. 56, pp. 411–430, 1934. 5.25 Goglia, G. L., and Van Wylen, G. J., Experimental determination of limit of supersaturation of nitrogen vapor expanding in a nozzle, J. Heat Transf., vol. 83, pp. 27–32, 1961. 5.26 Mason, B. J., The Physics of Clouds, 2nd ed., Oxford University Press, London, England, 1971. 5.27 Asai, A., Bubble dynamics in boiling under high heat flux pulse heating, J. Heat Transf., vol. 113, pp. 973–979, 1991. 5.28 Lin, L., Pisano, A. P., and Carey, V. P., Thermal bubble formation on polysilicon micro resistors, J. Heat Transf., vol. 120, pp. 735–742, 1998. 5.29 Andrews, J. R., and O’Horo, M. P., High speed stroboscopic system for visualization of thermal inkjet processes, Proc., Society of Photo-Optical Instrumentation Engineers (SPIE), J. Bares (editor), vol. 2413, pp. 176–181, 1995. 5.30 Rembe, C., aus der Wiesche, S., and Hofer, E. P., Thermal ink jet dynamics: Modeling, simulation and testing, Microelectron. Reliab., vol. 40, pp. 525–532, 2000. 5.31 Gerweck, V., and Yadigaroglu, G., A local equation of state for fluid in the presence of a wall and its application to rewetting, Int. J. Heat Mass Transf., vol. 35, pp. 1823–1832, 1992. 5.32 Carey, V. P., and Wemhoff, A. P., Thermodynamic analysis of near-wall effects on phase stability and homogeneous nucleation during rapid surface heating, Int. J. Heat Mass Transf., vol. 48, pp. 5431–5445, 2005. 5.33 Zhang, J. T., Peng, X. F., and Peterson, G. P., Analysis of phase-change mechanisms in microchannels using cluster nucleation theory, Microscale Thermophys. Eng., vol. 4, pp. 177–187, 2000. 5.34 Carey, V. P., Thermodynamic analysis of the intrinsic stability of superheated liquid in a micromechanical actuator with elastic walls, Microscale Thermophys. Eng., vol. 4, pp. 109–124, 2000. 5.35 Cavicchi, R. E., and Avedisian, C. T., Bubble nucleation and growth anomaly for a hydrophilic microheater attributed to metastable nanobubbles, Phys. Rev. Lett., vol. 98, pp. 124501-1–124501-4, 2007. 5.36 Li, C., Wang, Z., Wang, P.-I., Peles, Y., Koratkar, N., Peterson, G. P., Nano-structured copper interfaces for enhanced boiling, Small, vol. 4, pp. 1084–1088, 2008. 5.37 Lu, M-C., Chen, R., Srinivasan, V., Carey, V. P., and Majumdar, A., Critical heat flux of pool boiling on Si nanowire array-coated surfaces, Int. J. Heat Mass Transf., vol. 54, pp. 5359–5367, 2011. 5.38 Peng, X. F., Liu, D., Lee, D. J., Yan, Y., and Wang, B. X., Cluster dynamics and fictitious boiling in microchannels, Int. J. Heat Mass Transf., vol. 43, pp. 4259–4266, 2000. 5.39 Neumann, O., Urban, A. S.; Day, J., Lal, S., Nordlander, P., and Halas, N. J., Solar vapor generation enabled by nanoparticles, ACS Nano, vol. 7, pp. 42–49, 2013. 5.40 Fang, Z., Zhen, Y-R., Neumann, O., Polman, A., García de Abajo, A. F. J., Nordlander, P., and Halas, N. J., Evolution of light-Induced vapor generation at a liquid-immersed metallic nanoparticle, Nano Lett., vol. 13, pp 1736–1742, 2013. 5.41 Wagnera, D. S., Delka, N. A., Lukianova-Hlebb, E. Y., Hafnerc, J. H., Farach-Carsona, M. C., and Lapotko, D. O., The in vivo performance of plasmonic nanobubbles as cell theranostic agents in zebrafish hosting prostate cancer xenografts, Biomaterials, vol. 31, pp. 7567–7574, 2010. 5.42 Agarwal, A., Ng, W. J., and Liu, Y., Principle and applications of microbubble and nanobubble technology for water treatment, Chemosphere, vol. 84, pp. 1175–1180, 2011. 5.43 Maali, A., and Bhushan, B., Nanobubbles and their role in slip and drag, J. Phys.: Condens. Matter, vol. 25, p. 184003, 2013. DOI: 10.1088/0953-8984/25/18/184003. 5.44 Craig, V. S. J., Very small bubbles at surfaces—the nanobubble puzzle, Soft Matter, vol. 7, pp. 40–48, 2011. 5.45 Attard, P., The stability of nanobubbles, Eur. Phys. J. Spec. Top., 2013. doi: 10.1140/epjst/e2013-01817-0. 5.46 Seddon, J. R. T., Lohse, D., Ducker, W. A., and Craig, V. S. J., A deliberation on nanobubbles at surfaces and in bulk, Chem. Phys. Chem., vol. 13, pp. 2179–2187, 2012.

PROBLEMS 5.1 Use the relation for the spinodal limits derived in Example 5.1 to predict the theoretical limit of superheat for liquid oxygen as a function of pressure. Plot your results for pressures between 101 kPa and the critical pressure. 5.2 Propane (C3H8) is a hydrocarbon fuel that has a critical temperature of 370 K and a critical pressure of 4260 kPa. The molecular mass of propane is 44.097 kg/kmol. Use the relation for the spinodal limits derived in Example 5.1 to predict the theoretical limit of superheat for liquid propane as a function of pressure. Plot your results for pressures between 101 kPa and the critical pressure

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5.3 The pressure on a sample of saturated R-134a liquid is slowly decreased isothermally from an initial value of 500 kPa. Use the relation for the spinodal limits derived in Section 5.4 with constants for a RedlichKwong fluid to predict the intrinsic limit of superheat at which the sample must begin to vaporize. The critical constants Tc, Pc, and vc for R-134a can be obtained from Appendix II. How close to this limit do you expect the system to get before homogeneous nucleation actually begins? Explain briefly. 5.4 Using the relation for the spinodal limits derived in Section 5.4 with Redlich-Kwong constants, determine the minimum temperature at which vapor is stable, and the maximum temperature at which liquid is stable for methane (CH4) at atmospheric pressure. (For CH4, Tc = 191.1 K, Pc = 4640 kPa.) 5.5 Determine the variation of the equilibrium bubble size with liquid superheat for water at atmospheric pressure. If the kinetic limit of superheat is 280°C, determine the smallest possible bubble that can exist in this system. How many molecules does such a bubble contain? 5.6 Determine the variation of the equilibrium bubble size with pressure for water that is superheated 20°C above its normal saturation temperature. Plot your results for pressure values between 101 kPa and the critical point. 5.7 Determine the vapor pressure inside a water bubble in equilibrium with superheated liquid at atmospheric pressure and 280°C. (Note that this is near the kinetic limit of superheat and is therefore about the smallest possible bubble size that can exist in such a system in equilibrium.) 5.8 Determine the vapor pressure inside a vapor bubble in equilibrium with superheated liquid mercury at atmospheric pressure and 427°C. 5.9 Estimate the number of molecules needed to make up a bubble of critical size for superheated liquid nitrogen at atmospheric pressure and 100 K. 5.10 Use Eqs. (5.104) and (5.106) to estimate the kinetic limit of superheat for liquid nitrogen at atmospheric pressure. Compare your results with the spinodal limit prediction obtained in Example 5.1. 5.11 Use Eqs. (5.104) and (5.106) to estimate the kinetic limit of superheat for liquid oxygen at atmospheric pressure. 5.12 Determine the equilibrium vapor pressure for water vapor in equilibrium with droplets having a diameter of 0.5 μm at 250°C. 5.13 Mercury vapor at 630 K exists in equilibrium with droplets of liquid mercury having a diameter of 2.0 μm. Determine the equilibrium vapor pressure in the system. 5.14 Droplets of liquid water in an atmospheric fog have a mean diameter of 10 μm. If the air temperature is 15°C, estimate the equilibrium partial pressure of water vapor in the air. 5.15 Droplets of liquid water in an atmospheric fog at a location near Lake Tahoe, California have a mean diameter of 5 μm. If the air temperature is 5°C, estimate the equilibrium partial pressure of water vapor in the air if the atmospheric pressure at this location is 78.2 kPa. 5.16 Determine the critical radius of a mercury droplet for supersaturated mercury vapor at atmospheric pressure and a temperature of 600 K. 5.17 In the last stages of the vaporization process inside tubes of an evaporator, the two-phase flow consists of 5 μm droplets of R-22 liquid entrained in the flowing vapor. If the vapor was in equilibrium with the droplets at a (vapor) pressure of 218 kPa, by how much would the equilibrium temperature differ from the standard (flat interface) saturation temperature? 5.18 Estimate the kinetic limit supersaturation vapor pressure for mercury vapor at a temperature of 650 K. 5.19 Steam maintained at atmospheric pressure is cooled slowly below its normal saturation temperature of 100°C. Estimate the temperature corresponding to the kinetic limit of supersaturation for this process. 5.20 Nitrogen gas at atmospheric pressure is cooled slowly below its normal saturation temperature of 77.4 K. Estimate the temperature at which you would expect liquid droplets first to form in the system. Also determine the equilibrium size of the droplets that form at the limiting condition. Based on your computed size, how easy do you think it would be to see these droplets? 5.21 In the last stages of the vaporization process inside an R-134a evaporator, the two-phase flow consists of 0.1 μm diameter droplets entrained in the flowing vapor. As the flow exits the evaporator, a pressure transducer and thermocouple measure the temperature and pressure in the vapor. A test operator records a temperature of 277 K and 340 kPa at this exit condition. The technician believes that there is something wrong with his data. Do you agree with him? Quantitatively justify your answer. 5.22 R-134a is a refrigerant that is commonly used in air conditioning systems and refrigerators. A sealed aluminum 1 liter container contains 0.8 liters of liquid R-134a and 0.2 liters of R 134a vapor. The container is stored at a room temperature of 20°C. An engineer at a refrigeration system manufacturer says that if the seal on the container is broken, the rapid depressurization will cause the liquid inside to violently boil, spraying liquid out of the break in the seal. Do you agree with the engineer’s statement? Quantitatively justify your answer.

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5.23 Propane (C3H8) is a commonly used fuel that has a critical temperature of 370 K and a critical pressure of 4260 kPa. The molecular mass of propane is 44.097 kg/kmol. A sealed steel 20 liter container contains 18 liters of liquid propane and 2 liters of propane vapor. The container is stored (improperly) at a location at a construction site where the temperature reaches 60°C. An engineer says that if the seal on the container is broken at this temperature, the rapid depressurization will cause the liquid inside to violently boil, spraying liquid out of the break in the seal. Do you agree with the engineer’s statement? Quantitatively justify your answer. 5.24 For pure saturated water in contact with a solid flat gold surface, the Hamaker constant is estimated to be Āls = 22.0 × 10 −20 J. It is further estimated that for such a system Āll = 12.1 × 10 −20 J, and the closest approach distances for water-water and water-gold molecular pairs are Df = 0.28 nm and Dm = 0.31 nm, respectively. The bulk pressure of the saturated water is 101 kPa. Use the model of Carey and Wemhoff [5.32] to estimate the distance from the solid surface at which the spinodal temperature (in K) is 20% higher than the spinodal temperature for the bulk liquid water at 101 kPa. Critical constants for water can be obtained from Appendix II. 5.25 Liquid water at 101 kPa and 100 °C is suddenly depressurized to 12.4 kPa. Estimate the number density (#/m3) of bubbles in the resulting superheated liquid having a radius of 1.0 nm. 5.26 Liquid water is initially held in a container at 70 kPa and 90°C. If the pressure is held constant, estimate how high the temperature would have to be raised to raise the equilibrium concentration of bubbles with a radius of 5 nm to 1 bubble per cubic meter of water. 5.27 Liquid water is initially held in a container at 101 kPa and 100°C. If the temperature is held constant, estimate how low the pressure would have to be decreased to raise the equilibrium concentration of bubbles with a radius of 10 nm to 1 bubble per cubic meter of water.

Part II Boiling and Condensation Near Immersed Bodies

6

Heterogeneous Nucleation and Bubble Growth in Liquids

6.1  HETEROGENEOUS NUCLEATION AT A SMOOTH INTERFACE In many applications, vaporization of a liquid is made to occur by transferring heat through the solid wall of some containing structure. In such cases, the hottest liquid in the system will be in the region immediately adjacent to the wall. If enough heat is added to the system, the liquid near the wall may reach and slightly exceed the equilibrium saturation temperature. Since the temperature is highest right at the solid surface, formation of a vapor embryo is most likely to occur at the solid-liquid interface. As noted in Chapter 5, one possibility is homogeneous nucleation within the liquid closest to the solid surface. The alternative is formation of a vapor embryo as a heterogeneous nucleation process in which the embryo formed is in contact with both the solid and liquid phases at the interface. The analysis described in Sections 5.2 and 5.3 for homogeneous nucleation within a metastable liquid phase can be extended to heterogeneous nucleation at the solid-liquid interface if the solid surface is idealized as being perfectly smooth. In general, the shape of a vapor embryo at the surface will be dictated by the contact angle and interfacial tension (see Section 2.2), together with the shape of the surface itself. If the solid surface is flat, the vapor embryo will have a profile shape like that shown in Fig. 6.1. The formation of such an embryo in a system held at constant temperature Tl and pressure Pl is schematically shown in Fig. 6.2. If the embryo shape is idealized as being a portion of a sphere, the geometry dictates that the embryo volume Vv and the areas of the liquid-vapor (Alv) and the solid-vapor (Asv) interfaces are given by

Vv =

πr 3 (2 + 3 cos θ − cos3 θ) (6.1) 3



Alv = 2πr 2 (1 + cos θ) (6.2)



Asv = πr 2 (1 − cos2 θ) (6.3)

where θ is the contact angle and r is the spherical cap radius indicated in Fig. 6.1. Initially, the system shown in Fig. 6.2 contains only superheated liquid, and the solid walls are fully wetted by the liquid. The initial free energy G0 is therefore given by

G0 = Nˆ T gˆl (Tl , Pl ) + ( Asl )i σ sl (6.4)

where Nˆ T is the total number of moles in the system and, by definition, gˆl (Tl , Pl ) = uˆl − Tl sˆl + Pl vˆl . After formation of the embryo, the total free energy of the system G is equal to the sum of contributions associated with the liquid (Gl), vapor (Gv), and interfacial (Gi) regions:

G = Gl + Gv + Gi (6.5)

203

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FIGURE 6.1  An embryo vapor bubble formed at an idealized liquid-solid interface.

The three terms on the rights side of Eq. (6.5) are given by

Gl = ( Nˆ T − Nˆ v ) gˆl (Tl , Pl ) (6.6)



GV = Nˆ v [ gˆ v (Tl , Pv ) + ( Pl − Pv ) vv ] (6.7)



Gi = σ lv Alv + σ sv Asv + ( Asl ) f σ sl (6.8)

where Nˆ v is the number of moles of vapor and gˆ v (Tl , Pv ) = uˆv − Tl sˆv + Pv vˆv . We will also use the fact that

( Asl )i − ( Asl ) f = Asv (6.9)

and Young’s equation (resulting from a force balance at the contact line),

σ sv − σ sl = σ lv cos θ (6.10)

FIGURE 6.2  System considered in the thermodynamic analysis of the formation of an embryo bubble by heterogeneous nucleation.

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Combining Eqs. (6.2)–(6.10), the following relation is obtained for the change in system free energy ΔG associated with appearance of the embryo:



∆G = G − G0 = Nˆ v [ gˆ v (Tl , Pv ) − gˆl (Tl , Pl ) ] + ( Pl − Pv )Vv

(6.11) 1 1                           + 4 πr 2 σ lv  (1 + cos θ) + cos θ(1 − cos 2 θ)  4 2 

If the embryo radius is exactly the right size, (r = re) to be in thermodynamic equilibrium with the surrounding liquid, the gˆ v and gˆl terms in Eq. (6.11) are equal and the relation (6.11) for ΔG simplifies to ∆Ge =



4 2 1 3 1 πre σ lv  + cos θ − cos3 θ  (6.12) 3 4  2 4

Because Tl and Pl are fixed, if we assume that the vapor obeys the ideal gas law and that the embryo is in mechanical equilibrium and satisfies the Young-Laplace equation Pv = Pl + 2σlv/r regardless of size, the right side of Eq. (6.11) is only a function of r. Using these idealizations and appropriate thermodynamic relations, Eq. (6.11) can be used to derive the following Taylor series expansion for ΔG about the equilibrium condition r = re:

∆G =

4 2 4 πσ lv F   P  2 + l  (r − re )2 +… (6.13) πre σ lv F −   3   3 pve 

where

F = F (θ) =

1 3 1 + cos θ − cos3 θ (6.14) 2 4 4

Equation (6.13) is identical to the corresponding expansion (5.80) for the homogeneous nucleation case, except that σi in Eq. (5.80) has been replaced with σlv F. It directly follows from the arguments described in Section 5.2 that the equilibrium condition corresponds to a maximum value of ΔG and is a state of unstable equilibrium. The variation of ΔG with r is expected to look like that for the homogeneous nucleation case (see Fig. 5.9), leading once again to the conclusion that embryos having a radius less than re spontaneously collapse, while those having a radius greater that re spontaneously grow (see Section 5.2). As was done for homogeneous nucleation in Chapter 5, the expansion (6.13) for ΔG can be used to determine the kinetic limit of superheat for the heterogeneous nucleation process illustrated in Fig. 6.2. The details of the analysis are virtually identical to that presented in Section 5.3 for homogeneous nucleation, and hence they will not be repeated here. However, there are two important differences worth noting. First, at the start of the kinetic limit analysis, it is postulated that the equilibrium number density of embryos of n molecules per unit of interface area N n′′ is given by

 −∆G (r )  N n′′ = ρ2/3 N ,l exp   (6.15)  k B Tl 

where ρN,l is the number density of liquid molecules per unit volume and ΔG is the free energy of formation previously defined. For the heterogeneous nucleation process considered here, only liquid molecules near the surface can participate in embryo formation. The prefactor multiplying the exponential term in the relation (6.15) for N n′′ is therefore taken to be ρ2/3 N ,l , which is representative of the number of molecules immediately adjacent to the solid surface per unit of surface area.

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In addition, for the heterogeneous nucleation process considered here, the relationship between the number of molecules n in the embryo and its radius is n=



N A πr 3 2 + 3 cos θ − cos3 θ (6.16) 3 Mvv

(

)

This relation differs from that used in the analysis of homogeneous nucleation because the embryo geometries are different. Analysis of the kinetics of the heterogeneous nucleation process incorporates the modifications noted above and the expansion (6.13) for ΔG developed for this case. Otherwise the analysis is identical to that presented in Section 5.3 for homogeneous nucleation. Carrying the analysis to completion yields the following relation between the rate of embryo formation at the surface J (m−2s−1) and the system properties:

J=

ρ2/3 N ,l (1 + cos θ)  3Fσ lv    2F πm 

1/2

  −16πFσ lv3 exp  2  (6.17)  3k B Tl [ ηPsat (Tl ) − Pl ] 

where

 v [ P − Psat (Tl ) ]  η = exp  l l  (6.18) RTl  

and F is defined by Eq. (6.14). It should be noted that if θ is taken to be zero and ρ2/3 N ,l is replaced by ρN,l, then Eq. (6.17) becomes identical to the expression (5.105) obtained in Section 5.3 for homogeneous bubble nucleation. As in the homogeneous nucleation case, J is interpreted as the rate at which embryos of critical size are generated. As J increases, the probability that a bubble will exceed critical size and spontaneously grow becomes greater. If a threshold value of J is specified as corresponding to the onset of nucleation, the corresponding liquid temperature Tl = TSL can be determined from Eq. (6.17). For superheated liquid at atmospheric pressure, the variation of J with liquid temperature predicted by Eq. (6.17) is shown in Fig. 6.3 for contact angles of 20° and 108°. A contact angle θ of 20° is typical of water on metal surfaces, whereas θ = 108° is representative of the contact angle of water on low energy materials such as Teflon. Also shown is the variation of J with Tl for homogeneous nucleation in superheated water predicted by Eq. (5.105) (θ = 0). As illustrated in Fig. 6.3, J increases rapidly with Tl for both the homogeneous and the heterogeneous nucleation process. However, the curves shift as θ varies so that the rapid increase in J occurs over different temperature intervals for different values of θ. Relative to the homogeneous nucleation curve, for θ > 70° the curve shifts to lower temperatures, whereas for 0 < 65° the curve shifts to higher temperatures. For 65° ≤ θ ≤ 70°, the heterogeneous and homogeneous nucleation curves are at about the same location, implying that in a system of uniform temperature, the two modes are equally probable. Analytical treatments of the kinetics of heterogeneous nucleation similar to that described above have also been developed for other interface geometries, such as spherical and conical cavities and spherical projections. In each case, the analysis must be modified to account for the manner in which the embryo geometry affects the equilibrium conditions and the formation change in free energy. Further description of this type of analysis for other geometries can be found in references [6.1–6.4]. An excellent summary of the results for different geometries is provided by Cole [6.5]. It is noteworthy, however, that the models in these references do not take into account the alteration of embryo free energy that would result from the wall-fluid attractive forces very close to the solid surface that may be important in some systems (see the discussion in Section 5.7).

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FIGURE 6.3  Variation of the rate of embryo formation at a water-solid interface with liquid temperature as predicted for different contact angles by analysis of the kinetics of embryo bubble formation.

Jarvis et al. [6.6] have similarly analyzed heterogeneous nucleation at the interface between a volatile liquid (1) and non-volatile liquid (2) that are immiscible. The profile of the vapor embryo formed at such an interface depends on the interfacial tension at the liquid-liquid and liquid-vapor interfaces. Jarvis et al. [6.6] showed that nucleation will occur at the interface only if the following conditions are satisfied:

σ12 > σ 2 − σ1 (6.19)



σ12 > σ1 − σ 2 (6.20)

If Eq. (6.19) is not satisfied, the volatile liquid 1 spreads on the non-volatile liquid 2, and nucleation occurs entirely within the volatile liquid. If Eq. (6.20) is not satisfied, liquid 2 spreads on liquid 1, and nucleation occurs completely within the non-volatile liquid. This phenomenon is usually referred to as bubble blowing. When Eqs. (6.19) and (6.20) are satisfied, the analysis of Jarvis et al. [6.6] predicts the following relationship between J and the system properties:

  2/3 −16πFll σ13 1 − Z1   3σ1/2 1  exp J = ρN ,l ,1    2  (6.21)    2   πmFll   3k B Tl [ ηPsat (Tl ) − Pl ] 

where

Fll =

3  1  σ2  3 3 (2 − 3 Z1 + Z 2 ) +   (2 − 3 Z 2 + Z1 )  (6.22)  σ1  4  

208

Liquid-Vapor Phase-Change Phenomena



Z1 =

2 σ12 + σ12 − σ 22 (6.23) 2σ1σ12



Z2 =

2 σ 22 + σ12 − σ12 (6.24) 2σ 2 σ12

and η is given by Eq. (6.18). Using Eq. (6.17), which was developed for heterogeneous nucleation on a flat solid surface, it can be shown that for a given threshold J value, the superheat limit will decrease to the saturation as θ → 180°. Decreasing superheat limit as θ → 180° is also typical of the theoretical predictions for other solid surface geometries. However, in most real solid-liquid systems, the contact angle is virtually always less than 180°. Even for water on Teflon, the contact angle is only about 108°. Values above 108° are extremely rare, and contact angles for common liquids on metal surfaces are invariably much lower (see Chapter 3). Example 6.1 A vessel with a Teflon liner holds liquid N2 at atmospheric pressure. If the contact angle at the walls is 70°, estimate the superheat corresponding to the kinetic limit of superheat. For θ = 70°, F = 0.25(2 + 3 cos θ – cos3 θ) = 0.747. The kinetic limit of superheat is specified by Eq. (6.17). Substituting F = 0.747 and 1 + cos θ = 1.342, this equation becomes  0.713σ lv  J = 0.898ρN2/,3l    m



1/ 2

  −12.5σ 3lv exp  2  kBTl [ ηPsat (Tl ) − Pl ] 

where  v l [ Pl − Psat (Tl )]  η = exp   RTl  



For N2 liquid as specified, Pl = 101 kPa, R = 0.297 kJ/kg K, and m = M / NA = 28.0 / 6.02 × 10 26 = 4.65 × 10 −26 kg/molecule. Noting that 2/ 3  N  ρN ,l =  A   vl M 



2/ 3

and that vl , σlv , and Psat , are functions of temperature Tl , it is clear that variation of Tl causes these properties and ρN2/,3l to vary. Using the saturation tables for N2 to evaluate vl , σlv , and Psat with the values of Pl , R, and m noted above and kB = 1.38 × 10 −23 J/K, the values of Jθ=70 corresponding to different values of Tl can be determined. The values of Jθ=70 obtained in this manner for several values of Tl are given in the following table. Tl (K) 105 110 115

Psat(Tl) (kPa)

vl (m3/kg)

σlv (N/m)

log10 Jθ = 70

1083 1467 1940

0.00151 0.00160 0.00171

0.00279 0.00198 0.00118

–63.7 12.8 27.8

While an exact threshold J value corresponding to the kinetic limit of superheat is not known, Fig. 6.3 suggests that J = 1010 m−2s−1 might be about right. Interpolating the results in the above table to obtain the value of Tl corresponding to log10 Jθ=70 = 10 yields (Tl )SL = 109.1 K.

Heterogeneous Nucleation and Bubble Growth

209

6.2  NUCLEATION FROM ENTRAPPED GAS OR VAPOR IN CAVITIES For real systems with contact-angle values significantly less than 180°, the analyses described in Section 6.1 indicate that heterogeneous nucleation of bubbles on flat or projecting solid surfaces requires superheat levels that are only slightly different from those required for homogeneous nucleation. Well-wetted cavities also require superheat approaching the homogeneous nucleation value to initiate boiling at a liquid-solid interface. Wall temperatures on the order of 300°C would be necessary to initiate boiling in water at atmospheric pressure if these predictions were correct. Fortunately, the high wall superheat predicted by these analyses of heterogeneous nucleation is not usually observed experimentally. One exception is the rapid transient heating of a solid surface in contact with a liquid described in Section 5.7. For most other circumstances of interest, boiling is initiated at much lower superheat levels. For boiling of water in a metal pan on a stove, wall superheats of no more than 10–15°C are typically necessary to initiate boiling once the water has reached saturation conditions. Initiation of nucleate boiling at these much lower superheat levels is attributed to the presence of trapped gas in narrow crevices in the solid surface. Unlike the idealized smooth surfaces considered in the previously described models of heterogeneous nucleation, most real solid surfaces invariably contain natural or machine-formed pits, scratches, or other irregularities. The size of these imperfections may range from microscopic to macroscopic. If the surface is not completely wetted by the liquid, it is expected that many of the cavities will contain entrapped gas. Vaporization may occur at the liquid-gas interface in the cavity at relatively low temperatures. The mechanism that causes the entrapment of vapor in cavities in the surface is schematically illustrated in Fig. 6.4a. For a liquid sheet passing over a gas-filled groove, the contact angle with the downslope tends to be maintained as liquid begins to fill the groove. Because the liquid front is convex, it is clear that the “nose” of the liquid front will strike the opposite wall before the contact line reaches the bottom of the groove if the contact angle is greater than the groove angle 2γ. The condition for entrapment of gas by the advancing front can therefore be stated as

θa > 2 γ (6.25)

FIGURE 6.4  Schematic representation of vapor and liquid entrapment in a groove due to motion of the liquid front over the surface.

210

Liquid-Vapor Phase-Change Phenomena

It should be noted that in this case θa represents the advancing contact angle, which may be significantly greater than the static or equilibrium contact angle (see Chapter 3). Contact-angle hysteresis may thus serve to increase the tendency to trap vapor in surface cavities. We next consider a receding liquid front passing over a liquid-filled groove, as shown in Fig. 6.4b. Arguments similar to those above lead to the conclusion that some liquid will be left behind in the groove if θr < 180° – 2γ. Hence the condition for liquid entrapment in the groove may be stated as

θr < 180° − 2 γ (6.26)

Contact-angle hysteresis may result in θr being much lower than the static or advancing contact angles. Thus, contact-angle hysteresis may also act to enhance liquid entrapment in surface cavities. In an early study of liquid and vapor entrapment, Bankoff [6.7] noted that cavities may be classified into one of four categories: (1) cavities that trap gas only, (2) cavities that trap liquid only, (3) cavities that trap both liquid and vapor, and (4) cavities that trap neither liquid nor vapor. The category to which a particular cavity belongs depends on whether Eqs. (6.25) and (6.26) are satisfied. Contact angle hysteresis enhances the probability that cavities that trap gas (category 1 or 3) will be present on the solid surface. The above observations indicate that when the containment is initially filled with liquid, there is a high probability that gas will be trapped in some of the cavities on the surface of the containment wall. If the wall is heated and/or the liquid temperature is raised, vaporization of the liquid may occur first at the liquid-gas interface in cavities containing entrapped vapor. The degree to which entrapped gas in surface cavities can act to initiate vaporization may be affected by the rate at which the gas in the cavity dissolves into the liquid and diffuses away from the gas-liquid interface. The population density of cavities that can act as vaporization initiators may therefore depend on the initial gas concentration in the liquid and the elapsed time between filling the system and the application of heat. Although dissolving of entrapped gas may deactivate a cavity, the opposite can also occur. Gas in a liquid saturated with dissolved gas at room temperature may come out of solution and form gas bubbles in additional cavities as the liquid is heated to its boiling point due to the decrease in its solubility with increasing temperature. There exists an extensive body of evidence that supports the contention that entrapped gas in cavities serves to provide nuclei for the formation of vapor bubbles at the onset of vaporization. The validity of this conclusion has typically been demonstrated in one of two ways. One approach has been to use electron micrographs or high-speed photography to determine the number and location of nucleation sites and clearly document the vaporization and bubble growth process within surface cavities [6.8–6.12]. The fact that entrapped gas in surface cavities facilitates the initial nucleation process has also been confirmed by experiments in which steps were taken to eliminate entrapped gas in cavities [6.13–6.15]. In these experiments, the system was pressurized prior to heating to increase the solubility of gases in the liquid and drive liquid into cavities in the surface, thereby eliminating most of the cavities as nucleation sites. When this is done, the superheat required to initiate nucleation was found to be of the same order as that required for homogeneous nucleation. Once nucleation was initiated, the superheat required to sustain bubble formation typically dropped down to a much lower value. This drop apparently occurred because cavities in the surface refilled with vapor when nucleation commenced. Bubble formation could then be sustained at lower superheat levels because vaporization could occur at the liquid-vapor interface within these reactivated cavities. Once a cavity becomes active, as it emits vapor bubbles, a portion of the entrapped gas is carried away with each bubble. For cavities with simple conical or grooved geometries, this process can, over a few minutes or hours, remove the non-condensable gas from the cavity, leaving only vapor behind. Once this occurs, if the surface and surrounding liquid cools down, the vapor may condense completely, thereby deactivating the cavity.

Heterogeneous Nucleation and Bubble Growth

211

FIGURE 6.5  Growth of a bubble from a reentrant cavity.

Deactivation of nucleation sites in the manner described above may occur in heat-transfer equipment because the duty cycle requires starting and stopping of the heat input, or as a result of variations in the imposed pressure and temperature conditions. In some cases, cavities may deactivate due to penetration of colder bulk liquid into the cavity as a bubble releases. The stability of active cavities subjected to this type of cooling after each bubble is released has been considered in detail by Bankoff [6.16]. Cooling as a result of liquid microlayer evaporation within the cavity, subsequent vapor condensation and liquid penetration into the cavity, and their effect on site stability were also examined by Marto and Rohsenow [6.17]. The tendency for non-condensable gas to be replaced by vapor as bubbles are released varies with the cavity geometry. Particularly noteworthy in this regard are reentrant cavities like that schematically represented in Fig. 6.5. Cavities of this type are typically very stable and may generate bubbles for a very long period of time before the entrapped gas is completely eliminated. The enhanced stability of reentrant cavities is linked to the fact that the interface curvature reverses when liquid penetrates the mouth of the cavity. For such circumstances, thermodynamic considerations dictate that the equilibrium saturation pressure of the vapor will be higher than the flat interface value at the same temperature (see Section 5.5). Consequently, substantial subcooling would be required to condense vapor in the reentrant cavity, making it more difficult to deactivate the site. Unfortunately, most naturally occurring cavities are not reentrant. It is useful at this point to consider the growth of a vapor embryo within a surface cavity. Although the geometry of real cavities is highly irregular, we will idealize the cavity as being conical with a mouth radius R and a cone angle of 2γ. In general, the principal radii of curvature may vary over the interface, as dictated by the Young-Laplace equation, the hydrostatic pressure gradient, the cavity geometry, and the contact angle (see Chapter 2). However, here we will idealize the interface profile as being a section of a sphere, characterized by a single radius of curvature, r. The manner in which the interface profile changes as the vapor embryo grows is illustrated in Fig. 6.6 for three different ranges of contact angle. The qualitative variation of R/r as the embryo grows and emerges from the cavity is indicated in Fig. 6.6d. For a highly wetting liquid, (θ < γ < 90°), the interface radius of curvature starts at some initially small value and increases monotonically as the embryo grows up to and out of the mouth of the cavity (Fig. 6.6a). Consequently, the ratio R/r starts at some finite value and decreases monotonically toward zero. When examining the interface motion, it is useful to consider the apparent contact angle θapp of the interface measured relative to a horizontal line through the liquid. For θ < γ < 90°, the apparent contact angle is less than 90° both before and after the contact line emerges from the cavity, which facilitates the monotonic increase in the radius of curvature r. It is unlikely that the bubble growth shown in Fig. 6.6a would actually occur in the idealized conical cavity considered here because, as described above, entrapment of gas in a “v”-shaped

212

Liquid-Vapor Phase-Change Phenomena

FIGURE 6.6  Variation of bubble radius as the bubble grows within and out of an idealized surface cavity.

cavity will only occur if θ > 2γ. However, if a nominally conical cavity had a small reentrant bottom, this growth pattern may still be obtained. For contact angles in the range 2γ < θ ≤ 90°, r first increases as the interface moves up to the mouth of the cavity (Fig. 6.6b). However, as the contact line turns the corner at the mouth of the cavity, r decreases and then begins to increase again. The complicated r variation is associated with the fact that, as the interface emerges from the cavity, the apparent contact angle changes from a value greater than 90° inside the cavity to a value less than or equal to 90° outside. As a result, the R/r ratio decreases initially, then increases briefly, peaking as the interface emerges from the cavity, and then decreases again toward zero. The interface may initially be concave at even higher contact angles, where θ > 90° + γ. Such a circumstance is shown in Fig. 6.6c. The radius of curvature is then taken to be negative. As the embryo grows and the interface moves toward the mouth of the cavity, the interface becomes flatter and 1/r becomes less negative. As the contact line turns the corner at the mouth of the cavity, the curvature of the interface reverses and the ratio R/r changes sign. As the embryo grows further, R/r reaches a positive maximum and then begins to approach zero. The qualitative variation of R/r with embryo volume for each of the three cases described above is shown in Fig. 6.6d. If we assume that the superheat must exceed the equilibrium value for growth of the embryo to continue, then for the embryo to grow beyond the mouth of the cavity, the system superheat must exceed the equilibrium value for the minimum interface radius. Making use of the approximation that Pve − Pl ≅ Psat(Tl) − Pl (see Section 5.2) and combining the Clausius-Clapeyron and YoungLaplace equations, the condition for the cavity to be active can therefore be stated as

Tl − Tsat ( Pl ) >

2σTsat ( Pl ) vlv (6.27) hlv rmin

Heterogeneous Nucleation and Bubble Growth

213

FIGURE 6.7  Idealized model of vapor trapping process used to estimate initial radius of embryo vapor bubble in the cavity.

The minimum of r corresponds to the maximum value of R/r in Fig. 6.6d as the embryo grows up to the mouth and out of the cavity. The maximum value of R/r occurs near R/r = 1 if the contact angle is large and the embryo becomes large enough that R/r ≤ 1. For such circumstances, r min can be approximated by R. The required superheat for the site to be active is then given by

Tl − Tsat ( Pl ) >

2σTsat ( Pl ) vlv R   for ≤ 1 (6.28) hlv R r

For small contact angles or moderate contact angles in cavities with a large mouth radius, the maximum value of R/r is larger than 2 and corresponds to the initial radius of the embryo. In such instances, a prediction of the initial embryo radius is needed before Eq. (6.27) can be used to determine the conditions necessary for the site to be active. Because fluid-solid combinations in heat-transfer equipment often result in low to moderate contact angles, these circumstances may be commonly encountered in applications involving vaporization processes. A model to predict the initial radius of the vapor embryo has been developed by Lorenz et al. [6.18]. In this model, the cavity is idealized as being conical, and during the filling process the liquid front is assumed to be planar as it passes over the cavity, preserving the contact angle on the downsloping wall, as indicated in Fig. 6.7a. The volume of vapor in the initial gas embryo was taken to be equal to the volume of vapor sealed into the cavity when the planar liquid front contacts the mouth of the cavity (Fig. 6.7a). The radius of curvature is then geometrically determined for a section of the conical cavity bounded by a spherical cap that preserves the contact at the cavity wall and has a volume equal to that determined by the vapor-trapping model. This initial vapor embryo configuration is shown in Fig. 6.7b. The radius of curvature for the initial embryo determined from the model described above is a function of the contact angle and the cone angle 2γ of the cavity. The variation of r/R = Ω with θ and 2γ computed by Lorenz et al. [6.18] using this model is summarized in Fig. 6.8. For given values of θ and 2γ, r/R = Ω can be determined from this figure and the conditions for the site to be active can be specified from Eq. (6.27) as

Tl − Tsat ( Pl ) >

2σTsat ( Pl ) vlv R   for > 1 (6.29) hlv RΩ(θ, 2 γ ) r

214

Liquid-Vapor Phase-Change Phenomena

FIGURE 6.8  Variation of the initial radius of the vapor embryo with the cavity cone angle and contact angle as predicted by the model of Lorenz et al. [6.18].

For θ ≤ 2γ, the computed curves in Fig. 6.7 indicate that r/R is zero and no vapor is entrapped, which is consistent with the more general model of vapor entrapment described earlier. For a given cone angle 2γ, there is a maximum contact angle at which Ω = r/R = 1, Above this maximum value of θ, Ω(θ, 2γ) is taken to be unity, so that Eq. (6.29) becomes equivalent to Eq. (6.28). In most real systems, cavities typically have very small half-angles γ, and the contact angle θ is invariably less than 90°. As a result, it is often found that γ < θ < 90° for cavities in real systems. The variation of R/r = Ω with embryo volume will then exhibit a maximum near R/r = 1, similar to the middle curve in Fig. 6.6d. In addition, the model results plotted in Fig. 6.8 indicate that if the cavity angle 2γ is small, the initial interface radius of curvature is nearly the same as or larger than the mouth radius for all but the smallest contact angles. These observations suggest that in many real systems the minimum interface radius is equal to, or nearly the same as, the cavity mouth radius. This implies that the cavity mouth radius alone is often sufficient to characterize the nucleation behavior of a given site. (See reference [6.19] for a further discussion of this idealization.) If that were true, the nucleation characteristics of a surface would be dictated primarily by the size distribution of potentially active cavities on the surface. The above discussion clearly implies that the minimum interface radius during embryo growth is not always equal to the cavity mouth radius, but may vary with contact angle and cavity half-angle γ:

rmin = Ω(θ, γ ) (6.30) R

Hence, for a given surface having cavities with specified γ values, the associated values of rmin may depend on contact angle θ. Equation (6.27) implies that for a given level of imposed superheat, a cavity will be active if rmin is greater than a critical value r* given by

r* =

2σTsat ( Pl ) vlv (6.31) hlv [ Tl − Tsat ( Pl ) ]

Each cavity of a real surface has a specific rmin value dictated by its geometry and the contact angle θ. An idealized representation of the number distribution of cavities with respect to rmin for a

215

Heterogeneous Nucleation and Bubble Growth

FIGURE 6.9  Idealized representation of the number distribution of cavities with respect to rmin for a typical surface.

typical surface is shown in Fig. 6.9. Only those cavities having rmin values greater than r* will be active. The total number of active sites per unit area na′ is therefore given by ∞

na′ =



 dnc′   drmin min 

∫  dr

r*

As the wall superheat increases, r* decreases and the number of active sites having rmin values greater that r* increases. Example 6.2 For water at atmospheric pressure, estimate the critical bubble radius r* for liquid superheat levels of 2, 10, and 40°C. For water at atmospheric pressure, Tsat = 100°C, σ = 0.05878 N/m, vlv = 1.672 m3/kg, hlv = 2257 kJ/kg. The critical radius r* is determined by substituting these values into Eq. (6.31): r∗ = =

2σTsat (Pl )v lv hlv [Tl − Tsat (Pl )] 2(0.05878)(100 + 273)1.672 (2257 × 1000)[Tl − Tsat (Pl )]

Substituting 2, 10, and 40°C for [Tl – Tsat(Pl )] yields [Tl – Tsat(Pl)] (°C) 2 10 40

r* (μm) 16.20 3.25 0.81

The arguments described above suggest that the number of observed active sites per unit area on a given surface is a function primarily of θ and r*. Experimental data obtained by Lorenz et al. [6.18], Wang and Dhir [6.20], Qi and Klausner [6.21], and others, generally support this conclusion. Data presented by Lorenz et al. [6.18] for boiling of four different liquids on a #240 (sandpaper) finish copper surface are shown in Fig. 6.10. The number of observed active sites na′ was determined by visual counting at different superheat levels [Tl − Tsat,(Pl)]. The measured superheat values were converted to r* values using Eq. (6.31). The number of active sites increases rapidly as the superheat increases (and r* decreases), for all fluids tested. The organic fluids all have a very low contact angle

216

Liquid-Vapor Phase-Change Phenomena

FIGURE 6.10  The number density of active sites observed by Lorenz et al. [6.18] for boiling of different liquids on a copper surface.

on copper. The discussion above implies that they should therefore have almost identical variations of na′ with r*. This is, in fact, observed in Fig. 6.10. Water generally has a higher contact angle on copper, and we would therefore expect more effective vapor trapping, resulting in more sites with large rmin values. Relative to organic fluids, more sites should therefore be active at lower superheat levels (higher r*) and, as indicated in Fig. 6.10, the na′ versus r* curve should be shifted to the right. The na′ versus r* distributions on the log-log plot in Fig. 6.10 are very nearly parallel straight lines. Mikic and Rohsenow [6.22] postulated that the number of cavities having a mouth radius greater than a specified value R* obeys the power-law relation m



R nc′ =  s  (6.32)  R* 

where Rs and m are constants that depend on the surface geometry. If a single equivalent cone angle 2γ represents all surface cavities, then Eqs. (6.30) and (6.32) can be combined to obtain m



 R Ω(θ, 2 γ )  nc′ =  s ∗  (6.33)  rmin

This relation gives the number of sites per unit area having R values greater than R*, or, equiva* lently, rmin values greater than rmin (because rmin is presumed to be proportional to R). The number density of active sites is equal to the number of cavities with rmin values greater than r* given by * Eq. (6.31). Replacing rmin by Eq. (6.31) for r*, we therefore conclude that the number of active sites per unit area na′ is given by the relation  R Ω(θ, 2 γ )hlv [ Tl − Tsat ( Pl ) ]  na′ =  s  (6.34) 2σTsat ( Pl ) vlv   m



Heterogeneous Nucleation and Bubble Growth

217

The line of reasoning described above leads to the conclusion that the number of active sites varies proportional to [Tl − Tsat(Pl)]m for a given fluid and surface combination, which is consistent with the straight-line variation of na′ with [Tl − Tsat(Pl)] implied by the data in Fig. 6.10. The data in Fig. 6.10 further indicate that the exponent m is about the same for all fluids tested, which is consistent with the original assumption that it depends only on the geometry of the solid surface. Equation (6.34) appears to provide a means of characterizing the heterogeneous nucleation behavior of different solid-liquid combinations. However, this relation was developed by considering the behavior of a single isolated cavity. As a result, the accuracy of Eq. (6.34) becomes questionable when the nucleation site density becomes so large that adjacent sites may interact thermally and/or hydrodynamically. The results of this analysis nevertheless provide important physical insight into the mechanisms of the nucleation process during nucleate boiling at low superheat levels. The arguments described above provide a logical basis for expecting that vapor entrapment will play a major role in determining the onset conditions and the number of active heterogeneous nucleation sites per unit of surface area during nucleate boiling. Recent studies by Cornwell [6.10] and Wang and Dhir [6.20] have further explored the effects of cavity geometry on entrapment and its subsequent impact on nucleation. There is abundant evidence that entrapment of vapor has a major impact on the conditions for the onset of nucleate boiling and the active nucleation site density variation with temperature. Qi and Klausner [6.21] concluded however, that the assumption that entrapment is exclusively responsible for seeding heterogeneous nucleation should be questioned. They noted that experiments conducted by Theofanous et al. [6.23] indicated that superheats as low as 10°C resulted in the onset of nucleation for water on contact with a titanium film surface with a mean roughness of 4 nm. They pointed out that the conventional cavity vapor trapping nucleation theory described in this section would predict that a much higher superheat would be required to activate cavities with nanoscale dimensions. In their boiling experiments with ethanol on metal surfaces, Qi and Klausner [6.21] observed that the variation of nucleation site density with superheat was virtually the same for a polished and a rough brass surface. They argued that these experimental results also contradict the tendencies implied by entrapment nucleation theory, and the results suggest that a mechanism other heterogeneous cavity vapor trapping may be involved in seeding pool boiling nucleation sites. Qi and Klausner [6.21] also noted that Tyrrell and Attard [6.24] obtained atomic force microscope images of closely packed nanobubbles on a glass surface with 0.5 nm RMS roughness immersed in liquid water. Qi and Klausner [6.21] speculated that the presence of nanobubbles at some locations on a heated surface could provide active nucleation sites in addition to those resulting from vapor trapping. A second mechanism of this sort could explain some of the observations described above that seem to contradict the hypothesis that vapor entrapment is the sole seeding mechanism for heterogeneous nucleation. Thus, although there is ample evidence that vapor or gas entrapment in surface cavities can be a primary mechanism for the production of heterogeneous nucleation sites, the results of the studies cited here suggest that further research is needed to more fully understand the mechanisms of heterogeneous nucleation in boiling systems with widely different combinations of fluid and surface properties.

6.3  CRITERIA FOR THE ONSET OF NUCLEATE BOILING In Section 6.2, we considered the growth of a vapor embryo and the associated motion of the interface up to and out of the mouth of a cavity. In doing so, we implicitly assumed that the superheat temperature of the surrounding liquid was uniform. In real systems, this is almost never true, and the non-uniformity of the temperature field very often has a significant effect on the nucleation process within surface cavities.

218

Liquid-Vapor Phase-Change Phenomena

FIGURE 6.11  Model ebullition cycle.

The steady cyclic growth and release of vapor bubbles at an active nucleation site is usually termed as the ebullition cycle. In real systems, non-uniform superheat of the liquid surrounding an active cavity on a heated surface is a natural consequence of transient heat-transfer processes during the ebullition cycle. The semi-theoretical model proposed by Hsu [6.25] provides considerable insight into the effects of non-uniform liquid superheat resulting from transient conduction in the liquid during the bubble growth and release process. It also provides an indication of the roles of subcooling, pressure and physical properties in determining the range of surface cavity sizes that will be active. The system considered in Hsu’s model is shown schematically in Fig. 6.11. Initially, a small bubble embryo is assumed to exist at the mouth of the cavity. This bubble presumably was formed by residual vapor left behind after the release of the preceding bubble in an intermittent bubbling process. Cooler bulk liquid has replaced the just-departed bubble. At this initial stage of the process, the fluid adjacent to the wall and the embryo bubble is idealized as being entirely at the bulk temperature T∞. A period of time passes during which liquid adjacent to the wall is heated, and a thermal boundary layer (or penetration layer) grows near the wall as a result of transient conduction. This portion of the bubble growth and release cycle is often referred to as the waiting period. The treatment of the waiting period described by Hsu [6.25] is similar to that proposed in an earlier study by Hsu and Graham [6.26]. It is assumed in this analysis that convection and/or turbulence in the bulk fluid away from the wall limits the thickness of the thermal layer that develops as a result of transient conduction into the liquid near the wall. Although in real turbulent flows the eddy diffusivity generally increases with increasing distance from the wall, in this model, Hsu [6.25] assumed that a limiting thermal layer of thickness δt exists such that for y < δt transport of heat occurs by molecular diffusion alone, whereas for y ≥ δt, vigorous turbulent transport results in a uniform temperature of T∞. Although δt, in a real system may vary with time, we will assume here that δt, is a fixed constant. We will also consider the wall to be at a constant and uniform temperature Tw. The results for a uniform heat flux boundary condition were shown by Hsu [6.25] to be qualitatively the same. Heat transfer to the liquid during the waiting period is modeled as one-dimensional transient conduction, for which the governing equation may be written in the form

 ∂2 θ  ∂θ = α l  2  (6.35) ∂t  ∂y 

where θ = T – T∞. The appropriate initial and boundary conditions are:

θ = 0 at  t = 0 (6.36)

219

Heterogeneous Nucleation and Bubble Growth

FIGURE 6.12  Transient temperature profile near surface.

for t > 0:

θ = θ w = Tw − T∞ at y = 0 (6.37)



θ = 0 at  y = δ t (6.38)

Using elementary methods, the following solution for this system can be obtained (see, for example, Carslaw and Jaeger [6.27])

θ δt − y 2 = + θw δt π

∞





∑ cosnnπ sin nπ  δ δ− y   e t

− n 2 π 2 ( α l t / δ12 )

(6.39)

t

n =1

The qualitative behavior of the temperature profile predicted by this relation is shown in Fig. 6.12. This solution indicates that the region of heated liquid near the wall grows in thickness until its edge reaches y = δt. The temperature profile then adjusts to ultimately establish a linear profile between the wall and y = δt at steady state. The following relation between the equilibrium superheat and the bubble radius re can be derived by combining the Clausius-Clapeyron and Young-Laplace equations and approximating Pve − Pl as Psat(T) − Pl (see Section 5.2).

Tle − Tsat ( Pl ) =

2σTsat ( Pl ) (6.40) ρv hlv re

Hsu [6.25] postulated the following simple relations among the height of the embryo bubble b, the radius of the bubble embryo re, and the mouth radius of the cavity rc:

b = 2rc = 1.6re (6.41)

The exact relations among these variables may, of course, depend on the fluids and the contact angle with the solid. These relations, although simplistic, are at least consistent with idealization of the embryo geometry as a truncated sphere.

220

Liquid-Vapor Phase-Change Phenomena

Hsu [6.25] argued that the embryo bubble would grow and hence the cavity would be an active site if the equilibrium superheat was equaled or exceeded all around the perimeter of the embryo bubble. Because the temperature decreases with increasing y, this will be true if the temperature at y = b is greater than the value of Tle specified by Eq. (6.40). Given Eq. (6.41), and that b is the y value at which we will evaluate the expression for the equilibrium temperature, we define

θsat = Tsat ( Pl ) − T∞ (6.42)

and reorganize Eq. (6.40) to obtain the following relation for the equilibrium θ value θle = Tle – T∞ at y = b for the bubble

θle θsat 3.2σTsat ( Pl )  δ t    at  y = b = 2rc = 1.6re (6.43) = + θw θw θ w ρv hlv δ t  y 

Quantitatively, then, it may be stated that a site is active if

θ θle ≥   at  y = b = 2rc (6.44) θw θw

where θle / θ w is given by Eq. (6.43). This condition can be interpreted graphically by considering Fig. 6.12. The broken curve in this figure represents the equilibrium superheat requirement at y = b given by Eq. (6.43). Cavities having a given size rc and corresponding b value will become active if, during the conduction transient, the value of θ/θ w at y/δt = b/δt = 2rc / δt exceeds the value of θle / θ w determined from Eq. (6.43). The highest temperature achieved at any location corresponds to the steady-state value indicated by the linear profile in Fig. 6.12. There may exist minimum and maximum values of 2rc/δt between which the equilibrium superheat requirement at the corresponding value of y/δt = b/δt is eventually exceeded. These limiting values are determined by the intersection of the linear steady-state profile with the broken curve representing the superheat requirement. Only those cavities on the surface for which the value of 2rc/δt is within these limits will be active nucleation sites. Specifically, a cavity is active if

 2rc  2rc  2rc   δ  ≤ δ ≤  δ  (6.45) t t t min max

The above-specified condition implies that there may be minimum (rc,min) and maximum (rc,max) values of rc that define a range of active cavity sizes on the heated surface. These results imply that for values of rc above rc,max, the embryo bubble is sufficiently large that it protrudes beyond the superheated boundary layer, exposing the upper portion of the bubble to liquid that is below its equilibrium saturation temperature. Condensation of vapor may then occur at these locations, counteracting vaporization that occurs at portions of the bubble interface closer to the wall and preventing the bubble from growing large enough to release from the surface. Such a cavity is unable to sustain the ebullition and is therefore not an active site. For values of rc less than rc,min, the resulting embryo bubble is so small that the required equilibrium superheat cannot be supplied by the wall at the specified superheat level Tw – Tsat(Pl). Thus, cavities with rc values less than rc,min are inactive because the required superheat to make them active cannot be supplied by the system.

Heterogeneous Nucleation and Bubble Growth

221

Mathematically, the values of rc,min and rc,max can be determined by substituting Eq. (6.43) and the linear steady-state relation for the temperature profile

θ δt − y = (6.46) θw δt

into Eq. (6.44), setting y = 2rc, and solving for rc. The resulting quadratic equation can have two real solutions, corresponding to rc,min and rc,max. The relation used to compute these values can be written compactly as

2  rc ,min  δ t  θsat  +   θsat  12.8σTsat ( Pl )  1− − (6.47)   1−  = r 4  θw  −   θ w  ρv hlv δ t θ w   c ,max   

The plus and minus sign on the right side of this equation correspond to rc,min and rc,max, respectively. Equation (6.47) indicates that as the superheat θ w decreases, eventually the collection of terms inside the square root will go to zero, whereupon rc,min and rc,max will be the same. For the superheat level at which this occurs, only one specific cavity size can be active. For values of θ w below this value, the values of rc,min and rc,max computed from this relation are imaginary, implying that no cavities of any size will be active. The range of active sites predicted by Eq. (6.47) is indicated in Figure 6.13 for water at atmospheric pressure and δt = 0.2 mm. For the specified Pl and δt values, a finite range of active cavity sizes exists at higher superheat levels. However, as θ w decreases, the range of active sizes decreases in extent, vanishing completely at a certain threshold value of θ w . This threshold value of θ w is the minimum superheat necessary to initiate and sustain nucleate boiling. The arguments presented in the previous section indicate that a cavity is active if the minimum bubble radius during growth is greater than a limiting value dictated by the superheat of the surrounding liquid (rmin > r* given by Eq. (6.31)). If rmin is simply related to the cavity mouth radius, the arguments presented in the previous section imply that any cavity with a sufficiently large mouth

FIGURE 6.13  Prediction of the range of active cavity sizes using Hsu’s analysis.

222

Liquid-Vapor Phase-Change Phenomena

will be active. Hsu’s analysis indicates that with a finite thermal boundary layer thickness, there is, in fact, an upper bound to the range of cavity sizes that will be active. The model developed by Hsu [6.25] thus predicts two important features of the heterogeneous nucleation on a heated wall: 1. A certain minimum value of wall superheat must be attained before any cavities on the surface will become active nucleation sites. 2. Above the superheat required to initiate nucleation, a finite range of cavities can become active sites with sustained bubble growth and release. The extent of this range depends on the fluid properties, δt and the subcooling (if any) of the bulk fluid. Hsu [6.25] found that experimental observations of nucleate boiling reported by Clark et al. [6.9] were consistent with the trends predicted by this model analysis. Example 6.3 For turbulent flow forced convection of saturated liquid water at atmospheric pressure in a tube, the heat-transfer coefficient at a particular location is 11.0 kW/m2°C. Estimate the required wall superheat to initiate nucleate boiling at this location. Also estimate the range of active cavity sizes at a wall superheat of 5°C. For saturated liquid water, θw = Tsat – Tsat = 0. Substituting the appropriate properties into Eq. (6.47) yields

 rc ,min   rc ,max

 δ   +  2.08 × 10 −4 t  = 1  1− δt θw  4   − 

  

δt is approximated as δt = kl /h = 0.681/(11.0 × 103) = 6.09 × 10 −5 m. Setting the term inside the square-root sign equal to zero, it is found that

(θw )onset = 3.4°C

Since the onset of boiling occurs at θw = 3.4°C, only a few widely spaced sites are expected to be active at θw = 5°C and δt is likely to be close to that for turbulent single-phase convection. Substituting δt = 6.09 × 10 −5 m into the above relation with θw = 5°C yields

rc ,min = 6.65 × 10 −6 m, rc ,max = 2.38 × 10 −5 m

The significance of Hsu’s [6.25] model is not so much its explicit predictive capacity, but the insight it provides into the mechanisms that affect the nucleation process. The model provides a basis for understanding the experimentally determined effects of subcooling, fluid properties, surface finish, and wall superheat on nucleate boiling. In particular, it is clear from the results of this model that the degree to which the distribution of cavity sizes on the surface overlaps the range of potentially active sizes will directly dictate the density of active sites observed at a given superheat level. The results also indicate that decreasing θw will increase the threshold level of superheat θw required to initiate nucleation. This implies that enhancement of bulk convection, (thereby reducing δt) will suppress nucleation. As will be seen in later chapters, this latter effect has important consequences in convective boiling.

6.4  BUBBLE GROWTH IN AN EXTENSIVE LIQUID POOL Before considering further the bubble growth process during nucleate boiling at a heated surface, we will first consider the simpler case of bubble growth in an extensive uniformly superheated liquid. Many of the more complex features of bubble growth near a heated solid surface are absent

Heterogeneous Nucleation and Bubble Growth

223

FIGURE 6.14  Bubble growing in a superheated liquid.

for these circumstances. However, the spherical symmetry for this simpler case makes it possible to predict the nature of the bubble growth process using relatively simple analytical tools. In addition, since many of the fundamental mechanisms are the same, study of bubble growth in an extensive pool of liquid provides considerable insight into mechanisms that also play a role in bubble growth near a heated surface. The physical circumstances of interest here are shown in Fig. 6.14. At any instant during the growth process, the interface is located at r = R and is moving with a velocity dR/dt relative to the laboratory reference frame. The pressure and temperature are P v and Tv inside the bubble and P∞ and T∞ in the surrounding liquid, respectively. When the bubble first forms, the interface temperature will be nearly equal to the superheated liquid temperature, and the vapor generated at the interface will be at a pressure nearly equal to Psat(T∞ ). As the liquid superheat near the interface is consumed to provide the latent heat of vaporization, the temperature at the interface will drop toward Tsat(P∞). During the bubble growth process, the capillary pressure difference across the interface decreases as the radius increases, and the pressure inside the bubble drops toward P∞. During the growth process, P v and Tv will therefore lie in the ranges

P∞ ≤ Pv ≤ Psat (T∞ ) (6.48)



Tsat ( P∞ ) ≤ Tv ≤ T∞ (6.49)

The rate of bubble growth at any instant of time during the growth process is dictated by three factors: 1. The fluid momentum and pressure difference interactions. 2. The rate of heat transfer to the interface to supply the latent heat of vaporization. 3. The thermodynamic constraint (consistent with the assumption of local thermodynamic equilibrium) that Pv = Psat(Tv). There are two limiting cases of the bubble growth process: Inertia-controlled growth. In this regime, Pv is near its maximum value, Psat(T∞), and Tv ≅ T∞. Heat transfer to the interface is very fast and is not a limiting factor to growth. The growth rate is therefore governed by the momentum interaction between the bubble and the surrounding liquid (i.e., it is limited by how rapidly it can push back the surrounding liquid). These conditions usually exist during the initial stages of bubble growth, just after the embryo bubble forms and begins to grow.

224

Liquid-Vapor Phase-Change Phenomena

Heat-transfer-controlled growth. In this regime, Tv is near its minimum value, Tsat(P∞), and Pv ≅ P∞. Growth is limited by the relatively slower transport of heat to the interface. As a result, the interface motion is slow compared to that for inertia-controlled growth, and the momentum transfer between the bubble and the surrounding liquid is not a limiting factor. These conditions generally correspond to the later stages of bubble growth when the bubble is larger and the liquid superheat near the interface has been significantly depleted. The above limiting cases represent the initial and final stages of bubble growth. At intermediate times both heat transfer and liquid inertia effects come into play. This more general case is fairly complex to analyze. However, it is not too difficult to develop reasonably accurate analyses of the two limiting cases described above. For inertia-controlled growth, the treatment originally proposed by Rayleigh [6.28] is particularly illuminating. For the incompressible, radially symmetric flow of liquid near the bubble, conservation of mass requires that

( )  = 0  where u

2 1 ∂ r u  r 2  ∂r 

r=R



=

dR (6.50) dt

Integrating the above equation from r = R to and arbitrary location r yields dR  R  2 (6.51) dt  r 

u=



The total kinetic energy in the moving liquid surrounding the bubble (KE)l can be obtained from the above velocity variation as ∞



2

ρ dR ( KE )l =  l  u 2 dV = 2πρl   R3 (6.52) 2

∫

dt

R

The net work Wl done against the surrounding liquid as the bubble grows from R = 0 to R is given by R



∫ (

)

Wl = Pli 4 πR 2 dR − 0

4 πP∞ R3 (6.53) 3

In the above equation, Pli is the pressure at the interface in the liquid. The second term subtracts work done against the ambient surrounding liquid to accommodate the volume change of the bubble. The right side of Eq. (6.53) thus represents the net work on the liquid which provides the kinetic energy of the liquid motion. Setting the expression (6.52) for (KE)l equal to the relation (6.53) for Wl yields 2



R

dR  4 2πρl     R3 = Pli 4 πR 2   dR − πP∞ R3 (6.54)  dt  3

∫ (

)

0

We further note that Pli is related to the vapor pressure inside the bubble by the Young-Laplace equation

Pli = Pv −

2σ (6.55) R

225

Heterogeneous Nucleation and Bubble Growth

Substituting the relation (6.55) for Pli into Eq. (6.54) and differentiating the entire equation with respect to R, the following equation is obtained 2



R

1 2σ  d 2 R 3  dR   (6.56)   =  Pv − P∞ − 2 + dt 2 dt ρl R

This equation is sometimes referred to as the Rayleigh equation [6.28]. Although this equation was derived from a mechanical energy balance, it also can be interpreted as representing the forcemomentum balance between the bubble and the surrounding liquid during the growth process. For the inertia-controlled growth considered here, 2σ/R is usually much smaller than Pv – P∞. If 2σ/R is neglected compared to Pv – P∞, and the following linearized form of the Clapeyron equation is used to evaluate Pv – P∞, Pv − P∞ =



ρv hlv [ Tv − Tsat ( P∞ ) ] (6.57) Tsat ( P∞ )

Equation (6.56) can be written as d 2 R 3  dR  ρv  hlv [ Tv − Tsat ( P∞ ) ]    =  2 +  (6.58) dt 2 dt ρl  Tsat ( P∞ ) 2



R

Taking Tv to be equal T∞ for inertia-controlled growth, the above non-linear differential equation has the solution 1/2



 2  [T − T (P )]  h ρ  R(t ) =   ∞ sat ∞  lv v  Tsat ( P∞ )  ρl   3 

t (6.59)

which satisfies the initial condition

R = 0 at  t = 0 (6.60)

This result implies that the radius increases linearly with time during the initial inertia-controlled stage of bubble growth. The simplified analysis presented above obviously does not completely account for all mechanisms of momentum exchange during the bubble growth process. In particular, the effects of viscosity and the capillary pressure difference across the interface are neglected. Extended versions of Rayleigh’s analysis that include more complete treatments of the momentum transport have been developed by Forster and Zuber [6.29], Plesset and Zwick [6.30], and Scriven [6.31]. In the latter stages of bubble growth, the rate at which heat is transported to the liquid-vapor interface of the bubble becomes the factor that limits the growth rate. During bubble growth, the governing equation for the transport of heat in the liquid surrounding the bubble is

∂T ∂T  α l  ∂  2 ∂T  +u = r  (6.61) ∂t ∂r  r 2  ∂r  ∂r 

As in the analysis of inertia-controlled growth analyzed above, it follows from continuity and the imposed boundary condition at the interface that the velocity field in the above equation is given by

u=

dR  R  2 (6.62) dt  r 

226

Liquid-Vapor Phase-Change Phenomena

The boundary and initial conditions for Eq. (6.61) are

T (r , 0) = T∞



T ( R, t ) = Tsat ( Pv ) (6.63)



T (∞, t ) = T∞

In addition, the requirement of conservation of mass and thermal energy at the interface can be stated in the form

kl

∂T dR ( R, t ) = ρv hlv (6.64) ∂r dt

Equation (6.61) with conditions (6.63) and (6.64) have been solved exactly by Scriven [6.31] and approximately by Plesset and Zwick [6.30]. The latter investigators showed that for large Jakob number Ja, the solution has the relatively simple form R(t ) = 2C R α l t (6.65)

where

CR =

ρl c pl ( T∞ − Tsat ) 3 Ja     Ja = (6.66) π ρv hlv

Thus for heat-transfer-controlled growth, the bubble radius grows proportional to the square root of time for large values of the Jakob number. The results of the analyses described above imply that for radially symmetric bubble growth in a superheated liquid, the early stages of the bubble-growth process are primarily inertia-controlled, and the long-time growth is heat-transfer controlled. The complete bubble growth process must therefore be characterized by a smooth transition between these regimes. Based on this premise, Mikic et al. [6.32] used the asymptotic limiting behaviors described above to derive the following relation for the variation of the bubble radius with time

R+ =

3/2 2 + t + 1) − (t + )3/2 − 1 (6.67) (   3

where

R+ =

tA2 RA + ,    t = (6.68) B2 B2

 2 [ T∞ − Tsat ( P∞ ) ] hlv ρv  A=  (6.69) 3ρl Tsat ( P∞ )   1/2







 12α l  B=  π  Ja = 

1/2

Ja (6.70)

ρl c pl [ T∞ − Tsat ( P∞ ) ] (6.71) ρv hlv

227

Heterogeneous Nucleation and Bubble Growth

Although Eq. (6.67) looks considerably more complicated than either of the limiting cases individually, it does reduce to Eq. (6.59) for small values of t  + and Eq. (6.65) for large values of t  +. Mikic et al. [6.32] found good agreement between the R +(t +) variation predicted by Eq. (6.67) and data obtained by Lien [6.33] for bubble growth in uniformly superheated liquid water over a range of ambient pressure levels. At t + = 1, Eq. (6.67) predicts a value of R + = 1. Since R + increases monotonically with + t  , R + values much less than one correspond to t + « 1, indicating inertia-controlled growth. Likewise, R+ values much greater than one correspond to t + » 1, indicating heat-transfercontrolled growth. Consequently, the magnitude of R + relative to 1 is an indication of the regime of bubble growth. For a bubble growing in an infinite ambient, this provides a convenient means of approximately determining the regime of bubble growth for a given bubble size R and ambient conditions. While this formulation provides a convenient interpolation between the limiting bubble growth behaviors for large and small t +, its accuracy at intermediate times is by no means guaranteed. In addition, because this formulation uses a linearized form of the Clapeyron equation to relate Pv – P∞ to Tv – Tsat, it becomes less accurate at high superheat levels. Subsequent studies by Theofanous and Patel [6.34] and Prosperetti and Plesset [6.35] have resulted in more refined treatments of the bubble growth process that address these inaccuracies. Example 6.4 Assuming inertia-controlled growth, estimate the interface velocity of a 0.2 mm diameter bubble growing in water at atmospheric pressure and 120°C. For water at atmospheric pressure, Tsat, = 100°C, hlv = 2257 kJ/kg, ρl = 958 kg/m3, and ρv = 0.598 kg/m3. For inertia-controlled growth, Eq. (6.59) specifies that



 2  [T∞ − Tsat (P∞ )]  hlv ρv R=    3  Tsat ( P∞ )  ρl

1/ 2

  t 

It is clear from this relation that dR/dt is a constant, independent of time and bubble diameter. Thus, 1/ 2



dR  2  [T∞ − Tsat (P∞ )]  hlv ρv  =    ρ  dt Tsat (P ) l   3  

1/ 2

2 20  2257(1000)(0.598)  =      + 3 100 273 958   = 7.10 m/s

Example 6.5 For bubble growth in water at atmospheric pressure and 120°C, estimate the bubble size corresponding approximately to the transition between inertia-controlled and heat-transfer-controlled growth. Since the transition corresponds approximately to R + = 1 or, equivalently

Rtrans =

B2 A

228

Liquid-Vapor Phase-Change Phenomena

Using the saturation properties for water at atmospheric pressure (see Example 6.4), it follows that Ja =

ρl c pl [T∞ − Tsat (P∞ )] 958(4.22)(20)  = = 59.9 ρv hlv 0.598(2257)

 12α l  B =   π 

1/ 2

 12kl  Ja =    πρl c pl 

1/ 2

1/ 2

 12(0.681)  Ja =    π(958)(4220) 

(59.9)

= 0.0477 ms −1/ 2  2[T∞ − Tsat (P∞ )] hlv ρv  A =   ρl Tsat (P∞ )  

1/ 2

1/ 2

 2( 20 )( 2,257,000 )( 0.598)  =  958 (100 + 273)  

= 12.29 m/s Rtrans =

(0.0477)2 12.29

= 0.000185 m = 0.185 mm

6.5  BUBBLE GROWTH NEAR HEATED SURFACES The growth of vapor bubbles in the thermal boundary layer region near a superheated surface is more complex than the circumstances considered in the previous section because of the lack of spherical symmetry and the non-uniformity of the temperature field in the surrounding liquid. Despite these significant differences, bubble growth near a superheated surface exhibits regimes of inertia-controlled and heat-transfer-controlled growth similar to those for spherical bubble growth in an infinite uniformly superheated ambient. The bubble growth process near a heated wall can be idealized as consisting of the sequence of stages indicated schematically in Fig. 6.15. After the departure of a bubble, liquid at the bulk fluid temperature T∞ is brought into contact with the surface at temperature Tw > Tsat(P∞). A brief period of time then elapses during which transient conduction into the liquid occurs, but no bubble growth takes place. This time interval is referred to as the waiting period, designated here as t w. Once bubble growth begins, the thermal energy needed to vaporize liquid at the interface comes, at least in part, from the liquid region adjacent to the bubble that was superheated during the waiting period. During the initial stage of bubble growth, the liquid immediately adjacent to the interface is highly superheated, and transfer of heat to the interface is not a limiting factor. As the bubble embryo bubble emerges from the nucleation site cavity, a rapid expansion is triggered as a result of the sudden increase in the radius of curvature of the bubble. The resulting rapid growth of the bubble is resisted primarily by the inertia of the liquid. For this inertiacontrolled early stage of the bubble growth process, the bubble grows in a nearly hemispherical shape, as shown schematically in Fig. 6.15c. In this regime, as the bubble grows radially, a thin microlayer of liquid is left between the lower portion of the bubble interface and the heated wall (Fig. 6.15c). This film, which is sometimes referred to as the evaporation microlayer, varies in thickness from nearly zero near the nucleation site cavity to a finite value at the edge of the hemispherical bubble. Heat is transferred across this film from the wall to the interface, directly vaporizing liquid at the interface. This film may evaporate completely near the cavity where nucleation began, significantly elevating the surface temperature there. When this occurs, the surface temperature may fluctuate strongly during the repeated growth and release of bubbles as the surface cyclically dries out and then rewets. The liquid region adjacent to the interface is gradually depleted of its superheat as the bubble grows. This region is sometimes referred to as the relaxation microlayer. The nature of the temperature profile in this region at an intermediate stage of the bubble growth process is indicated in Fig. 6.15c. The interface is at the saturation temperature corresponding to the ambient pressure in

Heterogeneous Nucleation and Bubble Growth

229

FIGURE 6.15  The waiting period and subsequent growth and release of a vapor bubble at an active cavity site.

the liquid. The liquid temperature increases with increasing distance from the interface, reaches a peak and then decreases toward the ambient temperature. As growth continues, heat transfer to the interface may become a limiting factor, whereupon the bubble growth becomes heat-transfer controlled. If the bubble growth process does become heat-transfer controlled, pressure and liquid inertia forces become relatively smaller, and surface tension then tends to pull the bubble into a more spherical shape. Thus in undergoing the transition from inertia-controlled growth to heattransfer-controlled growth, the shape of the bubble is transformed from a hemispherical shape to a more spherical configuration, as indicated in Fig. 6.15d. Throughout the bubble growth process, interfacial tension acting along the contact line (where the interface meets the solid surface) tends to hold the bubble in place on the surface. Buoyancy, drag, lift, and/or inertia forces associated with motion of the surrounding fluid may act to pull the bubble away. These detaching forces generally become stronger as the bubble becomes larger. The bubble releases, as shown in Fig. 6.15e, when their net effect becomes large enough to overcome the retaining effect of surface tension forces at the contact line. While the above description admits the possibility of both inertia-controlled and heat-transfer-controlled growth regimes, the occurrence or absence of either one depends on the conditions under which bubble growth occurs. Specifically, very rapid, inertia-controlled growth is more likely to be observed if the following conditions exist: High wall superheat High imposed heat flux Highly polished surface having only very small cavities

230

Liquid-Vapor Phase-Change Phenomena

Very low contact angle (highly wetting liquid) Low latent heat of vaporization Low system pressure (resulting in low vapor density) The first four items on this list result in the build-up of high superheat levels during the waiting period. The last two items result in very rapid volumetric growth of the bubble once the growth process begins. The first item and the last two imply that inertia-controlled growth is likely for large values of the Jakob number Ja defined in Eq. (6.71) above. The shape of the bubble is likely to be hemispherical when these conditions exist. Conversely, heat-transfer-controlled growth of a bubble is more likely for: Low wall superheat Low imposed heat flux A rough surface having many large and moderate-sized cavities Moderate contact angle (moderately wetting liquid) High latent heat of vaporization Moderate to high system pressure All of the conditions specified above result in slower bubble growth, which makes inertia effects smaller, or result in a stronger dependence of bubble growth rate on heat transfer to the interface. The more of these conditions that are met, the greater is the likelihood that heat-transfer-controlled growth will result. Analytical treatments of heat-transfer-controlled bubble growth in the non-uniform temperature fields near a superheated wall have been presented by Savic [6.36], Griffith [6.37], Bankoff and Mikesell [6.38], Zuber [6.39], Han and Griffith [6.40], Cole and Shulman [6.41], van Stralen [6.42], and Mikic and Rohsenow [6.43]. The model of Mikic and Rohsenow [6.43] treats the heat-transfercontrolled growth as being governed by a one-dimensional transient conduction process that consists of two parts. During the waiting process, which begins at time t = –t w, transient conduction in the liquid is postulated to satisfy the following well-known one-dimensional transport equation with appropriate boundary and initial conditions:

 ∂2 T  ∂T = α l  2  (6.72) ∂t  ∂y 



T ( y, − t w ) = T∞ (6.73)



T (∞, t ) = T∞ (6.74)



T (0, t ) = Tw for − t w ≤ t < 0 (6.75)

This model problem has a well-known conjugate error function solution that qualitatively varies as shown in Fig. 6.16a. The second part of the overall model incorporates the idealization that conduction of heat to the bubble interface for t > 0 is again a one-dimensional transient process governed by Eq. (6.72). The solution is still subject to the far-field boundary condition (6.74). However, the boundary condition at y = 0 is taken to be the interface temperature Tsat ( P∞ ),

T (0, t ) = Tsat ( P∞ )  for  t > 0 (6.76)

and the initial condition is the temperature field which exists at the end of the first portion of the transient (at time t = 0). The variation of the temperature field during this second portion of the

231

Heterogeneous Nucleation and Bubble Growth

FIGURE 6.16  Transient temperature profiles in bubble growth model.

transient is shown qualitatively in Fig. 6.16b. For t > 0, Mikic and Rohsenow [6.43] obtained the following solution of the overall transient

   y  y T ( y, t ) = T∞ + ( Tw − T∞ ) erfc    − ( Tw − T∞ ) erfc    (6.77)  2 α l ( t + t w )   2 αlt 

The above relation (6.77) for the temperature field can be used to determine the rate of heat transfer and resulting vaporization rate at the bubble interface. Using the vaporization rate predicted in this manner, Mikic and Rohsenow [6.43] derived the following relation for the bubble radius as a function of time

R(t ) =

2Ja 3πα l t π

 Tw − T∞ 1 −  Tw − Tsat

 t w  1/2  t w  1/2    1 +  −     (6.78) t t   

where Ja is the Jakob number defined by Eq. (6.71). To evaluate the waiting time t w, these investigators considered the temperature at the top of a hemispherical bubble of radius rc covering a cavity of the same radius on the heated surface. They postulated that the temperature at this location in the non-uniform temperature field must exceed the equilibrium superheat for the bubble for growth to begin. The relation for the transient variation of the liquid temperature field for the waiting period was used to determine the time at which this condition was satisfied at a given y = rc. This yielded the following relation for the waiting time: 2



  1  rc  (6.79) tw =  ( vv − vl )   4α l  erfc −1  TTsatw−−TT∞∞ + 2(TσTsat w − T∞ )hlv rc     

232

Liquid-Vapor Phase-Change Phenomena

Equation (6.78) with Eqs. (6.79) and (6.71) provide a complete prediction of R(t) for heattransfer-controlled bubble growth near a superheated wall. Mikic and Rohsenow [6.43] reported good agreement between the predicted variation of R with t and observed variations in the experiments of Han and Griffith [6.40]. The R(t) relation for heat-transfer-controlled growth obtained by Mikic and Rohsenow [6.43] will not be accurate when inertia effects are significant and hemispherical growth occurs. For such conditions, vapor production at the interface of the evaporation microlayer at the base of the bubble plays a major role and must be incorporated into the bubble growth model. Example 6.6 Using Eq. (6.79), estimate the waiting period associated with ebullition from a cavity with a mouth radius of 0.1 mm during boiling of saturated water at atmospheric pressure with a wall superheat of 20°C. For water at atmospheric pressure, Tsat = 100°C, vl = 0.00104 m3/kg, vv = 1.673 m3/kg, hlv = 2257 kJ/kg, σ = 0.05878 N/m, and α l = kl / ρl c pl = 5.16 × 10 −7 m2/s. Thus,

2σTsat (vv − v l ) 2( 0.0588)( 373)(1.673 − 0.001) = = 0.0162 (Tw − T∞ ) hlv rc 20(2,257.000) 0.1× 10 −3

(

)

Substituting into Eq. (6.79) yields



tw =

  1 0.1× 10 −3   −1 −7 4 5.16 × 10  erfc ( 0 + 0.0162) 

(

2

)

From standard math tables, erfc−1(0.0162) = 1.70, from which it follows that

tw = 1.68 × 10 −3 s

Analytical models of the transport in the evaporation microlayer have been proposed by Cooper and Lloyd [6.44] and van Stralen et al. [6.45]. For pure fluids at low pressures, the latent heat of vaporization is high and the density of the vapor is low. Because of these tendencies, at low pressures hemispherical growth may persist long enough to significantly deplete the relaxation microlayer adjacent to the dome of the bubble. Evaporation and hemispherical growth may then be sustained solely by vaporization of the evaporation microlayer between the base of the bubble and the wall. van Stralen et al. [6.45] used the Pohlhausen [6.46] result for laminar boundary layer heat transfer to estimate the initial thickness of the microlayer formed as the bubble grows. The Pohlhausen analysis predicts that for laminar forced convection boundary layer flow, the local boundary layer thickness δ at a given downstream location x is given by

δ Ux = 3.012     v  x

−1/2

Pr −1/3 (6.80)

where U is the free stream velocity. In the model proposed by van Stralen et al. [6.45], the initial thickness of the microlayer δl0 at a given radial position is assumed to be given by Eq. (6.80) with δ replaced by δl0, x replaced by r, and U replaced by the interface velocity R = dR / dt at the instant it passes location r. The resulting expression for δl0 is

v r 1/2 δ l 0 = 3.012  l  Prl −1/3 (6.81)  R

233

Heterogeneous Nucleation and Bubble Growth

If we further postulate that R is a power-law function of time t: R = γ   t m (6.82)

it follows that

R = mγ   t m −1 (6.83)

Because R in Eq. (6.81) is evaluated at r = R, r t = (6.84)  R m

and

vt δ l 0 = 3.012  l   m

1/2

Prl−1/3 (6.85)

Because this equation applies at r = R = γ tm,

r t=  γ

1/ m

(6.86)

which, when substituted into Eq. (6.85) yields

 v  δ l 0 = 3.012Prl −1/3  1/l m   mγ 

1/2

r 1/2 m (6.87)

For growth of the hemispherical bubble solely as a result of evaporation of the microlayer under the base of the bubble, the following energy balance relation must be satisfied R



ρv hlv ( 2πR ) R =

∫ 0

kl ( Tw − Tsat )( 2πr ) dr (6.88) δl 0

Substituting Eq. (6.87) into Eq. (6.88) and using Eq. (6.82) to write the remaining R dependence in terms of t, the following equation is obtained

 m   α l Ja Prl1/3 R =    v  3.012(2 − 1 / 2 m )   l

1/2

t −1/2 (6.89)

where Ja is the Jakob number defined by Eq. (6.71). Integrating and using the condition that R = 0 at t = 0 yields the following relation for R(t):

 2α l Ja Prl1/3   m  R=    3.012(2 − 1 / 2m)   vl 

1/2

t1/2 (6.90)

Comparing this relation with Eq. (6.82), it is clear that m = 1/2, whereupon the above equation, after a bit of rearranging, becomes

R = 0.470 Ja Prl−1/6 (α l t )1/2 (6.91)

234

Liquid-Vapor Phase-Change Phenomena

The above analysis suggests that for growth controlled by vaporization of the evaporation microlayer, the variation of the bubble radius with time is proportional to t1/2, just as for heattransfer-controlled growth in which the latent heat is supplied to the bubble dome interface from the relaxation microlayer. As a consequence of this similarity, it can be difficult to separate the effects of one mechanism from the other. van Stralen et al. [6.45] also proposed the following relation for bubble growth in either the inertia-controlled or heat-transfer-controlled regime R(t ) =



R1 (t ) R2 (t ) (6.92) R1 (t ) + R2 (t )

where







R1 (t ) = 0.8165

{

ρv hlv (Tw − Tsat ) exp −(t / td )1/2 ρl Tsat

}  t (6.93)

   t  1/2  T∞ − Tsat  1/2 R2 (t ) = 1.9544 b * exp  −    +  Ja (α l t )   td   Tw − Tsat      t  1/2                +0.3730Prl−1/6 exp  −     Ja (α l t )1/2   td     b* = 1.3908

(6.94)

R2 (td ) − 0.1908 Prl−1/6 (6.95) Ja(α l t )1/2

In the above relations, td is the time of departure of the bubble from the surface. Equation (6.92) is basically a superposition of the growth rate relations R1(t) for inertia-controlled growth and R2(t) for heat-transfer-controlled growth. R2(t) represents the combined effects of vapor generation at the bubble dome interface at the expense of the relaxation microlayer, and vaporization of the evaporation microlayer at the base of the bubble. The bubble growth parameter b* accounts for the fact that only a portion of the vapor bubble dome may be in contact with superheated liquid. If the departure time td is known, b* can be determined from Eqs. (6.94) and (6.95). Equations (6.92)–(6.95) are actually the pure-liquid forms of equations given by van Stralen et al. [6.45], which apply to either pure liquids or binary systems. The more general forms of these relations can be obtained from reference [6.45]. In a companion study [6.47], these same investigators compared the predictions of these theoretically based relations with experimental bubble growth data for water over a range of temperatures. The agreement between the data and the predicted variation of R with t was generally found to be quite good. The models of bubble growth discussed above are mainly analytical, using appropriate idealizations and mathematical tools to extract relations for the bubble growth parameters of interest. However, the advent of rapidly increasing available computing power in recent years has made solving the fundamental governing equations for fluid flow and heat transfer around a growing bubble increasingly tractable. As a result, there have been increasingly detailed models of bubble growth formulated and computationally solved to provide detailed information about transport during bubble growth. Particularly noteworthy examples of such efforts include the studies of Lee and Merte [6.48], Mei et al. [6.49, 6.50], and Dhir and co-workers [6.51–6.53]. Lee and Merte [6.48] computationally solved the transport equations for a spherically symmetric bubble growing in a uniformly superheated liquid. The predictions of bubble radius variation with

Heterogeneous Nucleation and Bubble Growth

235

time for this model were found to agree well with experimental data, and they clearly reflected the inertia-controlled growth trend expected for short times and heat-transfer-controlled growth limit expected for long times. Mei et al. [6.49, 6.50] computed finite difference solutions of the governing equations for transport interactions among the bubble, liquid microlayer, and heater. With some adjustment of model parameters to match experimental data, the model predicted bubble growth rates that agreed well with experimental data over wide ranges of conditions. Son et al. [6.51] also numerically modeled heat transfer adjacent to a growing single bubble on a heated surface. Their model numerically solved the governing conservation equations for mass, momentum, and energy in the liquid and vapor phases and used a level-set method to capture the moving interface. The same investigators [6.52] then extended this type of modeling to model merging of successive bubbles growing at a single nucleation site. Mukherjee and Dhir [6.53] subsequently extended this approach to model lateral merging of bubbles growing at two adjacent active sites on a heated surface. The results of this study are remarkable for two reasons. First, they indicate that the merger process enhances heat transfer by trapping the liquid layer near the wall during merger and by drawing cooler liquid toward the wall during contraction after merger. The other remarkable aspect is the impressive agreement between the predicted bubble shape variations during the merge and experimental observations. Figure 6.17 shows a comparison of the model-predicted bubble shapes and photographed experimental observations of bubble shapes for saturated water at atmospheric pressure with a wall superheat of 5°C. The agreement between the model predictions and the shapes captured during the experiments is extremely good. As computing power continues to increase, it seems clear that models of this sort will be an increasingly useful tools for exploring the physics of bubble nucleation, growth, and release during nucleate boiling. The models of the ebullition cycle described earlier in this section indicate that for a given cavity there will be a specific bubble size at which bubble release occurs and there will be a specific frequency at which bubbles are generated. These quantities, which parameterize the vapor generation process in a useful manner, are discussed in more detail in the next section.

6.6 BUBBLE DEPARTURE DIAMETER AND THE FREQUENCY OF BUBBLE RELEASE The complete process of liquid heating, nucleation, bubble growth and release, collectively referred to as the ebullition cycle, is the central mechanism of heat transfer from a superheated wall during nucleate boiling. Two features of this process that impact the rate of heat transfer during the ebullition cycle are the bubble diameter at departure, dd, and the frequency, f, at which bubbles are generated and released. The bubble growth rate analyses described in previous sections suggest that the departure diameter and frequency of release must be related. The inverse of the frequency τ = 1/f, which is the time period associated with the growth of each bubble, must equal the sum of the waiting period and the time required for the bubble to grow to its departure diameter

1 = τ = t w + t2 R (t )= dd (6.96) f

The frequency of bubble release thus depends directly on how large the bubble must become for release to occur, and, as a consequence, on the rate at which the bubble can grow to this size. The bubble diameter at release is primarily determined by the net effect of forces acting on the bubble as it grows on the surface. Interfacial tension acting along the contact line invariably acts to hold the bubble in place on the surface. Buoyancy is often a major player in the force balance, although its effect depends on the orientation of the surface with respect to the accelerating or gravitational body force vector. For an upward-facing horizontal surface, buoyancy directly acts to

236

Liquid-Vapor Phase-Change Phenomena 00.0 ms

08.0 ms

14.4 ms

15.2 ms

Experimental 00.0 ms

08.0 ms

11.2 ms

11.8 ms

Numerical 16.0 ms

16.8 ms

17.6 ms

20.0 ms

Experimental 13.1 ms

13.5 ms

14.9 ms

15.8 ms

Numerical 24.0 ms

28.0 ms

40.0 ms

52.0 ms

Experimental 22.9 ms

26.1 ms

40.8 ms

52.0 ms

Numerical 60.0 ms

Experimental

62.8 ms

Numerical

FIGURE 6.17  Bubble profiles predicted by the computational model of Mukherjee and Dhir [6.53] and experimental observations of bubble profiles during merging of bubbles growing at two adjacent nucleation sites. Profiles shown are for saturated water at atmospheric pressure with a wall superheat of 5°C. (Reproduced from Mukherjee and Dhir [6.53] with permission, copyright © 2004, American Society of Mechanical Engineers.)

detach the bubble, whereas for a similar downward-facing surface, buoyancy acts to keep the bubble pressed against the wall. The effect of buoyancy will vary around the perimeter of a superheated horizontal cylinder. If the bubble grows very rapidly, the inertia associated with the induced liquid flow field around the bubble may also tend to pull the bubble away from the surface. When the liquid adjacent to the surface has a bulk motion associated with it, drag and lift forces on the growing bubble may also act to detach the bubble from the surface. In addition, because the rate of bubble growth and the

237

Heterogeneous Nucleation and Bubble Growth

shape of the bubble (hemispherical or spherical) may impact the conditions for bubble release, the departure diameter may be affected by the wall superheat or heat flux, the contact angle θ, and the thermophysical properties of the liquid and vapor phases. The departure diameter of bubbles during nucleate boiling has been the subject of numerous investigations over the past eighty years. In experimental studies, the departure diameter was typically determined from high-speed movies of the boiling process. Based on data obtained in this manner, a number of investigators have proposed correlation equations that can be used to predict the departure diameter of bubbles during nucleate boiling. A representative sample of correlations of this type is given in Table 6.1. In this table, many of the correlations are written in terms of the departure Bond number Bod defined as g(ρl − ρv )d d2 (6.97) σ

Bod =



TABLE 6.1 Departure Diameter Correlations (ref. [6.54])

Bo1/d 2 = 0.0208θ

(6.98)

where θ is the contact angle in degrees (ref. [6.55])

(ref. [6.56, 6.57])

  σ Bo1/d 2 =   g ( ρ − ρ ) l v  

−1/ 6

1/ 3

 6 kl (Tw − Tsat )    q′′  

(6.99)

1/ 3

 3π 2ρl α Tl2 g1/ 2 (ρl − ρv )1/ 2  Bo1/d 2 =   σ 3/ 2   where Ja =

Ja 4 / 3

(6.100a)

ρl c pl [Tw − Tsat ( P∞ )] ρv hlv

(6.100b)

(ref. [6.58])

Bo1/d 2 = 0.04Ja where Ja is computed using Eq. (6.100b)

(6.101)

(ref. [6.59])

Bo1/d 2 = C (Ja*)5/ 4

(6.102)

where Ja* =

Tcc pl ρl ρv hlv

(6.103)

C = 1.5 × 10−4 for water C = 4.65 × 10−4 for fluids other than water (ref. [6.60])

Bo1/d 2 = 0.25(1 + 10 5 K1 )1/ 2 for K1 < 0.06   Ja    gρl (ρl − ρv )   σ where K1 =    ρ − ρ   Prl    µ l2 g ( ) l v    2

(ref. [6.61])

(6.104) 3/ 2

  

−1

(6.105)

1/ 2

  Ja  2 10 5  σ d d = 0.25 1 +  g(ρl − ρv )   Prl  Ar 

(6.106)

2

 Ja  1 for 5 × 10 −7 ≤  ≤ 0.1  Prl  Ar where Ja is given by Eq. (100b) and Ar =

gρl2  σ  µ l2  gρl 

3/ 2

is the Archimedes number

(6.107) (Continued)

238

Liquid-Vapor Phase-Change Phenomena

TABLE 6.1 (Continued) Departure Diameter Correlations (ref. [6.62])

 PM dd  c   k BTc 

(ref. [6.63])

Bo1/d 2 = 0.19(1.8 + 10 5 K1 )2 / 3

1/ 3

 P = 5.0 × 10 5    Pc 

−0.46

(6.108) (6.109)

where K1 is given by Eq. (6.105) (ref. [6.65])

(ref. [6.66])

(ref. [6.67])

 Ja 4α l2  d d = C1   g   3 Bod =  6  2 

2/3

1/ 3

 2π  1 + 1 +  3Ja  

 ρv   ρl   ρ   ρ − 1 l v

4/3

(6.110)

2/3

tan −1/ 3θ

Bod = Ca 0.25 Ja 0.75 Ar 0.05

(6.111) (6.112)

µ vVv , Vv = q′′ / ρv hlv , σ cos θ ρ c [T − Tsat ( P∞ )] , and Ja = l pl w ρv hlv where Ca =

Ar =

 σ gρl (ρl − ρv )   g(ρ − ρ )  µ l2 l v

(6.113)

3/ 2

This same dimensionless group in Eq. (6.97) is also sometimes referred to as the Eötvös number. The first correlation (6.98) given in Table 6.1 was developed in early work by Fritz [6.54]. This relation reflects the assumption of a simple balance of surface tension forces and buoyancy at the instant of departure. The effect of the contact angle is taken into account in an empirical manner. Equation (6.99), proposed by Zuber [6.55], was developed from an analysis of the bubble growth in a non-uniform temperature field near a heated surface. The ratio kl (Tw − Tsat ) / q ′′ on the right side of this equation is a length scale that is representative of the superheated thermal layer thickness near the surface. Thus surface tension, buoyancy, and the size of the bubble relative to this layer thickness are represented in this correlation. Equation (6.100a) was proposed by Ruckenstein [6.56] (also cited in reference [6.57]). The form of this relation was obtained by considering the balance between buoyancy, drag, and surface tension forces. Cole [6.58] proposed Eq. (6.101), which indicates a simple functional dependence of departure Bond number on Jakob number Ja = ρl c pl [Tw − Tsat ( P∞ )] / ρv hlv . Cole and Rohsenow [6.59] proposed Eq. (6.102) as an evolutionary improvement of Eq. (6.101). In Eq. (6.102) the wall superheat was replaced by the critical temperature Tc because experimental data contradicted the proportionality between wall superheat and departure diameter implied by Eq. (6.101) Kutateladze and Gogonin [6.60] found that they could correlate a large body of data from the literature with the correlation given by Eqs. (6.104) and (6.105) which contains only the Bond number and the dimensionless group K1. Stephan [6.61] noted that the relation used by Kutateladze and Gogonin [6.60] implies that the departure bond number is a function of the Jakob number, Prandtl number and Archimedes number Ar = ( gρl2 / µ l2 )(σ / gρl )3/2. A plot presented by Stephan [6.61] demonstrates that departure diameter data for a variety of fluids agrees well with Eq. (6.106), which essentially is a relation specifying the functional dependence of departure Bond number on Ja, Prl and Ar. The correlation given by Eq. (6.108) was developed by Borishanskiy et al. [6.62] based on thermodynamic similitude. Jensen and Memmel [6.63] compared several of the correlations described above against available departure diameter data. As observed by Jensen and Memmel [6.63], there is considerable

Heterogeneous Nucleation and Bubble Growth

239

scatter in the available data from different references. Despite differences in the forms of the correlating equations, all of the correlations fit the data to some extent. This is undoubtedly a consequence of the scatter in the data and the empirical nature of the correlations. The correlation of Kutateladze and Gogonin [6.60] given by Eqs. (6.104) and (6.105) was found to provide the best overall fit to the data examined by Jensen and Memmel [6.63]. The average absolute deviation for this correlation was 45.4%. Jensen and Memmel [6.63] also proposed the correlation given by Eqs. (6.105) and (6.109) as an improved version of the correlation of Kutateladze and Gogonin [6.60]. This improved correlation fit the available data to an average absolute deviation of 44.4%. In a later investigation, an alternate model for predicting the bubble departure diameter in pool boiling was developed by Zeng et al. [6.64]. Their model postulates that at the point of departure, the buoyancy force is balanced by the growth force. In contrast, most previous studies of bubble departure in pool boiling from upward facing surfaces have assumed that departure corresponded to a balance between buoyancy and surface tension. The growth force is the reaction force on the bubble due to the force it exerts to displace the surrounding liquid as it grows. The model of Zeng et al. [6.64] links the departure diameter to the mean bubble growth rate. The model of Zeng et al. [6.64] was tested at pressures from 0.02 to 2.8 bar, Jakob number values from 4 to 869, and gravity levels from 1g to 0.014 g. These investigators found that the agreement between the predictions of their model and departure diameter data for a wide variety of boiling systems was slightly better than other available correlations. Because this prediction scheme requires concurrent prediction of the bubble growth rate, it cannot be expressed simply in terms of Bond number, Jakob number, Archimedes number, and contact angle, like other predictive correlations shown in Table 6.1. Gorenflo et al. [6.65] proposed Eq. (6.110) as a means of predicting bubble departure diameters in pool boiling at high heat flux levels. Based on a model of the force-balance in the contact line region of a bubble on a solid surface, Phan et al. [6.66] derived an equation for bubble departure diameter that can be rearranged to the form of Eq. (6.111). The predictions of this relation were compared to bubble departure diameter data for water, HFE-7100, R-11, R-113, FC-72, and n-pentane. The bubble departure diameter predictions of relation were found to agree with the experimental data for these fluids within ±30%. In another recent study, Hamzekhani et al. [6.67]) used a model analysis and a fit to experimental data to develop Eq. (6.112) as a predictor of bubble departure diameter in pool boiling. This relation was found to fit limited experimental data for pure water and pure ethanol to within ±7%. Generally, correlation relations for bubble departure diameter can be designed to provide a good fit to limited experimental data. However, when compared to large data collections from many studies, predictions for correlations like those described above generally exhibit larger mean deviations from the measured data. The resulting considerable scatter in the data about even the best available correlations suggests that there may be considerable uncertainty in reported measurements of bubble departure diameters. It also suggests that current understanding of factors that influence the bubble departure diameter may be incomplete. The frequency of bubble release depends directly on how large the bubble must become for release to occur and on the rate at which the bubble can grow to the release diameter. The frequency of release will therefore be a function of the departure diameter of the bubble and all the conditions and fluid properties that affect the waiting time and the bubble growth rate. As discussed previously, the size and nature of each cavity affects the nucleation and waiting-time behavior. This would seem to at least partially explain why visual observations of nucleate boiling generally indicate that individual sites emit bubbles with a nominally constant frequency, but the observed frequency varies from site to site. Although different cavities will bubble at different frequencies, it is useful to consider the mean bubbling frequency f associated with the boiling process for a given solid-liquid combination and imposed conditions. There have been a number of attempts to develop relations to predict the mean bubble frequency for growth at a heated surface. The accuracy of some of the resulting relations is open to question,

240

Liquid-Vapor Phase-Change Phenomena

however. In a very early study, Jakob and Fritz [6.68] proposed the following relation for hydrogen and water vapor bubbles f d d = 0.078 (6.114)



In subsequent studies, Peebles and Garber [6.69] and Zuber [6.70, 6.71] also proposed relations that suggest that the bubble frequency f is inversely proportional to departure diameter dd. Peebles and Garber [6.69] proposed the relation 1/4



 t g   σg(ρl − ρv )  f d d = 1.18    (6.115) ρl2  t g + t w   

where t w is the waiting time and tg is the growth time to the departure diameter. Based on an analogy between the bubble release process and natural convection, Zuber [6.71] suggested the following relation 1/4



 σg(ρl − ρv )  f d d = 0.59   (6.116) ρl2  

Cole [6.58] showed that 1.18tg /(tg + t w) can vary between 0.15 and 1.4, which implies that Eq. (6.16) is consistent with Eq. (6.115), but that the assumption of a constant multiplier of the term in square brackets for all possible circumstances is an oversimplification. Malenkov [6.72] proposed the following predictive relation for the f d d product that accounted for the effects of heat flux q′′:

f dd =

Vd (6.117) π 1 − [1 + Vd ρv hlv / q ′′]−1

(

)

where 1/2



  d g(ρ − ρv ) 2σ Vd =  d l +  (6.118) ρ + ρ ( ρ + ρ ) 2( ) d l v d l v  

Ivey [6.73] argued that the relation between ƒ and dd is dependent on the regime of bubble growth, with

f 2 d d = constant for dynamically (inertia) controlled growth f 1/2 d d = constant for thermally (heat-transfer) controlled growth

For the intermediate range between these limits, the exponent of ƒ is postulated to change from 2 to 12 . This suggests that Eqs. (6.114)–(6.116) apply to conditions in the intermediate regime between heat-transfer and inertia-controlled growth. For the inertia-controlled regime, Cole [6.74] suggested the relation

4  g ( ρl − ρv ) f 2 d d =   (6.119)  3  Cd ρl

where Cd is a bubble drag coefficient, estimated to equal 1.0 for water at 1 atm.

Heterogeneous Nucleation and Bubble Growth

241

Mikic and Rohsenow [6.43] used their model of the heat-transfer-controlled growth of a bubble in the non-uniform temperature field near a heated surface to evaluate the waiting and growth times and derived the following relation for the bubble frequency 4 f 1/2 d d =    Ja  3πα l  π



1/2  t g  1/2   tg   + + − 1 (6.120) 1     tw + tg   t w + t g  

These investigators further showed that for 0.15 < t w /(t w + tg) < 0.8, this expression is well approximated by the following simpler relation f 1/2 d d = 0.83 Ja πα l (6.121)



where Ja is the Jakob number defined by Eq. (6.71) The relations described above for predicting the bubble frequency are based on theoretical arguments and a limited quantity of experimental data. Because their accuracy has not been extensively verified, and there are differences among the trends in different sets of data, they should be treated as being approximate, and used over the limited range of conditions where supporting data have been obtained. Developing simple relations for bubble release frequency f that are applicable over broad ranges of operating conditions is challenging because it is influenced by so many factors: duration of waiting and growth periods, bubble diameter at departure, and bubble growth rate, which, in turn are dependent on the mechanisms of heat transfer, adjacent nucleation site interaction, microlayer evaporation contribution, and bubble merger. Despite their limited accuracy as predictive tools, the departure diameter and frequency relations described above provide some useful insight into important aspects of nucleate boiling heat transfer. However, recent research explorations of the fundamental mechanisms of nucleation, bubble growth, and release at heated surfaces have reflected a shift in research strategy away from simple model and data correlation development to the more frequent use of two different modern strategies. The first is the increased use of microstructured, microcontrolled, and microinstrumented surfaces to explore the details of the linkage between heat transfer, bubble growth rate, and bubble departure diameter at higher time and space resolutions. Examples of this type of approach include the studies of Rule and Kim [6.75], Bae et al. [6.76], Demiray and Kim [6.77, 6.78], Myers et al. [6.79], Chao et al. [6.80], and Hutter et al. [6.81]. Modern research in this area is also embracing the development and use of more powerful computational fluid dynamics and heat-transfer tools to model the mechanisms of nucleate boiling. Continual increases in available computational power together with development of advanced computational schemes to model multiphase flow, heat transfer and phase change have improved capabilities to explore the physics and predict boiling heat-transfer performance for applications. The Volume of Fluid (VOF) method [6.82] is particularly well suited to problems involving bubbles or droplets, but use of other types of models has also been explored. Examples of studies of this type include the work of Son et al. [6.52], Son [6.83], Yang et al. [6.84], Stephan and Kunkelmann [6.85], and Lamas et al. [6.86]. The physics of nucleate boiling and strategies for modeling it and predicting the resulting heat transfer are discussed further in Chapter 7. Example 6.7 Estimate the bubble departure diameter and bubbling frequency for saturated water at atmospheric pressure for a wall superheat of 20°C. Compare the period between bubbles with the waiting time determined in Example 6.6.

242

Liquid-Vapor Phase-Change Phenomena

For water at atmospheric pressure, Tsat = 100°C, vl = 0.00104 m3/kg, vv = 1.673 m3/kg, hlv = 2257 kJ/kg, σ = 0.0588 N/m, and cpl = 4.22 kJ/kgK. Thus,

Ja =

ρl c pl Tw − Tsat ( P∞ )  ρv hlv

=

( 4.22 / 0.00104)( 20 ) = 60.2 (1/ 1.673)( 2257)

Using Eq. (6.101), it follows that Bo1/d 2 = 0.04 Ja = 2.40



and using Eq. (6.97), it can be shown that 1/ 2



 σBod  dd =    g ( ρl − ρv ) 

1/ 2

  0.0588 =   9.8 (1/ 0.00104 − 1/ 1.673) 

( 2.40 )

= 6.00 × 10 −3 m = 6.0 mm Using Eq. (6.112),  σg ( ρl − ρv )  fdd = 0.059   ρl2  

1/ 4

1/ 4



 0.0588 ( 9.8)(1/ 0.00104 − 1/ 1.673)  = 0.59   (1/ 0.00104 )2   = 0.0923 m / s

Substituting the value of dd obtained above

f =

0.0923 0.0923 = = 15.4 s −1 0.006 dd

The period τb between bubble releases is therefore τb = 1/f = 1/15.4 = 0.065 s, which is significantly longer than the 0.0017 s waiting period determined in Example 6.6.

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6.11 Wei, C. C., and Preckshot, G. W., Photographic evidence of bubble departure from capillaries during boiling, Chem. Eng. Sci., vol. 19, pp. 838–839, 1964. 6.12 Kosky, P. G., Nucleation site instability in nucleate boiling, Int. J. Heat Mass Transf., vol. 11, pp. 929–930, 1968. 6.13 Harvey, E. N., McElroy, W. D., and Whiteley, A. H., On cavity formation in water, J. Appl. Phys., vol. 18, pp. 162–172, 1947. 6.14 Knapp, R. T., Cavitation and nuclei, Trans. ASME, vol. 80, pp. 1315–1324, 1958. 6.15 Sabersky, R. H., and Gates, C. W., Jr., Effect of pressure on start of nucleation in boiling heat transfer, Jet Propul., vol. 25, p. 67, 1955. 6.16 Bankoff, S. G., The prediction of surface temperatures at incipient boiling, Chem. Eng. Prog. Symp. Ser., vol. 55, no. 29, pp. 87–94, 1959. 6.17 Marto, P. J., and Rohsenow, W. M., Nucleate boiling instability of alkali metals, J. Heat Transf., vol. 91, p. 315, 1966. 6.18 Lorenz, J. J., Mikic, B. B., and Rohsenow, W. M., The effect of surface conditions on boiling characteristics, Proc. Fifth Int. Heat Transfer Conf., vol. IV, p. 35, 1974. 6.19 Griffith, P., and Wallis, J. D., The role of surface conditions in nucleate boiling, Chem. Eng. Prog. Symp. Ser., vol. 561, no. 30, pp. 49–63, 1960. 6.20 Wang, C. H., and Dhir, V. K., Effect of surface wettability on active nucleation site density during pool boiling of saturated water, J. Heat Transf., vol. 115, pp. 659–669, 1993. 6.21 Qi, Y., and Klausner, J. F., Comparison of nucleation site density for pool boiling and gas nucleation, J. Heat Transf., vol. 128, pp. 13–20, 2006. 6.22 Mikic, B. B., and Rohsenow, W. M., A new correlation of pool-boiling data including the effect of heating surface characteristics, J. Heat Transf., vol. 91, pp. 245–250, 1969. 6.23 Theofanous, T. G., Tu, J. P., Dinh, A. T., and Dinh, T. N., The boiling crisis phenomenon. Part 1: Nucleation and nucleate boiling heat transfer, Exp. Therm. Fluid Sci., vol. 26, pp. 775–792, 2002. 6.24 Tyrrell, J. W. G., and Attard, P., Images of nanobubbles on hydrophobic surfaces and their interactions, Phys. Rev. Lett., vol. 87, no. 176104, pp. 1–4, 2001. 6.25 Hsu, Y. Y., On the size range of active nucleation cavities on a heating surface, J. Heat Transf., vol. 84, pp. 207–213, 1962. 6.26 Hsu, Y. Y., and Graham, R. W., An analytical and experimental study of the thermal boundary layer and the ebullition cycle in nucleate boiling, NASATN-D-594, 1961. 6.27 Carslaw, H. S., and Jaeger, J. C., Conduction of Heat in Solids, 2nd ed., Oxford University Press, Oxford, 1978. 6.28 Rayleigh, L., On the pressure developed in a liquid during the collapse of a spherical cavity, Philos. Mag., vol. 34, pp. 94–98, 1917 (also in Scientific Papers, vol. 6, Cambridge University Press, Cambridge, 1920). 6.29 Forster, H. K., and Zuber, N., Dynamics of vapor bubbles and boiling heat transfer, AIChE J., vol. 1, pp. 531–535, 1955. 6.30 Plesset, M. S., and Zwick, S. A., The growth of vapor bubbles in superheated liquids, J. Appl. Phys., vol. 25, pp. 493–500, 1954. 6.31 Scriven, L. E., On the dynamics of phase growth. Chem. Eng. Sci., vol. 90, pp. 1–13, 1959. 6.32 Mikic, B. B., Rohsenow, W. M., and Griffith, P., On bubble growth rates, Int. J. Heat Mass Transf., vol. 13, pp. 657–666, 1970. 6.33 Lien, Y., Bubble growth rates at reduced pressure, Sc.D. Thesis, Mechanical Engineering Dept., M.I.T., 1969. 6.34 Theofanous, T. G., and Patel, P. D., Universal relations for bubble growth, Int. J. Heat Mass Transf., vol. 2, pp. 83–98, 1976. 6.35 Prosperetti, A., and Plesset, M. S., Vapour-bubble growth in a superheated liquid, J. Fluid Mech., vol. 85, pp. 349–368, 1978. 6.36 Savic, P., Discussion on bubble growth rates in boiling, J. Heat Transf., vol. 80, pp. 726–728, 1958. 6.37 Griffith, P., Bubble growth rates in boiling, J. Heat Transf., vol. 80, pp. 721–726, 1958. 6.38 Bankoff, S. G., and Mikesell, R. D., Growth of bubbles in a liquid of initially non-uniform temperature, ASME Paper no.58-A-105, New York, NY, 1958. 6.39 Zuber, N., Vapor bubbles in non-uniform temperature fields, Int. J. Heat Mass Transf., vol. 2, pp. 83–98, 1961. 6.40 Han, C. Y., and Griffith, P., The mechanism of heat transfer in nucleate pool boiling, Parts I and II, Int. J. Heat Mass Transf., vol. 8, pp. 887–914, 1965.

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6.41 Cole, R., and Shulman, H. L., Bubble growth rates of high Jakob numbers, Int. J. Heat Mass Transf., vol. 9, pp. 1377–1398, 1966. 6.42 van Stralen, J. D., The mechanism of nucleate boiling in pure liquids and in a binary mixture, Parts I and II, Int. J. Heat Mass Transf., vol. 9, pp. 995–1046, 1966. 6.43 Mikic, B. B., and Rohsenow, W. M., Bubble growth rates in non-uniform temperature fields, Prog. Heat Mass Transf., vol. II, pp. 283–292, 1969. 6.44 Cooper, M. G. and Lloyd, A. J. P., The microlayer in nucleate pool boiling, Int. J. Heat Mass Transf., vol. 12, pp. 895–913, 1969. 6.45 van Stralen, S. J. D., Sohal, M. S., Cole, R., and Sluyter, W. M., Bubble growth rates in pure and binary systems: Combined effect of relaxation and evaporation microlayers, Int. J. Heat Mass Transf., vol. 18, pp. 453–467, 1975. 6.46 Pohlhausen, E., Der Warmeaustauch Zwishen Festen Korpern und Flussigkeiten mit kleiner Treibung und Warmeleitung, Z. Angew. Math. Mech., vol. 1, pp. 115–121, 1921. 6.47 van Stralen, S. J. D., Cole, R., Sluyter, W. M., and Sohal, M. S., Bubble growth rates in nucleate boiling of water at subatmospheric pressures, Int. J. Heat Mass Transf., vol. 18, pp. 655–669, 1975. 6.48 Lee, H. S., and Merte, H., Jr., Spherical vapor bubble growth in uniformly superheated liquids, Int. J. Heat Mass Transf., vol. 39, pp. 2427–2447, 1996. 6.49 Mei, R., Chen, W., and Klausner, J. F., Vapor bubble growth in heterogenous boiling – I. Formulation, Int. J. Heat Mass Transf., vol. 38, pp. 909–919, 1995. 6.50 Mei, R., Chen, W., and Klausner, J. F., Vapor bubble growth in heterogenous boiling – II. Growth rate and thermal fields, Int. J. Heat Mass Transf., vol. 38, pp. 921–934, 1995. 6.51 Son, G., Dhir, V. K., and Ramanujapu, N., Dynamics and heat transfer associated with a single bubble during nucleate boiling on a horizontal surface, J. Heat Transf., vol. 121, pp. 623–631, 1999. 6.52 Son, G., Ramanujapu, N., and Dhir, V. K., Numerical simulation of bubble merger process on a single nucleation site during pool nucleate boiling, J. Heat Transf., vol. 124, pp. 51–62, 2002. 6.53 Mukheijee, A., and Dhir, V. K., Study of lateral merger of vapor bubbles during nucleate pool boiling, J. Heat Transf., vol. 126, pp. 1023–1039, 2004. 6.54 Fritz, W., Berechnung des Maximalvolume von Dampfblasen, Phys. Z, vol. 36, pp. 379–388, 1935. 6.55 Zuber, N., Hydrodynamic aspects of boiling heat transfer, U.S. AEC Report AECU 4439, June, 1959. 6.56 Ruckenstein, E., Physical model for nucleate boiling heat transfer from a horizontal surface, Bul. Institutului Politech. Bucaresti, vol. 33, no. 3, pp. 79–88, 1961; Appl. Mech. Rev., vol. 16, Rev. 6055, 1963. 6.57 Zuber, N., Recent trends in boiling heat transfer research. Part I: Nucleate pool boiling, Appl. Mech. Rev., vol. 17, pp. 663–672, 1964. 6.58 Cole, R. Bubble frequencies and departure volumes at subatmospheric pressures, AIChE J., vol. 13, pp. 779–783, 1967. 6.59 Cole, R., and Rohsenow, W. M., Correlation of bubble departure diameters for boiling of saturated liquids, Chem. Eng. Prog. Symp. Ser., vol. 65, no. 92, pp. 211–213, 1968. 6.60 Kutateladze, S. S., and Gogonin, I. I., Growth rate and detachment diameter of a vapor bubble in free convection boiling of a saturated liquid, High Temp. Sci., vol. 17, pp. 667–671, 1979. 6.61 Stephan, K., Heat Transfer in Condensation and Boiling, Chapter 10, Springer-Verlag, New York, NY, 1992. 6.62 Borishanskiy, V. M., Danilova, G. N., Gotovskiy, M. A., Borishanskiy, A. V., Danilova, G. P., and Kupriyanova, A. V., Correlation of data on heat transfer in, and elementary characteristics of the nucleate boiling mechanism, Heat Transf.-Sov. Res., vol. 13, no. 1, pp. 100–116, 1981. 6.63 Jensen, M. K., and Memmel, G. J., Evaluation of bubble departure diameter correlations, Proc. Eighth Int. Heat Transf. Conf., vol. 4, pp. 1907–1912, 1986. 6.64 Zeng, L. Z., Klausner, J. F., and Mei, R., A unified model for prediction of bubble detachment diameters in boiling systems – I. Pool boiling, Int. J. Heat Mass Transf., vol. 36, 2261–2270, 1993. 6.65 Gorenflo, D., Knabe, V., and Bieling, V., Bubble density on surfaces with nucleate boiling – Its influence on heat transfer and burnout heat flux at elevated saturated pressures, Proc. 8th Int. Heat Transfer Conf., Hemisphere, Washington, DC, vol. 4, pp. 1995–2000, 1986. 6.66 Phan, H. T., Caney, N., Marty, P., Colasson, S., and Gavillet, J., A model tonpredict the effect of contact angle on the bubble departure diameter during heterogeoues boiling, Int. Comm. Heat Mass Transf., vol. 37, pp. 964–969, 2010. 6.67 Hamzekhani, S., Falahieh, M. M., and Akbari, A., Bubble departure diameter in nucleate pool boiling at saturation: Pure liquids and binary mixtures, Int. J. Refrig., vol. 46, pp. 50–58, 2014.

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6.68 Jakob, M., and Fritz, W., Versuche über den Verdampfungsvorgang, Forsch. Ingenieurwes, vol. 2, pp. 435–447, 1931. 6.69 Peebles, F. N., and Garber, H. J., Studies on motion of gas bubbles in liquids, Chem. Eng. Prog., vol. 49, pp. 88–97, 1953. 6.70 Zuber, N., Hydrodynamic aspects of boiling heat transfer, Ph.D. Thesis, UCLA, USAEC Rept. AECU4439, 1959. 6.71 Zuber, N., Nucleate boiling – The region of isolated bubbles – Similarity with natural convection. Int. J. Heat Mass Transf., vol. 6, pp. 53–65, 1963. 6.72 Malenkov, I. G., Detachment frequency as a function of size of vapor bubbles, Inzh.-Fiz. Zh., vol. 20, pp. 704–708, 1971. 6.73 Ivey, H. J., Relationships between bubble frequency, departure diameter and rise velocity in nucleate boiling, Int. J. Heat Mass Transf., vol. 10, pp. 1023–1040, 1967. 6.74 Cole, R., Photographic study of boiling in region of critical heat flux, AIChE J., vol. 6, pp. 533–542, 1960. 6.75 Rule, T. D., and Kim, J., Heat transfer behavior on small horizontal heaters during pool boiling of FC-72, J. Heat Transf., vol. 121, pp. 386–393, 1999. 6.76 Bae, S., Kim, M., and Kim, J., Improved technique to measure time and space resolved heat transfer under single bubbles during saturated pool boiling of FC-72, Exp. Heat Transf. vol. 12, pp. 265–268, 1999. 6.77 Demiray, F., and Kim, J., Heat transfer from a single nucleation site during saturated pool boiling of FC-72 using an array of 100 micron heaters, Proc. 2002 AIAA/ASME Joint Thermophysics Conference, St. Louis, MO, 2002. 6.78 Demiray, F., and Kim, J., Microscale heat transfer measurements during pool boiling of FC-72: Effect of subcooling, Int. J. Heat Mass Transf., vol. 47, pp. 3257–3268, 2004. 6.79 Myers, J. G., Yerramilli, V. K., Hussey, S. W., Yee, G. F., and Kim, J., Time and space resolved wall temperature and heat flux measurements during nucleate boiling with constant heat flux boundary conditions, Int. J. Heat Mass Transf., vol. 48, pp. 2429–2442, 2005. 6.80 Chao, David F., Sankovic, John M., Motil, Brian J., Yang, W-J., and Zhang, Nengli, Bubble departure from metal-graphite composite surfaces and its effects on pool boiling heat transfer, J. Flow Vis. Image Process., vol. 17, pp. 1–11, 2010. 6.81 Hutter, C., Kenning, D. B. R., Sefiane, K., Karayiannis, T. G., Lin, H., Cummins, G., and Walton, A. J., Experimental pool boiling investigations of FC-72 on silicon with artificial cavities and integrated temperature microsensors, Exp. Therm. Fluid Sci., vol. 34, pp. 422–433, 2010. 6.82 Hirt, C.W., and Nichols, B. D., Volume of fluid (VOF) method for the dynamics of free boundaries, J. Comp. Phys., vol. 39, pp. 201–225, 1981. 6.83 Son, G., Numerical study on a sliding bubble during nucleate boiling, KSME Int. J., vol. 15, pp. 931–940, 2001. 6.84 Yang, J., Guo, L., and Zhang, X., A numerical simulation of pool boiling using CAS model, Int. J. Heat Mass Transf., vol. 46, pp 4789–4797, 2003. 6.85 Stephan, P., and Kunkelmann, C., CFD simulation of boiling flows using the Volume-of-Fluid method within OpenFoam, Proc. ECI Int. Conf. on Boiling Heat Transfer, Florianópolis, Brazil, 3-7 May, 2009. 6.86 Lamas, M. I., Sáiz Jabardo, J. M., Arce, A., and Fariñas, P., Numerical analysis of the bubble detachment diameter in nucleate boiling, J. Phys. Conf. Ser., vol. 395, pp. 012174-1–012174-8, 2012.

PROBLEMS 6.1 A vessel made of stainless steel holds liquid acetone at atmospheric pressure. If the contact angle at the walls is 10°, estimate the wall superheat corresponding to the kinetic limit of superheat. Use J = 1010 m−2s−1 as a threshold value of J corresponding to the superheat limit. 6.2 A vessel made of stainless steel holds saturated liquid nitrogen at atmospheric pressure. If the contact angle at the walls is 5°, estimate the wall superheat corresponding to the kinetic limit of superheat. Use J = 1010 m−2s−1 as a threshold value of J corresponding to the superheat limit. 6.3 Determine and plot the variation of the critical bubble radius r* with pressure for water between atmospheric pressure and the critical pressure for a wall superheat of 10°C. Based on your results how do you expect nucleation behavior to be affected by an increase in system pressure? Briefly explain.

246

Liquid-Vapor Phase-Change Phenomena

6.4 Determine and plot the variation of the critical bubble radius r* with pressure for liquid potassium at atmospheric pressure with wall superheat values of 5, 10, 20, and 40°C. Relevant potassium saturation properties at atmospheric pressure are: Tsat = 755.9°C, vl = 0.00151 m3/kg, vv = 1.996 m3/kg, hlv = 1960 kJ/kg, and σ = 0.0673 N/m. Based on your results how do you expect nucleation behavior to be affected by an increase in the wall superheat? Briefly explain. 6.5 For a thermal boundary layer thickness of 5.0 × 10 −6 m, use Hsu’s analysis [6.25] to determine the size range of active nucleation site cavities for saturated R-134a at 201 kPa flowing over a wall at a temperature of 10°C. 6.6 Using Hsu’s analysis [6.25], compute for saturated water the variation of the range of active nucleation site sizes as a function of pressure for a thermal boundary layer thickness of 3.0 × 10 −6 m and a wall superheat of 15°C. Plot the results for pressures between atmospheric pressure and the critical point. 6.7 For forced convection flow of liquid nitrogen at 360 kPa over a surface held at 100 K, the thermal boundary layer is estimated to be 2.0 × 10 −5 m thick. Using Hsu’s analysis [6.25], determine the size range of active nucleation sites on the surface for (a) saturated liquid and (b) liquid at a bulk temperature of 85 K. 6.8 For water at atmospheric pressure, use Hsu’s analysis [6.25] to plot the variation of the range of active cavity sizes with bulk subcooling for a thermal boundary layer thickness of 2.0 × 10 −6 m and a wall superheat of 15°C. 6.9 Saturated liquid nitrogen at atmospheric pressure flows over a surface in turbulent flow resulting in a thermal boundary layer thickness of 8.0 × 10 −6 m. The surface is held at a wall superheat of 15°C. Use Hsu’s analysis [6.25] to determine if active sites may exist on the surface. If they do exist, determine the expected range of active cavity sizes on the surface. 6.10 Using the relation δ = kl / h to estimate the thermal boundary layer thickness, determine the minimum value of the heat transfer coefficient h that will ensure that nucleate boiling does not occur on a surface held at 67°C over which saturated liquid R-134a is flowing. The pressure in the system is 1190 kPa. 6.11 For saturated liquid, water is flowing over a surface at a local pressure of one atmosphere, estimate the thermal boundary layer thickness that is just sufficiently small enough to ensure that nucleate boiling does not occur if the surface is held at 120°C. 6.12 For R-l 34a liquid at 537 kPa superheated to a bulk temperature of 300 K, estimate the time required for a bubble to grow from essentially zero radius to a radius of 1 mm. 6.13 A specific bubble radius Rtrans corresponds approximately to the transition between the inertiacontrolled growth regime and the heat-transfer-controlled growth regime. Determine and plot the variation of Rtrans with pressure for water superheated above its normal saturation temperature by 10°C. (Hint: Assume that Rtrans corresponds to R+ ≅ 1.) 6.14 A specific bubble radius Rtrans corresponds approximately to the transition between the inertiacontrolled growth regime and the heat-transfer-controlled growth regime. Water at atmospheric pressure is superheated to a temperature 27°C above the saturation temperature. (a) Determine Rtrans for these conditions. (b) A very small amount of a surfactant is added to the water, which reduces the surface tension to half of its original value. Other properties for the water are unchanged. What effect does this have on the transition size Rtrans? 6.15 Using the results of the Mikic and Rohsenow [6.39] model discussed in Section 6.5, determine the waiting period associated with ebullition from a cavity with a mouth radius of 0.1 mm during boiling of saturated liquid nitrogen at atmospheric pressure. Compare your result to that obtained for boiling of saturated liquid mercury from a cavity of the same size at atmospheric pressure. The wall superheat is 10°C. 6.16 Use Eqs. (6.98) and (6.111) to estimate the bubble departure diameter for boiling of saturated liquid nitrogen at atmospheric pressure with a wall superheat of 10°C. Summarize in a table the departure diameter predicted by each equation and the percent difference for contact angles of 10°, 25°, 45° and 60°. 6.17 Use Eqs. (6.100) and (6.116) to estimate the bubble departure diameter and bubble frequency for boiling of saturated liquid nitrogen at atmospheric pressure with a wall superheat of 10°C. Repeat the calculations for wall superheats of 2, 5 and 20°C, and plot the variations of these quantities with superheat. 6.18 Use Eqs. (6.109) and (6.116) to estimate the bubble departure diameter and bubble frequency for boiling of saturated liquid water at atmospheric pressure with a wall superheat of 15°C. Repeat the calculations for wall superheats of 2, 5 and 20°C, and summarize the variations of these quantities with superheat in a table.

247

Heterogeneous Nucleation and Bubble Growth

6.19 (a) Use Eqs. (6.100) and (6.116) to estimate the bubble departure diameter and bubble frequency for boiling of saturated liquid water at atmospheric pressure with a wall superheat of 10°C. (b) A very small amount of a surfactant is added to the water, which reduces the surface tension to 25% of its original value. Other properties for the water are unchanged. What effect does this have on the bubble frequency and departure diameter? 6.20 The bubble frequency-diameter product is a lower bound for the upward velocity of the bubbles departing form the surface. For water boiling at atmospheric pressure on a wall superheated by 70°C, computed the fdd product and dd using Eqs. (6.112) and (6.101). Compare the results with the rise velocity vr for spherical bubbles of diameter dd predicted by the relation vr = 1.79 gd d (ρl − ρv ) / ρl . (This relation is valid for bubbles modeled as solid spheres at Reynolds numbers between 1000 and 105.) 6.21 A flat polished copper surface is immersed in a horizontal position in saturated liquid water at a pressure of 571 kPa. The liquid flows over the surface with a free-stream velocity of 1.0 m/s. The surface is 0.2 m long in the direction of flow and is held at a constant and uniform temperature of 435 K. Cavities with a mouth radius of 0.03 mm have been added over the entire surface. Because of the polishing, naturally occurring cavities on the surface have mouth radii that are less than 5.0 × 10 −7 m. For singlephase forced convection under these conditions, the thermal boundary layer thickness is given by



 µ  δ l = 24.9 x 0.2  l   ρl u∞ 

0.8

where x is the distance downstream of the leading edge of the surface. (a) At the locations x = 0.05 m, 0.1 m, and 0.2 m, are active nucleation sites present or absent? Justify your answer quantitatively. (b) With the system initially at the conditions specified above, the pressure is suddenly decreased to atmospheric pressure. Describe the changes in the nucleation behavior that you expect to observe immediately after the drop in pressure. (c) Estimate the maximum growth rate dR/dt for bubble growth at the surface immediately following the drop in pressure described in part (b).

7

Pool Boiling

7.1  REGIMES OF POOL BOILING Boiling at the surface of a body immersed in an extensive pool of motionless liquid is generally referred to as pool boiling. This type of boiling process is encountered in a number of applications including metallurgical quenching processes, flooded tube and shell evaporators (with boiling on the shell side), immersion cooling of electronic components, and boiling of water in a pot on the burner of a stove. The nature of the pool boiling process varies considerably depending on the conditions at which boiling occurs. The level of heat flux, the thermophysical properties of the liquid and vapor, the surface material and finish, and the physical size of the heated surface all may have an effect on the mechanisms of the boiling process. The regimes of pool boiling are most easily understood in terms of a plot of heat flux q″ versus wall superheat Tw – Tsat for the circumstances of interest. Most of the features of this type of classical pool boiling curve were determined in the early investigations of pool boiling conducted by Nukiyama [7.1], Jakob and Linke [7.2], and Drew and Mueller [7.3]. Strictly speaking, the classical pool boiling curve defined by the work of these and other investigators applies to well-wetted surfaces for which the characteristic physical dimension L is large compared to the bubble or capillary length scale Lb defined as



Lb =

σ g(ρl − ρv )

The discussion in this section is limited to pool boiling of wetting liquids on surfaces with dimensions large compared to Lb. In subsequent sections, features of the boiling curve when the liquid poorly wets the surface or when L/Lb is not large will be examined more closely. For the purposes of this discussion, we will specifically consider boiling at the surface of a body immersed in liquid at the saturation temperature for the ambient pressure. If the surface temperature of the immersed body is controlled and slowly increased, the boiling curve will look similar to that shown in Fig. 7.1. The regimes of pool boiling encountered for a temperaturecontrolled, horizontal, upward-facing, flat surface as its temperature is increased are indicated schematically in Fig. 7.2. The lateral extent of the surface is presumed to be much larger than Lb. As described in the previous chapter, at very low wall superheat levels, no nucleation sites may be active and heat may be transferred from the surface to the ambient liquid by natural convection alone. The heat transfer coefficient associated with natural convection is relatively low, and q″ increases slowly with Tw – Tsat. When the superheat becomes large enough to initiate nucleation at some of the cavities on the surface, the onset of nucleate boiling (ONB) condition occurs. This is designated as point c in Fig. 7.1. For controlled surface temperature, the sudden appearance of this added heat transfer mechanism increases the heat flux, with the result that the system operating point moves vertically upward from point c to point d in Fig. 7.1. Once nucleate boiling is initiated, any further increase in wall temperature causes the system operating point to move upward along section d–f of the curve in Fig. 7.1. This portion of the curve corresponds to the nucleate boiling regime. The active sites are few and widely separated at low wall superheat levels. This range of conditions, corresponding to segment d–e of the curve, is sometimes referred to as the isolated bubble regime. 249

250

Liquid-Vapor Phase-Change Phenomena

FIGURE 7.1  Pool boiling regimes for an independently controlled surface temperature.

With increasing surface superheat, the number of active sites per cm2 of surface increases, and the bubble frequency at each site generally increases. Eventually the active sites are spaced so closely that bubbles from adjacent sites merge together to form columns of vapor slugs that rise upward in the liquid pool. This higher range of wall superheat, corresponding to segment e–f of the boiling curve in Fig. 7.1, is referred to as the regime of slugs and columns. Increasing the wall superheat and heat flux within the regime of slugs and columns, produces an increase in the flow rate of vapor away from the surface. Eventually, mechanisms that facilitate vapor flow away from the surface and/or liquid flow toward the surface reach their maximum limits. When this happens, vapor accumulates near the surface at some locations, and evaporation of the liquid between the surface and some of these adjacent regions of vapor dries out portions of the surface. If the surface temperature is held constant and uniform, dry portions of the surface covered with a vapor film will locally transfer a much lower heat flux than wetted portions of the surface where nucleate boiling is occurring. Because of the reduction in heat flux from intermittently dry portions of the surface, the mean overall heat flux from the surface is reduced. Thus, increasing the wall temperature within the slugs and columns region ultimately results in a peaking and rollover of the heat flux. The peak value of heat flux is termed the critical heat flux (CHF), designated as point f in Fig. 7.1. If the wall temperature is increased beyond the critical heat flux condition, a transition boiling regime, is encountered in which the mean overall heat flux decreases as the wall superheat increases. This regime corresponds to segment f–g on the boiling curve shown in Fig. 7.1. The transition boiling regime is typically characterized by rapid fluctuations in the local surface heat flux and/or temperature values (depending on the imposed boundary condition). These fluctuations

Pool Boiling

251

FIGURE 7.2  Schematic representation of the regimes on the pool boiling curve.

occur because the dry regions are generally unstable, existing momentarily at a given location before collapsing and allowing the surface to be rewetted. The vapor film generated during transition boiling can be sustained for longer intervals at higher wall temperatures. Because the intermittent insulating effect of the vapor blanketing is maintained longer and over a larger fraction of the area at higher wall superheat levels, the time-averaged contributions of the vapor-blanketed locations to the mean heat flux are reduced. The mean heat flux from the surface thus decreases as the wall superheat is increased in the transition regime. As this trend continues, eventually a point is reached at which the surface is hot enough to sustain a stable vapor film on the entire surface for an indefinite period of time. The entire surface then becomes blanketed with a vapor film, thus making the transition to the film boiling regime. This transition occurs at point g in Fig. 7.1. Within the film boiling regime, the heat flux monotonically increases as the superheat increases. This trend is a consequence of the increased conduction and/or convection transport due to the increased driving temperature difference [Tw − Tsat ( Pl )] across the vapor film. Radiative transport across the vapor layer may also become important at higher wall temperatures. In general, once a surface is heated to a superheat level in the film boiling regime, if the surface temperature is slowly decreased, the system will progress through each of the regimes described above in reverse order. Experimental evidence indicates, however, that the path of the boiling curve may differ significantly from that observed for increasing wall superheat. Differences between the

252

Liquid-Vapor Phase-Change Phenomena

FIGURE 7.3  Pool boiling curve exhibiting two transition curves.

increasing and decreasing superheat curves may arise in the transition boiling regime and at the transition between nucleate boiling and natural convection, as indicated in Fig. 7.3. Experimental evidence summarized by Witte and Lienhard [7.4] implies that the path of the transition boiling curve is determined, to a large degree, by the wetting characteristics of the liquid on the solid surface. Witte and Lienhard [7.4] noted that when transition boiling is initiated by increasing the wall superheat beyond the critical heat flux condition, the observed liquid contact angle is usually near the receding contact angle θr for the system. If transition boiling results from decreasing the wall superheat to the point that film boiling collapses, the observed liquid contact angle for liquid intermittently touching the surface is usually near the advancing contact angle θa for the system. For systems in which θa is significantly larger than θr , the transition boiling curves obtained for decreasing and increasing wall superheat may therefore be quite different. The transition boiling curve for decreasing wall superheat may be significantly below that for increasing superheat for such circumstances, as illustrated in Fig. 7.3. These trends are discussed in more detail in Section 7.7. The boiling curves in Fig. 7.3 also differ near the onset of nucleate boiling. For increasing wall superheat, nucleation is not initiated until the onset condition is reached, and then the system jumps upward to the nucleate boiling curve (Fig. 7.1). Once activated, nucleation sites may remain active below the superheat required for onset. As a result, the boiling curve for decreasing wall superheat often simply follows the nucleate boiling curve downward, merging into the natural convection curve near the location where they cross. Thus the boiling curve for rising and dropping wall temperature may differ substantially near the onset condition. Note that the sequence of boiling regime trends discussed above are specific to a surface for which the surface temperature is spatially uniform and constant over time. The boiling curve for a surface subjected to a uniform and controlled heat flux generally takes on a different character. If the applied surface heat flux is slowly increased, the boiling curve typically looks like that shown

Pool Boiling

253

FIGURE 7.4  Boiling curve for increasing controlled heat flux.

in Fig. 7.4. The natural convection, onset of boiling, and nucleate boiling regimes are essentially the same as those observed for the temperature-controlled case shown in Fig. 7.1. One difference between the temperature-controlled and heat-flux-controlled boundary condition is that, at the onset condition, the system state jumps horizontally to the nucleate boiling curve for the heat-fluxcontrolled case. Another difference is that when the heat flux is increased beyond the critical heat flux, the surface temperature must jump to a much higher temperature on the film boiling curve to deliver the increased heat flux. As a result, the boiling curve jumps from point f to point g in Fig. 7.4 before further increase in heat flux can be accommodated. Hence, the system never encounters the transition regime observed when the surface temperature is controlled. For an electrically heated surface, the rise in temperature associated with the jump from nucleate to film boiling at the critical heat flux is very often large enough to melt component materials and burn out the component. As a result, the critical heat flux is often referred to as the burnout heat flux to acknowledge the potentially damaging effects of applying this heat flux level to components cooled by nucleate boiling. Once the jump to film boiling has been made, any further increase in applied heat flux increases the wall superheat, and the system follows basically the same film boiling curve as in the temperature-controlled case. For a surface at a very high heat flux, already in the film boiling regime, if the heat flux is slowly reduced, the system generally tracks down the film boiling curve to point b in Fig. 7.5, which corresponds to the minimum heat flux that can sustain stable film boiling. The system operating point must then jump to the nucleate boiling curve at point c before further reduction in heat flux can be accommodated. As indicated in Fig 7.5, this sequence bypasses the transition boiling regime. Further reduction then typically follows the nucleate boiling curve and the natural convection curve as for the temperature-controlled case previously described. The discussion in this section has been primarily at an overview level. It can be seen, however, even from this limited discussion that the mechanisms of pool boiling are complex. The following sections of this chapter will attempt to describe these mechanisms in more detail.

254

Liquid-Vapor Phase-Change Phenomena

FIGURE 7.5  Boiling curve for decreasing controlled heat flux.

7.2 MECHANISMS AND MODELS OF TRANSPORT DURING NUCLEATE BOILING The importance of nucleate boiling in a wide variety of applications has provided the incentive for numerous investigations of its basic mechanisms over the past 90 years. A substantial number of such efforts have been devoted to understanding and modeling of the transport during the nucleate boiling process. A complete account of all attempts to model the heat transfer and fluid motion during nucleate boiling is beyond the scope of this text. A few of the modeling efforts are described here, however, to illustrate the different types of models that have been considered, and to point out models that illuminate the physics particularly well.

Microconvection Models Rohsenow’s Model Many of the very early models of the nucleate boiling process were based on the assumption that the process of bubble growth and release induced motions of the surrounding liquid that facilitated convective transport of heat from the adjacent surface. Jakob and Linke [7.2] were apparently the first to propose this line of reasoning. Perhaps the most successful application of this approach was made by Rohsenow [7.5] who postulated that heat flows from the surface first to the adjacent liquid, as in any single-phase convection process, and that the high heat transfer coefficient associated with nucleate boiling is a result of local agitation due to liquid flowing behind the wake of departing bubbles. The above reasoning suggests that it may be possible to adapt a single-phase forced-convection heat transfer correlation to nucleate pool boiling, if we could specify the appropriate length and velocity scales associated with the convection process. This reasoning implies that it may be possible to correlate pool boiling heat transfer data with a relation of the form



Nu b =

hLb = A Re nb Prlm kl

(7.1)

255

Pool Boiling

where Lb is an appropriate bubble length scale. The Reynolds number Reb is given by

Re b =

ρvU b Lb µl

(7.2)

where Ub is an appropriate characteristic velocity. Rohsenow [7.5] took the length scale and the velocity scale to be the bubble departure diameter dd and the vapor superficial velocity, defined as 1/ 2



  2σ Lb = d d = C b θ   g ( ρ − ρ ) l v   Ub =

q ′′ ρv hlv

(7.3) (7.4)

where θ is the contact angle and Cb is a constant specific to the system. Note that Eq. (7.3) is essentially equivalent to the departure diameter correlation of Fritz [7.6] (see Section 6.6). This correlation indicates that the departure diameter of a bubble during nucleate pool boiling is a function of surface tension and contact angle. Reducing surface tension generally also reduces contact angle. Thus Eq. (7.3) predicts that reducing surface tension generally reduces the departure diameter. Figure 7.6 illustrates this behavior. This figure shows bubbles releasing at a diameter of about 2 mm for boiling of pure water at a heat flux of 20 W/cm2. 2-propanol is a surface-active agent, so at the small concentration of alcohol in the solution in Fig. 7.6b, the surface tension is about 30% of that for the pure water. Because the concentration is very low, the other properties of the fluid are virtually the same as the pure water case. Thus the only significant change is the reduction in surface tension, and the departure diameter of departing bubbles is clearly smaller. The convective effect of the bubble release process is therefore expected to have a much smaller length scale than the pure water case. The bubble departure diameter is therefore a logical choice for the characteristic length scale for the convection associated with bubble growth and release during nucleate boiling. By using the vapor superficial velocity and vapor density in the numerator of the right side of Eq. (7.2), the bubble Reynolds number so defined can be interpreted as a ratio of vapor inertia to liquid viscous forces.

FIGURE 7.6  Pool boiling from a 1.27 cm square upward-facing surface at atmospheric pressure for (a) pure water and (b) a water/2-propanol solution with a low bulk liquid 2-propanol concentration. In both cases, the pool is saturated liquid and the surface heat flux is 20 W/cm2. (Archive photos, Energy and Multiphase Transport Laboratory, UC Berkeley.)

256

Liquid-Vapor Phase-Change Phenomena

Experimental data indicate that the effect of subcooling on heat transfer disappears rapidly with increasing heat flux or wall superheat. The heat transfer coefficient in the bubble Nusselt number was defined as h=



q ′′ Tw − Tsat ( Pl )

(7.5)

where Tw is the surface temperature and Pl is the ambient pressure. The relationship between the Nusselt number and the Reynolds and Prandtl numbers is postulated to be of the form Nu b = A Re1b− r Prl1− s



(7.6)

Substitution of Eqs. (7.2)–(7.5) into this relation and rearranging yields 1/ 2



 q ′′  σ   µ l hlv  g(ρl − ρv ) 

 1  =  Csf 

1/ r

1/ r

 c pl [Tw − Tsat ( Pl )]  Prl− s / r   hlv  

(7.7)

where Csf =



2Cb θ A

(7.8)

Equation (7.7) is of the form of the well-known Rohsenow correlation [7.5] for pool boiling heat transfer. Recommended values of the constants in this equation are described in the next section. Other Convection Analogy Models Reasoning similar to that followed by Rohsenow [7.5] was used by Forster and Zuber [7.7] in the development of their microconvection model. They also postulated that the heat transfer could be represented with a correlation of the form given by Eq. (7.1). In evaluating the length and velocity scales, however, they made use of bubble growth relations that they had developed in an earlier investigation. For heat-transfer-controlled growth, they showed that the bubble growth radius R and the velocity of the interface are given by

R = Ja ( πα l t )



πα R = Ja  l   4t 

1/ 2

1/ 2

(7.9) (7.10)

where

Ja = 

[T∞ − Tsat ( P∞ )]c pl ρl ρv hlv

(7.11)

Using 2R as the length scale and R as the velocity scale, the bubble Reynolds number and Nusselt number are



Re b =

2 Rρl R = πJa 2 Prl−1 µl

(7.12)

q ′′(2 R) (Tw − Tl ) kl

(7.13)

Nub =

257

Pool Boiling

where T∞ in the Ja definition (Eq. (7.11)) is replaced by Tw. Equation (7.12) indicates that the bubble Reynolds number is independent of bubble radius. If this Reynolds number is interpreted as an indicator of the level of agitation in the liquid, the above expression for Reb suggests that more rapidly growing small bubbles produce about the same agitation of the surrounding liquid as larger bubbles that grow more slowly. Direct determination of a characteristic length scale from Eq. (7.9) is not possible, since the radius varies with time. Based on theoretical arguments, Forster and Zuber [7.7] proposed the following definition for the length scale 2R: 1/ 4



 ρ R 2   R 2   2 R = Rc  l  2   2σ / Rc   Rc  



 4 π 2 α l2 ρl2 σ 2  = Ja  3   [ Psat (Tw ) − Pl ] 

(7.14)

1/ 4

(7.15)

where



Rc =

2σ Psat (Tw ) − Pl

(7.16)

Using this characteristic length scale in the Nusselt and Reynolds numbers defined in Eqs. (7.12) and (7.13), Forster and Zuber [7.7] found that pool boiling heat transfer data could be correlated using the forced convection relation (7.1) with A = 0.0015, n = 0.62, and m = 0.33. Unlike the Rohsenow model described above, the heat transfer coefficient in the Nusselt number defined by Eq. (7.13) is based on Tw – Tl. This suggests that the difference between the wall and liquid bulk temperatures is the temperature difference that characterizes the driving potential for heat transfer. Data for saturated pool boiling, where Tl = Tsat(Pl), generally support this conclusion. However, experimental data for nucleate pool boiling generally correlate better in terms of the wall superheat Tw – Tsat(Pl) even when the liquid pool is significantly subcooled below the saturation temperature. This is particularly surprising, because for subcooled boiling, the agitation mechanism may be considerably different from that for saturated boiling. For saturated boiling the bubbles will simply grow and release, whereas for subcooled boiling, bubbles tend to collapse as they rise after release and some bubbles may grow only until they extend far enough into the subcooled ambient for condensation to cause them to collapse again. As noted by Forster and Greif [7.8], the microconvection model implies that for subcooled pool boiling, the heat flux should be a function of the temperature difference between the wall and the bulk fluid. This is not observed in experimental data, which suggest that the microconvection model outlined above is not a completely accurate model of the mechanism. Experiments and subsequent analysis by Gunther and Kreith [7.9] indicate that in many instances the latent heat of vaporization associated with bubble formation during subcooled nucleate boiling represents only a small fraction of the total heat transfer rate from the surface. Based on more recent studies, other investigators (see, e.g., Jung et al. [7.10]) have also concluded that the majority of the heat transfer from the surface occurs by convection to the liquid on the surface in pool boiling processes. These conclusions appear to justify the use of microconvection models as a framework for modeling nucleate boiling processes. Vapor-Liquid Exchange Models The vapor-liquid exchange model proposed by Forster and Greif [7.8] provides a means of explaining the seemingly contradictory trends noted above. This model postulates that bubbles act as microscopic pumps, which draw cold ambient fluid to the surface as the bubble releases

258

Liquid-Vapor Phase-Change Phenomena

or collapses. As a new bubble grows, the pumping action pushes heated liquid from the nearwall region out into the cooler ambient. Forster and Greif [7.8] further assumed that each bubble pumps a quantity of liquid equal to its volume at release. Based on this postulated behavior, if the bubble grows hemispherically to a maximum radius R max, and the bubble frequency is f the sensible contribution to the heat transfer rate from the surface at a given site qs is given approximately by



 2π 3  Tw + Tl  2π 3  1  qs = ρl c pl   Rmax − Tl  f = ρl c pl   Rmax   ( Tw − Tl ) f  2   3   3  2

(7.17)

If na′ is the density of active nucleation sites on the surface, and natural convection between the sites is neglected, the mean heat flux from the surface is given by



 2π 3  1  ( − ) q ′′ = ρl c pl   Rmax   Tw Tl f na′  3  2

(7.18)

Equation (7.17) appears to imply a strong dependence of heat transfer rate on Tw – Tl. However, Forster and Greif [7.8] demonstrated using experimental data that the right side of Eq. (7.18) is, in fact, largely independent of subcooling. As the subcooling is increased, the temperature difference Tw – Tl obviously increases. But increasing the subcooling also was shown to decrease the maximum radius Rmax and increase the bubble frequency f. The net result is that these three effects approximately compensate for each other, resulting in very little change in the right side of Eq. (7.18). A change of over 300% in subcooling was found to produce only about a 20% change in the heat transfer rate per nucleation site. The question then arises: even if the heat transfer per site is insensitive to subcooling, why should the total heat flux from the surface depend primarily on wall superheat? One reason is that in Eq. (7.19) both Rmax and na′ increase with increasing wall superheat. As described in Chapter 6, higher superheat will produce more rapid bubble growth, causing the bubble to achieve a larger radius before releasing or collapsing. Likewise, more (and smaller sites) can become active as the wall superheat increases and the critical radius gets smaller. As a result of these trends, the heat flux varies with superheat even though its dependence on subcooling is weak. Using the vapor-liquid exchange concept developed for subcooled boiling, Han and Griffith [7.11] developed a more detailed model of saturated boiling. A similar model was subsequently used by Mikic and Rohsenow [7.12] to derive a correlation for saturated pool boiling heat transfer. They postulated that in the isolated bubble regime, the surface can be envisioned as consisting of regions of natural convection between active nucleation sites, and regions immediately adjacent to active sites that are cooled by microconvection resulting from the liquid-vapor exchange mechanism. Microconvection regions were postulated to correspond to an area of influence within a circle around each active site with a radius equal to twice the bubble departure radius. After departure of a bubble, colder fluid at Tsat(Pl) is brought into contact with the surface in the region of influence, initiating transient conduction of heat into the liquid. The Mikic and Rohsenow [7.12] model incorporates the following additional idealizations: i. The actual conduction process was modeled as the simple one-dimensional transient conduction process into a semi-infinite medium considered in the bubble growth model of Mikic and Rohsenow [7.13]. ii. The mean heat flux density over the entire surface is expected to be equal to the areaweighted average of the microconvection and natural-convection contribution. The natural convection heat flux is expected to be so small that its contribution may be neglected.

259

Pool Boiling

iii. The variations of bubble frequency f, departure diameter dd were determined from relations of the type discussed in Chapter 6 (see references [7.14, 7.15]). iv. Mikic and Rohsenow [7.12] argued that the cavity mouth radius alone characterizes the nucleation behavior of a given site and the number of active nucleation sites should vary as a power-law function of cavity mouth radius R (see reference [7.16] for a further discussion of this idealization). Developing submodel relations based on the idealizations listed above, Mikic and Rohsenow [7.12] derived a relation linking mean surface heat flux q′′ and wall superheat. This relation can be written in the form:



q* =

q ′′ µ l hlv

1/ 2

  σ  g(ρ − ρ )  l v  

= B[φ(Tw − Tsat )]m +1

(7.19)

where 1/( m +1)



m − 23/8 m −15/8   kl1/ 2 ρ17/8 c19/8 ρv l pl hlv φ= 9/8 m −11/8 m −15/8  Tsat  µ l (ρl − ρv ) σ 

(7.20)

and B is a system-dependent constant. Equation (7.19) has basically the same form as Rohsenow’s [7.5] original correlation as specified in Eq. (7.7). The advantage of Eq. (7.19) is that the model provides a way to predict the constants B and m if the properties of the fluids, the cavity size distribution, and the bubble departure diameter and frequency relations are known. Once B and m are determined, the above relation would specify the nucleate boiling curve. Mikic and Rohsenow [7.12] found that nucleate boiling data for several different systems were well correlated in terms of q* and (Tw – Tsat), although the values of B and m varied from one system to the next. The data of Addoms [7.17] for water are plotted in terms of these variables in Fig. 7.7. Natural-Convection Analogy Model Zuber [7.18] proposed a model of the microconvective effect during nucleate boiling based on an analogy between nucleate boiling and single-phase turbulent natural convection. Beginning with the form of a heat transfer correlation for turbulent natural convection, Zuber [7.18] substituted appropriate length, velocity and time scales into the correlation and evaluated the fluid density as the mean density of the two-phase system, accounting for the vapor void fraction at the surface. The resulting correlation has the form 1/3



q ′′L d f (ρl − ρv )   π  gL3  = C0  β(Tw − Tsat ) + na′ d d2 d   kl 6 ρl uT    vl α l 

(7.21)

Note in the above relation that the length scale L cancels out. C0 is an undetermined constant. In the square bracket on the right side, the second term can be evaluated only if the active site density na′ , bubble departure diameter dd, bubble frequency f, and (terminal) rise velocity of the vapor bubbles uT are known. Using experimental data and appropriate correlations, Zuber [7.18] demonstrated that Eq. (7.21) is consistent with nucleate boiling heat transfer data in the isolated bubble regime, supporting the idea of an analogy between nucleate boiling and turbulent natural convection.

260

Liquid-Vapor Phase-Change Phenomena

FIGURE 7.7  Pool boiling data obtained by Addoms [7.17] for water on a heated horizontal wire. (Adapted from Mikic and Rohsenow [7.12] with permission, copyright © 1969, American Society of Mechanical Engineers.)

The Inverted Stagnation Flow Model Tien [7.19] noted the analogy between the induced flow associated with a rising bubble column and an inverted stagnation flow, as shown in Fig. 7.8. For laminar axisymmetric stagnation flow (Fig. 7.8b) it is known that the heat transfer coefficient h can be computed from



u r hr = 1.32  ∞   vl  kl

0.5

Prl0.33

(7.22)

where u∞ = ar



(7.23)

with a being a constant and r being the radial distance from the center of the axisymmetric flow. For these conditions, the thermal boundary-layer thickness δt = kl/h is constant (independent of r). The heat transfer coefficient associated with an active site is evaluated over a circle of diameter s, where s is the spacing between sites. Replacing r with s/2 in the heat transfer correlation yields



h = 1.32

kl s

 as 2   v  l

0.5

Prl0.33

(7.24)

261

Pool Boiling

FIGURE 7.8  Actual and idealized flow fields. (Adapted from Tien [7.19] with permission, copyright © 1962, Pergamon Press.)

If the density of active sites na′ is taken as being equal to s–2, this relation can be written as



 a  h = 1.32 kl (na′ )0.5   na′ vl 

0.5

Prl0.33

(7.25)

Based on the active site density data of Yamagata et al. [7.20] and Gaertner and Westwater [7.21], Tien argued that a / na′ vl is about equal to 2,150 and approximately constant. Substituting this value in Eq. (7.25), it is found that



h=

q ′′ = 61.3Prl0.33 kl (na′ )0.5 Tw − Tsat

(7.26)

Tien showed that Eq. (7.26) is in general agreement with data at moderate to high active site densities for a wide variety of fluids. This model is yet another demonstration that single-phase convection heat transfer concepts can, in some instances, be extended to nucleate boiling. A similar inverted stagnation flow model was proposed by Kutateladze et al. [7.22] a bit later.

Latent Heat and Microlayer Evaporation Effects As intricate as they are, the models described above only account for the microconvective effects of bubble growth and release. This type of model is plausible for circumstances in which microconvective effects dominate. For some conditions, however, the latent heat associated with the vapor bubble and/or evaporation of a liquid microlayer under the bubble may also play an important role. Analytical models similar to that of Mikic and Rohsenow [7.12] have also been developed which attempt to account for these effects. van Stralen [7.23] developed a theoretical treatment of nucleate boiling heat transfer for pure and binary liquids that accounted for both microconvective and latent heat transfer using a superposition approach. This model treats the microconvective effect in a manner similar to the models of Han and Griffith [7.11] and Mikic and Rohsenow [7.12], and considers the total heat transfer rate to equal the sum of heat flow due to the microconvection and latent heat mechanisms. Microconvection due to bubble agitation appears to account for most of the heat transfer at low heat flux densities in the isolated bubble regime. However, experimental data suggest that microconvection does not completely account for the total heat flow at higher heat flux levels where closer spacing of active sites results in interference that reduces the bubbles ability to convect superheated liquid into the ambient. More of the superheat energy then supplies the latent heat of vaporization to the bubble, resulting in a larger fraction of the total energy flow being carried away from the surface

262

Liquid-Vapor Phase-Change Phenomena

by this mechanism. van Stralen [7.23] presents comparisons with experimental data that appear to confirm the superposition behavior postulated for these circumstances. As described in Chapter 6, at relatively low pressures, bubble growth during the ebullition cycle may be in the inertia-controlled regime, resulting in hemispherical bubble growth. For such circumstances, microlayer evaporation may also contribute significantly to the total heat transfer from the surface. A model that incorporates the natural convection and microconvection mechanisms, as well as a treatment of the microlayer evaporation effect was developed by Judd and Hwang [7.24].

Model Formulation Based on Thermodynamic Similitude The general approach of accounting for fluid thermodynamic and transport properties as functions of reduced pressure and/or temperature is sometimes referred to as invoking the principle of corresponding states or the principle of thermodynamic similitude. The principle stems from the fact that some models of thermodynamic properties suggest that liquid and vapor properties for different pure substances can be treated in a unified way if the properties are normalized with values associated with the critical point (see, e.g., references [7.25,7.26]). This concept is sometimes extended to transport properties as well (see Poling et al. [7.25]). At saturation conditions, even properties that depend primarily on temperature can be converted to functions of reduced saturation pressure because Tsat is a known function of pressure alone at saturation. This approach generally leads to a relation in which the non-dimensionalized property is a function of reduced pressure (Pr = P/Pc). The transport models described above ultimately yield a relation linking heat flux, superheat, and properties of the surface and the liquid and vapor. The principle of thermodynamic similitude suggests that the properties in the resulting relations could, in principle, be replaced with relations that reflect their dependence on reduced pressure Pr and reduced temperature. If this is done, the relation linking heat flux and superheat for nucleate boiling could be cast in the form

q ′′ = f1 ( Pr ) f2 (Tw − Tsat ( Pl ))

(7.27)

where f1 and f2 are functions of reduced pressure and superheat, respectively. Note that the principle of thermodynamic similitude usually does not provide a means to model the mechanisms of the transport, but it can result in a predictive tool that is simpler to use because it does not require detailed property information for the system. A predictive correlation method of this type is discussed in the next section.

General Observations While the models described above all capture some of the important elements of the transport during nucleate boiling, idealizations incorporated into the models limit their accuracy and/or range of applicability. Virtually all the models proposed to date consider only the isolated bubble regime. The applicability of these models to the regime of slugs and columns, where bubbles growing on the surface interact, is highly suspect. The modeling efforts to date invariably require knowledge of how the bubble frequency, departure diameter, and density of active sites vary with wall superheat. Despite many years of research, these aspects of the boiling process are not well understood. As noted in Section 6.6, correlations for the bubble departure diameter, at best, fit available experimental data to an average absolute deviation that is typically on the order of 30–40%. Using such correlations in a heat transfer model is obviously going to introduce considerable uncertainty into its predictions. Bubble frequency correlations are often linked to the departure diameter (see Section 6.6). Hence, predictions of the

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Pool Boiling

bubble frequency are likely to suffer from the same high uncertainty level as the departure diameter. In addition, most of the models described above do not account for the fact that the bubble frequency relation should change depending on whether inertia-controlled or heat-transfer-controlled growth is expected (again, see Section 6.6). Similar difficulties arise in attempting to incorporate in these models an appropriate prediction of the active nucleation site density na′. As discussed in Chapter 6, the number of active sites depends in a complicated way on the surface morphology, the thermophysical properties of the fluids, the imposed pressure and flow conditions, and the wetting characteristics of the liquid-surface combination. The simplistic relations between na′ and the wall superheat adopted in most models cannot account for the complex dependence of na′ on all these factors. The inability of such simplistic relations to account for factors other than wall superheat may be the source of some of the scatter in experimental data when compared with model predictions. Further complicating this issue is the fact that different models suggest different dependencies of q′′ on na′ and Tw – Tsat. The modeling efforts described above suggest that overall, the nucleate boiling heat flux exhibits a power-law dependence on the nucleation site density and wall superheat

q ′′ ∝ (na′ )a (Tw − Tsat )b

(7.28)

The microconvection model of Mikic and Rohsenow [7.12] implies a linear dependence of q′′ on na′. Zuber’s [7.18] natural convection analogy model implies that q′′ is proportional to (na′ )1/3 and Tien’s [7.19] model indicates that q′′ is proportional to (na′ )1/ 2. The relations for all three of these models imply an almost linear variation of q′′ with Tw – Tsat. Several experimental studies, such as those of Gaertner and Westwater [7.21] and Heled et al. [7.27], also indicate power-law dependencies of q′′ on nucleation site density and superheat. These studies suggest that a lies somewhere between 0.3 and 0.5 and b is between 1.0 and 1.8. This appears to support the values of a suggested by the models of Zuber [7.18] and Tien [7.19]. Mikic and Rohsenow [7.12] noted, however, that very seldom in these experimental studies was only na′ changed, without simultaneously changing the wall superheat, bubble departure diameter, and/or the bubble frequency. Hence, the correct interrelationship among q′′, na′ , and Tw − Tsat can be difficult to deduce from pool boiling experimental data. The exact values of the exponents in the power-law relation (7.28) continue to be a subject of investigation and debate. However, for many common systems, the variation of na′ (and any other proportionality factors not indicated in (7.28)) with wall superheat is usually such that q′′ varies in an overall manner about proportional to (Tw – Tsat)3. It seems likely that this fortuitous situation is a consequence of the fact that common manufacturing practices used on metals, glass, and other frequently used materials produce cavity distributions which lie within specific bounds. The probability of the distribution for a given surface being within these limits is usually high, and as a result, the increase in the number of active sites with increasing wall superheat is likely to be comparable to that for other surfaces subjected to similar manufacturing methods. The models of nucleate boiling heat transfer described above are, at best, crude idealizations. The development of better models has been hindered by an inability to accurately predict basic features of the boiling process such as the number density of active sites, and the mean bubble frequency and departure diameter. Fortunately, the nearly identical power-law dependence of heat flux on superheat for many wall material and liquid combinations makes it possible to develop relatively simple nucleate boiling heat transfer correlations that can be used with fair accuracy for a wide variety of systems. Correlations of this type are described in more detail in the next section. However, as will be discussed in later sections of this text, the effect of wetting is poorly understood and its representation is nucleate boiling predictive tools remains a challenging task.

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Liquid-Vapor Phase-Change Phenomena

Numerical Simulation of Pool Boiling The models of nucleate pool boiling discussed to this point in this section are mainly analytical. They typically use a physical model with appropriate idealizations and mathematically manipulate the model to extract a prediction of the functional dependence of heat flux on superheat and temperature. The rapid increase in accessible computing power has made it progressively easier to solve the governing equations for single-phase convective and conduction transport and the moving interface problems that arise as part of nucleate boiling. As a result, increasingly detailed model simulations of pool boiling heat transfer have been formulated and computationally solved to explore their use as a predictive tool and to explore the mechanisms of pool boiling and their interaction. A common feature of this category of model is that detailed mechanism models are constructed and then linked in a way that reflects their physical interaction. In an early study, Yang et al. [7.28] developed a methodology to numerically model nucleate boiling that combined a cellular automata technique with a finite difference computational fluid dynamics model of the transport. In this model, the cellular automata technique handled the microscopic dynamic interactions of bubbles while the traditional computational fluid dynamics algorithm was used to determine microscopic system parameters such as pressure and temperature. Yang et al. [7.28] found that their computational model can reproduce most of the basic features of boiling and capture the fundamental characteristics of boiling phenomena. The heat transfer coefficient predicted by their model was found to be in agreement with the experimental data and existing empirical correlations. Dhir and Liaw [7.29] developed a detailed model of high heat flux pool boiling on a vertical surface. At high heat flux levels the pool boiling is expected to be in the slugs and columns regime in which large slug bubbles are fed by vapor by vapor stems that emanate from the surface. These investigators modeled the heat transfer through the macrolayer adjacent to a vapor stem and allowed for multiple stem locations at active nucleation sites over the surface. These investigators used experimentally determined information about the void fraction near the surface, together with a specified contact angle to determine the geometry of the stems and their spacing. The conduction transfer of heat from the solid heater surface to the liquid-vapor interface in the stem was determined by solving the heat conduction equation numerically. Dhir and Liaw [7.29] demonstrated that this type of model could predict the heat transfer in the nucleate boiling and transition boiling ranges, as well as the critical heat flux condition. Comparisons indicated good agreement between the model predictions and heat transfer data for systems with two different contact angles. Stephan and Hammer [7.30] used a detailed computational model of heat transfer in the thin film under the bubble and transport surrounding the bubble in the liquid and solid to predict the mean heat transfer rate associated with bubble growth to the departure diameter. He et al. [7.31] constructed a model that was similar to that of Dhir and Liaw [7.29] but included additional features. The model of He et al. [7.31] also incorporated treatment of surface roughness and nucleation site density on the surface and it modeled transient conduction in the solid surface, which facilitated numerical studies of transient wall temperature variations during boiling. Sanna et al. [7.32] also developed a numerical model of nucleate boiling heat transfer on a surface with a large number of artificial nucleation sites. As noted in Chapter 6, detailed computational models of bubble growth and release from surfaces have been developed, some of which have been extended to include interaction of bubbles growing at adjacent nucleation sites [7.33–7.38]. For bubble growth near surfaces, these models typically use one detailed model for transport in the microregion of thin liquid film transport near the contact line under the bubble and a second model is used to predict transport in the region within a few bubble diameters of the bubble interface. Models of this type developed by Dhir and co-workers [7.36–7.38] have numerically solved the equations for mass, momentum, and energy transport in the liquid and vapor phases and used a level set method to capture the moving interface. Solutions in

Pool Boiling

265

the two regions were linked using appropriate boundary conditions. Studies by Son et al. [7.37] and Mukherjee and Dhir [7.38] have explored bubble merging using these types of simulations. These simulations indicate that the merging process enhances heat transfer during the bubble growth and release process during nucleate boiling. The simulations have also been used to explore the effects of variable gravity on bubble growth, release, and the associated heat transfer. It is a natural next step to combine multiple models of this type for multiple active sites on a heated surface into a complete numerical simulation of nucleate pool boiling on a heated surface. Dhir [7.39, 7.40] has noted the potential advantages of this type of simulation. An alternate approach to modeling pool boiling using chaos theory has also been explored. Early studies by Yanagita [7.41], Nelson et al. [7.42], Shoji and Tajima [7.43], and others suggest that boiling is a type of spatio-temporal chaotic phenomenon. This type of modeling requires characterizing boiling in terms of parameters that facilitate analysis of the system in the framework of chaos theory. Studies by Yanagita [7.41] and Ghoshdastidar et al. [7.44] have used the coupled map lattice method to model boiling as a chaotic system. Ghoshdastidar et al. [7.44] presented boiling curve predictions generated using this methodology. While this is an interesting approach to modeling boiling transport, it has not yet provided a viable pathway to a means of predicting nucleate boiling heat transfer.

7.3  CORRELATION OF NUCLEATE BOILING HEAT TRANSFER DATA As noted in the previous sections of this chapter, nucleate pool boiling heat transfer data have been obtained for a number of pure fluids at different system pressure levels. Correlations of such data have typically been used as a tool to predict nucleate boiling heat transfer in engineering systems and heat exchangers. Many investigators have proposed methods of correlating data of this type – so many, in fact, that a complete discussion of them all could easily fill a major portion of this chapter. In this section, three specific heat transfer correlation methods will be described in detail: (1) Rohsenow’s correlation [7.5], (2) the Borishansky-Mostinski correlation [7.45, 7.46], and (3) the correlations of Stephan and Abdelsalam [7.47]. Other correlations have been proposed by Jakob and Linke [7.2], Forster and Zuber [7.7], Fritz [7.48], Kutateladze [7.49], Kruzhilin [7.50], Forster and Greif [7.8], Levy [7.51], Alad’yev [7.52], Labuntsov [7.53], Katuteladze et al. [7.54], Lienhard [7.55], and Mikic and Rohsenow [7.12]. The first three correlations noted above are discussed in detail here because either they are widely used or they are convenient, yet seem to provide reasonably reliable results. These correlations were selected because they are representative of the spectrum of different logical approaches used to correlate data of this type. Rohsenow’s correlation [7.5] is based on analogies with forced convection processes. In contrast, the Borishansky-Mostinski correlation [7.45, 7.46] is based on the principle of thermodynamic similitude, and the correlations proposed by Stephan and Abdelsalam [7.47] were obtained from dimensional analysis and regression fits of available data. Before further discussing correlation techniques, two aspects of the interpretation of such correlations are worth noting. First, since the subcooling of the liquid pool has virtually no effect on the resulting heat transfer rate (see Section 7.2), the pool boiling correlations are generally regarded as being valid for both subcooled and saturated nucleate boiling. Second, it has also been observed that a pool boiling heat transfer correlation developed for one heated surface geometry, in one specific orientation often works reasonably well for other geometries and/or other orientations. Hence, although a correlation was developed for a specific geometry and orientation, it may often be used, at least as a good approximation for others as well. The effects of surface orientation on nucleate boiling heat transfer can be assessed somewhat more directly by examining the results of a study by Nishikawa et al. [7.56]. These investigators obtained pool boiling heat transfer data for a flat heated surface in an extensive liquid pool. The

266

Liquid-Vapor Phase-Change Phenomena

FIGURE 7.9  Data of Nishikawa et al. [7.56], and the predicted transitions from isolated bubbles to slugs and columns. (Adapted from Lienhard [7.59] with permission, copyright © 1985, American Society of Mechanical Engineers.)

test surface was specifically constructed so that it could be rotated to change the orientation of the surface with respect to the gravity vector. The resulting boiling curves for water at atmospheric pressure for different surface orientations are shown in Fig. 7.9. The data in this figure indicate that above a certain threshold heat flux level, the pool boiling curves for all orientations are virtually identical. Below this threshold value the curves differ significantly for the different surface orientations. This threshold value corresponds approximately to the transition from the isolated bubble regime to the regime of slugs and columns. This transition is sometimes referred to as the Moissis-Berenson transition [7.57]. Based on arguments that can be traced back to Zuber [7.18], these investigators developed a semi-empirical model that predicts that bubbles releasing from adjacent sites on an upward-facing flat plate will touch one another, merging into vapor columns, when the heat flux reaches the level given by



 σg  qMB ′′ = 0.11ρv hlv θ1/ 2   ρl − ρv 

1/ 4



(7.29)

where θ is the liquid contact angle in degrees. This relation was found to agree well with visual observations of the transition between the isolated bubble regime and the regime of slugs and columns. A corresponding relation for this transition on horizontal cylinders has been obtained by Bhattacharya and Lienhard [7.58]. Lienhard [7.59] showed that the transition suggested in the data of Nishikawa et al. [7.56] is consistent with that indicated by Eq. (7.29), if the contact angle is between 35° and 85° (Fig. 7.9). These results imply that in the slugs and columns regime, surface orientation has little impact on the pool boiling heat transfer performance. However, in the isolated bubble regime, at lower heat flux, surface orientation appears to affect the boiling curve in a systematic way. Interestingly

267

Pool Boiling

enough, the upward-facing surface resulted in the highest superheat for a given heat flux, whereas in the downward-facing position, the surface superheat was the lowest. At lower heat flux levels, the enhancement of heat transfer for the downward-facing orientations would appear to be due, at least in part, to two effects. First, the natural convection boundary layer for a downward-facing surface is thicker than for upward-facing or vertical surface orientations. Hsu’s analysis described in Section 6.3 implies that the wall superheat required to initiate nucleate boiling will be lower for the thicker thermal boundary layer associated with the downward-facing surface. In addition, when a bubble grows and releases from an inclined or horizontal downward-facing surface, the bubble must travel along the surface to its lateral edge before escaping to the ambient. This sweeping of the surface may serve to enhance heat transfer by facilitating vaporization of the liquid film between the interface of the bubble and the wall as it moves along the surface. The enhancement provided by this mechanism together with the earlier onset of boiling due to the mechanism described above may account for the upward shift in the nucleate boiling curve for a downward-facing surface relative to that for a vertical or upward-facing surface. Further discussion of the mechanisms that may be responsible for these trends can be found in references [7.60–7.62]. The mechanisms that give rise to the trends in the boiling curves in Fig. 7.9 are not fully understood at this time. These results clearly imply, however, that at low heat flux levels, nucleate pool boiling correlations are not universally applicable to all surface geometries, as is often assumed. At higher heat flux levels, the general applicability is expected to be better. Having taken heed of the above warning, we will now consider in detail the three correlations noted earlier in this section. As described in the previous section, Rohsenow’s [7.5] correlation is of the form



q ′′ µ l hlv

1/ 2

  σ  g(ρ − ρ )  l v  

 1  =  Csf 

1/ r

1/ r

 c pl [Tw − Tsat ( Pl )]  Prl− s / r   hlv  

(7.30)

Originally, values of r = 0.33 and s = 1.7 were recommended for this correlation. Subsequently, Rohsenow recommended that for water only, s be changed to 1.0. Values of Csf recommended for different liquid-solid combinations are listed in Table 7.1. These values were tabulated by Vachon et al. [7.63] based on fits to pool boiling data available in the literature.

TABLE 7.1 Values of Csf in the Rohsenow Correlation (Eq. (7.30)) for Different Liquid-surface Combinations Liquid-surface Combination Water on Teflon pitted stainless steel Water on scored copper Water on ground and polished stainless steel Water on emery polished copper Water on chemically etched stainless steel Water on mechanically polished stainless steel Water on emery polished, paraffin-treated copper n-Pentane on lapped copper n-Pentane on emery polished nickel n-Pentane on emery polished copper Carbon tetrachloride on emery polished copper Source:

Data from [7.63].

Csf 0.0058 0.0068 0.0080 0.0128 0.0133 0.0132 0.0147 0.0049 0.0127 0.0154 0.0070

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Liquid-Vapor Phase-Change Phenomena

TABLE 7.2 Values of Csf in the Rohsenow Correlation (Eq. (7.30)) for Different Liquid-surface Combinations Liquid-surface Combination

Csf

Water on nickel (vertical tube) Water on stainless steel (horizontal tube) Water on stainless steel (horizontal tube) Water on copper (vertical tube) Carbon tetrachloride on copper (vertical tube) Isopropyl alcohol (vertical tube) n-Butyl alcohol on copper (vertical tube) Source:

0.006 0.015 0.020 0.013 0.013 0.0022 0.0030

Data from [7.65]–[7.68].

Additional values of Csf have been obtained for other liquid-surface combinations from subcooled forced-convective boiling data. Following a superposition model proposed by Rohsenow [7.64], the total heat transfer rate was postulated as being equal to the sum of a forced convection contribution and a nucleate boiling contribution, with the latter given by the Rohsenow correlation. Subtracting the forced convective contribution from the total, values of Csf have been determined that best fit the implied nucleate boiling contribution in the data of Rohsenow and Clarke [7.65], Kreith and Sommerfield [7.66], Piret and Isbin [7.67], and Bergles and Rohsenow [7.68]. These values of Csf are listed in Table 7.2. The accuracy of Csf values obtained in this manner is thus limited by the accuracy of the superposition method. This type of model will be discussed further in Chapter 12. The values of Csf listed in Tables 7.1 and 7.2 are useful indicators of typical values. However, it is generally recommended that, whenever possible, an experiment be conducted to determine the appropriate value of Csf for the particular solid-liquid combination of interest. If this is not possible, and the combination is not listed in Table 7.1 or 7.2, a value of Csf = 0.013 is recommended as a first approximation. Based on thermodynamic similitude, Borishansky [7.45] proposed a correlation that can be written as q ′′ =  ( A *)

3.33



Tw − Tsat ( Pl )

3.33

 F ( Pl )

3.33



(7.31)

where q″ is in W/m2 and F(Pr) is a function of the reduced pressure Pr = Pl /Pc. For this correlation, Mostinski [7.44] proposed the following relations for A* and F(Pr)

A* = 0.1011Pc0.69 (with Pc  in bar)

(7.32)



F ( Pr ) = 1.8 Pr0.17 + 4 Pr1.2 + 10 Pr10

(7.33)



Pr = Pl / Pc

(7.34)

In a more recent study, Stephan and Abdelsalam [7.47] proposed the following correlations based on dimensional analysis and optimal fits to experimental data: For water:

q ′′ = {C1 [Tw − Tsat ( Pl )]}

1/ 0.327



(7.35)

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Pool Boiling

For hydrocarbons: q ′′ = {C2 [Tw − Tsat ( Pl )]}

1/ 0.330



(7.36)



For cryogenic fluids:

{

}

q ′′ = C3 (ρc p k )c0.117 [Tw − Tsat ( Pl )]



1/ 0.376



(7.37)

For refrigerants:



q ′′ =  {C4 [Tw − Tsat ( Pl )]}1/ 0.255

(7.38)

Values of the constants C1–C4 for materials of the indicated types are shown in Figs. 7.10–7.13. In the above relations, the units to be used are kg/m3 for ρ, kJ/kg°C for cp, W/m°C for k, °C for Tw − Tsat ( Pl ) , and W/m2 for q″.

FIGURE 7.10  Variation of C1 with pressure for water. (Adapted with permission from [7.47]; plots provided by K. Stephan, copyright © 1980, Pergamon Press.)

270

Liquid-Vapor Phase-Change Phenomena

FIGURE 7.11  Variation of C2 with pressure. (Adapted with permission from [7.47]; plots provided by K. Stephan, copyright © 1980, Pergamon Press.)

Stephan and Abdelsalam [7.47] also developed mathematical forms of their correlation scheme cast in terms of several dimensionless groups. The resulting correlations are listed in Table 7.3. For cryogenic fluids, (ρcpk)c represents the indicated properties of the heated surface or the surface cover material, which were found by Stephan and Abdelsalam [7.47] to affect the heat transfer performance. The authors also noted that in developing these equations, the mean surface roughness Rs was assumed to be 1 μm. For other roughness values in the range 0.1 ≤ Rs ≤ 10 μm, they suggest that the heat transfer coefficient predicted by the above relations be corrected by multiplying it by Rs0.133 , where Rs is the mean surface roughness in microns (μm).

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Pool Boiling

TABLE 7.3 Equation Forms of the Boiling Heat Transfer Correlations Proposed by Stephan and Abdelsalam [7.47] For water in the pressure range 10–4 ≤ P/Pc ≤ 0.886 with θ = 45°:



 q′′Db  q′′Db = 2.46 × 10 6   kl Tsat  kl (Tw − Tsat )

0.673

 hlv Db2   α 2  l

−1.58

 c pl Tsat Db2   α 2 

1.26

l

 ρl − ρv   ρ  l

5.22

(7.39)

For hydrocarbons in the pressure range 5.7 × 10–3 ≤ P/Pc ≤ 0.9 with θ = 35°:



 q′′D   ρ  q′′Db b v = 0.0546   k T   ρ  kl (Tw − Tsat )  l sat  l 

0.5 0.67

  

 hlv Db2   α 2  l

0.248

 ρl − ρv   ρ 

4.33

l

(7.40)

For cryogenic liquids in the pressure range 4.0 × 10–3 ≤ P/Pc ≤ 0.97 with θ = 1°:



 q′′Db  q′′Db = 4.82   kl Tsat  kl (Tw − Tsat )

0.624

 (ρc p k )c     (ρc p k )l 

 c pl Tsat Db2                          ×   α 2 

0.374

 hlv Db2   α 2  l

l

0.117

 ρv   ρ 

0.257

l

(7.41)

−0.329



For refrigerants in the pressure range 3.0 × 10–3 ≤ P/Pc ≤ 0.78 with θ = 35°:



 q′′Db  q′′Db = 207   kl Tsat  kl (Tw − Tsat )

0.745

 ρv   ρ  l

0.581

 νl   α l 

0.533

(7.42)

FIGURE 7.12  Variation of C3 with pressure. (Adapted with permission from [7.47]; plots provided by K. Stephan, copyright © 1980, Pergamon Press.)

272

Liquid-Vapor Phase-Change Phenomena

FIGURE 7.13  Variation of C4 with pressure. (Adapted with permission from [7.47]; plots provided by K Stephan, copyright © 1980, Pergamon Press.)

Stephan and Abdelsalam [7.47] also developed the following correlation that can be applied, with more limited accuracy, to any liquid:



 q ′′Db  q ′′Db = 0.23   kl Tsat  kl (Tw − Tsat )

0.674

 ρv   ρ  l

0.297

 hlv Db2   α 2  l

0.371

 ρl − ρv   ρ  l

−1.73

 α l2 ρl   σD  b

0.35

(7.43)

In the above equation, Db is the bubble diameter predicted by the bubble departure diameter correlation (6.98) developed by Fritz [6.54].

273

Pool Boiling

The pool boiling curves for water at atmospheric pressure predicted by the three correlation methods described here are shown in Fig. 7.14. For Rohsenow’s correlation, Csf was taken to be 0.013. Although the four curves in this figure are somewhat different, the overall variation is within the typical scatter in nucleate pool boiling heat transfer data (Fig. 7.7). The heat flux predicted by these correlations may vary by a factor of two or more for a given wall superheat, whereas the difference in the wall superheat predicted for a specified heat flux is at most only a few degrees. Example 7.1 Compare the heat flux values predicted by the Rohsenow correlation and the Stephan and Abdelsalam correlation for nucleate pool boiling of n-butyl alcohol (n-butanol) on a copper surface at atmospheric pressure for a wall superheat of 10°C. From the saturation table for n-butanol for Psat = 101 kPa: ρl = 712 kg/m3, ρv = 2.30 kg/m3, hlv = 591.3 kJ/kg, cpl = 3.20 kJ/kgK, μl = 4.04 × 10 –4 Ns/m2, kl = 0.127 W/mK, Prl = 10.3, and σ = 0.0171 N/m. The Rohsenow correlation for fluids other than water can be written in the form 1/ 2



 g(ρl − ρv )  q′′ = µ l hlv   σ 

 c pl[Tw − Tsat (Pl )]    Csf hlv  

3.0

Prl−5.15

Using Csf = 0.0030 for n-butanol/copper from Table 7.2 and substituting the physical properties 1/ 2

 9.8(712 − 23)  q′′ = 4.04 × 10 −4 (591.3)   0.0171 

3.20[10]    (0.0030)(591.3)   

3.0

(10.3)−5.15

= 5.43 kW/m 2

For hydrocarbons, the appropriate form of the Stephan and Adelsalam correlation is

q′′ =  {C2[Tw − Tsat (Pl )]}1/ 0.330

FIGURE 7.14  Comparison of the nucleate boiling curves predicted by the three indicated correlations for water at atmospheric pressure.

274

Liquid-Vapor Phase-Change Phenomena

From Fig. 7.11, the C2 value for n-butanol at 1.0 bar is C2 = 2.7. This was obtained from the curve labeled “Ethanol” which agrees with the one data point on this figure for n-butanol. Substituting in the above equation,

q′′ =  {2.7[10]}1/ 0.330 = 2.18 × 10 4 W/m2 = 21.8 kW/m 2

Thus, the predicted heat flux values for these two correlations differ by about a factor of 4.

The pool boiling behavior of cryogenic fluids such as liquefied oxygen, nitrogen, and hydrogen is generally consistent with that of non-cryogenic fluids. Appropriate forms of the correlations described above should therefore be applicable to these fluids. Clarke [7.69] has also proposed the following modified form of the Rohsenow correlation as a correlating equation for nucleate boiling of cryogenic liquids 1/ 2



 q ′′  σ   µ l hlv  g(ρl − ρv ) 

1.8  c pl [Tw − Tsat ( Pl )]  T   = 3.25 × 10   T   hlv Prl1.8 c  

2.89

(7.44)

5



The T/Tc term in this relation incorporates an additional pressure effect, since at saturation the system temperature T depends on the ambient pressure. This relation also differs from the original Rohsenow correlation in that Csf is fixed at a specific value, and the exponents r and s are different. Some aspects of nucleate pool boiling in liquid metals are significantly different from the corresponding behavior of non-metallic fluids. Because of the high thermal conductivity of the liquid, heat can be transferred to the vapor-liquid interface very rapidly during bubble growth. Bubbles therefore grow rapidly, resulting in inertia-controlled growth over most of the growth process. Although bubble growth times are shorter, the waiting period between bubbles in liquid metals is typically considerably longer. When a bubble departs and cold liquid metal is brought into contact with the wall, locally, the surface temperature will drop more than for a non-metal under comparable circumstances because the non-metal has a much lower value of klρlcpl than the liquid metal. This localized drop in the surface temperature lengthens the waiting period, resulting in much lower bubble frequencies. In addition, the low contact angles of liquid metals on solid metal surfaces often result in large cavities being fully wetted and inactive as nucleation sites. As a result, a high wall superheat may be required to initiate boiling at smaller cavity sites. Once bubbling is initiated, the in-rush of cold liquid as the bubble releases may drop the temperature of the surface so low that the cavity cannot sustain the boiling process. A relatively long waiting period must then pass before the wall and adjacent fluid build up enough superheat to initiate boiling again. At high superheat, the bubbles grow almost explosively, generating a shock wave in the liquid that makes an audible sound. The resulting intermittent violent growth of vapor bubbles in liquid metals is often referred to as bumping. The following correlation for pool boiling heat transfer has been proposed by Subbotin et al. [7.70] as a best fit to experimental data for potassium, sodium, and cesium



(q ′′)1/3 kh ρ  = Cs  l lv 2 l   [Tw − Tsat ( Pl )] σT 

1/3

s

 Pl   P  c

(7.45)

where

Cs = 8.0,

s = 0.45

for Pl / Pc <  0.001

(7.46a)



Cs = 1.0,

s = 0.15

for Pl / Pc ≥  0.001

(7.46b)

275

Pool Boiling

and T is a mean system temperature. Further discussion of the special features of pool boiling in liquid metals may be found in Dwyer’s [7.71] comprehensive book on the subject. Example 7.2 Using Clark’s correlation and the Stephan and Abdelsalam correlation, determine the heat flux for nucleate boiling of liquid nitrogen at atmospheric pressure on a copper surface for a wall superheat of 5°C. For saturated nitrogen at atmospheric pressure, Tsat = 77.4 K, ρl = 807.1 kg/m3, ρv = 4.62 kg/m3, hlv = 197.6 kJ/kg, cpl = 2.06 kJ/kgK, μl = 1.63 × 10 –4 Ns/m2, kl = 0.137 W/mK, Prl = 2.46, σ = 0.00885 N/m, and Tc = 126.3 K. Clarke’s correlation can be written in the form 1/ 2

 g(ρl − ρv )  q′′ = 3.25 × 10 µ l hlv   σ  5



 c pl[Tw − Tsat (Pl )]  T  1.8    T   hlv Prl1.8 c  

2.89

Substituting the above properties and using T = (1/2)(Tw + Tsat) = 79.9 K. q′′ = 3.25 × 105(1.63 × 10 −4 )(197.6) 1/ 2

 9.8(807 − 4.6)  ×  0.00885 

 2.06(5)  1.8  197.6(2.46)

 79.9    126.3 

1.8

  

2.89

= 1.61 kW/m 2

For a copper surface,



(ρc p k )c = 8954

kg kJ W WkJ × 0.384 × 398 = 1.37 × 106 4 2 m3 kgK mk mK

For cryogenic liquids, the Stephan and Abdelsalam correlation is

{

}

q′′ = C3(ρc p k )c0.117[Tw − Tsat ( Pl )]

1/ 0.376

From Fig 7.11 at 1.0 bar pressure, C3 = 2.6. Substituting,

{

}

q′′ = 2.6(1.37 × 106 )0.117[5]

1/ 0.376

= 7.45 × 10 4 W/m 2 =  74.5 kW/m 2

These two correlations predict widely different heat flux levels for these conditions.

7.4 LIMITATIONS OF NUCLEATE BOILING PROCESSES AND THE MAXIMUM HEAT FLUX TRANSITION To sustain steady nucleate boiling at high heat flux levels, the six processes depicted in Fig. 7.15 must be sustained to transfer the prescribed heat flux without interruption. At the solid-liquid interface, heat must be steadily transferred across this interface (mechanism (M1)) with a low enough driving temperature difference that it does not adversely limit the other mechanisms. Often the thermal resistance at the solid-liquid interface is small and the temperature difference to drive the heat flux is small and not an important limit. However, if thermal resistance at this interface due

276

Liquid-Vapor Phase-Change Phenomena

FIGURE 7.15  Transport processes in nucleate pool boiling.

to scattering of thermal energy carriers (e.g., phonons) is significant (sometimes referred to as the Kapitza effect [7.72, 7.73]), the temperature difference across the interface may be larger, and could affect the system’s ability to sustain the applied surface heat flux. A flow of heat through the liquid from the sold-liquid interface to the liquid-vapor interface (M2) must also be sustained at the rate dictated by the applied heat flux. The required temperature difference to drive the heat flow may be small if the path length through the liquid is small or convective effects are strong. Vapor generation at the liquid-vapor interface (M3) must proceed fast enough that latent heat removal during vapor generation can carry away the prescribed heat flux per unit of surface area. At a greater distance from the heated surface, the per-unit-of-surface-area flow of vapor away from the surface (M4) and liquid toward the surface (M5) must be large enough to replace vaporized liquid and prevent vapor accumulation at the surface. Finally, liquid moving toward the surface from the far field must be delivered to the near-contact line region between the growing bubbles and the solid wall at a rate that keeps the surface wetted as liquid evaporates (M6). It is clear that all six of these processes must be sustained at specific rates to sustain a given heat flux from the heated surface. Each of these processes may have upper limits for their transport rate that cannot be exceeding in a given system at specific conditions. Increasing the heat flux at the surface may push the system beyond the limit of one of these processes, making it impossible for the nucleate boiling process to operate steadily at the required heat flux, and resulting in a critical heat flux transition to film boiling. The question is: which of these mechanisms in the process chain will fail first and trigger a CHF transition as the heat flux increases? To develop model relations to predict the critical heat flux condition for pool boiling, investigators have generally postulated that one of these processes is the weak link in the process chain and that raising the heat flux above the level that exceeds its maximum rate for this mechanism triggers the CHF transition. Analysis of the Kapitza limit [7.72, 7.73] indicates that it is not likely to prevent sustained heat flux levels near the CHF condition in most typical systems of engineering

277

Pool Boiling

interest. Likewise, the analysis of CHF data by Gambill and Lienahard [7.74] (see Section 4.6) indicated that kinetic theory maximum vapor flux generation rates at a liquid-vapor interface are usually much higher than the rates needed to transfer the corresponding latent heat for heat flux levels near the CHF for common boiling circumstances. The mechanisms (M1) and (M3) therefore appear unlikely to be triggers of CHF. Consistent with the above observation, the history of research in this area is that most often, efforts to model the CHF trigger mechanism have focused either on the far field counterflow of vapor and liquid normal to the surface, or the sustaining of delivery of liquid to the near-contact line region near the surface as critical mechanisms for the CHF transition. CHF modeling efforts based on these strategies are discussed below.

Onset of Flooding Kutateladze [7.75] apparently was among the first investigators to notice the similarity between flooding phenomena and the vapor/liquid counterflow near the CHF condition in pool boiling. In a counterflow distillation column, vapor rich in the more volatile components flows upward while liquid rich in the less volatile components flows downward. If the relative velocity of the two streams (mechanisms (M4) and (M5) in Fig. 7.15) becomes too large, the flows become Helmholtz unstable, causing the vapor drag on the downward moving liquid to increase to the point that the flow of liquid is impeded. Liquid that is then unable to move downward accumulates at the top of the column, which is then said to be “flooded.” Based on the similarity between the CHF condition and column flooding, Kutateladze [7.75] used dimensional analysis arguments to derive the following relation for the maximum heat flux. If ′′ is postulated to be a function of the thermal properties ρv , ρl − ρv , hlv , σ, and gravitational the qmax acceleration g, dimensionless variables formed by multiplying and/or dividing these properties can be defined and experimental data can be analyzed to determine whether a functional relation exits among the dimensionless variables. Using this specific set of variables, the simplest approach would be to form a single dimensionless group and determine if a critical value of this parameter = CCHF corresponds to the CHF threshold. This leads to a relation of the form



Ku =

qmax ′′ ρv hlv

1/ 4

  ρ2v  σ (ρ − ρ ) g  l v  

= CCHF

(7.47)

where the defined dimensionless parameter Ku is sometimes referred to as the Kutateladze number. Note that this neglects any dependence on surface wetting (contact angle), which now is acknowledged to have a significant effect, as will be discussed later. This could be expected to yield useful predictions for surfaces of a single geometry and characteristic length with liquid having a contact angle in a limited range. If additional parameters are important, such as contact angle, surface orientation, surface length scale, or liquid thermal conductivity, additional dimensionless parameters can be defined and their effects of the CHF condition could be deduced from experimental data. This approach also could provide a means of developing improved CHF correlation methods if used in conjunction with more accurate and detailed experimental information on boiling processes obtained with advanced instrumentation. Kutateladze’s [7.75] flooding mechanism model and dimensional analysis lead to a CHF relation of the form

qmax ′′ = CK ρ1/v 2 hlv [ g(ρl − ρv )σ ]1/ 4

(7.48)

which is consistent with the concept of a threshold Kutateladze number discussed above. Strictly speaking, the flooding analogy used to obtain this relation is applicable only to one-dimensional flow associated with boiling from a flat heated surface of infinite lateral extent. Kutateladze [7.75]

278

Liquid-Vapor Phase-Change Phenomena

nevertheless concluded that CK was equal to 0.131 based on maximum heat flux data for horizontal cylinders and other configurations different than the infinite surface for which this relation was developed. Although this approach was somewhat successful empirically, the validity of the postulated mechanism has not been supported by theoretical arguments or experimental observations, and this ′′ does not account for surface effects or accurately reflect observed trends in qmax ′′ relation for qmax with pressure or gravitational acceleration. These issues are discussed further below.

Critical Bubble Packing Early CHF models proposed by Rohsenow and Griffith [7.76] and Chang and Snyder [7.77] postulated that as heat flux and nucleation site density increases, a critical bubble-packing density is eventually reached that blocks liquid flow area and inhibits liquid flow to the surface (mechanism (M6) in Fig. 7.15) to such a degree that a vapor blanket forms over portions of the surface. There is some logic to the idea that packing of bubbles at the surface could ultimately shut-off the liquid flow to the surface and induce vapor blanketing over portions of the surface. However, this line of reasoning has two pitfalls. First, visual evidence does not support this model’s basic premise that round bubbles are packed against the surface more and more tightly as the heat flux increases. Vapor is generated so rapidly at high heat flux levels near the CHF condition that successive bubbles from a given cavity tend to merge into what would better be described as a small jet of vapor leaving the surface. The small vapor jets formed at an upward-facing flat surface immediately merge into slugs that rise in columns (in the slugs and columns regime discussed earlier in this chapter). Furthermore, to quantitatively pursue this type of model it is necessary to predict the bubble generation frequency and departure diameter accurately at high heat flux levels. As described in Chapter 6, accurately predicting these quantities is extremely difficult, particularly at high heat flux levels where bubbles at the surface interact and merge. Consequently, bubble-packing models have been largely abandoned in favor of hydrodynamic and vapor recoil CHF models.

Helmholtz Instability of Large Vapor Columns At high heat flux levels, bubbles generated at the surface generally coalesce, as shown in Fig. 7.16, to form vapor columns or jets. In this model, the CHF condition occurs when Helmholtz instability of the large vapor jets leaving the surface distorts the jets, blocking liquid flow to portions

FIGURE 7.16  Postulated vapor column Helmholtz instability CHF mechanism.

279

Pool Boiling

of the heated surface (mechanism (M5)). Continued vaporization of liquid at locations on the surface that are starved of replacement liquid then leads to formation of a vapor blanket over part or all of the surface. The Helmholtz instability mechanism was first incorporated into a formal model by Zuber [7.78] for a flat horizontal surface. The interface stability analysis developed by Taylor [7.79] in 1950 was a key factor in the subsequent development of more detailed versions of hydrodynamic models of the CHF mechanism. Chang [7.80] appears to have been the first to suggest a link between Taylor wave motion and pool boiling processes. Chang’s observations, together with arguments in the Soviet literature linking flooding phenomena to the CHF condition, apparently influenced Zuber [7.78] to include Taylor wave motion and Helmholtz instability as key elements in his model of the CHF mechanism. (See Chapter 4 for a detailed discussion of Taylor and Helmholtz instabilities.) Zuber’s model analysis, as refined and extended to other geometries by Lienhard, Dhir, and coworkers [7.81–7.85] and others, is based on the following idealizations: 1. The critical heat flux is attained when the interface of the large vapor columns leaving the surface becomes Helmholtz unstable. 2. The columns leave the surface in a rectangular array as shown in Fig. 7.17. The centerline spacing of the columns coincides with nodes of the most dangerous wavelength associated with the two-dimensional wave pattern for Taylor instability of the horizontal interface between a semi-infinite liquid region above a layer of vapor. 3. The column radius is equal to λD/4, where λD is the spacing of the columns as predicted by Taylor instability analysis (see Section 4.2). 4. The Helmholtz unstable wavelength imposed on the columns is equal to the Taylor wave node spacing λD. Using the relation for the critical Helmholtz velocity uc for vertical vapor and liquid flow described in Section 4.3, and assuming that ρl » ρv , leads to the following relation for the mean velocity of the vapor in the columns as the CHF condition: 1/ 2



 2πσ  uc =    ρv λ 

FIGURE 7.17  Vapor column spacing in the Zuber critical heat flux model.

(7.49)

280

Liquid-Vapor Phase-Change Phenomena

Using conservation of mass and energy for steady boiling, and the area ratio for the column geometry shown in Fig. 7.17, the mean vapor velocity in the columns must also be equal to



uc =

qmax ′′  Asurf   16  qmax ′′ =  ρv hlv  Acol   π  ρv hlv

(7.50)

Setting the relations for uc equal to each other, using λ = λD where

λ D = 2π ( 3σ / [(ρl − ρv ) g]) 1/ 2

(7.51)

(Eq. (4.56) from Chapter 4), and solving for qmax ′′ yields 1/ 4



qmax ′′ =

 σ (ρl − ρv ) g  π ρv hlv   16(3)1/ 4 ρ2v  

(7.52)

1/ 4



 σ (ρl − ρv ) g  = 0.149ρv hlv   ρ2v  

(7.53)

The model analysis described above basically follows Zuber’s original model, except for the idealization that the Helmholtz unstable wavelength is equal to λD, which was later proposed by Lienhard and Dhir [7.81]. The assumption that the Helmholtz unstable wavelength is equal to the Taylor unstable wavelength is perhaps the least transparent of the idealizations in this model. However, this is a plausible position if we consider approaching the maximum heat flux from the transition boiling side (Fig. 7.1). For film boiling on an upward-facing surface, release of vapor bubbles is expected to occur at the nodes of waves having the most dangerous two-dimensional wavelength with spacing λD since waves having this configuration grow most rapidly (this is discussed further in the next section). The existence of vapor bubble columns with a similar spacing is also expected in the transition regime near the minimum heat flux since the departure from film boiling is small. If the spacing of the jets does not significantly change as the heat flux increases, then the assumption that ′′ is reasonable. this spacing exists near qmax Zuber’s original model resulted in a relation for qmax ′′ identical to Eq. (7.53) except that the constant on the right side was π/24 = 0.131: 1/ 4



 σ (ρl − ρv ) g  qmax, ′′ Z = 0.131ρv hlv   ρ2v  

(7.54)

This relation is identical to that obtained by Kutateladze [7.75] using dimensional analysis. However, Lienhard and Dhir [7.81] found that Eq. (7.53) provides a better fit to available data for large flat surfaces. Initial criticism of Zuber’s CHF model, as reflected in the works of Bernath [7.86], Costello and Frea [7.87], and Chang [7.88], centered on the fact that the model did not account for possible effects of the geometry, surface condition and wetting characteristics of the heater surface on the CHF condition. Experimental data available at that time suggested that these factors could significantly affect the CHF condition. Unfortunately, in these early studies the data were obtained in such a way that it was impossible to separate the effects of these different influences. The role of heater geometry in hydrodynamic models of the CHF condition has been clarified somewhat by the systematic studies of Lienhard and co-workers [7.82–7.85]. These investigators adapted Zuber’s model for the critical heat flux mechanism to saturated pool boiling for square and round heated surfaces of finite size, horizontal cylinders, horizontal ribbons, and spheres. For each of these configurations, the heated surface is finite in extent in at least one dimension. Hence, the

281

Pool Boiling

length scale characterizing the size of the heater becomes an important parameter. Correlations for the critical heat flux for these finite-sized surfaces have typically been written in the form

qmax ′′ = f ( L / Lb ) qmax, ′′ Z

(7.55)

where Lb is the bubble (capillary) length scale defined as



Lb =

σ g(ρl − ρv )

(7.56)

and qmax, ′′ Z is the Zuber maximum heat flux defined by Eq. (7.54). Since Lb = λ D / (2π 3) , the ratio L/Lb indicates the size of the heater relative to the expected spacing of the vapor columns carrying vapor away from the surface near the critical condition. For heaters of finite size, variation of the value of this dimensionless group (i.e., the Bond number) is expected to significantly alter the CHF condition, particularly if its value is near or below one. CHF correlations and the range of applicability suggested by the developers are given in Table 7.4 for ′′ correlation for the infinite flat plate devela number of finite heater configurations. The improved qmax ′′ / qmax, ′′ Z = 1.14 . oped by Lienhard and Dhir [7.82] can also be written in the form of Eq. (7.55): qmax

TABLE 7.4 Correlations Indicating Geometry Effects on the Maximum Pool Boiling Heat Flux Geometry

Correlation

Range of Applicability

Reference

Infinite heated flat plate

qmax ′′ = 1.14 qmax, ′′ Z

L > 30 Lb

[7.82]

Small heater of width or diameter L with vertical side walls

qmax 1.14 λ 2D ′′ = qmax, Aheater ′′ Z

Horizontal cylinder of radius R

 qmax R  ′′ = 0.89 + 2.27 exp −3.44  qmax, Lb  ′′ Z 

R > 0.15 Lb

[7.83]

Large horizontal cylinder of radius R

qmax ′′ = 0.90 qmax, ′′ Z

R > 1.2 Lb

[7.84]

Small horizontal cylinder of radius R

 R qmax ′′ = 0.94    Lb  qmax, ′′ Z

Large sphere of radius R

qmax ′′ = 0.84 qmax, ′′ Z

Small sphere of radius R

 R qmax ′′ = 1.734    Lb  qmax, ′′ Z

Small horizontal ribbon oriented vertically with side height H – both sides heated

H qmax ′′ = 1.18    Lb  qmax, ′′ Z

Small horizontal ribbon oriented vertically with side height H – back side insulated

H qmax ′′ = 1.4    Lb  qmax, ′′ Z

−1/ 4

Small, slender, horizontal cylindrical body of arbitrary cross section with transverse perimeter Lp

L  qmax ′′ = 1.4  p   Lb  qmax, ′′ Z

−1/ 4

Small bluff body with characteristic dimension L

 L qmax ′′ = C0    Lb  qmax, ′′ Z

9
8 (small diameters), the heat flux became almost independent of diameter. Heat transfer for such conditions again deviated from the Bromley theory, apparently being determined by the bubble size and release mechanism only. As a best fit to available experimental data for a broad spectrum of cylinder sizes, Breen and Westwater [7.149] proposed the following empirical correlation for film boiling on horizontal cylinders hcon =



[0.59 + 0.069(λ D / D)]F (λ D )1/ 4

(7.137)

where 1/ 4

(7.138)



 k 3 gρ (ρ − ρv )hlv′  F= v v l   µ v (Tw − Tsat ) 

(7.139)



0.34c pv (Tw − Tsat )   hlv′ = hlv 1.0 +  hlv  

This correlation is recommended for any λD/D value. Alternatively, one of the following relations can be used, depending on the specific value of λD/D: For λD/D ≤ 0.8:

hcon = 0.6 Fλ −D1/ 4

(7.140)

hcon = 0.6 FD −1/ 4

(7.141)

For 0.8 < λD/D < 8: For 8 ≤ λD/D:



λ  hcon = 0.16  D   D

0.83

Fλ −D1/ 4



(7.142)

310

Liquid-Vapor Phase-Change Phenomena

Large Horizontal Surfaces Heat transfer during film boiling from a large flat horizontal heated surface, is dictated, to a large degree, by the vapor release mechanism which determines the mean vapor film thickness. Chang [7.136] proposed a model of the film boiling process on a horizontal surface that linked the wave action associated with Taylor instability to the breakup of the interface associated with the vapor release process. This model predicts a heat transfer relation of the form 1/3



 k 3 g(ρl − ρv )  h∞ = Cl  v  µ v a0  

(7.143)

where



a0 =

k v (Tw − Tsat ) 2hlv ρv

(7.144)

Chang [7.132] proposed 0.43 as a value of Cl that provided good agreement with available data. The vapor escape model used by Zuber to analyze the minimum heat flux condition (see Section 7.5) has been extended by Berenson [7.137] to model steady film boiling from a flat horizontal surface. From observations of the boiling process, Berenson [7.137] again postulated that bubbles were released at the nodes of the most dangerous Taylor wave, with the nodes spaced in a rectangular array, as shown in Fig. 7.21. The conceptual model on which Berenson’s analysis is based is shown in Fig. 7.29. For the postulated spacing of the bubble-release locations, the unit cell shown in Fig. 7.21 has an area equal to λ 2D / 2. For a radially symmetric unit cell of the same area, r2 = (2π)−1/ 2 λ D. Vapor is generated and flows with a radial velocity uv from a distance r2 = (2π)−1/ 2 λ D to the bubble, which is characterized by the radius r = r1. Based on experimental observation of bubble departure diameters, Berenson argued that r1 is given approximately by 1/ 2



  σ r1 = 2.35   g ( ρ − ρ ) l v  

(7.145)

FIGURE 7.29  Conceptual model of vapor release during film boiling on a horizontal, upward-facing surface.

311

Pool Boiling

Conservation of mass requires that the mass flow rate of vapor m v toward the bubble at any location between rl and r2 be given by

m v = ρv (2πr )δuv

(7.146)

Assuming that heat is transferred across the thin portion of the film by conduction alone, an energy balance also requires that the vapor mass flow rate satisfy



k m v hlv′ = π  v  (r22 − r 2 )(Tw − Tsat )  δ

(7.147)

hlv′ = hlv + 0.5c pv (Tw − Tsat )

(7.148)

where

The use of hlv′ as defined above approximately accounts for the sensible heating of the vapor film. Combining Eqs. (7.145)–(7.147), the following relation is obtained for the vapor velocity



 k (T − T )  λ 2 − 2πr 2 uv =  v w 2sat  D 4 πr  ρv hlv′ δ 

(7.149)

The approximate momentum balance for the inward flow of vapor in the film requires that

dPv =

bµ v uv dr δ2

(7.150)

where b = 12 if the liquid surface is taken to be stationary, and b = 3 if the shear stress is zero at the interface. Substituting Eq. (7.149) into (7.150) yields



 bµ k (T − T )   λ 2 − 2πr 2  dPv =  v v w 4 sat   D dr 4 πr  ρv hlv′ δ  

(7.151)

Integrating this relation from rl to r2 and using Eq. (7.52) to evaluate λD in the relations for r2 yields



 8bµ v k v (Tw − Tsat )    σ P1 − P2 = 1.36     4 πρv hlv′ δ    g(ρl − ρv ) 

(7.152)

At a height z = S + δ above the surface, the local liquid pressure Pl is a constant at all horizontal locations. In the vapor near the surface, the pressures Pl at r = rl and P2 at r = r2 are given by



P1 = Pl + ρv gS +

2σ Rb

P2 = Pl + ρl gS

(7.153) (7.154)

The last term on the right side of Eq. (7.153) accounts for the capillary pressure difference across the curved interface of the bubble. Combining Eqs. (7.153) and (7.154), we find that



P2 − P1 = (ρl − ρv ) gS −

2σ Rb

(7.155)

312

Liquid-Vapor Phase-Change Phenomena

In this equation, S and Rb were evaluated using empirical relations derived from bubble release data published by Borishansky [7.152] 1/ 2

  σ S = 1.36 Rb = 3.2   g ( ρ − ρ ) l v  



(7.156)

Combining Eqs. (7.152) and (7.155) to eliminate P2 – P1 and using Eq. (7.156), the following equation for δ is obtained 1/ 4



1/ 2   1.09bµ v (Tw − Tsat )     σ δ = 1.36      gρv (ρl − ρv )hlv′   g(ρl − ρv )   

(7.157)

Assuming that heat is transferred across the film by conduction only, the heat transfer coefficient is given by hcon = kv/δ. It follows directly from Eq. (7.157) that 1/ 4

hcon

1/ 2   k 3 gρ (ρ − ρv )hiv′   g(ρl − ρv   = C2   v v l     σ   µ v (Tw − Tsat )   

(7.158)

where the numerical constants and b have been absorbed into C2. The form of this relation can be obtained from Eq. (7.122) derived for a vertical surface, if λD is substituted for the characteristic length x. A value of 0.425 was proposed for C2 because it correlated Berenson’s [7.137] data for the minimum film boiling heat flux within ±10%. In a later study, Klimenko [7.153] extended the models described above for laminar film boiling on a horizontal flat plate to circumstances for which turbulent transport occurs in the vapor film. He noted that the previous predictions of laminar film boiling heat transfer from a horizontal flat surface developed by Berenson [7.137] and others have typically been of the form m



 gλ 3 (ρ − ρv )Prv hlv′  Nu λ = C3  D l   vv c pv (Tw − Tsat ) 

(7.159)

where Nu λ =



(7.160)

hlv′ = hlv (1 + C4 Ja)

(7.161)

c pv (Tw − Tsat ) hlv

(7.162)

Ja =



q ′′λ D k v (Tw − Tsat )

In the above expressions, C3, m, and C4 are constants that vary somewhat from one correlation to the next. Klimenko [7.153] proposed (as did Berenson [7.137]) that heat is transferred from the surface to the interface across a forced convection flow of vapor in the film, with the vapor flow being driven by hydrostatic pressure differences generated by the bubble release process. He argued that the vapor flow would be turbulent if



R=

gλ 3D (3)3/2 vv3/ 2

 (ρl − ρv )  8  ρ  > 10 v

(7.163)

313

Pool Boiling

For laminar flow, Klimenko [7.153] proposed the correlation Nu λ = 0.19 3 R1/3 Prv1/3 fl (Ja)



(7.164)

where fl = 1



for Ja ≥ 0.714

= 0.89Ja −1/3



for

Ja < 0.714

(7.165a) (7.165b)



Note that the function f l results in a more complex variation of the heat transfer coefficient with wall superheat than is predicted by Berenson’s correlation (7.159). However, this variation agrees better with available data than Eq. (7.159). For turbulent flow, using Reynolds analogy arguments, Klimenko [7.149] developed the following correlation for the film boiling heat transfer coefficient Nu λ = 0.0086 3 R1/ 2 Prv1/3 f2 (Ja)



(7.166)

where

f2 = 1

for Ja ≥ 0.5

(7.167a)

= 0.71Ja −1/ 2

for Ja ≥ 0.5

(7.167b)

Ramilison and Lienhard [7.154] found that the trends in their data for film boiling on a flat horizontal surface and Berenson’s [7.137] earlier data are well represented by Klimenko’s [7.153] turbulent film boiling correlation for Ja > 0.5. However, they replaced the constant 0.0086 in that correlation with the following specific values for each liquid: 0.0057 for R-113, 0.0066 for acetone, and 0.0154 for benzene. Example 7.8 Use Berenson’s correlation (7.158) together with the qmin ′′ values predicted in Example 7.5 to estimate the minimum wall superheat required to maintain stable film boiling on an upward-facing infinite horizontal surface for pool boiling of water at atmospheric pressure. Berenson’s correlation can be written in the form 1/ 4



1/ 2 q′′   k 3 gρ (ρ − ρv )hlv′   g(ρl − pv )   = 0.425   v v l     Tw − Tsat σ   µv (Tw − Tsat )   

where hl′v = hlv + 0.5 c pv (Tw − Tsat )



Because cpv is low (2.034 kJ/kgK for water at atmospheric pressure), as a first approximation, we will take hl′v = hlv . It is then possible to rearrange the above equation to solve for Tw − Tsat , yielding Tw − Tsat

 (q′′ )4 µv = 3.130  3 k gρ (ρ − ρl )hlv  v v l

1/ 2 1/ 3

  σ   ( ρ − ρ ) g l v  

  

314

Liquid-Vapor Phase-Change Phenomena

For water at atmospheric pressure (taking properties at saturation), kv = 0.0249 W/mK, μv = 1.21 × 10 –5 Ns/m2. The other needed properties are given in Example 7.5. Substituting in the above relation, using q′′ = qmin ′′ = 19.0 kW/m 2 from Example 7.6, 1/ 3

Tw − Tsat

1/ 2  (19.0 × 103 )4 (1.21× 10 −5 )  0.0589   = 3.130  3 3     (0.0249) (9.8)(0.598)(958)(2257 × 10 )  (9.8)(958)   = 85.2°C

Note that these results imply that for water,



0.5c pv (Tw − Tsat ) 0.5(2.034)(85.2) = = 0.038 hlv 2257

Since Tw – Tsat is proportional to (hlv′ )1/ 3, the fractional error in Tw – Tsat associated with taking hlv′ = hlv is only one third of the value of 0.5c pv (Tw − Tsat ) / hlv indicated above for each case. Thus, hlv′ = hlv is quite good for water. Because of the large temperature difference across the vapor film, it would be more accurate to repeat the calculation with properties evaluated at the mean film temperature.

Finite Horizontal Surfaces The modeling efforts in the previous subsection specifically focus on film boiling on an upwardfacing heated surface with large lateral dimensions. The models generally are based on the premise that bubbles release at nodes of the Taylor waves that lie on square unit cells that have a characteristic length proportional to λD. If the size of the heated surface is smaller than the Taylor most dangerous wavelength, λD = 2π[3σ/(ρl – ρv)g]1/2, bubble release cannot occur from Taylor wave node locations. An example of such circumstances is shown in Fig. 7.30. This figure shows photos of film boiling of water at 10 kPa on a square heated surface with a side length of 1.3 cm. For saturated water at 10 kPa, λD is about 2.9 cm. In this system bubbles generally depart from the film at the center of the heated surface. Note that in this system the thickness of the vapor film is zero at the edges of the heated surface. This implies that in the near edge region, the film thickness will be less than in regions far from the edge, and the heat transfer rate from the surface will be higher in the near edge region. As the heater size gets smaller, the fraction of the total surface affected by edge effects increases. It is therefore expected that as the size of the heated surface becomes comparable to and then less than λD, the mean film boiling heat transfer coefficient will progressively increase above that predicted by the large-surface models described above. These observations suggest that the large surface model predictions are useful when the characteristic lateral dimensions of the surface are large compared to λD.

Corrections for Sensible Heat and Convective Effects A number of the correlations for the film boiling heat transfer coefficient described in this section make use of an effective latent heat term hlv′ . The definition of this term typically has the general form

hlv′ = hlv (1 + C L Ja)

(7.168a)

c pv (Tw − Tsat ) hlv

(7.168b)

where Ja is the Jakob number defined as



Ja =

315

Pool Boiling

FIGURE 7.30  Photos taken at two different times during film boiling from a square 1.3 cm by 1.3 cm copper surface in water at a pressure of 10 kPa. (Archive photos, Energy and Multiphase Transport Laboratory, UC Berkeley.)

The recommended value of the constant CL varies from one correlation to another. Heat transfer correlations involving a corrected latent heat term evolve most directly from integral boundarylayer treatments of laminar film boiling heat transfer. For laminar film boiling on a vertical flat plate, the integral analysis in this section led to the following equation for the heat transfer coefficient in the absence of radiation: 1/ 4



hcon

 k 3 gρ (ρ − ρv )hlv  k = v = v v l δ con  4µ v (Tw − Tsat ) x 

(7.122)

No correction to the latent heat appears in this relation. However, in deriving this relation, terms accounting for superheating of the vapor in the energy were neglected, as were the convective transport terms in the momentum and energy equations. Inclusion of the convective terms in the momentum and energy transport equations adds considerable complexity to the analysis. However, a term accounting for superheating of the vapor in the energy balance can be handled without much difficulty. Doing so and neglecting radiation converts Eq. (7.118) to the form δ



 ∂T  d = ρv uv [hlv + c pv (Tw − Tsat )] dy − kv   ∂ y  y =δ dx

∫ 0

(7.169)

316

Liquid-Vapor Phase-Change Phenomena

It is left as an exercise for the reader to show that proceeding with the integral analysis described above results in the following equation for h: 1/ 4



hcon

 k 3 gρ (ρ − ρv )hlv′  k = v = v v l δ con  4µ v (Tw − Tsat ) x 

(7.170)

where hlv′ is defined by Eq. (7.168a) with C L = 83 . Including the convective transport terms in the analysis tends to further increase C L. Bromley [7.141] suggested a value of 0.5 for C L. Because of the high wall temperatures associated with some film boiling processes, this correction can significantly affect the predicted heat transfer coefficient. As a result, this type of correction to the latent heat is also usually found in correlations for film boiling heat transfer on cylinders, spheres, and flat horizontal surfaces. For laminar film boiling on a vertical surface, Sadasivan and Lienhard [7.155] have examined this issue in some detail. These investigators determined the variation of CL necessary to make Eq. (7.170) match the predictions of Koh’s [7.140] similarity solution described above. In computing the similarity solutions, they took (ρµ) v / (ρµ)l to be zero, which implies that the upward vapor velocity is essentially zero at the interface. Computed results for Prandtl numbers between 0.6 and 1000 and Ja values up to 0.8 indicated that CL was virtually independent of Ja, but did vary with Prandtl number. Based on these results, they recommended that CL be evaluated using the following relation, which closely fit their computed variation of CL:



C L = 0.968 −

0.163 Prv

(7.171)

For a vapor with a Prandtl number of 1.0, this relation indicates that CL should be 0.81, which is well above the value of 3/8 obtained from the approximate integral analysis. Sadasivan and Lienhard [7.155] noted that this relation will not be accurate when (ρµ) v is not small compared to (ρµ)l (i.e., at high pressures), or when bulk motion of the liquid exerts significant shear on the interface. Within the appropriate constraints, they argued that the inaccuracy for non-flat-plate geometries should be minimal, implying that this relation for CL is applicable to cylinders and spherical bodies as well. Further discussion of film boiling heat transfer can be found in the review articles by Jordan [7.156] and Bressler [7.157].

7.7  TRANSITION BOILING As described in Section 7.1, the range of wall superheat levels between the critical heat flux and the minimum heat flux conditions on the boiling curve is usually termed the transition boiling regime. Transition boiling has traditionally been interpreted as a combination of nucleate and film boiling alternately occurring over the heated surface. Compared with the other regimes of pool boiling discussed in this chapter, relatively fewer investigations of transition boiling have been conducted. Much of the transition boiling data that are available have been obtained in transient quenching experiments. Quenching studies typically have been initiated by heating a solid body to a high temperature and suddenly immersing the body into a pool of liquid. The body temperature is typically high enough that immediately after immersion, film boiling occurs at its surface. As the body cools, its surface temperature drops to the point that the boiling process shifts to the transition boiling regime, followed by transitions to nucleate boiling and finally to natural convection.

Pool Boiling

317

The heat flux versus wall superheat for this transient process would typically traverse a curve similar to a-b-c-d-e-f in Fig. 7.3. Transition boiling heat transfer data obtained in this manner correspond to the transition boiling curve obtained when the transition region is entered from the film boiling side. Based on analysis of available data, Witte and Lienhard [7.4] concluded that for a given liquid and heater surface combination, two distinctly different transition boiling curves are possible. The data suggest that the two different transition boiling curves correspond to the extreme limits of surface wettability – one corresponding to conditions where the liquid wets the surface (low contact angle), and the other corresponding to circumstances for which the liquid wets the surface to a lesser degree (higher contact angle). Witte and Lienhard [7.4] also note that in quenching of a hot body, a transition from a higher advancing contact angle to a lower receding one can occur abruptly, causing a dramatic jump in the transition boiling heat flux. The transition boiling conditions obtained by increasing the surface temperature beyond that for the critical heat flux is generally presumed to correspond to a well-wetted surface condition. For these circumstances, when liquid contacts the surface as a dry patch rewets, the low contact angle apparently facilitates a rapid spread of liquid over the surface, resulting in vigorous nucleation, which tends to blow liquid away from the surface. The slugs and columns structure tends to persist beyond the critical heat flux, with vapor supplied to the columns mainly by the vigorous nucleate boiling occurring when a dry patch is rewetted. Because it is an extension of the nucleate boiling curve beyond the critical heat flux, the q″-versus-superheat curve for these conditions is sometimes referred to as the nucleate transition boiling curve. Beginning in the film boiling regime and decreasing the wall superheat, eventually the surface temperature drops to the point that vapor is not produced rapidly enough to sustain a stable vapor layer. The breakdown of the film at these high wall temperatures is postulated to result from liquid being brought into contact with the hot surface by interfacial wave action. The contact occurs over small areas near the nodes of waves, as indicated schematically in Fig. 7.31. The liquid that contacts the solid surface during this breakdown usually exhibits the advancing contact angle θα for the system along the contact line (see Chapter 3). If θα is large, the liquid contacting the surface may spread very little, with the result that the wetted area over which nucleate boiling can occur is small. For such circumstances, the heat transfer from the surface may be only slightly enhanced above that for film boiling alone.

FIGURE 7.31  Onset of transition boiling due to breakdown of the vapor film at node locations in the Taylor wave motion of the interface.

318

Liquid-Vapor Phase-Change Phenomena

FIGURE 7.32  Transition and film boiling data for different surface conditions. Also shown is the film boiling prediction of Klimenko’s correlation. (Adapted from Ramilison and Lienhard [7.154] with permission, copyright 1987, American Society of Mechanical Engineers.)

The effect of surface wettability on the departure from film boiling is indicated by the data in Fig. 7.32. The data in this figure were obtained by Ramilison and Lienhard [7.154] for pool boiling of n-pentane. The data represent the pool boiling heat transfer obtained for three different surfaces as the wall temperature was gradually decreased beginning in the film boiling regime. In tests conducted prior to the boiling experiments, the Teflon was found to have the lowest advancing contact angle of 30°, while the mirror and rough surfaces had higher measured values of 40° and 50°, respectively. As the superheat is decreased, departure from film boiling is observed first in Fig. 7.32 for the surface with the lowest contact angle, and successively later departures are observed for increasingly large values of θα. A similar trend is evident in Fig. 7.33, which shows data obtained by Berenson [7.103] for pool boiling of pentane on a copper surface. The surface with some oxidation, which is better wetted by the liquid, resulted in a higher transition boiling curve and departure from film boiling at a higher temperature and heat flux than for the more poorly wetted clean copper surface. As described above, enhancement of the heat flux above the film boiling correlation value is expected to be highest for the best wetting condition. The portion of the boiling curve which has departed from the film boiling curve but has not yet reached qmin ′′ was referred to by Ramilison and Lienhard [7.154] as the film-transition regime. The trends described above imply that if a system exhibits significant contact angle hysteresis (i.e., θα > θr), the transition boiling curve obtained for increasing wall superheat may be very different from that obtained for decreasing superheat. For increasing superheat, the nucleate transition boiling is encountered first. For such conditions, the liquid-solid contact angle is generally near the receding contact angle θr. This lower contact angle apparently persists as the wall superheat is increased, resulting in the higher transition boiling curve associated with better wetting liquids. For decreasing superheat, the film-transition boiling regime is encountered first, and, as noted above, the liquid-solid contact is initially near the advancing contact angle value for the system. If the advancing contact angle is significantly higher than the receding value, then the transition

Pool Boiling

319

FIGURE 7.33  Berenson’s [7.103] pool boiling data for different surface conditions. The solid lines are best fits to the data in the nucleate boiling and film boiling regimes.

boiling curve for decreasing superheat will, at least initially, be lower than that for increasing superheat, as indicated in Fig. 7.34. If the contact angle remains near θα as the wall temperature drops, the entire transition boiling curve may be lower than for the increasing superheat case, as indicted in Fig. 7.34. However, if the contact angle suddenly changes to a value near the receding contact angle, the heat flux may suddenly rise, and the system may jump to the upper curve in Fig. 7.34.

FIGURE 7.34  Relative locations of the nucleate transition and film transition portions of the boiling curves.

320

Liquid-Vapor Phase-Change Phenomena

The inability to control the contact angle may thus be responsible for considerable variability and/ or scatter in transition pool boiling data. Based on a model of the heat transfer mechanism, Ramilison and Lienhard [7.154] recommended the following correlation for predicting the heat transfer in the transition film boiling regime

Bi* = 3.74 × 10 −6 ( Ja *) K

(7.172)

Bi* =

(q ′′ − q ′′fb ) α h τ kh [Tw − Tsat ( Pl )]K

(7.173)

Ja* =

(ρc p )h (Tdfb − Tw ) ρv hlv

(7.174)

2

where



1/ 4



  σ τ= 3   g (ρl − ρv )  K=

(7.175a)

kl / α1/l 2 kl / α1/l 2 + kh / α1/h 2

(7.175b)

In these relations, q ′′fb is the heat flux predicted for film boiling alone at the specified wall superheat, τ is the characteristic period of the Taylor wave action at the interface, and Tdfb is the wall temperature at which the boiling process departs from film boiling due to the onset of liquid contact with the surface. The variables kh and αh are the thermal conductivity and thermal diffusivity of the heated surface, respectively. Ramilison and Lienhard [7.154] gave a graphical correlation for determining Tdfb, which is well approximated by the relation



Tdfb − Tsat = 0.97 exp −0.00060θ1.8 a Thn − Tsat

(

)

(7.176)

where θa is the advancing contact angle in degrees and T hn is the homogeneous nucleation temperature. The latter can be determined by the corresponding states correlation proposed by Lienhard [7.158]



9   Tsat   Thn =  0.932 + 0.077   Tc  Tc   

(7.177)

where Tc is the critical temperature of the working fluid. The above correlation was found to provide a good fit to transition boiling heat transfer data obtained by Ramilison and Lienhard [7.154] and Berenson [7.103], which spanned a wide variety of fluid and surface combinations. The variation of the transition boiling curve with liquid contact angle was also observed in the experimental data obtained by Chowdhury and Winterton [7.159] for vertical cylinders and Bui and Dhir [7.160] for a vertical flat surface. In both cases the transition boiling curve was found to shift upwards as the liquid became more wetting. Bui and Dhir [7.160] also indicated that for the transient cooling typical of quenching processes, the maximum heat fluxes were as much as 60% lower than the maximum steady-state heat fluxes for the same system. This reduction apparently is a direct consequence of the lower transition boiling curve which results when transition boiling is entered from the film boiling side.

321

Pool Boiling

Dhir and Liaw [7.29] modeled transition boiling from a vertical surface as a rectangular array of vapor stems carrying vapor across a thin region of wall-dominated flow near the surface. Based on this model, they proposed that the heat flux in the transition boiling regime be predicted by adding together the volume-fraction-weighted contributions to the total heat transfer associated with the dry and wet areas on the surface.

q ′′ = h(Tw − Tsat ) = hl (1 − α w )(Tw − Tsat ) + hv α w (Tw − Tsat )

(7.178)

where α w is the vapor volume fraction at the wall. The heat transfer coefficient in the dry region was taken to be that given by a correlation for film boiling heat transfer from a vertical surface developed by Bui and Dhir [7.145]: 1/ 4



  k 3 gρ (ρ − ρv )hlv   g(ρl − ρv ) 1/ 2  hv = 0.47   v v l    σ   µ v (Tw − Tsat )   

(7.179)

The heat transfer to the wet portions of the surface was idealized as being simple conduction from the heated surface to the liquid-vapor interface near the base of the vapor stem. From an analysis of this conduction problem, Dhir and Liaw [7.29] derived the following relation for hl : hl =

2 L2 (1 − α w )

π/4

∫ C k (L sec ψ + D )dψ l

w

(7.180)

0

where ∞

C=

∑ n =1

2sin 2 (λ n b) λ n b + sin(λ n b) cos(λ n b)

(7.181)

L sec ψ − Dw 2

(7.182)

λ n b tanh ( λ n b ) = θ

(7.183)

b=

and the λn values are roots of the equation

where θ is the liquid contact angle. Although the computation is complex, in principle, the above relations can be used to determine hl if the wall void fraction αw, the center-to-center spacing of the vapor stems L, and the diameter of the vapor stems Dw at the wall are specified. Dhir and Liaw [7.29] argued that L should equal the nucleation site spacing for nucleate boiling near the maximum heat flux and they used measured void fraction data to evaluate α w. Dw can then be determined from the following relation, which follows from the geometry of the vapor stem model



αw =

πD 2 4 L2

(7.184)

Dhir and Liaw [7.29] found good agreement between measured heat transfer data and the predictions of the above model. Although many aspects of the model are highly idealized, it does appear to provide a useful theoretical framework for predicting transition boiling heat transfer.

322

Liquid-Vapor Phase-Change Phenomena

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7.56 Nishikawa, K., Fujita, Y., Uchida, S., and Ohta, H., Effect of heating surface orientation on nucleate boiling heat transfer, Proc. ASME-JSME Thermal Eng. Joint Conf, Honolulu, March 20–24, 1983, ASME, New York, NY, vol. 1, pp. 129–136, 1983. 7.57 Moissis, R., and Berenson, P. J., On the hydrodynamic transitions in nucleate boiling, ASME J. Heat Transf., vol. 85, pp. 221–229, 1963. 7.58 Bhattacharya, A., and Lienhard, J. H., Hydrodynamic transition in electrolysis, ASME J. Basic Eng., vol. 94, 804–810,1972. 7.59 Lienhard, J. H., On the two regimes of nucleate boiling, ASME J. Heat Transf., vol. 107, pp 262–264, 1985. 7.60 Dhir, V. K., and Tung, V.X., A thermal model for fully developed nucleate boiling of saturated liquids, in Collected Papers in Heat Transfer 1988 – Volume Two, K. T. Yang (editor), ASME HTD, vol. 104, pp. 153–164, New York, NY, 1988. 7.61 Tong, W., Simon, T. W., and Bar-Cohen, A., A bubble sweeping heat transfer mechanism for low flux boiling on downward-facing inclined surfaces, in Collected Papers in Heat Transfer 1988 – Volume Two, K. T. Yang (editor), ASME HTD, vol. 104, pp. 173–178, New York, NY 1988. 7.62 Merte, H., Jr., Combined roles of buoyancy and orientation in nucleate pool boiling, in Collected Papers in Heat Transfer 1988 – Volume Two, K. T. Yang (editor), ASME HTD, vol. 104, pp. 179–186, New York, NY, 1988. 7.63 Vachon, R. I., Nix, G. H., and Tanger, G. E., Evaluation of constants for the Rohsenow pool-boiling correlation, ASME J. Heat Transf., vol. 90, pp. 239–247, 1968. 7.64 Rohsenow, W. M., Heat transfer with evaporation, Heat Transfer – Symp. Held at the University of Michigan During the Summer of 1952, University of Michigan Press, Ann Arbor, MI, pp. 101–150, 1953. 7.65 Rohsenow, W. M., and Clarke, J. A., Heat transfer and pressure drop data for high heat flux densities to water at high subcritical pressures, 1951 Heat Transfer and Fluid Mechanics Institute, Stanford University Press, Stanford, CA, 1951. 7.66 Kreith, F., and Sommerfield, M., Heat transfer to water at high flux densities with and without surface boiling, Trans. ASME, vol. 71, pp. 805–815, 1949. 7.67 Piret, E. L., and Isbin, H. S., Natural circulation evaporation two-phase heat transfer, Chem. Eng. Prog., vol. 50, p. 305-311, 1954. 7.68 Bergles, A. E. and Rohsenow, W. M., The determination of forced-convection surface-boiling heat transfer, ASME J. Heat Transf., vol. 86, pp. 365–372, 1964. 7.69 Clarke, J. A., in Cryogenic Technology, R. W. Vance (editor), Chapter 5, Heat transfer in cryogenic systems, J. Wiley and Sons, New York, NY, 1963. 7.70 Subbotin, V. I., Sorokin, D. N., and Kudryavtsev, R. R., Generalized relationship for heat transfer in developed boiling of alkali metals, At. Energy, vol. 29, p. 45, 1970. 7.71 Dwyer, O. E., Boiling Liquid Metal Heat Transfer, American Nuclear Society, Hinsdale, IL, 1976. 7.72 Pollack, G. L., Kapitza resistance, Rev. Mod. Phys., vol. 41, pp. 48–81, 1969. 7.73 Pham, A., Barisik, M., and Kim, B., Pressure dependence of Kapitza resistance at gold/wáter and silver/ wáter interface, J. Chem. Phys., vol. 139, 244702-1–244702-10, 2013. 7.74 Gambill, W. R., and Lienhard, J. H., An upper bound for the critical boiling heat flux, Proc. 1987 ASME-JSME Thermal Engineering Joint Conf., vol. 3, pp. 621–626, 1987. 7.75 Kutateladze, S. S., On the transition to film boiling under natural convection, Kotloturbostroenie, no. 3, p. 10, 1948. 7.76 Rohsenow, W. M., and Griffith, P., Correlation of maximum heat transfer data for boiling of saturated liquids, Chem. Eng. Prog. Symp. Ser., vol. 52, no. 18, p. 47–49, 1956. 7.77 Chang, Y. P., and Snyder, N. W., Heat transfer in saturated boiling, Chem. Eng. Prog. Symp. Series, vol. 56, no. 30, pp. 25–38, 1960. 7.78 Zuber, N., Hydrodynamic aspects of boiling heat transfer, AEC Report AECU-4439, 1959. 7.79 Taylor, G. I., The instability of liquid surfaces when accelerated in a direction perpendicular to their planes. I,Proc. R. Soc. Lond. Ser. A, vol. 201, pp. 192–196, 1950. 7.80 Chang, Y. P., A theoretical analysis of heat transfer in natural convection and in boiling, Trans. ASME, vol. 79, pp. 1501–1513, 1957. 7.81 Lienhard, J. H., and Dhir, V. K., Extended hydrodynamic theory of the peak and minimum pool boiling heat fluxes, NASA CR-2270, 1973. 7.82 Lienhard, J. H., Dhir, V. K., and Riherd, D. M., Peak pool boiling heat flux measurements on finite horizontal flat plates, ASME J. Heat Transf., vol. 95, pp. 477–482, 1973. 7.83 Sun, K. H., and Lienhard, J. H., The peak pool boiling heat flux on horizontal cylinders, Int. J. Heat Mass Transf., vol. 13, pp. 1425–1439, 1970.

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7.84 Lienhard, J. H., and Dhir, V. K., Hydrodynamic prediction of peak pool-boiling heat fluxes from finite bodies, ASME J. Heat Transf., vol. 95, pp. 152–158, 1973. 7.85 Ded, J. S. and Lienhard, J. H., The peak pool boiling heat flux from a sphere, AIChE J., vol. 18, pp. 337–342, 1972. 7.86 Bernath, L., A theory of local burnout and its application to existing data, AIChE Symp. Ser., vol. 56, p. 95–116, 1960. 7.87 Costello, C. P., and Frea, W. J., A salient non-hydrodynamic effect on pool boiling burnout of small semi-cylindrical heaters, AIChE Preprint No. 15, 6th Nat. Heat Transfer Conf., Boston, Aug. 11–14, 1963. 7.88 Chang, Y. P., Some possible critical conditions in nucleate boiling, J. Heat Transf., vol. 85, pp. 90–100, 1963. 7.89 Lienhard, J. H., and Hasan, M. M., On predicting boiling burnout with the mechanical energy stability criterion, ASME J. Heat Transf., vol. 101, pp. 276–279, 1979. 7.90 Lienhard, J. H., and Witte, L. C., An historical review of the hydrodynamic theory of boiling, Rev. Chem. Eng., vol. 3, pp. 187–277, 1985. 7.91 Lienhard, J. H., Things we don’t know about boiling heat transfer: 1988, Int. Commun. Heat Mass Transf., vol. 15, pp. 401–428, 1988. 7.92 Theofanous, T. G., Tu, J. P., Dinh, A. T., and Dinh, T. N., The boiling crisis phenomenon Part I: Nucleation and nucleate boiling heat transfer, Exp. Therm. Fluid Sci., vol. 26, pp. 775–792, 2002. 7.93 Kazakova, E. A., Effect of pressure on first crisis in water boiling on a horizontal plate, in: Issues in Heat Transfer Under Phase Change, S. S. Kutateladze (editor), GEI, Moscow, pp. 92–101, 1953. 7.94 Kutateladze, S. S., and Mamontova, N. N., Investigation of critical heat fluxes with pool boiling of liquids under reduced pressure conditions, Inzh. Fiz. J. vol. 12, pp. 181–186, 1967. 7.95 Labulsov, D. A., Yagov, V. V., and Gorodov, A. K., Critical heat fluxes in boiling at low pressure region, Proceedings 6th Int. Heat Transfer Conf., vol. 1, 1978, pp. 221–225. 7.96 Avksentyuk, B. P., and Mesrakesshivili, Z. S., The effect of subcooling on critical heat fluxes with liquid boiling in the region of subatmospheric pressures, in: Boiling and Condensation, Novosibirsk, Institute of Thermal Physics, USSR Academy of Sciences, pp. 46–51, 1986. 7.97 Samokhin, G. I. and Yagov, V. V., Heat transfer and critical fluxes with liquids boiling in the region of low reduced pressures, Therm. Eng., vol. 35, pp. 115–118, 1988. 7.98 Yagov, V. V., A physical model and calculation formula for critical heat fluxes with nucleate pool boiling of liquids, J. Thermal Eng., vol. 35, pp. 333–339, 1988. 7.99 Straub, J., Zell, M., and Vogel, B., Pool boiling in reduced gravity field, Proceedings of the 9th International Heat Transfer Conference, Jerusalem, Israel, vol. 1, pp. 91–112, Hemisphere, NY, 1990. 7.100 Abe, Y., Oka, T., Mori, Y., and Nagashima, A., Pool boiling of a non-azeotropic mixture under microgravity, Int. J. Heat Mass Transf., vol. 37, pp. 2405–2413, 1994. 7.101 Oka, T., Abe, Y., Mori, Y. H., and Nagashima, A., Pool boiling of n-pentane, CFC-113, and water under reduced gravity: Parabolic flight experiments with a transparent heater, ASME J. Heat Transf., vol. 117, pp. 408–417, 1995. 7.102 Shatto, D. P., and Peterson, G. P., Pool boiling critical heat flux in reduced gravity, ASME J. Heat Transf., vol. 121, pp. 865–873, 1999. 7.103 Berenson, P. J., Transition boiling heat transfer from a horizontal surface, AIChE Paper 18, ASME-AIChE Heat Transfer Conference, Buffalo, NY (also MIT Heat Transfer Laboratory Tech. Report No. 17), 1960. 7.104 Hahne, E., and Diesselhorst, T., Hydrodynamic and surface effects on the peak heat flux in pool boiling, Proc 6th Int. Heat Transf. Conf., vol. 1, pp. 209–214, 1978. 7.105 Nagai, N., and Carey, V. P., Assessment of surface wettability and its relation to boiling phenomena, Therm. Sci. Eng., vol. 10, pp. 1–9, 2002. 7.106 Kandlikar, S. G., A theoretical model to predict pool boiling CHF incorporating effects of contact angle and orientation, J. Heat Transf., vol. 123, pp. 1071–1079, 2001. 7.107 Kirishenko, Y. A., and Cherniakov, P. S., Determination of the first critical thermal heat flux on flat heaters, J. Eng. Phys., vol. 20, pp. 699–702, 1973. 7.108 Diesselhorst, T., Grigull, U., and Hahne, E., Hydrodynamic and surface effects on the peak heat flux in pool boiling, on Heat Transfer in Boiling, E. Hahne, and U. Grigul (editors), Hemisphere Publishing Corporation, Washington, 1977. 7.109 Noyes, R. C., and Lurie, H., Boiling sodium heat transfer, Proc. 3rd Int. Heat Transfer Conf, Chicago, IL, vol. 5, p. 92, 1966. 7.110 Theofanous, T. G., Dinh, T. N., Tu, J. P. and Dinh, A. T., The boiling crisis phenomenon Part II: Dryout dynamics and burnout, Exp. Therm. Fluid Sci., vol. 26, pp. 793–810, 2002.

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7.111 Guan, C-K., Klausner, J. F., and Mei, R., A new mechanistic model for pool boiling CHF on horizontal surfaces, Int. J. Heat Mass Transf., vol. 54, pp. 3960–3969, 2011. 7.112 Haramura, Y., and Katto, Y., A new hydrodynamic model of the critical heat flux, applicable widely to both pool and forced convective boiling on submerged bodies in saturated liquids, Int. J. Heat Mass Trans., vol. 26, pp. 389–399, 1983. 7.113 Reyes, R., and Wayner, J. P. C., An adsorption model for the superheat at the critical heat flux, J. Heat Transf., vol. 117, pp. 779–782, 1995. 7.114 Galloway, J. E., and Mudawar, I., CHF mechanism in flow boiling from a short heated wall – I. Examination of near-wall conditions with the aid of photomicrography and high-speed video imaging, Int. J. Heat Mass Transf., vol. 36, pp. 2511–2526, 1993. 7.115 Galloway, J. E., and Mudawar, I., CHF mechanism in flow boiling from a short heated wall – II. Theoretical CHF model, Int. J. Heat Mass Transf., vol. 36, pp. 2527–2540, 1993. 7.116 Sturgis, J. C., and Mudawar, I., Critical heat flux in a long rectangular channel subjected to one-sided heating – II. Analysis of critical heat flux data, Int. J. Heat Mass Transf., vol. 42, pp. 1849–1862, 1999. 7.117 Mudawar, I., Howard, A. H., and Gersey, C. O., An analytical model for near-saturated pool boiling critical heat flux on vertical surfaces, Int. J. Heat Mass Transf., vol. 40, pp. 2327–2339, 1997. 7.118 Kim, T. H., Kommer, E., Dessiatoun, S., and Kim, J., Measurement of two-phase flow and heat transfer parameters using infrared thermometry, Int. J. Multiphase Flow, vol. 40, pp. 56–67. 2011. 7.119 O’Hanley, H., Coyle, C., Buongiorno, J., McKrell, T., Hu, L.- W., Rubner, M., Cohen, R. Separate effects of surface roughness, wettability, and porosity on the boiling critical heat flux, Appl. Phys. Lett., vol. 103, p. 024102, 2013. 7.120 Zou, A., and Maroo, S. C., Critical height of micro/nano structures for pool boiling heat transfer enhancement, Appl. Phys. Lett., vol. 103, p. 221602, 2013. 7.121 Chen, R., Lu, M.-C., Srinivasan, V., Wang, Z., Cho, H. H., and Majumdar, A., Nanowires for enhanced boiling heat transfer, Nano Lett., vol. 9, pp. 548−553, 2009. 7.122 Li, C., Wang, Z., Wang, P. I., Peles, Y., Koratkar, N., and Peterson, G. P., Nanostructured copper interfaces for enhanced boiling, Small, vol. 4, pp. 1084−1088, 2008. 7.123 Lu, M.-C., Chen, R., Srinivasan, V., Carey, V. P., and Majumdar, A., Critical heat flux of pool boiling on Si nanowire array-coated surfaces, Int. J. Heat Mass Transf., vol. 54, pp. 5359−5367, 2011. 7.124 Yao, Z., Lu, Y.-W., and Kandlikar, S. G., Pool boiling heat transfer enhancement through nanostructures on silicon microchannels, J. Nanotechnol. Eng. Med., vol. 3, p. 031002-1 to 031002-8, 2012. 7.125 Betz, A. R., Jenkins, J., Kim, C. J., and Attinger, D., Boiling heat transfer on superhydrophilic, superhydrophobic, and superbiphilic surfaces, Int. J. Heat Mass Transf., vol. 57, pp. 733−741, 2013. 7.126 Betz, A. R., Xu, J., Qiu, H., and Attinger, D., Do surfaces with mixed hydrophilic and hydrophobic areas enhance pool boiling?, Appl. Phys. Lett., vol. 97, p. 141909, 2010. 7.127 Guan, C. K., Bon, B., Klausner, J., and Mei, R. W., Comparison of CHF enhancement on microstructured surfaces with a predictive model, Heat Transf. Eng., vol. 35, pp. 452−460, 2014. 7.128 Kim, B. S., Lee, H., Shin, S., Choi, G., and Cho, H. H., Interfacial wicking dynamics and its impact on critical heat flux of boiling heat transfer, Appl. Phys. Lett., vol. 105, pp. 191671-1–191671-4, 2014. 7.129 Rahman, M. M., Ölcerogl˘u, E., and McCarthy, M., Role of wickability on the critical heat flux of structured superhydrophilic surfaces, Langmuir, vol. 30, pp. 11225−11234, 2014. 7.130 Ahn, H. S., and Kim, M. H., A review on critical heat flux enhancement with nanofluids and surface modification, J. Heat Transf., vol.134, pp. 024001-1–024001-13, 2012. 7.131 Dhillon, N. S., Buongiorno, J., and Varanasi, K. K., Critical heat flux maxima during boiling crisis on textured surfaces, Nat. Commun., vol. 6, 8247. 7.132 Borishansky, V. M., Novikov, I. I., and Kutateladze, S. S., Use of thermodynamic similarity in generalizing experimental data of heat transfer, paper no. 56, Int. Heat Transfer Conf., Univ. of Colorado, Boulder, Co, 1961, Int. Dev. in Heat Transfer, ASME, New York, NY, pp. 475–482, 1963. 7.133 Sharan, A., Lienhard, J. H., and Kaul, R., Corresponding states correlations for pool and flow boiling burnout, ASME J. Heat Transf., vol. 107, pp. 392–397, 1985. 7.134 Stephan, K., Heat Transfer in Condensation and Boiling, Springer-Verlag, Berlin, 1992. 7.135 Nikolaev, G. P., and Skripov, V. P., Calculation of the critical heat flux based upon thermodynamic similarity (in Russian), Inzh. Fiz. Zh., vol. 15, pp. 46–51, 1968. 7.136 Chang, Y. P., Wave theory of heat transfer in film boiling, ASME J. Heat Transf., vol. 81, p. 112, 1959. 7.137 Berenson, P. J., Film boiling heat transfer from a horizontal surface, ASME J. Heat Transf., vol. 83, p. 351, 1961. 7.138 Lienhard, J. H., and Wong, P. T. Y., The dominant unstable wavelength and minimum heat flux during film boiling on a horizontal cylinder, ASME J. Heat Transf., vol. 86, pp. 220–226, 1964.

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7.139 Gunnerson, F. S., and Cronenberg, A. W., On the minimum film boiling conditions for spherical geometries, ASME J. Heat Transf., vol. 102, pp. 335–341, 1980. 7.140 Koh, J. C. Y., Analysis of film boiling on vertical surfaces, ASME J. Heat Transf., vol. 84, p. 55, 1962. 7.141 Bromley, L. A., Heat transfer in stable film boiling, Chem. Eng. Prog., vol. 46, no. 5, pp. 221–227, 1950. 7.142 Lubin, B. T., Analytical derivation for total heat transfer coefficient in stable film boiling from vertical plate, ASME J. Heat Transf., vol. 91, pp. 452–453, 1969. 7.143 Sparrow, E. M., The effect of radiation on film-boiling heat transfer, Int. J. Heat Mass Transf., vol. 7, pp. 229–238, 1964. 7.144 Greitzer, E. M., and Abernathy, F. H., Film boiling on vertical surfaces, Int. J. HeatMass Transf., vol. 15, pp. 475–491, 1972. 7.145 Bui, T. D., and Dhir, V. K., Film boiling heat transfer on an isothermal vertical surface, ASME J. Heat Transf., vol. 107, pp. 764–771, 1985. 7.146 Hsu, Y. Y., and Westwater, J. W., Approximate theory for film boiling on vertical surfaces, Chem. Eng. Prog. Symp. Ser., vol. 56, no. 30, p. 15, 1960. 7.147 Hsu, Y. Y., and Westwater, J. W., Film boiling from vertical tubes, AIChE J., vol. 4, pp. 58–62, 1958. 7.148 Suryanarayana, N. Y., and Merte, H., Film boiling on vertical surfaces, ASME J. Heat Transf., vol. 94, pp. 377–384, 1972. 7.149 Breen, B. P., and Westwater, J. W., Effect of diameter of horizontal tubes on film heat transfer, Chem. Eng. Prog., vol. 58, no. 7, p. 67, 1962. 7.150 Schlichting, H., Boundary Layer Theory, Chapter XI, McGraw-Hill, New York, NY, 1968 7.151 Frederking, T. H. K., and Clark, J. A., Natural convection film boiling on a sphere, Adv. Cryog. Eng., vol. 8, pp. 501–506, 1963. 7.152 Borishansky, V. M., Heat transfer to a liquid freely flowing over a surface heated to a temperature above the boiling point, in Problems of Heat Transfer During a Change of State, S. S. Kutateladze (editor) (in translated form as AEC-tr-3405), pp. 109–144, US Atomic Energy Commission, Washington, DC, 1959. 7.153 Klimenko, V. V., Film boiling on a horizontal plate – new correlation, Int. J. Heat Mass Transf., vol. 24, pp. 69–79, 1981. 7.154 Ramilison, J. M., and Lienhard, J. H., Transition boiling heat transfer and the film transition regime, ASME J. Heat Transf., vol. 109, pp. 746–752, 1987. 7.155 Sadasivan, P., and Lienhard, J. H., Sensible heat correction in laminar film boiling and condensation, ASME J. Heat Transf., vol. 109, pp. 545–546, 1987. 7.156 Jordan, D. P., Film and transition boiling, Adv. Heat Transf., vol. 5, pp. 55–128, 1968. 7.157 Bressler, R. G., A review of physical models and heat transfer correlations for free convection film boiling, Adv. Cryog. Eng., vol. 17, pp. 382–406, 1972. 7.158 Lienhard, J. H., Corresponding states correlations for the spinoidal and homogeneous nucleation temperatures, ASME J. Heat Transf., vol. 104, pp. 379–381, 1982. 7.159 Chowdhury, S. K. R., and Winterton, R. H. S., Surface effects in pool boiling, Int. J. Heat Mass Transf., vol. 28, pp. 1881–1889, 1985. 7.160 Bui, T. D., and Dhir, V. K., Transition boiling heat transfer on a vertical surface, ASME J. Heat Transf., vol. 107, pp. 756–763, 1985.

PROBLEMS 7.1 Compute the heat flux value predicted by the Borishansky nucleate boiling correlations for n-butanol at atmospheric pressure for a wall superheat of 10°C. Compare the results with those in Example 7.1 for the Rohsenow and Stephan-Abdelsalam correlations. 7.2 Use the Forster-Zuber correlation to predict the heat flux for nucleate boiling of liquid nitrogen at atmospheric pressure at a wall superheat of 5°C. Compare the result with those for the Clark and StephanAbdelsalam correlations given in Example 7.2. 7.3 You are asked to design an enhanced pool boiling heat transfer surface made of copper with artificial nucleation sites. This surface must transfer as high a heat flux as possible to a saturated liquid pool at atmospheric pressure and a wall superheat of 5°C. To estimate the mouth size for the cavities, proceed as follows: (a) Use the Rohsenow correlation for a rough copper surface to estimate q″ at the designated operating conditions. Then estimate δt as δt = kl(Tw – Tsat)/q″. (b) Use the results of Hsu’s analysis described in Section 6.3 to estimate the range of cavity mouth radii that will be active for these conditions. This provides an estimate for the mouth size of the artificial cavities on the enhanced surface.

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7.4 Pool boiling of liquid mercury at atmospheric pressure occurs on an upward-facing horizontal surface immersed in saturated liquid mercury. The wall superheat for the surface is 10°C. Determine the resulting heat flux using (a) the Rohsenow correlation with Csf = 0.013 and (b) the correlation proposed by Subbotin et al. [7.70]. 7.5 Determine the heat flux corresponding to the Moissis-Berenson transition for water boiling at atmospheric pressure, and at pressures of 571, 2185, 6124, and 14,044 kPa. The heated surface is upward facing and essentially infinite in extent, and the contact angle is 20°. Also, compute the maximum heat flux condition at these same pressures. From the computed results, determine and plot the fraction of the maximum heat flux corresponding to the Moissis-Berenson transition as a function of reduced pressure (P/Pc). What do you conclude about the effect of pressure on this transition? 7.6 Determine the heat flux corresponding to the Moissis-Berenson transition for water boiling at atmospheric pressure, and at pressures of 70.2, 31.2, and 12.35 kPa. The heated surface is upward facing and essentially infinite in extent, and the contact angle is 20°. Also, compute the maximum heat flux condition at these same pressures. From the computed results, determine and plot the fraction of the maximum heat flux corresponding to the Moissis-Berenson transition as a function of reduced pressure (P/Pc). What do you conclude about the effect of subatmospheric pressure on this transition? 7.7 For a horizontal cylinder with a diameter of 2 mm, determine the critical heat flux for pool boiling of methanol at a pressure of 1 atm. Using the appropriate form of the Stephan-Abdelsalam correlation, estimate the wall temperature corresponding to the critical heat flux condition. 7.8 For a horizontal cylinder with a diameter of 2 mm, determine the critical heat flux for pool boiling of water at a pressure of 1 atm and 70.2 kPa. Using the appropriate form of the Stephan-Abdelsalam correlation, estimate the wall temperature corresponding to the critical heat flux condition. 7.9 Use the Zuber Helmholtz instability model to estimate the critical heat flux for pool boiling from a 13 mm square stainless steel surface facing upward in a pool of saturated liquid water at atmospheric pressure. Assuming that the nucleate boiling correlation applies up to the critical heat flux condition, use the Rohsenow correlation with Csf = 0.013 and the Stephan-Abdelsalam correlation to estimate the temperature of the heated surface at the maximum heat flux. 7.10 Use the Zuber Helmholtz instability model to estimate the critical heat flux for pool boiling from a flat, infinite, stainless steel surface facing upward in a pool of saturated liquid water at atmospheric pressure and 70.2 kPa. Assuming that the nucleate boiling correlation applies up to the critical heat flux condition, use the Rohsenow correlation with Csf = 0.013 and the Stephan-Abdelsalam correlation to estimate the temperature of the heated surface at the maximum heat flux. 7.11 Use the liquid macrolayer liftoff model of Guan et al. [7.111] to estimate the critical heat flux for pool boiling from a flat, infinite, stainless steel surface facing upward in a pool of saturated liquid water at atmospheric pressure and 70.2 kPa. Compare these results to the CHF predicted by the Kandlikar [7.106] correlation at these pressures. 7.12 A 30 mm square computer processor is mounted on a circuit board that is immersed in a pool of saturated R-113 liquid at atmospheric pressure to cool it during bum-in. The board is positioned so the processor surface faces upward. The saturation properties of R-113 are: Tsat = 320.8 K, ρv = 7.46 kg/m3 ρl, = 1507 kg/m3, hlv = 143.8 kJ/kg, and σ = 0.0169 N/m. (a) Is L/Lb large enough to use the infinite plate relation for qmax ′′ in Table 7.4? (b) Assuming the correlation can be used, predict the maximum heat flux that can be removed from the surface of the processor. (c) Use the Stephan-Abdelsalam correlation and the Borishansky-Mostinski correlation to estimate the surface temperature of the chip at the maximum heat flux. How well do they agree? 7.13 Estimate the critical heat flux for pool boiling from a 1 mm diameter sphere immersed in saturated liquid oxygen at atmospheric pressure. By how much is the value estimated to change if the sphere diameter is increased to 5 mm? 7.14 Estimate the percent change in CHF for an upward-facing heated infinite surface at atmospheric pressure if the contact angel changes from 40° to 2°. 7.15 Estimate how large the wickability V0′′ must be to increase the critical heat flux by a factor of two above that for a flat surface with zero wickability. 7.16 For pressures between 1 atm and the critical point, determine and plot the variation of the maximum heat flux with pressure for pool boiling from an infinite upward-facing horizontal surface immersed in saturated liquid nitrogen. Estimate the pressure corresponding to the highest maximum heat flux value for these conditions. 7.17 For pressures between 1 atm and the critical point, determine and plot the variation of the minimum heat flux with pressure for pool boiling from an infinite upward-facing horizontal surface immersed in saturated liquid water.

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7.18 For values of cylinder radius between 1 and 10 mm, determine and plot the variation of the minimum heat flux for pool boiling from a horizontal cylinder immersed in saturated liquid nitrogen at atmospheric pressure. 7.19 For pressures between 1 atm and the critical point, determine and plot the variation of the minimum heat flux with pressure for pool boiling from an infinite upward-facing horizontal surface immersed in saturated liquid mercury. 7.20 A vertical flat surface held at 190°C is immersed in saturated liquid water at atmospheric pressure. Assuming that laminar film boiling occurs at the surface, use Bromley’s superposition technique to estimate the total heat flux from the surface due to the combined effects of convection and radiation. Assume that the wall radiates as a gray body with an emittance of 0.9 and that the interface radiates as a blackbody. (Neglect any radiation interaction with water vapor in the film.) 7.21 (a) For an infinite, upward-facing flat, horizontal surface, determine the minimum heat flux for pool boiling of saturated R-22 at 619kPa. (b) Use Berenson’s correlation for laminar film boiling, with the radiation effect included, to estimate the wall superheat corresponding to the minimum heat flux condition determined in part (a). Assume that the wall radiates as a gray body with an emittance of 0.85 and the interface acts like a blackbody. 7.22 For laminar, natural-convection film boiling over an isothermal horizontal cylinder, use an integral analysis to derive an expression for the local heat transfer coefficient near the bottom of the cylinder. Assume that the interface is smooth, neglect radiation effects, and take all thermophysical properties to be constant. Compare your results to the correlation represented in Eqs. (7.135) and (7.136). (Hint: Follow the analysis presented for the sphere in Example 7.6. Make similar idealizations, but allow for the differences in geometry.) 7.23 Use an integral analysis of laminar, natural-convection film boiling over a vertical, isothermal, flat surface to show that, when radiation from the surface dominates over conduction and convection, the film thickness varies with downstream location as 1/3



 3µ h (T − T ) x  δ =  v rad w sat   ρv g(ρl − ρv )hlv′ 

where



3   hlv′ = hlv  1 + Ja  ,  8 

Ja =

c pv (Tw − Tsat ) hlv

and



hrad =

2 σ SB (Tw2 + Tsat )(Tw + Tsat ) (1 / ε w ) + (1 / ε l ) − 1

7.24 Film boiling of liquid nitrogen at atmospheric pressure occurs over an infinite, upward-facing flat plate held at 120 K. For these circumstances, determine whether the flow is laminar or turbulent, and use the appropriate form of Klimenko’s correlation to predict the heat transfer coefficient. 7.25 Use Berenson’s correlation (7.158) together with the qmin ′′ value predicted by Eq. (7.75) to estimate the minimum wall superheat required to maintain stable film boiling on an upward-facing infinite horizontal surface for pool boiling of liquid nitrogen at atmospheric pressure. 7.26 Consider pool boiling of water under earth-normal gravity. (a) Use the appropriate correlation in Table 7.4 to determine the critical heat flux for water boiling on a large (L/Lmax » 1) heated horizontal upwardfacing surface at atmospheric pressure. (b) Use the Rohsenow correlation with Csf = 0.013 and the ′′ to qmax Stephan-Abdelsalam correlation to determine the pool boiling curve from 0.1 qmax ′′ . Plot the resulting nucleate boiling curves for both correlations. (c) Now consider the same boiling process in a base on the moon’s surface at atmospheric pressure. Take the gravity body force to be one-sixth of earth-normal gravity. Repeat (a) and (b) for these circumstances, plot the boiling curves, and assess the effect of reduced gravity on the boiling process.

8

Other Aspects of Boiling and Evaporation in an Extensive Ambient

8.1  ADDITIONAL PARAMETRIC EFFECTS ON POOL BOILING Subcooling Subcooling of the ambient pool below saturation generally shifts the boiling curve as shown in Fig. 8.1. The natural convection portion of the boiling curve will shift upward because the driving temperature difference increases with increasing subcooling. As noted in Section 7.2, subcooling has little, if any, effect on nucleate boiling heat transfer. As a result, the nucleate boiling portion of the boiling curve usually changes very little with increasing subcooling. Saturated nucleate boiling correlations may therefore be used with reasonable accuracy, in most cases, even for subcooled conditions. The maximum heat flux is strongly influenced by subcooling. The vapor leaving the region near the heated surface will tend to condense as it rises through the subcooled pool, making it easier for liquid to flow toward the surface. This suggests that a higher heat flux can be attained before the critical condition is reached. This trend is, in fact, observed experimentally. Kutateladze [8.1] apparently was the first to derive an equation for the critical heat flux that accounts for the effect of liquid subcooling. He postulated that the critical heat flux for subcooled conditions should exceed that for saturation by the amount of heat needed to bring the subcooled liquid approaching the surface to the saturation temperature. Based on this line of reasoning, he proposed the correlation 1/ 4



 gσ (ρl − ρv )  qmax ′′ = 0.16ρv hlv   ρ2v  

m   ρl   c pl ( Tsat − Tl )   1 + C0      (8.1) hlv  ρv    

where Tl is the bulk liquid temperature. Values of the constants C0 and m equal to 0.065 and 0.8, respectively, were recommended as a best fit to available data. Somewhat later, Ivey and Morris [8.2] recommended the same correlation, but proposed constant values C0 = 0.1 and m = 0.75. Zuber et al. [8.3] proposed the relation 1/ 4



 gσ (ρl − ρv )  qmax ′′ = 0.16ρv hlv   ρ2v  

1/8 1/ 4    g(ρl − ρv )  ( kl c pl ρl )1/ 2 (Tsat − Tl )  ρ2v × 1 + 5.32   gσ (ρ − ρ    σ ρv hlv  l v    

(8.2)

Transition boiling is also strongly affected by subcooling. In general, the transition boiling curve is expected to shift upward as the subcooling increases, but at the present time there is virtually no quantitative information on its effect.

331

332

Liquid-Vapor Phase-Change Phenomena

FIGURE 8.1  The effects of subcooling on the boiling curve.

Film boiling heat transfer is also enhanced by increased subcooling of the liquid pool. However, the enhancement tends to diminish as the heat flux and wall superheat increase because convection of sensible heat into the subcooled liquid pool becomes progressively smaller relative to the energy convected away in the vapor. Boundary-layer analyses of film boiling in a subcooled liquid pool have been presented by Sparrow and Cess [8.4] and Frederking and Hopenfeld [8.5]. Example 8.1 Compute qmax ′′ using Kutateladze’s [8.1] correlation and the correlation of Zuber et al. [8.3] for water boiling on an infinite horizontal surface at atmospheric pressure and subcooling levels of 10°C and 50°C. For water at atmospheric pressure, ρv = 0.597 kg/m3, ρl = 958 kg/m3, hlv = 2257 kJ/kg, cpl = 4.22 kJ/kgK, kl = 0.679 W/mK, and σ = 0.0589 N/m. With the recommended C0 and m values, Kutateladze’s [8.1] correlation is 1/ 4



 gσ(ρl − ρv )  qmax ′′ = 0.16ρv hlv   ρv2  

0.8   ρ   c (T − Tl )   1+ 0.065  l   pl sat   hlv  ρv    

Substituting for 10°C subcooling, 1/ 4



 9.8(0.0589)(958 − 0.597)  qmax ′′ = 0.16(0.597)(2257)   (0.597)2  

0.8   958   4.22(10)   2 × 1+ 0.065   = 1953 kW/m  0.597   2257    

333

Other Aspects of Boiling and Evaporation Repeating the calculation for Tsat – Tl = 50°C yields q′′max = 4353 kw/m 2



The correlation of Zuber et al. [8.3] can be written as 1/ 4

 gσ(ρl − ρv )  qmax ′′ = 0.16ρv hlv   ρv2  



(1+ ξ )

where 1/ 4

 g(ρl − ρv )  ξ = 5.32   σ 



1/8

 (kl cρl ρl )1/ 2(Tsat − Tl )  ρv2   ρv hlv  gσ(ρl − ρv ) 

Substituting for 10°C subcooling, 1/ 4



 9.8(957)  ξ = 5.32   0.0589 

1/8

 [(0.679)(4220)(958)]1/ 2(10)  (0.597)2   0.597(2,257,000) 9.8(0.0589)(957)  

= 0.522

1/ 4

 9.8(0.0589)(957)  q′′max = 0.16(0.597)(2257)   (0.597)2  

(1+ 0.522) = 2059 kW/m 2

Repeating the calculation for Tsat – Tl = 50°C yields

q′′max = 4884 kw/m 2

It can be seen that the predicted qmax ′′ values for these correlations differ only slightly. Increasing the subcooling produces a substantial increase in qmax ′′ .

Forced Convection The introduction of a forced convection effect generally increases the single-phase heat transfer coefficient above that for natural convection alone, resulting in an upward shift in the single-phase portion of the curve. The associated reduction in the thermal boundary-layer thickness may also suppress the onset of nucleation until a higher wall superheat is attained, as indicated schematically in Fig. 8.2. In most instances, nucleate boiling is such a strong heat transfer mechanism that introduction of a forced convection effect has little effect on the nucleate boiling portion of the boiling curve. The exception would be systems with weak nucleate boiling (at low wall superheat levels) and/or very strong convective effects. When both mechanisms are important and the amount of vapor generated is small, Bergles and Rohsenow [8.6] recommend the following empirical relation for predicting the heat flux:

q ′′pb q ′′ = q ′′fc 1 + q ′′fc

2

 qi′′   1 − q ′′  (8.3) pb

where q ′′fc is the heat flux for single-phase liquid forced convection alone for the heater at the specified wall temperature, q ′′pb is the heat flux for pool boiling alone for the surface at the actual wall superheat, and qi′′ is the heat flux for pool boiling at the threshold superheat where nucleate boiling just begins. The wall superheat for incipient boiling can be estimated using Hsu’s

334

Liquid-Vapor Phase-Change Phenomena

FIGURE 8.2  The effects of forced convection on the boiling curve.

analysis (see Section 6.2) or from empirical correlations. While this is a useful interpolation scheme, it is strictly applicable to subcooled flow or low-quality flow, where the amount of vapor generated is small. As indicated in Fig. 8.2, the maximum pool boiling heat flux is generally strongly increased by the addition of a forced convection effect. For parallel forced-flow of saturated liquid over a flat plate, Haramura and Katto [8.7] recommend the correlation

ρ  qmax ′′ = 0.175  v  Ghlv  ρl 

0.467

1/3

 σρl  (8.4)  2  G L

where L is the length of the plate in the flow direction, and G is the bulk mass flux G = ρlul, ul being the bulk flow velocity parallel to the plate. For a cylinder of diameter d in a saturated liquid cross flow, these investigators recommended

ρ  qmax ′′ = 0.151 v  Ghlv  ρl 

0.467

1/3

 σρl  (8.5)  2  G d

For cross flow over a cylinder at high velocities, Lienhard and Eichhorn [8.8] recommended the relation

1/ 4 1/6 1/3 qmax 1  1  ρv  1  ρv   σ   ′′ =  +  (8.6a) 19.2  ρl   ρl u∞2 d   ρl u∞ hlv π  169  ρl   

335

Other Aspects of Boiling and Evaporation

while at low velocities they recommended 2/3 1/3 ρ   σ   qmax 1  ρ  ′′ =  v  + (4)1/3  v    (8.6b) ρl u∞ hlv π  ρl   ρl   ρl u∞2 d   



In the above relations, u∞ is the cross-flow velocity and d is the diameter of the cylinder. The highand low-velocity regimes were specified as

Low velocity: u∞ ≤

πqmax ′′ ρv hlv  0.275(ρl / ρv )1/ 2 + 1



High velocity: u∞ >

πqmax ′′ 1/ 2  ρv hlv 0.275 ( ρl / ρv ) + 1  

Other correlations for qmax ′′ have also been developed for specific forced flow circumstances. For a cylindrical jet of saturated liquid with diameter djet impinging on a heated disk of diameter D, Sharan and Lienhard [8.9] proposed qmax ′′  d jet  = ( 0.21 + 0.0017 γ )   D  ρl hlv u jet



1/3

A

 1000   γWe  (8.7) D

where A = 0.486 + 0.06052 ln γ − 0.0378(ln γ )2 + 0.00362(ln γ )3 (8.8)



γ=

ρl (8.9) ρv



WeD =

ρl u 2jet D (8.10) σ

and ujet is the mean velocity of the jet. Relations predicting the critical heat flux have also been developed for a submerged jet of saturated liquid flowing parallel to a small rectangular heater [8.10], for a disk heater flush mounted in a rectangular flow channel [8.11], and for subcooled boiling from an isolated small square heater element on the wall of a channel [8.12]. Example 8.2 Determine qmax ′′ for flow of saturated R-113 at atmospheric pressure over a flat plate 5 mm long (in the flow direction) for free stream flow rates of 0.5, 5.0, and 50 m/s. Equation (8.4) recommended by Haramura and Katto [8.7] can be written as

ρ  qmax ′′ = 0.175ρl u∞ hlv  v   ρl 

0.467

 σ   ρ u 2 L 

1/ 3

l ∞

For saturated R-113 at atmospheric pressure, ρl = 1507 kg/m3, ρv = 7.46 kg/m3, hlv = 143.8 kJ/kg, and σ = 0.0169 N/m. Substituting these properties with L = 0.005 m for u∞ = 0.5 m/s yields



 7.46  qmax ′′ = 0.175 (1507)( 0.5)(143.8)   1507  = 330 kW/m 2

0.467

1/ 3

  0.0169   2  1507 ( 0.5) ( 0.005) 

336

Liquid-Vapor Phase-Change Phenomena

Repeating the calculation for u∞ = 5.0 and 50 m/s yields

qmax ′′ = 712 kW/m 2



qmax ′′ = 1534 kW/m

for u∞ = 5.0 m/s 2

for u∞ = 50 m/s

The addition of a forced convection effect is generally expected to enhance transition boiling heat transfer relative to that for pool boiling under similar conditions. However, at the present time there is very little information regarding the effects of forced convection on transition boiling. Forced convection film boiling over immersed bodies will result in a higher heat flux for a given wall superheat level than for ordinary pool boiling under the same conditions. Experimental data for forced convection film boiling over cylinders have been reported by Bromley et al. [8.13] and Yilmaz and Westwater [8.14]. Bromley et al. [8.13] recommended the following relation for film boiling heat transfer during forced flow of saturated liquid over a cylinder  k ( T − Tsat ) ρv hlv′ u∞  q ′′ = C0  v w  (8.11) D   1/ 2

where

c pv ( Tw − Tsat )   hlv′ = hlv 1 + 0.5  (8.12) hlv  



A value of C0 = 2.70 was found to provide a best fit to experimental data. Witte [8.15] recommended the same relation for saturated forced convection film boiling over a sphere, but recommended 2.98 as a better value of C0 for spherical bodies. In a recent study, Orozco and Witte [8.16] obtained additional heat transfer data for forcedconvection film boiling over a sphere, and found good agreement between their data and predictions of an integral boundary-layer analysis. Analytical treatments of forced-convection film boiling have also been developed by Motte and Bromley [8.17], Cess and Sparrow [8.18, 8.19], and Cess [8.20]. Example 8.3 Use the correlation of Bromley et al. [8.13] to predict the heat flux for film boiling over a cylinder with a diameter of 4 mm in a cross flow of saturated liquid nitrogen at atmospheric pressure. Compare the heat flux for a wall superheat of 40°C with cross-flow velocity values of 0.5, 2.0, and 5.0 m/s. For saturated nitrogen at atmospheric pressure, ρv = 4.62 kg/m3, cpv = 1.12 kJ/kgK, hlv = 197.6 kJ/kg, and kv = 0.00754 W/mK. It follows that



 c pv (Tw − Tsat )  hlv′ = hlv 1+ 0.5  hlv    0.5 (1.12)( 40 )  = (197.6) 1+  = 220.0 kJ/kg 197.6  

The correlation (8.11) proposed by Bromley et al. [8.13] is  kv (Tw − Tsat ) ρv hlv′ u∞  q′′ = 2.70   D  

1/ 2



337

Other Aspects of Boiling and Evaporation Substituting for u∞ = 0.5 m/s yields  ( 0.00754 )( 40 )( 4.62)( 220,000 )( 0.5)  q′′ = 2.70   0.004  

1/ 2



= 16,700 W/m 2 = 16.7 kW/m 2 Repeating the calculation for u∞ = 2.0 and 5.0 m/s results in

q′′ = 33.4 kW/m 2  for u∞ = 2.0 m/s



q′′ = 52.8 kW/m 2  for u∞ = 5.0 m/s

Size and Wettability of the Surface The classical boiling curves described in Section 7.1 are characteristic of heater surfaces that satisfy two conditions: (1) they must be at least partially wetted by the liquid in the surrounding pool and (2) the characteristic dimension of the heater L must be large compared with the capillary length scale Lb = σ /g ( ρl − ρv ) . If the surface does not satisfy these conditions, the resulting boiling curve can be very different from the classical curves described in Chapter 7. If the liquid does not wet the heated surface, vapor and/or air will be trapped in virtually every cavity on the surface when the body is immersed. Because of the abundance of vapor-filled cavities, vaporization is initiated immediately when the surface temperature begins to exceed the saturation temperature. If the liquid does not wet the surface, vapor produced at one location on the surface will displace liquid adjacent to the surface and spread laterally to form a vapor blanket over the surface. Thus once boiling is initiated, the boiling process immediately enters the film boiling regime. The nucleate boiling regime, critical heat flux condition, and transition boiling regimes are not observed for a non-wetting liquid. The pool boiling curve for a non-wetting liquid is presented schematically in Fig. 8.3. The surface heat flux increases monotonically with superheat, eventually merging with the “classical” film boiling curve for the particular fluid and surface geometry at high superheat levels. This behavior is consistent with the reduction in qmin ′′ with increasing contact angle described in Section 7.7, and the reduction of qmax ′′ with increasing contact angle described in Section 7.4. In the limit of θ = 180°, reduction of qmin ′′ to zero is equivalent to the entire boiling curve being in the film boiling regime. Circumstances in which the liquid does not wet the heated surface are not usually encountered in common applications. This type of condition can arise, however, when boiling of water occurs on a heated surface that has adsorbed or has been coated with a material that is hydrophobic. It can also occur if a high surface tension fluid such as mercury is vaporized on a low-energy surface such as Teflon. As discussed in Chapter 3, the very high interfacial tension will result in a contact angle that is so high that it is practically non-wetting on Teflon. For the more common circumstances of water or organics boiling on typical metal surfaces in heat transfer equipment, the liquid does wet the surface, usually with a contact angle well below 90°, and the immediate jump to film boiling indicated in Fig. 8.3 does not occur. The “classic” boiling curves discussed in Section 7.1 also may be significantly altered if the characteristic length scale of the heated surface is small compared to the capillary length scale Lb = σ /g ( ρl − ρv ) . The ebullition cycle associated with nucleate boiling was considered in Chapter 6 under the implicit assumption that the extent of the heated surface was large compared to the departure diameter of the bubble dd. The departure diameter correlations discussed in Chapter 6 indicate that dd /Lb is typically of the order of one in most systems. A heated surface that is small compared to Lb is small compared to the expected departure diameter of bubbles that may grow from active cavities on the surface. For such a surface a growing

338

Liquid-Vapor Phase-Change Phenomena

FIGURE 8.3  Boiling curve in the limit of a completely non-wetting liquid. (Note that this is a linear plot of q″ versus superheat.)

bubble may completely cover the surface of the heater, evaporating all liquid in contact with the surface and thereby inducing a transition to film boiling. Thus, for very small heaters, the possibility exists that the onset of boiling may initiate a film-type boiling process. The “classical” nucleate boiling, transition boiling, and the maximum heat flux behavior may be absent for small heaters, or if present, they may be much different in character. The departure from “classical” boiling behavior discussed above has been observed in experiments conducted by Bakhru and Lienhard [8.21]. These investigators determined the pool boiling curves for very small diameter horizontal wires in a variety of fluids. An example of their results is shown in Fig. 8.4. The data in this figure were obtained for boiling of benzene at atmospheric pressure on a wire with a diameter r w of 12.7 × 10−6 m. For these conditions the ratio r w /Lb is 0.0076. The boiling curve indicated by the data in Fig. 8.4 is obviously markedly different from the “classical” pool boiling curves described in Chapter 7. The regimes of nucleate boiling and transition boiling are absent and q″ increases monotonically without passing through a maximum and minimum. At the onset of boiling, the first bubble reportedly grew and spread horizontally until the wire was partially blanketed with a vapor patch. Bakhru and Lienhard [8.21] referred to this as “patchy” boiling. As the heat flux was increased, the fraction of the wire covered by vapor increased until the wire was totally blanketed and film boiling was established. In organic liquids, the vapor patches would rapidly grow, release a bubble, and then collapse. For water, however, the patches were more stable, spreading steadily as the heat flux was increased. Based on their experimental results, Bakhru and Lienhard [8.21] concluded that the local minimum and maximum in the boiling curve vanish for r w /Lb ≤ 0.01. The range 0.01 < r w /Lb < 0.15 corresponds to a transition regime in which the hydrodynamic mechanisms responsible for the qmax ′′ and qmin ′′ conditions establish themselves. For r w /Lb ≥ 0.15 the “classical” pool boiling behavior is observed. The departures from “classical” pool boiling behavior observed in this study are unlikely to be observed in large-scale heat transfer equipment used on earth. It may be observed, however, in microgravity environments in space, where Lb is very large, or in immersion cooling of microelectronic devices that are very small.

Other Aspects of Boiling and Evaporation

339

FIGURE 8.4  Pool boiling data obtained by Bakhru and Lienhard [8.21] for a horizontal wire with a diameter much smaller than Lb. (Adapted with permission, copyright © 1972, Pergamon Press.)

Surface Roughness A number of investigators (e.g., [8.22, 8.23]) have examined the effects of surface roughness on pool boiling data. From the discussion of nucleation phenomena in the previous chapter, rougher surfaces are generally expected to provide a higher heat flux for a given wall superheat because of the generally higher density of potential nucleation sites. It can also be seen in Table 7.1 that for Rohsenow’s correlation, a rougher surface finish generally corresponds to a lower value of Csf, which has the effect of shifting the nucleate pool boiling curve to the left on a q″ versus Tw – Tsat plot. This again implies a higher heat flux at a given superheat level for a rough surface relative to a smoother one. In general, surface finish may also affect transition boiling because it affects the wettability of the surface by the liquid. Measured data suggest that the critical heat flux and minimum heat flux are somewhat dependent on roughness. Roughness can affect surface wetting (see Chapter 3), and, as discussed in Chapter 7, surface wetting characteristics have been demonstrated to affect qmin ′′ and qmax ′′ . Hence, roughness effects on these parameters may be a consequence of roughness variations resulting in altered surface wetting characteristics. The effect of surface roughness on film boiling is usually small. It can, however, have an effect on the radiation transport, since it may affect the emissivity of the surface.

Gravitational Acceleration The vast majority of boiling research experiments have been done at earth normal gravity. Interest in boiling processes that may arise in spacecraft power and thermal management systems has stimulated research on the effects of reduced gravity on boiling. The earliest studies of this type were conducted in the late 1950s and 1960s (see references [8.24, 8.25] for a summary of the early literature). Early experiments were conducted in drop towers under NASA sponsorship. More recently, NASA sponsored experiments have been conducted in aircraft executing parabolic flights and aboard the

340

Liquid-Vapor Phase-Change Phenomena

space shuttle. Recent experiments using drop towers, and aircraft and rockets in parabolic flight have also been conducted by Japanese and European investigators. Before assessing the results of reduced gravity experiments, it is useful to examine the prediction of theories and correlations developed for earth-normal gravity conditions. Examination of the Rohsenow pool boiling correlation discussed in Chapter 7 1/ 2



 q ′′  σ   µ l hlv  g ( ρl − ρv ) 

3

 c pl Tw − Tsat ( Pl )   1  = Prl−3s    (7.30)  hlv  Csf    3

indicates that the heat flux for a fixed superheat will vary proportional to g1/ 2 . This implies that the heat flux vanishes as gravitational acceleration approaches zero. On the other hand, the correlations (7.35)–(7.38) of Stephan and Abdelsalam [8.26] that use graphically obtained constants do not contain gravitational acceleration as a parameter, so it predicts no change in the heat flux as g is reduced. The dimensionless correlations (7.39)–(7.42) proposed by Stephan and Abdelsalam [8.26] predict a varying dependence of heat flux on gravitational acceleration due to the dependence of bubble departure diameter on g. For the pool boiling critical heat flux (CHF), the Zuber correlation,  σ ( ρl − ρv ) g  = 0.131ρv hlv   (7.54) ρ2v   1/ 4



qmax, ′′ Z

and other model relations developed for earth-normal gravity generally predict that the maximum heat flux for pool boiling is proportional to g1/4. This implies that the critical heat flux will diminish proportional to g1/4 as the gravitational acceleration becomes smaller. Many of the critical heat flux predictions are based on models in which the Taylor most dangerous wavelength 1/ 2



  3σ λ D = 2π   (7.51)  ( ρl − ρv ) g 

is the characteristic length scale of wave disturbances that play a central role in the onset of the critical heat flux condition. As discussed in Chapter 7, Lb, the bubble length scale, is proportional to λ D ( Lb = λ D /[2π 3]). Hence, as gravity becomes small, this wavelength becomes larger, proportional to g−1/2. As discussed in Chapter 7, small heater effects become important as the length scale of the heater L becomes comparable to Lb. This implies that heaters that would be categorized as large at earth normal gravity will be subject to small heater effects under microgravity conditions. The above observations indicate that the extrapolation of nucleate boiling heat transfer relations to microgravity conditions leads to conflicting predictions. DiMarco [8.27] noted that Staub [8.28] identified three regimes of boiling in his microgravity experiments. For boiling liquid at or near saturation, bubbles formed at the surface merge into large bubbles that reside at short distance from the surface. The large bubbles apparently sustain their size by collecting small vapor bubbles on the side near the wall and condensing vapor on the side away from the wall. The second regime corresponds to subcooled boiling, with bubbles that form, but remain attached to the surface, functioning as heat pipes with vaporization at the interface near the wall and condensation at the interface away from the wall. In this regime, variation of the liquid temperature with distance from the wall may give rise to thermocapillary flow along the bubble interface. The third regime identified by Staub [8.28] corresponds to highly subcooled boiling. For such conditions very small bubbles grow and collapse near the wall, acting as pumps. Thermocapillary flows apparently do not play an important role in this third regime. Overall, the heat transfer coefficient q″/(Tw – Tsat)

Other Aspects of Boiling and Evaporation

341

for saturated or subcooled nucleate boiling at low-to-moderate heat fluxes generally did not decrease appreciably with decreasing gravity. Experiments generally indicate that the critical heat flux for nucleate boiling is substantially reduced at low gravity, compared to earth-normal gravity at the same conditions. Physically, the reduced CHF is due to weaker buoyancy forces at reduced gravity, which diminishes the system’s ability to deliver liquid to the heated surface to replace liquid converted to vapor there. Vapor blanketing of the surface due to inadequate liquid flow to the surface may then occur at a lower heat flux level. While critical heat flux values for boiling on a flat surface were found to be substantially lower than those at earth-normal gravity, they were higher than values predicted by the Zuber correlation (7.54). As discussed in Chapter 7, the Zuber correlation applies to a heated surface that is large compared to Lb. Because Lb increases as gravitational acceleration decreases, the increase of the critical heat flux above the Zuber value may be a consequence of the same mechanism that causes the CHF to increase at earth-normal gravity when the length scale of the heater L is smaller than Lb (see Section 7.4). The above discussion considered only investigations of the basic mechanisms of microgravity boiling in pure fluids. Other aspects of microgravity have also been explored, including transition boiling, boiling of mixtures, and electric field effects. Of particular note are the investigations of microgravity nucleate boiling in a water-alcohol mixture by Abe et al. [8.29] and Ahmed and Carey [8.30]. In nucleate boiling in binary mixtures, concentration gradients can arise along the interface at the heated surface as a result of preferential vaporization of the more volatile component. When the more volatile component has a lower surface tension, this can result in a Marangoni driven flow that draws liquid toward the solid surface. This type of Marangoni flow can suppress the onset of the critical heart flux condition. In experimental studies of pool boiling in water and alcohol solutions at earth-normal gravity, investigators such as McGillis and Carey [8.31] found that some dilute alcohol solutions exhibited substantially higher critical heat flux values that pure water or the pure alcohol alone at the same pressure. McGillis and Carey [8.31] argued that the higher critical heat flux observed in their dilute water and 2-propanol solutions is a consequence of strong Marangoni driven flow near the surface. Ahmed and Carey [8.30] experimentally studied the effects of gravity on boiling of water and 2-propanol mixtures on parabolic aircraft flights. All mixtures showed higher CHF values than for pure water, and at low-alcohol concentrations the CHF under microgravity was reduced only slightly below that for earth-normal gravity. The CHF was found to correlate directly with the surface tension gradient at the bubble interface near the wall, suggesting that the high CHF for the mixtures was a consequence of Marangoni effects that helped sustain liquid flow to the surface. Abe et al. [8.29] obtained similar results for water and ethanol mixtures. Further information on the results of reduced gravity boiling research in Japan, Europe, and the US can be obtained in the reviews by Ohta [8.32], DiMarco [8.27], and Kim [8.33].

Other Factors The technical literature contains a number of publications that report the results of experimental studies of additional factors that affect pool boiling heat transfer. These include the effects of system pressure [8.34, 8.35], surface aging [8.36], surface coatings or deposits [8.22, 8.37, 8.38], the presence of particulates [8.39, 8.40], dissolved gases [8.41], vapor entrapment [8.42], agitation [8.43], and dissolved lubricating oil [8.44–8.47].

8.2  THE LEIDENFROST PHENOMENON Simply defined, the Leidenfrost phenomenon refers to the film boiling of small liquid masses on a hot surface. When a small droplet of liquid is brought into the proximity of a highly superheated surface, the liquid may vaporize so rapidly that the production of vapor on the side of the droplet

342

Liquid-Vapor Phase-Change Phenomena

  FIGURE 8.5  The Leidenfrost phenomenon – film boiling of a droplet near a heated surface: (a) schematic depiction, (b) a water spheroid at 1 atm on an aluminum surface at 247°C. (Archive photo. Multiphase Transport Laboratory, UC Berkeley.)

facing the surface establishes a pressure field that acts to repel the droplet away from the surface. If the droplet is above a solid surface, the repelling force may just balance gravity, allowing the droplet to hover over the surface on a film of vapor, as shown in Fig. 8.5. Heat is transferred from the hot surface to the droplet by conduction through the vapor film and by radiation. For small liquid masses, interfacial tension acts to pull the mass into a spheroidal shape. Consequently, the term spheroidal state is sometimes used to describe liquid in these circumstances. This phenomenon is named after J. G. Leidenfrost, a German medical doctor who observed the film boiling of water droplets on a red-hot spoon. His report of his observations [8.48] document what is generally acknowledged to be the first experimental investigation of boiling phenomena. Interest in the Leidenfrost phenomenon stems from the fact that impingement of liquid droplets on hot surfaces can arise in a number of technological applications. These include spray cooling of hot metal during metallurgical processing, the design of quick-response steam generators which spray liquid on a hot surface, film cooling of a rocket nozzle, post-dryout mist flow heat transfer in evaporators, vaporization of fuel droplets in fuel-injected engines, and reflooding of a nuclear reactor core after a loss-of-coolant accident. Of central interest in these applications is whether an impinging droplet will contact the surface, wetting it and vaporizing via nucleate or transition boiling, or whether the droplet will undergo film boiling, hovering over the surface or being repelled by it. If the droplet does hover over the surface, the heat transfer rate from the surface and the time required to vaporize the droplet are of major interest. There are two central research issues associated with the Leidenfrost phenomenon. The first is determination of the threshold minimum temperature or minimum heat flux at which the droplet is levitated and stable film boiling is established under the droplet. This is often termed the Leidenfrost point. The minimum temperature to achieve this condition is the Leidenfrost temperature. The second issue of interest is heat transfer rate that results when stable film boiling is achieved.

Leidenfrost Film Boiling Heat Transfer The mechanisms of Leidenfrost droplet levitation and film boiling heat transfer can be more clearly understood by considering the idealized model of the process indicated schematically in Fig. 8.6. For the purposes of determining the heat transfer, the droplet is modeled as a cylinder of radius R

343

Other Aspects of Boiling and Evaporation

FIGURE 8.6  Idealized model of Leidenfrost vaporization of a liquid droplet.

and height H. The laminar vapor flow in the thin film between the droplet and the surface is governed by the following continuity and u-momentum equations ∂( ur ) ∂( vr ) + = 0 (8.13) ∂r ∂z

ρv u



 ∂2 u  ∂u dP ∂u + ρv v =− + µ v  2  (8.14) ∂r ∂z dr  ∂z 

Consistent with typical thin-film approximations, the pressure P is taken to be independent of z, and the r derivatives of u are neglected compared to z derivatives. Boundary conditions for these equations are taken to be u (r , δ ) = 0

u ( r ,0 ) = 0



T ( r ,0 ) = Tw



v ( r ,0 ) = 0 (8.15)

T ( r , δ ) = Tb (8.16)

 ∂T  ρv hlv vz =δ = − k v  (8.17)  ∂ z  z =δ



 ∂P   =0  ∂r  r = 0



P ( R, z ) = Ps (8.18)

Integrating Eqs. (8.13) and (8.14) from z = 0 to z = δ, and using boundary conditions (8.15) yield the following forms of the continuity and u-momentum equations: ∂ ∂r

δ



1 ∂



δ

∫ (ur ) dz + rv

z =δ

= 0 (8.19)

0

 ∂u 

∫  r ∂r (ρ ru ) dz = −δ  dr  + µ  ∂z  0

v

2

 dP 

v

z =δ

 ∂u  − µ v   (8.20)  ∂z  z = 0

Assuming that transport of thermal energy across the vapor film is by conduction alone, the temperature profile is linear for the specified boundary conditions and given by

T − Tb z = 1 − (8.21) Tw − Tb δ

344

Liquid-Vapor Phase-Change Phenomena

We will also assume the existence of a parabolic velocity profile δ



 z z2  u 1 udz (8.22) = 6  − 2  , um = δ um δ δ 

∫ 0

Using Eq. (8.21), boundary condition (8.17) can be written as

k ( T − Tb )  ∂T  hlv ρv vz =δ = − k v  = v w (8.23)  ∂ z  z =δ δ

Combining the continuity relation (8.19) with Eqs. (8.22) and (8.23), and using the fact that um = 0 at r = 0 yields the following relation for um um =



α v Ja r (8.24) 2δ 2

where Ja =



c pv ( Tw − Tb ) (8.25) hlv

Substituting the parabolic profile given by Eq. (8.22) into the integral momentum equation (8.20) yields a differential equation for the radial pressure variation under the droplet that can be solved, using the boundary conditions (8.18), to obtain the following relation for the pressure variation:

P − Ps =

3ρv vv2  Ja   3  Ja   2 2 1 +  R − r (8.26) δ 4  Prv   4  Prv  

(

)

For steady film boiling evaporation of the droplet, the weight of the droplet must be supported by the pressure field, which requires that

(ρl − ρv ) gHπR2 = 2π

R

∫ ( P − P )rdr (8.27) s

0

Substituting Eq. (8.26) into Eq. (8.27) and evaluating the integral yields an equation for the vapor film thickness δ: 1/ 4



 3ρv vv2 R 2  Ja   3  Ja    δ=   1 +     (8.28)  2 ( ρl − ρv ) gH  Prv   4  Prv   

For conduction alone, the heat transfer coefficient is given by h = k v/δ. Using this fact together with Eq. (8.28) leads to a relation for h that can be written in dimensionless form as

 hR  2 Nu R =   =     kv  3

1/ 4

 H  R

1/ 4

 Ra R    Ja 

1/ 4

 3  Ja   1 +    4 Prv  

−1/ 4

(8.29)

where

Ra R =

(ρl − ρv ) gR3 Prv (8.30) ρv vv2

345

Other Aspects of Boiling and Evaporation

In a number of experimental studies of the Leidenfrost phenomenon, the time for complete vaporization of the liquid droplet as it levitates above a heated surface has been measured. A prediction of the time for complete vaporization of such a droplet can easily be developed from the analysis described above. Assuming that all heat transferred to the droplet goes into the latent heat of vaporization, an energy balance requires that hlv ρv



dV = hAb ( Tw − Tb ) (8.31) dt

where V and Ab are the volume and base area of the droplet, respectively. Using the fact that for a cylindrical droplet V = πR 2 H



Ab = πR 2 (8.32)

and

and using the fact that h = k / δ , Eq. (8.31) can be written as dV = −C0V 7/12 (8.33) dt

where



k π 2/3 C0 = − v hlv ρl

1/ 4

 2 ( ρl − ρv ) g Pr  3  Ja   −1   v  1+      2 v πρ 3 Ja   4  Prv    v v  

1/3  R  T − T (8.34) ( w b)  H

Integrating Eq. (8.33) from an initial droplet volume V0 at t = 0 to V = 0 at t = t v yields tv =



12V05/12 (8.35) 5C0

For small droplets, the droplet may be closer to spherical in shape than cylindrical, making the meaning of R/H for such conditions unclear. However, for the cylinder shown in Fig. 8.6, H represents the ratio of the cylinder volume to its base area. If we adopt this same interpretation of H for the spherical droplet case, it follows simply from the spherical geometry that R 3 = (8.36) H 4



Taking R/H = 3/4 for small droplets and substituting Eq. (8.34), Eq. (8.35) can be written as

( )

t * = 1.81 V0*



5/12

(8.37)

where

 ρ1/ 2 µ h 3 σ 5/ 2  t = t v  3 7/l 2 v lv 3   k v g ρv ( Tw − Tb ) 



 σ  V0* = V   ρl g 

*

−1/ 4

(8.38)

−3/ 2

(8.39)

346

Liquid-Vapor Phase-Change Phenomena

This result differs only slightly from that obtained from a more detailed model analysis developed by Baumeister et al. [8.49]. Their relation is

( )

t *B = 1.21 V0*



5/12

(8.40)

where V0* is given by Eq. (8.39), and t *B is given by  t  ρ1/ 2 µ h 4 σ 5/ 2 t = v  3 7/ 2l v* lv 3  f  k v g ρv hlv ( Tw − Tb ) 

−1/ 4

* B



hlv* = hlv +



(8.41)

7 c pv ( Tw − Tb ) (8.42) 20

and f is a correction factor for radiation effects given by −3



  hr / 4 f = 1 +  (8.43)  hc 1 + ( 7 / 20 ) c pv ( Tw − Tb ) / hlv  



2 hr = εl σ SB Tw2 + Tsat ( Tw + Tsat ) (8.44)

(

)

In the above expressions, εl is the emissivity of the liquid at the droplet surface, and hc is the heat transfer coefficient for convective effects alone given by 1/ 4



  k v3 hlv* gρv ρl hc = 1.1   (8.45) 1/3  µ v (V0 / 2 ) ( Tw − Tb ) 

Baumeister et al. [8.49] found fairly good agreement between measured droplet evaporation times and times predicted using Eqs. (8.49)–(8.54) for the small droplet regime. These investigators also found that data for larger liquid masses for a wide variety of fluids also correlated well in terms of t *B and V0* , although different correlations in terms of these parameters were proposed as best fits to data in the large drop and extended drop regimes. Note that in applying this framework to evaporating droplets, the base temperature of the droplet Tb is generally taken to be the saturation temperature Tsat at the ambient pressure.

The Leidenfrost Transition In the limit of the droplet radius approaching infinity, the Leidenfrost phenomenon becomes equivalent to stable film boiling on a horizontal upward-facing surface. Because of this link, for either large or small liquid masses, the minimum temperature that supports stable film boiling is often referred to as the Leidenfrost temperature. It should be noted that in real systems there is often a gradual transition from stable film boiling to nucleate boiling over which the liquid increasingly wets the surface. This is reflected in the droplet evaporation time measurements for droplets of liquid deposited on a hot surface at different superheat levels. Figure 8.7 shows data obtained by Huang et al. [8.50] for 10 μl droplets of pure water deposited on an aluminum surface. At very low superheats (Tw – Tsat < 50°C), the evaporation time is very low because the liquid fully wets the surface and nucleate boiling results. At very high wall superheat levels (Tw – Tsat > 200°C), the evaporation time is substantially larger. As superheat decreases from about 200°C to about 50°C, a gradual decrease in evaporation time is observed, due to the progressively increased wetting of the

Other Aspects of Boiling and Evaporation

347

FIGURE 8.7  Evaporation time data obtained by Huang et al. [8.50] for 10 μl water droplets deposited on an aluminum surface at different superheat levels.

surface. For such circumstances, the Leidenfrost transition temperature is identified as the temperature corresponding to the first departure from stable film boiling. As the wall superheat is raised through the transition, first establishment of stable film boiling produces the slowest evaporation rate because the low heat transfer coefficient for film boiling is achieved and the wall superheat is the lowest possible for film boiling. Further increasing of the superheat results in film boiling at an increased heat flux. Consequently, for the data in Fig. 8.7, incipient stable film boiling corresponds to the peak in the evaporation time curve. It can be seen in the figure that for a freshly machined aluminum surface this peak occurs at a lower superheat than for an oxidized surface. Generally, oxidized metal surfaces have been found to be better wetted (lower contact angle) than newly machined metal surfaces. This suggests that the minimum stable film boiling temperature (corresponding to the peak evaporation time) is higher for the oxidized surface because its affinity for the liquid phase requires that it be driven to a higher temperature to de-wet the surface and establish stable film boiling. The above discussion implies that there are two categories of mechanisms that may influence the Leidenfrost temperature. The first is: 1. The hydrodynamic instability of the liquid-vapor interface. Models of this mechanism generally are based on the assumption that Taylor wave motion facilitates vapor bubble release from the film and the Leidenfrost point corresponds to the lowest heat flux that can sustain the vapor release mechanism without waves growing to the point that liquid contacts the surface. This instability mechanism may result in liquid-solid contact, but its effect is dictated by the physics of the system before contact occurs. The second category includes mechanisms that are associated with the consequences of solid-liquid contact. These include: 2. Violation of intrinsic stability. If the temperature of the surface exceeds the spinodal temperature, stable liquid cannot exist in contact with it, making it impossible for the liquid to wet the surface, thus assuring stable film boiling. 3. The surface temperature reaches the homogeneous nucleation (kinetic) limit. This dictates that any liquid that comes in contact with the surface immediately flashes to vapor, making it impossible for the liquid to wet the surface and thus assuring stable film boiling. 4. Temperature effects on contact angle. The argument in support of this mechanism is based on the hypothesis that wettability varies with temperature and that at a critical threshold temperature, the wettability is altered in a way that initiates the Leidenfrost transition.

348

Liquid-Vapor Phase-Change Phenomena

The mechanisms of the incipient onset/departure from stable film boiling have been the subject of much debate. Models based on these mechanisms have been developed and explored as a means of predicting the Leidenfrost temperature. One way of modeling the effects of interface instability (mechanism 1) is to assume that the minimum heat flux at the Leidenfrost transition corresponds to the minimum heat flux qmin ′′ that will sustain the Taylor wave action that facilitates the escape of vapor. If a correlation for the film boiling heat transfer coefficient hfb in terms of the wall superheat is available, the Leidenfrost temperature can be estimated by solving the following relation for the wall temperature: qmin ′′ = h fb ( Tw , L − Tsat ) (8.46)



Using such arguments, Berenson [8.51] obtained the following relation for the minimum film boiling temperature Tw,m:

Tw ,m − Tsat

ρ h  g ( ρl − ρv )  = 0.127 v lv   k v  ρl + ρv 

2/3

1/ 2

  σ    g ( ρl − ρv ) 

1/3

  µv   (8.47)  g ( ρl − ρv ) 

While this type of model is plausible for pool boiling on a horizontal upward-facing surface, for droplet evaporation, the size of the droplet is often of the same order as the Taylor most dangerous wavelength, implying that wave motion of this type is not possible on such a small droplet. Furthermore, the transition and film boiling data discussed in Section 7.7 indicate that surface wetting and roughness characteristics affect the Leidenfrost transition temperature in pool boiling from an upward-facing flat surface. The interface stability model described above cannot account for these effects, and this implies that, at least in some cases, a predictive model should be based on one or more of the other mechanisms listed above. Yao and Henry [8.52] reasoned that the minimum wall superheat for film boiling is actually not necessarily dictated by the need to sustain the Taylor wave action. They argued that in some cases, the limiting condition may be dictated by the homogeneous nucleation of liquid brought into proximity of the surface by the collapse of the film under a detached bubble. Of these two mechanisms, the one that requires the lowest wall temperature to establish stable film boiling will be the one that is expected to dictate the Leidenfrost temperature. Cryogenic liquids generally require a relatively small level of superheat above the normal boiling point to initiate homogeneous nucleation. The arguments of Yao and Henry [8.52] suggest that the Leidenfrost point for cryogenic fluids is generally dictated by the homogeneous nucleation condition, whereas for most other liquids, such as water, hydrocarbons and liquid metals, the stability of the Taylor wave action appears to be the more likely limiting mechanism. As discussed above and in Section 7.7, as the heat flux is decreased in the film boiling regime, some partial liquid-surface contact may initially result in only a small departure from the film boiling conditions. This implies that the initial collapse of the Taylor wave action may not correspond exactly to the qmin ′′ condition, and the qmin ′′ condition may depend somewhat on the heated surface characteristics. To address this issue, Henry [8.53] developed a model of the Leidenfrost phenomenon that included the effects of transient wetting and subsequent liquid microlayer evaporation on the Leidenfrost temperature. Based on arguments regarding the role of this mechanism, he proposed the following correlation for the Leidenfrost temperature Tw,m: 0.6



 kl ρl c pl   Tw ,m − TwmB hlv = 0.42     (8.48) Twmb − Tl  k w ρw c pw  c pw ( TwmB − Tsat )  

where TwmB is the value of Tw,m given by Berenson’s [8.51] correlation, Eq. (8.47). This correlation was found to provide good agreement with available experimental data over a wide range of

349

Other Aspects of Boiling and Evaporation

conditions. It was not recommended, however, for small liquid drops because such droplets tend to move about the surface, resulting in a local surface cooling effect that is much different. Other studies have more directly linked the onset of homogeneous nucleation or the kinetic limit of superheat to the Leidenfrost or minimum heat flux condition. Nishio [8.54] found that for several pure fluids, he could correlate the superheat at the minimum heat flux condition as a function of reduced pressure alone. Nishio [8.54] developed the following relation for the minimum heat flux temperature that closely matched measured data for water, ethanol, nitrogen, and several hydrocarbons at reduced temperatures between 0.65 and 0.85:



 T − Tsat   hlv*  * 0.6302 * 1.008 Tw ,m = Tsat + 3.449 × 10 −4  c (ρr ) (Prl ) *  *   Tc − Tsat   c pl   4.94T   ×  (lr* )0.2056 exp   Tc  

(8.49)

where

ρr = ρv /ρl (8.50a)



lr = σ 3 /g ( ρl − ρv ) vl4 (8.50b) 3

and the superscript “*” denotes parameters evaluated with properties at T/Tc = 0.7. As discussed in Section 7.7, Ramilison and Lienhard [8.55] also developed a correlation relation for the minimum heat flux temperature that directly reflects its dependence on the homogeneous nucleation temperature and the advancing contact angle. Transition models based on the temperature effects on contact angle (mechanism 4) have been proposed by Segev and Bankoff [8.56] and Olek et al. [8.57]. Segev and Bankoff [8.56] proposed that beyond a threshold temperature, the precursor adsorbed film necessary to facilitate wetting of the surface drops sharply in thickness. This makes it impossible to wet the surface, leading to the onset of stable film boiling. Bernardin and Mudawar [8.58] conducted an extensive examination of the different proposed mechanisms listed above and the Leidenfrost temperature predictions of the models developed from them. In this investigation, Bernardin and Mudawar [8.58] explored the potential role of each of the four mechanisms described above in determining the Leidenfrost transition temperature for evaporating droplets. Because the size of the droplet is usually of the same order as λD, the most dangerous wavelength for Taylor waves, they concluded that wave instability was not likely to be a primary mechanism for such circumstances. In this investigation, Bernardin and Mudawar [8.58] compared Leidenfrost transition data for droplets of various liquids with the predictions of intrinsic stability and homogeneous nucleation (kinetic) limit models of the Liedenfrost temperature. Their overall conclusion was that predictions of these types of models agree well in some cases, but poorly in others. Bernardin and Mudawar [8.58] also concluded that although wetting is expected to play some role in boiling heat transfer, available data suggest that it is not a controlling Leidenfrost point mechanism. For polished surfaces, Bernardin and Mudawar [8.59] subsequently proposed a Leidenfrost transition temperature model that is based on the premise that the transition mechanism is rapid growth and merging of bubbles that nucleate as liquid initially contacts the solid surface. The postulated sequence of events in this model is indicated in Fig. 8.8. In the Bernardin and Mudawar [8.59] model, liquid contacts the solid surface when the droplet is initially deposited on the surface. When the surface temperature is at or above the Leidenfrost point temperature, bubbles nucleate at crevices on the surface when this initial contact occurs and the rapid growth of bubbles quickly blankets

350

Liquid-Vapor Phase-Change Phenomena

FIGURE 8.8  Vapor layer formation process.

the surface with vapor to establish a vapor film. Bernardin and Mudawar [8.59] developed a model of the growth rate of bubbles as the thermal boundary layer in the liquid grows following initial contact of the liquid with the surface. From this model, they developed a relation for the time variation of the percent area coverage of the wall by vapor as a result of the nucleation and growth of bubbles. This model predicts that coverage by vapor becomes more rapid as the wall temperature approaches the Leidenfrost temperature. Bernardin and Mudawar [8.59] found that for water droplets on an aluminum surface, the Leidefrost point corresponded to a rate of vapor coverage of 0.5% per microsecond. This rate of coverage was taken to be the threshold at which the Leidenfrost point was achieved. Bernardin and Mudawar [8.59] used their model of the rate of surface coverage by vapor to determine the surface temperature at which this threshold rate was achieved for several liquids on polished metallic surfaces. The predictions of this model were found to agree well with experimentally determined Liedenfrost transition temperatures for droplets of a variety of liquids on polished surfaces. In the limit of a perfectly smooth surface, this model predicts that the Leidenfrost transition temperature is infinite. This is physically unrealistic since liquid heated to the kinetic limit of superheat at the solid surface would be expected to flash to vapor. The model therefore is not expected to be applicable to perfectly smooth surfaces. Roughness greater than the small nucleation sites on polished surfaces is expected to increase the Liedenfrost transition temperature because a thicker vapor film must be established to prevent contact of the liquid with the more irregular surface profile. These observations imply that the model of Bernardin and Mudawar [8.59] is a good predictor of the Leidenfrost transition temperature for polished surfaces, but is not expected to be accurate for rougher surfaces and for nearly perfectly smooth surfaces.

351

Other Aspects of Boiling and Evaporation

The investigations described in this section have provided useful insight into the mechanisms of the Leidenfrost transition and their importance in different types of systems. These studies suggest, however, that the theoretical understanding of these mechanisms and how they interact, is not yet complete. Further discussion of the features of the Leidenfrost phenomenon can be found in references [8.60–8.66]. Example 8.4 Bernardin and Mudawar [8.59] report a measured Leidenfrost transition temperature of 162°C for water droplets deposited on a polished aluminum surface at atmospheric pressure. Their model also predicts a Leidenfrost transition temperature of 162°C for these conditions. Compare this value to the Leidenfrost temperature predicted by the relations proposed by Berenson [8.51] and Ramilison and Lienhard [8.55] for these circumstances. Assume that the advancing contact angle for water on polished aluminum is 60°. For saturated water at atmospheric pressure, Tsat = 100°C, ρl = 958 kg/m3, ρv = 0.597 kg/m3, cpl = 4.22 kJ/kgK, hlv = 2257 kJ/kg, kv = 0.025 W/mK, μv = 1.255 × 10−5 Ns/m, and σ = 0.0589 N/m. Substituting in the Eq. (8.47) proposed by Berenson [8.51], we obtain

Tw ,m

ρ h  g(ρl − ρv )  = Tsat + 0.127 v lv  kv  ρl + ρv  = 100 + 0.127

2/ 3

1/ 2

  σ   g ( ρ − ρ ) l v  

(0.597)(2,257,000)  9.8(957)   959  0.025

2/ 3

1/ 3

  µv   g ( ρ − ρ ) l v  

1/ 2

 0.0589   9.8(957)   

1/ 3

 1.255 × 10 −5   9.8(957)   

= 186°C To use the correlation of Ramilison and Lienhard [8.55], we first determine the homogeneous nucleation temperature using Eq. (7.177):



9 9   T    373   Thn = 0.932 + 0.077  sat   Tc = 0.932 + 0.077   647   647    Tc       = 603 K = 330°C

The Leidenfrost point (departure from film boiling) temperature is then computed as



(

)

Tdfb = Tsat + 0.97exp −0.00060θ1.8 (Thn − Tsat ) a

(

= 100 + 0.97exp −0.00060 (60 )

1.8

) (330 − 100) = 186°C

Thus, the correlations of Berenson [8.51] and Ramilison and Lienhard [8.55] both predict Leidenfrost point temperature values that are more than 20°C higher than the experimental and model values reported by Bernardin and Mudawar [8.59] for these circumstances.

8.3  FLUID-WALL INTERACTIONS AND DISJOINING PRESSURE EFFECTS As described in Chapter 3, for ultrathin films, attractive forces between the liquid molecules and molecules of the solid surface effectively produces a pressure difference across the liquid-vapor interface referred to as the disjoining pressure difference. The disjoining pressure difference can have important effects on the heat transfer performance during vaporization of ultrathin films. Examples of circumstances in which this may be important include vaporization under a bubble during nucleate boiling and evaporation of the working fluid in the evaporators of micro heat pipes.

352

Liquid-Vapor Phase-Change Phenomena

FIGURE 8.9  Cross section of an evaporator with thin liquid films on top surfaces of ribs.

Molecular Force Interactions and Disjoining Pressure Effects To explore the link between disjoining pressure and intermolecular forces, we will consider the evaporator shown schematically in Fig. 8.9. This figure shows a cross section of an evaporator with metal walls that is heated from below. Metal rib structures in the evaporator protrude upward from the bottom. Along the top surface of each rib is a thin film of liquid with film thickness δ. Most of the liquid in the evaporator is in a deeper region between the fins. At locations where the flat interface separates the deeper liquid from the vapor, at equilibrium we expect that

Pve = Pl 0 (8.51)

where Pl0 is the liquid pressure at the interface in the absence of wall attractive forces and Pve is the vapor pressure. In the thin liquid film at location B, the pressure in the liquid film is altered by force interactions between liquid molecules and atoms or molecules in the solid. To analyze the role of intermolecular forces, we will consider the system shown in Fig. 8.10. Long-range attractive forces between solid and fluid molecules can result from several different mechanisms (i.e., van der Waals, hydration, and electrostatic forces). Here we will not develop detailed models of different force mechanisms.

FIGURE 8.10  Schematic of fluid molecules near a solid metal wall.

353

Other Aspects of Boiling and Evaporation

Instead, we will adopt the simpler idealization that we can adequately model the fluid and metallic solid interactions with a Lennard-Jones interaction potential having the form φ fs ( r ) = −



Cφ, fs r6

 Dm6  1 − r 6  (8.52)  

We further assume that a similar functional form, with different constants, models the long-range attraction between a pair of two fluid molecules. φ ff ( r ) = −



Cφ, ff r6

 D 6f  1 − r 6  (8.53)  

In the liquid and solid phases of interest here, the interactions between pairs of molecules are treated as independent and additive. Cφ, fs and Cφ, ff are constants that depend on the species involved, Dm is the closest approach distance of fluid to solid molecules, and Df is the closest approach distance of two fluid molecules. Dm and Df are on the order of mean diameters for the molecular species involved. Note that the Lennard-Jones type potential used here is not expected to precisely model specific attractive or repulsive force interactions, but it is expected to at least qualitatively represent these types of interactions in real solid-fluid systems. To get the total effect of all the solid metallic molecules on a given free fluid molecule, we integrate the product of the density and molecular potential to sum the contributions of all the solid molecules. It follows that the mean-field potential energy felt by the free fluid molecule due to interactions with all the metallic solid molecules is ∞



Φ fmf =

∞

∫ ∫ρ

n ,s

φ fs ( 2πx ) dxdzs (8.54)

zs = z x = 0

Substituting the right side of Eq. (8.52) above for the molecular potential, with r 2 replaced by x 2 + z2 , integrating, and reorganizing the resulting equation in terms of a modified Hamaker constant Als defined as Als = π 2 ρn ,l ρn ,s Cφ, fs (8.55)

Eq. (8.54) is converted to the form

Φ fmf =

Als 6πρn.l Dm3

 2  Dm  9  Dm  3     −   (8.56)  15 z   z  

Here, ρn,l = NA/vlM is the number density of liquid molecules. As noted by Israelachvili [8.67], this definition links the Hamaker constant to the force interaction between the fluid molecules and atoms in the metal surface. This potential Φ fmf is equivalent to a body force that produces a variation of pressure in the fluid similar to the hydrostatic variation produced by gravity. We determine the pressure field by using the wall potential to determine the body force in the Navier-Stokes equations with no velocity terms,

0=−

 1 ∇P + f fs (8.57) (ρn,l M /N A )

354

Liquid-Vapor Phase-Change Phenomena

 In the above equation, f fs is the force per unit mass on the fluid in the system. The force exerted on a single molecule by the entire wall is given by

 Ffs = −∇Φ fmf =

Als 2πρn , f Dm4

 2  Dm  10  Dm  4      −    z (8.58) z    5 z 

 where z is the unit vector in the z-direction, and the corresponding force per unit mass is

  N A Ffs N A Als f fs = = 2πMρn , f Dm4 M

 2  Dm  10  Dm  4      −    z (8.59) z    5 z 

The force per unit mass specified by Eq. (8.59) is substituted into Eq. (8.67), and since the force only acts in the z-direction, the relation simplifies to dP Als = dz 2πDm4



 2  Dm  10  Dm  4     −    (8.60) z    5 z 

Integrating both sides of Eq. (8.60) from a position z to ∞, and taking the pressure at ∞ to be Pl0, the resultant expression for the pressure profile close to the wall is

P ( z ) = Pl 0 −

Als 6πDm3

 2  Dm  9  Dm  3     −   (8.61)  15 z   z  

Because D m is on the order of a molecular diameter, and we are interested in z values larger than that, the z −9 term in Eq. (8.61) may be neglected. The relation for the pressure profile then simplifies to

P ( z ) = Pl 0 +

Als (8.62) 6πz 3

In the thin liquid film at location B in Fig. 8.9, we therefore expect the pressure to vary with distance from the lower wall according to Eq. (8.62). It follows that at the interface (z = δ), the pressure in the liquid Pl,i must be

Pl ,i = P ( δ ) = Pl 0 +

Als (8.63) 6πδ 3

and solving for Pl0 – Pl,i yields

Pl 0 − Pl ,i = −

Als (8.64) 6πδ 3

The left side of Eq. (8.64) equals the disjoining pressure Pd = Pl,0 – Pl,i and we can write this equation in the form

Pd = −

Als (8.65) 6πδ 3

Thus, for the circumstances depicted in Fig. 8.9, the pressure difference Pl0 – Pl,i is equal to the disjoining pressure, which increases rapidly in magnitude as the film grows thinner. The result

355

Other Aspects of Boiling and Evaporation

provided by Eq. (8.65) is also given in many texts (e.g., Israelachvili [8.67]). Note that Eq. (8.65) is consistent with the following general relation between disjoining pressure and film thickness described in Chapter 3

Pd = − Aδ − B (8.66)

if A and B are taken to be Als/6π and 3, respectively. It is clear from the above analysis that altering the exponent in the r6 terms in the molecular force interaction potential function will alter B in the disjoining pressure relation. It also can be seen that altering the prefactor in the potential function will change A in the disjoining pressure relation. Thus, the values of the constants A and B clearly depend on the nature of the force interaction between the solid atoms and the fluid molecules. Experimental data indicate that the values of the constants that best represent fluid behavior do vary with the fluid and solid surface types. Disjoining pressure effects can cause the equilibrium vapor pressure adjacent to an ultrathin liquid film to differ significantly from the normal bulk-fluid equilibrium value in standard saturation tables. We will next examine these effects on the equilibrium conditions for the thin film and adjacent vapor shown in Fig. 8.11. The conventional thermodynamic analysis of disjoining pressure effects on thermodynamic equilibrium begins by integrating the Gibbs-Duhem equation

dµ = − sdT + vdP (8.67)

at constant temperature Te from saturation conditions to an arbitrary state in the liquid and vapor phases. In doing so, we follow the usual conventions of treating the liquid as incompressible and the vapor as an ideal gas. This yields the following relations for the vapor and liquid chemical potentials in the absence of wall attraction effects:

µ v = µ v ,sat + RTe ln( Pve /Psat ) (8.68)



µ l = µ l ,sat + vl ( Pl − Psat ) (8.69)

For a pure fluid, the chemical potential is equivalent to the specific Gibbs function

µ l = gl

In the presence of wall attraction effects in the liquid film, the Gibbs function is augmented by the potential energy associated with interaction between fluid molecules and surface molecules or

FIGURE 8.11  An ultrathin liquid film on a solid surface.

356

Liquid-Vapor Phase-Change Phenomena

atoms. Denoting the potential energy per fluid molecule due to interaction with the wall as a whole as Φ fmf, the expression for the liquid chemical potential becomes

µ l = µ l ,sat + vl ( Pl − Psat ) +

N A Φ fmf (8.70) M

At the flat interface, thermodynamic equilibrium requires that the liquid pressure must equal the equilibrium vapor pressure Pl ,i = Pve (8.71)



and the chemical potentials must be equal there. Equating the right sides of Eqs. (8.68) and (8.69), and setting Pl equal to Pve as dictated by Eq. (8.71) yields

RTe ln ( Pve / Psat ) = vl ( Pve − Psat ) +

N A Φ fmf (8.72) M

which can be rearranged to the form

 ( P − Psat ) vl N A Φ fmf  Pve = exp  ve +  (8.73) Psat RTe MRTe  

For the circumstances of interest here, Pve is close to Psat, with the net result that the first term in the curly brackets above is negligible compared to the second. It is often a good approximation to neglect the first term, but here we will retain it for completeness. Substituting the relation (8.56) derived previously for the wall potential, neglecting the z−9 term, and setting z equal to δ at the interface, and using the fact that ρn,l = NA/vlM, the following relation is obtained

 ( P − Psat ) vl Pve Als vl  = exp  ve −  (8.74) 3 Psat 6 πδ RT RTe  e 

Equation (8.74) predicts how the equilibrium vapor pressure at the liquid-vapor interface of thin liquid films is altered by disjoining pressure effects. By taking A = Als/6π and B = 3, we can write Eq. (8.74) in the more generalized form:

 vl  Pve − Psat ( Te ) − Aδ − B   Pve = Psat ( Te ) exp    (8.75) RTe  

For a specified film thickness, this relation can be used to determine the equilibrium vapor pressure for a given system temperature. Alternatively, for a specified system vapor pressure and film thickness, the equilibrium temperature can be determined. Rearranging this relation, an expression for the equilibrium film thickness for a specified set of vapor pressure and temperature conditions can also be obtained. 1/ B



 A  δ=   Psat ( Te ) 

 Pve  RTe   Pve   −1−    ln    vl Psat ( Te )   Psat ( Te )    Psat ( Te )

−1/ B

(8.76)

At a given vapor pressure, this equation indicates that if the film is thin enough, it can be superheated (Te > Tsat(Pve)) and be in equilibrium with the vapor. This means that heating the solid surface, and hence the liquid film, to a temperature above its normal saturation temperature will not

Other Aspects of Boiling and Evaporation

357

FIGURE 8.12  Predicted variation of the equilibrium vapor pressure with liquid film thickness due to disjoining pressure effects.

necessarily evaporate the liquid film completely. Instead, an equilibrium thin film may remain on the surface whose thickness satisfies Eq. (8.76). Equation (8.76) can also be interpreted as meaning that a thin film can exist on a solid surface even when its vapor pressure in the surrounding gas is below the normal saturation pressure for the system temperature. This is consistent with the adsorption of thin liquid films onto high energy surfaces (such as metals) discussed in Chapter 3. The film thickness must be exceedingly small for the equilibrium saturation conditions to differ significantly from normal flat-interface conditions. Values of A and B in the disjoining pressure relation (8.66) are not available for all surface and fluid combinations. For carbon tetrachloride on glass, however, Potash and Wayner [8.68] recommend the values A = 1.782 Pa mB and B = 0.6. Using these values with thermodynamic data for CCl4, Eq. (8.75) was used to predict the fractional change in the equilibrium saturation pressure with film thickness at a system temperature of 76.7°C. The resulting variation is shown in Fig. 8.12. For a liquid film thickness of 1 mm (1000 μm) the fractional change in the vapor pressure is less than 10−4, or less than 0.01%. A 1% change in the vapor pressure would require a film thickness of about 0.1 μm. For films thicker than 1 μm, the deviation from normal equilibrium conditions is insignificant.

Disjoining Pressure Effects on Evaporation Heat Transfer The disjoining pressure effects discussed above may directly affect vaporization of thin liquid films on solid surfaces. The alteration of the equilibrium vapor pressure and liquid pressure in the film associated with disjoining pressure effects has been shown to alter the thermodynamic equilibrium conditions at the liquid-vapor interface of thin liquid films. The resulting shift in the vapor-pressure versus temperature relation must be taken into account when modeling thin-film evaporation or condensation processes in applications. As indicated in Fig. 8.11, the net effect of the altered thermodynamic equilibrium condition is that the liquid in the film must be slightly superheated to be in equilibrium with vapor at a specified local pressure Pv . To vaporize liquid at the interface, the wall temperature must be raised above this higher equilibrium liquid temperature. Because the film is very thin, usually a laminar flow and heat transfer analysis can be used to model the transport. However, for ultrathin liquid films, specification of the correct temperature boundary conditions on the film interface requires that disjoining pressure effects be taken into account.

358

Liquid-Vapor Phase-Change Phenomena

FIGURE 8.13  The extended meniscus region near a heated wall.

One commonly encountered example of the manner in which disjoining pressure effects may come into play in applications is shown in Fig. 8.13. This figure depicts the location where a liquid-vapor interface meets the wall of a container. If the wall temperature is above the normal saturation temperature, heat may be conducted across the liquid film in this region, resulting in evaporation at the interface. However, because of the curved interface in the intrinsic meniscus region, the wall superheat must exceed that required for equilibrium with the interface curvature present, as indicated by the analysis in Section 5.2. In addition, because of the disjoining pressure effects described above, beyond the intrinsic meniscus a thin film can exist on the wall at equilibrium. This thin film can exist without evaporating, provided that its thickness is equal to that given by Eq. (8.86) above. Example 8.5 Determine the thickness of an equilibrium thin film of carbon tetrachloride on glass at atmospheric pressure for a superheat of 10°C. For these circumstances, Eq. (8.76) can be used to estimate the equilibrium film thickness: 1/ B



 A  δ=   Psat (Te ) 

 RTe   Pve    Pve − 1−    ln   ( ) P T  νl Psat (Te )   Psat (Te )    sat e

−1/ B

For CCl4 under these conditions, Te = 360 K, Psat(Te) = 142.5 kPa, vl = 6.84 × 10−4 m3/kg, and R = 0.0541 kJ/kgK. Taking A = 1.782 Pa mB and B = 0.6 as indicated by Potash and Wayner [8.68], substitution in the above equation yields 1/ 0.6



 1.782  δ=   142,500 

= 8.28 × 10 −10  m

 101  (0.0541)(360)   101   − 1−    −3  ln  (6.84 × 10 )(142.5)   142.5    142.5

−1/ 0.6

Other Aspects of Boiling and Evaporation

359

For the circumstances shown in Fig. 8.13, at equilibrium there is a region between the intrinsic meniscus and the equilibrium film over which the film varies in thickness and curvature to accommodate the transition between these two regions. This so-called interline region is the thinnest portion of the extended meniscus over which vaporization can occur. Because it is thinnest, it is also the location where the evaporation rate is the highest. The high local heat transfer and evaporation rates in the interline region have stimulated interest in the possibility of promoting these circumstances in heat transfer equipment to enhance evaporation heat transfer. Fundamental experimental and theoretical studies of heat transfer in the interline region have been conducted by Wayner and co-workers [8.68–8.71], Mirzamoghadam and Catton [8.72, 8.73], Swanson and Peterson [8.74], and Hallinan et al. [8.75]. Studies by Xu and Carey [8.76], Stephan and Busse [8.77], Swanson and Herdt [8.78], Peterson et al. [8.79], Gerner et al. [8.80], Khrustalev and Faghri [8.81], Kandlikar [8.82], and Park and Lee [8.83] have also explored the effects of disjoining pressure in circumstances typically encountered in micropassages of micro heat pipes and micro evaporators.

Near-Wall Nucleation Recent investigations also suggest that the intermolecular force interactions that give rise to disjoining pressure effects can also have an important impact on bubble nucleation during rapid heating of a liquid at a solid surface. The investigations by Gerweck and Yadigaroglu [8.84] and Carey and Wemhoff [8.85] described in Section 5.7 indicate that long-range attractive forces between fluid molecules and atoms or molecules in the solid alter the thermodynamic properties of the liquid phase in a region very close to the solid surface. The Redlich-Kwong statistical thermodynamics fluid model and the molecular dynamic simulations of Carey and Wemhoff [8.85] predict that these intermolecular forces result in a rise in pressure very near the surface. The near-wall region in which intermolecular attraction produces this pressure rise was determined to be on the order of about 10 nm thick. The model of Carey and Wemhoff [8.85] further predicts that in the near-wall region, the local spinodal temperature increases with decreasing distance from the wall. This implies that the nearwall region fluid would have to be heated to a higher temperature than the bulk fluid outside this region to initiate homogeneous nucleation. The model of Gerweck and Yadigaroglu [8.84] similarly predicts that the near-wall fluid would have to be heated to a higher temperature to initiate homogeneous nucleation than the bulk fluid. The existence of a layer of high-spinodal temperature fluid near the wall has important implications for rapid transient heating of liquid near a solid surface. For such circumstances, we expect transient conduction to produce a temperature field in which the hottest fluid is right at the surface with temperature diminishing with distance from the surface asymptotically toward the ambient temperature. The surface temperature elevates and the penetration layer of heated fluid increases in thickness with time, as dictated by the well-known transient solution for a semi-infinite solid. It can be shown [8.85] that if a constant heat flux ql′′ is suddenly delivered to the liquid adjacent to a solid surface, the transient temperature field very near the surface is well represented by the series

T = T∞ +

2ql′′ α l t ql′′z z3 − + +  (8.77) π kl kl 6 πα l t

This indicates that the temperature profile will ramp up over time, as shown in Fig. 8.14. For high-heat-flux pulse heating of wires and thin-film heaters in contact with wetting fluids, experiments in a variety of systems [8.86–8.90] indicate that the fluid near the solid surface generally approaches the superheat limit and the dominant bubble generation mechanism is homogeneous nucleation. The increase in spinodal temperature very near the wall and the nature of the transient temperature field imply that when liquid near the wall is subjected to rapid transient heating, nucleation will occur when the transient temperature profile first intersects the spinodal temperature

360

Liquid-Vapor Phase-Change Phenomena

FIGURE 8.14  Schematic of near-wall temperature variation and approach to the spinodal condition during rapid transient heating at a constant applied flux.

variation with temperature. This scenario, which is depicted schematically in Fig. 8.14, suggests that homogeneous nucleation is most likely to occur a small distance away from the surface, just outside the near-wall region where the spinodal temperature is elevated. In their experimental study of rapid transient heating of a film heater in contact with water, Andrews and O’Horo [8.89] reported bubble formation by homogeneous nucleation sometimes appeared to occur on or near the surface. The onset of nucleation away from the surface would seem to contradict the expectation that nucleation, if it occurs, will take place first at the surface where the liquid is superheated the most. The discussion of near-wall effects above considered a perfectly flat surface. In applications, the surface is likely to be rough, and the near-wall region presumably would follow the surface contour. Very smooth surfaces would have a surface roughness on the order of nanometers, and the near-wall region thickness would be comparable in size to the dimensions of surface irregularities. This is depicted in Fig. 8.15. In such cases, onset of homogeneous nucleation in rapid transient heating is

FIGURE 8.15  Schematic of the wall-affected region near a surface with nanoscale roughness.

361

Other Aspects of Boiling and Evaporation

expected to occur first at the location where the fluid just outside the wall region is first heated to the bulk spinodal temperature. Of particular note is the crevice at location A in Fig. 8.15 in which a finger of low spinodal temperature fluid protrudes into the crevice. When the wall is suddenly heated at a constant flux level, Carey et al. [8.91] have shown that homogeneous nucleation conditions will be achieved earlier at the apex of the bulk fluid intrusion in crevices like that at location A compared to shallow crevices and flat locations. Although the temperature in the cavity at the apex is only slightly higher than that at other locations just outside the wall region, Carey et al. [8.91] showed that the extreme sensitivity of embryo production rate (see Eq. (5.105)) makes first nucleation much more probable at the apex inside the cavity. In experimental studies, Rembe et al. [8.92] and Balss et al. [8.90] reported observing repeatable first homogeneous nucleation of bubbles at specific locations on their heaters during repeated rapid pulse heating. The model analysis of Carey et al. [8.91] indicated that first homogeneous nucleation at the apex of a bulk fluid intrusion into a cavity is a plausible explanation for the localized repeatable first nucleation observed in the experiments. The existence of this alternate mechanism suggests that localized repeatable first nucleation during rapid transient heating is not necessarily an indication of heterogeneous nucleation on the heater surface. It may, instead, be homogeneous nucleation at preferred cavity sites where the homogeneous nucleation conditions are achieved first during the transient heating process. Example 8.6 A thin, perfectly flat metal film heater on the wall of a micro reservoir is suddenly pulsed with a voltage step, delivering a heat flux of ql′′= 2.3 × 108 W/m 2 to liquid water in contact with the heater. Initially the water is at atmospheric pressure and a uniform temperature of 40°C. For such circumstances, Carey and Wemhoff [8.85] estimated that the thickness of the wall-affected region is about 2 nm. Estimate the time required for the temperature of the water to reach the spinodal limit of superheat limit (a) at the surface of the heater, and (b) just outside the wall-affected region. Take the superheat limit to be 306°C. For water under these conditions we use the properties at 400 K: kl = 0.674 W/mK, and αl = 1.72 × 10−7 m2/s. Since we are interested in the temperature variation very near the wall, we use Eq. (8.77) and omit the cubic term in z:

T = T∞ +

2ql′′ α l t ql′′z − kl kl π

Rearranging this relation to solve for the elapsed time, we obtain:

t spin

 π  (Tspin − T∞ )kl = + z 4α l  ql′′ 

2

a. Substituting with z = 0 yields 2



t spin ,z = 0 =

π  (306 − 40)0.674  −6   = 2.77 × 10 s 4(1.72 × 10 −7 )  2.3 × 108

b. Repeating the calculation with z = 2.0 × 10−9 m yields

t spin ,z = 2nm = 2.79 × 10 −6 s

Thus the tendency for the onset of nucleation to occur slightly away from the wall, due to molecular attractive force interactions, has a small effect on the time at which nucleation occurs during the rapid transient heating process.

362

Liquid-Vapor Phase-Change Phenomena

8.4 POOL BOILING HEAT TRANSFER ON MICRO AND NANO STRUCTURED SURFACES Despite the high heat transfer coefficients often associated with pool boiling, considerable effort has been expended developing ways of enhancing pool boiling heat transfer. These efforts have been motivated by cryogenics, electronics cooling, and other applications in which efficient heat removal at high heat flux levels is desired. From the discussion of the physics of surface wetting and bubble nucleation in earlier chapters, and the discussion of basic mechanisms of nucleate boiling and the critical heat flux (CHF) in Sections 7.2 and 7.4, it is clear that nucleate boiling heat transfer is more effective and the critical heat flux is higher when: i. there exists an abundance of cavities on the surface to act as nucleation sites that effectively trap vapor and/or gas in them when the system is filled and during periods when the system is inactive; ii. cavities on the surface span a size range that facilitates onset of nucleation at low superheat levels and rapid increase in the number of active sites with increasing superheat; iii. the surface is highly wetted by the liquid, exhibiting a low advancing contact angle, tending to draw liquid toward the contact line regions under bubbles, and tending to rewet dry surface areas under bubbles; iv. the surface area of direct contact between the solid surface and the liquid is increased so as to enhance heat transport from the surface to the liquid-vapor interface. Strategies to enhance nucleate boiling have generally aimed to strengthen one of the four enhancing mechanisms described above by altering the surface at the macro, micro, or nano scale level. Early efforts to enhance nucleate boiling focused on roughness, wetting, and nucleation site cavity creation. Commonly used manufacturing processes used to fabricate metal boiling heat transfer surfaces typically result in surface roughness levels in the micron range. As discussed in Chapter 3, if the intrinsic contact angle of the liquid on the surface material is less than 90°, increasing surface roughness will tend to reduce the apparent contact angle. The enhanced wetting will enhance nucleate boiling heat transfer and CHF, as discussed above. In addition, enhancing roughness can also alter the size distribution and morphology of potential bubble nucleation sites. Taking boiling of water at atmospheric pressure as an example, Hsu’s [6.25] model analysis discussed in Chapter 6 indicates that for moderate superheat levels of say 10–30°C, naturally occurring cavities that may become active nucleation sites typically have effective mouth radii on the order of 10–100 µm. Adding cavities with larger mouth radii will trigger onset of boiling at lower wall superheat, and adding more cavities with moderate to small mouth radii may enhance the number density of active sites as superheat increases, ensuring that the surface is thoroughly covered with active nucleation sites, removing heat through the bubble growth and release process. The net conclusion from the above observations is that altering the surface by increasing roughness can enhance nucleate boiling heat transfer by altering wetting and nucleation in ways that enhance heat transport from the surface and suppress the CHF transition. These observations also suggest that altering the roughness at the macro-to-micro scale of 1–500 µm would have the most positive effect on nucleation. Given these observations, it is not surprising that early efforts to enhance nucleate boiling focused on (1) increasing the surface roughness and (2) creation of special surfaces having artificially formed cavities designed to efficiently trap vapor. Evidence confirming the positive effect of increasing the surface roughness on nucleate boiling is abundant in the literature. As demonstrated in Fig. 8.16 from the study by Young and Hummel [8.93], the nucleate pool boiling curve generally shifts to the left as the surface roughness increases. The same trend is reflected in the fact that as the surface roughness increases, the Csf constant in Rohsenow’s correlation (7.30) decreases.

Other Aspects of Boiling and Evaporation

363

FIGURE 8.16  Enhancement of nucleate boiling in water by addition of pits and/or Teflon patches to the surface. (Adapted from Young and Hummel [8.93] with permission, copyright © 1964. American Institute of Chemical Engineers.)

Some of the past attempts to enhance nucleate boiling have considered more sophisticated methods. For nucleate boiling in water, Young and Hummel [8.93] enhanced the heat transfer by adding spots of Teflon either on the heated surface or in pits on the surface. As indicated in Fig. 8.16, this strongly shifted the boiling curve to the left, implying a significant increase in the heat transfer coefficient. Apparently the enhancement observed by Young and Hummel [8.93] is associated with the fact that water poorly wets the Teflon spots. Gas or vapor is therefore easily trapped and held in crevices of the Teflon coating. Nucleation can then be initiated at very low superheat levels, and since there is an abundance of available sites, the number of active sites increases rapidly as the superheat increases. The overall result is an increase in the effectiveness of the nucleate boiling process. Note, however, that this technique is not generally effective with chloro-fluoro hydrocarbon refrigerants because they wet virtually all substances, including Teflon, to a much greater degree. Many of the subsequent efforts to enhance nucleate boiling have employed surface structures designed to produce re-entrant or doubly re-entrant cavities. The main objective in designing the cavities in this manner is to produce a convex liquid-vapor interface, as seen from the liquid side, when liquid penetrates into the cavity. This assures that the vapor trapped in the cavity will withstand at least some subcooling of the surrounding solid surface without condensing. Results of the thermodynamic analysis in Chapter 5 indicate that the saturation temperature of the vapor in the cavity Tv for a given system (liquid) pressure Pl is related to the interface radius of curvature by the relation

Pl −

 v [ P − Psat (Tv )]  2σ = Psat (Tv ) exp  l l  (8.78) r RTv  

For a given liquid pressure, this relation requires that the equilibrium value of Tv decreases monotonically as r decreases. This implies that the maximum subcooling that can be sustained without condensing the vapor and flooding the cavity corresponds to the minimum interfacial radius of curvature that occurs at the mouth of the re-entrant cavity.

364

Liquid-Vapor Phase-Change Phenomena

FIGURE 8.17  Liquid penetration into a re-entrant and a doubly re-entrant cavity.

Note in Fig. 8.17 that an ordinary re-entrant cavity could fill with liquid if the contact angle is near zero. Establishing this contact angle as the contact line goes around the corner at the mouth of the re-entrant chamber forms a concave interface and allows a liquid film to flow down along the walls of the chamber, displacing the vapor and flooding the cavity. However, in the doubly reentrant cavity, the low contact angle can be established just beyond the mouth of the chamber while maintaining a convex interface, thus depressing the effective saturation temperature of the vapor inside the cavity slightly and preventing flooding of the cavity. Artificially produced enhanced surfaces for boiling are generally fabricated in one of two ways. One approach is to fuse metal particles or small drops of molten metal to a flat metal surface to create a thin layer which provides an irregular matrix of re-entrant cavities. The second method first produces a tube or surface having fin-like protrusions which are subsequently flattened, upset or bent over to form re-entrant cavity structures. Surfaces created by methods falling into the first category described above have been studied by Marto and Rohsenow [8.35], O’Neil et al. [8.94], Danilova et al. [8.95], Oktay and Schmeckenbecher [8.96], and Fujii et al. [8.97]. Examples of surfaces produced by methods in the second category include those described by Kun and Czikk [8.98], Webb [8.99], Zatell [8.100], and Nakayama et al. [8.101]. Yilmaz et al. [8.102] systematically compared the performance of

Other Aspects of Boiling and Evaporation

365

FIGURE 8.18  Schematic representation of the cross sections of three enhanced surfaces for nucleate boiling (a) and comparison of their pool boiling curves with that for a plain tube (b). (Adapted from Yilmaz et al. [8.102] with permission, copyright © 1980, American Society of Mechanical Engineers.)

three enhanced surfaces of these types. The surface constructions and experimentally determined pool boiling curves for the three surfaces considered by these investigators are shown in Fig. 8.18. It can be seen that the boiling curves for all three of the enhanced surfaces are shifted well to the left of that for the plain tube, implying a substantial enhancement. Superheat reductions of up to a factor of 10 have been observed for enhanced boiling surfaces like those described above, relative to corresponding results for a plain heater surface. Available

366

Liquid-Vapor Phase-Change Phenomena

evidence implies that the mechanism of vaporization for these surfaces is different from that for naturally occurring cavities (which are generally much smaller than the artificial cavities). Apparently, liquid flows partially down into the porous layer or re-entrant cavity structure, where thin-film evaporation takes place on the large interior surface provided by these structures. The vapor generated in this manner leaves as a bubble that emerges from an opening in the structure or cavity. Because, as depicted in Fig. 7.15, the mechanisms of boiling transport span a range of physical scales from macro scale to sub micro (nano) scale, it is not surprising that more recent surface alteration strategies to enhance boiling have spanned that entire range from macroscopic to nanoscale. More recent studies have explored use of more ordered microstructured surfaces for enhancing boiling processes that can be produced using micromachining, microetching, or lithographic methods developed for electronics fabrication. These advanced manufacturing methods have made it possible to tightly control the surface morphology and produce surfaces with spatially periodic variations of surface roughness, cavity size, and/or wettability, with characteristic length scales for the surfaces being on the order of microns. New methodologies for nanostructuring surfaces using advanced deposition, etching, and thermal growth processes developed in the past decade have also facilitated manipulation of the wetting, wickability, and nucleation mechanisms for boiling surfaces. Recent enhancement strategies that affect the process at the largest scales (microns to mm) typically include those that structure the surface to increase surface area thereby enhancing heat flow from solid surface to liquid vapor interface. An example is the recent work by Pastuszko and Piasecka [8.103]. Another approach is to add a surface structure that controls vapor and liquid pathways to suppress hydrodynamic instability that may limit liquid delivery to surface (see, e.g., the surfaces proposed in studies by Jaikumar and Kandlikar [8.104–8.106]). Examples of studies that have explored microscale surface modifications (1–100 µm) to provide additional nucleation sites include the work of Li and Peterson [8.107], Chu et al. [8.108], and Dong et al. [8.109]. An alternate approach is to use different types of surface structuring to created surfaces having an alternating patchwork of hydrophilic and hydrophobic areas. This type of enhanced surface has been explored, for example by Betz et al. [8.110]. Some recent investigations have considered boiling surfaces that are entirely nanostructured (see, e.g., the studies by Takata et al. [8.111], Heitich et al. [8.112], Kim et al. [8.113], Rahman et al. [8.114], Carey et al. [8.115], and Cabrera and Carey [8.116]). An example of a nanoporous surface structure of this type is the stochastic nanopillar surface shown in Fig. 8.19, which was used in water pool boiling experiments by Carey et al. [8.115] and Cabrera and Carey [8.116]. Studies of boiling on this type of surface have shown that the resulting enhanced wetting and wickability can improve the heat transfer and increase the CHF. However, there appears to be an advantage to using a surface morphology that has a combination of microstructured and nanostructured features. Combined micro/nano structures can result when nanostructures are grown on the surface and the growth process also produces microstructure features in the surface. An example is the nano/microstructured surface fabricated and tested by Lu et al. [8.117], which had regions of nanopillars bounded by microscale size trenches. The second possibility is hierarchical surfaces like those tested by Chu et al. [8.108] and Rahman et al. [8.114] which have a fabricated microstructure, with nanoscale fibers, pillars, or other features added to portions of the microstructure. The microscale elements of these hybrid surfaces can enhance nucleation, while the nanostructured features enhance wetting wickabililty. Generally, the enhancement of the nucleate boiling heat transfer process on the types of micro and nano structured surfaces described above is similar to that observed for enhanced roughened surfaces in Fig. 8.16. The boiling curve shifts to the left, with the process requiring less driving superheat than for a plain metal surface at a given heat flux. The enhancement varies with surface type. Jaikumar et al. [8.118], for example, found that some microstructured surfaces reduce the required superheat (or equivalently, increased the heat transfer coefficient) by as much as a factor of 4. Although microstrucured surface effects on nucleation are not expected to alter the onset

Other Aspects of Boiling and Evaporation

367

FIGURE 8.19  Nanoporous enhanced boiling surface (ZnO nanopillars) performance tested by Carey et al. [8.115] and Cabrera and Carey [8.116].

of the CHF, if the microstructured surface enhances surface wetting (reduces apparent contact angle) the microstructured surface may have the effect of raising the critical heat flux. Although information regarding the critical heat flux for the enhanced macro and microstructure surfaces was not always obtained, the data of O’Neill et al. [8.94] suggest that it is usually as high or higher than that for plain metal surfaces. However, the enhanced wickability associated with some nanostructured surfaces may improve nucleate boiling heat transfer and substantially increase the critical heat flux. While microstructured and nanostructured surfaces offer clear advantages for nucleate boiling heat transfer applications, their suitability for a given application must be weighed against the added cost of producing the surface. Subsequent degradation of the surface performance due to fouling, alteration of wetting due to adsorbed materials, and surface alteration due to thermal stresses, resulting from cycling operation, are also important concerns. Other methods of enhancing boiling heat transfer are also possible. The experiments of Chuah and Carey [8.40] demonstrated that an unconfined layer of metal beads on an upward-facing horizontal surface can enhance nucleate boiling heat transfer. Raiff and Wayner [8.119] have shown that film boiling heat transfer can be enhanced by suction of the vapor through a porous heated wall. Experiments described in the literature further suggest that liquid additives, mechanical aids, surface or fluid vibration, and electrostatic fields can enhance pool boiling heat transfer. For further discussion of enhancement methods, the interested reader is referred to the review articles by Bergles [8.120, 8.121] or the books by Thome [8.122], Webb [8.123], and Kandlikar et al. [8.124], and the discussion in Jaikumar et al. [8.118].

8.5  FUNDAMENTALS OF POOL BOILING IN BINARY MIXTURES Despite the common occurrence and importance of boiling of multicomponent liquid mixtures in petrochemical and cryogenic processing, at the present time, its mechanisms are not well understood. The simplest and most widely studied example of such a process is the boiling of a binary mixture. Even for this simplest type of mixture, the mechanisms may be quite complex, and accurate prediction of heat transfer coefficients is difficult. As a prelude to considering the overall boiling process, we will therefore first discuss some of the fundamental aspects of each of these features.

368

Liquid-Vapor Phase-Change Phenomena

Thermodynamics of Binary Mixtures For a binary mixture, the equilibrium conditions at which phase-change is initiated are represented in a phase equilibrium diagram. For a simple binary mixture with no azeotropic points, the phase diagram looks qualitatively like that shown in Fig. 8.20. For a mixture that forms an azeotrope at one concentration, the phase diagram looks like that shown in Fig. 8.21. Note that these diagrams represent the mixture behavior at a fixed pressure. In these diagrams, xˆ1 and yˆ1 are the concentrations of species 1 in the liquid and vapor, respectively, specified as mole fractions. Here species 1 is taken to be the more volatile component (lower pure component boiling temperature) in the mixture. The dew-point curve shown in Fig. 8.20 represents the locus of points at which condensation is first observed as the binary mixture, with a specified concentration, is cooled at constant pressure. Similarly, the bubble point curve is the locus of points at which vaporization begins as the binary mixture, with a given concentration, is heated at constant pressure. Note that these transitions correspond to systems in thermodynamic equilibrium. The phase diagram shown in Fig. 8.20 also reflects the fact that for a system with specific temperature and pressure, at saturation, the concentration of the more volatile component l is higher in the vapor than in the liquid. If the vapor is idealized as a mixture of independent ideal gases, Dalton’s law dictates that the partial pressure of each component Pvi is equal to the total pressure multiplied by the mole fraction yˆi of component i:

Pvi = Pyˆi (8.79)

Two idealized models that relate the vapor partial pressure of a component to its concentration in the liquid phase are Henry’s law and Raoult’s law. Henry’s law states that the partial pressure Pvi is proportional to the mole fraction of the component in the liquid xˆi :

Pvi = C H xˆi (8.80)

FIGURE 8.20  Typical form of the equilibrium phase diagram for a binary mixture that does not form an azeotrope.

369

Other Aspects of Boiling and Evaporation

FIGURE 8.21  Typical form of the equilibrium phase diagram for a binary mixture that forms an azeotrope.

In this relation CH designates the Henry’s law proportionality constant. Henry’s law is usually a good approximation for components having low concentrations (i.e., small xˆi ) Raoult’s law states that the partial pressure for the ith component is given by Pvi = Ppi xˆi (8.81)



where Ppi is the saturation pressure for pure component i at the specified system temperature T, and xi is the mole fraction of component i in the liquid. Raoult’s law is generally a good approximation for values of xˆi near 1. If Raoult’s law is assumed to apply to a binary system, it follows that

Pv1 = Pp1 xˆ1

Pv 2 = Pp 2 xˆ 2 (8.82)

where the subscript “1” designates the more volatile component. Since, by definition the sum of the partial pressures must equal the total system pressure

P = Pv1 + Pv 2 = Pp1 xˆ1 + Pp 2 xˆ 2 (8.83)

Rearranging and combining these relations and using the fact that xˆ1 + xˆ 2 = 1 the following relation for xˆ1 can be obtained

xˆ1 =

P − Pp 2 (T ) (8.84) Pp1 (T ) − Pp 2 (T )

Note that in this expression, the functional dependence of the pure component vapor pressures on temperature has been indicated. Thus, if the binary mixture conforms to Raoult’s law, the

370

Liquid-Vapor Phase-Change Phenomena

variation of xˆ1 with T on the phase diagram can be predicted from the pure component vapor curves using Eq. (8.84). If, in addition, Dalton’s law applies to the vapor mixture at equilibrium, then

Pv1 = Pp1 xˆ1 = Pyˆ1 (8.85)

Rearranging the above result yields yˆ1 =



Pp1 (T ) xˆ1 (8.86) P

Thus, having determined xˆ1 (T ) as described above, Eq. (8.86) can be used to determine yˆ1 (T ), completing the phase diagram shown in Fig. 8.20. The phase diagram shown in Fig. 8.22 was constructed by applying this idealized method for a mixture of nitrogen and oxygen at atmospheric pressure. The predictions of this model are actually fairly close to the actual observed mixture behavior for these fluids at atmospheric pressure. However, not all binary mixtures conform to the Raoult’s law and Dalton’s law idealizations described above. This idealized model does, however, provide some insight into the reasons for the qualitative trends observed in phase diagrams like that shown in Fig. 8.22. It should be noted that for a binary mixture, the concentration of a component can be specified as mole fraction or mass fraction, and it is easy to convert between these specifications. The mass fraction of the more volatile component can be computed from its mole fraction as

x1 =

xˆ1 M1 (8.87a) xˆ1 M1 + M 2 (1 − xˆ1 )

FIGURE 8.22  Equilibrium phase diagram for a nitrogen/oxygen mixture determined from the ideal mixture model.

Other Aspects of Boiling and Evaporation

371

where M1 and M2 are the molecular mass (kg/kmol) for species 1 and 2, respectively. Similarly, the mole fraction can be computed from the mass fraction as

xˆ1 =

x1 M 2 (8.87b) x1 M 2 + M1 (1 − x1 )

If the phase diagrams in Figs. 8.20–8.22 are replotted with mass fraction as the abscissa, they look essentially the same. The only difference will be that the horizontal axis is rescaled. As discussed in the next subsection, methodologies used to predict heat transfer in nucleate boiling of binary mixtures have sometimes specified concentration as mole fraction, and sometimes as mass fraction. For binary mixtures these specifications are interchangeable since one can easily be converted to the other.

Nucleate Boiling Heat Transfer A number of investigators have experimentally determined the heat transfer coefficients associated with nucleate boiling of a binary mixture [8.125–8.132]. Such experiments usually consist of a series of steady-state boiling tests at liquid compositions ranging from all one pure fluid to all of the other while keeping other conditions constant. For these experiments, a heat transfer coefficient hbl is often defined as

hbl =

q ′′ (8.88) [Tw − Tbp ( Pl , xˆ1 )]

where Tw is the wall temperature and Tbp ( Pl , xˆ1 ) is the bubble point temperature at the liquid pressure Pl and bulk concentration xˆ1 indicated in the equilibrium binary phase diagram shown in Fig. 8.20. In this relation, the concentration dependence could be quantified in terms of mass fraction instead of mole fraction, since in binary mixtures one is directly related to the other through Eq. (8.87a) or Eq. (8.87b). This also applies to concentrations specified in the subsequent sections of this chapter. Although mole or mass fractions may be used in the specific case considered, either can be used in practice. When vaporization occurs in a non-azeotropic binary mixture, the vapor generated is richer in the more volatile component than the bulk liquid, and the remaining liquid in the vicinity of the interface has a correspondingly lower concentration of the more volatile component. Consequently, in the liquid phase, the more volatile component diffuses toward the interface, and the excess less volatile component diffuses away from the interface into the bulk liquid. As a result of the behavior noted above, the actual temperature at the interface must therefore be equal to the bubble point for the interface concentration of the more volatile component, which is lower than that in the ambient liquid. As can be seen from Fig. 8.20, this implies that the interface temperature is somewhat higher than Tbp ( Pl , xˆ1 ). It follows that the actual driving temperature difference for vaporization at the interface Tw – Ti is somewhat less than Tw − Tbp ( Pl , xˆ1 ) on which the heat transfer coefficient is often based. The amount by which Tw – Ti is less than Tw − Tbp ( Pl , xˆ1 ) increases as the difference between the equilibrium vapor and liquid concentrations yˆ1 − xˆ1 increases. At the interface, the vapor and liquid concentrations must differ by the amount specified by the phase diagram (Fig. 8.20) at the interface temperature. The difference between the bulk concentrations yˆ1 − xˆ1 may, in general, be slightly different than the concentration difference at the interface yˆ1,i − xˆ1,i , due to additional concentration differences established to facilitate transport of one component or the other away from the interface by diffusion and/or convection. Very often, diffusion in the liquid phase has the greatest effect on the overall concentration difference during vaporization processes.

372

Liquid-Vapor Phase-Change Phenomena h hbℓi

hbℓ

0

x1

1.0

FIGURE 8.23  Qualitative variation of the nucleate boiling heat transfer coefficient with the bulk liquid concentration of a binary mixture at a fixed pressure.

It can be seen in Fig. 8.20 that the difference between the liquid and vapor concentrations at the interface yˆ1,i − xˆ1,i must go to zero at bulk liquid concentrations of zero and one, but increases and passes through a maximum as the concentration varies between the pure fluid limits. Concentration differences driving mass transfer to and from the interface will also go to zero for these limiting cases. At some intermediate bulk liquid concentration, the overall bulk concentration difference will achieve a maximum value. It follows directly from the above arguments that the amount by which Tw – Tl is less than Tw − Tbp ( Pl , xˆ1 ) is zero at bulk liquid concentrations of zero and one, achieving a maximum value at some intermediate concentration. Consequently, for nucleate boiling at a given pressure and wall temperature, the heat transfer coefficient hbl defined by Eq. (8.88) generally varies with bulk liquid concentration in the manner indicated in Fig. 8.23. The minimum in the hbl variation is a direct consequence of the maximum in the difference between the liquid and vapor concentrations at some intermediate bulk liquid concentration, as described above. If the diffusion of the more volatile component in the liquid were very rapid during the vaporization process, then its concentration at the interface would be virtually identical to that of the bulk liquid, and the equilibrium temperature of the interface at the given system pressure would simply equal Tbp ( Pl , xˆ1 ). Thus the fact that Tw – Ti is less than Tw − Tbp ( Pl , xˆ1 ) is a direct consequence of the resistance to mass diffusion in the liquid. For this reason, the depression of the heat transfer coefficient when yˆ1 − xˆ1 is nonzero is sometimes referred to as resulting from a diffusion resistance to heat transfer. One obvious alternative to the above approach would be to base the heat transfer coefficient on the actual temperature difference Tw – Ti between the wall and the interface. However, the value of Ti established in a given circumstance depends on the mass transport from the interface, and consequently, is difficult even in the simplest systems to accurately predict (see, e.g., the discussion by Cooper and Stone [8.133]). If diffusion in the liquid were infinitely fast, the interface concentration would equal that in the bulk liquid. As far as heat transfer is concerned, the mixture would then behave essentially like an azeotropic mixture (i.e., like a pure fluid with properties that are averages of pure fluid properties for the components in the mixture). For such idealized circumstances, a plausible estimate of the heat transfer coefficient hbli would be a weighted average of the heat transfer coefficients for the pure components. There are at least three ways that this has been done in previous studies. A straightforward method would be to define hbli as the mole fraction weighted average of single component nucleate boiling heat transfer coefficients for the pure components at the system pressure.

hbli = hb1 xˆ1 + hb 2 (1 − xˆ1 ) (8.89)

373

Other Aspects of Boiling and Evaporation

In the above relation, hb1 and hb2 are the pure component heat transfer coefficients at the same pressure and heat flux for the low boiling point and high boiling point constituents, respectively. Variations of this approach include weighting with the mass fraction instead of mole fractions. Note that this relation predicts a linear variation of the heat transfer coefficient with liquid molar concentration for the idealized case of infinitely fast mass diffusion. An alternate approach is to define an average ideal heat transfer coefficient based on a mole fraction weighted average of the wall superheats for the pure component fluids at the specified heat flux and the same system temperature or pressure. ∆Tsi = ∆Ts1 xˆ1 + ∆Ts 2 (1 − xˆ1 ) (8.90)



The associated ideal heat transfer coefficient is the reciprocal mole fraction weighted average of the pure coefficients 1 (1 − xˆ1 ) xˆ = 1 + (8.91) hbli hb1 hb 2

which implies that

−1

(1 − xˆ1 )   xˆ hbli =  1 + (8.92) h hb 2   b1



A variation of this approach is to define similar weighted averages using the mass fractions instead of the mole fraction. This leads to ∆Tsi = ∆Ts1 x1 + ∆Ts 2 (1 − x1 ) (8.93)

and

−1



(1 − x1 )  x hbli =  1 + (8.94) hb 2   hb1

A third approach to predicting the ideal heat transfer coefficient hbli is to use appropriately defined mixture properties in a pure fluid nucleate boiling correlation. As discussed above, the deviation from this idealized behavior due to finite mass transfer rates will be greatest when yˆ1 − xˆ1 or y1 − x1 is largest. This suggests that the variation of the actual mixture heat transfer coefficient hbl with concentration can be represented by a relation of the form

hbl = hbli FD

(8.95)

where FD is a correction factor for mass diffusion effects that has the qualities: FD is minimum at maximum yˆ1 − xˆ1 (or maximum y1 − x1 ) FD → 1 when yˆ1 − xˆ1 → 0 (or equivalently, when y1 − x1 → 0) Efforts to developed predictive relations for binary mixture nucleate boiling heat transfer have used one of three approaches to obtain relations for FD: (1) develop an empirical relation from a fit to experimental data; (2) develop a relation from a theoretical model; or (3) develop a semi-theoretical relation based on theoretical arguments and a best fit to data.

374

Liquid-Vapor Phase-Change Phenomena

Using variations of the general strategy outline above, useful methods for predicting binary mixture nucleate boiling heat transfer have been derived by Happel [8.129], Stephan and Körner [8.134], Calus and Rice [8.135], Calus and Leonidopoulos [8.128], Schlünder [8.136], Thome and Shakir [8.137], Wenzel et al. [8.138], Fujita et al. [8.139], and Kandlikar [8.140]. The correction factor relations developed by Happel [8.129], Stephan and Körner [8.134], and Calus and Leonidopoulos [8.128] were relatively simple. For nucleate boiling of a benzene-toluene mixture at atmospheric pressure at a heat flux of 100 kW/m2, Happel [8.129] used the mole fraction weighted definition of hbli (8.89) and obtained good agreement between his experimental data and a relation of the form (8.95) for

n

FD = 1 − Cc yˆ1 − xˆ1 (8.96)

with Cc = 1.5 and n = 1.4. Values of Cc and n are expected to vary with the substances and system pressure. Although a general method for predicting values of Cc and n for a given set of conditions is not available, they could be determined from experiments for a given system. Based on the available data, Happel [8.129] observed that for an equal concentration difference yˆ1 − xˆ1 , the reduction in hbl generally increases as system pressure increases. This suggests that Cc and n will likely be larger at higher pressures. Stephan and Körner [8.134] used the reciprocal mole fraction average for hbli, (8.92) and proposed the following relation for FD:

FD = [1  + A y1 − x1 (0.88 + 0.12 P)]−1 (8.97)

where P is the pressure in bar and A is a constant specific to the mixture, typically in the range 0.43–0.56. Calus and Leonidopoulos [8.128] used the reciprocal mass fraction average for hbli (8.104) and proposed the following relation for FD: −1



  α   c pl   dTbp   FD = 1 + ( y1 − x1 )  T ,l     (8.98)  D12l   hlv   dx1   

where dT bp /dx1 is the slope of the bubble point curve in the equilibrium phase diagram for the mixture, αT,l is the liquid thermal diffusivity, and D12l is the mass diffusivity for the liquid mixture. Thome and Shakir [8.137] proposed a modified version of an earlier correlation developed by Schlünder [8.136]. This correlation used the reciprocal mole fraction average for hbli (8.92) and proposed the following relation for FD: −1



 Tdp ( xˆ1 ) − Tbp ( xˆ1 )   B0 q ′′    FD = 1 +  1 − exp     (8.99a) ∆Tsi    β L ρl hlv   

where

B0 = 1,

β L = 3.0 × 10 −4 m/s (8.99b)

In Eq. (8.99a), Tdp ( xˆ1 ) and Tbp ( xˆ1 ) are the dew point and bubble point temperatures at the liquid concentration, and ΔTsi is the ideal superheat determined using Eq. (8.90). Recent studies by Fujita et al. [8.139] and Kandlikar [8.140] have produced more complex predictive relations for binary mixture nucleate boiling heat transfer that fit data for a variety of mixtures

375

Other Aspects of Boiling and Evaporation

fairly well. Fujita et al. [8.139] proposed using the reciprocal mole fraction average for hbli (8.92) together with the following relation for FD: −1

    Tdp ( xˆ1 ) − Tbp ( xˆ1 )    ∆Tsi FD = 1 +  1 − exp −2.8    (8.100) ∆Tsi Tsat 2 − Tsat1      



In the above relation, Tsat1 and Tsat2 are the pure fluid saturation temperatures of species 1 and 2, respectively, at the system pressure. Instead of an ideal weighted-average coefficient, the predictive model developed by Kandlikar [8.140] is based on an average pseudo single-component coefficient hpsc,avg defined as

h psc ,avg

−1   x1 (1 − x1 )   = 0.5  x1hb1 + (1 − x1 )hb 2 +  +  (8.101)  hb1 hb 2   

This average coefficient is then corrected to account for real mixture properties using the relation

h psc

 Tbp,m  = h psc ,avg   Tsat ,avg 

−0.674

 hlv ,m   h  lv ,avg 

0.371

 ρv , m   ρ  v ,avg 

0.297

 σm   σ  avg

−0.317

 kl ,m   k  l ,avg 

0.284

(8.102)

In the above relation, the “m” subscript designation denotes actual mixture properties, whereas the “avg” subscript designation is the mass fraction average of the pure component properties

Tsat ,avg = x1Tsat ,1 + (1 − x1 )Tsat ,2 (8.103a)



hlv ,avg =   x1hlv ,1 + (1 − x1 )hlv ,2 (8.103b)



ρv ,avg = x1ρv ,1 + (1 − x1 )ρv ,2 (8.103c)



σ avg =   x1σ1 + (1 − x1 )σ 2 (8.103d)



kl ,avg = x1 kl ,1 + (1 − x1 ) kl ,2 (8.103e)

The true mixture properties must be determined from mixture property correlations or databases. The binary mixture nucleate boiling heat transfer coefficient hbl is then obtained by multiplying by the correction factor for mass diffusion effects, FD: hbl = h psc FD (8.104)

where





1 for V1 ≤ 0.03  − V for 0.03 < V1 ≤ 0.2 1 64 l  FD =  (8.105) −1   α T ,l  1/ 2  c pl   Tbp ( x l,i ) − Tbp ( x l )      for V1 > 0.2  0.678 1 +  Dl2l   hlv   gK    α  V1 =  T .l   D12l 

1/ 2

 c pl   dTbp   h   dx  ( yl − xl ) (8.106) lv l

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Liquid-Vapor Phase-Change Phenomena

gK =



x l − x l,i (8.107) yl,i − x l,i

α  2.13 Ja 0  T .l  x l,i = x l −  Dl2l  π



Ja 0 =



1/ 2

 ρv   ρ  ( yl .i − xl ) (8.108) l

Tw − Tbp ( x l ) 1/ 2  ρv   hlv   α T ,l   Tbp ( x l,i ) − Tbp ( x l )   (8.109) +     ρ   c   D   gK l2 l l  pl 

To computationally predict the nucleate boiling heat transfer coefficient with this scheme, the following iterative method is recommended by Kandlikar [8.140]: 1. Calculate Tsat = Tbp(xl) corresponding to the bulk liquid concentration xl and system pressure P. 2. Assume the liquid concentration at the interface xl.i. Determine the corresponding bubble point temperature Tsat,i = Tbp(x1,i) and equilibrium vapor concentration yl,i for the interface from thermodynamic property data for the mixture. 3. Assume hbl and compute the wall temperature for the specified heat flux: Tw = Tbp + q ′′/hbl . 4. Calculate gK and Ja0 using Eqs. (8.107) and (8.109). 5. Recalculate xl,i using Eq. (8.108). 6. Calculate V1 from Eq. (8.106). 7. Calculate FD using Eq. (8.105) and recalculate hbl using Eq. (8.104). 8. Iterate steps 2–7 until xl,i and hbi are converged. Kandlikar [8.140] presented comparisons of the predictions of this correlation and those of the models of Fujita et al. [8.139] and Calus and Leonidopoulos [8.128] with experimental data for different mixtures. These comparisons suggest that although more complicated, the Kandlikar correlation method agrees somewhat better with available data for different binary mixtures. Example 8.7 Nucleate pool boiling in a water/methanol mixture occurs at atmospheric pressure with a heat flux of 20 W/cm2. The alcohol mole fraction in the bulk liquid is 0.1. Use the correlation of Fujita et al. [8.139] to predict the wall superheat for these conditions. For these circumstances, we will use the Stephan-Abdelsalam correlations (7.35) and (7.36) described in Chapter 7 for pure component nucleate boiling. For methanol at 101 kPa = 1.0 bar, Fig. 7.11 indicates that C2 = 2.4. From Eq. (7.36) it follows that

(Tw −  Tsat )1 = (q′′ )0.330 /C2 = (20 × 10 4 )0.330 / 2.4 = 23.4°C

The pure fluid heat transfer coefficient is then given by

hbl = q′′ /(Tw − Tsat ) = 20 × 10 4 / 23.4 = 8550 W/m 2 °C

For water at 101 kPa = 1.0 bar, Fig. 7.10 indicates that C1 = 3.7. From Eq. (7.35) it follows that

(Tw − Tsat )2 = (q′′ )0.327 /C1 = (20 × 10 4 )0.327 /3.7 = 14.6°C hb 2 = q′′ /(Tw − Tsat ) = 20 × 10 4 /14.6 = 13,700 W/m 2 °C

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The ideal average wall superheat and heat transfer coefficient are then computed using Eqs. (8.90) and (8.92)

∆Tsi = ∆Ts1xˆ1 + ∆Ts 2(1− xˆ ) = 23.4(0.1) + 14.6(1− 0.1) = 15.5°C



(1− xˆ1)  (1− 0.1)   0.1  xˆ hbli =  1 + = +  hb 2   hb1  8.550 13,700 

−1

−1

= 12,900 W/m 2 °C

The binary mixture phase diagram dictates that for xˆ1 = 0.1, the boiling point and dew point temperatures are

Tbp ( xˆ1) = 87.4°C ,

Tdp ( xˆ1) = 97.5o C

FD is computed using Eq. (8.100):



     Tdp ( xˆl ) − (Tbp ( xˆl )   ∆Tsi FD = 1+  1− exp −2.8    Tsat 2 − Tsat1   ∆Tsi    

−1

−1

  (15.5)   97.5 − 87.4    = 1+  1− exp −2.8     = 0.683 100 15.5  − 65      The binary mixture boiling heat transfer coefficient is then given by

hbl = hbliFD = 12,900(0.683) = 8830 W/m 2°C

and the superheat for boiling in the binary mixture is

Tw − Tbp ( xˆ1) = q′′ /hbl = (20 × 10 4 )/8830 = 22.6°C

The predicted value of 22.6°C is only slightly different from the reported experimental value of 24.0°C [8.140].

Critical Heat Flux A number of investigations have examined the critical heat flux conditions for pool boiling of binary mixtures. Early studies by van Stralen and co-workers [8.141–8.144] extensively studied pool boiling in binary mixtures. Subsequent investigations by a number of investigators [8.145–8.150] added to the body of CHF data for pool boiling of various binary liquid mixtures with a variety of heater geometries. Among the more interesting findings of these studies was the observation that the critical heat flux for water-alcohol mixtures at low alcohol concentrations were significantly higher than the corresponding values for pure water under comparable conditions. Many of these investigators also found that, in some instances, variation of the critical heat flux with concentration exhibits a maximum at low concentrations of one component. Early efforts to develop correlations to predict the critical heat flux for heaters in binary liquid mixtures included investigations by Kutateladze et al. [8.151], Matorin [8.152], Gaidarov [8.153], Stephan and Preusser [8.154], and Yang [8.155]. In these early studies, modeling of the CHF mechanism usually focused on McEligot’s [8.156] observation that evaporation from a liquid-vapor interface in a binary system leaves the liquid phase at a higher temperature than the dew point temperature of the vapor generated. The surrounding liquid, at the bulk concentration, is colder than the temperature at which the liquid evaporates, effectively being subcooled relative to the vaporization condition. This is often referred to as induced subcooling. Because of the shape of the equilibrium phase diagram (Fig. 8.20), this induced subcooling is greatest where the quantity xˆ l − yˆl is highest.

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Given that subcooling of the surrounding bulk liquid pool is known to increase the critical heat flux for pool boiling of pure fluids, these comments suggest that the peak in the critical heat flux qcr′′ is at least partially a consequence of the high induced subcooling that occurs where xˆ l − yˆl is largest. Reddy and Lienhard [8.157] pointed out that overall trends in earlier CHF data were almost impossible to identify at the time these data were obtained, since effects of heater geometry variations were not well understood until more recently. They explored the CHF for boiling of binary mixtures at the heated surface of small wires. Reddy and Lienhard [8.157] used pure fluid correlations for the critical heat flux during saturated and subcooled pool boiling to quantitatively assess the induced subcooling in their experiments. For the ethanol-water mixture considered in their study, they noted that xˆ l − yˆl varies about linearly with Tdb ( xˆ l , P) − Tbp ( xˆ l , P). They were consequently able to correlate their data in terms of this temperature difference instead of the concentration difference xˆ l − yˆl used by Kutateladze et al. [8.151] and others. They separately accounted for the effect of heater size and the effect of induced subcooling discussed above. This approach to correlation development was a key contribution in this area. The numerical constants in their correlation equation were chosen to obtain a best fit to their data for saturated ethanol-water mixtures. The resulting correlation matched their data for wires and tubes to an RMS deviation of 15%. The results of these early studies seemed to indicate that for nucleate boiling of binary mixtures, concentration variations near the interface at the heater surface result in effective subcooling that enhances heat removal from the surface, and consequently, the critical heat flux. While this appears to explain the increase in the critical heat flux in some cases, results of subsequent experiments conducted by McGillis and Carey [8.31] and Fujita et al. [8.158] suggest that in some systems, this mechanism alone cannot be responsible for the increase. CHF data obtained by McGillis and Carey [8.31] for pool boiling of water and 2-propanol mixtures on a 1.27 cm square surface are shown in Fig. 8.24. For the data shown, the system pressure was adjusted as concentration varied to hold a fixed vapor dew point temperature of 38.9°C. The experiments of McGillis and Carey [8.31] indicated that the effects of the induced subcooling mechanism implied by earlier studies are not consistent with critical heat flux data they obtained for ethylene glycol and water mixtures and 2-propanol and water mixtures. Instead, these investigators found that the variation of critical heat flux correlates more strongly with the magnitude of surface tension gradients near the heated surface. This connection had been suggested earlier by Hovestreijdt [8.159], who obtained experimental CHF data for a horizontal wire in binary organic mixtures with differences in surface tension. To explain trends in the data, Hovestreijdt [8.159] speculated that variations in surface tension affected bubble stability on the wire and that, in turn, affected the onset of the CHF condition.

FIGURE 8.24  The effect of concentration of 2-propanol in water on the critical heat flux condition for pool boiling on a 1.27 cm square surface. These data were obtained by McGillis and Carey [8.31].

Other Aspects of Boiling and Evaporation

379

FIGURE 8.25  Concentration variations generated as vapor slugs depart from the heated surface: xˆ l,i is the liquid concentration of the more volatile component at the interface away from the wall, xˆ l.iw is the liquid concentration of the more volatile component very near the wall.

For water/2-propanol mixtures and water/methanol mixtures, the more volatile component (the alcohol) has the lower surface tension of the two components. In mixtures of this type, the concentration difference due to preferential vaporization of the more volatile component at the portion of the interface nearest the heated surface will result in a surface tension gradient along the liquid-vapor interface. For water/2-propanol solutions at low 2-propanol concentrations, the variation of surface tension with more-volatile component concentration exhibits a strong negative slope ( − ∂σ / ∂ xˆl >> 1). For a methanol and water mixture, ∂σ / ∂ xˆl is also negative, but the gradient is less steep. In both cases, however, the negative value of the surface tension gradient will produce stronger surface tension near the heated wall and weaker surface tension further away, where bulk conditions apply. This imbalance will tend to pull liquid near the interface from the bulk, where xˆ l,i is higher and surface tension is smaller, to the near-wall region, where xˆ l,iw is lower and surface tension is larger (Fig. 8.25). The higher temperature near the wall will tend to reduce the surface tension compared to the cooler bulk liquid. However, in the alcohol/water mixtures considered by McGillis and Carey [8.31], changes in surface tension resulting from temperature changes are negligible compared to those induced by concentration differences. In the absence of surface tension gradients, liquid is delivered to the surface to replace liquid converted to vapor during the boiling process by gravitational force effects. This gravitational body force effect manifests itself as bubble buoyancy, which causes generated vapor to move upward with a corresponding downward motion of liquid to replace the vapor. This acts to keep the surface wetted, thereby preventing surface dryout and the onset of the critical heat flux condition. Beginning with the well-known Zuber correlation for the critical heat flux for an upward-facing flat heated surface, McGillis and Carey [8.31] replaced the liquid-restoring effect of gravitational buoyancy in the Zuber relation with the combined effect of buoyancy and liquid motion induced by surface tension gradients. The resulting form of the McGillis and Carey [8.31] correlation for the critical heat flux of binary mixtures can be written as:

1  ∂σ   qm′′,bm = qm′′,sf 1 + Cm   ( yˆl − xˆ l )  (8.110)   ˆ ∂ σ x l  

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where qm′′,sf is the critical heat flux for a single component fluid evaluated with mixture properties for the surface geometry of interest. In Eq. (8.110), σ −1 (∂σ / ∂ xˆ l )( yˆl − xˆ l ) is the parameter that quantifies the influence of the surface tension gradient on the critical heat flux of binary mixtures. The derivative ∂σ / ∂ xˆ l is the slope of the mixture surface tension curve with respect to the more volatile component, ( yˆl − xˆ l ) is the difference between the equilibrium vapor concentration of the more volatile component and the bulk liquid concentration at the system temperature and pressure. McGillis and Carey [8.31] found that by setting the constant Cm equal to 1.14, the correlation predictions matched critical heat flux data for the water and 2-propanol mixtures shown in Fig. 8.24 as well as their data for water and methanol mixtures, and the CHF data of Reddy and Lienhard [8.157] for ethanol and water mixtures. Agreement with the data is quite good, both qualitatively and quantitatively, over the entire range of concentrations tested for each mixture. The CHF model analysis of McGillis and Carey [8.31] is based on the premise that when ∂σ / ∂ xˆ l for a binary mixture is negative (species 1 being the more volatile component), gravitational body forces and Marangoni effects may both act to sustain a flow of liquid to the surface during boiling of the binary mixture. The validity of this conjecture was supported by results of later experiments by Ahmed and Carey [8.30] that explored the effects of varying gravity on binary mixture boiling. The experiments of these investigators indicated that high critical heat flux levels were attained in binary mixture boiling, even under microgravity conditions. The magnitudes of the critical heat flux determined in the binary mixture boiling experiments were consistent with the predictions of the McGillis and Carey [8.31] model. This supports the contention that when ∂σ / ∂ xˆ l is negative for the binary mixture, concentration difference Marangoni effects will act in tandem with gravity to deliver liquid to the heated surface under normal earth gravity, and under reduced gravity, the Marangoni effects alone may sustain high CHF levels during binary mixture boiling. Although induced subcooling may have an effect on the critical heat flux during binary mixture boiling, the experimental evidence cited above implies that for some common binary systems, concentration Marangoni effects may be the most important concentration-related CHF mechanism. For binary mixture boiling, an effective CHF model-based prediction method that includes Marangoni effects, has also been developed by Yagov [8.160].

Other Features Nucleation, bubble growth and other detailed facets of nucleate boiling in binary liquids are, as one might expect, significantly more complex than similar phenomena in pure component systems. The interested reader is referred to the studies of van Stralen [8.126, 8.127], van Ouwerkerk [8.161], Calus and Leonidopoulos [8.128], Cooper [8.162], or Cooper and Stone [8.133] for more information on these aspects of boiling of binary mixtures. Film boiling of a binary mixture on a horizontal surface has been considered in an early study by Kautzky and Westwater [8.163], These investigators obtained film boiling curves for pure CCl4, pure R-113 and for mixtures of these two components. Their results clearly demonstrate that a simple interpolation between the boiling curves for the pure components does not adequately represent the results for binary mixtures of these components. In subsequent studies, Yue and Weber [8.164, 8.165] and Marschall and Moresco [8.166] have presented analyses of film boiling of binary mixtures over a vertical surface. It is noteworthy that these analyses incorporate the effects of induced (or effective) subcooling on heat transfer. As in the case of nucleate boiling, this results from alteration of the interface bubble point temperature due to finite mass transfer rates in the liquid near the interface. The past investigations discussed above have substantially increased the under-standing of binary mixture boiling. However, the added complexities of binary mixture thermodynamics, mass diffusion, and Marangoni effects in binary mixture boiling have not been fully explored, and further research is needed to develop a more complete understanding of how these mechanisms affect binary mixture boiling.

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381

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8.54 Nishio, S., Prediction technique for minimum-heat-flux(MHF)-point condition of saturated pool boiling, Int. J. Heat Mass Transf., vol. 30, pp. 2045–2057, 1987. 8.55 Ramilison, J. M., and Lienhard, J. H., Transition boiling heat transfer and the film transition regime, ASME J. Heat Transf., vol. 109, pp. 746–752, 1987. 8.56 Segev, A., and Bankoff, S. G., The role of adsorption in determining the minimum film boiling temperature, Int. J. Heat Mass Transf., vol. 23, pp. 637–642, 1980. 8.57 Olek, S., Zvirin, Y., and Elias, E., The relation between the rewetting temperature and the liquid-solid contact angle, Int. J. Heat Mass Transf., vol. 31, pp. 898–902, 1988. 8.58 Bernardin, J. D., and Mudawar, I., The Leidenfrost Point: Experimental study and assessment of existing models, ASME J. Heat Transf., vol. 121, pp. 894–903, 1999. 8.59 Bernardin, J. D., and Mudawar, I., A cavity activation and bubble growth model of the Leidenfrost point, ASME J. Heat Transf., vol. 124, pp. 864–873, 2002. 8.60 Baumeister, K. J., Hamill, T. D., Schwartz, F.L., and Schoessow, G. J., Film boiling heat transfer to water drops on a flat plate, NASA TM-X-52103, 1965. 8.61 Gottfried, B. S., Lee, C. J., and Bell, K. J., The Leidenfrost phenomenon: Film boiling of liquid droplets on a flat plate, Int. J. Heat Mass Transf., vol. 9, pp. 1167–1187, 1966. 8.62 Gottfried, B. S., and Bell, K. J., Film boiling of spheroidal droplets, Ind. Eng. Chem. Fundam., vol. 5, pp. 561–568, 1966. 8.63 Patel, B. M., and Bell, K. J., The Leidenfrost phenomenon for extended Liquid masses, AIChE Chem. Eng. Prog. Symp. Ser., vol. 62, no. 64, pp. 62–71, 1966. 8.64 Wachters, L. H. J., Bonne, H., and van Nouhius, H. J., The heat transfer from a hot horizontal plate to sessile water drops in the spheroidal state, Chem. Eng. Sci., vol. 21, pp. 923–936, 1966. 8.65 Goleski, E. S., and Bell, K. J., The Leidenfrost phenomenon for binary liquid solutions, Proc. 3rd Int. Heat Transf. Conf., vol. 4, pp. 51–58, 1966. 8.66 Bell, K. J., The Leidenfrost phenomenon: A survey, Chem. Eng. Prog. Symp. Ser., vol. 63, no.79, pp. 73–82, 1967. 8.67 Israelachvili, J., Intermolecular & Surface Forces, 2nd ed., Academic Press, London, England, 1992. 8.68 Potash, M. L., Jr., and Wayner, P. C., Jr., Evaporation from a two-dimensional extended meniscus, Int. J. Heat Mass Transf., vol. 15, pp. 1851–1863, 1972. 8.69 Wayner, P. C., Jr., The effect of the London-van der Waals dispersion forces on interline heat transfer, ASME J. Heat Transf., vol. 100, pp. 155–159, 1978. 8.70 Wayner, P. C., Jr., Kao, Y. K., and Lacroix, L. V., The interline heat transfer coefficient of an evaporating wetting film, Int. J. Heat Mass Transf., vol. 19, pp. 487–492, 1976. 8.71 Wayner, P. C., Jr., A constant heat flux model of the evaporating interline region, Int. J. Heat Mass Transf., vol. 21, pp. 362–364, 1978. 8.72 Mirzamoghadam, A., and Catton, I., A physical model of the evaporating meniscus, ASME J. Heat Transf., vol. 110, pp. 201–207, 1988. 8.73 Mirzamoghadam, A., and Catton, I. Holographic interferometry investigation of enhanced tube meniscus behavior, ASME J. Heat Transfer, vol. 110, pp. 208–213, 1988. 8.74 Swanson, L., and Peterson, G. P., The evaporating extended meniscus in a V-shaped channel, AIAA J. Thermophys. Heat Transf., vol. 8, pp. 172–181, 1994. 8.75 Hallinan, K. P., Chebaro, H. C., Kim, S. J., and Change, W. S., Evaporation from an extended meniscus for nonisothermal interfacial conditions, AIAA J. Thermophys. Heat Transf., vol. 8, pp. 709–716, 1994. 8.76 Xu, X., and Carey, V. P., Film evaporation from a micro-grooved surface – an approximate heat transfer model and its comparison with experimental data, AIAA J. Thermophys. Heat Transf., vol. 4, pp. 512–520, 1991. 8.77 Stephan, P., and Busse, C. A., Analysis of the heat transfer coefficient of grooved heat pipe evaporator walls, Int. J. Heat Mass Transf., vol. 35, pp. 383–391, 1992. 8.78 Swanson, L., and Herdt, G. C., Model of the evaporating meniscus in a capillary tube, ASME J. Heat Transf., vol. 114, pp. 434–441, 1992. 8.79 Peterson, G. P., Duncan, A. B., and Weichold, M. H., Experimental investigation of micro heat pipes fabricated in silicone wafers, ASME J. Heat Transf., vol. 115, pp. 751–756, 1993. 8.80 Gerner, F. M., Badran, B., Henderson, H. T., and Ramadas, P., Silicon-water micro heat pipes, Therm. Sci. Eng., vol. 2, pp. 90–97, 1994. 8.81 Khrustalev, D., and Faghri, A., Heat transfer during evaporation on capillary grooved structures of heat pipes, ASME J. Heat Transf., vol. 117, pp. 740–747, 1995. 8.82 Kandlikar, S. G., Fundamental issues related to flow boiling in minichannels and microchannels, Exp. Therm. Fluid Sci., vol. 26, pp. 389–407, 2002.

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8.110 Betz, A. R., Xu, J., Qiu, H., and Attinger, D., Do surfaces with mixed hydrophilic and hydrophobic areas enhance pool boiling?, Appl. Phys. Lett., vol. 97, p. 141909, 2010. 8.111 Takata, Y., Hidaka, S., Cao, J. M., Nakamura, T., Yamamoto, H., Masuda, M., and Ito, T., Effect of surface wettability on boiling and evaporation, Energy, vol. 30 pp. 209–220, 2005. 8.112 Heitich, L. V., Passos, J. C., Cardoso, E. M., da Silva, M. F., Klein, A. N., Nucleate boiling of water using nanostructured surfaces, J Braz. Soc. Mech. Sci. Eng., vol. 36, pp. 181–192, 2014. 8.113 Kim, B. S., Lee, H., Shin, S., Choi, G., and Cho, H. H., Interfacial wicking dynamics and its impact on critical heat flux of boiling heat transfer, Appl. Phys. Letters, vol. 105, pp. 191671-1–191671-4, 2014. 8.114 Rahman, M. M., Ölçero˘glu, E., and McCarthy, M., Role of wickability on the critical heat flux of structured superhydrophilic surfaces, Langmuir, vol. 30, pp. 11225–11234, 2014. 8.115 Carey, V. P., Wemp, C. K., McClure, E. R., and Cabrera, S., Mechanism interaction during droplet evaporation on nanostructured hydrophilic surfaces, paper IMECE2018-87991, Proc. ASME 2018 International Mechanical Engineering Congress and Exposition, IMECE2018, November 9-15, 2018, Pittsburgh, PA, USA. 8.116 Cabrera, S. and Carey, V. P., Comparison of droplet evaporation and nucleate boiling mechanisms on nanoporous superhydrophilic surfaces, paper SHTC2019-3539, Proc. ASME 2019 Summer Heat Transfer Conf., SHTC2019, July 14-17, 2019, Bellevue, WA, USA. 8.117 Lu, M.-C., Chen, R., Srinivasan, V., Carey, V. P., and Majumdar, A., Critical heat flux of pool boiling on si nanowire array-coated surfaces, Int. J. Heat Mass Transf., vol. 54, pp. 5359−5367, 2011. 8 .118 Jaikumar, A., Rishi, A., Gupta, A., and Kandlikar, S. G., microscale morphology effects of copper–graphene oxide coatings on pool boiling characteristics, J. Heat Transf., vol. 139, pp. 111509-1–111509-11, 2017. 8.119 Raiff, R. J., and Wayner, P. C., Jr., Evaporation from a porous flow control element on a porous heat source, Int. J. Heat Mass Transf., vol. 16, pp. 1919–1930, 1973. 8.120 Bergles, A. E., Survey and evaluation of techniques to augment convective heat and mass Transfer, in Progress in Heat and Mass Transfer, U. Grigull, and E. Hahne (editors), Pergamon Press, Oxford, 1969, vol. 1, pp. 331–334. 8.121 Bergles, A. E., Enhancement of boiling and condensation, in Two-Phase Flow and Heal Transfer, China-U.S. Progress, X.-J. Chen, and T. N. Veziroglu (editors), Hemisphere Publishing Company, New York, NY, 1985. 8.122 Thome, J. R., Enhanced Boiling Heat Transfer, Hemisphere Publishing Company, New York, NY, 1990. 8.123 Webb, R. L., Principles of Enhanced Heat Transfer, John Wiley & Sons, New York, NY, Chapter 11, 1994. 8.124 Kandlikar, S., Shoji, M., and Dhir, V. K., editors. Handbook of Phase Change – Boiling and Condensation, Taylor and Francis, Philadelphia, PA, Chapter 5, 1999. 8.125 Bonnilla, C. F., and Perry, C. W., Heat transmission to boiling binary liquid mixtures, Trans. AIChE, vol. 37, pp. 269–290, 1941. 8.126 van Stralen, S., The mechanism of nucleate boiling in pure liquids and binary mixtures, Parts I & II, Int. J. Heat Mass Transf., vol. 9, pp. 995–1046, 1966. 8.127 van Stralen, S., The mechanism of nucleate boiling in pure liquids and binary mixtures, Parts III & IV, Int. J. Heat Mass Transf., vol. 10, pp. 1469–1498, 1967. 8.128 Calus, W. F., and Leonidopoulus, D. J., Pool boiling – binary liquid mixtures, Int. J. Heat Mass Transf., vol. 17, pp. 249–256, 1974. 8.129 Happel, O., Heat transfer during boiling of binary mixtures in the nucleate and film boiling ranges, chapter 9 in Heat Transfer in Boiling, E. Hane, and U. Grigull (editors), Academic Press/Hemisphere Publishing, New York, NY, 1977. 8.130 Thome, J. R., Nucleate boiling of binary liquids, AIChE Prog. Symp. Ser., vol. 77, pp. 238–250, 1981. 8.131 Stephan, K., Natural convection boiling in multicomponent mixtures, in Heat Exchangers, S. Kakac, A. E. Bergles, and F. Mayinger (editors), Hemisphere Publishing, New York, NY, pp. 315–336, 1981. 8.132 van Stralen, S. J. D., Heat transfer to boiling binary and ternary systems, chapter 2 in Boiling Phenomena, by S. van Stralen, and R. Cole, Hemisphere/McGraw Hill, New York, NY, pp. 33–65, 1979. 8.133 Cooper, M. G., and Stone, C. R., Boiling of binary mixtures – study of individual bubbles, Int. J. Heat Mass Transf., vol. 24, pp. 1937–1950, 1981. 8.134 Stephan, K., and Kömer, M., Berechnung des Wärmeübergangs Verdamp Fender Binärer Flüssigkeitsgemische, Chem. Ing. Tech., vol. 41, pp. 409–417, 1969.

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8.135 Calus, W. F., and Rice, P., Pool boiling – binary liquid mixtures, Chem. Eng. Sci., vol. 27, pp. 1687–1697, 1972. 8.136 Schlünder, E. U., Heat transfer in nucleate pool boiling of mixtures, Proc. 7th Int. Heat Transfer Conf., vol. 4, pp. 2073–2079, 1982. 8.137 Thome, J. R., and Shakir, S., A new correlation for nucleate pool boiling of binary mixtures, AIChE Symp. Ser., vol. 83, pp. 46–51, 1987. 8.138 Wenzel, U., Balzer, F., Jamialahmadi, M., and Müller-Steinhagen, H., Pool boiling heat transfer coefficients for binary mixtures of acetone, isopropanol and water, Heat Transf. Eng., vol. 16, pp. 36–43, 1995. 8.139 Fujita, Y., Bai, Q., and Tsutsui, M., Heat transfer of binary mixtures in nucleate boiling, 2nd European Thermal Sciences and 14th UIT National Heat Transfer Conf., G. P. Celata, P. DiMarco, and A. Mariani (editors), pp. 1639–1646, 1996. 8.140 Kandlikar, S. G., Boiling heat transfer with binary mixtures: Part I – a theoretical model for pool boiling, ASME J. Heat Transf., vol. 120, pp. 380–387, 1998. 8.141 van Stralen, S. J. D., Heat transfer to boiling binary liquid mixtures at atmospheric and subatmospheric pressures, Chem. Eng. Sci., vol. 5, pp. 290–296, 1956. 8.142 van Stralen, S. J. D., Heat transfer to boiling liquid mixtures, Br. Chem. J., Part I, vol. 4, pp. 8–17; Part II, vol. 4, pp. 78–82; Part III, vol. 6, pp. 834–840; Part IV, vol. 7, pp. 90–97, 1959. 8.143 Van Wijk, W. R., Vos, A. S., and van Stralen, S. J. D., Heat transfer to boiling binary liquid mixtures, Chem. Eng. Sci., vol. 5, pp. 68–80, 1956. 8.144 Vos, A. S., and van Stralen, S. J. D., Heat transfer to boiling water-methylethylketone mixtures, Chem. Eng. Sci., vol. 5, pp. 50–56, 1956. 8.145 Came, M., Studies of the critical heat flux for some binary mixtures and their components, Can. J. Chem. Eng., vol. 41, pp. 235–241, 1964. 8.146 Dunskus, T., and Westwater, J. W., The effect of trace additives on the heat transfer to boiling isopropanol, Chem. Eng. Symp. Ser., vol. 57, no. 32, pp. 173–181, 1961. 8.147 Grigoriev, L. N., Khairullin, I. Kh., and Usmanov, A. G., An experimental study of the critical heat flux in the boiling of binary mixtures, Int. Chem Eng., vol. 8, no. 1, pp. 39–42, 1968. 8.148 Jordan, D. P., and Leppert, G., Nucleate boiling characteristics of organic reactor coolants, Nuc. Sci. Eng., vol. 5, pp. 349–359, 1959. 8.149 Sterman, L. S., Vilemas, J. V., and Abramov, A. I., On heat transfer and critical heat flux in organic coolants and their mixtures, Proc. Int. Heat Transfer Conf., Chicago, vol. 4, pp. 258–270, 1966. 8.150 van Stralen, S. J. D., and Sluyter, W. M., Investigations on the critical heat flux of pure liquids and mixtures under various conditions, Int. J. Heat Mass Transf., vol. 12, pp. 1353–1384, 1969. 8.151 Kutateladze, S. S., Brobrovich, G. I., Gogonin, I. I., Mamontova, N. N., and Moskvichova, V. N., The critical heat flux at the pool boiling of some binary liquid mixtures, Proc. 3rd Int. Heat Transf. Conf., Chicago, vol. 3, pp. 149–159, 1966. 8.152 Matorin, A. S., Correlation of experimental data on heat transfer crisis in pool boiling of pure liquids and binary mixtures, Heat Transf. – Sov. Res., vol. 5, pp. 85–89, 1973. 8.153 Gaidarov, S. A., Evaluation of the critical heat flow in the case of a boiling mixture of large volume, J. Appl. Mech. Tech. Phys., vol. 16, pp. 601–603, 1975. 8.154 Stephan, K., and Preusser, P., Heat transfer and critical heat flux in pool boiling of binary and ternary mixtures, Ger. Chem. Engr., vol. 2, pp. 161–169, 1979. 8.155 Yang, Y. M., An estimation of pool boiling critical heat flux for binary mixtures, Proc. 2nd ASME/ JSME Joint Thermal Engineering Conf, Honolulu, Hawaii, vol. 5, pp. 439–446, 1987. 8.156 McEligot, D. M., Generalized peak heat flux for dilute binary mixtures, AlChE J.., vol. 10, pp. 130–131, 1964. 8.157 Reddy, R. P., and Lienhard, J. H., The peak boiling heat flux in saturated ethanol-water mixtures, ASME J. Heat Transf., vol. 111, pp. 480–486, 1989. 8.158 Fujita, Y., Bai, Q., and Tsutsui, M., Critical heat flux of binary mixtures in pool boiling, Proc. ASME/ JSME Thermal Engineering Joint Conf. 1995, vol. 2, pp. 193–200, 1995. 8.159 Hovestreijdt, J., The influence of the surface tension difference on the boiling of mixtures, Chem Eng. Sci., vol. 18, pp. 631–639, 1963. 8.160 Yagov, V. V., Critical heat flux prediction for pool boiling of binary mixtures, Chem. Eng. Res. Des., vol. 82, pp. 457–461, 2004. 8.161 van Ouwerkerk, H. J., Hemispherical bubble growth in a binary mixture, Chem. Eng. Sci., vol. 27, 1957–1960, 1972.

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8.162 Cooper, M. G., The binary microlayer – a double diffusion problem, Chem. Eng. Sci., vol. 37, pp. 27–35, 1982. 8.163 Kautzky, D. E., and Westwater, J. W., Film boiling of a mixture on a horizontal plate, Int. J. Heat Mass Transf., vol. 10, pp. 253–256, 1967. 8.164 Yue, P.-L., and Weber, M. E., Film boiling of saturated binary mixtures, Int. J. Heat Mass Transf., vol. 16, pp. 1877–1888, 1973. 8.165 Yue, P.-L., and Weber, M. E., Minimum film boiling flux of binary mixtures, Trans. Inst. Chem. Eng., vol. 52, pp. 217–222, 1974. 8.166 Marschall, E., and Moresco, L. L., Analysis of binary film boiling. Int. J. Heat Mass Transf., vol. 20, pp. 1013–1018, 1977.

PROBLEMS 8.1 Compute the critical heat flux using Kutateladze’s [8.1] correlation and the correlation of Zuber et al. [8.3] for liquid nitrogen boiling on an infinite horizontal surface (facing upward) at atmospheric pressure for ambient pool subcoolings ranging from 5°C to 40°C and plot the variation of qmax ′′ with subcooling over this range. Where do these two correlations differ the most? 8.2 Determine the critical heat flux for subcooled boiling of R-134a on a large upward-facing flat surface at pressures of 101, 815, and 2926 kPa for a pool subcooling of 20°C. Where does the subcooling have the greatest effect on qmax ′′ ? 8.3 For bulk flow velocities of 0.1, 1.0, and 10 m/s, determine the critical heat flux for saturated water at atmospheric pressure flowing over a flat plate 3.0 cm long in the direction of flow. Repeat the calculations for the same flow rates at pressure of 6124 kPa. At which pressure does increasing the flow velocity have the greatest effect? 8.4 Equation (8.4) was proposed by Haramura and Katto [8.7] for predicting the critical heat flux for flow of a saturated liquid over a finite-sized flat plate. This relation predicts that qmax ′′ will not equal zero, but instead will equal the pool boiling value, as predicted, for example, by the appropriate equation from Table 7.4. Assume that the lateral width of the plate is essentially infinite and its length L in the flow direction is 2.0 cm. (a) Using an appropriate pool boiling equation from Table 7.4, find the flow velocity at which Eq. (8.4) predicts a value of qmax ′′ that equals the pool boiling value for water and liquid nitrogen at atmospheric pressure. (b) What is your interpretation of the values of qmax ′′ predicted by Eq. (8.4) for flow velocities below the values determined in part (a)? 8.5 A small heated cylinder with an outside diameter of 5 mm is immersed in saturated liquid nitrogen in a system at a pressure of 1083 kPa. The liquid nitrogen flows normal to the cylinder axis with a bulk velocity of 50 cm/s. Determine the critical heat flux for the cylinder. What happens to the boiling process if the pressure in the system suddenly drops to atmospheric pressure (101 kPa)? 8.6 Determine and plot the variation of the critical heat flux with jet velocity for a cylindrical jet of saturated liquid R-22 impinging on a heated disk with a diameter of 1.0 cm flush-mounted on an otherwise adiabatic surface. Specifically consider jet velocities between 0.1 and 10 m/s. The pressure in the system is 376 kPa, and the jet diameter is 5 mm. 8.7 It is proposed to cool a flush-mounted electronic chip containing high-temperature superconducting elements with an impinging jet of liquid nitrogen at atmospheric pressure. The chip is modeled as a circular element with a diameter of 1.0 cm dissipating heat uniformly over its surface. To operate properly, this chip must be maintained at a temperature near 80 K while dissipating a heat flux of 80 W/cm 2 . (a) Determine the jet velocity to meet the heat flux requirement if the jet diameter is 2 mm and the impinging liquid is saturated. (b) Using an appropriate nucleate boiling correlation from Chapter Seven, estimate the surface temperature of the element for the specific heat flux condition. (c) If the wall temperature exceeds the design specification of 80 K, propose a modification to the cooling scheme that will make it possible to meet this condition without exceeding the maximum heat flux. 8.8 Estimate the film boiling heat transfer coefficient for film boiling of saturated liquid nitrogen flowing normal to a cylinder with a velocity of 30 cm/s. The cylinder wall is held at a temperature of 110 K and the system is at atmospheric pressure. 8.9 Determine the film boiling heat transfer coefficient for film boiling of liquid nitrogen on a sphere over which liquid nitrogen flows with a bulk velocity of 40 cm/s. The system pressure is 778 kPa, the sphere diameter is 3 cm, and the surface of the sphere is held at 130 K. What happens if the system pressure suddenly drops to atmospheric pressure?

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8.10 Estimate the time to completely vaporize a droplet of saturated liquid nitrogen undergoing Leidenfrost vaporization over a solid surface at a temperature of 20°C. The emissivity of the surface is 0.8, and the emissivity of the liquid interface is 0.95. Does the radiation have a significant effect? Justify your answer. 8.11 Droplets of liquid R-134a sprayed from a leaking refrigeration system at atmospheric pressure fall 2 cm to a copper plate at 20°C. Do you expect the droplets to strike and wet the surface? Justify your answer quantitatively. 8.12 Pure saturated liquid nitrogen at atmospheric pressure is spilled from an insulated vessel, with the result that liquid nitrogen droplets fall onto the upward-facing palm of a lab worker’s hand. The skin temperature is 35°C. Do you expect the droplets to undergo Leidenfrost evaporation, floating on a film of vapor, or do you expect them to wet the skin surface and vaporize by nucleate boiling? Quantitatively justify your answer. 8.13 As part of a metallurgical process, it is proposed that a 2 cm thick stainless steel plate be cooled with a spray of water droplets. The plate, which is heated to 1200 K, is oriented vertically with the droplets sprayed horizontally at the plate. How effective do you expect this spray to be in cooling the plate? Does radiation play a significant role in the process? Justify your answers quantitatively where possible. 8.14 For liquid nitrogen at atmospheric pressure, estimate the Leidenfrost temperature for a solid copper surface. Compare your result to the value of the kinetic limit of superheat predicted by Eq. (5.106). Which is larger? Based on these results, what do expect to observe if liquid nitrogen droplets fall into a container with liquid water in it at 20°C? Explain briefly. 8.15 Use the relation (8.47) proposed by Berenson [8.51] to estimate the Leidenfrost temperatures for acetone, methanol, and R-134a at atmospheric pressure on a solid copper surface. 8.16 Develop an integral analysis of forced-convection-dominated laminar film boiling over a vertical surface exposed to a saturated liquid flowing upward over the surface at a uniform free-stream velocity u∞. Assume that a constant heat flux q″ is dissipated to the flow at the surface and adopt the following idealizations: Neglect downstream convection of momentum and energy compared to transport across the film. Neglect radiation effects. Assume that the interface is smooth. Assume that the upward velocity u at the liquid-vapor interface is equal to the free-stream velocity u∞. From the analysis, derive relations for the temperature and velocity profiles in the film, the variation of the film thickness with downstream location x, and show that the local heat transfer coefficient h is given by



h=

ρv k vu∞ hlv 2 xq′′

8.17 Consider the problem of cooling a circular flush-mounted heated element on a solid surface by boiling of an impinging jet of saturated liquid nitrogen at atmospheric pressure. Use the correlation of Sharan and Lienhard [8.9] to predict the value of the jet velocity that would yield a value of qmax ′′ equal to the maximum possible flux as specified by the curve in Fig. 4.13. What is your opinion of using such a jet velocity in a real system for cooling electronic components? 8.18 Assuming that the A and B values in Eq. (8.76) recommended by Potash and Wayner [8.68] for CCl4 also apply to acetone, estimate the thickness of an equilibrium film of acetone on glass at atmospheric pressure and a superheat of 5°C. 8.19 Example 8.8 considers boiling of a methanol and water liquid mixture on a large flat upward-facing surface at atmospheric pressure. The bulk alcohol mole fraction in the liquid is 0.1. For nucleate boiling in this system, use the correlation of Fujita et al. [8.139] to predict the superheat for 4 heat flux values between 10 and 80 W/cm2. Plot these points to define the boiling curve for this mixture. Then determine the superheat at the same heat flux levels for pure water using the same pure fluid nucleate boiling correlation used in the mixture calculation. Does the presence of the alcohol increase or decrease the heat transfer coefficient? 8.20 Nucleate pool boiling occurs on a flat upward-facing surface in a mixture of liquid oxygen and liquid nitrogen at atmospheric pressure. The concentration of nitrogen in the bulk liquid is 0.2 and the surface temperature is 95 K. Determine the heat flux for this boiling process using the correlation of

Other Aspects of Boiling and Evaporation

389

Fujita et al. [8.139] together with the Stephan-Abdelsalam correlation from Section 7.3 for the pure fluid nucleate boiling coefficients. For this analysis, use Fig. 8.21 for bubble point and dew point information for the mixture and use the property information in Appendix II for pure fluid properties. 8.21 Nucleate pool boiling occurs on a flat upward-facing surface in a mixture of liquid oxygen and liquid nitrogen at atmospheric pressure. The concentration of nitrogen in the bulk liquid is 0.2 and the surface temperature is 95 K. Determine the heat flux for this boiling process using the correlation of Stephan and Kömer [8.134] together with the Stephan-Abdelsalam correlation from Section 7.3 for the pure fluid nucleate boiling coefficients. Use Fig. 8.21 for bubble point and dew point information for the mixture and use the property information in Appendix II for pure fluid properties. For the purposes of this calculation take A = 0.5.

9

External Condensation

9.1  HETEROGENEOUS NUCLEATION IN VAPORS In most applications involving condensation, the process is initiated by removing heat through the walls of the structure containing the vapor to be condensed. If enough heat is removed, the vapor near the wall may be cooled below its equilibrium saturation temperature for the specified system pressure. Since the heat removal process will establish a temperature field in which the temperature is lowest right at the wall of the containment, the formation of a liquid droplet embryo is most likely to occur right on the solid-vapor surface. The formation of a liquid embryo at the interface between a metastable supersaturated vapor and another solid phase is one type of heterogeneous nucleation (see Chapter 5 for further description of the differences between heterogeneous and homogeneous nucleation). As in the case of heterogeneous nucleation of vapor bubbles, the analysis of homogeneous nucleation of liquid droplets can be extended to heterogeneous nucleation at a solid-vapor interface. Since the analysis of the kinetics of the heterogeneous nucleation process is very similar to that described in Chapter 5 for the comparable homogeneous nucleation process, the analysis for heterogeneous nucleation will be only briefly summarized here. If the solid surface is idealized as being perfectly smooth, in general, the shape of a droplet at the surface will be dictated by the shape of the surface itself, the interfacial tension σlv, and the contact angle θ. For a flat solid surface, the embryo liquid droplet will have a profile like that shown in Fig. 9.1. We will specifically consider the heterogeneous nucleation process in which formation of a droplet embryo occurs in a system held at constant temperature Tv and pressure Pv, as shown schematically in Fig. 9.2. If the embryo shape is idealized as being a portion of a sphere, it follows directly from its geometry that the embryo volume Vl and the areas of the liquid-vapor (Alv) and the solidliquid interfaces (Asl) are given by

 πr 3  (2 − 3 cos θ + cos3 θ) (9.1) Vl =   3 



Alv = 2πr 2 (1 − cos θ) (9.2)



Asl = πr 2 (1 − cos2 θ) (9.3)

In the above relations, θ is the liquid contact angle and r is the spherical cap radius indicated in Fig. 9.1. The system shown in Fig. 9.2 initially contains only supersaturated vapor. The initial free energy of the system at Tv and Pv , is therefore given by

G0 = Nˆ T gˆ v (Tv , Pv ) + ( Asv )i σ sv (9.4)

where Nˆ T is the total number of moles in the system and, by definition,

gˆ v (Tv , Pv ) = uˆv − Tv sˆv + Pv vˆv (9.5)

391

392

Liquid-Vapor Phase-Change Phenomena

FIGURE 9.1  An embryo liquid droplet formed at an idealized fluid-solid interface.

After the formation of the embryo, the total free energy for the system is the sum of contributions associated with the bulk liquid, bulk vapor, and the interfaces between the three phases. Accounting for the thermodynamic contributions of each to the free energy, and subtracting the initial free energy of the system, the following relation is obtained for the change in free energy associated with the formation of an embryo of radius r:

∆G = G − G0 (9.6)



= Nˆ l [ gˆl (Tv , Pl ) − gˆ v (Tv , Pv )] − Vl ( Pl − Pv ) + Alv σ lv + Asl (σ sl − σ sv ) (9.7)

where

gˆl (Tv , Pl ) = uˆl − Tv sˆl + Pl vˆl (9.8)

FIGURE 9.2  System considered in the thermodynamic analysis of the formation of an embryo liquid droplet by heterogeneous nucleation.

393

External Condensation

Note that this holds for the formation of an embryo of volume Vl with Nˆ l moles of liquid and Nˆ v moles in the system remaining as vapor, where Nˆ v = Nˆ T − Nˆ l (9.9)



Using the requirement of a tangential force balance along the interline, or minimization of free energy (see Chapter 3), σ sl + σ lv cos θ = σ sv (9.10)



Eqs. (9.1)–(9.3) and Eqs. (9.7) and (9.10) can be combined to obtain the following expression for the change in the free energy associated with formation of the embryo

∆G =

πr 2 πr 2 Fθ [ gˆl (Tv , Pl ) − gˆ v (Tv , Pv )] − Fθ ( Pl − Pv ) + 4 πr 2 σ lv Fθ (9.11) 3vˆl 3

where Fθ =



2 − 3 cos θ + cos3 θ (9.12) 4

If the embryo has exactly the right radius r = re to be in thermodynamic equilibrium with the surrounding vapor, the ĝl and ĝv terms in Eq. (9.11) are equal, the Young-Laplace equation dictates that Pl – Pv = 2σlv/re, and the relation for ΔG = ΔGe becomes ∆Ge =



4 2 πre σ lv Fθ (9.13) 3

In the same manner as for the homogeneous nucleation case (see Chapter 5), ΔG can be expanded in a power series in terms of r – re about the equilibrium radius. The first two terms of the expansion are

∆G =

4 2 πre σ lv Fθ − 4 πσ lv Fθ (r − re )2 +  (9.14) 3

It follows directly from the same arguments presented for the homogeneous nucleation case in Section 5.5 that the equilibrium condition corresponds to a maximum value of ΔG and is therefore an unstable equilibrium. We therefore expect that ΔG increases to a maximum and then decreases with increasing radius r. This once again leads to the conclusion that embryos having a radius less that re spontaneously disappear, while those having a radius greater that re spontaneously grow (see Section 5.5). The above expansion for ΔG is used to determine the kinetic limit of supersaturation in a manner similar to that for the homogeneous nucleation case considered in Section 5.6. The details are virtually identical to those of the homogeneous nucleation analysis presented in Section 5.6, and hence they will not be presented here. There are, however, two important differences in the heterogeneous nucleation analysis. First, as an initial step in the analysis, it is postulated that at equilibrium, the number density of embryos containing n molecules per unit of interface area N″(n) is given by  ∆G (r )  N ′′(n) = ρ2/3 N , v exp  −  (9.15)  k B Tv  where ρN,v is the number of vapor molecules per unit volume, and ΔG(r) is the free energy change for formation of the embryo of radius r previously defined. For the heterogeneous nucleation process

394

Liquid-Vapor Phase-Change Phenomena

considered here, only vapor molecules near the solid surface can participate in embryo droplet formation. To account for this condition, the factor multiplying the exponential term in Eq. (9.15) is taken to be ρ2/3 N , v , which is representative of the number of vapor molecules immediately adjacent to the solid surface per unit of surface area. The second different aspect of the heterogeneous analysis is the relationship between the number of molecules n in the embryo and its radius.

n=

N A πr 3 (2 − 3 cos θ + cos3 θ) (9.16) 3 Mvv

This relation differs from that used in the analysis of homogeneous nucleation because the embryo geometries are different. Analysis of the kinetics of the heterogeneous nucleation process incorporates these two changes and makes use of the expansion for ΔG developed for this case. Otherwise the analysis is identical to that presented in Section 5.6 for homogeneous nucleation. Carrying the analysis to completion yields the following relation between the rate of embryo formation J (m−2s−1) and the system conditions and properties



2σ F N  J =  lv θ A   πM 

1/2

 Pv   RT  v

5/3

 NA   M

2/3

1 − cos θ  vl Fθ    2 

 −16π(σ lv Fθ / k B Tv )3 ( Mvl / N A )2         × exp  2  3 {ln[ Pv / Psat (Tv )]}  

(9.17)

where Fθ is defined by Eq. (9.12). If θ is taken to be 180° and ρ2/3 N , v is replaced by ρN,v , Eq. (9.17) becomes identical to the expression (5.130) obtained in Section 5.6 for homogeneous droplet nucleation. As in the homogeneous nucleation case, J is interpreted as the rate at which embryos of critical size are generated. As J increases, the probability that a bubble will exceed critical size and grow spontaneously becomes greater. If a threshold value of J is specified as corresponding to the onset of nucleation, the corresponding vapor temperature Tv = TSSL for the specified system pressure can be determined from Eq. (9.17). Alternatively, for the specified threshold J value, the limiting supersaturation pressure can be determined for a given system temperature. For water vapor at 100°C, the variation of J with vapor pressure as predicted by Eq. (9.17) is shown in Fig. 9.3 for several values of liquid contact angle. Assuming a fixed threshold value of J would apply for all contact angles, it is clear that the predicted value of (Pv)SSL decreases with decreasing contact angle toward the normal saturation vapor pressure. At a liquid contact angle of 10°, the difference between the predicted (Pv)SSL value and Psat(Tv) is negligible for virtually any threshold value of J between 10 −5 and 1010. Contact angles for virtually all real systems with flat ordinary surfaces lie between zero and about 110°, and for metal surfaces with nonmetallic liquids, the contact angle is often below 50° (see Chapter 3). The results of the above analysis therefore suggest that condensation can be initiated at a solid surface in contact with the vapor at supersaturation levels significantly below those required for homogeneous nucleation, if the liquid phase of the vapor wets the surface reasonably well. As discussed in Chapters 3 and 8, it is quite possible for a thin microfilm of liquid to be adsorbed on all or part of a solid surface. This is particularly true for high-energy surfaces like metals. In addition, when water is the fluid, its polar nature can enhance the tendency of water molecules to attach to portions of the solid surface. (Many oxides and corrosion-produced compounds on metals surfaces are hydrophilic.) Patches of adsorbed liquid molecules on the solid surface can thus serve as nuclei for condensation of the liquid phase when the vapor is supersaturated. Condensation on

External Condensation

395

FIGURE 9.3  Variation of the rate of embryo formation at a water-solid interface with vapor pressure as predicted for different contact angles by analysis of the kinetics of embryo droplet formation.

the surface can begin as the formation of very small droplets on the surface at the sites of these nuclei. This so-called dropwise condensation process is, in fact, commonly observed when water vapor in air condenses on a cold beverage glass. This is usually interpreted as being a direct consequence of the fact that the liquid poorly wets the glass, except at nuclei locations where water molecules have adsorbed to crevices (scratches), or foreign matter (such as dust particles) on the surface. Dropwise condensation is discussed further in the next section.

9.2  DROPWISE CONDENSATION As described in the previous section, dropwise condensation may occur on a solid surface cooled below the saturation temperature of a surrounding vapor when the surface is poorly wetted except at locations where well-wetted contaminant nuclei exist. The poorly wetted surface condition can result from contamination or coating of the surface with a substance which is poorly wetted by the liquid phase of the surrounding vapor. In practice, a poorly wetted surface condition can be achieved for steam condensation by (1) injecting a nonwetting chemical into the vapor which subsequently deposits on the surface, (2) introducing a substance such as a fatty (i.e., oleic) acid or wax onto the solid surface, (3) by permanently coating the surface with a low-surface-energy polymer or a noble metal, or by (4) micro or nanostructuring the surface to increase the apparent contact angle. The effects of the first two methods are generally temporary, since the resulting surface films eventually are dissolved or eroded away. Although this type of process is most commonly encountered in steam condensation, it can occur in condensation of some organic fluids [9.1, 9.2] and liquid metals. The third and fourth methods of promoting dropwise condensation described above are of particular interest because they hold the prospect of providing continuous dropwise condensation in applications. The third method will be discussed in this section, and micro and nanostructuring will be discussed in Section 9.7. Dropwise condensation is generally the preferred mode of condensation because the resulting heat transfer coefficient may be as much as an order of magnitude higher than that for film condensation under comparable circumstances. Studies by Westwater and co-workers [9.3, 9.4] have demonstrated that dropwise condensation of steam can be consistently obtained on gold and silver surfaces. The occurrence of dropwise condensation on gold and silver surfaces would appear to contradict the reasoning that high-surface-energy metal surfaces should be well-wetted by the

396

Liquid-Vapor Phase-Change Phenomena

liquid phase, producing film condensation instead of dropwise condensation. It has been demonstrated in recent experiments [9.5], however, that a gold surface applied under highly controlled ultra-clean conditions will spontaneously wet with liquid water as suggested by the above arguments. Apparently because of its high surface energy, the gold tends to attract and retain organics that render the surface hydrophobic, thereby producing dropwise condensation. The studies of Woodruff and Westwater [9.3] also indicate that the promotion of dropwise condensation on gold-plated surfaces is somewhat affected by the presence of very small amounts of carbon, copper, aluminum, and oxygen in the coating. During dropwise condensation, the condensate is usually observed to appear in the form of droplets that grow on the surface and coalesce with adjacent droplets. When droplets become large enough, they are generally removed from the surface by the action of gravity or drag forces resulting from the motion of the surrounding gas. As the drops roll or fall from the surface they merge with droplets in their path, effectively sweeping the surface clean of droplets. Droplets then begin to grow anew on the freshly exposed solid surface. This sweeping and renewal of the droplet growth process is responsible for the high heat transfer coefficients associated with dropwise condensation. The visual appearance of this process is depicted in Fig. 9.4. Despite numerous studies of dropwise condensation over the years, its mechanism remains the subject of debate. Two different types of models have been proposed. The first model type is based on the premise that droplet formation is a heterogeneous nucleation process like that described in the previous section. Droplet embryos are postulated to form and grow at nucleation sites, while portions of the surface between the growing droplets remain dry. This type of model apparently was first proposed by Eucken [9.6] in 1937. Experimental evidence supporting this physical model of the condensation process has emerged from several experimental investigations. In an early study, McCormick and Baer [9.7] presented experimental results supporting the contention that microscopic droplets are nucleated at active nucleation sites on the cooled surface. These active sites were identified as wetted crevices and grooves in the surface that were repeatedly re-exposed to supersaturated vapor as a result of the coalescence of droplets and their removal from the surface by drag or gravity body forces. Using an optical technique to indicate changes in the thickness of very thin liquid films, Umur and Griffith [9.8] found that at least for low temperature

FIGURE 9.4  Droplet array appearance for dropwise condensation.

External Condensation

397

differences, the area between growing droplets on the surface was, in fact, dry. Their results indicate that no film greater than a monolayer existed between the droplets, and that no condensation took place in those areas. In the second type of dropwise condensation model, it is postulated that condensation occurs initially in a filmwise manner, forming an extremely thin film on the solid surface. As condensation continues, this film eventually reaches a critical thickness, estimated to be about 1 μm, at that point it ruptures and droplets form. Condensation then continues on the surface between the droplets that form when the film ruptures. Condensate produced in these regions is drawn to adjacent drops by surface tension effects. Droplets also grow by direct condensation on the droplet surfaces themselves. This second model of the dropwise condensation process apparently was proposed by Jakob [9.9] as early as 1936. Modified versions of this model have also been proposed by Kast [9.10] and Silver [9.11]. The results of several investigations seem to support this type of interpretation of the condensation process. Results presented by Welch and Westwater [9.12] and Sugawara and Katsuta [9.13] indicate that condensation occurs entirely between droplets on a very thin liquid film. Welch and Westwater [9.12] observed the process by taking high speed movies through a microscope. It appeared to them that droplets large enough to be visible grew mainly by coalescence, leaving behind a “lustrous bare area” that quickly took on a faded appearance. They interpreted the lustrous appearance as corresponding to thin film which, upon rupturing at a thickness of around 1 μm, took on a more dull appearance. In contrast, it is postulated in the first model described above that condensation occurs only on the droplets, and not on the surface between them. The rate of condensation on the larger droplets is less than on the smaller ones because of the higher resistance to heat conduction through larger drops. The large drops therefore grow primarily through coalescence. This model implies that during dropwise condensation, most of the heat transfer is to that portion of the surface covered with the smallest droplets. Detailed modeling of dropwise condensation heat transfer based on the first model hypothesis has, in fact, been attempted by several investigators. These models generally idealize the heat transfer process as indicated in Fig. 9.5. Droplets in a wide range of sizes are expected to exist on the surface. The sizes are designated by the diameter of the droplets D, all of which are taken to be hemispheres. The smallest droplet size possible corresponds to the equilibrium radius of curvature re for the specified wall subcooling Tsat – Tw:

Dmin 2σ = re = (9.18) 2 ( RTw / vl ) ln[ Pv / Psat (Tw )] − Pv + Psat (Tw )

where Tw is the wall temperature. The highest supersaturation exists at the cold surface where the subcooling is greatest. As described in Chapter 5, the equilibrium radius is the smallest droplet that can exist in the supersaturated system without spontaneously collapsing. Thus droplets on the surface cannot have a radius of curvature less than re. If we neglect, Pv – Psat(Tw) compared with 2σ/re, the above equation can be simplified (see Chapter 5) to

Dmin 2σ = re = (9.19) 2 ( RTw / vl ) ln[ Pv / Psat (Tw )]

Combining the Clapeyron equation with ideal gas law for the vapor, integrating between Pv and Psat, and approximating the product Tw Tsat as Tw2 and vlv as vv , the following relation is obtained



 Pv  hlv [Tsat ( Pv ) − Tw ] ln   (9.20) = RTw2  Psat (Tw ) 

398

Liquid-Vapor Phase-Change Phenomena

FIGURE 9.5  Model of heat transfer resistance components for dropwise condensation.

Substituting this relation into Eq. (9.19) yields

rmin =

2 vl σTw Dmin = (re )min = (9.21) 2 hlv (Tsat − Tw )

For larger droplets, the mean conduction path through the droplet from its surface to the solid wall is longer than for small droplets, which implies that the resistance to heat transfer is larger. For very large droplets, the larger resistance results in a heat transfer rate so low that portions of the surface covered by such drops contribute very little to the total heat transfer. It is useful, then, to think of the cold solid surface as consisting of thermally insulated regions under large droplets surrounded by thermally active regions covered with smaller droplets through which virtually all the heat is transferred. Figure 9.5 illustrates such a configuration. For the idealized circumstances in Fig. 9.5, there are four factors that may contribute to the overall resistance to heat flow: (1) interfacial resistance Ri, (2) resistance as a result of loss of driving temperature potential due to droplet interface curvature, (3) resistance associated with conduction of heat through the droplet Rc, and (4) constriction resistance Rs associated with the flow of heat around the thermally inactive portions of the surface under the large droplets. In some cases, a thin film of a promoter material may coat the surface to increase the contact angle and promote dropwise condensation. This may also impose additional resistance to heat flow and should be included as a resistance term in such cases. If the vapor is one component in a mixture, there may also be resistance associated with the mass transport of the condensing vapor to the surface. For the purposes of further discussion here, however, condensation of pure vapors will

399

External Condensation

be considered, and this effect will not arise. We will also assume that promoter resistance is small enough to neglect. Constriction resistance, associated with conduction in the solid, can also play a role. However, its effect is often small in common systems. We will therefore initially neglect its effects, and we will return to it later in this section. If the solid surface is assumed to be held at a constant mean temperature Tw , the overall temperature difference ΔTt = Tsat – Tw between the vapor and the wall must equal the sum of temperature differences associated with the resistances to heat flow described above. The overall temperature difference can thus be written as

∆Tt = ∆Ti + ∆Tcap + ∆Tcon (9.22)

where ΔTi, ΔTcap and ΔTcon are the temperature differences associated with interfacial resistance, capillary depression of the equilibrium saturation temperature, and conduction resistance through the droplet, respectively. In Section 4.6, the following relation for the interfacial heat transfer coefficient hi was derived for small values of the parameter a = qi′′( M / 2 RTv )1/2 / ρv hlv:

 2σˆ  hlv2  M  hi =   2 − σˆ  Tv vl  2πRTv 

1/2

 Pv vlv  1 − 2h  (9.23) lv  

In the above relation, σˆ is the accommodation coefficient (see Chapter 4). The second term in the square brackets is very small compared to one for most systems and can be neglected, reducing this relation to 1/2



 2σˆ  hlv2  M  hi =  (9.24)  2 − σˆ  Tv vl  2πRTv 

From the definition of this coefficient, it follows that the temperature drop associated with the interface resistance ΔTi is given by

∆Ti =

qd (9.25) hi πD 2 / 2

where qd is the heat transfer rate from the droplet. The depression of the equilibrium interface temperature below the normal saturation temperature for a droplet of radius r = D/2 can be estimated by replacing Tsat – Tw by ΔTcap and Dmin by the droplet diameter D in Eq. (9.21). The resulting relation for ΔTcap is

∆Tcap =

4 vl σTw (9.26) hlv D

Combining this relation with Eq. (9.21), the relation for ΔTcap can be written as

∆Tcap =

(Tsat − Tw ) Dmin (9.27) D

where Dmin is given by Eq. (9.21). The contribution of conduction resistance through the droplets has typically been modeled using an approximate model of conduction heat transfer in the droplets. Graham and Griffith [9.14], for example, used this type of approach to argue that the conduction resistance through a liquid droplet

400

Liquid-Vapor Phase-Change Phenomena

from the wall to the liquid-vapor interface is such that the effective temperature drop associated with this resistance is given by ∆Tcon =



qd ( D / 2) (9.28) 4 πkl ( D / 2)2

Substituting Eqs. (9.25), (9.27), and (9.28) into Eq. (9.22) yields the following relation for the total temperature drop for these three resistances

∆Tt = Tsat − Tw =

2qd (Tsat − Tw ) D qd + (9.29) 2 + hi πD Dmin 2πkl D

where hi and Dmin are given by Eqs. (9.24) and (9.21), respectively. Rose and co-workers [9.15–9.20] pioneered the development of this type of model for dropwise condensation heat transfer. In doing so, these investigators constructed a relation for the total temperature drop similar to Eq. (9.29), but they adopted slightly more generalized relations for the conduction and capillary contributions to the overall temperature drop, and the relation they used for the interfacial resistance was slightly different. This yielded the following relation for the overall temperature drop:

∆Tt =

2 vl σTw  K1r K 2 vv Tsat  σˆ + 1  + + rhlv hlv2  σˆ − 1   kl

RTsat 2π

  qd (9.30)

In the model, developed by Rose and co-workers [9.15, 9.20], the droplet base (footprint) radius size distribution was specified in terms of a cumulative distribution function f(r) (in terms of droplet radius) that was defined to be the fraction of the surface area covered by droplets with base radii larger than r. This function was postulated to have the form nr



 r  f (r ) = 1 −  (9.31)  rmax 

This distribution form, with nr = 1/3 is consistent with results of experimental studies of droplet sizes in dropwise condensation (e.g., Rose and Glicksmann [9.16] and Tanasawa and Ochiai [9.21]). The related distribution function A(r) is defined such that A(r)dr is the fraction of the area occupied by droplets with radius values between r and r + dr. It follows from the definition of f that

A(r ) = −

df nr nr −1 = nr (9.32) dr rmax

The heat flux, (heat flow rate per unit area of surface) is equal to the integral of the heat flow for each droplet size multiplied by the fraction of the one square meter area occupied by that size A(r)dr. The integration is over all possible droplet sizes: rmax



q ′′ =

∫q

d

A(r )dr (9.33a)

rmin

Based on a balance between surface tension and gravity forces, LeFevre and Rose [9.15] estimate the maximum droplet radius in Eq. (9.33a) to be 1/2



 σ rmax = K 3   (9.33b)  ρg 

401

External Condensation

where K 3 is a constant of order unity. By definition, the heat transfer coefficient is determined as

hdc =

q ′′ 1 = ∆Tt ∆Tt

rmax

∫q

d

A(r )dr (9.34)

rmin

Solving Eq. (9.30) for qd , substituting the resulting relation and the relation for A(r) specified by Eq. (9.32) into the integral yields the following relation for the heat transfer coefficient:

 n  hdc =  nr  rmax ∆Tt 

rmax

∫(

rmin

( ∆Tt − 2σvl Tsat / rhlv ) K1r kl

+

K 2 vv Tsat hlv2

( ) ˆ 1 σ+ ˆ 1 σ−

RTsat 2π

)

  r nr −1dr (9.35a)

The integral in Eq. (9.35a) was later corrected to account for more recent kinetic theory results and to include a correction for polyatomic molecules (see reference [9.22]). This resulted in the following relation for the heat transfer coefficient

 n  hdc =  nr  rmax ∆Tt 

rmax

∫(

rmin

( ∆Tt − 2σvl Tsat / rhlv ) K1r kl

+

0.627 K 2 vv Tsat 0.664 hlv2

( ) γ c +1 γ c −1

RTsat 2π

)

  r nr −1dr (9.35b)

where γ c is the ratio of specific heats for the vapor. The second term in the numerator of the integrand in Eqs. (9.35a) and (9.35b) is the amount of subcooling required to condense on the convex droplet surface. The first term in the integrand denominator is the conduction resistance of the drop, and the second term accounts for the interface temperature drop associated with molecular transport and accommodation there. The constants in Eqs. (9.35a) and (9.35b) (including K3 with Eq. (9.33b)) to evaluate rmax ) were evaluated to provide a best fit to a large body of steam data (LeFevre and Rose [9.15]). A best fit was provided by the values K1 = 2/3, K2 = 1/2, K3 = 0.4 and nr   = 1/3. In later studies, Stylianou and Rose [9.19] recommended alternative values to provide a best fit to data for dropwise condensation of ethylene glycol, and Niknejad, and Rose [9.23] determined values that best fit data for dropwise condensation of mercury. Variations of this type of model have been developed by Graham and Griffith [9.14], Maa [9.24], and Abu-Orabi [9.25]. While this type of model provides a useful framework for exploring the mechanisms of dropwise condensation, the droplet size distribution must be known for the conditions of interest before computation of the heat transfer coefficient is possible. As a result, determination of this distribution has been the focus of research efforts by a number of investigators. From visual observations, Graham and Griffith [9.14] found that for water condensing on a mirror-smooth copper surface, the droplet size number distribution, N (r ) = (− df /dr )/(πr 2 ) , varied about proportional to D −1.7 for droplet diameters between 10 and 1000 μm. However, they were unable to observe and count droplets with diameters smaller than 10 μm. They also demonstrated that a significant fraction of the heat is transferred by droplets with diameters smaller than 10 μm. The results obtained by Graham and Griffith [9.14] indicated that the success of this type of model requires accurate knowledge of the distribution of droplet sizes at small diameters (less than 10 μm). Investigations by Rose and Glicksman [9.16], Tanaka [9.26], Wu and Maa [9.27], Maa [9.24], Rose [9.20], and Abu-Orabi [9.25] have developed alternate relations for the drop size distribution. The more recent efforts have focused on small droplet sizes. The heat transfer model pioneered by Rose and co-workers [9.15–9.20] does not include constriction effects associated with conduction in the solid wall (due to the flow of heat around large droplets). As noted above, these are small is some systems, but can be important in others. Based on

402

Liquid-Vapor Phase-Change Phenomena

analysis of the associated conduction problem, Mikic [9.28] argued that the constriction resistance to heat flow is given approximately by

 1  Rs =   3πkl 

Dmax

∫

Dmin

nD′′ D 2 dD (9.36) [1 − f ( D)]

where f(D) is the fraction of the surface area covered by droplets having a diameter greater than D. (Note that the radius size distribution (9.31) can be easily converted to a diameter distribution.) If this resistance is considered to be in series with the additional resistance mechanisms considered above, the heat transfer coefficient with constriction effects hdcwc is then given by −1



 1  hdcwc =  + Rs  (9.37)  hdc 

where hdc and Rs are given by Eqs. (9.35) and (9.36), respectively. Mikic [9.28] found that the constriction resistance for stainless steel as the condensing surface was about 84% of the total resistance associated with the dropwise condensation process. For a copper surface his results implied that the contribution of the constriction resistance was about 20% of the total. The effects of constriction resistance, coating layer resistance, and/or solid surface thermal properties on dropwise condensation have also been explored in studies by Aksan and Rose [9.29], Tsuruta and Tanaka [9.30], Tsuruta and Togashi [9.31], Griffith and Lee [9.32], Hannemann and Mikic [9.33], and Hannemann [9.34]. The theoretical model of dropwise condensation heat transfer developed by Tanaka [9.35, 9.26] considers the transient change of the droplet size distribution as a result of growth and coalescence of droplets after an area is swept clear by a departing droplet. From the model of the growth rates of droplets during this transient process, a relation for the heat transfer coefficient was developed. Further information on this model can be found in references [9.26, 9.35]. Yet another approach was adopted in a recent study by Mei, et al. [9.36, 9.37]. These investigators used a fractals-based model to predict the droplet size distribution in steady dropwise condensation, and showed that the resulting number distribution with size agreed with experimental data obtained by Tanasawa and Ochiai [9.21]. This innovative approach provides an interesting alternative perspective on the character of the droplet distribution on the surface during dropwise condensation processes. The discussion above reflects the fact that most recent investigations of dropwise condensation have been based on the premise that condensation occurs primarily on the droplets, with little or no condensation on portions of the surface between droplets. At this point, the absence of condensation between the droplets remains open to question. However, the success of models that ignore this effect suggests that it only weakly affects the heat transfer in many common systems. The mechanism of initial droplet formation (heterogeneous nucleation or film rupture) has also not been completely resolved. As suggested by Collier [9.38], it may be that droplet nucleation dominates at low condensation rates, with the film disruption mechanism taking over at higher condensation rates. The model embodied in Eqs. (9.31)–(9.33) indicates very directly that the heat transfer resistance of a single droplet decreases as its size decreases, and the overall heat transfer coefficient will be higher if the size distribution is skewed to have more smaller droplets instead of fewer larger droplets on the surface. Another viewpoint of this is that keeping the mean droplet size on the surface smaller will enhance dropwise condensation heat transfer. The heat transfer resistance for a droplet of given size and contact angle is more or less fixed by the conduction process for that geometry. This suggests that the best enhancement strategy is to modify the system to better control the liquid inventory to get liquid off the surface and keep the mean droplet size small. The merging process described above that produces large high contact angle droplets that sweep downward by gravity and fall off the surface, serves to accomplish this objective. Surface modifications that further

403

External Condensation

FIGURE 9.6  Comparison of dropwise and filmwise condensation heat transfer data for steam at atmospheric pressure. The lines are approximate best fits to the data.

exploit this enhancement strategy will be discussed in Section 9.7. The noteworthy point here is that in addition to changing wetting conditions, the best enhancement strategies also enhance the liquid removal from the surface, which can reduce the mean droplet size on the surface, and consequently, enhance the heat transfer coefficient. The use of organic coatings to promote dropwise condensation of steam is discussed in some detail by Holden et al. [9.39]. As shown in Fig. 9.6, use of a promoter can cause a transition from filmwise condensation to dropwise condensation. This type of transition has been explored in investigations by Utaka et al. [9.2], Takeyama and Shimizu [9.40], and Stylianou and Rose [9.41]. The effects of varying the system pressure on dropwise condensation are described by O’Bara et al. [9.42] and Tanaka [9.43]. Correlations for the heat transfer coefficient associated with dropwise condensation have been proposed by a number of investigators [9.1, 9.12, 9.44–9.46]. One example is the following correlation proposed by Peterson and Westwater [9.1] for dropwise condensation of steam and ethylene glycol:

Nu = 1.46 × 10 −6 (Re* )−1.63 ∏1.16 Prl0.5 (9.38) k

where

Nu =

2hdc σTsat (9.39) ρl hlv kl (Tsat − Tw )



Re* =



∏k = −

kl (Tsat − Tw ) (9.40) µ l hlv 2σTsat µ l2 hlv

 dσ    (9.41) dT 

404

Liquid-Vapor Phase-Change Phenomena

and Prl is the liquid Prandtl number. This modified version of the correlation proposed by Isachenko [9.46] was recommended for 1.75 ≤ Prl ≤ 23.6, 7.8 × 10 −4 ≤ Πk ≤ 2.65 × 10 −2, and 2 × 10 −4 ≤ Re* ≤ 3 × 10 −2. While correlations of this type can be made to agree quite well with data for a specific surface and fluid combination, their general applicability has not been demonstrated. For dropwise condensation of steam at pressures below one atmosphere, Rose et al. [9.47] recommended the following empirical correlation for the heat transfer coefficient hdc = Tv0.8 [5 + 0.3(Tsat − Tw )] (9.42)



Note that Eq. (9.42) is dimensional. Tv is the saturated vapor temperature in Celsius, Tsat – Tw is the wall subcooling in Kelvin, and the heat transfer coefficient hdc is determined in kW/m2K. This dimensional correlation was found to be a fairly good match to data for dropwise condensation of steam from a variety of studies. Additional information on correlations for dropwise condensation is provided in the review article by Merte [9.48]. Further discussion of the foundation concepts and recent developments associated with dropwise condensation can be found in the review publications by Rose [9.47, 9.49–9.51]. The prior research cited above indicates that promoting dropwise condensation can significantly enhanced condensation heat transfer, and promotion strategies must have a net positive effect on nucleation, liquid wetting, droplet size distribution, and liquid removal mechanisms. Strategies to promote dropwise condensation are discussed further in Section 9.7 Example 9.1 In Fig. 9.6, for a surface subcooling of 8 K the heat transfer coefficient for dropwise condensation of steam at atmospheric pressure is indicated to be about 4.0 × 105 W/m2K. Assuming the steam is saturated, compare the prediction of Rose’s [9.47] correlation (9.42) to the value indicated in Fig. 9.6 for these conditions. Assuming the vapor is saturated, the vapor temperature Tv is equal to Tsat = 100°C. Substituting this value and Tsat – Tw = 8 K in Eq. (9.42) yields

hdc = Tv0.8[5 + 0.3(Tsat − Tw )] = (100)0.8[5 + 0.3(8)] = 295 kW/m 2K = 2.95 × 105 W/m 2K

The value predicted by Rose’s correlation differs slightly from the value of 4.0 × 105 W/m2K indicated by the data in Fig. 9.6.

9.3  FILM CONDENSATION ON A FLAT, VERTICAL SURFACE If the liquid phase fully wets a cold surface in contact with a vapor near saturation conditions, the conversion of vapor to liquid will take the form of film condensation. As the name implies, the condensation takes place at the interface of a liquid film covering the solid surface. Because the latent heat of vaporization must be removed at the interface to sustain the process, the rate of condensation is directly linked to the rate at which heat is transported across the liquid film from the interface to the surface.

Integral Analysis of Laminar Film Condensation The classic Nusselt integral analysis [9.52] of laminar film condensation on a vertical surface considers the physical circumstances shown in Fig. 9.7. The surface exposed to a motionless ambient of saturated vapor is taken to be isothermal with a temperature below the saturation temperature. Note that although a vertical surface is considered here, the analysis is identical for an inclined surface, except that the gravitational acceleration g is replaced by g sin Ω, where Ω is the angle between the

405

External Condensation

FIGURE 9.7  Model system for analysis of falling-film condensation on a vertical surface.

surface and the horizontal. Because the liquid film flows down the surface due to gravity, this situation is sometimes referred to as falling-film condensation. A simple force balance on the shaded film element in Fig. 9.7 requires that

 du  (δ − y)dx (ρl − ρv ) g = µ l   dx (9.43)  dy 

Note that this force balance includes the effects of gravity body forces and viscosity, but neglects inertia effects. Integrating this equation from y = 0 to y = δ and using the fact that u = 0 at y = 0, we obtain the following relation for the velocity profile

u=

(ρl − ρv ) g  y2  yδ −  (9.44)  µl 2 

Integrating this velocity across the film yields the following relation for the mass flow rate per unit width of surface m ′ δ



∫

m ′ = ρl udy = 0

ρl (ρl − ρv ) gδ 3 (9.45) 3µ l

Differentiating the above relation with respect to δ yields

dm ′ ρl (ρl − ρv ) gδ 2 = (9.46) dδ µl

Assuming heat transfer across the film is mainly due to conduction, convective effects are neglected, whereupon the heat flux to the interface is given by

q ′′ =

kl ∆T , δ

∆T = Tsat − Tw (9.47)

406

Liquid-Vapor Phase-Change Phenomena

This implies that the heat flow per unit width of surface dq across the differential element shown in Fig. 9.7 is given by dq =



kl ∆T dx (9.48) δ

Ignoring sensible subcooling of the liquid film compared to latent heat effects, conservation of mass and energy dictates that dq = hlv dm ′ (9.49)



Combining Eqs. (9.46), (9.48), and (9.49), the following differential equation is obtained for δ: dδ kl µ l ∆T = (9.50) dx ρl (ρl − ρv ) ghlv δ 3



Integrating the above equation using the condition that δ = 0 at x = 0 yields 1/4

 4 kl µ l x∆T  δ=  (9.51)  ρl (ρl − ρv ) ghlv 



Since heat transfer across the film is by conduction alone, the local heat transfer coefficient is given by hl = kl/δ. It follows from Eq. (9.51) that the local Nusselt number is given by 1/4

Nu x =



hl x  ρl (ρl − ρv ) ghlv x 3  =  (9.52) kl  4 kl µ l (Tsat − Tw ) 

We define a mean heat transfer coefficient as xe

hl =



1 hl ( x )dx (9.53) xe

∫ 0

Using Eq. (9.52) to execute the integration reveals that hl is related to the local liquid flow rate in the film per unit width of surface m ′ as 1/3



 hl  µ l2  kl  ρl (ρl − ρv ) g 

= 1.47 Re −L1/3 ,

Re L =

4 m ′ (9.54) µl

The relation for the mean Nusselt number can also be written in the form 1/4

Nu x =



 ρ (ρ − ρv ) ghlv x 3  hl x = 0.943  l l  (9.55) kl  kl µ l (Tsat − Tw ) 

The classic Nusselt analysis described above incorporates the following idealizations: 1. Laminar flow. 2. Constant properties.

407

External Condensation

3. Subcooling of liquid is negligible in the energy balance. 4. Inertia effects are negligible in the momentum balance. 5. The vapor is stationary and exerts no drag. 6. The liquid-vapor interface is smooth. 7. Heat transfer across film is only by conduction (convection is neglected). Modified versions of this analysis have been subsequently developed that relax many of these assumptions. For example, idealization 3 can be relaxed by including the effect of liquid subcooling in the energy balance δ

d m ′ d dq kl ∆T = = hlv + ρl c pl u(Tsat − T )dy (9.56) dx dx dx δ

∫



0

If we use the velocity profile given by Eq. (9.44) and the linear temperature profile, Tsat − T y = 1 − (9.57) Tsat − Tw δ



to evaluate the integral in Eq. (9.56), the energy balance can be written as d m ′ kl ∆T = hlv* (9.58) dx δ

where

 3  c pl (Tsat − Tw )   hlv* = hlv 1 +    (9.59) hlv   8



Equation (9.58) is identical to the original energy balance given by Eqs. (9.48) and (9.49) in the original analysis, except that hlv has been replaced by hlv* . Consequently, the effect of liquid subcooling can be incorporated by simply replacing hlv with hlv* in Eqs. (9.52) and (9.55) for the heat transfer coefficient. More detailed integral analyses of laminar film condensation on a vertical surface have been developed by Koh [9.53] and Rohsenow [9.54]. The analysis of Rohsenow [9.54] indicated that the effect of subcooling and energy convection can be incorporated by replacing hlv in the above relations by hlv′ given by   c pl (Tsat − Tw )   hlv′ = hlv 1 + 0.68    (9.60) hlv   



Example 9.2 Use Eq. (9.55) to predict the mean heat transfer coefficient for film condensation of steam at atmospheric pressure on a vertical flat plate. The plate, which is 10 cm high, is held at a uniform temperature of 80°C. Compare the resulting hl with the dropwise condensation result from Example 9.1. Equation (9.55) can be written as 1/ 4



3  k   ρ (ρ − ρv )ghlv x  hl = 0.943  l   l l  x   kl µ l (Tsat − Tw ) 

408

Liquid-Vapor Phase-Change Phenomena

For water at atmospheric pressure, ρl = 958 kg/m3, ρv = 0.597 kg/m3, hlv = 2257 kJ/kg, kl = 0.679 W/mK, and μl = 2.78 × 10 −4 Ns/m2. Substituting in the above equation, 1/ 4



3  0.679   958(958 − 0.597)(9.8)(2, 257,000)(0.1)  hl = 0.943      −4  0.10   0.679(2.78 × 10 )(100 − 80) 

= 9.75 × 103 W/m 2K This value for the film condensation heat transfer coefficient is about a factor of 2 smaller than the predicted dropwise heat transfer coefficient.

Boundary-layer Analysis of Film Condensation Laminar film condensation on a vertical surface can also be analyzed with a full boundary-layer formulation. In terms of the Cartesian coordinates and the corresponding velocities shown in Fig. 9.8, the governing two-dimensional boundary-layer equations are ∂u ∂ v + = 0 (9.61) ∂x ∂ y







u

 ∂2 u  g(ρl − ρv ) ∂u ∂u +v = vl  2  + (9.62) ∂x ∂y ρl  ∂y 

u

 ∂2 T  ∂T ∂T +v = α T ,l  2  (9.63) ∂x ∂y  ∂y 

where vl and αT,l are the kinematic viscosity and thermal diffusivity of the liquid, respectively. In this analysis, shear exerted by the vapor on the falling liquid film will be neglected. The temperature in the vapor is assumed to be equal to Tsat(Pυ) everywhere since the vapor in the ambient is at the saturation temperature.

FIGURE 9.8  System model for similarity analysis of film condensation.

409

External Condensation

The boundary conditions at the cooled wall (y = 0), and at the liquid-vapor interface (y = δ) are



at y = 0: u = v = 0, T = Tw (9.64) ∂u = 0, T = Tsat ( Pv ) (9.65) ∂y

at y = δ:

At the wall, the boundary conditions are a consequence of the no-slip condition and the isothermal wall specification. The relations (9.65), which apply for y = δ, specify continuity of the temperature profile across the interface and a negligible shear stress exerted by the surrounding vapor on the liquid film. Following the analysis presented by Sparrow and Gregg [9.55], these equations and boundary conditions can be cast into a similarity formulation. Defining a stream function in the liquid film so that u=



∂ψ ∂ψ , v=− (9.66) ∂x ∂y

it follows that the continuity relation (9.61) is automatically satisfied. The following similarity transformation is then imposed

η = Cl yx −1/4 (9.67)



ψ = 4α T ,l Cl x 3/4 f ( η) (9.68) θ( η) =



Tsat − T (9.69) Tsat − Tw

where 1/4



 gc pl (ρl − ρv )  Cl =   (9.70) 4 vl kl  

The velocity components are related to the similarity variables as

u = 4Cl2α T , l x1/2 f ′( η) (9.71)



v = Cl α T ,l x −1/4 [ ηf ′( η) − 3 f ( η)] (9.72)

where the primes denote differentiation with respect to η. In terms of the similarity variables, the governing momentum and energy equations and associated boundary conditions become

f ′′′ +

1 [3 ff ′′ − 2( f ′)2 ] + 1 = 0 (9.73) Prl



θ′′ + 3 fθ′ = 0 (9.74)



f (0) = f ′(0) = 0, θ(0) = 1 (9.75)



f ′′( ηδ ) = θ( ηδ ) = 0 (9.76)

410

Liquid-Vapor Phase-Change Phenomena

The mathematical problem posed by Eqs. (9.73) and (9.74) with boundary conditions (9.75) and (9.76) is a fifth-order system of nonlinear ordinary differential equations with five boundary conditions. The system is therefore closed if we can specify the location of the interface y = δ. However, the location of the interface is not known a priori. The interface location is dictated by the transport, and it therefore must be determined as part of the solution process. The additional relation needed to allow determination of the interface location is obtained from an energy balance over a segment of the film. If we neglect sensible cooling of the condensate, this balance can be written as x

∫



0

δ   ∂T    kl     dx = ρl uhlv dy (9.77)    ∂ y  y =δ  0

∫

In terms of the similarity variables defined above, Eq. (9.77) can be written as −



c pl (Tsat − Tw ) 3 f ( ηδ ) = Ja l = (9.78) θ′( ηδ ) hlv

where, as before, ηδ is η evaluated at y = δ. The problem is now completely closed mathematically. If a value of ηδ is assumed, Eqs. (9.73) and (9.74) can be solved numerically using boundary conditions (9.75) and (9.76). Using the results of the computed solution and Eq. (9.78), the corresponding value of Jal = cpl(Tsat – Tw)/hlv, can be determined. Thus, ηδ is a function of Jal. Also, if we begin with the relation h=



 ∂T  q ′′ kl = (9.79) Tsat − Tw Tsat − Tw  ∂ y  y = 0

applying the similarity transformation to the temperature gradient term yields the relation 1/4



Nu x =

 gρl (ρl − ρv ) x 3 hlv  hx = [−θ′(0)] Ja 1/4 l   (9.80) kl  4 kl µ l (Tsat − Tw ) 

Thus, for a given set of physical conditions, the value of θ′(0) can be computed using the scheme described above and then the variation of the Nusselt number or heat transfer coefficient along the surface can be determined from Eq. (9.80). Note that this relation is the same form as that obtained from the Nusselt analysis except that the term in the large square brackets now has the prefactor [−θ′(0)] Ja 1/4 l . The variation of the Nusselt number with the system parameters, as determined by the similarity solution calculations of Sparrow and Gregg [9.55] is indicated in Fig. 9.9. These results indicate that for Prl ≥ 100, the variation of the Nusselt number is identical to that predicted by an integral analysis neglecting the inertia terms. For such conditions, the variation of θ′(0) is closely approximated by the relation

−θ′(0) = (0.68 + Ja l−1 )1/4

Substituting this result into Eq. (9.80), the resulting relation for the local Nusselt number can be written as

1/4

 gρ (ρ − ρv ) x 3 hlv′  Nu x =  l l   4 kl µ l (Tsat − Tw ) 

where hlv′ = (1 + 0.68Ja l )hlv (9.81)

411

External Condensation

FIGURE 9.9  Nusselt number dependence on subcooling and fluid properties determined from similarity solution calculations. (Adapted from reference [9.55] with permission, copyright © 1959, American Society of Mechanical Engineers.)

In this form, it is clear that the similarity solution prediction for Prl → ∞ is virtually identical to the correlation obtained by Rohsenow [9.54] based on an integral analysis which included subcooling and energy convection, but neglected inertia terms. Sadasivan and Lienhard [9.56] showed that the similarity solution values for Nux could be accurately predicted for all likely vales of Prl and Jal using Eq. (9.81) with

hlv′ = hlv (1 + Cc Jal ) (9.82)



Cc = 0.683 −

0.228 (9.83) Prl

Equation (9.83) for Cc was developed as a best fit to the variation of Cc with Prl inferred from results computed numerically using the similarity analysis of Sparrow and Gregg [9.55]. The analyses described above predict the main features of laminar film condensation on a vertical or inclined surface reasonably well. There are, however, two physical mechanisms not included in these analyses which can significantly affect the transport: (1) the effects of waves on the liquidvapor interface and (2) interfacial vapor drag on the interface. The effects of interfacial vapor drag were examined analytically by Koh et al. [9.57]. These investigators presented a similarity solution of the governing boundary-layer transport equations in the liquid film and surrounding vapor that was basically an extension of the similarity solution of Sparrow and Gregg [9.55] described above. The results of Koh et al. [9.57] indicate that interfacial shear has very little effect on heat transfer for large liquid Prandtl numbers (≥ 10). For Prl = 1.0, the effect is less than 10% for most practical circumstances. On the other hand, for very small Prl values characteristic of liquid metals, the vapor drag has a strong effect, significantly reducing the heat transfer below that predicted when the interfacial shear is neglected. The reduction of the heat transfer data for film condensation of liquid metals below that predicted by the classical Nusselt theory may also be due, at least in part, to interfacial resistance [9.58] (see Section 4.6).

412

Liquid-Vapor Phase-Change Phenomena

FIGURE 9.10  Data for film condensation of steam on a vertical surface obtained by Spencer and Ibele [9.60].

The effects of surface waves on laminar film condensation are more difficult to incorporate into theoretical analyses. It was shown in Chapter 4 that a falling film on an inclined (or, in the limit, vertical) wall is unstable at any finite film Reynolds number. However, the disturbance amplification rate increases rapidly as the film Reynolds number increases, with the result that there typically exists a quasi-critical Reynolds number beyond which waves of finite amplitude are so highly probable that they invariably are found. Based on experimental observation, surface waves are expected when Re L = 4 m ′ / µ l is greater than 33 [9.59]. In general, interfacial waves are expected to enhance convective heat transport in the film since it intermittently thins the film, increases the interfacial area, and induces mixing. Because of these effects, laminar film condensation heat transfer data are often significantly higher than the values predicted by Eq. (9.54) or Eq. (9.55). In some cases, however, the data have also been found to be significantly lower than the values predicted by these relations, as indicated in Fig. 9.10 from a study by Spencer and Ibele [9.60]. Deviations of ±50% from the prediction of the Nusselt relation (9.54) or (9.55) for data reported in the literature are not uncommon. The results of the above laminar analyses for falling film condensation on a flat vertical surface can also be applied to condensation on the exterior of a vertical tube if the liquid film thickness is small relative to the tube diameter. However, for such circumstances, McAdams [9.61] recommends that the coefficient on the right side of Eqs. (9.54) and (9.55) be changed to 1.88 and 1.13, respectively, to better match available data.

Turbulent Film Condensation As for any boundary-layer flow, when the film Reynolds number Re L = 4 m ′ / µ l becomes large enough, it is expected that a transition to turbulent flow will occur. In general, there may exist a regime of laminar flow over a portion of the surface 0 < x ≤ xL near the leading edge of the film flow, with turbulent flow for x > xL, as indicated in Fig. 9.11. The length of the laminar regime compared to the overall surface length dictates whether the flow is all laminar, partially laminar and partially turbulent, or mostly turbulent. In the turbulent regime, the governing two-dimensional boundary-layer forms of the mass, momentum, and energy balance equations for the film in terms of mean velocity and temperature values can be written as

∂u ∂ v + = 0 (9.84) ∂x ∂ y

413

External Condensation

FIGURE 9.11  System model for analysis of turbulent film condensation.





u

∂u ∂u ∂  ∂u  g(ρl − ρv ) +v = ( vl + ε M )  + (9.85)  ∂x ∂y ∂y  ∂y  ρl u

∂T ∂T ∂  ∂T  +v = (α T ,l + ε H ) (9.86) ∂x ∂ y ∂ y  ∂ y 

where εM and εH are the eddy diffusivities of momentum and heat, respectively. The other variables are identical to those for the laminar case described above. Neglecting interfacial shear effects and assuming the surrounding vapor is saturated, the corresponding boundary conditions are

at y = 0: u = v = 0, T = Tw (9.87) ∂u = 0, T = Tsat ( Pv ) (9.88) ∂y

at y = δ:

The convection terms in the energy and momentum equations are often neglected because the transport rate across the film is much greater than downstream convection. Ignoring the downstream (x) variation in the u and temperature fields compared to the cross-stream variation, the energy and momentum equations simplify to

∂  ∂T  = 0 (9.89) (α T ,l + ε H ) ∂ y  ∂ y 



∂  ∂u  g(ρl − ρv ) = 0 (9.90) ( vl + ε M )  + ∂ y  ∂y  ρl

Integrating across the film and applying appropriate mass and energy balances at the interface, the continuity equation becomes δ



q ′′y =δ d u dy − = 0 (9.91) dx ρl hlv

∫ 0

414

Liquid-Vapor Phase-Change Phenomena

In a similar fashion, integrating the energy Eq. (9.89) first from 0 to y and then from 0 to δ yields Tsat − Tw kl = = qw′′ / kl h



δ

dy

∫ 1 + (Pr / Pr )(ε l

0

t

M

/ vl )

(9.92)

where h is the local heat transfer coefficient and Prt is the turbulent Prandtl number defined as Prl = εM/εH. Before the above system of Eqs. (9.90)–(9.92) can be analyzed further, some means of determining the eddy diffusivities must be specified. If the variations of the turbulent Prandtl number and εM with y are known, the heat transfer coefficient h at a given x location can be determined by the following sequence of steps: 1. The velocity profile u(y) can be determined by integration of Eq. (9.90). 2. Using the velocity profile determined in step 1, the mass flow rate per unit width of surface is determined as δ

∫

m ′ = ρl u dy (9.93)



0

3. Differentiating the expression for m ′ obtained in step 2 with respect to δ, a relation for dm ′ / dδ is obtained. 4. An energy balance at the interface requires that

hlv

d m ′ = h(Tsat − Tw ) (9.94) dx

This equation is solved for dm ′ / dx. 5. Executing the integration in Eq. (9.92) yields a relation for the heat transfer coefficient h as a function of δ. 6. The relations for dm ′ /dδ , dm ′/dx, and h(δ) obtained in steps 3–5 are combined to obtain a relation for dδ/dx. The value of δ for a given x location is obtained by integrating the resulting relation for dδ/dx. 7. Substituting the value of δ obtained in step 6 into the relation for h(δ) obtained in step 5, the value of the local heat transfer coefficient h for a given x location is obtained. Seban [9.62] apparently was the first to propose using known information about fully developed turbulent pipe flows to evaluate the eddy diffusivities in the falling-film governing equations. He postulated that turbulent flow in the film consisted of a viscous sublayer near the wall and a fully turbulent region farther away, as indicated in Fig. 9.12, with a buffer layer between. Using the reduced momentum equation (9.90), it can easily be shown that the shear stress must vary linearly from a finite value τw = ρlgδ at the wall to zero at the interface (Fig. 9.12).

y τ = τ w  1 −  (9.95)  δ

Using Eq. (9.95) for the shear stress together with the universal velocity profiles for turbulent flow,

0 ≤ y + ≤ 5: u + = y + (9.96a)

415

External Condensation

FIGURE 9.12  Temperature and shear stress distributions in a turbulent falling liquid film.



5 ≤ y + ≤ 30: u + = −3.05 + 5 ln y + (9.96b)



30 < y + : u + = 5.5 + 2.5 ln y + (9.96c)



y+ =

y vl

τw , ρl

u+ = u

ρl τw

(9.96d)

and the definition of the eddy diffusivity

εM =

τ / ρl − vl (9.97) ∂u / ∂ y

Seban [9.62] obtained the following relations for the eddy diffusivity

0 ≤ y + ≤ 5: ε M = 0 (9.98a)



5 ≤ y + ≤ 30: ε M =

y gδ   1 −  y − vl (9.98b) δ 5



30 < y + : ε M =

y gδ   1 −  y (9.98c) δ 2.5

416

Liquid-Vapor Phase-Change Phenomena

In the fully turbulent regime 30 < y+, the molecular diffusivity has been neglected compared to the turbulent eddy diffusivity. Assuming that the viscous sublayer and buffer regions are thin, the total mass flow per unit length can be computed using the velocity profile in the fully turbulent range. Following steps 1–4, Seban [9.62] obtained the following relation for dm ′ / dδ: dm ′ = µ l (5.5 + 2.5 ln δ + ) (9.99) dδ +



The integral relation (9.92) for the heat transfer coefficient can then be written in terms of y+ and broken down into three parts ρl c pl gδ = h

5

∫ 0

dy + + (1 / Prl ) + (ε M / Prt vl )

30

∫ 5

dy + (1 / Prl ) + (ε M / Prt vl ) (9.100)

δ+

dy +                       + (1 / Prl ) + (ε M / Prt vl )

∫

30

Seban [9.62] took Prt = 1 and used the appropriate relation for εM to evaluate each of the integrals in (9.100). The resulting relation for the heat transfer coefficient is h  v2  kl  g 



1/3

= Prl (δ + )1/3 [5 Prl + 5 ln(5 Prl + 1) + I *]−1 (9.101)

where

I* =

2.5  (1 + S )(60 / δ + ) − 1 − S  , S  (1 − S )(60 / δ + ) − 1 + S 

S = 1+

10 Prl δ +

(9.102)

Following step 6, Eqs. (9.90), (9.95), and (9.97) are combined to obtain the differential equation

d δ+ kl Prl (δ + g / vl2 )1/3 (Tsat − Tw ) = (9.103) dx µ l hlv (5.5 + 2.5 ln δ + )[5 Prl + 5 ln(5 Prl + 1) + I *]

Integrating from the point of transition to turbulence (xL and δ +L at Re L = 4 m ′ / µ l = 1600 ) to any location of interest (x and δ+), the following implicit relation for δ+ is obtained δ+

∫

δ +L



µ l hlv (5.5 + 2.5 ln δ + )[5 Prl + 5 ln(5 Prl + 1) + I *] + dδ (δ + )1/3  g = 2  vl 

1/3

 c pl (Tsat − Tw )    (x − xL ) hlv  

(9.104)

Finally, in step 7, upon integrating Eq. (9.104), the resulting value of δ+ can be substituted into Eq. (9.101) to determine the local value of h at the specified x location. Note that to do the integration indicated in Eq. (9.104), we must first use the laminar solution to get δL and δ +L for ReL = 1600. Although subsequent investigations by Dukler [9.63] and Lee [9.64] refined this method of predicting the heat transfer during falling film condensation on a flat vertical surface, the structure of the methods, as indicated in steps 1–7 above, was basically unchanged from that in Seban’s [9.62]

417

External Condensation

original study. This methodology was also subsequently extended to evaporation of a falling liquid film by Dukler [9.63], Kunz and Yerazunis [9.65], Chun and Seban [9.66], Mills and Chung [9.67], and Mostofizadeh and Stephan [9.68]. More recent studies (see, e.g., Mills and Chung [9.67]) have suggested that the presence of the interface tends to damp larger turbulent eddies near the interface in the liquid film. This implies that a viscous sublayer exists at the interface as well as at the wall. Recent efforts to model falling film evaporation and condensation processes by Kutateladze [9.69] and Sandall et al. [9.70] have therefore included a variation of the eddy viscosity in which it goes to zero at both the wall and the interface. Kutateladze [9.69] used the following variation of εM with y+: 0 < y + < 6.8: ε M = 0 (9.105a)



6.8 ≤ y + ≤ 6.8 + 0.2(δ + − 6.8): ε M = 0.4 ( y + − 6.8) 1 −

y+ (9.105b) δ+



6.8 + 0.2(δ + − 6.8) < y + < δ + : ε M = 0.08( y + − 6.8) 1 −

y+ (9.105c) δ+

The variation of the mean Nusselt number with ReL and Prl predicted using these equations in the analysis described above is shown in Fig. 9.13. The variation for fully turbulent flow is qualitatively similar to that predicted by the following empirical correlation recommended by Colburn [9.71] h  vl2  kl  g 



1/3

= 0.056 Prl1/3 Re 0.2 L (9.106)

The dotted lines in Fig. 9.13 indicate the interpolated variation predicted by the relation

Nu =

h  v2  kl  g 

1/3

 Re   Re − Re L ,trans  = (Nu)lam  L ,trans  + (Nu) turb  L  (9.107)  Re L   Re L

where the transition Reynolds number ReL,trans is taken to be 2000 and (Nu)lam is the value predicted by the laminar Nusselt correlation (9.54) at ReL,trans.

FIGURE 9.13  Variation of the Nusselt number with Reynolds and Prandtl numbers as predicted by the analysis of Kutateladze [9.69] for turbulent film condensation. (Adapted with permission from reference [9.69], copyright © 1982, Pergamon Press.)

418

Liquid-Vapor Phase-Change Phenomena

Based on experimental data for liquid Prandtl numbers between l and 5, Grober et al. [9.72] proposed a film condensation heat transfer correlation that can be cast in the form h  vl2  kl  g 



1/3

= 0.0131Re1/3 L (9.108)

This relation was recommended at ReL values above the transition value which was taken to be ReL = 1400. Example 9.3 Estimate the mean heat transfer coefficient using Eqs. (9.54), (9.106), and (9.107) for condensation of stream at atmospheric pressure on a 1.0 m by 1.0 m vertical plate with a heat removal rate of 500 kW/m2. For saturated water at atmospheric pressure, Tsat = 100°C, ρl = 958 kg/m3, ρv = 0.597 kg/m3, hlv = 2257 kJ/kg, kl = 0.679 W/mK, Prl = 1.72, and μl = 2.78 × 10 −4 Ns/m2. It follows that

 L′ = m



Re L =

q′′L (5 × 105 )(1.0) = = 0.222 kg/ms hlv 2, 257,000  L′ 4m 4(0.222) = = 3,187 2.78 × 10 −4 µl

Equation (9.54) can be written approximately as

Nulam =

hl kl

1/ 3

 µ l2   2   ρl g 

= 1.47Re L−1/ 3

Substituting for ReL,trans = 2000,

Nulam = 1.47(2000)−1/ 3 = 0.117

Using Eq. (9.106) for turbulent flow,

Nuturb = 0.056Prl1/ 3 Re L0.2 = 0.056(1.72)1/ 3 (3187)0.2 = 0.337

The overall Nusselt number is given by Eq. (9.107):  Re − 2000   2000  Nu = (Nu)lam  + (Nu)turb  L   Re L   Re L

 3187 − 2000   2000  = (0.117)  + (0.337)     3187  3187 = 0.199

It follows that the mean heat transfer coefficient is given by  gρ2  h = kl  2l   µl 

1/ 3

Nu 1/ 3



2   958   = (0.679) 9.8    −4   2.78 × 10  

= 6596 W/m 2K = 6.60 kW/m 2K

(0.199)

419

External Condensation

The analysis tools and correlations described above work reasonably well for Prl values above 1. However, deviation of the predictions using these methods from heat transfer data for liquid metals can be quite significant. Generally, Dukler’s [9.63] somewhat idealized treatment for film condensation has been found to agree reasonably well with falling-film condensation heat transfer data for liquid metals. On the other hand, computed results obtained using Lee’s [9.64] more detailed analysis were found to deviate more from available liquid metal data. Further experimental and theoretical investigation appears to be necessary to resolve the differences between trends predicted by analytical models of turbulent falling-film condensation heat transfer and the current body of experimental data for liquid metals.

9.4  FILM CONDENSATION ON CYLINDERS AND AXISYMMETRIC BODIES Because of its importance to the design of tube and shell condensers [9.73], condensation on the outside of horizontal tubes has been the subject of numerous studies. The length of the tube perimeter over which the condensate flows is usually small for commonly-used tubes. Consequently, the film Reynolds number is usually low and the flow in the liquid film is laminar. With slight modification, the Nusselt [9.52] analysis of laminar falling-film condensation over a flat plate can be adapted to film condensation on an isothermal horizontal cylinder, as shown in Fig. 9.14. The analysis is, in fact, very similar to that for laminar film boiling on a horizontal cylinder discussed in Section 7.6. (Historically, however, Nusselt’s [9.52] analysis of film condensation preceded the comparable analysis for film boiling.) As for vertical surfaces, if convective terms are neglected, an energy balance on a differential element of the film requires that

hlv

d m ′ kl [Tsat ( Pv ) − Tw ] = (9.109) dx δ

The component of the gravitational body force acting tangential to the surface of the cylinder is g sin Ω. Assuming that the momentum balance is dominated by viscous and gravity body forces, Eq. (9.43) again applies, except that g is replaced by g sin Ω. Integrating this equation and using the boundary condition u = 0 at y = 0 yields

u=

(ρl − ρv ) g sin Ω  y2  y δ − (9.110)  µl 2 

FIGURE 9.14  Model used to analyze film condensation on a horizontal cylinder.

420

Liquid-Vapor Phase-Change Phenomena

The condensate mass flow rate per unit of tube length m ′ is obtained, as before, by integrating the above velocity profile δ

∫

m ′ = ρl udy =



0

ρl (ρl − ρv )δ 3 g sin Ω (9.111) 3µ l

Using the fact that Ω = x/R and substituting Eq. (9.111) into (9.109) yields, after some manipulation 1/3



(m ′)1/3 dm ′ =

Rkl (Tsat − Tw )  (ρl − ρv ) g   3v  hlv l  

sin1/3 Ω   dΩ (9.112)

This relation can be integrated from Ω = 0 to Ω = π to obtain the liquid condensed over half the cylinder per unit length (in the axial direction) 1/4

 R3 kl3 (Tsat − Tw )3 (ρl − ρv ) g  m ′ = 1.924   (9.113) hlv3 vl  



An overall energy balance over a control volume surrounding the cylinder requires that 2hlv m ′ = 2πRh (Tsat − Tw ) (9.114)



Substituting Eq. (9.113) for m ′ and solving the above equation for h , we obtain the following relation for the Nusselt number hD Ra  = 0.728    kl Ja 

Nu D =



1/4

(9.115a)

where Ra =



g(ρl − ρv ) Prl D 3 (9.115b) ρl vl2

Ja =



c pl (Tsat − Tw ) (9.115c) hlv

and D = 2R. Alternatively, Eqs. (9.113) and (9.115) can be combined to eliminate (Tsat – Tw), which allows the relation for the mean heat transfer coefficient to be written in the form



h kl

1/3

  µ l2  ρ (ρ − ρ ) g  v  l l 

= 1.92 Re −L1/3 ,

Re L =

4 m ′ (9.116) µl

Selin [9.74] found that better agreement with film condensation data for horizontal tubes was obtained by replacing the constants in Eqs. (9.115a) and (9.116) by 0.61 and 1.27, respectively. If the film flow around each half of the tube is considered to be equivalent to film flow over a flat plate of equal length, the flow is expected to become turbulent when ReL is greater than 1400. For tubes in most condensers, Reynolds numbers this high are usually not reached, and the condensate film flow is rarely turbulent. As in the case of vertical plates, film condensation heat transfer coefficients for liquid metals on horizontal tubes are typically 10–70% below the values predicted by these correlations [9.75].

421

External Condensation

By replacing g with g cosϕ, where ϕ is the angle of inclination, Selin [9.74] found that the correlation (9.115) for horizontal round tubes (with his recommended coefficient) matched inclined tube data within 15% for 0° ≤ ϕ ≤ 60°. This simple modification works well for predicting the mean heat transfer coefficient, apparently because most of the condensation occurs near the top of the tube where the film is thin and the effects of flow axially along the tube are small compared to transport across the film. At the bottom of the tube, where the film is thicker, although axial effects may be significant, this region contributes little to the overall heat transfer. Hence the inaccurate treatment of this region does not strongly affect the accuracy of the overall prediction of the mean heat transfer coefficient. A more detailed model for film condensation on inclined tubes has been proposed by Sheynkman and Linetskiy [9.76]. As noted by Nusselt [9.52] in his pioneering study of laminar film condensation, the film-flow analysis for horizontal tubes can be extended in a straightforward manner to an in-line bank of tubes (Fig. 9.15). The analysis of each individual tube in the bank is identical to that for a single tube, as described above, up to the derivation of Eq. (9.112). Integration of that equation then requires the use of a different boundary condition at Ω = 0 for each tube, since condensate from the tube immediately above is assumed to attach to the tube below at that location. It can easily be shown by integrating Eq. (9.112) that for tube i in the bank

4/3 = ( m ′ )i ,top + M ′ (9.117) ( m ′ )i4/3 ,bottom

where 1/3



 R3 kl3 ( Tsat − Tw )3 ( ρl − ρv ) g  M ′ = 2.393    (9.118) hlv3 vl  

FIGURE 9.15  Idealized model of film condensation on an in-line bank of round tubes.

422

Liquid-Vapor Phase-Change Phenomena

Conservation of liquid also requires that 4/3 = ( m ′ )i −1,bottom (9.119) ( m ′ )i4/3 ,top

Combining these relations yields

4/3 − ( m ′ )i −1,bottom = M ′ (9.120) ( m ′ )i4/3 ,bottom

If i is interpreted as a continuous variable instead of an integer, the above expression can be considered to be equivalent to

d   = M ′ (9.121) ( m ′ )4/3 bottom   di

Integrating this relation, using the fact that ( m ′ )4/3

i ,bottom



= M ′ at i = 1 yields

= iM ′ (9.122) ( m ′ )i4/3 ,bottom

Neglecting sensible cooling of the film, the mean heat transfer coefficient for the bank of tubes h is then found from conservation of mass and energy as

2πRih ( Tsat − Tw ) = 2hlv ( m ′ )i ,bottom (9.123)

Combining the above relations, the following relation for h can be obtained  g ( ρl − ρv ) ( nD )3 hlv  h (nD) = 0.728    (9.124) kl  kl vl ( Tsat − Tw )  1/4



Nusselt’s [5.52] treatment of this problem is obviously highly idealized, and is perhaps best viewed as a crude first cut at modeling this type of condensation process. Effects such as liquid subcooling, splashing, misalignment of the tubes, vibration of the tubes, and film waviness usually make it necessary to alter the coefficient on the right side of this correlation to achieve a best fit to data. Analytical treatment of laminar film condensation on a sphere is virtually the same as that for a horizontal cylinder. The only differences result from the angular variation of the body perimeter because of the spherical geometry. This effect can be included in the analysis in the manner described in Section 7.6 for film boiling from a sphere. Dhir and Lienhard [9.77] have shown that analytical prediction of the local heat transfer coefficient for laminar film condensation on arbitrary axisymmetric bodies can be generated by replacing g in the vertical surface correlation with an effective value geff given by

geff =

x ( gR )

∫

x

4/3

g1/3 R 4/3 dx

(9.125)

0

In this formula, g = g(x) is the local component of the gravity force in the direction of the streamwise coordinate along the body (x), and R is the body’s local radius of curvature in a vertical plane cutting through the center of the body. Based on the above discussion, this method is expected to work for all fluids except liquid metals.

423

External Condensation

Example 9.4 Use Eq. (9.124) to estimate the mean heat transfer coefficient for condensation of R-134a on the outside of a row of 10 tubes with an outside diameter of 2.0 cm. Motionless saturated R-134a vapor at 1190 kPa surrounds the tubes. The wall temperature of the tubes is 27°C. For saturated R-134a at 1190 kPa, Tsat, = 319 K, ρl = 1, 121 kg/m3, ρv = 59.3 kg/m3, hlv = 156.5 kJ/kg, μl = 1.51 × 10 −4 Ns/m2, and kl = 0.0721 W/mK. Equation (9.124) can be written as 1/ 4

3  k   gρ (ρ − ρv )(nD) hlv  h = 0.728  l     l l  nD   kl µ l (Tsat − Tw ) 

Substituting yields

1/ 4



3  0.0721   9.8 (1121)(1121− 59.3) 10 ( 0.02)  156,500    h = 0.728     0.0721) 1.51 × 10 −4 ( 319 − 300 ) (  10 ( 0.02)    

(

)

= 760.7 W/m K 2

Note that hl = hn1/ 4 =  760.7 (10 ) = 1353 W/m2K . It can be seen that the mean heat transfer coefficient for the first tube is substantially higher than the mean for the entire bank of tubes. 1/ 4

9.5  EFFECTS OF VAPOR MOTION AND INTERFACIAL WAVES The effects of downward motion of the surrounding vapor on falling-film condensation on vertical plates or tubes were examined analytically in the early study of Rohsenow et al. [9.78]. For laminar flow under such circumstances, the assumption of a linear variation of the shear stress across the film is still justifiable. However, at the interface, the shear stress in the liquid film approaches a constant value, dictated mainly by the vapor flow field near the interface (Fig. 9.16). The integral analysis of falling film condensation with downward interfacial shear basically follows the Nusselt-type analysis for zero interfacial shear described in Section 9.4. However, considering the shaded differential element in Fig. 9.7 with interfacial shear τi present, the momentum balance (neglecting the inertia and downstream diffusion terms) requires that

(δ − y ) dx  ρl g −

 du  dP   + τ i dx = µ l   dx (9.126) dx   dy 

FIGURE 9.16  Model shear stress variation in a liquid film subjected to interfacial shear.

424

Liquid-Vapor Phase-Change Phenomena

Note that in this momentum balance the pressure gradient along the surface dP/dx may, in general, be equal to the hydrostatic gradient plus an imposed gradient that drives the vapor motion

dP  dP   dP  = ρ g +  dP  (9.127) = v  +    dx  dx  hyd  dx  m dx  m

If a fictitious vapor density ρ∗v is defined such that dP  ρ∗v g = ρv g +  (9.128)  dx  m



the momentum balance (9.126) can be integrated using u = 0 at y = 0 to obtain

u=

(ρ − ρ ) g  yδ − y ∗ v

l

µl

 τi y + (9.129) 2  µ l



2

Using this velocity profile to evaluate the integral relation (9.45) for m ′ yields

m ′ =

ρl ( ρl − ρ∗v ) gδ 3 3µ l

+

τi ρl δ 2 (9.130) 2µ l

Differentiating with respect to δ, the following relation is obtained

∗ 2 dm ′ ρl ( ρl − ρv ) gδ τρδ = + i l (9.131) µl dδ µl

The energy and mass balance relations (9.48) and (9.49) are still valid for these circumstances. Combining these with Eq. (9.131) yields

kl µ l ( Tsat − Tw ) dδ = (9.132) dx ρl ( ρl − ρ∗v ) ghlv δ 3 + τ i ρl hlv δ 2

which can be integrated using the condition that δ = 0 at x = 0 to obtain

4 xkl µ l ( Tsat − Tw ) 4τi δ3 (9.133) = δ4 + ∗ ρl ( ρl − ρv ) ghlv 3 ( ρl − ρ∗v ) g

It can easily be shown that if convective effects (in the subcooled liquid film) are included, the above analysis results in the same equation, except that hlv is replaced by hlv′ given by Eq. (9.59)

4 xkl µ l ( Tsat − Tw ) 4τi δ3 (9.134) = δ4 + ∗ ρl ( ρl − ρv ) ghlv′ 3 ( ρl − ρ∗v ) g

Rohsenow et al. [9.78] defined the following dimensionless parameters



δ∗ =

δ (9.135) LF

 x  4  c pl ( Tsat − Tw ) x∗ =  (9.136)  LF  Prl hlv′

425

External Condensation

τ∗i =



τi (9.137) LF ( ρl − ρ∗v ) g

where 1/3

  µ l2 LF =   (9.138) ∗  ρl ( ρl − ρv ) g 



In terms of these dimensionless parameters, Eq. (9.134) becomes x ∗ = ( δ∗ ) + 4



4 ∗ 3 ∗ (δ ) τi (9.139) 3

and it can be shown that ∗ ∗ ∗ hF LF 4 ( δ ) 2 ( δ ) τi NuF = = + (9.140) 3 x∗ kl x∗ 3



Re L =



2

2 4 m ′ 4 ∗ 3 = ( δ ) + 2 ( δ∗ ) τ∗i (9.141) kl 3

If the length of the surface x, its temperature, and the physical properties of the fluids are specified, then x* and τ∗i can be computed from Eqs. (9.136)–(9.138). For the x* and τ∗i values thus determined, the three nonlinear equations (9.139)–(9.141) can be solved for δ*, NuF and ReL. Alternatively, if τ∗i and ReL are specified, the equations can be solved for x*, NuF and δ*. Rohsenow et al. [9.78] presented plots of NuF as a function of τ∗i and ReL computed in this manner. For low values of interfacial shear stress, Rohsenow et al. [9.78] also presented theoretical arguments from which they concluded that transition to turbulent film flow occurs at a transition Reynolds number ReL,tr given by

Re L ,tr

 ρ  = 1800 − 246  1 − v  ρl  

1/3

 ρ  3 τ∗i + 0.667  1 − v  ( τ∗i ) (9.142) ρl  

For condensation at film Reynolds numbers beyond this transition point, Rohsenow et al. [9.78] extended Seban’s [9.62] falling-film condensation analysis for turbulent flow with no vapor shear to include the effect of vapor shear at the interface. The results of this extended analysis were combined with those for the above laminar analysis to predict the variation of hL over a wide range of ReL values. Specifically, the laminar analysis was used to predict δ and m ′ at the transition point. These values were used as boundary conditions in the integration of the equation governing the turbulent transport. Computed results for Prl = 10 are shown in Fig. 9.17. For 2 ≤ Prl ≤ 3 and 5 ≤ τ∗i ≤ 50 , Rohsenow et al. [9.78] found that their computed results agreed well with the relation obtained by Carpenter and Colburn [9.79] which can be cast in the form

hL  vl2  kl  g 

1/3

= 0.065Prl1/2 ( τ∗i ) (9.143) 1/2

The later analysis by Dukler [9.63] similarly treated falling-film condensation with interfacial shear present. However, for fully turbulent flow conditions, Dukler used Deissler’s [9.80]

426

Liquid-Vapor Phase-Change Phenomena

FIGURE 9.17  Variation of the mean film condensation heat transfer coefficient with Reynolds number and τ*i as predicted by the analytical model of Rohsenow et al. [9.78). (Adapted from [9.78] with permission, copyright © 1956, American Society of Mechanical Engineers.)

equation for the eddy viscosity variation near the solid wall (y+ ≤ 20) and von Karman’s relation farther away

  − n 2uy   for y + ≤ 20:     ε M = ε H = n 2uy 1 − exp   (9.144a)  vl   



 du   d 2u  for y + > 20:     ε M = ε H = 0.4    2  (9.144b)  dy   dy 

3

2

where n is an experimentally determined constant equal to 0.124. Dukler’s [9.63] analysis of laminar film flow was essentially equivalent to the Nusselt-type analysis used by Rohsenow et al. [9.78]. The governing equations were solved numerically to determine the local heat transfer coefficient as a function of downstream position and film Reynolds number. Dukler’s computed results for Prl = 1.0 are plotted in nondimensional form in Fig. 9.18. One questionable aspect of Dukler’s analysis is that it neglects the molecular diffusion relative to the turbulent diffusivity in the fully turbulent region away from the wall. This is generally appropriate for high-Prandtl-number fluids, but could result in significant error for low-Prandtl-number, high-conductivity fluids. The analyses of Dukler [9.63] and Rohsenow et al. [9.78] both rely heavily on the assumption that the eddy diffusivity variation across the film is essentially the same as that near the wall in a turbulent pipe flow. In particular, they do not explicitly account for the damping effect that the interface has on turbulent transport. These analyses nevertheless predict film condensation heat transfer coefficients that are reasonably close to measured data, at least for Prl values greater than 1. Forced convection laminar film condensation on a horizontal tube can also be analyzed in a manner similar to that for condensation on a vertical surface described above. Such an analysis has been developed by Denny and Mills [9.81]. The effects of interfacial shear on film condensation in internal flow circumstances will be discussed in detail in Chapter 11. The effects of interfacial waves on condensation heat transfer are more difficult to analyze exactly. However, consideration of such effects by Brauer [9.82] and Kutateladze [9.69] has shed some light on these matters. Data presented by Brauer [9.82] imply that waves begin to affect laminar film condensation for

Re L > 2.3Ar1/5 (9.145)

427

External Condensation

FIGURE 9.18  Variation of the local film condensation heat transfer coefficient with Reynolds number and τi, as predicted by the analytical model of Dukler [9.63]. (Adapted from [9.63] with permission, copyright; © 1960, American Institute of Chemical Engineers.)

where Ar is the Archimedes number defined as Ar =



ρl2 σ 3/2 3/2 (9.146) µ l2 g1/2 ( ρl − ρv )

For 0 < ReL < 400 with wave effects present, Kutateladze [9.69] recommends the following correlation for the mean heat transfer coefficient for falling-film condensation as a best fit to data h  ρv vl2  NuL = L  kl  g ( ρl − ρv ) 



1/3

= 1.23 Re −L1/4 (9.147)

While this relation appears to agree well with data for liquid Prandtl numbers greater than one, it has not been tested against data for liquid metals. Example 9.5 For the film condensation process considered in Example 9.3, estimate where waves on the interface are likely to be observed. The Archimedes number Ar is given by Eq. (9.146):

Ar =

ρl2σ 3/ 2

µ l2 g1/ 2 ( ρl − ρv )

3/ 2

For steam at atmospheric pressure, σ = 0.0589 N/m, µ l = 2.78 × 10 −4 Ns/m2, ρl = 958 kg/m3, and ρv = 0.597 kg/m3. Substituting in the above equation, Ar is determined to be

Ar =

(958)2 (0.0589)3/ 2 ( 2.78 × 10−4 ) (9.8)1/ 2 (958 − 0.597)3/ 2

= 509

428

Liquid-Vapor Phase-Change Phenomena

Brauer [9.82] argued that waves will affect laminar film condensation for

Re L > 2.3Ar1/ 5

This implies that waves begin to appear for

Re L ,w ≅ 2.3(509)1/ 5 = 8.0

 x′ = q ′′x / hlv , and therefore Re L = 4m  x′ / µ l vary For the uniform-flux heat removal on this surface, m linearly with x. It follows that the location xw where waves first have a significant effect is given by



 Re  xw = L  l ,w   Re L   8.0  = (1.0 )  = 0.0025 m = 2.5 mm  3,187 

Thus, waves are expected to be observable and to have a significant effect on film condensation at only a short distance downstream of the leading (upper) edge of the plate.

9.6  CONDENSATION IN THE PRESENCE OF A NONCONDENSABLE GAS In nature and a number of technological applications, condensation of one component vapor in a mixture may occur in the presence of other noncondensable components. The most common example is the condensation of water vapor in the air on a cold solid surface. If the component gases are considered to be a mixture of independent substances, condensation of one component vapor will occur if the temperature of the surface is below the saturation temperature of the pure vapor at its partial pressure in the mixture. This temperature threshold is referred to as the dew point of the mixture. The general effects of a noncondensable gas on a film condensation process can be envisioned by considering Fig. 9.19. Once steady-state is achieved, condensation occurs at the interface of a

FIGURE 9.19  System model for analysis of film condensation in the presence of a noncondensable gas.

429

External Condensation

liquid film on the wall. Due to the condensation process at the interface, there is a bulk velocity of the gas toward the wall, as if there were suction at the interface. Because only the vapor is condensed, the concentration of the noncondensable gas at the interface Wi is higher than its value W∞ in the far ambient. This, in turn, decreases the partial pressure of the vapor at the interface below its ambient value. The corresponding saturation temperature at the interface is therefore lower than the dew point of the bulk fluid. At equilibrium, the interface concentration of the noncondensable gas is high enough so that the resulting diffusion and/or convection of this component away from the interface into the ambient just balances the rate at which its concentration increases due to the condensation process. In systems where the noncondensable gas is a low-concentration contaminant, the phenomena described above can lead to a high concentration of the noncondensable contaminant at the interface. The resulting depression of the interface temperature generally reduces the condensation heat transfer rate below that which would result for pure vapor alone under the same conditions. For concreteness, we will first consider the forced convection film condensation process shown in Fig. 9.20. The flat wall, taken to be isothermal, is exposed to a forced flow of a mixture of vapor and noncondensable gas, which results in film condensation on the surface. The ambient flow is assumed to have uniform values of velocity u∞, temperature T∞ and noncondensable gas concentration W∞ . For the liquid film, if the energy convection and inertia terms are neglected, the boundary-layer forms of the governing mass, momentum, and energy conservation equations become

∂ul ∂ vl + = 0 (9.148a) ∂x ∂ y



∂2 ul = 0 (9.148b) ∂ y2



∂2 Tl = 0 (9.148c) ∂ y2

Appropriate boundary conditions are

at y = 0: ul = vl = 0, Tl = Tw (9.149a)



at y = δ:    Tl = Tl ,i (9.149b)

FIGURE 9.20  System model for analysis of forced-convection film condensation in the presence of a noncondensable gas.

430

Liquid-Vapor Phase-Change Phenomena

For the gas-vapor boundary layer adjacent to the film, the continuity and momentum equations are ∂u ∂ v + = 0 (9.150a) ∂x ∂ y





u

∂u ∂u ∂2 u +ν = νm 2 (9.150b) ∂x ∂y ∂y

At the interface, the streamwise velocity u must be continuous. However, the liquid velocity at the interface is generally very much smaller than the free stream velocity u ∞ of the vapor. Consequently, for the vapor, the u velocity can be taken to be zero at the interface with very little loss in accuracy. With this approximation, the boundary conditions on the gas velocity fields are

at y  = δ: 

u = 0 (9.151a)



 dδ  ml′′= ρm  u − ν (9.151b)  dx 



 ∂u   ∂u  = µ m   (9.151c) µl  l   ∂ y  y =δ  ∂ y  y =δ



at y → ∞: u → u∞ (9.151d)

where ml′′ is the mass flux into the film due to condensation. The transport of mass and heat in the two-component gas mixture is generally more complex than the transport in the liquid film. For the purposes of the present analysis, the local concentrations of the vapor and noncondensable gas will be specified in terms of their mass fractions Wv and Wg , respectively

Wv =

ρv ρm

Wg =

ρg ρm

(9.152)

where ρm is the local density of the mixture and ρg and ρv are the local densities of the gas and vapor, respectively. It follows directly from these definitions that Wv + Wg = l. Since these concentrations are not independent, one can be eliminated from the problem. For this analysis, we will therefore deal strictly with the concentration of the noncondensable gas Wg. In a mixture of the type considered here, the diffusive mass flux of the noncondensable gas jg is given by

jg = −ρm D

∂Wg (9.153) ∂y

The right side of the above expression is the well-known Fickian diffusion term, with D being the binary diffusion coefficient. In general, the diffusive flux may be altered by the presence of a temperature gradient in the system. This additional mass transport mechanism is the so-called thermal diffusion or Soret effect. Because this effect is small in many systems of interest, we will not include it here.

431

External Condensation

A relation similar to Eq. (9.153) can also be written for the diffusion flux of the vapor species

∂Wv (9.154) ∂y

jv = −ρm D

from which it follows directly that jv = –jg. At any location in the flow, the heat flux in the direction normal to the surface is given by the Fourier conduction relation

q ′′ = − ks

∂T (9.155) ∂y

In systems with a concentration gradient, an additional energy flux contribution occurs due to species diffusion. This additional energy transport effect is termed as the diffusion thermo effect or the Dufour effect. This effect is small in most systems of interest and we will therefore neglect it in the model analysis here. The boundary-layer equations governing the species and heat transport in the gas mixture near the interface are

 ∂Wg ∂Wg  ∂ jg =− ρm  u +v (9.156)  ∂y  ∂y  ∂x



 ∂T ∂T  ∂T ∂q ′′ + (c pg − c pv ) jg ρm c pm  u +v =− (9.157)  ∂y  ∂y ∂y  ∂x

where cpm , cpg, and cpv are the specific heat of the mixture, the gas, and the vapor, respectively, and jg and q′′ are given by Eqs. (9.153) and (9.155). The second term on the left side of Eq. (9.153) accounts for the net enthalpy flux associated with the diffusive flux of the individual components. For low to moderate concentrations of the noncondensable gas, the sensible heat transfer to the interface is usually negligible compared to the heat transferred as a result of the transport and condensation of the vapor at the interface. Restricting our attention to circumstances of this type in which the transport of heat is mass-transfer dominated, the equation governing sensible transport of thermal energy will be neglected. Of primary concern is the resulting reduced form of the equation for mass transport

 ∂Wg ∂Wg  ∂2 Wg = u + v D (9.158)  ∂ x ∂ y  ∂ y2

Boundary conditions on the concentration field are



at y  = δ: jg = −ρm D

∂Wg = −ρg v (9.159a) ∂y

at y → ∞: Wg → Wg ,∞ (9.159b)

The relations (9.148)–(9.151), (9.158) and (9.159) form a mathematically complete system of equations and boundary conditions. However, it is presumed in this formulation that the location y = δ is

432

Liquid-Vapor Phase-Change Phenomena

known. The additional relation that makes it possible to determine δ comes from the requirement of an energy balance at the liquid-gas interface:  ∂T   ∂T  = kl  l  (9.160) m l′′ hlv + km  m  ∂ y   y =δ  ∂ y  y =δ



Consistent with the arguments given above, the contribution of conduction from the gas mixture, represented by the second term on the left side of Eq. (9.160), is taken to be negligibly small. The above relation can therefore be simplified to  ∂T  dδ   ml′′ hlv =  ρl vl − ρl ul hlv = − kl  l  (9.161)   dx  y =δ  ∂ y  y =δ



With the addition of Eq. (9.161), the mathematical problem is completely closed. To facilitate solution of this problem, Sparrow et al. [9.83] introduced the following similarity variables: In the liquid film: η= y



ψ =   f ( η) u∞ vl x     





u∞ (9.162a) vl x

u=

∂ψ = u∞ f ( η)     ∂y

v=−

θ=

Tl − Tw (9.162b) Tl ,i − Tw

∂ψ 1 = ∂x 2

u∞ ( ηf ′ − f ) (9.162c) vl x

In the vapor: ξ = ( y − δ)



ψ = F ( ξ ) u∞ vm x   



u∞ (9.163a) vl x φ  =

Wg − Wg.∞ (9.163b) Wg ,i − Wg.∞

In terms of these variables, the governing equations and boundary conditions become In the liquid film: f ′′′ = 0



f (0) = 0

f ′ (0) = 0

θ′′ = 0 (9.164a) θ(0) = 0

θ ( ηδ ) = 1 (9.164b)

In the gas boundary layer: 1 FF ′′ = 0 (9.165a) 2



F ′′′ +



1 φ′′ + ScFφ′ = 0 (9.165b) 2

433

External Condensation



F ′(0) = 0

φ(0) = 1

F ′ (∞) = 1

φ(∞) = 0 (9.165c)

The required conditions at the interface are

F (0) = Rf ( ηδ )    F ′′(0) = Rf ′′( ηδ ) (9.166a)



f ( ηδ ) Ja = l (9.166b) 2θ′( ηδ ) Prl



−

Wg ,∞ ScF (0) = 1− (9.166c) 2φ′(0) Wg ,i

where 1/2



 ρµ  R= l l   ρm µ m 

      Ja l =

c pl ( Ti − Tw ) (9.166d) hlv

and ηδ = δ



u∞ (9.166e) vl x

It can be easily shown that solutions of Eqs. (9.164a) that satisfy the boundary conditions (9.164b) are

f =

1 η f ′′(0) η2       θ  = (9.167) 2 ηδ

Substituting these relations, the interface conditions (9.166a)–(9.166b) can be combined to obtain 1/2



3 RJa l  [ F (0) ]  =  (9.168a) Prl  2 F ′′(0) 



 2 F (0)  ηδ =   (9.168b)  F ′′(0) 

1/2

With these relations, the complete solution to the problem can be obtained as follows. For a chosen value of F(0), Eq. (9.165a) with appropriate boundary conditions (9.165c) can be solved numerically. Then, using the chosen F(0) value and the numerically determined value of F″(0), Eqs. (9.168a) and (9.168b) are used to determine the corresponding values of R Jal/Prl and ηδ. This implies that ηδ is a unique function of R Jal/Prl. By doing such calculations for a sequence of F(0) values, the relation between ηδ and R Jal/Prl can be determined for a wide range of conditions. Figure 9.21 shows the predicted variation of ηδ with R Jal/Prl as determined numerically by Sparrow et al. [9.83]. Once this is accomplished, for any real circumstances of interest, ηδ can be determined from the established ηδ variation with R Jal/Prl. F(0) and F″(0) can then be determined from Eqs. (9.168a) and (9.168b), and Eq. (9.165a) can be integrated to determine the F(ξ) and F′(ξ) variations. With the solution for F(ξ) and the value of ηδ determined, Eq. (9.165b) can be solved numerically for a specific value of the Schmidt number Sc = vm/D, subject to the boundary conditions on ϕ given in Eq. (9.165c). The numerically determined value of ϕ′(0) with the corresponding value of F(0) can then be used in Eq. (9.166c) to determine the interface condition Wg,i . This solution scheme implies

434

Liquid-Vapor Phase-Change Phenomena

FIGURE 9.21  Variation of ηδ with R Jal / Prl as determined numerically by Sparrow et al. [9.83]. (Adapted from [9.83] with permission, copyright © 1967, Pergamon Press.)

that Wg ,i / Wg ,∞ is a function only of Sc and R Jal//Prl. The computed variation of Wg ,i / Wg ,∞ for Sc = 0.55 determined numerically by Sparrow et al. [9.83] is shown in Fig. 9.22. Treating the vapor and noncondensable gas as a mixture of ideal gases, the partial pressure of the vapor at the interface Pv,i can be computed as

1 − Wg ,i Pv ,i = (9.169) P∞ 1 − Wg ,i (1 − M v / M g )

The interface temperature is the saturation temperature at the vapor partial pressure there Ti = Tsat ( Pv ,i ) (9.170)



It also follows directly from the above formulation that

Nux =

hx  Ti − Tw  Re1/2 x = (9.171) kl  T∞ − Tw  ηδ

FIGURE 9.22  Variation of Wg,l/Wg,∞ with R Jal/Prl for Sc = 0.55 as determined numerically by Sparrow et al. [9.83]. (Adapted from [9.83] with permission, copyright © 1967, Pergamon Press.)

435

External Condensation

where Re x =



u∞ x q ′′ ,     h = (9.172) vl T∞ − Tw

Thus, once Ti and ηδ are determined for a given flow condition, the heat transfer coefficient and/or the Nusselt number for the condensation process at a specified downstream location can be determined from these results. Although several idealizations are included, the above analysis is a compact formulation that is general enough to be applicable to a broad range of circumstances. More exact analyses must often be tailored to a specific set of fluids, particularly if variable property effects are included. The results of such analyses are therefore limited to a particular set of flow circumstances. The analysis of Sparrow et al. [9.83] indicates that the thickening of the condensate film with downstream distance leads to a decrease in the condensation rate such that the effective suction velocity vi at the interface decreases proportional to x−1/2. It further implies that for such circumstances, the interface gas composition and the corresponding interface temperature Ti are independent of x. Consequently, the mass transport problem in the vapor-gas boundary layer is identical to the case of heat transfer for flow over an isothermal plate with a surface suction velocity varying proportional to x  −1/2. Rose [9.84, 9.85] showed that this similarity makes it possible to apply an approximate relation developed for the heat transfer problem to the mass transfer for the condensation process considered above. The resulting relation for the mass transfer coefficient is

(

hm x  0.93 = ζ 1 + 0.941β1.14 x Sc ρm D 

)

−1

+ β x Sc  Re1/2 (9.173a)  x

where

ζ = Sc1/2 (27.8 + 75.9Sc 0.306 + 657Sc)−1/6 (9.173b)



 v  β x = −  i  Re1/2 x (9.173c)  u∞ 



hm =

−ρm D  ∂Wg  (9.173d) (Wg ,i − Wg ,∞ )  ∂ y  y =δ

A mass balance for the gas at the interface also requires that

hm = −

ρm viWg ,i ρm vi =− (9.174) (Wg ,i − Wg ,∞ ) (1 − Wg ,∞ / Wg ,i )

Substituting this relation into Eq. (9.173a) for hm and rearranging yields the following explicit relation for the interface condition

Wg ,i β Sc (1+0.914 Sc 0.93 ) = 1+ x (9.175) Wg ,∞ ζ

For a given set of free stream and wall temperature conditions, this equation relates the interface velocity vi and Wg,i directly. Rose [9.85] found that this relation predicts values of Wg ,i / Wg ,∞ that agree well with the numerical solutions of Sparrow et al. [9.83] for Sc = 0.55. If it is presumed that

436

Liquid-Vapor Phase-Change Phenomena

FIGURE 9.23  Variation of F″(0) with F(0) as determined from numerical solution of Eq. (9.165a) with boundary conditions F′(0) = 0 and F′(∞) = 1.

the heat transfer at the interface is entirely due to condensation of vapor, an energy balance at the interface also requires that

vi = −

h(T∞ − Tw ) (9.176) ρm hlv (1 − Wg ,i )

Equations (9.175) and (9.176) can be used to greatly simplify the computation of the condensation heat transfer coefficient for the flat plate forced convection condensation process considered above. As noted above, Eq. (9.165a) can be solved numerically subject to the boundary conditions F ′(0) = 0 and F ′(∞) = 1 for various values of F(0). The resulting numerically computed variation of F ″(0) with F(0) is shown in Fig. 9.23. Using the computed results represented in Fig. 9.23, the heat transfer coefficient for a given set of flow circumstances can be iteratively computed as follows (It is assumed that Tw, T∞, u∞, Wg ,∞,P∞ and x have been specified and that the fluid properties are known.): 1. A value of vi is guessed and the corresponding value of F(0) = –2vi( x u∞ /vm )1/2 is calculated. 2. F ″(0) can then be determined from interpolation of tabulated numerical results or graphically from Fig. 9.23. 3. Using the F(0) and F ″(0) values, Eqs. (9.168a) and (9.168b) can be used to compute R Jal /Prl and ηδ. 4. Equation (9.175) can then be used to determine Wgi, the value of which can be substituted into Eq. (9.169) to determine Pv,i. 5. Ti is calculated as Ti = Tsat(Pv,i) from thermodynamic data for the pure vapor. 6. The heat transfer coefficient h is calculated using Eq. (9.171). 7. Finally, Eq. (9.176) is used to recalculate vi using the value of h computed in step 6. If this value of vi agrees with that initially guessed, the solution is complete. If it does not agree to an acceptable degree, an improved guess for vi is generated and the process is repeated, beginning with step 2, until convergence is achieved. While the above computational scheme to determine h is iterative, it does not require integration of the differential equations governing the transport, if the variation of F"(0) with F(0) is known. It is therefore somewhat easier to use than the full similarity solution analysis of Sparrow et al. [9.83], and yet has comparable accuracy.

437

External Condensation

Example 9.6 Forced-convection film condensation of water occurs on a 50 cm long flat plate exposed to a flow of a mixture of steam and air. The ambient conditions are T∞ = 95°C, u∞ = 5 m/s , P∞ = 101 kPa, and Wg ,∞ = 0.02. The plate is held at constant temperature of 60°C. Experimentally, the heat transfer coefficient at a distance x = 15 cm from the leading edge of the plate is determined to be 1.5 kW/m2K. Use Eqs. (9.175) and (9.176) to determine the interface velocity vi and interface concentration Wg,i. For air and water vapor at the specified conditions, ρm = 0.60 kg/m3, Sc = 0.9, vm = 2.10 × −5 10 m2/s, and at Tw = 60°C, hlv for the condensate water is 2359 kJ/kg. Equations (9.175) and (9.176) must be satisfied simultaneously to determine vi and Wg,i. Initially taking Wg ,i = Wg ,∞, vi is determined from Eq. (9.176) to be

vi = −

h(T∞ − Tw ) 1.5(95 − 60) = = −0.0378 m/s ρm hlv (1− Wg ,i ) 0.6(2359)(1− 0.02)

Using Eqs. (9.173b) and (9.173c),





ζ = Sc1/ 2 (27.8 + 75.9 Sc0.306 +  657 Sc)−1/6 = (0.9)1/ 2  27.8 + 75.9(0.9)0.306 +  657 (0.9)   v  u x βx = −  i   ∞   u∞   v m 

1/ 2

−1/6

 0.0378   5.0(0.15)  =  5.0   2.10 × 10 −5 

= 0.336

1/ 2

  = 1.43

Substituting in Eq. (9.175),



 β  Sc(1 + 0.914 Sc0.93 )  Wg ,i = Wg ,∞ 1+ x  ζ    1.43(0.9)(1+ 0.914(0.9)0.93 )  = (0.02) 1+  = 0.162 0.336  

Substituting back into Eq. (9.176),

vi = −

1.5(95 − 60) = −0.0443 m/s 0.6(2359)(1− 0.162)

Repeating the above calculations of βx and Wg,i yields βχ = 1.67, Wg,i = 0.186. After a few iterations, the following converged values are obtained:

v i = −0.046  m /s     Wg ,i = 0.19

Thus, there is a “suction velocity” of 4.6 cm/s at the interface and the concentration of noncondensable gas is almost 10 times the ambient concentration.

Analyses similar to those described above have also been developed for falling film condensation on a vertical plate in the presence of a noncondensable gas [9.86–9.89]. Combined forced and free convection film condensation on a vertical plate immersed in a steam-air mixture has also been analyzed by Denny et al. [9.90]. Film condensation of liquid metals from a metal vapor and gas mixture on a vertical surface for forced and free convection conditions has also been treated theoretically by Turner et al. [9.91]. Where necessary, their analysis also included the effects of interfacial resistance.

438

Liquid-Vapor Phase-Change Phenomena

Experimentally determined heat transfer coefficients for film condensation on vertical flat surfaces in the presence of a noncondensable gas reported by Al-Diwany and Rose [9.92] and Slegers and Seban [9.93] generally are consistent with the predictions of the boundary-layer analyses proposed in the studies noted above. Film condensation on a horizontal round tube in the presence of a noncondensable gas may occur in tube and shell condensers when a contaminant such as air is introduced into the system. Berman [9.94] has proposed a correlation to predict the mass transfer coefficient for such circumstances. A more complete analysis of such processes has been presented by Al-Diwany and Rose [9.92]. Their model analysis was found to agree well with experimental data reported by Mills et al. [9.95].

9.7  ENHANCEMENT OF CONDENSATION HEAT TRANSFER In many circumstances of practical interest, most, if not all of the resistance to heat flow during the condensation process is due to conduction and/or convection through the liquid condensation on the cooled surface. Consequently, most techniques for enhancing condensation heat transfer involve controlling the condensate so as to reduce the heat transfer resistance associated with its presence on the surface.

Surface Modification Enhancement of Dropwise Condensation As discussed in Section 9.2, strategies to enhance dropwise condensation generally aim to modify the system to better control the liquid inventory to get liquid off the surface and keep the mean droplet size small. A fundamental component of such strategies has typically been to make most of the surface hydrophobic, so the system tends to minimize liquid-solid contact to minimize its free energy. Early strategies for making the surface hydrophobic have included introducing a fatty acid, wax, or other substance onto the cooled solid surface, or by permanently coating the surface with a low-surface-energy polymer or noble metal. Each of these methods has its drawbacks and/or limitations. Fatty acids and other injected promoters are washed off condenser surfaces under common industrial operating conditions. To sustain dropwise condensation, it is therefore necessary to periodically reinject the promoter into the system. Using permanent coatings of the condenser surface avoids this difficulty, but generally adds considerable cost to the fabrication of the condenser. In practice, the above techniques for promoting dropwise condensation have been successful primarily for condensation of steam. There are a number of substances that are poorly wetted by liquid water, as a consequence of water’s relatively high interfacial tension. For liquids having lower interfacial tension, such as the fluorocarbon refrigerants R-ll, R-22, R-113, R-134a, and many hydrocarbons, no dropwise condensation promoters have been reported in the literature (see the survey by Iltscheff [9.96]). Hence, promotion of dropwise condensers for refrigeration systems using fluorocarbon working fluids awaits the discovery of substances that are poorly wetted by these liquids. Further information on techniques for promoting dropwise condensation of steam can be found in the survey paper by Tanasawa [9.97]. Because nano and microstructuring can also render a surface hydrophobic, numerous studies in the past two decades have explored how surfaces of this type may affect the mechanisms of dropwise condensation. The basic objective of using these surfaces is to increase the free energy of the surface when fully wetted, so the system will be thermodynamically favored to spontaneously take-on a less wetted, lower free energy state. As discussed in Chapter 3, it can be shown from thermodynamic analysis that if the intrinsic contact angle of the surface material is more than 90°, roughening the surface will result in a higher apparent contact angle, and thermodynamically favor transition of droplet states to a less wetted, lower free energy state. Nanostructuring or microstructuring can be viewed as specific forms of roughening the surface, and therefore are expected to make the surface less wetting if the surface material intrinsic contact angle is more than 90°.

External Condensation

439

Some of the recent studies of nano and microstructured surfaces have focused on how the hydrophobic character of these surfaces can affect wetting and liquid transport during dropwise condensation (see, e.g., references [9.98–9.102]). It is clear, however, that optimal design of nano/ microstructured surfaces to enhance dropwise condensation requires an understanding of how the morphology affects nucleation and growth of liquid droplets, as well as how it affects droplet removal/drainage from the surface. This has motivated studies that have focused on nano/ microstructure alteration of droplet nucleation [9.103–9.112], droplet growth or merging on nano/ microstructure surfaces [9.109, 9.113–9.117], droplet mobility, or gravity draining/removal of droplets on nano/microstructured surfaces [9.118–9.121], and droplet jumping on nano/microstructure surfaces [9.122–9.125]. Examples of the types of nanostructured surfaces explored in these studies are shown in Figs. 9.24 and 9.25. The recent studies described above have substantially expanded the knowledge of how surface microstructuring can affect droplet behavior, and have demonstrated that by using materials having intrinsic contact angles above 90°, micro and nanostructured surfaces can achieve extremely hydrophobic surface conditions that enhance merging of droplets and removal of liquid from the surface by gravity drainage. However, studies of these issues also clearly indicate that complex structures and/or localized irregularities in the structure or wetting can give rise to droplet pinning that can slow droplet transport, merging and drainage from the surface, which may lead to more liquid retention on the surface, larger maximum droplet size, resulting in a lower heat transfer coefficient.

FIGURE 9.24  Hydrophobic stochastic nanostructured surfaces tested in the study by Miljkovic et al [9.124]. Field emission scanning electron microscopy (FESEM) images of a CuO surface with (a) top view, no silane, (b) side view, no silane, (c) high magnification showing the blade structure of the oxide, no silane, and (d) high magnification after silane deposition. Silane deposition thickens the oxide blades but maintains the general nanostructure morphology (with thanks to Prof. Nenad Miljkovic who provided the photos). The sharp, knifelike CuO structures have characteristic heights of about 1 μm, solid fraction of about 0.023, and roughness factor of about 10.

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FIGURE 9.25  Hydrophobic ordered nanostructured surfaces tested in the study by Enright et al [9.120]. Scanning electron micrographs of fabricated pillar geometries spanning a range of length scales from 100 nm to 10 μm. Left: Si micropillars fabricated using optical lithography and deep reactive-ion etching (DRIE); Center: Si nanopillars fabricated using e-beam written mask and DRIE; Right: Si nanopillars fabricated using interference lithography and metal-assisted wet etching (with thanks to Prof. Nenad Miljkovic who provided the photos).

On the other hand, studies of droplet nucleation on micro and nanostructured surfaces indicate that some better-wetted locations on the generally hydrophobic surfaces are needed to ensure that the condensation process can be initiated without requiring high surface subcooling. Wetted regions randomly produced in the manufacture of the surface, or hydrophilic particles from the environment that deposit on the surface may serve as droplet nucleation sites. Alternatively the surface may be designed with a hybrid nano/microstructure that provides a mostly hydrophobic surface, with a repeated pattern of localized hydrophilic areas that can serve as nucleation sites. This type of hybrid surface design was developed, for example, by Vanisari et al. [9.104, 9.107]]. The studies of pinning effects on droplet motion and drainage indicate, however, that an optimized surface design requires enough hydrophilic sites to adequately support nucleation, but not so many that droplet pinning slows liquid drainage to the point of reducing heat transfer performance. One of the more interesting ideas to emerge from recent studies is designing nanostructured surfaces that exhibit an extreme free energy increase associated with surface wetting. As a consequence, dewetting of such surfaces releases a relatively large amount of free energy, which can be converted to droplet kinetic energy of motion. Liquid motion during dewetting under such conditions can eject droplets from the surface. As demonstrated in the studies by Narhe et al. [9.123], Miljkovic et al. [9.124], and Mulroe et al. [9.125], the “jumping droplets” resulting from this effect, can enhance liquid removal from the surface. By designing condenser surfaces to maximize the benefits of jumping droplets, the mean droplet size on the surface the surface can be reduced, thereby increasing the heat transfer coefficient. This research suggests that promoting droplet jumping can be another useful strategy for enhancing dropwise condensation.

Enhancement of Film Condensation Heat Transfer For film condensation, the use of formed or extended surfaces as a means of enhancing the heat transfer coefficient has received considerable attention. Gregorig [7.126] pioneered the use of a wavy surface to enhance film condensation heat transfer. The effect of such a surface profile on the condensate can be seen by considering Fig. 9.26. As indicated in this figure, the curvature of the liquid-vapor interface will reverse at the “peak” and “valley” locations to allow the film to conform

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FIGURE 9.26  Schematic of a wavy surface used to enhance film condensation heat transfer.

to the wavy surface. For a two-dimensional surface of this type, it follows from Young’s equation (see Chapter 2) that

Pl = Pv +

σ (9.177a) r1



Pv = P2 +

σ (9.177b) r2

where r1 and r2 are the radii of curvature of the interface at 1 and 2, respectively. These equations are easily combined to show that

 1 1 P1 − P2 = σ  +  (9.178)  r1 r2 

Thus interfacial tension induces a pressure difference in the liquid film that causes liquid to flow from the peak regions to the valley regions of the surface. This thins the film in the peak regions, allowing more rapid condensation there, at the expense of the valley regions where the film is thicker. The overall mean heat transfer coefficient for the surface can be substantially greater than if the surface were covered with a film of uniform thickness. This effect of surface tension on film condensation on a wavy surface is sometimes referred to as the Gregorig effect. A number of variations of the above theme have been proposed as means of enhancing film condensation heat transfer. Thomas [9.127], for example, proposed attaching vertical wires onto vertical condenser tubes to take advantage of this effect. Liquid condensate is pulled into the crevice between the round tube wall and the wire, thinning the liquid film between the wires, as shown in Fig. 9.27. The tendency for circumferentially finned condenser tubes to similarly modify the condensate distribution has also been studied by a number of investigators (see, e.g., Winniarachchi et al. [9.128], Zozulya et al. [9.129], Rudy and Webb [9.130], and Hirasawa et al. [9.131]). Excellent summaries of the works in this area can be found in the review articles by Marto and Nunn [9.132], Webb [9.133], Cooper and Rose [9.134], and Marto [9.135]. Enhancement of external film condensation using mechanical aids such as rotating cylinders, disks, and wiper blades has also been examined [9.136–9.139]. Other enhancement methods that have been considered include surface vibration [9.140], electrostatic fields [9.141], and suction [9.142]. Obviously, combinations of these different methods can also be employed. Further discussion of such methods is provided in the review article by Bergles [9.143] and the review presented by Webb [9.144].

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FIGURE 9.27  Schematic representation of wires used to enhance film condensation on vertical tubes.

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9.108 Thickett, S. C., Neto, C. and Harris, A. T., Biomemetic surface coatings for atmospheric water capture prepared by dewettting of polumer films, Adv. Mater., vol. 23, pp. 3718–3722, 2011. 9.109 Anderson, D. M., Gupta, M. K., Voevodin, A. A., Hunter, C. N., Putnam, S .A. Tsukruk, V. V. and Fedorov, A. G., Using amphiphilic nanostructures to enable long-range ensemble coalescence and surface rejuvenation in dropwise condensation, ACS Nano, vol. 6, pp. 3262–3268, 2012. 9.110 Her, E. K., Ko, T. J., Lee, K. R., Oh, K. H., and Moon, M. W., Bioinspired steel surfaces with extreme wettability contrast, Nanoscale, vol. 4, pp. 2900–2905, 2012. 9.111 Miljkovic, N., Enright, R., and Wang, E. N., Modeling and optimization of superhydrophobic condensation, J. Heat Transf., vol. 135, pp. 111004-1–111004-14, 2013. 9.112 Yao, C. W., Garvin, T. P., Alvarado, J. L., Jacobi, A. M., Jones, B. G. and Marsh, C. P., Droplet contact angle behavior on a hybrid surface with hydrophobic and hydrophilic properties, Appl. Phys. Lett., vol. 101, p. 111605, 2012, DOI: 10.1063/1.4752470. 9.113 Miljkovic, N., Enright, R., and Wang, E. N., Effect of Droplet morphology on growth dynamics and heat transfer during condensation on superhydrophobic nanostructured surfaces, ACS Nano, vol. 6, pp. 1776–1785, 2012. 9.114 Narhe, R. D., and Beysens, D. A., Growth dynamics of water drops on a square-pattern rough hydrophobic surface, Langmuir, vol. 23, pp. 6486–6489, 2007. 9.115 Rykaczewski, K., Microdroplet growth mechanism during water condensation on superhydrophobic surfaces, Langmuir, vol. 28, pp. 7720–7729, 2012. 9.116 Rykaczewski, K., Scott, J. H. J., Rajauria, S., Chinn, J., Chinn, A. M., and Jones, W., Three dimensional aspects of droplet coalescence during dropwise condensation on superhydrophobic surface, Soft Matter, vol. 7, pp. 8749–8752, 2011. 9.117 Rykaczewski, K., Paxson, A. T., Anand, S., Chen, X., Wang, Z., and Varanasi, K. K., Multimode multidrop serial coalescence effects during condensation on hierarchical superhydrophobic surfaces, Langmuir, vol. 29, pp. 881−891, 2013. 9.118 Wier, K. A., and McCarthy, T. J. Condensation on ultrahydrophobic surfaces and its effect on droplet mobility: Ultrahydrophobic surfaces are not always water repellant, Langmuir, vol. 22, pp. 2433–2436, 2006. 9.119 Boreyko, J. B., and Chen, C.-H. Self-propelled dropwise condensate on superhydrophobic surfaces, Phys. Rev. Lett., vol. 103, p. 184501, 2009. 9.120 Enright, R., Miljkovic, N., Al-Obeidi, A., Thompson, C. V., and Wang, E. N. Condensation on superhydrophobic surfaces: The Role of local energy barriers and structure length scale, Langmuir, vol. 28, pp. 14424–14432, 2012. 9.121 Dietz, C., Rykaczewski, K., Fedorov, A. G., and Joshi, Y., Visualization of droplet departure on a superhydrophobic surface and implications to heat transfer enhancement during dropwise condensation, Appl. Phys. Lett., vol. 97, p. 033104, 2010. 9.122 Boreyko, J. B., and Chen, C. H., Self-propelled dropwise condensate on superhydrophobic surfaces, Phys. Rev. Lett., vol. 103, pp. 184501−184504, 2009. 9.123 Narhe, R. D., Khandkar, M. D., Shelke, P. B., Limaye, A. V., and Beysens, D. A., Condensation-induced jumping water drops, Phys. Rev. E, vol. 80, pp. 031601−031605, 2009. 9.124 Miljkovic, N., Enright, R., Nam, Y., Lopez, K., Dou, N., Sack, J., and Wang, E. N., Jumping-dropletenhanced condensation on scalable superhydrophobic nanostructured surfaces. Nano Lett., vol. 13, pp. 179–187, 2013. 9.125 Mulroe, M. D., Srijanto, B. R., Ahmadi, S. F., Collier, C. P., and Boreyko, J. B., Tuning superhydrophobic nanostructures to enhance jumping-droplet condensation, ACS Nano, vol. 11, pp 8499–8510, 2017. 9.126 Gregorig, R., Film condensation on finely rippled surfaces with consideration of surface tension, Z. Angew. Math. Phys., vol. 5, pp. 36–49, 1954. 9.127 Thomas, D. G., Enhancement of film condensation rates on vertical tubes by vertical wires, Ind. Eng. Chem. Fundam., vol. 6, pp. 97–102, 1967. 9.128 Winniarachchi, A. S., Marto, P. J., and Rose, J. W., Film condensation of steam on horizontal finned tubes: effects of fin spacing, J. Heat Transf., vol. 108, pp. 960–966, 1986. 9.129 Zozulya, N. V., Karkhu, V. A., and Borovkov, V. P., An analytical and experimental study of heat transfer in condensation of vapor on finned surfaces, Heat Transf. Sov. Res., vol. 9, no. 2, pp. 18–22, 1977. 9.130 Rudy, T. M., and Webb, R. L., An analytical model to predict condensate retention on horizontal integral-fin tubes, J. Heat Transf., vol. 107, pp. 361–368, 1985. 9.131 Hirasawa, S., Hijikata, K., Mori, Y., and Nakayama, W., Effect of surface tension on condensate motion in laminar film condensation. Int. J. Heat Mass Transf., vol. 23, pp. 1471–1478, 1980.

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9.132 Marto, P. J., and Nunn, R. H., The potential for heat transfer enhancement in surface condensers, Inst. Chem. Engineers Symp. Ser., no. 75 (ISBN 0 85295 152 3), pp. 23–39, 1982. 9.133 Webb, R. L., The use of enhanced surface geometries in condensers: An overview, in Power Condenser Heat Transfer Technology, P. J. Marto, and R. H. Nunn (editors), Hemisphere Publishing Company, New York, NY, pp. 287–324, 1981. 9.134 Cooper, J. R., and Rose, J. W., Condensation heat transfer enhancement by vapour-side surface geometry modification, Proc. 1980 HTFS Research Symp., Oxford, Paper RS402, pp. 642–672, 1981. 9.135 Marto, P. J., Recent progress in enhancing film condensation heat transfer on horizontal tubes, Heat Transfer 1986, Proc. 8th Int. Heat Transfer Conf, vol. 1, pp. 161–170, 1986. 9.136 Nicol, A. A., and Gacesa, M., Condensation of steam on a rotating vertical cylinder, J. Heat Transf., vol. 92, pp. 144–152, 1970. 9.137 Sparrow, E. M., and Gregg, J. L., A theory of rotating condensation, J. Heat Transf., vol. 81, p. 113, 1959. 9.138 Tleimat, B. W., Performance of a rotating flat-disk wiped-film evaporator, Paper no.71-HT-37, ASMEAIChE Heat Transfer, Tulsa, OK, August 15–18, 1971. 9.139 Lustenader, E. L., Richter, R., and Neugebauer, F. J., The use of thin films for increasing evaporation and condensation rates in process equipment, J. Heat Transf., vol. 81, p. 297, 1959. 9.140 Brodov, Y. M., Savel’yev, R. Z., Permyakov, V. A., Kuptsov, V. K., and Gal’perin, A. G., The effect of vibration on heat transfer and flow of condensing steam on a single tube, Heat Transf. Sov. Res., vol. 9, no. 1, pp. 152–156, 1977. 9.141 Seth, A. K., and Lee, L., The effect of an electric field on the presence of noncondensable gas on film condensation heat transfer, J. Heat Transf., vol. 96, pp. 257–258, 1974. 9.142 Lienhard, J., and Dhir, V., A simple analysis of laminar film condensation with suction, J. Heat Transf., vol. 94, pp. 334–336, 1977. 9.143 Bergles, A. E., Enhancement of boiling and condensation, in Two-Phase Flow and Heat Transfer, China-U.S. Progress, X.-J. Chen, and T. N. Veziroglu (editors), Hemisphere Publishing Company, New York, NY, 1985. 9.144 Webb, R. L., Principles of Enhanced Heat Transfer, Chapter 12, John Wiley & Sons, New York, NY, 1994.

PROBLEMS 9.1 Use the correlations of Rose et al. [9.47] to predict the heat transfer coefficient for dropwise condensation of steam at a surface subcooling of 5°C for pressures ranging between atmospheric pressure and 9460 kPa. Plot the variation of the heat transfer coefficient over this range of pressures. What fluid properties do you expect to affect this predicted variation the most? How do they vary with saturation pressure? 9.2 Using the correlations of Peterson and Westwater [9.1] and Rose et al. [9.47], predict the heat transfer coefficient for dropwise condensation of steam at atmospheric pressure for wall subcooling values of 2, 5, 10, 20, and 50°C. Plot the values of the heat transfer coefficient as a function of Tsat – Tw and compare the resulting variations for the two different correlations. Discuss possible reasons for the differences. 9.3 Film condensation of acetone occurs on a vertical flat plate. The plate is 30 cm long by 100 cm wide and is held at a constant temperature of 40°C. The plate is surrounded by motionlesssaturated acetone vapor at atmospheric pressure. Determine the mean heat transfer coefficient for the plate. 9.4 Film condensation of nitrogen occurs on a flat plate surrounded by motionless saturated nitrogen vapor at a pressure of 540 kPa. The plate is 20 cm long by 60 cm wide and is inclined at an angle of 45° to the horizontal. The plate is held at a constant temperature of 80 K. Determine the mean heat transfer coefficient for the plate. 9.5 Film condensation of ethanol occurs at atmospheric pressure on a 50 cm long, vertical plate held at 20°C. Determine and plot the estimated variation of the local heat transfer coefficient with local position along the plate. Assuming that transition to turbulent flow occurs at Ret = 2000, be sure to account for the laminar or turbulent conditions at each location. 9.6 Derive a modified form of Eq. (9.78) that also accounts for sensible cooling of the liquid in the overall energy balance. Describe how you would organize the overall computational scheme to use this improved relation to solve the system of similarity equations that govern the flow and temperature fields.

448

Liquid-Vapor Phase-Change Phenomena

9.7 Laminar film condensation of R-134a occurs at a pressure of 1604 kPa over a flat plate that is 20 cm long. The plate is held at a constant temperature of 40°C. For this process, determine the mean heat transfer coefficient over the surface of the plate using (a) the Nusselt correlation Eq. (9.55), and (b) Eq. (9.81) from the similarity analysis of Sparrow and Gregg [9.55]. Discuss the reasons for the difference in the results. 9.8 Film condensation of ammonia occurs on a flat, upward-facing surface, 50 cm long and 50 cm wide, inclined at 45° to the horizontal. The surface is exposed to a motionless ambient of ammonia gas at a pressure of 775 kPa. If heat is removed uniformly over the surface (i.e., a uniform heat flux), estimate the highest heat flux (into the surface) for which the condensate flow will be laminar over the entire surface. 9.9 Estimate the mean heat transfer coefficient for laminar film condensation of R-22 on a horizontal round tube with an outside diameter of 2 cm. The tube wall is held at 20°C. The tube is surrounded by motionless, saturated R-22 vapor at 1420 kPa. For the same wall subcooling, how does the heat transfer coefficient change as the pressure increases toward the critical point? 9.10 Equation (9.115a), which predicts the mean heat transfer coefficient for laminar film condensation on a horizontal cylinder, was derived using an integral boundary-layer analysis of the transport in the liquid film. The boundary-layer approximations in the analysis are generally expected to be valid if the film thickness is small compared to the diameter of the cylinder. Film condensation of steam occurs at atmospheric pressure on a horizontal cylinder with a wall temperature of 80°C. Use the results of the analysis presented in Section 9.4 to estimate the cylinder diameter below which the accuracy of the boundary-layer result is questionable. 9.11 Estimate the mean heat transfer coefficient for laminar film condensation of oxygen on a vertical row of 10 horizontal tubes. The tubes are exposed to saturated oxygen vapor at atmospheric pressure. The walls of the tubes are held at 77 K. The diameter and length of the tubes are 1.0 cm and 1.5 m, respectively. For the specified conditions, also estimate the fraction of the total heat duty for the row done by each tube. 9.12 In Example 9.5 it was estimated that for steam condensing on a vertical plate at atmospheric pressure, waves were expected to affect the transport at distances greater than 1.3 cm downstream of the leading edge. As the pressure of the system is increased, how does this location for the onset of wave-affected transport change? What happens to this location if a surfactant is added to the water? Explain briefly. 9.13 In Example 9.5 it was estimated that for steam condensing on a vertical plate at atmospheric pressure, waves were expected to affect the transport at distances greater than 1.3 cm downstream of the leading edge. Use Eq. (9.147) recommended by Kutateladze [9.69] to estimate the mean heat transfer coefficient for film condensation of water on a vertical surface with a heat removal rate of 500 kW/m 2. Also compute the mean heat transfer coefficient for these conditions using the classical Nusselt relation for film condensation and compare the results for the two methods. 9.14 Forced-convection film condensation of water occurs on a 50 cm long flat plate exposed to a flow of a mixture of steam and air. The ambient conditions are T∞ = 95°C, ux = 7 m/s, P∞ = 101 kPa, and Wg ,∞ = 0.02. The plate is held at a constant temperature of 50°C. For air and water vapor at the specified conditions, ρm = 0.60 kg/m3, Sc = 0.9, vm = 2.10 × 10 −5 m2/s, and at Tw = 50°C, hlv for the condensate water is 2383 kJ/kg. Experimental measurements of conditions at the film interface indicate that the interface concentration Wg,i at a distance x = 15 cm from the leading edge of the plate is equal to 0.25. Use the model analysis embodied in Eqs. (9.173)–(9.176) to determine the heat transfer coefficient and interface velocity vi at this x location. 9.15 Laminar film condensation occurs on a semi-infinite, vertical, isothermal surface as shown in Fig. P9.1. The cold surface at temperature Tw is surrounded by saturated steam at atmospheric pressure. An infinite hot surface at temperature Th is positioned parallel to the cold surface. The liquid-vapor interface and the hot surface radiate like blackbodies so that the net radiative heat flux to the interface at any location is given by



4 qR′′ = σ SB (Th4 − Tsat )

where



σ SB = 5.67 × 10 −8 W/m 2 K 4

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External Condensation

a. Write down an appropriate set of governing equations and boundary conditions that could be solved to predict the variation of the film thickness and heat transfer coefficient with x for the condensation process on the cooled surface. Be sure to include all necessary relations to close the system, (b) Assuming that the film remains laminar and that heat transfer across the liquid film is by conduction only, determine the asymptotic variation of the liquid film thickness far from the leading edge (large x) for Tw = 50°C and Th = 400°C.

FIGURE P9.1 9.16 Laminar film condensation occurs on a horizontal isothermal cylinder with a diameter of D surrounded by motionless saturated nitrogen vapor at atmospheric pressure. The cold surface of the cylinder is at a temperature Tw. The surrounding walls, which contain the vapor, are far from the surface, but they are at a temperature Th, which is much higher than the saturation temperature of the nitrogen. Consequently, these walls radiate a net heat flux to the liquid-vapor interface that is given by the blackbody relation



4 qR′′ = σ SB (Th4 − Tsat )

where σSB is the Stephan-Boltzmann constant. Extend the approximate integral analysis of laminar film condensation on a horizontal tube presented in Section 9.4 to include the radiation heat flux to the interface. Show that the mean heat transfer coefficient is given by h = m ′hlv / πR(Tsat − Tw ) , where R = D/2 and m ′ is obtained by solving the following equation with the initial condition m ′ = 0 at Ω = 0:



dm ′ kl R(Tsat − Tw )sin1/3 Ω Rq ′′ = − R dΩ hlv (m ′)1/3 [ 3µ l / ρl g(ρl − ρv ) ]1/3 hlv

9.17 Inside a shell and tube condenser, essentially motionless saturated R-12 vapor at 1602 kPa surrounds the outside of 1.8 cm diameter tubes. Use Eq. (9.124) to estimate the mean heat transfer coefficient for condensation of R-12 on the outside of 8 vertically aligned tubes. Repeat the calculation for R-134a at 1604kPa. Based on your results, if the system refrigerant was switched from R-12 to R-134a, do you think the condenser heat transfer performance would be better, worse or about the same? Quantitatively justify your answer.

Part III Internal Flow Convective Boiling and Condensation

10

Introduction to Two-Phase Flow

10.1  TWO-PHASE FLOW REGIMES In internal convective vaporization and condensation processes, the vapor and liquid are in simultaneous motion inside the channel or pipe. The resulting two-phase flow is generally more complicated physically than single-phase flow. In addition to the usual inertia, viscous and pressure forces present in single-phase flow, two-phase flows are also affected by interfacial tension forces, the wetting characteristics of the liquid on the tube wall, and the exchange of momentum between the liquid and vapor phases in the flow. Before describing convective condensation and boiling processes, it is both useful and necessary to first develop a framework for treating the associated two-phase flow. As will be seen in later chapters, the morphology of the two-phase flow very often plays a critical role in the determination of the heat and mass transfer during vaporization and condensation processes. For concreteness, we will begin by considering the very simple two-phase flow shown in Fig. 10.1. Although we will refer specifically to this flow configuration, the basic definitions and terminology developed here will be applicable to any gas-liquid flow circumstance. The total mass flow rate through the tube m is equal to the sum of the mass flow rates of gas m v and liquid m l ,

m = m v + m l (10.1)

The ratio of vapor flow to total flow x,

x=

m v (10.2) m

is sometimes called the dryness fraction or the quality, since it is often taken to be equal to the thermodynamic quality during convective vaporization and condensation processes. In similar fashion, the value of 1 − x = m l / m is sometimes referred to as the wetness fraction. For a channel with cross-sectional area A, the mass flux or mass velocity G is defined as

G=

m (10.3) A

The void fraction α is defined as the ratio of the gas flow cross-sectional area Av to the total crosssectional area A,

α=

Av (10.4) A

where A must equal the sum of the cross-sectional areas occupied by the two phases

A = Av + Al (10.5)

It follows directly that the liquid area fraction αl is given by

αl = 1 − α =

Al (10.6) A 453

454

Liquid-Vapor Phase-Change Phenomena

FIGURE 10.1  Idealized model of two-phase liquid-vapor flow in an inclined tube.

Based on the assumption that α and α l are average cross-section fractions, they are often assumed to equal the respective average volume fractions of each phase in the flow. It is also useful to define superficial gas and liquid fluxes, jv and jl , respectively, as

jv = jl =

Gx (10.7a) ρv

G (1 − x ) (10.7b) ρl

Although these parameters have units of velocity, they can also be thought of as the volume flux of each phase through the channel. Numerically they are equal to the velocity that each phase would have if it flowed at its specified mass flow rate through the channel alone. In the two-phase flow shown in Fig. 10.1, the arrangement of the phases is relatively simple. In general, however, the morphology of the two phases can be quite complex, and it can vary depending on the fluid properties and flow conditions.

Upward Vertical Flow For co-current upward flow in a vertical round tube, the possible observed flow regimes are indicated in Fig. 10.2. At very low quality, the flow is usually found to be in the bubbly flow regime, which is characterized by discrete bubbles of vapor dispersed in a continuous liquid phase. In bubbly flow, the mean size of the bubbles is generally small compared to the diameter of the tube.

FIGURE 10.2  Schematic representations of flow regimes observed in vertical upward co-current gas-liquid flow.

Introduction to Two-Phase Flow

455

At slightly higher qualities, smaller bubbles may coalesce into slugs that span almost the entire cross section of the channel. The resulting flow regime is usually referred to as slug flow. At much higher quality levels, the two-phase flow generally assumes an annular configuration with most of the liquid flowing along the wall of the tube, and the gas flowing in the central core. For obvious reasons, this regime is termed the annular flow regime. When the vapor flow velocity is high, the interface of the liquid film may become Helmholtz unstable, leading to the formation of waves at the interface. Liquid droplets formed by breaking waves may then be entrained in the vapor core flow. At intermediate qualities, one of two additional regimes may be observed. If both the liquid and vapor flow rates are high, an annular-type flow is observed with heavy “wisps” of entrained liquid flowing in the vapor core. Although this is a form of annular flow, it is sometimes designated as a separate regime, referred to as wispy annular flow. For intermediate qualities and lower flow rates, the vapor shear on the liquid-vapor interface may be near the value where it just balances the combined effects of the imposed pressure gradient and the downward gravitational body force on the liquid film. As a result, the liquid flow tends to be unstable and oscillatory. The vapor flow in the center of the tube flows continuously upward. Although the mean velocity of the liquid film is upward, the liquid experiences intermittent upward and downward motion. The flow for these conditions is highly agitated, resulting in a highly irregular interface. This oscillatory flow is referred to as churn flow. The conditions corresponding to the flow regimes described above can be represented on the flow regime map shown in Fig. 10.3. The form of this map was proposed by Hewitt and Roberts [10.1]. The vertical coordinate is equal to the superficial momentum flux of the vapor, and the horizontal coordinate is the superficial momentum flux of liquid through the tube. The boundaries between the flow regimes have been established from visual observation of the two-phase flow in a series of experiments (using a transparent tube) that spanned the entire flow regime map. Since the flow regime for a given set of conditions is a matter of judgment regarding the apparent morphology of the flow, the boundaries should be interpreted as specifying the middle of a transition between two regimes. From theoretical considerations, analytical expressions for the transition conditions between the two-phase flow regimes have also been obtained. Radovcich and Moissis [10.2] presented arguments about the frequency of bubble collisions which suggest that the transition from bubbly to slug

FIGURE 10.3  Flow regime map of the type proposed by Hewitt and Roberts [10.1].

456

Liquid-Vapor Phase-Change Phenomena

flow is highly probable at void fractions above α = 0.3. Based on a more detailed analysis, Taitel and Dukler [10.3] proposed the relation

jl [ g(ρl − ρv )σ ]1/ 4 = 2.34 − 1.07 (10.8) jv jv ρ1/l 2

as defining the incipient conditions for the transition from bubbly to slug flow. As noted above, increasing quality can lead to a transition from slug flow to churn flow. This breakdown of slug flow is a consequence of the interaction between the rising slug bubble and the liquid film between the slug and the wall. In a flow of this type, this liquid film actually moves downward as the slug moves upward at a velocity higher than the mean velocity of the two-phase flow, due to its buoyancy. As the quality and void fraction increase, this countercurrent flow becomes unstable in a manner similar to the Helmholtz instability described in Chapter 4. This instability eventually leads to the break up of the large bubbles characteristic of slug flow, initiating a transition to churn flow. Porteus [10.4] presented theoretical arguments that suggest that this transition corresponds to conditions defined by the relation

jl [ gD(ρl − ρv )]1/ 2 = 0.105 − 1 (10.9) jv jv ρ1/v 2

where D is the tube diameter. Taitel and Dukler [10.3], on the other hand, argued that for ( jl + jv ) / ( gD)1/ 2 greater than 50, the slug to churn transition occurs at conditions that correspond to jl /jv = 0.16 . The transition from churn flow to annular flow occurs at conditions where the upward shear stress of the vapor core flow plus the imposed pressure gradient just balances the downward gravitational force on the liquid film. These conditions correspond to the lower vapor velocity limit for which steady upward annular flow can be sustained. Based on theoretical arguments, Wallis [10.5] concluded that this transition occurred approximately at conditions specified by the relation

  jv2 ρv  gD(ρ − ρ )  l v  

0.5

= 0.9 (10.10)

Taitel and Dukler [10.3] proposed the following relation as a means of predicting the transition from churn to annular flow

jv ρ0.5 (1 + 20 X + X 2 )0.5 − X v = 3.09 (10.11a) 0.25 (1 + 20 X + X 2 )0.5 [ g(ρl − ρv )σ ]

where X is the Martinelli parameter defined as 1/ 2



 (dP /dz )l  X=  (10.11b)  (dP /dz ) v 

In Eq. (10.11b), (dP/dz)l and (dP/dz)v are the frictional pressure gradients for the liquid and vapor phases flowing alone in the pipe, respectively. These frictional gradients can be computed as

2 fl G 2 (1 − x )2  dP  = − (10.12a)   dz  l ρl D

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Introduction to Two-Phase Flow



2 fv G 2 x 2  dP  = − (10.12b)   ρv D dz  v fl = B Rel− n fv = B Re −v n

Rel =

G (1 − x ) D (10.13a) µl

Re v

GxD (10.13b) µv

In the above friction factor relations, for round tubes the constants can be taken to be B = 16 and n = 1, respectively, for laminar flow (Rel or Rev < 2000), or B = 0.079 and n = 0.25 for turbulent flow (Rel or Rev ≥ 2000). The transition between wispy-annular flow and annular flow is difficult to distinguish precisely because the regimes are so similar. Based on experiments that used a probe to detect wispy filaments in the core flow, Wallis [10.5] proposed the following correlation for the transition condition

jv  ρ  =  7 + 0.06 l  (10.14) jl  ρv 

This relation is recommended for jl ρl0.5 [ gD(ρl − ρv )]−0.5 > 1.5.

Horizontal Flow For two-phase flow in horizontal round tubes, the flow regimes that may be encountered are shown in Fig. 10.4. One of the main differences between the regimes observed for horizontal flow and those for vertical flow is that there is often a tendency for stratification of the flow. Regardless of

FIGURE 10.4  Schematic representation of flow regimes observed in horizontal, co-current gas-liquid flow.

458

Liquid-Vapor Phase-Change Phenomena

the flow regime, the vapor tends to migrate toward the top of the tube while the lower portion of the channel carries more of the liquid. At very low quality, bubbly flow is often observed for horizontal flow. However, as indicated in Fig. 10.4, the bubbles, because of their buoyancy, flow mainly in the upper portion of the tube. As the quality is increased in the bubbly regime, coalescence of small bubbles produces larger plug-type bubbles which flow in the upper portion of the tube (Fig. 10.4). This referred to as the plug flow regime. At low flow rates and somewhat higher qualities, stratified flow may be observed in which liquid flowing in the bottom of the pipe is separated from vapor in the upper portion of the pipe by a relatively smooth interface. If the flow rate and/or the quality is increased in the stratified flow regime, eventually the interface becomes Helmholtz unstable, whereupon the interface becomes wavy. This type of flow is categorized as wavy flow. The strong vapor shear on the interface for these circumstances, together with the formation and breaking of waves on the interface may lead to significant entrainment of liquid droplets in the vapor core flow. At high liquid flow rates, the amplitude of the waves may grow so that the crests span almost the entire width of the tube, effectively forming large slug-type bubbles. Because of their buoyancy, the slugs of vapor flowing along the tube tend to skew toward the upper portion of the tube. In other respects it is identical to slug flow in vertical tubes, and hence it too is referred to as slug flow. At high vapor velocities and moderate liquid flow rates, annular flow is observed for horizontal gas-liquid flow. For such conditions, buoyancy effects may tend to thin the liquid film on the top portion of the tube wall and thicken it at the bottom. However, at sufficiently high vapor flow rates, the vapor flow is invariably turbulent, and strong lateral Reynolds stresses and the shear resulting from secondary flows may serve to distribute liquid more evenly around the tube perimeter against the tendency of gravity to stratify the flow. The strong vapor shear may also result in significant entrainment of liquid in the vapor core. Since gravitational body forces are often small compared to inertia effects and turbulent transport of momentum, the resulting flow for these circumstances is generally expected to differ little from annular flow in a vertical tube under similar flow conditions. Flow regime maps for gas-liquid flow in horizontal or slightly inclined round tubes have been proposed by Baker [10.6], Mandhane et al. [10.7], and Taitel and Dukler [10.8]. Of these, the map proposed by Taitel and Dukler [10.8], shown in Fig. 10.5, has perhaps the most carefully conceived

FIGURE 10.5  Flow regime map for horizontal co-current gas-liquid flow of the type proposed by Taitel and Dukler [10.8].

459

Introduction to Two-Phase Flow

theoretical basis. Although this map is computationally a bit more difficult to use than the others, it at least attempts to account for the different combinations of physical parameters that affect different regime transitions on the map. The horizontal coordinate on the Taitel-Dukler [10.8] map is the Martinelli parameter X defined by Eqs. (10.11b) and (10.12). The value of this parameter fixes the horizontal position on this map regardless of the flow regime. However, the second dimensionless parameter used to determine the flow regime varies depending on the specific transition being considered. For the stratified-to-wavy-flow transition, the vertical position of the corresponding point in Fig. 10.5 is specified in terms of the parameter KTD defined as 0.5



  ρv jv2 jl K TD =   (10.15a) v ( ρ − ρ ) g cos Ω v  l l 

where vl is the kinematic viscosity of the liquid and Ω is the angle of inclination between the tube axis and the horizontal. The wavy-annular and wavy-intermittent (plug or slug) transitions in Fig. 10.5 are evaluated in terms of X and the parameter FTD defined as 0.5



  ρv jv2 FTD =  (10.15b)   (ρl − ρv ) Dg cos Ω 

The transition from bubbly flow to intermittent flow is specified in terms of X and yet a third parameter T TD. 0.5



 −(dP /dz )l  TTD =   (10.15c)  (ρl − ρv ) g cos Ω 

where (dP/dz)l is given by Eq. (10.12a). The transition between intermittent and annular flow or bubbly and annular flow corresponds simply to X = 1.6 on this map.

General Observations Co-current upward and co-current horizontal two-phase flows are by far the most commonly encountered configurations, and a round tube is the most common geometry used in technological applications. However, a few other flow circumstances and channel geometries, that arise in other applications, have also been investigated. Studies of vertical upward and horizontal co-current gas-liquid flow in plain rectangular channels [10.9], in channels with offset strip fins [10.10], and in cross-ribbed channel geometries [10.11] have indicated that the flow regimes and transitions in these geometries are similar to those in round tubes. The two-phase flow regimes for upward or horizontal co-current flow in tubes with internal grooves [10.12], expansions and contractions [10.13] and helical inserts [10.14], and in bends and helically wrapped coils [10.15–10.17] have also been explored. Although flow regimes similar to those observed in corresponding round tube flows are often observed, in some cases these geometry variations can produce significant differences in the flow behavior. Some additional aspects of geometry effects will be discussed in the last section of this chapter. Based on the results of experimental studies, Oshinowo and Charles [10.18] have proposed flow regime maps for vertical downward co-current gas-liquid flow in a round tube. For such circumstances, regimes of bubbly, slug, churn, and annular flow similar to the regimes observed in upward co-current flow may be observed.

460

Liquid-Vapor Phase-Change Phenomena

FIGURE 10.6  Additional flow regimes observed under special conditions.

Additional flow regimes can arise in upward or horizontal co-current flow when the tube is very small or when the liquid poorly wets the tube wall. For very small tubes, the capillary bubble flow shown in Fig. 10.6 can result. In this regime the vapor bubble completely fills the tube cross section, leaving the wall at that location completely dry. In larger tubes, if the liquid poorly wets the wall, so-called capillary flow may result in which the liquid flows in rivulets on the wall, driven by vapor shear forces. If the liquid on the wall breaks up into droplets, the drop-flow regime may be encountered, in which the vapor flow drags individual droplets along the wall (Fig. 10.6). An overview of two-phase flow patterns encountered in applications and flow-pattern maps can be found in the review article by Cheng et al. [10.19]. When a phase change occurs as the two-phase mixture flows along the channel, different flow regimes are generally observed at different positions along its length. The sequence of flow regimes observed will depend primarily on the flow rate, channel orientation, fluid properties, and the distribution and magnitude of the heat flux into or out of the flow at the channel wall. The sequence of flow regimes for upward flow boiling in a vertical heated tube at low to moderate heat flux levels is shown in Fig. 10.7. Boiling may be initiated before the bulk liquid reaches the saturation temperature. At this initial stage of the boiling process, the void fraction is low and bubbly flow results. As the vaporization process continues, and liquid is converted to vapor, the void fraction increases, jv increases, and jl decreases. Consequently, the flow regime progressively changes from bubbly flow to slug flow, slug flow to churn flow, and from churn flow to annular flow. On the flow regime map in Fig. 10.3, the sequence of system state points would trace a curve similar to curve A

461

Introduction to Two-Phase Flow

FIGURE 10.7  Sequence of flow regimes observed during upward flow boiling in a vertical tube.

in that figure. Note that since the total mass flux is the same everywhere along the tube, the path of the system on this map is defined by the relation

ρv jv + ρl jl = G = constant

In a similar fashion, horizontal co-current flow with vaporization or condensation results in the sequences of flow regimes indicated in Figs. 10.8 and 10.9, respectively. It should be noted, however, that the sequence of flow regimes encountered may differ from those indicated by an adiabatic flow

FIGURE 10.8  Horizontal co-current flow with evaporation.

462

Liquid-Vapor Phase-Change Phenomena

FIGURE 10.9  Horizontal co-current flow with condensation.

pattern map for the local conditions in the tube. Some examples include locally strong vaporization effects drying out a portion of the wall in annular flow, or vapor recoil effects in a vaporizing flow altering the morphology of the two-phase mixture. This has motivated some recent efforts to develop two-phase flow pattern maps for flows with condensation or vaporization effects present (see, e.g., the studies by Kattan et al. [10.20] and Nema et al. [10.21]). As discussed in Chapters 11 and 12, the sequence of flow regimes that occur along the tube has a very strong impact on the heat transfer characteristics associated with convective boiling and condensation processes in tubes and ducts. Example 10.1 A two-phase mixture of saturated nitrogen liquid and vapor at atmospheric pressure flows upward in a vertical tube. The tube inside diameter is 1.0 cm and the mass flux is 300 kg/m 2s. Estimate the values of quality at which the transitions from bubbly to slug and churn to annular flow will occur. For saturated nitrogen at atmospheric pressure, ρl = 807 kg/m3, ρv = 4.62 kg/m3, and σ = 0.00885 N/m. Combining Eqs. (10.7) and (10.8), the bubble-to-slug transition is estimated to occur at a quality that satisfies

 ρv   1− x  [ g(ρl − ρv )σ ]1/ 4  ρ   x  = 2.34 − 1.07 ρ1/ 2Gx / ρ l

l

v

Substituting the property and G values yields

0.00572(1– x ) = 2.34 x – 0.00168

Solving for x = xb–s yields

xb – s = 0.0032

Combining Eqs. (10.7a) and (10.10), the churn-annular transition is predicted to occur at a quality predicted approximately by the relation

  G2 x  ρ ( ρ − ρ ) gD v l v  

0.5

= 0.9

463

Introduction to Two-Phase Flow Substituting the appropriate values of G, D, and the densities, and solving for x = xc–a yields

xc – a = 0.057

Thus, the flow is predicted to be in the annular flow regime at qualities as low as 0.06.

10.2 BASIC MODELS AND GOVERNING EQUATIONS FOR ONE-DIMENSIONAL TWO-PHASE FLOW General Considerations Generalized analytical treatment of liquid-vapor two-phase flow is an extremely challenging task. Because of the physical complexity of such flows, simplified models of the mechanisms of phase interaction and transport have often been used as component parts of this type of analysis. The dramatic increase in the speed of computers over the past 40 years has aided efforts to computationally model two-phase flows. The availability of increased computing power has allowed development of two-phase flow models that rely more directly on the fundamental governing equations of fluid dynamics. However, even these more advanced models frequently incorporate simplified submodels of some mechanisms in the flow. As a result, research efforts continue to explore the development of simplified models of two-phase flow mechanisms, and they continue to explore how such simplified models can be incorporated into fundamental model frameworks to generate computationally efficient, yet accurate predictions of transport in two-phase flow. A full exploration of detailed modeling of two-phase flow is beyond the scope of the coverage here. Instead, we will explore the simplest model of two-phase transport, which treats the flow as steady and one-dimensional. Developments of the full governing equations for three-dimensional, time-varying two-phase flow can be found in works by Ishii [10.22] and Boure [10.23]. The form of the governing equations can be simplified by invoking time and/or space averaging. However, in doing so, information regarding the instantaneous localized behavior of the flow is lost. In the treatment of internal gas-liquid flows considered here, the flow is considered to be steady and onedimensional in the sense that all dependent variables are idealized as being constant over any cross section of the tube or duct, varying only in the axial direction. To facilitate development of a one-dimensional analysis of gas-liquid flow, we will consider the system shown in Fig. 10.10. Although a stratified flow is shown for concreteness, the relations derived for conservation of mass and momentum will be generally applicable to gas liquid flows in any of the regimes described in the previous section, within the one-dimensional approximations adopted here. The individual mean phase velocities indicated in this diagram are related to the void fraction as

uv = ul =

Gx (10.16a) ρv α

G (1 − x ) (10.16b) ρl (1 − α)

Conservation of mass requires that

m = m v + m l (10.17a)



m = GxA + G (1 − x ) A (10.17b)

464

Liquid-Vapor Phase-Change Phenomena

FIGURE 10.10  Idealized model of momentum transport during gas-liquid two-phase flow in an inclined tube.

Differentiating these relations, the conservation of mass requirement can also be expressed in the forms

dm v = − dm l (10.18)

or

d d (GxA) + (G (1 − x ) A) = 0 (10.19) dz dz

Alternate forms of the continuity relation can also be derived by combining one of the above equations with the variable definitions (10.2)–(10.7) or (10.16). The second governing equation for the flow in Fig. 10.10 is derived from a force-momentum balance in the axial direction. A balance of this type on the vapor phase only gives



PAv − ( P + dP)( Av + dAv ) − dFv − dFi , v − Av dzρv g sin Ω (10.20) = (m v + dm v )(uv + duv ) − m v uv − dm v ul

The first two terms on the left side of this relation represent pressure forces on the element. The terms dFv and dFi,v represent the frictional effect of the vapor on the channel wall and the interfacial shear force, respectively. The first two terms on the right side of the equation account for the momentum change due to acceleration (or deceleration) of the vapor. The last term on the right side represents the momentum exchange between the liquid and vapor due to a phase change at the interface that transfers liquid molecules into the vapor. (An equivalent term could be included for transfer in the opposite direction instead, if necessary. Terms of this type cancel later anyway.) In a completely analogous way, a momentum balance for the liquid can be written as

PAl − ( P + dP)( Al + dAl ) − dFl − dFi ,l − Al dzρl g sin Ω (10.21) = (m l + dm l )(ul + dul ) − m l ul − dm l ul

465

Introduction to Two-Phase Flow

For steady flow in a duct of uniform cross-sectional area, the interfacial shear forces must balance so that dFi ,l = – dFi , v (10.22)

and Eq. 10.5 implies that

dA = dAv + dAl = 0 (10.23)

Adding Eqs. (10.20) and (10.21) and using Eqs. (10.18), (10.22), and (10.23), the following relation for the overall momentum balance is obtained

− AdP − dFl − dFv − ( Al ρl + Av ρv ) gdz sin Ω = d (m v uv + m l ul ) (10.24)

Defining a fictitious pressure gradient (dP/dz)fr to account for the combined frictional effect of the two phases on the channel wall

 dP  dFl + dFv = −  Adz (10.25)  dz  fr

and using Eqs. (10.4) and (10.16), the momentum balance relation (10.24) can be written as

d  G 2 x 2 G 2 (1 − x )2   dP   dP  g = − + − α ρ + αρ Ω + + − [(1 ) ] sin (10.26) l v    dz  dz  fr dz  ρv α ρl (1 − α) 

There are three contributions to the overall pressure gradient for the two-phase flow in the channel represented on the right side of the above equation. The first and second terms account for frictional and gravitational head effects, respectively, and the last term represents acceleration (or deceleration) of the flow. If no phase change is occurring, x and α will be constant and the last term will be zero. To further develop this one-dimensional model of momentum transport we will invoke the following additional idealizations: 1. At a given axial location in the channel, the velocities of the liquid (ul) and vapor (uv) phases are uniform over the portion of the channel occupied by each, but are not necessarily equal. 2. The two phases are in local thermodynamic equilibrium. 3. Empirical correlations or relations derived from simplified theories are available to predict the void fraction α and one of the two-phase multipliers ϕ l, ϕ lo, or ϕ v from parameters which quantify the local flow conditions. The two-phase multipliers mentioned above are defined as 1/ 2



 (dP /dz ) fr  φl =   (10.27)  (dP /dz )l 



 (dP /dz ) fr  φlo =   (10.28)  (dP /dz )lo 



 (dP /dz ) fr  φv =   (10.29)  (dP /dz ) v 

1/ 2

1/ 2

466

Liquid-Vapor Phase-Change Phenomena

In the above expressions the subscript designations are as follows: fr denotes the frictional component of the two-phase pressure drop in Eq. (10.26). l denotes the frictional pressure gradient that would result if the liquid flowed alone through the channel (at a mass flow rate equal to G(1 – x)A). lo denotes the frictional pressure gradient that would result if liquid only flowed through the channel at the same total mass flow rate (GA). v denotes the frictional pressure gradient that would result if the vapor flowed alone through the channel (at a mass flow rate equal to GxA). In terms of the two-phase multiplier ϕ l, the momentum equation (10.26) can be written as d  G 2 x 2 G 2 (1 − x )2   dP   dP  = −φl2  + [(1 − α )ρl + αρv ]g sin Ω +  − + (10.30)    dz   dz  l dz  ρv α ρl (1 − α) 



This force-momentum balance relation can also be written in terms of the multipliers ϕ lο and ϕ ν as d  G 2 x 2 G 2 (1 − x )2   dP   dP  = −φlo2  + [(1 − α )ρl + αρv ]g sin Ω +  − + (10.31)    dz   dz  lo dz  ρv α ρl (1 − α) 

and

d  G 2 x 2 G 2 (1 − x )2   dP   dP  = −φ2v  + [(1 − α)ρl + αρv ]g sin Ω +  − + (10.32)    dz   dz  v dz  ρv α ρl (1 − α) 

Homogeneous Flow The so-called homogeneous model of momentum transport in gas-liquid flows can be considered to be a special case of the separated flow analysis described thus far. For reasons that will become obvious, this model is also sometimes called the friction factor model or fog flow model. The idealizations adopted under this model, in addition to those noted above for separated flow, are: 1. The vapor and liquid velocities are equal. 2. The two-phase flow behaves like a single phase having fluid properties whose values are, in some sense, mean values for the flow. Equating the expressions (10.16a) and (10.16b) for uv and ul, the following relation between the void fraction α and the quality is obtained:

α=

x / ρv xvv = (10.33) [(1 − x ) / ρl ] + ( x / ρv ) (1 − x ) vl + xvv

where vv and vl denote the specific volumes of the vapor and the liquid, respectively. Treating the two-phase flow as an equivalent single-phase flow, the frictional contribution to the overall pressure gradient can be determined using a conventional friction factor.

2 ftp G 2  dP  = − (10.34)  dz  fr ρd h

467

Introduction to Two-Phase Flow

In the above expression, ftp is an effective (Fanning) friction factor for the two-phase flow, d h is the hydraulic diameter, and ρ is the mean density of the flow given by the usual thermodynamic definition

1 x 1− x = v = vv x + (1 − x ) vl = + (10.35) ρ ρl ρv

The fictitious single-phase pressure gradient for the entire flow as liquid can be similarly evaluated in terms of a friction factor

2 f G2  dP  = lo (10.36) −   dz  lo ρl d h

Substituting Eqs. (10.34) and (10.36) into Eq. (10.27), the following relation can be obtained for φlo2 :

φlo2 =

ftp flo

  ρl   1 +  − 1 x  (10.37)     ρv

The friction factors can usually be expressed as power-law functions of the Reynolds number −m



flo = M Re

−m lo

 Gd  = M  h  (10.38a)  µl 

−n tp

 Gd  = N  h  (10.38b)  µ 

−n



ftp = N Re

where dh is the hydraulic diameter and µ is the effective mean viscosity for the two-phase flow. The ratio of the friction factors is then given by

ftp  N   Gd h  = flo  M   µ l 

m−n

n

 µ  µ  (10.39) l

Substituting this result into Eq. (10.37), the relation for φlo2 becomes

N  Gd  φlo2 =    h   M   µl 

m−n

n

   µ    ρl  µ  1 +  ρ − 1 x  (10.40) v l  

In the above treatment, the friction factor relations for the two-phase flow and the entire flow as liquid have been assumed to be different. This is appropriate if the flow of the two-phase mixture is turbulent whereas the entire flow as liquid would be laminar. If both flows are in the same regime, the relation should be the same (i.e., M = N, n = m) and the relation for φlo2 simplifies to n



   µ  ρ φlo2 =   1 +  l − 1 x  (10.41)  µ l    ρv  

This simplification removes the dependence of φlo2 on the mass flux G.

468

Liquid-Vapor Phase-Change Phenomena

The only remaining detail to be completed at this point is the evaluation of the mean viscosity µ. In general, a weighted average with respect to the quality of the flow is perhaps the most logical means of defining µ. A mean value defined in this manner should approach the liquid viscosity and vapor viscosity as x → 0 and x → 1, respectively. Three proposed relations that satisfy these criteria are:

1 x 1− x = + (10.42) µ µv µl



µ = xµ v + (1 − x )µ l (10.43)



µ xµ v (1 − x )µ l = + (10.44) ρ ρv ρl

Equations (10.42)–(10.44) were proposed by McAdams et al. [10.24], Cicchitti et al. [10.25], and Dukler et al. [10.26], respectively. Equation (10.42) is probably the most commonly used definition of µ. For turbulent single-phase flow in round tubes, the Fanning friction factor can be determined from the well-known Blasius correlation  Gd  f = 0.079  h   µ 



−0.25

(10.45)

If both the two-phase flow and the entire flow as liquid are turbulent, then M = N = 0.079 and n = m = 0.25. Combining Eq. (10.41) with Eq. (10.42) for such circumstances yields  µ   φlo2 = 1 +  l − 1 x      µv



−0.25

  ρl   1 +  − 1 x  (10.46)     ρv

Substituting the void-fraction relation (10.33), the acceleration term on the right side of Eq. (10.31) simplifies to d  G 2 x 2 G 2 (1 − x )2  d = [G 2 v ] (10.47) + dz  ρv α ρl (1 − α)  dz

where from Eq. (10.35)

v = vl + ( vv − vl ) x (10.48)

Applying the derivative to the product of terms in the parentheses on the right side of Eq. (10.47), and assuming the variation of the liquid density due to changes in temperature and pressure along the channel is negligible, the following relation is obtained



d  G 2 x 2 G 2 (1 − x )2  d dv   dx = G 2 [ v ] = G 2  vlv + x v  (10.49) + dz  ρv α dz dz  ρl (1 − α)   dz

where vlv has been used to designate vv – vl. Using the chain rule to evaluate dvv /dz as

dvv dvv = dz dP

 dP    (10.50) dz 

469

Introduction to Two-Phase Flow

Eq. (10.49) can be written as  dx d  G 2 x 2 G 2 (1 − x )2  dv  dP   = G 2  vlv +x v  +  (10.51)   dz  ρv α dP  dz   ρl (1 − α)   dz



This decomposes the acceleration term into two parts. The two resulting terms account for acceleration or deceleration of the flow due to changes in the flow density, either due to changes in quality or due to expansion of the vapor as the pressure varies along the duct. Substituting Eq. (10.51) into Eq. (10.31), using Eq. (10.33) to evaluate α in the gravitational head terms, and solving for –dP/dz yields

dv  dP   2  dP   dx  g sin Ω   1 + G 2 x v  (10.52) =  −φlo  + G 2 vlv   + −    dz    dz  lo  dz  vl + vlv x   dP 

Using Eq. (10.36) to evaluate (dP/dz)lo then yields

2 dv  dx  g sin Ω    dP   2  2 flo G vl  + G 2 vlv   + 1 + G 2 x v  (10.53) = φlo  −   dz  vl + vlv x    dz    d h  dP 

Given appropriate information about the flow conditions along the channel, this equation can be integrated to determine the pressure variation along the duct, as predicted by the homogeneous model.

Separated Flow Returning to the more general separated flow case, the acceleration term in Eq. (10.30) can be expanded by simple application of the chain rule to obtain



d  G 2 x 2 G 2 (1 − x )2  dx   2 xvv 2(1 − x ) vl + = G2 −    dz  ρv α dz   α (1 − α) ρl (1 − α) 

 dα  (1 − x )2 vl x 2 vv    + dx  (1 − α)2 − α 2      

dP  x 2  dvv  dα  (1 − x )2 vl x 2 vv   − 2  +G 2   + α   dz  α  dP  dP  (1 − α)2

(10.54)

As for the homogeneous model above, the frictional pressure drop for the liquid phase alone in Eq. (10.30) can be evaluated in terms a friction factor f l: 2 f G 2 (1 − x )2  dP  = l − (10.55)   dz  l ρl d h



Substituting the expressions (10.54) and (10.55) into Eq. (10.30) and solving for –dP/dz yields



1   2 f G 2 (1 − x )2   dP  − =  φl2  l   + [(1 − α)ρl + αρv ]g sin Ω  dz  Λ  ρl d h  dx   2 xvv 2(1 − x ) vl +G −  (1 − α) dz   α 2

 dα  (1 − x )2 vl x 2 vv    + dx  (1 − α)2 − α 2      

(10.56a)

470

Liquid-Vapor Phase-Change Phenomena

where

 x 2 dv  dα  (1 − x )2 vl x 2 vv   − 2   (10.56b) Λ = 1 + G 2   v  +  dP  (1 − α)2  α    α dP

Note that Eq. (10.56a) can be equivalently be written in terms of f lo and φlo2 as



1   2 f G2   dP  − =  φlo2  lo  + [(1 − α)ρl + αρv ]g sin Ω   dz  Λ   ρl d h  dx   2 xvv 2(1 − x ) vl  dα  (1 − x )2 vl x 2 vv   +G − + − 2    (1 − α)  dx  (1 − α)2 α   dz   α

(10.57)

2

Integration of Eq. (10.56a) or Eq. (10.57) to determine the pressure variation along the duct can be accomplished only if the two-phase multiplier and the void fraction can be determined from the local conditions all along the duct. Methods of determining these parameters are discussed in detail in the next section.

10.3 DETERMINATION OF THE TWO-PHASE MULTIPLIER AND VOID FRACTION The separated flow analysis described in the Section 10.2 can be used to predict the pressure gradient for a gas-liquid flow only if some means of determining the two-phase multiplier and void fraction are available. The methodologies developed to predict these quantities have typically been based on a mixture of semi-theoretical arguments and empirical evidence. The pioneering work of Martinelli and his coworkers in the 1940s provided the first widely successful methods for predicting ϕ l and α, and established an analytical foundation on which much of the subsequent work on two-phase flow has been built. Lockhart and Martinelli [10.27] proposed a generalized correlation method for determining the two-phase multiplier ϕ l or ϕ v from which the frictional pressure gradient can be predicted for adiabatic gas-liquid flow in a round tube. This correlation is based on the premise that ϕ l and ϕ ν are, to a first approximation, functions only of the Martinelli parameter X defined by Eq. (10.11b). Lockhart and Martinelli [10.27] showed that this type of correlation provided a good match to data from a series of studies of adiabatic two-phase flow in horizontal tubes. Chisholm and Laird [10.28] subsequently formulated the Lockhart and Martinelli [10.27] correlation in terms of the following relations for ϕ l and ϕυ: 1/ 2



C 1  φl =  1 + + 2  (10.58a)  X X 



φ v = (1 + CX + X 2 )1/ 2 (10.58b)

In the above equations, the value of the constant C differs, depending on the flow regime associated with the flow of the vapor and the liquid alone in the tube. The constant values recommended by Chisholm and Laird [10.28] for each of the four possible combinations are indicated in Table 10.1. For round tubes, the liquid flow is generally regarded as being turbulent for Rel > 2000, and laminar for Rel ≤ 2000. For the gas flow, the transition from laminar to turbulent may similarly be taken to occur at Rev = 2000.

471

Introduction to Two-Phase Flow

TABLE 10.1 Separated Flow Regimes Liquid

Gas

Turbulent Viscous Turbulent Viscous

Turbulent Turbulent Viscous Viscous

Subscript Designation

C

tt vt tv vv

20 12 10 5

At first glance, it may not be apparent that ϕl and ϕυ should depend on the Martinelli parameter X. However, by considering the variation of these parameters with quality, the link can be made more obvious. First, as the quality increases, for a given total flow rate, (dP/dz)l decreases and (dP/dz)v increases. As a result, X monotonically decreases as the quality increases. It also follows directly from the definitions of φl and φ v that as x → 0 (X →∞), φl → 1, φ v  → X , and as x → 1 (X → 0), φl  → 1/X , φ v → 1. This asymptotic behavior requires that φl vary from 1 at low quality to 1/X at high quality. It is not unreasonable, as a first hypothesis, to presume that φl varies solely as a function of X between these limits. Similar arguments can be applied to the variation of φ v. The dependence of φl and φ v on X can be made more concrete by considering the separate cylinders model of gas-liquid flow described by Wallis [10.5]. In this model, an actual two-phase flow such as that shown in Fig. 10.11a is taken to be equivalent to the model flow circumstances shown in Fig. 10.11b, in which the vapor and liquid phases flow at the same flow rates through separate cylinders. The radii of the vapor (r ve) and liquid (rle) cylinders are required to satisfy the relations

rve2 = α (10.59a) r02



rle2 = 1 − α (10.59b) r02

FIGURE 10.11  System model for separate cylinders analysis of two-phase flow.

472

Liquid-Vapor Phase-Change Phenomena

This ensures that the combined total flow area of the two cylinders is the same as that for the actual pipe, and that the cross-sectional area associated with each phase is the same. The pressure gradients in each of the model cylinders are assumed to be equal to each other, and their value is taken to be equal to the two-phase frictional pressure gradient in the actual flow. With the idealizations described above, the pressure gradient in the separate cylinder carrying the vapor flow is given by 2 2 2 2 2 2 2  dP  2 fve  (πr0 G ) x /(πrve )  fve G x = = − (10.60)    2  dz  2rve  ρv  ρv rve α



where f ve is presumed to be a function of the Reynolds number Reve defined as Re ve =



(πr02 G ) x (2rve )/(πrve2 ) 2rve Gx = (10.61) µv µvα

For the vapor flowing alone through the actual pipe, the pressure gradient is given by f G2 x2  dP  = v − (10.62)   dz  v ρv r0



where the friction factor f v is a function of Rev defined as Re v =



2r0 Gx (10.63) µv

Assuming that f v and f ve both obey the same power-law dependence on Reynolds number f = BRe–n, and making use of the assumption that dP/dz in the separate cylinders equals (dP/dz)fr in the actual cylindrical pipe, Eqs. (10.29) and (10.60)–(10.63) can be combined to obtain the following expressions for φ2v: r  φ2v =  0   rve 



5− n

=

1 (10.64) α (5− n )/ 2

Based on similar arguments, it can be shown that for the liquid flowing in the separate model cylinder that

2 2 2 2 2 2 2  dP  2 fle  (πr0 G ) (1 − x ) /(πrle )  fle G (1 − x ) = = − (10.65)    2  dz  2rle  ρl  ρl rle (1 − α)

where f le is a function of

Rele =

(πr02 G )(1 − x )(2rle )/(πrle2 ) 2rle G (1 − x ) = (10.66) µl µ l (1 − α)

For the liquid flowing alone in the actual pipe

f G 2 (1 − x )2  dP  = l − (10.67)   dz  l ρl r0

473

Introduction to Two-Phase Flow

where f l depends on Rel given by Rel =



2r0 G (1 − x ) (10.68) µl

Assuming that both liquid model flows obey the same power-law variation of f with Reynolds number as for the vapor flows, the above equations can be combined with Eq. (10.27) to obtain the following relation for φl2:

r  φl2 =  0   rle 

5− n

=

1 (10.69) (1 − α)(5− n )/ 2

Equations (10.64) and (10.69) can be combined to eliminate α, which yields the relation

1 1 + = 1 (10.70) φl4 /(5− n ) φ4v /(5− n )

Multiplying through by φl4 /(5− n ) and using the fact that

X2 =

(dP /dz )l φ2v = (10.71) (dP /dz ) v φl2

the following relation is obtained for φl2:

4 /(5− n )   1 φl2 = 1 +      X 

(5− n )/ 2

(10.72)

It can be similarly be shown that φ2v is given by

φ2v = [1 + X 4 /(5− n ) ](5− n )/ 2 (10.73)

If both the liquid and vapor flows are laminar, n = 1 for a round tube, and the relation for φl2 becomes 2

  1  φ = 1 +    (10.74)   X vv   2 l



If the flows associated with the two phases are both turbulent, n = 0.25 for a round tube, which reduces Eq. (10.72) to 19/8



  1  16/19  φ = 1 +      X tt   2 l

(10.75)

The above relations for φl differ only slightly from the corresponding relations in the LockhartMartinelli correlation. Perhaps of greater significance is the fact that this analysis directly demonstrates that different φ( X ) relations are expected to apply, depending on the flow regimes associated with each phase.

474

Liquid-Vapor Phase-Change Phenomena

Lockhart and Martinelli [10.27] also correlated the vapor void fraction as a function of the Martinelli parameter X. As noted by Butterworth [10.29], their correlation is well represented by the relation

α = [1 + 0.28 X 0.17 ]–1 (10.76)

The Lockhart-Martinelli correlations for the two-phase multipliers and the void fraction are plotted in Fig. 10.12. Martinelli and Nelson [10.30] later developed a correlation technique for predicting the pressure drop during flow boiling inside tubes. Their correlation is based on the Lockhart-Martinelli methodology described above, with the following additional idealizations: 1. Thermodynamic equilibrium exists at each location along the tube in which flow boiling is taking place. 2. For forced-flow boiling, the two-phase flow corresponds to the turbulent-turbulent case. 3. The frictional contribution to the overall pressure gradient for flow boiling is equal to that predicted by a correlation for horizontal adiabatic gas-vapor flow under comparable conditions. Based on the above idealizations, it should be possible to predict the frictional contribution to the flow boiling pressure gradient using the Lockhart-Martinelli correlation. For such circumstances, the two-phase multiplier φlo should approach 1 as the system saturation pressure approaches the

FIGURE 10.12  Graphical representation of the Lockhart-Martinelli correlation.

475

Introduction to Two-Phase Flow

critical point. However, Martinelli and Nelson [10.30] found that the values of φlo predicted by the Lockhart-Martinelli correlation near the critical point were about 5. From these results, they surmised that the two-phase multiplier must vary with pressure as well as with the Martinelli parameter X. Assuming that the predictions of the Lockhart-Martinelli correlation were correct at atmospheric pressure and adopting a variation of the two-phase multiplier that was consistent with the critical point limit and available data at intermediate pressures, Martinelli and Nelson [10.30] developed the correlations for (φl )tt and 1 – α as functions of pressure and a parameter χtt defined as

ρ  χtt =  v   ρl 

0.571

 µl   µ  v

0.143

 1− x   (10.77)  x 

These correlations for flow boiling in a round tube are shown in Fig. 10.13. The definition of χtt differs only slightly from that of the Martinelli parameter for turbulent-turbulent flow Xtt as given by Eqs. (10.11b)–(10.13) with n = 0.25:

ρ  X tt =  v   ρl 

0.5

 µl   µ  v

0.125

 1− x    x 

0.875

(10.78)

It can be seen in Fig. 10.13 that increasing pressure at a given χtt value tends to decrease α and (φl )tt . From Eqs. (10.27), (10.28), (10.36), and (10.55) it can easily be shown that

φlo2 f = l (1 − x )2 (10.79) φl2 flo

Using Eq. (10.45) to evaluate f l and f lo for round tubes, the relation between φlo2 and φl2 becomes

φlo2 = φl2 (1 − x )1.75 (10.80)

FIGURE 10.13  Graphical representation of the Martinelli-Nelson correlation. (Adapted from [10.30] with permission, copyright © 1948, American Society of Mechanical Engineers.)

476

Liquid-Vapor Phase-Change Phenomena

2 FIGURE 10.14  Variation of φlo with quality and pressure as predicted by the Martinelli-Nelson correlation. (Adapted from [10.30] with permission, copyright © 1948, American Society of Mechanical Engineers.)

Martinelli and Nelson [10.30] used this relation together with the correlation for φl2 represented in Fig. 10.13 to obtain a prediction of φlo2 for flow boiling of water in round pipes as a function of quality and pressure. Their predicted variation of φlo2 with quality and pressure is shown graphically in Fig. 10.14. In general, integration of Eq. (10.52) or Eq. (10.57) to determine the pressure variation along the pipe is so complex that closed-form integration is impossible. In the most general circumstances, numerical integration is necessary to predict the pressure variation. However, under some special circumstances, calculation of the total pressure drop can be greatly simplified. In particular, for separated flow, simplified calculation of the total pressure drop is possible if the following conditions are satisfied: 1. The cross-sectional area of the duct is constant. 2. Λ ≈ 1, which implies that the compressibility of the gas phase is negligible. 3. The friction factor f lo and the liquid and vapor densities ρl and ρv are constant along the entire flow passage. 4. Liquid is evaporated from saturated liquid at the inlet (x = 0 at z = 0) to some arbitrary quality xe at z = ze, and the quality varies linearly with z along the tube (dx/dz = constant). Invoking the above conditions and integrating Eq. (10.57) from z = 0 to z = ze, the following relation for the overall pressure drop ΔP is obtained xe



2 flo G 2 ze  1  2 G2 −∆P = φ dx + lo d h ρl  xe  ρl

∫ 0

xe

 x 2 ρl (1 − x )2   αρ + (1 − α) − 1  v  x = xe

z g sin Ω [ρv α + ρl (1 − α)]dx + e xe

∫ 0

(10.81)

477

Introduction to Two-Phase Flow

The correlations for the two-phase multiplier and void fraction described above can be used to evaluate the terms xe



 1 I φ =   φlo2 dx (10.82a)  xe 



 x ρl (1 − x )2  aα =  + − 1 (10.82b) (1 − α)  αρv  x = xe



1 [ρv α + ρl (1 − α)]dx (10.82c) Iρ = xe

∫ 0

2

xe

∫ 0

as functions of exit quality and pressure alone. The variations of Iϕ and aα as functions of xe and pressure were determined by Martinelli and Nelson [10.30] using their correlations for ϕ lo and α for water and steam. For flow boiling of water under the conditions described above, these two computed variations are sufficient to compute the overall pressure drop if the gravitational head term is small. Thom [10.31] later published alternative variations of α, ϕ lo, Iϕ, aα, and Iρ developed from a more extensive body of two-phase pressure drop data for steam-water flow in horizontal and vertical tubes. The resulting variations of I φ, aα, and Iρ are plotted in Figs. 10.15–10.17. The simplistic Lockhart-Martinelli-Nelson correlations for void fraction and two-phase multiplier clearly do not account for effects other than the fluid properties and quality. Comparison of the

FIGURE 10.15  Variation of Iϕ with exit quality and pressure for water as predicted by the correlation of Thom. (Adapted from [10.31] with permission, copyright © 1964, Pergamon Press.)

478

Liquid-Vapor Phase-Change Phenomena

FIGURE 10.16  Variation of aα with exit quality and pressure for water as predicted by the correlation of Thom. (Adapted from [10.31] with permission, copyright © 1964, Pergamon Press.)

FIGURE 10.17  Variation of Iρ with exit quality and pressure for water as predicted by the correlation of Thom. (Adapted from [10.31] with permission, copyright ©, 1964, Pergamon Press.)

479

Introduction to Two-Phase Flow

predictions of these correlations with data suggests that there is a systematic variation of the twophase multiplier with mass flux that these correlations do not take into account. Several attempts have been made to develop improved correlations for the two-phase multiplier and void fraction. However, it has been generally found that improved accuracy can be obtained only with more complicated correlation techniques or by developing correlations only for a specific gas-liquid pair over a limited range of conditions. A subsequent correlation proposed by Baroczy [10.32] attempted to account for the effects of mass flux on the two-phase multiplier, and to develop a broadly based correlation technique that could be used for a variety of gas-liquid combinations. This correlation technique is fundamentally the same as the Martinelli-Nelson (separated flow) method with the following modifications: 1. The two-phase multiplier φlo2 corresponding to a reference mass flux of 1355 kg/m2s was correlated as a function of quality x and a property index defined as 0.2



ρ  µ  Property Index =  v   l  (10.83)  ρl   µ v  2. The effect of mass flux on the two-phase multiplier was taken into account by providing a separate correlation for the correction factor γ, defined as



γ=

(φ ) 2 l0

φlo2

(10.84)

G =1355kg/m 2 s

This correlation technique is represented graphically in Figs. 10.18 and 10.19. At a given flow condition, the property index can be evaluated, and it and the quality can be used with Fig. 10.18 to determine φlo2 at the reference mass flux. Figure 10.19 is used to determine the correction factor that, when multiplied by the value of φlo2 for the reference mass flux, yields the predicted value of φlo2 for the conditions of interest. This value can then be used to evaluate the frictional contribution to the overall pressure gradient in Eq. (10.57). Baroczy [10.33] also proposed a void fraction correlation in which the void fraction was correlated in terms of the turbulent-turbulent Martinelli parameter

FIGURE 10.18  Correlation for φlo2 at G = 1335 kg/m2s proposed by Baroczy. (Adapted from [10.32] with permission, copyright © 1965, American Institute of Chemical Engineers.)

480

Liquid-Vapor Phase-Change Phenomena

FIGURE 10.19  Correlation for the mass flux and property effects correction factor proposed by Baroczy. (Adapted from [10.32] with permission, copyright © 1965 American Institute of Chemical Engineers.)

Xtt and the property index used in his two-phase multiplier correlation. This correlation is shown graphically in Fig. 10.20. The complex behavior of the multiplier correction (γ) curves has been perhaps the most controversial aspect of this correlation method. No interpretation of the complex tangle of curves in Fig. 10.19 has been offered. However, since the conditions represented in this figure span several quite different flow regimes, it is possible that some of the complex changes in the correction factor variations may be associated with flow regime transitions. Friedel [10.34] subsequently used a database of 25,000 points to develop the following correlation for predicting the two-phase multiplier φlo2 for vertical upward and horizontal flow in round tubes.

φlo2 = CF 1 +

3.24CF2 (10.85) Fr 0.045 We 0.035

where

 ρ  f  CFl = (1 − x )2 + x 2  l   vo  (10.86)  ρu   flo 

481

Introduction to Two-Phase Flow

FIGURE 10.20  Void fraction correlation proposed by Baroczy. (Adapted from [10.33] with permission, copyright © 1965, American Institute of Chemical Engineers.)



CF 2 = x

0.78

(1 − x )

0.24

 ρl   ρ  v

0.91

 µv   µ  l

0.19

0.7

 µv   1 − µ  (10.87) l

G2 (10.88) gd h ρtp2



Fr =



We  =  

G 2 dh (10.89) ρtp σ −1



 x 1− x  ρtp =  + (10.90) ρl   ρv

In the above correlation, f vo and f lo are friction factors for the total mass flowing as vapor and liquid, respectively. For single component two-phase flows, Friedel [10.34] found that the standard deviation of the data relative to the correlation was about 30%. This correlation has been recommended for use when (μl /μv) < 1000 (see reference [10.35]). For large diameter tubes Chenoweth and Martin [10.36] found that a good fit to available data was provided by correlating φlo2 as a function of the inverse of the property index used by Baroczy [10.32] (defined by Eq. (10.83)) and the liquid volume fraction. As discussed in Section 10.1, the vapor volume fraction β is usually taken to be equal to the void fraction α. The liquid volume fraction is equal to 1 – β. This correlation is shown graphically in Fig. 10.21. In this figure, the property index, rather than its inverse, are indicated for each curve. This correlation is usually recommended for tubes having inside diameters greater than 5 cm. A method for predicting the void fraction is essential for predicting the acceleration and gravitational head components of the pressure gradient in two-phase flows. Butterworth [10.29] has shown that several of the available void fraction correlations can be cast in the general form −1



n2 n3 n  1 − x  1  ρv   µ l    α = 1 + BB   (10.91)  x   ρl   µ v    

482

Liquid-Vapor Phase-Change Phenomena

2 FIGURE 10.21  Correlation for φlo proposed by Chenoweth and Martin [10.36] for large-diameter tubes.

where the values of the unspecified constants in this relation corresponding to different correlations are listed in Table 10.2. This formulation facilitates comparison of the different models and makes using them somewhat easier. It also illustrates some inconsistencies. When ρl = ρv and μl = μv the void fraction should equal the quality, implying that BB and n1 should equal 1 in all the correlations. In those correlations where these constants are not 1, some inaccuracy may be expected when the densities and the viscosities of the two phases are nearly the same (i.e., near the critical point). For engineering design calculations, selection of a two-phase flow model should be done carefully. The homogeneous model yields reasonably accurate results only for limited circumstances. The best agreement is expected for bubbly or dispersed droplet (mist) flows, where the slip velocity between the phases is small. The Lockhart-Martinelli [10.27] correlation methodology has been shown to yield reasonably accurate results for a wide variety of two-phase flow circumstances in round tubes and simple channel geometries. This type of correlation has also been adapted to fit data for complex finned channel geometries [10.38]. Given the database from which this correlation was derived, it is expected to yield best results at low pressures for adiabatic or boiling flows at low heat flux levels. Baroczy’s [10.32] correlation, although more complex than some of the others, is expected to give good agreement with experimental data for a wide variety of circumstances. Baroczy [10.32] found that this correlation agreed with a large set of experimental data within ±20%. Because it is a fit to a very large database, the Friedel [10.34] correlation is likely to provide the best accuracy of the methods discussed here when the circumstances of interest are within its recommended range of use.

TABLE 10.2 Void Fraction Correlation Parameters Correlation or Model Homogeneous model Zivi [10.37] model Wallis [10.5] separate-cylinders model Lockhart and Martinelli [10.27] Thom correlation [10.31] Baroczy correlation [10.33]

BB

n1

n2

n3

1 1 1 0.28 l 1

1 1 0.72 0.64 1 0.74

1 0.67 0.40 0.36 0.89 0.65

0 0 0.08 0.07 0.18 0.13

483

Introduction to Two-Phase Flow

Other useful methods for computing predictions of two-phase frictional pressure drop are discussed by Muller-Steinhagen and Heck [10.39]. Further discussion of the models and correlations discussed in this section can be found in the books by Wallis [10.5], Chisholm [10.40], and Hetsroni [10.35]. Example 10.2 Saturated R-134a at a pressure of 338 kPa flows adiabatically from an expansion valve to the inlet of an evaporator in a tube with a length of 0.7 m and an inside diameter of 1.0 cm. The mass flux is 250 kg/m2s, and the quality is 0.25. From the valve to the evaporator inlet, the flow rises 0.5 m in elevation. Determine (a) the flow regime and void fraction, and (b) the gravitational and frictional pressure drop. a. For saturated R-134a at 338 kPa, ρl = 1281 kg/m3, ρv = 16.6 kg/m3, μl = 258 × 10 –6 Ns/m2, and μv = 10.9 × 10 –6 Ns/m2. For the specified conditions, ρv jv2 = G 2x 2 / ρv = (250)2(0.25)2 / 16.6 = 235 kg/s 2m

and

ρl jl2 = G 2(1− x )2 / ρl = (250)2(1− 0.25)2 / 1281 = 27.4 kg/s 2m



2 2 From Fig. 10.3, and the above values of ρv jv and ρl jl , the flow is predicted to be in the annular flow regime. Using Eq. (10.91) with the constants from Table 10.2 for the Lockhart and Martinelli model, 0.36 0.07 0.64   ρv   µ l    1− x  α = 1+ 0.28    ρ   µ    x    l v



–1

Substituting and solving for α yields –1

0.64 0.36 0.07   0.75   16.6   258   α = 1+ 0.28   = 0.871       0.25   1281  10.9    



b. Using Eq. (10.57), neglecting compressibility effects (Λ = 1), and setting dx/dz = 0, the gravitational contribution – (dP/dz)G is given by –(dP / dz )G = [(1– α )ρl + αρv ]g sin  Ω



Since α is constant, it follows that – ∆PG = [(1– α )ρl + αρv ]g∆z sin Ω

Substituting,

– ∆PG = [(1– 0.871)(1281) + 0.871(16.6)](9.8)(0.7)(0.5/0.7) = 881 Pa = 0.881 kPa For the respective phases flowing alone,



Rev =

GxD 250(0.25)(0.01) = = 57,300 µv 10.9 × 10 −6



Re l =

G(1− x )D 250(0.75)(0.01) = = 7270 µl 258 × 10 −6

484

Liquid-Vapor Phase-Change Phenomena

Thus the flow is turbulent-turbulent, and the Martinelli parameter is given by Eq. (10.78): ρ  Xtt =  v   ρl 



0.5

 16.6  =   1281

 µl   µ  v 0.5

0.125

 258    10.9 

 1− x    x  0.125

0.875

 0.75    0.25 

0.875

= 0.442

From Eq. (10.58a) with C = 20, 1/ 2

1  C φl = 1+ + 2   X X 



1/ 2

20 1   = 1+ + 2 0.442 (0.442)  

= 7.17

Using Eqs. (10.45) and (10.80),  GD  flo = 0.079   µ l 



φlo2 = φl2 (1− x )

1.75

−0.25

 250(0.01)  = 0.079   258 × 10 −6 

= (7.17) ( 0.75) 2

1.75

−0.25

= 0.00796

= 31.1

From Eq. (10.57), it follows that the frictional component of the pressure drop –ΔPF is given by  2f G 2   dP  ∆z = φlo2  lo  ∆z −∆PF = −    dz  F  ρl D 



 2(0.00796)(250)2  = (31.1)   (0.7) = 1691 Pa  =  1.69 kPa  (1281)(0.01) 

Example 10.3 Water leaves an evaporator at 90% quality and flows along a horizontal tube 3.0 m long and 2.0 cm in inside diameter. Heat is added uniformly along the tube so that the quality at the exit is exactly 100%. The flow conditions are such that G = 600 kg/m2s and the pressure at the tube inlet is 3773 kPa. Determine the pressure drop along the tube and compare it to that if pure saturated vapor alone flowed through it. For saturated water at 3773 kPa, ρl = 804 kg/m3, ρv = 18.9 kg/m3, μl = 127 × 10 –6 Ns/m2, and μ ν = 17.0 × 10 –6 Ns/m2. Writing the frictional component of the pressure gradient in terms of ϕv, if Λ = 1, the separated-flow pressure gradient relation for a horizontal tube (Ω = 0) can be written as  

 2f G 2x 2  d  x 2vv (1− x )2v l   dP  = φv2  v − + G2 −   dz  dz  α (1− α )   ρvD 



Integrating this equation from z = z1 to z = z 2 yields the following relation for the pressure drop – ΔP:

−∆P =

 x 2vv (1− x )2v l   x 2vv (1− x )2v l  2G 2 Iz + G 2  − − G2  −  ρvD (1− α )  z = z2 (1− α )  z = z1  α  α

where z = z2



Iz =

∫ f x φ dz v

z = z1

2 2 v

485

Introduction to Two-Phase Flow

Note that in obtaining the above relations, the densities of the vapor and liquid have been taken to be constant along the tube. At x = 0.95,

Re l =

G(1− x )D 600(0.05)(0.02) = = 4724 µl 127 × 10 −6



Rev =

GxD 600(0.95)(0.02) = = 6.71 × 105 µv 17 × 10 −6





fv = 0.079Rev−0.25 = 0.079(6.71× 105 )−0.25 = 0.00276 ρ  Xtt =  v   ρl 

0.5

 18.9  =   804 

 µl   µ  v 0.5

0.125

 127    17 

 1− x   x 

0.125

0.875

 0.05    0.95 

0.875

= 0.0150

φv2 = 1+ CX + X 2 = 1+ 20 Xtt + Xtt2 = 1+ 20(0.015) + (0.015)2 = 1.300

Since both Reynolds numbers are above 2000, turbulent relations for fv and x have been used, and the turbulent-turbulent Martinelli correlation was used to determine ϕv. It can similarly be shown that



fv = 0.00280,  φv2 = 1.578  at  x = 0.9 fv = 0.00273,  φv2 = 1.0       at  x   = 1.0

Using the trapezoidal rule and the results of the above calculations, Iz is approximated numerically as

(

1 Iz =  fv x 2φv2 2

)

z=0

(

+ fv x 2φv2

)

z =1.5

+

(

1 fv x 2φv2 2

)

z = 3.0

  ∆z

Since heat is applied uniformly along the tube, z = 0, 1.5, 3.0 correspond to x = 0.9, 0.95, 1.0, respectively. It follows that



1 Iz =  (0.00280)(0.9)2(1.578) + (0.00276)(0.95)2(1.300) 2 1      + (0.00273)(1)2 (1)] (1.5) = 0.00959 2

At x = 1, α = 1 and the second term in the above expression for –ΔP reduces to G 2v v. At x = 0.9, α is obtained by using Eq. (10.91) with the constants from Table 10.2 for the Lockhart and Martinelli model:



0.36 0.07 0.64   ρv   µ l    1− x  α = 1+ 0.28   ρ   µ    x  l v  

−1

−1

0.64 0.36 0.07   0.1  18.9   127   = 1+ 0.28       = 0.980  0.9  804   17   

486

Liquid-Vapor Phase-Change Phenomena

Substituting into the relation for –ΔP yields −∆P =

 (0.9)2 2(600)2(0.00959) (600)2 (0.1)2  – (600)2  – +  18.9(0.02) 18.9  0.98(18.4) 0.080(804) 

= 18, 267 + 19,048 – 15,520 = 21,800 Pa  =  21.8 kPa

The pressure drop for pure saturated vapor flow is determined in the following sequence of steps:

Re =

GD 600(0.02) = = 7.06 × 105 µv 17 × 10 −6

fv = 0.079Rev−0.25 = 0.079(7.06 × 105 )−0.25 = 0.00273 −∆P =

2LfG 2 2(3)(0.00273)(600)2 = = 15,500 Pa  =  15.6 kPa ρvD 18.9(0.02)

Thus, the pressure drop for the vaporizing flow is more than for the pure vapor flow. Note that this is due to the higher frictional contribution and the added acceleration effect.

Example 10.4 Saturated R-410A flows adiabatically through a horizontal tube with an inside diameter of 1.5 cm. At a specific location in the tube, x = 0.4 and P = 600 kPa. Compute the pressure gradient at this location using the Baroczy and Martinelli correlations if G = 339 kg/m2s. For saturated R-410A at 600 kPa, Tsat(P) = 264.3 K = −8.7°C, ρl = 1204 kg/m3, ρv = 23.0 kg/m3, μl = 191 × 10 –6 Ns/m2, and μν = 11.8 × 10 –6 Ns/m2. The property index used in the Baroczy correlation is determined as

 µl   µ  v

( )

From Fig. 10.18, at x = 0.4: φlo2 1.29. Thus

0.2

1335

 ρv   191  0.2  23.0   ρ  =  11.8   1204  = 0.0333 l = 17.5. Using Fig. 10.19, the correction factor γ for G = 339 is

φlo2 = (φlo2 )1335   γ = (17.5)(1.29) = 22.6 Re lo =

GD 339(0.015) = = 2.66 × 10 4 µl 191× 10 −6

flo = 0.079Re lo−0.25 = 0.079(2.66  ×  10 4 )−0.25 = 0.00619  2f G 2   2(0.00619)(339)2   dP  = φlo2  lo  = 22.1 −  = 1740 Pa/m  dz  Bar  ρl D   (1204)(0.015) 

Using the Martinelli correlation,

Re l =

G(1− x )D 339(0.6)(0.015) = = 1.79  ×  10 4 µl 191× 10 −6



Rev =

GxD 339(0.4)(0.015) = 1.72  ×  105 = µv 11.8 × 10 −6

487

Introduction to Two-Phase Flow (thus the flow is turbulent-turbulent)



ρ  Xtt =  v  ρ  l

0.5

 µl   µ  v

 23.0  =   1204 

0.5

0.125

 1− x    x 

 191    11.8 

0.125

0.875

 0.6    0.4 

0.875

= 0.279

C 1 20 1 + = 1+ + = 85.5  Xtt Xtt2 0.279 (0.279)2



φl2 = 1+



fl = 0.079Re l−0.25 = 0.079(1.79  ×  10 4 )−0.25 = 0.00683



 2f G 2(1− x )2   dP  = φl2  l −    dz  Mar ρl D    2(0.00683)(339)2(0.6)2  = (85.5)   = 2675 Pa/m (1204)(0.015)  

Thus the two correlations agree within about 40% at G = 339 kg/m2s.

10.4  ANALYTICAL MODELS OF ANNULAR FLOW During convective vaporization and condensation processes inside tubes, the large difference between the liquid and vapor densities away from the critical point can result in high vapor void fractions even at relatively low quality. As an example, consider the evaporation of R-134a in the tubes of an air-conditioning evaporator at a saturation pressure of 387 kPa (corresponding to a saturation temperature of 8°C). For turbulent-turbulent flow in a round tube at a quality of 20%, the Martinelli parameter Xtt is about 0.6 and the Lockhart-Martinelli correlation (10.76) predicts that the corresponding void fraction is 0.83. At the moderate wall superheat levels typical of air-conditioning evaporators, this would very likely correspond to the annular flow regime with a thin film of liquid flowing along the tube walls, while vapor and possibly some liquid droplets flow in the central core region. It is noteworthy that the 20% quality condition considered above would correspond to the inlet condition of a typical evaporator, after the refrigerant completes the throttling process in the expansion valve. Thus the entire vaporization process occurs at void fractions greater than 0.83. Vaporliquid flows for such conditions will generally correspond to the annular flow regime, up to the point where the liquid film on the wall begins to dry out. Hence annular flow is the predominant flow regime over most of the length of the flow passage. Similar arguments for the convective condensation process in the condenser of a typical air-conditioning system indicate that annular flow will also exist for most of the length of the condensing-side passage. The importance of annular flow to air-conditioning systems and other applications has stimulated efforts to develop accurate methods to predict the transport for such circumstances. In this section, analytical treatments of annular flow will be considered in detail. Although annular flow may occur in channels of any cross section, the development here will focus on annular flow in round tubes. Extensions of the treatment for round tubes are often adequate to analyze the transport for annular flow in other geometries.

488

Liquid-Vapor Phase-Change Phenomena

FIGURE 10.22  Idealized model system used in analysis of annular flow without entrainment.

Annular Flow with No Entrainment To develop the model analysis of annular flow in a vertical round tube, we consider the configuration shown in Fig. 10.22. The model assumes (a) that the flow is steady, (b) that the downstream pressure gradients felt in the core and the liquid film are the same, and (c) that liquid flows in an annular film on the inside wall of the tube that has a uniform thickness δ and a smooth liquid-vapor interface. In most cases, the interface is expected to be wavy. However, the effects of the wavy interface on heat transfer and pressure drop are often small or can be treated by empirically modifying a smoothinterface model. As a first approximation, these effects will be neglected here. In addition, entrainment of liquid in the vapor core will, at least initially, be assumed to be zero. Modifications of the analysis to include entrainment effects will discussed later in this section. As shown by Hewitt and Hall-Taylor [10.41], the stream-wise force-momentum balance for the vapor core flow in Fig. 10.22 can be expressed as

2 2  ri  dP  ro  d  2  ro  x 2  τi = −  +  G    + ρv g  (10.92) 2  dz  ri  dz   ri  ρv   

where τi is the shear stress at the liquid-vapor interface. The terms inside the curly brackets on the right side represent, from left to right, the pressure gradient, acceleration, and body force terms. Using the fact that the core void fraction α = (ri/ro)2, and expanding the derivative with respect to z, the acceleration term can be written in the form

2 2 2  1  d  G x  =  2G x   dx  1 − x  dα   (10.93)         2 α dz  αρv   α ρv   dz   2α  dx  

For most systems of practical interest, the term (x/2α)(dα/dx) is small compared to one. This term will therefore be neglected, whereupon Eq. (10.92) can be written as where D = 2ro.

dP 4 τ 2 xG 2  dx  = − i − 2     − ρv g (10.94) dz D α ρv  dz 

489

Introduction to Two-Phase Flow

Neglecting inertia effects, the steady-state balance of shear stress, pressure, and gravitational body forces at a given r location in the liquid film requires that

2 2 r 1  dp r −r  τ(r ) = τi  i  +  + ρl g   i (10.95)  r  2  dz   r 

The cylindrical geometry requires that

ri D / 2 − δ = (10.96) r D / 2− y

where y is the distance measured from the tube wall toward the centerline. Making use of the relation

du τi = (10.97) dy µ l + ερl

where ε is the eddy diffusivity, Eq. (10.95) can be rearranged to obtain

2   D / 2 − δ  1  dP τi du   D / 2 − y   D / 2 − δ  − 1 (10.98) = + + ρ g  l          µ l + ερl   D / 2 − y  dy µ l + ερl  D / 2 − y  2 dz  

In the above relations ε is the eddy diffusivity for turbulent momentum transport in the liquid film. An additional conservation requirement that must be satisfied is that the total mass flow rate in the film must be equal to π(D/2)2G(1 – x). This condition is expressed mathematically as δ



 D  G (1 − x ) = ρ udy (10.99) l  4

∫ 0

Conservation of energy (neglecting sensible cooling of the film) also requires that dx/dz is related to the wall heat flux as dx 4q″ = (10.100) dz GDhlv

Boundary conditions for the liquid film are

at   y = 0 :   u   =  0 (10.101a) at  y = δ :  

du τi = (10.101b) dy µ l + ερl

The second condition (10.101b) is automatically satisfied by Eq. (10.98). Equation (10.101a) thus provides the needed boundary condition for the first-order differential equation for u. For specified values of q ″, G, x, D, and fluid properties, the differential equation (10.98) with boundary condition (10.101a) can be solved for the velocity profile in the film only if relations are available to determine (dP/dz), τi, ε, and δ simultaneously. There are, however, only two additional relations, Eqs. (10.94) and (10.99), available, since Eq. (10.94) requires the use of Eq. (10.100) to evaluate dx/dz. Two additional relations are therefore required to close the system of equations mathematically. Closure of the system is usually achieved by including empirical relations for determining the interfacial shear stress and the eddy diffusivity.

490

Liquid-Vapor Phase-Change Phenomena

The interfacial shear stress is usually evaluated in terms of an interfacial friction factor fi defined such that τi =



fi G 2 x 2 (10.102) 2ρv α 2

In general the interfacial friction factor fi may be a function of the film thickness, quality, mass flux, and the tube diameter fi = fi (δ, x , G , D) (10.103)



Perhaps the simplest correlation of this type is that proposed by Wallis [10.5] δ  fi = 0.005  1 + 300  (10.104)  D



The eddy diffusivity in the liquid film for turbulent flow conditions has typically been evaluated using relations like those developed for single-phase turbulent flow in tubes. The Deissler correlation is one example of such a correlation 2

ε = n 2 uy(1 + e −ρl n uy / µl ) (10.105)



where n is a constant equal to 0.1. More recently, for thin films, variations of the eddy diffusivity have been used in which the eddy diffusivity goes to zero at both the wall and the interface, reflecting the damping effect of the interface on turbulence. An example of a variation of this type is the relation used by Blanghetti and Schlunder [10.42]

(

)

1/ 2    + 2 ε = −0.5 + 0.5 1 + 0.64( y + )2 1 − e − ( y ) /26    vl



( y ≤ δ ′ ) (10.106a)

  ε τ δ y+   + = 0.0161 Ka1/3 Re1.34 + + 1 (δ − y + )  ( y > δ ′ ) (10.106b) l  2 1/3 2 1/3  +    ρ − ρ δ vl g ( )( v / g ) ( v / g ) l v l l   where

Ka  =

ρ3l g 3 ( vl2 /g)2 y(τ w ρl )1/ 2 δ(τ w ρl )1/ 2 ,   y + = ,   δ + = (10.106c) σ vl vl

and Rel is the Reynolds number defined by Eq. (10.68). Note that τw is the wall shear stress (at y = 0) that can be determined from Eqs. (10.97) and (10.98). The delimiting value of y = δ’ is the location where the above two expressions for ε/vl, (10.106a) and (10.106b), intersect, which is usually near y = δ/2. In practice, both expressions can be evaluated and the smaller value of ε/vl is taken to be correct. It can be seen that since ε/vl is a function of the local shear stress, Eq. (10.97) must be used to eliminate τ, resulting in a very complex relation for du/dy. With the addition of relations for fi and ε/vl, solution of the governing equations for specified fluid properties and values of G, q″ and D may proceed as follows: 1. A value of δ is guessed. 2. Equations (10.102) and (10.104) can be used to determine τi. 3. Equations (10.94) and (10.100) can be used to compute dP/dz.

491

Introduction to Two-Phase Flow

4. Using a relation such as Eq. (10.105) to evaluate the eddy diffusivity, Eq. (10.98) is numerically integrated using boundary condition (10.101a) to determine the u velocity across the liquid film. The integral of u from y = 0 to y = δ is also determined numerically as part of this computation. 5. If the numerically determined integral of u across the film satisfies Eq. (10.99) for the specified conditions, then the solution is complete. If this condition is not satisfied, a new value of δ must be guessed and the sequence of calculations must be repeated beginning with step 2. This process is repeated until Eq. (10.99) is satisfied. Programming of the algorithm described above on a computer is relatively straightforward. The converged solution thus provides a prediction of the two-phase pressure gradient for the imposed flow conditions. In addition, since the void fraction can be computed directly from the film thickness, the computed solution also predicts the void fraction.

Modeling Annular Flow with Entrainment The model analysis described above can be significantly in error when liquid is entrained into the vapor core flow. Perhaps the most commonly used analytical treatment of annular flow with entrainment makes use of a modified form of the separated flow analysis described in Section 10.2. The total flow is postulated to consist of three “separated” flows: (1) the liquid flow in the film on the walls, (2) the vapor flow in the core, and (3) the liquid entrained in the core flow. In the one-dimensional force-momentum balance, frictional, gravitational, and acceleration contributions to the overall pressure gradient arise, just as in the separated flow model without entrainment. The frictional and gravitational contributions to the overall pressure gradient for separated flow with entrainment are identical to those for separated flow without entrainment (see Eq. (10.31)). As formulated here, the only difference between these two cases is in the acceleration term. Following the annular flow model analysis of Hewitt and Hall-Taylor [10.41], we designate the volume fraction of the liquid in the film on the walls as βf, and the mass fraction of the liquid phase entrained in the core is denoted as E. The mean stream-wise velocity associated with the vapor (uv), liquid film (ulf ) and entrained liquid (ule ) are given by uv =



Gx (10.107a) αρv



ulf =

G (1 − x )(1 − E ) (10.107b) β f ρl



ule =

G (1 − x ) E (10.107c) (1 − α − β f )ρl

The acceleration contribution to the overall pressure gradient –(dP/dz)acc can be written in terms of these velocities as

d  dP  αρ u 2 + β f ρl ulf2 + (1 – α – β f )ρl ule2  (10.108) = –   dz  acc dz  v u

Substituting the relations (10.107) yields

d  x2 (1 − E )2 (1 − x )2 E 2 (1 − x )2  dP  = G2  + + −   dz  acc dz  αρv β f ρl (1 − α − β f )ρl

  (10.109) 

492

Liquid-Vapor Phase-Change Phenomena

If it is further assumed that the droplets in the core flow at the same velocity as the gas, setting ule = uv yields the following relation for βf : βf = 1− α −



αE (1 − x )ρv (10.110) xρl

This relation can be substituted into Eq. (10.109) to eliminate βf :

d  x2 (1 − E )2 (1 − x )2 x E (1 − x ) x   dP  = G2  − + + (10.111)   dz  acc dz  αρv ρl (1 − α) − ρv αE (1 − x ) αρv 

Magiros and Dukler [10.43] proposed the following additional idealizations: (1) acceleration of the liquid film is negligible compared to acceleration of the core flow, and (2) the void fraction is, to a first approximation, equal to 1.0. With these additional idealizations, Eq. (10.111) can be further simplified to the following approximate form:

 G2  d 2  dP   x + x (1 − x ) E  (10.112) = −   dz  acc  ρv  dz 

Here, however, we will retain the full form of Eq. (10.111). Replacing the last term on the right side of Eq. (10.30) with the acceleration contribution specified by Eq. (10.111) yields the following separated-flow expression for the total pressure gradient:



 dP   dP  – = φl2  + [(1 – α)ρl + αρv ]g sin Ω   dz   dz  l E (1 – x ) x  d  x2 (1 – E )2 (1 – x )2 x +G + +  dz  αρv ρl (1 – α) – ρv αE (1 – x ) αρv 

(10.113)

2

Similarly, for the core flow, the stream-wise force-momentum balance obtained by modifying Eq. (10.92) to account for entrainment yields



4 τ i ρv g [ x + E (1 − x ) ]  dP  − = +  dz  D x + E (1 − x )ρv / ρl +G 2

E (1 − x ) x  d  x2 (1 − E )2 (1 − x )2 x + +  dz  αρv ρl (1 − α ) − ρv αE (1 − x ) αρv 

(10.114)

In addition, for conditions under which entrainment occurs, conservation of mass in the liquid film requires that δ



 D  G (1 − x )(1 − E ) = ρ u   dy (10.115) l  4

∫ 0

If a means is available to predict the entrainment E, the pressure gradient and void fraction for annular flow with entrainment can be determined using the same iterative algorithm described above for the no-entrainment case. Methods used to predict the entrainment are generally empirical in nature. Several of the earlier efforts to develop entrainment models have been reviewed in the

493

Introduction to Two-Phase Flow

paper by Govan et al. [10.44]. Upon reviewing the available entrainment and deposition data, these investigators proposed an improved correlation scheme for predicting the amount of entrainment in vertical upward annular flow in round tubes. Using their methodology, the equilibrium entrainment is computed as follows. First, the minimum liquid film mass flux at which entrainment will occur Glfo is computed from the relation

µ  ρ  Glfo D  = exp 5.8504 + 0.429  v   l  µl  µ l   ρv  

0.5

  (10.116) 

The rate of entrainment E″ (mass of droplets entrained per second, per unit of wall area) is then computed from the relation

  Dρ   E″ = 5.75  ×  10 −5 (Glf − Glfo )2  2l   Gx  σρv   

 0.316

 (for)  Glf > Glfo   (10.117a)

where Glf is the liquid film mass flux given by

Glf = G (1 – x )(1 – E ) (10.117b)

The entrainment rate correlation embodied in Eq. (10.117a) is compared to the envelope of measured annular flow entrainment data summarized by Govan et al. [10.44] in Fig. 10.23. Because the flow is assumed to be in equilibrium, the rate of entrainment must equal the rate of deposition, which is postulated to equal a deposition coefficient kd multiplied by the droplet mass concentration in the core flow Ce (in kg/m3):

E ″ = kd Ce (10.118)

FIGURE 10.23  Plot of entrainment rate correlation proposed by Govan et al. [10.44].

494

Liquid-Vapor Phase-Change Phenomena

Based on a fit to deposition rate data, Govan et al. [10.44] proposed the following correlation for the deposition rate constant

kd

kd



ρv D = 0.18 σ

C  ρv D = 0.083  e  σ  ρv 

C  for   e   ρv  −0.65

< 0.3 (10.119a)

C  for   e  ≥ 0.3 (10.119b)  ρv 

The entrainment fraction E is related to the mass concentration of droplets Ce as E=



Ce x /ρv (10.120) (1 – x ) [1 – Ce /ρl ]

Equations (10.117a)–(10.118) can be combined to obtain   Dρ   k d Ce = 5.75 × 10 −5 [G (1 − x )(1 − E ) − Glfo ]2  2l   Gx  σρv   



0.316

(10.121)

Once Glfo is computed using Eq. (10.116) the non-linear system of Eqs. (10.119)–(10.121) must be solved simultaneously to determine kd , Ce, and E. The solution of this system thus provides a prediction of the entrainment of droplets in the core flow at the specified local conditions. While, in general, simultaneous solution of these relations requires an iterative procedure, it is possible to simplify the computation in some cases (see Example 10.5). Analysis of heat transfer during annular film-flow transport will be considered in detail in Chapters 11 and 12. For further information about annular film flow transport, the interested reader is referred to the excellent treatise by Hewitt and Hall-Taylor [10.41]. Example 10.5 A two-phase flow of saturated nitrogen at atmospheric pressure flows upward through a vertical tube with an inside diameter of 0.5 cm at G = 200 kg/m2s. The nitrogen enters the tube at a quality of 20% and slowly vaporizes (at low heat flux) as it flows along the passage. Estimate the equilibrium droplet mass concentration and the fraction of liquid entrained for x = 0.5 and x = 0.7. For saturated nitrogen at atmospheric pressure, ρl = 807 kg/m3 ρv = 4.62 kg/m3, μl = 163 × 10 –6 Ns/m2, μv = 5.41 × 10 –6 Ns/m2, and σ = 0.00885 N/m. Using Eq. (10.116), the minimum liquid mass flux for entrainment to occur, Gflo, is determined as

Glfo =

0.5   µ   ρ   µl exp 5.8504 + 0.429  v   l   D  µ l   ρv    

0.5   163 × 10 −6   5.41  807   2 exp 5.8504 + 0.429  =   = 13.7 kg/m s         163 4.62  0.005   

Inverting Eq. (10.120) to solve for Ce yields

 x (1− x )  Ce = [(1− x )E ]/  + E ρl   ρv

495

Introduction to Two-Phase Flow Since 0 ≤ E ≤ 1 and ρl >> ρv , this relation for Ce is well approximated as

Ce =

ρvE (1− x ) x

E=

(Ce / ρv )x 1− x

which implies that

It is further noted that for the specified flow conditions, kd, as given by Eqs. (10.119a) and (10.119b), is a function only of Ce:



kd = 0.18

kd = 0.083

σ ρvD

σ  Ce  ρvD  ρv 

C  for  e  < 0.3  ρv  −0.65

C  for  e  ≥ 0.3  ρv 

Substituting the relation obtained above for E in Eq. (10.121), after some rearrangement, the following equation is obtained:

2   x  Ce  5.75 × 10 −5Gx     Dρl   − 0 = Ce − G(1− x ) 1−    Glfo   2  kd 1− x  ρv       σρv   

0.316

Substituting the specified values of G, x, D, σ, ρl, ρv, and the corresponding Glfo, and using the above relations for kd, the right side of this equation can be evaluated iteratively for different values of Ce to determine the value that makes the right side equal to zero. Doing so for x = 0.5 and x = 0.7 and determining the corresponding E values yields

Ce = 1.90 kg/m3 , E = 0.411 at  x = 0.5



Ce = 1.04 kg/m 3 , E = 0.525 at  x = 0.7

Note that the mass concentration of entrained liquid droplets is lower at the higher quality, where the liquid inventory is lower. However, the fraction of liquid entrained is actually larger at higher quality. Linear extrapolation of these results further suggests that Ce = 0 at about x = 0.94, presumably because the liquid film Reynolds number becomes too small for entrainment of occur (i.e., it falls below the values specified by Eq. (10.116)).

Advanced Analysis and CFD Methods The equations for one-dimensional two-phase flow were derived earlier in this chapter using simple force-momentum balance and mass conservation principles. Alternatively, a more fundamental approach can be taken in which equations embodying conservation of mass, and transport of momentum and energy are developed to fully account for transient effects and the two- or threedimensional character of two-phase flow. Different formulations of the governing equations can be derived by volume and/or time averaging basic governing equations for fluid flow and allowing for flow and interaction of two distinct fluid phases. The general governing equations derived are often cast in a more useful form by time or space averaging, which can generate a set of governing equations for time-averaged and/or volumeaveraged velocities, temperatures, and void fraction (phase volume fraction). If the volume averaging is done over the volume occupied by each phase, equations in terms of mean quantities for each phase can be obtained.

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Even with simplified formulations that incorporate space and/or time averaging, it is usually necessary to incorporate accurate submodel treatments of the transport of momentum, heat, and mass at the interface between phases, and at wall boundaries. If the flow is turbulent, appropriate models of eddy viscosity or turbulent kinetic energy and dissipation must be included. These often incorporate models of how turbulent transport varies from the near wall or near interface regions to core flow regions far from such boundaries. The type of two-phase flow model development described above generally leads to a set of differential equations and accompanying submodel closure relations that can be solved using numerical methods like those typically used for computational fluid dynamics (CFD) simulation of single phase turbulent flows. A full exploration of the different variations of this approach is beyond the scope of this text. Indeed, it would take another large textbook to do justice to the developments in this area. The remainder of this summary will therefore be limited to mentioning a few commonly used model approaches, and providing references where interested readers can find further information on more advanced methods of modeling two-phase flow. An introduction to derivation of general governing equations for gas-liquid two-phase flow can be found in the book chapter by Boure and Delhaye [10.23] and the second edition of the advanced text of Ishii and Hibiki on Thermo-Fluid Dynamic Theory of Two-Phase Flow [10.22]. Two-fluid and drift flux models are developed in the reference article by Taitel and Barnea [10.45]. Time varying two-phase flows can be challenging to predict accurately, and are often simulated by using a two-fluid model or a drift flux model. Both models are simplified as they are based on averaging in space, but they can be effective for predicting transient two-phase flow behavior for applications. In the two-fluid model, each fluid phase is considered separately, and the model incorporates two sets of conservation equations governing the balance of mass, momentum, and energy of each phase together with constitutive relations. The two-fluid model is well-suited to treating separated flows in the stratified and annular flow regimes. It may also be used for intermittent and bubbly flows provided proper adjustments are used so that it will also provide reasonable answers for intermittent and bubbly flow. The separated flow model presented earlier in this chapter is an example of this category of model for steady flow. In contrast, the drift flux model treats the two phases as a single mixture allowing a slip between the gas and the liquid. When the slip is set to zero, this model is identical to the homogenous model. It is formulated using three conservation equations of mixture continuity, mixture momentum, and mixture energy. The fourth equation is the conservation of one of the flow species (usually the gas). The solution also requires a submodel of the slip between the gas and the liquid. Although the drift flux model is best suited to handle dispersed flows, it can be used for all flow patterns by choosing appropriate drift flux correlations and pressure drop coefficients. One example is the recent drift flux model proposed by Fu and Klausner [10.46], which also incorporates an alternate approach to modeling entrainment effects in annular flow. For evaporating annular flow, they suggested use of the following relation for the entrainment fraction:

α  x  ρl   E = 1 − 2    (10.122)  (1 − δ / ro )  1 − x  ρv 

This relation is dictated by the geometric arrangement of the phases and definitions of the void fraction and entrainment parameters for the system. Fu and Klausner [10.46] argued that this relation can be used to predict the entrainment when combined with the Zuber and Findlay [10.47] model for void fraction:

α=

1 (10.123)  1 − x  ρv   ρv Co 1 +  + Vvj x  ρl   Gx 

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Note that this relation contains the empirically determined constant Co and the empirically determined drift velocity Vvj. Choices of the constants can be tailored to a specific flow circumstance. Fu and Klausner [10.46] recommended Co = 0.98 and Vvj = 1.12 m/s for upflow and Co = 1.02 and Vvj = –0.11 m/s for downflow. With these constants, Eqs. (10.122) and (10.123) can be used to predict the entrainment at local conditions. With a method of determining the entrainment such as those described above, computation of the pressure gradient and void fraction for specified fluid properties and values of q″, G, x, and D may be achieved with the following algorithmml: 1. A value of δ is guessed. 2. Equations (10.102) and (10.104) can be used to determine τi. 3. Equations (10.119)–(10.121), can be solved simultaneously using an iterative scheme to determine kd, Ce and E (or an alternative entrainment model can be used to determine E). 4. Equations (10.100) and (10.114) can be used to compute dP/dz. 5. Using a relation such as Eq. (10.105) to evaluate the eddy diffusivity, Eq. (10.98) is integrated numerically using boundary condition (10.101a) to determine the u velocity across the liquid film. The integral of u from y = 0 to y = δ is also determined numerically as part of this computation. 6. If the numerically determined integral of u across the film satisfies Eq. (10.115) for the specified conditions, then the solution is complete. If this condition is not satisfied, a new value of δ must be guessed and the sequence of calculations must be repeated beginning with step 2. This process is repeated until Eq. (10.115) is satisfied. In form, this algorithm is quite similar to that described above for the case of no entrainment. With suitable closure models for the interfacial shear τi, the eddy diffusivity ε, and the entrainment fraction E, the above scheme can, in principle, be used to iteratively determine the pressure drop and void fraction for the flow. In practice, recommended closure models for these quantities are available only for a limited range of circumstances. Further details of the drift flux model can be found in references [10.5, 10.45]. Methods for modeling liquid-gas two-phase flows that combine finite-volume modeling and interface tracking have also been explored for a variety of applications-related systems. Generally, this type of model is formulated to use a numerical finite volume representation of the momentum transport and energy equations to march the velocity and temperature fields forward in time for a two-phase flow, while concurrently using a method to track the resulting motion of gas-liquid interfaces over time in the flow. Several methods for tracking the interphase surfaces can be found in the literature. Among the most popular are: the front tracking method [10.48] (interface modeled as a set of connected markers), the Level Set method [10.49] (interface captured implicitly as the zero level set of a signed distance function) and the Volume of Fluid method (VOF) method [10.50]. The Volume of Fluid (VOF) is one of the most widely successful interface tracking methods currently in use. A large number of applications have been reported for incompressible as well as for compressible flows (see, e.g., references [10.51, 10.52]). Pioneered by Hirt and Nichols [10.53], the VOF method is based on the idea of a phase volume fraction (e.g. α l ) that is treated as a transportable scalar property in the system. Different approaches have been developed for modeling the interface within each cell (see, e.g., references [10.53–10.56]). Although development of the VOF modeling methodology is still a work in progress, it is a promising approach for predicting two-phase flow regime behavior and transport in applications. Kishor et al. [10.57] found, for example, that for air/water two-phase flows, the frictional pressure drop predicted by a VOF method agreed with the Lockhart-Martinelli correlation predictions to a mean absolute deviation of 26%, and was capable of prediction two-phase flow behavior in T-junction flows. They also summarize a large list of more than 25 recent studies of the hydrodynamics, heat, and mass transfer for slug flow in microchannels based on the VOF method. Studies,

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such as that by De Schepper et al. [10.58] have shown that the VOF method can predict two-phase flow regimes in tubes with reasonable accuracy. Other recent investigations have demonstrated that Lattice Boltzmann methods can be used to simulate two-phase flow behavior and heat transport in liquid-vapor two-phase flow (see, e.g., the discussion in Schaefer [10.59]).

10.5  EFFECTS OF FLOW PASSAGE SIZE AND GEOMETRY Earlier sections of this chapter have described the features and physics of two-phase flow based on observations and analysis of flows in round tubes with moderate to large hydraulic diameters. In many applications of modern interest, the flow passage geometry is substantially different from a tube with a simple round cross section. A complete exploration of the effects of passage geometry on two-phase flow is beyond the scope of this text. However, in this section, some of the noteworthy findings of recent investigations of two-phase flow in different flow passage geometries will be summarized.

Two-Phase Flow in Micro- and Nanochannels Over the past decade, there has been rising interest in applications that involve two-phase flow in ultra-miniature flow passages. Examples include ultra-miniature evaporators for localized electronics cooling and evaporator and condenser sections of ultra-miniature heat pipes for electronics thermal management and biosystem applications. Because of these emerging applications, there have been numerous studies of the special features of two-phase flow in ultra-miniature channels. Ghiaasiaan and Abdel-Khalik [10.60] have summarized the results of 13 experimental studies of two-phase flow in microchannels with hydraulic diameters ranging from 0.18 to 12.3 mm. The majority of these studies have examined two-phase flows of liquid water and air in transparent tubes, usually with glass walls. Ghiaasiaan and Abdel-Khalik [10.60] concluded that the twophase flow regimes in microchannels are morphologically similar to flow regimes observed in large channels. They note, however, that surface tension effects often play a stronger role in two-phase flow in ultra-miniature passages compared with the impact of such effects in larger flow passages. Wettability of the passage wall was also noted to have a strong impact on two-phase flow in some instances (see reference [10.61]). The two-phase flow in microchannels was often found to be insensitive to gravity, and hence, insensitive to channel orientation. This insensitivity to gravity appears to be a consequence of that fact that two-phase flows in microchannels are often dominated by the interaction of pressure and fluid inertia with surface tension and viscous forces, with two or more of these effects being much stronger than gravitational body forces. The Bond number. Bod, which characterizes the ratio of gravitational buoyancy to surface tension forces, is defined as

Bod =

(ρl − ρv ) gd h2 (10.124) σ

This group is also sometimes referred to as the Eötvös number, Eo. In two-phase flows in ultraminiature passages, this number is often very small, reflecting the fact that gravitational effects are often small compared to surface tension forces. Because pressure, surface tension, and viscosity often dominate over gravitational body forces, stratification of the flow is generally not observed, even when the flow is horizontal, and the flow morphology usually resembles one of the regimes associated with vertical flow (see Section 10.1). The net effect of these observed trends is that the two-phase flow regimes observed in microchannels are similar to those observed in larger round tubes, but the conditions at which regime transitions occur may vary somewhat from transition conditions in larger round tube passages. This general observation is consistent with the early two-phase flow regime observations of Damianides and Westwater [10.62] for round tubes with diameters between 1 and 5 mm, and

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FIGURE 10.24  Comparison of the two-phase flow regime observations of Triplett et al. [10.65] for a 1 mm diameter tube with transitions observed by Zhao and Rezkallah [10.66] for a 9.5 mm diameter tube.

Lowry and Kawaji [10.63] and Wambsganss et al. [10.64] for flow between parallel plates with separations between 0.5 and 2 mm. Figure 10.24 shows the flow regime observations of Triplett et al. [10.65] for liquid water and air flows through a 1 mm diameter microchannel. Also shown are the flow regime transitions indicated by the microgravity two-phase flow observations of Zhao and Rezkallah [10.66] for larger channels. The microchannel data exhibit the same four major flow patterns as the larger tube microgravity flows: bubbly, slug, frothy slug-annular (churn), and annular. The microchannel data of Triplett et al. [10.65] indicate regime transitions at conditions that are slightly different from those indicated by the larger channel data. Ghiaasiaan and Abdel-Khalik [10.60] also compared the microchannel flow regime observations of Triplett et al. [10.65] to the large tube regime transitions indicated in the study of Bousman et al. [10.67] and reached similar conclusions. Thus, in most cases, the flow pattern maps deduced from observations of two-phase flow in ultra-miniature passages typically exhibit the same flow regimes as flows in larger tubes, but the boundaries that separate them are in different locations. These conclusions are further supported by the observations of Ohtake et al. [10.68] in their recent investigation of argon and water two-phase flow in round tubes and large aspect ratio rectangular channels with hydraulic diameters between 0.18 and 0.6 mm. They observed the same flow patterns indicated in Fig. 10.24, and the indicated regime transitions were slightly different from those indicated by flow pattern maps for larger tubes. Ohtake et al. [10.68] also found that two-phase multiplier data inferred from their pressure drop measurements for circular tubes agreed fairly well with the conventional Lockhart-Martinelli correlation. Their two-phase multiplier data for rectangular minichannels agreed well with predictions of the correlation proposed by Mishima and Hibiki [10.69] for small tubes. Other recent studies have assessed a variety of prediction methods for two-phase flow pressure drop in mini channels [10.70], explored the effects of surface tension on flow regime transitions in miniature and micro tubes [10.71], and developed methods for predicting void fraction in two-phase refrigerant flows in minichannels [10.72]. While previous investigations have shed some light on the features of two-phase flow in ultraminiature passages, methods for predicting flow regime transitions in large channels are often not good predictors of the flow regime in microchannels. Some correlations developed for microgravity data appear to be a fairly good match to some of the available data for microchannels. Although methods for predicting void fraction and two-phase pressure drop have been proposed, they have not been widely tested and their reliability is uncertain. Some of the experimental data for microchannels is for air and water or gas and water flow, and more experimental data for other fluids

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would contribute to a more complete understanding of the effects of fluid properties, interfacial tension, and surface wetting on two-phase flow and pressure drop. Studies also indicate that flow regimes for diabatic two-phase flow with condensation or vaporization in microchannels [10.21, 10.73] can exhibit different transitions than adiabatic two-phase flows under similar flow conditions. Developing widely applicable methods to predict flow regime and pressure drop in micropassages is challenging because of the wide variations in channel geometry, fluids, and flow conditions that are possible. Some progress in that direction has been made by Thome et al. [10.74]. Further information regarding two-phase flow in mini and microchannels can be found in the review article by Kandlikar [10.75].

Two-Phase Flow in Complex Finned and Ribbed Passages Complex flow passage geometries that have fins, ribs or corrugated walls may also be encountered in high performance compact evaporator and condenser designs. The flow in these types of enhanced passage geometries may differ substantially from the behavior discussed earlier in this chapter for two-phase flow in larger round tubes. These differences are frequently a consequence of two factors: (1) The flow passages are relatively small; and (2) the presence of ribs, fins, or other internal structures alters the two-phase flow behavior. The length scales that characterize enhanced heat transfer surfaces are typically much smaller than the round tube geometries used in the power and chemical processing industries. The small characteristic dimension of the passages (small hydraulic diameter) results in the same flow characteristics observed in microchannel flows: pressure and fluid inertia interactions with surface tension and viscosity dominate the flow, with these mechanisms typically being much stronger than gravitational body forces. This is one reason that the two-phase flow may be significantly different from that predicted for larger round tubes under comparable conditions. In two-phase flow, the presence of ribs, fins or other structures in the enhanced passages may produce recirculation zones and may promote vortex shedding and mixing, just as in single-phase flows. The induced turbulence and recirculation can break up large slug bubbles into smaller bubbles. In annular flow, entrained liquid droplets may impinge on the upstream end of rib or fin structures, only to be shed back into the core flow at the trailing edge. Irregularities in the interfacial shear stress produced by core velocity variations in the matrix may result in lateral variations of the liquid film thickness over the surfaces in the matrix at a given downstream location. The results of earlier studies [10.10–10.14, 10.35, 10.38, 10.76, 10.77] of the two-phase flow in more complex finned and ribbed channel geometries generally indicate that although the flow regimes encountered are basically the same as for larger round tubes under comparable conditions, the exact transition conditions between regimes can be significantly different. Further discussion of two-phase flow regimes in finned and ribbed surfaces used in compact evaporators and condensers can be found in the review of Carey [10.78], and studies of two-phase flow in specific passage types by Michel and Lebouché [10.79], Cognata et al. [10.80], and Ansari and Arzandi [10.81].

REFERENCES 10.1 Hewitt, G. F., and Roberts, D. N., Studies of two-phase flow patterns by simultaneous x-ray and flash photography, AERE-M 2159, Her Majesty’s Stationery Office, London, 1969. 10.2 Radovcich, N. A., and Moissis, R., The transition from two-phase bubble flow to slug flow, Report No. 7–7673–22, Mechanical Engineering Department, MIT, 1962. 10.3 Taitel, Y., and Dukler, A. E., Flow regime transitions for vertical upward gas-liquid flow: A preliminary approach through physical modeling, Paper presented at Session on Fundamental Research in Fluid Mechanics at the 70th AIChE Annual Meeting, New York, 1977. 10.4 Porteus, A., Prediction of the upper limit of the slug flow regime, Br. Chem. Eng., vol. 14, no. 9, pp. 117–119, 1969. 10.5 Wallis, G. B., One-Dimensional Two-Phase Flow, John Wiley, New York, NY, 1965.

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10.6 Baker, O., Simultaneous flow of oil and gas, Oil Gas J., vol. 53, pp. 185–195, 1954. 10.7 Mandhane, J. M., Gregory, G. A., and Aziz, K., Flow pattern map for gas-liquid flow in horizontal pipes, Int. J. Multiphase Flow, vol. 1, pp. 537–553, 1974. 10.8 Taitel, Y., and Dukler, A. E., A model for predicting flow regime transitions in horizontal and near horizontal gas-liquid flow, AIChE J., vol. 22, pp. 47–55, 1976. 10.9 Holser, E. R., Flow patterns in high pressure two-phase (steam-water) flow with heat addition, AIChE preprint 22, Paper presented at the 9th National Heat Transfer Conference, Seattle, August, 1967. 10.10 Carey, V. P., and Mandrusiak, G. D., Annular film-flow boiling of liquids in a partially heated vertical channel with offset strip fins, Int. J. of Heat Mass Transf., vol. 29, pp. 927–939, 1986. 10.11 Xu, X., and Carey, V. P., Heat transfer and two-phase flow during convective boiling in a partiallyheated cross-ribbed channel, Int. J. of Heat Mass Transf., vol. 30, pp. 2385–2397, 1987. 10.12 Shedd, T. A., and Newell, T. A., Visualization of two-phase flow through microgrooved tubes for understanding enhanced heat transfer, Int. J. Heat Mass Transf., vol. 46, pp. 4169–4177, 1999. 10.13 Richardson, B. L., Some problems in horizontal two-phase two-component flow, report ANL-5949, Argonne National Lab, 1958. 10.14 Zarnett, G. D., and Charles, M. E., Co-current gas-liquid flow in horizontal tubes with internal spiral ribs, Paper presented at the International Symposium on Research in Co-current Gas-Liquid Flow, University of Waterloo, 1968. 10.15 Zahn, W. R., A visual study of two-phase flow while evaporating in horizontal tubes, J. Heat Transf., vol. 86, pp. 417–429, 1964. 10.16 Boyce, B. E., Collier, J. G., and Levy, J., Hold up and pressure drop measurements in the two-phase flow of air-water mixtures in helical coils, Paper presented at the International Symposium on Research in Co-current Gas-Liquid Flow, University of Waterloo, 1968. 10.17 Banerjee, S., Rhodes, E., and Scott, D. S., Film inversion of co-current two-phase flow in helical coils, AIChE J., vol. 13, pp. 189–191, 1967. 10.18 Oshinowo, T., and Charles, M. E., Vertical two-phase flow. Part I. Flow pattern correlations, Can. J. Chem. Eng., vol. 52, pp. 25–35, 1974. 10.19 Cheng, L., Ribatski, G., and. Thome, J.R., Two-Phase Flow Patterns and Flow-Pattern Maps: Fundamentals and Applications, Appl. Mech. Rev., vol. 61, pp. 050802-1–050802-27, 2008. 10.20 Kattan, N., Thome, J. R., Favrat, D., Flow boiling in horizontal tubes: Part 1 – development of a diabatic two-phase flow pattern map, J. Heat Transf., vol. 120, pp. 140–147, 1998 10.21 Nema, G., Garimella, S., and Fronk, B. M., Flow regime transitions during condensation in microchannels, Int. J. Refrig., vol.40, pp. 227–240, 2014. 10.22 Ishii, M., and Hibiki, T., Thermo-Fluid Dynamic Theory of Two-Phase Flow, 2nd ed., Springer, New York, NY, 2011. 10.23 Boure, J. A., and Delhaye, J. M., General equations and two-phase flow modeling, Handbook of Multiphase Systems, G. Hestroni (editor), Hemisphere Publishing, New York, pp. 1.36–1.87, 1982. 10.24 McAdams, W. H., Woods, W. K., and Heroman, L. C., Jr., Vaporization inside horizontal tubes – II – Benzene-oil mixtures, Trans. ASME, vol. 64, pp. 193, 1942. 10.25 Cicchitti, A., Lombardi, C., Silvestri, M., Soldaini, G., and Zavattarelli, R., Two-phase cooling experiments – pressure drop, heat transfer and burnout measurements, Energ. Nucl., vol. 7, no. 6, pp. 407–425, 1960. 10.26 Dukler, A. E., Wicks, M., III, and Cleveland, R. G., Pressure drop and hold-up in two-phase flow, Part A – A comparison of existing correlations, and Part B – An approach through similarity analysis, AIChE J., vol. 10, pp. 38–51, 1964. 10.27 Lockhart, R. W., and Martinelli, R. C., Proposed correlation of data for isothermal two-phase, two-component flow in pipes, Chem. Eng. Prog., vol. 45, no. 1, pp. 39–48, 1949. 10.28 Chisholm, D., and Laird, A. D. K., Two-phase flow in rough tubes, Trans. ASME, vol. 80, pp. 276–286, 1958. 10.29 Butterworth, D., A comparison of some void-fraction relationships for co-current gas-liquid flow, Int. J. Multiphase Flow, vol. 1, pp. 845–850, 1975. 10.30 Martinelli, R. C., and Nelson, D. B., Prediction of pressure drop during forced-circulation boiling of water, Trans. ASME, vol. 70, pp. 695–702, 1948. 10.31 Thom, J. R. S., Prediction of pressure drop during forced circulation boiling of water, Int. J. Heat Mass Transf., vol. 7, pp. 709–724, 1964. 10.32 Baroczy, C. J., A systematic correlation for two-phase pressure drop, AIChE reprint 37, presented at the 8th National Heat Transfer Conference, Los Angeles, 1965. 10.33 Baroczy, C. J., Correlation of liquid fraction in two-phase flow with applications to liquid metals, Chem. Eng. Prog. Symp. Ser., vol. 61, no. 57, pp. 179–191, 1965.

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10.34 Friedel, L., Improved friction pressure drop correlations for horizontal and vertical two-phase pipe flow, paper E2, European Two Phase Flow Group Meeting, Ispra, Italy, 1979. 10.35 Hetsroni, G. (editor), Handbook of Multiphase Systems, Hemisphere, New York, NY, 1982. 10.36 Chenoweth, J. M., and Martin, M. W., Turbulent two-phase flow, Pet. Refin., vol. 34, no. 10, pp. 151–155, 1955. 10.37 Zivi, S. M., Estimation of steady-state steam void-fraction by means of the principal of minimum entropy production, J. Heat Transf., vol. 86, pp. 247–252, 1964. 10.38 Mandrusiak, G. D., and Carey, V. P., Pressure drop characteristics of two-phase flow in a vertical channel with offset strip fins, Exp. Therm. Fluid Sci., vol. 1, pp. 41–50, 1988. 10.39 Muller-Steinhagen, H., and Heck, K., A simple friction pressure drop correlation for two-phase flow in pipes, Chem. Eng. Process, vol. 20, pp. 291–308, 1986. 10.40 Chisholm, D., Two-Phase Flow in Pipelines and Heat Exchangers, George Godwin of Longman Group Limited, New York, NY, 1983. 10.41 Hewitt, G. F., and Hall-Taylor, N. S., Annular Two-Phase Flow, Pergamon Press, Oxford, U.K., 1970. 10.42 Blanghetti, F., and Schlunder, E. U., Local heat transfer coefficients on condensation in a vertical tube, Proc. 6th Int. Heat Transfer Conf., vol. 2, pp. 437–442, 1978. 10.43 Margiros, P. G., and Dukler, A. E., Entrainment and pressure drop in concurrent gas-liquid flow: II, Liquid property and momentum effects, Developments in Mechanics, vol. 1, Plenum Press, New York, NY, 1961. 10.44 Govan, A. H., Hewitt, G. F., Owen, D. G., and Bott, T. R., An improved CHF modelling code, Proc. Second U.K. Conf. on Heat Transfer, Institute of Mechanical Engineers, London, U.K., vol. 1, pp. 33–48, 1988. 10.45 Taitel, Y., and Barnea, D., Modeling of gas liquid flow in pipes, Ch. 6&7, vol. 1, Encyclopedia of Two-Phase Heat Transfer and Flow I – Fundamentals and Methods, J. R. Thome (editor), World Scientific Publishing Co. Pte. Ltd., 2016. 10.46 Fu, F., and Klausner, J. F., A separated flow model for predicting two-phase pressure drop and evaporative heat transfer for vertical annular flow, Int. J. Heat Fluid Flow, vol. 18, pp. 541–549, 1997. 10.47 Zuber, N., and Findlay, J. A., Average volumetric concentration in two-phase flow systems, J Heat Transf., vol. 87, pp. 453–468, 1965. 10.48 Tryggvason, G., Bunner, B., Juric, D., Tauber, W., Nas, S., Han, J., Al-Rawahi, N., and Jan, Y-J., A front tracking method for the computations of multiphase flow, J. Comput. Phys., vol. 169, pp. 708–759, 2001. 10.49 Osher, S., and Sethian, J. A., Fronts propagating with curvature dependent speed: Algorithms based on Hamilton-Jacobi formulations, J. Comput. Phys., vol. 79, pp. 12–49, 1988. 10.50 Sussman, M., and Puckett, E. G., A coupled level set and volume-of-fluid method for computing 3D and axisymmetric incompressible two-phase flows, J. Comput. Phys., vol. 162, pp. 301–337, 2000. 10.51 Annaland, M., Deen, N. G., and Kuipers, J. A. M., Numerical simulation of gas bubbles behavior using a three-dimensional volume of fluid method, J. Chem. Eng. Sci., vol. 60, pp. 2999–3011, 2005. 10.52 Puckett, E. G., and Saltzman, J. S., A 3D adaptive mesh refinement algorithm for multimaterial gas dynamics, Physica D, vol. 60, pp. 84–93, 1992. 10.53 Hirt, C. W., and Nichols, B. D., Volume of fluid (VOF) method for the dynamics of free boundaries, J. Comput. Phys., vol. 39, pp. 201–225,1981. 10.54 Unverdi, S. O. and Tryggvason, G., A front-tracking method for viscous, incompressible, multi-fluid flows, J. Comput. Phys., vol. 100, pp. 25–37, 1992. 10.55 Mulder, W., Osher, S. J., and Sethian, J. A., Computing interface motion in compressible gas dynamics, J. Comput. Phys., Vol. 100, pp. 209–228, 1992. 10.56 Sethian, J. A., Level Set Methods, Cambridge University Press, New York, NY 1996. 10.57 Kishor, K., Chandra, A. K., Khan, W., Mishra, P. K., and Siraj Alam, M., Numerical study on bubble dynamics and two-phase frictional pressure drop of slug flow regime in adiabatic T-junction square microchannel, Chem. Biochem. Eng. Q., vol. 31, pp. 275–291, 2017. 10.58 De Schepper, S. C. K., Heynderickx, G. J., and Marin, G. B., CFD modeling of all gas-liquid and vaporliquid flow regimes predicted by the Baker chart, Chem. Eng. J., vol. 138, pp. 349–357, 2008. 10.59 Yuan, P., and Schaefer, L., A thermal lattice Boltzmann two-phase flow model and its application to heat transfer problems – Part 1. Theoretical foundation, J. Fluids Eng., vol. 128, pp. 142–150, 2005. 10.60 Ghiaasiaan, S. M., and Abdel-Khalik, S. I., Two-phase flow in microchannels, Advances in Heat Transfer, vol. 34, pp. 145–254, 2001. 10.61 Barajas, A. M., and Panton, R. L., The effect of contact angle on two-phase flow in capillary tubes, Int. J. Multiphase Flow, vol. 19, pp. 337–346, 1993.

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10.62 Damianides, C. A., and Westwater, J. W., Two-phase flow patterns in a compact heat exchanger and in small tubes, Proc. 2nd UK National Conf. on Heat Transfer, vol. II, pp. 1257–1268, 1988. 10.63 Lowry, B., and Kawaji, M., Adiabatic two-phase flow in narrow flow channels, AIChE Symp. Ser., vol. 84, no. 263, pp. 133–139, 1988. 10.64 Wambsganss, M. W., Jendrzejczyk, J. A., and France, D. M., Two-phase flow patterns and transitions in a small, horizontal, rectangular channel, Int. J. Multiphase Flow, vol. 17, pp. 327–342, 1991. 10.65 Triplett, K. A., Ghiaasiaan, S. M., Abdel-Khalik, S. I., and Sadowski, D. L., Gas-liquid two- phase flow in microchannels. Part I: Two-phase flow patterns, Int. J. Multiphase Flow, vol. 25, pp. 377–394, 1999. 10.66 Zhao, L., and Rezkallah, K. S., Gas-liquid flow patterns at microgravity conditions, Int. J. Multiphase Flow, vol. 19, pp. 751–763, 1993. 10.67 Bousman, W. S., McQuillen, J. B., and Witte, L. C., Gas-liquid flow patterns in microgravity: Effects of tube diameter, liquid viscosity, and surface tension, Int. J. Multiphase Flow, vol. 22, pp. 1035–1053, 1996. 10.68 Ohtake, H., Ohtaki, H., and Koizumi, Y., Frictional pressure drop and two-phase flow pattern of gasliquid two-phase flow in circular and rectangular minichannels, Proc. 4th Int. Conf. on Nanochannels. Microchannels and Minichannels, Paper ICNMM2006–96201, Limerick, Ireland, 2006. 10.69 Mishima, K., and Hibiki, T., Some characteristics of air-water two-phase flow in small diameter vertical tubes, Int. J. Multiphase Flow, vol. 22, pp. 703–712, 1996. 10.70 Sun, L., and Mishima, K., Evaluation analysis of prediction methods for two-phase flow pressure drop in mini-channels, Int. J. Multiphase Flow, vol. 35, 47–54, 2009. 10.71 Tabatabai, A., and Faghri, A., A new two-phase flow map and transition boundary accounting for surface tension effects in horizontal miniature and micro tubes, J. Heat Transf., vol. 123, pp. 958–968, 2001. 10.72 Xu, Y., and Fang, X., Correlations of void fraction for two-phase refrigerant flow in pipes, Appl. Therm. Eng., vol. 64, pp. 242–251, 2014. 10.73 Revellin, R., Dupont, V., Ursenbacher, T., Thome, J. R., and Zun, I., Characterization of diabatic two-phase flows in microchannels: Flow parameter results for R-134a in a 0.5 mm channel, Int. J. Multiphase Flow, vol. 32, pp. 755–774, 2006. 10.74 Thome, J. R., Bar-Cohen, A., Revellin, R., and Zun, I., Unified mechanistic multiscale mapping of twophase flow patterns in microchannels, Exp. Therm. Fluid Sci., vol. 44, pp. 1–22, 2013. 10.75 Kandlikar, S. G., Two-phase flow patterns, pressure drop and heat transfer during boiling in minichannel and microchannel flow passages of compact heat exchangers, in Compact Heat Exchangers and Enhancement Technology for the Process Industries-2001, Begell House, New York, NY, pp. 319–334, 2001. 10.76 Mandrusiak, G. D., Carey, V. P., and Xu, X., An experimental study of convective boiling in a partiallyheated horizontal channel with offset strip fins, J. Heat Transf., vol. 110, pp. 229–236, 1988. 10.77 Mandrusiak, G. D., and Carey, V. P., A finite difference computational model of annular film-flow boiling and two-phase flow in vertical channels with offset strip fins. Int. J. Multiphase Flow, vol. 16, pp. 1071–1098, 1990. 10.78 Carey, V. P., Two-phase flow in small-scale ribbed and finned passages for compact evaporators and condensers, Nucl. Eng. Des., vol. 141, pp. 249–268, 1993. 10.79 Michel, G., and Lebouché, M. Two-phase gas–liquid flow in horizontal corrugated channels, Int. J. Multiphase Flow, vol. 26, pp. 435–443, 2000. 10.80 Cognata, T. J., Hollingsworth, D. K., and Witte, L. C., High-Speed visualization of two-phase flow in a micro-scale pin-fin heat exchanger, Heat Transf. Eng., vol. 28, pp. 861–869, 2007. 10.81 Ansari, M., and Arzandi, B., Two-phase gas–liquid flow regimes for smooth and ribbed rectangular ducts, Int. J. Multiphase Flow, vol. 38, pp. 118–125, 2012.

PROBLEMS 10.1 A two-phase mixture of saturated oxygen liquid and vapor at a pressure of 196 kPa flows upward in a vertical tube. The tube inside diameter is 8 mm and the mass flux is 200 kg/m2s. Determine the flow regime at qualities of 0.1, 0.5, and 0.9. 10.2 Solve problem 10.1 for a horizontal tube. 10.3 A two-phase mixture of saturated liquid and vapor water at atmospheric pressure flows through a round tube with a diameter of 1.2 cm. The flow rate is such that the mass flux is 400 kg/m 2s. Estimate the values of quality at which transitions from bubbly to slug and churn to annular flow are expected to occur.

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Liquid-Vapor Phase-Change Phenomena

10.4 Saturated ammonia at 165 kPa flows from the expansion valve of a refrigeration system to the inlet of an evaporator through a horizontal tube with a diameter of 2.2 cm and a length of 0.5 m. The mass flux is 500 kg/m2s and the quality is 0.22. Determine (a) the flow regime, (b) the void fraction, and (c) the total pressure drop using the separated flow model. 10.5 Solve parts (b) and (c) of problem 10.4 using the homogeneous flow model. 10.6 Adiabatic two-phase flow of steam and water at 6124 kPa flows upward in a vertical tube 4.0 m long and 2.5 cm in diameter. The mass flux is 700 kg/m2s and the quality is 0.7. Determine the pressure drop through the tube using (a) the homogeneous flow model, (b) the Martinelli correlation, and (c) the Baroczy correlation. 10.7 R-134a leaves an evaporator at 90% quality and flows through a vertical tube 2.0 m long and 1.0 cm in inside diameter. Heat is added uniformly so that the quality at the exit of the tube is 95%. The mass flux through the tube is 400 kg/m2s and the pressure at the tube inlet is 338 kPa. Determine the pressure drop along the tube using the separated flow model by dividing the tube into four segments and assuming that the pressure gradient is uniform over each segment. 10.8 For water at 3773 kPa, plot and compare the µ /µ l -versus-x variations predicted by Eqs. (10.42)–(10.44). Based on these variations, what conclusions do you draw regarding the suitability of each of these relations for bubbly and droplet mist flow? 10.9 Saturated two-phase flow of nitrogen occurs adiabatically in a vertical tube with an inside diameter of 1.7 cm. At a specific location in the tube, x = 0.3 and P = 778 kPa. The mass flux is 350 kg/m2s. Compute the void fraction and pressure gradient at this location using (a) the Baroczy correlation and (b) the Martinelli correlation. 10.10 For two-phase flow of liquid nitrogen at atmospheric pressure, compute and plot the variation of void fraction with quality predicted by each of the void fraction models listed in Table 10.2. 10.11 A water evaporator consists of a horizontal metal tube 1.5 m long with an inside diameter of 2.0 cm. A uniform heat flux of 800 kW/m2 is applied to the tube wall. Saturated liquid water enters the tube at a pressure of 2185 kPa. Using the Martinelli-Nelson correlation, determine and plot the pressure drop in the tube for flow rates between 21 and 800 g/s. You have been asked to select a mass flow rate that (1) assures that the exit flow is either in the bubbly or slug flow regime (but is not in the churn or annular flow regimes) and (2) minimizes the overall pressure drop within the limits imposed by constraint (1). What flow do you recommend? 10.12 A vaporizing flow of water flows downward in a tube having an inside diameter of 1.0 cm and a length of 1.5 m. The flow enters as saturated liquid at 1172 kPa. A heat flux of 600 kW/m2 (based on inside wall area) is applied uniformly to the wall of the tube. Determine the pressure drop over the length of the tube for mass flux values between 50 and 300 kg/m2s and plot the pressure drop for the system over this range of G. 10.13 Saturated liquid R-134a enters a tube at a pressure of 338 kPa. The tube inside diameter is 1.2 cm and its length is 1.0 m. The mass flux into the tube is 400 kg/m2s, and a uniform heat flux of 15 W/cm2 is delivered to the flow uniformly over the inside of the tube. Determine the void fraction variation and sequence of flow regimes encountered along the tube (a) if the tube is vertical and (b) if the tube is horizontal. 10.14 Saturated ammonia vapor enters a tube at a pressure of 2422 kPa and a mass flux of 400 kg/m2s. The tube inside diameter is 1.5 cm and the tube length is 2.0 m. The tube wall is cooled in such a way that a heat flux of 75 W/cm2 is removed uniformly along the inside wall of the tube. Determine the void fraction variation along the tube and the sequence of flow regimes encountered if (a) the tube is horizontal and (b) the tube is inclined at an angle of 40° to the horizontal. 10.15 Use the algorithm given in Section 10.4 for annular flow with no entrainment to write a computer program to iteratively compute the film thickness and interfacial shear stress for annular flow in vertical tube. Make use of Eqs. (10.94)–(10.105) as appropriate. Use the model to compute the film thickness and two-phase frictional pressure gradient for two-phase flow of saturated R-134a in a tube with an inside diameter of 1.0 cm at 338 kPa for x = 0.8 and G = 300 kg/m2s. Compare the results to those predicted by the Martinelli correlation. 10.16 A two-phase flow of R-134a at 201 kPa flows upward through a vertical tube with an inside diameter of 7.0 mm at G = 300 kg/m2s. The R-134a enters the tube at a quality of 20% and slowly vaporizes (at low heat flux) along the passage. Estimate the equilibrium droplet concentration and the fraction of liquid entrained for x = 0.5 and x = 0.75. 10.17 Use the algorithm given in Section 10.4 for annular flow with entrainment to write a computer program to iteratively compute the film thickness and interfacial shear stress for annular flow in vertical tube. Make use of the equations in that section and the entrainment correlation, as appropriate. Use

Introduction to Two-Phase Flow

505

the model to compute the film thickness and two-phase frictional pressure gradient for two-phase flow of saturated nitrogen in a tube with an inside diameter of 0.8 cm at 540 kPa for x = 0.7 and G = 300 kg/m 2s. Compare the results to those predicted by the Baroczy correlation. 10.18 For the two-phase flow conditions specified in Example 10.2, determine the gravitational and frictional components of the pressure drop if the fluid is saturated R-12 and the tube inlet pressure is 333 kPa. Compare your results to those in Example 10.2. What is your assessment of the effects of replacing R-12 with R-134a on two-phase pressure drop? 10.19 The separate cylinders model discussed in Section 10.3 predicts that for turbulent- turbulent flow (n = 0.25), φl2 = (1 − α)−2.375 (see Eq. (10.69)). A two-phase mixture of saturated nitrogen liquid and vapor at a pressure of 229 kPa flows upward in a vertical tube. The tube inside diameter is 8 mm and the mass flux is 200 kg/m2s. For qualities of 0.2, 0.5 and 0.8, compute φl2 and α using the LockhartMartinelli correlations for these flow conditions. For each quality, compare φl2 and (1 – α) –2.375 and determine the percent by which they differ. What is your assessment of the accuracy of the implied relationship φl2 = (1 − α)−2.375?

11

Internal Convective Condensation

11.1 REGIMES OF CONVECTIVE CONDENSATION IN CONVENTIONAL (MACRO) TUBES As noted in Chapter 10, the two-phase flow regime for gas-liquid flow in a tube generally depends on the mean velocities of the two phases and their properties. During convective condensation in a tube, the properties often vary only slightly over the flow length. However, because the phase change produces an appreciable variation in the relative flow velocities of the two phases, the flow regime can change dramatically over the length of the passage. Perhaps the most common flow configuration in which convective condensation occurs is flow in a horizontal circular tube. This configuration is encountered in air-conditioning and refrigeration condensers as well as condensers in Rankine power cycles. Although convective condensation is also sometimes contrived to occur in co-current vertical downward flow, horizontal flow is often preferred because the flow can be repeatedly passed through the heat exchanger core in a serpentine fashion, as indicated in Fig. 11.1, without trapping liquid or vapor in the return bends. If the straight sections of the serpentine tubes are oriented vertically, intermittent trapping of liquid or vapor in the bends can cause the flow to oscillate in a manner that may deteriorate the system’s performance. If the flow to the tubes is fed by a header or inlet manifold, as shown in Fig. 11.2, downflow condensation can also be accomplished without this problem. The sequence of flow regimes that can be encountered during condensation in a horizontal tube are shown schematically in Fig. 11.3. For water, if the tube wall is poorly wetted (high contact angle) near the onset of condensation, it is possible for a region of dropwise condensation to exist there. For horizontal tubes at moderate to high vapor velocities, droplets growing via dropwise condensation would tend to be simultaneously dragged downstream and pulled by gravity toward the bottom of the tube. If the condensation rate is high, the wall may be flooded with liquid, resulting in film condensation instead. For moderate condensation rates liquid accumulation in the bottom of the tube is likely to lead to film condensation there, while the upper portion of the tube may exhibit dropwise condensation. Eventually, as the quality becomes lower, the increasing liquid inventory would be expected to lead to an annular flow with film condensation. Unlike water, low surface tension refrigerants usually wet ordinary metal tube walls well, and dropwise condensation is unlikely. However, by using microporous wall structures infused with a poorly wetted material, some researchers have been able to achieve dropwise condensation even for some organic fluids that have low contact angle (see references [11.1, 11.2] for a description of these types of surfaces). The general sequence of flow regimes expected for horizontal tubes can be inferred from a two-phase flow pattern map. Although an adiabatic map, such as that developed by Taitel and Dukler [11.3], can be used to predict the sequence, a flow pattern map for condensing flows in horizontal passages, such as that developed by Coleman and Garimella [11.4], would likely provide a more accurate prediction. An indication of the transitions in the Coleman and Garimella [11.4] regime map are shown in Fig. 11.4. This flow regime diagram was based on regime

507

508

Liquid-Vapor Phase-Change Phenomena

FIGURE 11.1  Schematic of a typical round-tube cross-flow air-cooled condenser.

FIGURE 11.2  Schematic of a typical downflow air-cooled condenser.

FIGURE 11.3  Horizontal co-current flow with condensation.

Internal Convective Condensation

509

FIGURE 11.4  Estimated flow regime transitions for condensing refrigerant flow in a horizontal tube with a diameter of 4.9 mm, based on data obtained by Coleman and Garimella [11.4].

transitions determined experimentally for horizontal condensing flows in round and rectangular passages having hydraulic diameters of a few millimeters. Note that for a condensing flow in a tube of fixed cross section, the local regime traverses a horizontal line at fixed mass flux G, from right to left, on this diagram. The indicated mass flux values of 100–500 kg/m 2s span the range of conditions typically encountered in air conditioning and refrigeration applications. At low to moderate flow rates, Fig. 11.4 indicates that a region of somewhat stratified and/ or wavy flow may be encountered at moderate qualities. However, even for conditions at which Fig. 11.4 predicts a somewhat stratified or wavy flow, condensation on the upper portion of the inside wall of the tube will cover it with a liquid film. In addition, for pressure conditions typical of applications, the void fraction is high at moderate to low qualities, and the generally annular character of the flow may persist to qualities as low as 0.2-0.3. For a constant mass flux process line in Fig. 11.4, flow regimes without this annular character may occur, but they typically exist during only a small portion of the heat transfer process at low quality. At the very end of the condensation process, small portions of the flow may be in the slug, plug, and/or bubbly flow regimes. Since most of the heat transfer associated with the condensation process takes place under annular flow conditions, prediction of the condensation heat transfer coefficient for annular flow is obviously of primary importance to the design of condensers operating at low to moderate pressures. It is not surprising, therefore, that most efforts to develop methods for predicting internal convective condensation heat transfer have focused on annular flow. Condensation during slug, plug, or bubbly flow at the end of the condensation process has received much less attention. Annular flow is also somewhat easier to model analytically than the intermittent slug or plug flows. Analytical models and empirical prediction techniques for condensation in these regimes are discussed in detail in the next two sections.

510

Liquid-Vapor Phase-Change Phenomena

Example 11.1 R-410a condenses at 2800 kPa in a horizontal tube with an inside diameter of 5 mm. Determine the condensation regime for x = 0.9, 0.4, and 0.05 if G = 400 kg/m2s. For saturated R-410a at 2800 kPa, ρl = 936 kg/m3, ρv = 125 kg/m3, μl = 88.6 × 10 –6 Ns/m2, μν = 15.2 × 10 –6 Ns/m2, and σ = 0.0025 N/m. Substituting for x = 0.9,

Re l =

G (1− x ) D 400 ( 0.1)( 0.005) = = 2258 µl 88.6 × 10 −6



Rev =

GxD 400 ( 0.9)( 0.005) = 1.19 × 105 = µv 15.2 × 10 −6

Similar calculations for the other qualities yield

Re l = 13,500, Rev = 5.28 × 10 4 for x = 0.4



Re l = 21,400, Rev = 10,600 for  x = 0.05

All three qualities correspond to the turbulent-turbulent flow regime, and using Eq. (10.78), it follows that

ρ  X = Xtt =  v   ρl 

0.5

 µl   µv 

0.125

 1− x    x 

0.875

Substituting for x = 0.9,

 125  Xtt =   936 

0.5

 88.6    15.2 

0.125

 0.1   0.9 

0.875

= 0.0666

For the other qualities, it can be similarly shown that

Xtt = 0.355

for x = 0.4



Xtt = 5.99

for x = 0.05

For a horizontal tube, Ω = 0, and combining Eqs. (10.7a) and (10.15b) yields

FTD

  G2 x 2 =  Dg ρ ρ − ρ ( ) v  v l 

0.5

Substituting for x = 0.9, we find

  ( 400 )2 (0.9)2 FTD =    (125)( 936 − 125)( 0.005) 9.8 

0.5

= 5.10

Using this value of FTD with Xtt = 0.0666, the adiabatic flow regime map in Fig. 10.5 thus predicts that the flow is in the annular flow regime. This is consistent with the Coleman and Garimella [11.4] regime map as shown in Fig. 11.4 which also predicts that annular film-flow condensation is expected for x = 0.9. Substituting in a similar manner yields

FTD = 2.27

for x = 0.4



FTD = 0.178

for x = 0.05

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Internal Convective Condensation

Using FTD = 2.27 with Xtt = 0.355, Fig. 10.5 indicates that the flow is in the annular flow regime. Thus annular flow film condensation is also expected at x = 0.4. Using FTD = 0.178 with Xtt = 5.99, Fig. 10.5 indicates that the flow may be in the intermittent or dispersed bubble regime. We proceed as follows to evaluate TTD using Eqs. (10.12a) and (10.15c) for x = 0.05:

fl = 0.079Re −l 0.25 = 0.079 ( 21,400 )



= 0.00653

2fl G 2 (1− x ) 2( 0.00653)( 400 ) ( 0.95)  dP  = = = 382 Pa / m −   dz  l ρl D 991( 0.005) 2



−0.25

 − ( dP / dz )l TTD =   ( ρl − ρv ) g

2

  

0.5

  382 =   ( 936 − 125) 9.8 

2

0.5

= 0.220

For Xtt = 5.99 and TTD = 0.220, Fig. 10.5 indicates that the state point for the flow is below the intermittent/dispersed-bubble transition. The flow is therefore in the intermittent regime for these conditions. Thus slug flow or plug flow condensation is expected for x = 0.05. Note that these conditions are typical of those encountered in the horizontal tubes of the condenser in a heat pump system used for heating and/or air conditioning. The Coleman and Garimella [11.4] regime map shown in Fig. 11.4 predicts a wavy annular flow at x = 0.4, which is in general agreement with the Taitel-Dukler map prediction, and both maps predict intermittent flow at x = 0.05.

11.2 ANALYTICAL MODELING OF DOWNFLOW INTERNAL CONVECTIVE CONDENSATION As noted in Section 11.1, for many applications, the two-phase flow for convective condensation inside tubes has an annular character over much of the tube length. The flow pattern map of Coleman and Garimella [11.4], for example, predicts that wavy flow of annular character persists to qualities as low as about 0.3 for R-134a condensing in a horizontal tube at mass flux levels of about 400 kg/m 2s. If the flow is downward and laminar with no entrainment, the Nusselt-type analysis for film condensation can be adapted to internal convective condensation in a round tube. The circumstances of interest are shown schematically in Fig. 11.5. An integral analysis of the condensation heat transfer in the tube basically follows the Nusselt analysis for falling-film condensation described in Section 9.3. Neglecting the inertia and downstream diffusion contributions, the momentum balance for the differential element in Fig. 11.5 requires that

(δ − y ) dz ρl g − 

 du  dP  + τi dz = µ l   dz (11.1) dz   dy 

The total pressure gradient along the tube in the vapor (dP/dz) is equal to the hydrostatic gradient plus contributions due to friction and deceleration effects

dP  dP   dP   dP   dP   dP  + + = ρv g +  + = (11.2)      dz  fr  dz  dec dz  dz  hyd  dz  fr  dz  dec

The frictional pressure gradient in the vapor is due to the interfacial shear stress

4τi  dP  (11.3)  =−  dz  fr ( D − 2δ )

512

Liquid-Vapor Phase-Change Phenomena

FIGURE 11.5  Model system for analysis of downflow internal condensation in a tube.

Based on the results of the one-dimensional two-phase separated flow analysis presented in Chapter 10, the deceleration pressure gradient is given by



2 (1 − x )2   dP  2 d  x = − + G   (11.4)   dz  dec dz  ρv α ρl (1 − α ) 

If the vapor density is low compared to the liquid density, and the film is thin, the second term in the square brackets is small compared to the first. Assuming that the variation of the void fraction along the tube is small compared to the variation in x, the expression for (dP/dz)dec can be further simplified to the following approximate form

2 xDG 2  dx   dP  = −     (11.5) dz  dec ρv ( D − 2δ )  dz 

We will also adopt the usual idealization that the pressure gradient in the vapor is equal to that felt by the liquid film. Analysis of the momentum transport in the liquid film is facilitated by defining a fictitious vapor density ρ*v such that

ρ*v g = ρv g −

4τi 2 xDG 2  dx  − (11.6) ( D − 2δ ) ρv ( D − 2δ )  dz 

The gradient in the quality can be evaluated from an energy balance as

4 h ( Tsat − Tw ) dx 4 q ′′ = = (11.7) dz DGhlv DGhlv

513

Internal Convective Condensation

where q″ is the local heat flux. Combining Eqs. (11.1)–(11.3), (11.5), and (11.6) yields

(

)

* du ( δ − y ) ρl − ρv g τ i = + (11.8) µl µl dy



In this form, the momentum balance can be integrated using u = 0 at y = 0 to obtain

u=

(ρ − ρ ) g  yδ − y * v

l

 τi y + (11.9) 2  µ l



µl

2

Integrating this velocity profile across the liquid film, the following relation for the total liquid flow rate m l = G(1 – x)πD2/4 is obtained

(

)

 ρl ρl − ρ*v gδ 3 τi ρl δ 2  m l = πD  +  (11.10) 3µ l 2µ l   

Differentiating with respect to δ yields

dm l πD ρl ρl − ρ*v gδ 2 + τ i ρl δ  (11.11) =  dδ µl 

(

)

As in the case of falling film condensation, if subcooling of the film is neglected, an overall mass and energy balance requires that dm l kl ( Tsat − Tw ) πD = (11.12) dz hlv δ



Combining this with Eq. (11.11) yields

kl µ l ( Tsat − Tw ) dδ = (11.13) dz ρl ρl − ρ*v ghlv δ 3 + τ i ρl hlv δ 2

(

)

This equation can be integrated using the condition that δ = 0 at z = 0 to obtain

4 zkl µ l ( Tsat − Tw ) 4τi δ3 4 (11.14) = δ + ρl ρl − ρ*v ghlv 3 ρl − ρ*v g

(

)

(

)

If convective effects in the subcooled film are included, the results of the integral analysis are the same except that hlv is replaced by hlv′

4 zkl µ l ( Tsat − Tw ) 4τi δ3 (11.15) = δ4 + * ρl ρl − ρv ghlv′ 3 ρl − ρ*v g

(

)

(

)

where

  3  c pl ( Tsat − Tw )  hlv′ = hlv 1 +    (11.16) hlv   8 

514

Liquid-Vapor Phase-Change Phenomena

For laminar conduction-dominated transport, the heat transfer coefficient is then given by

h=

kl (11.17) δ

To close the computational scheme, a means of evaluating τi is required. Because the viscosity of the liquid is much larger than that of the vapor and the film is thin, the mean vapor velocity in the core is generally much larger than the liquid velocity at the interface. As a first approximation, the vapor core could be treated as a single-phase flow in a round tube, with the velocity of the vapor taken to be zero at the interface. The interfacial shear τi can then be computed using a conventional single-phase correlation for the friction factor f v:

   ρ u2  G2x2 τ i = fv  v v  = fv   (11.18) ρ − δ 2 1 8 / D  2  ( )  v 

For round tubes, f v can be evaluated from the correlation

 Gx ( D − 2δ )  fv = 0.079    µ v (1 − 4δ / D ) 

−0.25

(11.19)

Although prediction of the interfacial shear in this idealized manner is very approximate, it does provide a means of closing the problem mathematically. Alternatively, we could use the following simple relation for the interfacial friction factor discussed in Chapter 10:

 300δ  fv = fi = 0.005  1 +  (11.20)  D 

Even with the use of a simplified method of computing the interfacial shear, determination of the heat transfer coefficient is possible only with the use of an iterative technique. An appropriate scheme of this type (for specified G, Tw, P, and thermophysical properties) is as follows: 1. Guess a value of δ. 2. Use Eqs. (11.7) and (11.17) to evaluate dx/dz. 3. Use Eqs. (11.18) and (11.20) to compute τi . 4. Determine ρ*v using Eq. (11.6). 5. Substitute the guessed δ value and the computed ρ*v and τi values into Eq. (11.15). If this equation is satisfied to an acceptable level of accuracy, the guessed value of δ and the computed h value are correct. If this relation is not satisfied, a new δ value is guessed and the process is repeated, beginning with step 2. This sequence of computations is repeated until convergence. Because the thermophysical properties are assumed to be constant, this analysis applies to circumstances in which the pressure drop along the tube is small compared to the absolute pressure. When the film and vapor core Reynolds numbers are high enough, part or all of the flow may be turbulent. For such circumstances, the analysis described above is not appropriate, and must be modified to account for turbulent flow in the film. In addition, at higher liquid and vapor flow rates, entrainment of liquid droplets into the vapor core may also be significant. Prediction of the transport for such circumstances makes use of the analytical treatment of annular two-phase turbulent flow described in Section 10.4. Such a prediction can be achieved in the following manner. For a given downstream location, Eqs. (10.98), (10.100), (10.101a), (10.102), (10.104), (10.105), (10.114), and (10.115) are first

515

Internal Convective Condensation

solved iteratively, using the correlation represented by Eqs. (10.116)–(10.121) to determine the entrainment fraction E. Solution of these equations yields the axial velocity profile u(y) across the liquid film and the film thickness δ at a specified location along the length of the channel. For steady turbulent flow, the transport of thermal energy in the liquid film is governed by the equation

u

∂T ∂T ∂ ∂T ( α T ,l + ε H )  +v = (11.21) ∂x ∂y ∂y ∂y

where εH is the eddy diffusivity of heat. Since the film is thin compared to the downstream flow length, the usual boundary-layer approximations have been adopted. The boundary conditions on the temperature profile are at y = δ: T = Tsat ( Pv ) (11.22)

at the interface and

at y = 0: T = Tw (11.23)



The convection terms in the energy equation are often neglected because the transport rate across the film is much greater than downstream convection. If these terms are neglected, the energy equation can be simplified to  ∂  (α Τ ,l + ε H ) ∂∂Ty  = 0 (11.24) ∂ y  



Integrating the energy equation (11.25) first from 0 to y and then from 0 to δ yields Tsat − Tw k = l = qw′′ / ( ρl c pl α T ,l ) h



δ

dy

∫ 1 + ( Pr / Pr )( ε 0

l

t

M

/ vl )

(11.25)

where h is the local heat transfer coefficient, qw′′ is the heat flux applied at the wall, and Prt is the turbulent Prandtl number defined as Prt = εM/εH. Before the above form of the energy transport equation can be analyzed further, a means of determining the turbulent Prandtl number and the eddy diffusivity εM as function of y must be known. Appropriate relations for εM/vl that can be used to compute the heat transfer coefficient using Eq. (11.25) are discussed in Section 10.4. The manner in which such relations can be integrated into this type of analysis is illustrated in Example 11.2. Example 11.2 Develop an approximate heat transfer correlation from an integral analysis for shear dominated annular film condensation in a tube (as shown in Fig. 11.5) at moderate vapor flow rates. For the purposes of this approximate analysis, it will be assumed that entrainment of liquid in the vapor core is negligible and the liquid flows in a thin film with a smooth interface along the tube wall. Effects of interfacial waves are neglected. The thickness of the film δ is assumed to be very small compared to the tube inside diameter D. Downstream convection in the film is

516

Liquid-Vapor Phase-Change Phenomena

neglected compared to cross-film diffusion in the momentum and energy balances. The downstream momentum equation is therefore given by

−

1 ∂P ∂  ∂u  + ( νl + ε M )  + g = 0 ρl ∂ z ∂y  ∂y 

where εM is the turbulent eddy diffusivity in the liquid film. This equation can be rearranged to obtain

∂  ∂u   ∂P  ρl ( νl + ε M )  =   − ρl g ∂y  ∂y  ∂z

In general, the two-phase pressure gradient and body force terms on the right side of the above equation will have some effect on the momentum balance. However, in the present analysis it will be assumed that these terms are small compared with viscous and turbulent shear stresses (note that the term in square brackets represents the combined effects of these stresses). Setting the right side of the equation to zero, it can be integrated once to obtain ρl ( νl + ε M )



∂u = τ0 ∂y

The constant in the integration is denoted as τ0 because the left side is clearly the local shear stress in the layer. Thus, with these idealizations, the film becomes a constant shear layer. Defining

u+ =

y τ 0 / ρl u , y+ = νl τ 0 / ρl

the above equation can be cast in the form  ε M  du +  1+ ν  dy + = 1 l



It is further assumed here that the film is weakly turbulent so that εM/νl is small compared to 1, but not 0. Hence to a first approximation

du + = 1 which implies that u + = y + dy +

(since u = 0 at y = 0). This is consistent with the assumption that δ is small, resulting in δ+ being so small that most of the film is in the viscous sublayer. Mass conservation of liquid requires that δ



 πD 2   4  G (1− x ) = π D ρl u  dy

∫ 0

This relation can be written in terms of u+ and y+ as δ+

GD (1− x ) = u + dy + 4µ l

∫



0

Substituting u+ = y+ and integrating yields

δ+ =

GD (1− x ) 1 1/ 2 Re l , Re l = 2 µl

517

Internal Convective Condensation

For transport of heat in the film, neglecting downstream convection compared to cross-stream diffusion, the governing equation becomes

∂  ∂T  ρl c pl ( αT ,l + εH )  = 0 ∂y  ∂y 

where εH is the eddy diffusivity for heat. Integrating this equation once, we obtain

∂T = q0′′ ∂y

ρl c pl ( αT ,l + εH )

The constant obtained in this integration is denoted as q0′′ because the left side represents the local heat flux in the film. The film is therefore a constant heat flux layer. Rearranging and integrating across the film yields

Tsat − Tw = q0′′ / ρl c pl αT ,l

(

)

δ

dy

∫ 1+ Pr ε l

0

H

/ νl

where Tw is the wall temperature. Noting that ρlcplαT,l = kl and εH = εM/Prt (where Prt is the turbulent Prandtl number), the above equation can be written in terms of y+ as

kl τ 0 / ρl = hνl

δ+

∫ 0

dy + 1+ (Prl / Prt )( ε M / νl )

Because the film is a constant-heat-flux layer, q0′′ is equal to the heat flux removed at the wall of the tube and the local heat transfer coefficient h is taken to be

h=

q0′′ Tsat − Tw

To use the integral relation described above to predict the heat transfer coefficient, means of evaluating τ0, εM, and Prt are needed. In the momentum balance relation, εM /νl was assumed to be small compared to 1. Prt is taken to be 0.9. However, it is also assumed here that the liquid Prandtl number Prl is large enough that (Prl /Prt)(εM /νl) is not negligible compared to 1. If von Karman’s mixing-length model,

εM = l 2

du , l = κy dy

is combined with the result u+ = y+, it can easily be shown that εM 2 + κ y νl

( )



2

In Section 10.4, it was noted that investigations involving analysis of liquid film turbulent transport have used variations of the eddy diffusivity that go to zero at both the wall and the interface to account for the damping effect of the interface on turbulence. Consistent with this line of reasoning, the following continuous variation of εM /νlPrt is postulated here:  2 + 2 κ y  Prt εM = νl Prt  κ 2 δ + − y +  Prt 

( )



(

for 0 ≤ y + ≤

)

2

for

δ+ 2

δ+ ≤ y + ≤ δ+ 2

518

Liquid-Vapor Phase-Change Phenomena

Using this variation to evaluate the integral in the relation for h derived above, the following result is obtained:

h=

κkl τ 0 / ρl 2νl I

 Prl κδ +  Prl , I = tan−1   Prt  Prt 2 

For large values of the argument in parentheses, tan–1 approaches π/2. Consistent with the assumption of large Prl, I will be replaced with π/2 here. Also substituting κ = 0.4 (the usual recommended value of the von Karman constant) and Prt = 0.9 yields

h = 0.134

κ l Prl νl

τ0 ρl

The frictional component of the two-phase flow pressure gradient (dP/dz)F must be related to the wall shear stress τ0 as

2  dP  πD πDτ 0 = −    dz  F 4

Using the Martinelli correlation to evaluate (dP/dz)F yields

2fl G 2 (1− x )  dP  = φl2 , −   dz  F Dρl 2

 G (1− x ) D  fl = 0.046   µl  

−0.2

Using these relations to evaluate τ0, the equation for h can be written as

 G (1− x ) D  hD = 0.134φl fl   Prl kl µl  

Substituting the above relation for fl (for turbulent flow) and the Martinelli turbulent-turbulent correlation for ϕl:

 20 1 φl =  1+ +  Xtt Xtt2 

1/ 2

the correlation for h can be cast in the form

20 1 hD 1/ 2  = 0.028Re 0.9 l Prl  1+ X + X 2  kl tt tt

1/ 2

where

Re l =

G (1− x ) D µl

Perhaps the most interesting aspect of this result is that this relation is very similar to that proposed by Traviss et al. [11.5]. Their relation, which is described in detail in the next section, has the form

hD = f1 (Prl ,Re l ) f2 ( Xtt )Re 0.9 l kl

where f1 is a relatively weak function of Rel. This model analysis also demonstrates the direct link between the heat transfer performance and the shear stress in the liquid film during annular-flow convective condensation.

519

Internal Convective Condensation

11.3 CORRELATION METHODS FOR CONVECTIVE CONDENSATION HEAT TRANSFER It was demonstrated in Section 11.1 that for many applications involving internal flow condensation, the flow is expected to be in the annular regime over most of the passage length. While detailed analytical models of annular flow, such as those discussed in the previous section, can be used as a means of predicting the heat transfer performance for annular flow condensation, such models generally require a high level of computational effort, and they can be used only if reliable closure relations for the interfacial shear and entrainment are available. As a result, somewhat simpler empirical relations that more directly correlate condensation heat transfer data have also been developed by a number of investigators.

The Correlation of Ananiev et al. [11.6] In an early study, Ananiev et al. [11.6] proposed to correlate the local heat transfer coefficient for convective condensation with the relation

h = h0

ρl (11.26) ρm

where

 1 1  1 = (1 − x ) +   x (11.27) ρm  ρl   ρv 

and h0 is given by one of the available correlations for the single-phase heat transfer coefficient for the entire flow as liquid. (We could, e.g., use the well-known Dittus-Boelter equation to evaluate h0 for round tubes.) Boyko and Kruzhilin [11.7] evaluated this type of an approach for condensation of steam using the relation proposed by Miropolsky [11.8] for the single-phase coefficient h0:

 Pr  h0 D = 0.021Re 0.8 Prb0.43  b   Prw  k

0.25

(11.28)

where Re is the Reynolds number and Prb and Pr w are values of the Prandtl number evaluated at the bulk and wall temperatures, respectively. Boyko and Kruzhilin [11.7] found that the local condensation heat transfer data obtained by Miropolsky [11.8] for condensation of steam could be correlated reasonably well with this scheme. However, a best fit to data for a steel tube was obtained by replacing the constant 0.021 in Eq. (11.28) by 0.024. The data for condensation of steam in a copper tube differed from that for the steel tube, even though both had a diameter of 2 cm. A best fit to the copper tube data required that the constant in Eq. (11.28) be changed to 0.032. In both cases the data matched the correlation prediction within about ±20%. While this approach is simple to use, the apparent need for a different constant value in Eq. (11.28) for each fluid-solid combination makes its general applicability somewhat questionable. In addition, this correlation method ignores the effect of flow regime changes that are expected to occur along the channel as the flow proceeds downstream. In more recent studies, investigators have proposed correlation techniques for the local condensing heat transfer coefficient that allow for variation of the two-phase flow regime along the passage. In horizontal tubes at low vapor velocities, liquid that condenses on the upper portion of the inside tube wall tends to run down the wall toward the bottom, as indicated schematically in Fig. 11.6. This stratified-annular flow condition is observed most commonly at low condensation rates and/or short tube lengths. In an early study, Chato [11.9] developed a detailed analytical model

520

Liquid-Vapor Phase-Change Phenomena

FIGURE 11.6  Horizontal co-current annular flow with condensation.

of the heat transfer for these circumstances. His results implied that this type of flow configuration exists for inlet vapor Reynolds numbers (Rev) less than 35,000. Predictions of the analytical model and experimental data were found to agree well with an equation of the form  gρl ( ρl − ρv ) kl3 hlv′  h = 0.728 KC   (11.29)  µ l ( Tsat − Tw ) D  1/4

where hlv′ is given by

c pl ( Tsat − Tw )   hlv′ = hlv 1 + 0.68  (11.30) hlv  

The above relation for h is identical to the relation obtained from the classic Nusselt analysis for a vertical flat plate of height D, except that the multiplying prefactor has been changed. This relation is consistent with the interpretation that the condensation process over the top portion of the inside tube wall is very similar to falling-film condensation over a flat plate. Chato [11.9] found that test data for R-113 agreed well with this relation for KC = 0.76. He further argued that heat transfer through the liquid pool at the bottom of the tube (Fig. 11.6) is negligible compared to transport across the thin film on the upper portion of the tube wall. It follows, therefore, that the coefficient KC in Eq. (11.29) must vary with the void fraction. In a subsequent study, Jaster and Kosky [11.10] suggested the following simple relations as a means of predicting the void fraction and the factor KC in Eq. (11.29):

K c = α 3/4 (11.31)



 1 − x  ρv   α = 1 +  (11.32) x  ρl   

−1

This correlation technique applies to the stratified-annular flow conditions that may exist over much of the tube length if the inlet vapor velocity is low. This circumstance has also been examined in detail by Rufer and Kezios [11.11]. Flow circumstances of this type may be encountered near the end of the condensation process when the void fraction is still large, but deceleration of the flow has dropped the vapor velocity enough to produce stratified flow with a thin film of condensate on the upper portion of the tube wall.

521

Internal Convective Condensation

At moderate to high inlet vapor velocities, annular flow is established almost immediately at the inlet, and persists over most of the condensation process. A number of investigators have proposed ways of predicting the condensation heat transfer coefficient for annular flow conditions. Two of the more useful correlations are those developed by Soliman et al. [11.12] and Traviss et al. [11.5].

The Correlation of Soliman et al. [11.12] In the annular flow analysis described in the previous section, it became clear that the shear at the interface and at the tube wall were linked directly to the transport of heat across the liquid film. The correlation proposed by Soliman et al. [11.12] is cast in a form that explicitly acknowledges the importance of these shear parameters in determining the transport. Their correlation is given by hµ l = 0.036 Prl0.65 τ1/2 w (11.33) kl ρ1/2 l

where

τ w = τi + τ z + τ a (11.34)





τi =

D  dP  dP   dP  2 0.523 −  , −  =   φ g  −  , φ g = 1 + 2.85 X tt (11.35) 4  dz  F  dz  F dz  v −1





 1 − x  ρv  2/3  D τ z = (1 − α)(ρl − ρv ) g sin θ, α = 1 +  (11.36) 4 x  ρl    

τa =

D  G 2   dx    4  ρv   dz 

5

ρ  an  v   ρl  n =1

∑

n /3

(11.37)



a1 = 2 x − 1 − β S x (11.38a)



a2 = 2(1 − x ) (11.38b)



a3 = 2(1 − x − β S + β S x ) (11.38c)



1 a4 =   − 3 + 2 x (11.38d)  x



1   a5 = β S  2 −   − x  (11.38e)   x   interface velocity mean film velocity (11.38f) = 1.25 for turbulent film flow = 2.0 for laminar film flow

βS =

522

Liquid-Vapor Phase-Change Phenomena

Note that in the above correlation equations, (dP/dz)v is the single-phase pressure gradient for the vapor phase flowing alone, and Xtt is the turbulent-turbulent Martinelli parameter defined in Chapter 10. For flow in a round tube these can be evaluated as



 GxD  fv = 0.046   µ v 

2 fv G 2 x 2  dP  ,  =  − dz v Dρv 1− x  X tt =   x 



0.9

 ρv   ρ  l

0.5

−0.2

(11.39)

0.1

 µl   µ  (11.40) v

The value of βS is determined based on the Reynolds number for the liquid phase flowing alone in the passage. If it is greater than 2000, the turbulent value is used. Otherwise the laminar value is used. Although the scheme proposed by Soliman et al. [11.12] is complex, it can easily be implemented as an explicit calculation in a simple computer program.

The Correlation of Traviss et al. [11.5] Traviss et al. [11.5] proposed the following relation for the local heat transfer coefficient for annular flow convective condensation:

hD 0.15 Prl Rel0.9 = kl FT

2.85   1  X + X 0.476  , tt  tt 

Rel =

G (1 − x ) D (11.41) µl

where Xtt is given by Eq. (11.40) and FT is given by

{

FT = 5 Prl + 5 ln {1 + 5 Prl } + 2.5 ln 0.0031Rel0.812

{

}

}

for Rel > 1125 (11.42a)



= 5 Prl + 5 ln 1 + Prl (0.0964 Rel0.585 − 1)               for 50 < Rel ≤ 1125



= 0.707 Prl Rel0.5                                           

(11.42b)

Rel ≤ 50 (11.42c)

The Correlation of Shah [11.13, 11.14] In an early study, Shah [11.13] proposed an empirical correlation as a best fit to available convective condensation heat transfer data for round tubes. This correlation was found to agree well with film condensation data for a wide variety of fluids and flow conditions. This correlation was recommended for 11 ≤ G ≤ 211 kg/m2s, 0 ≤ x ≤ 1.0, 0 ≤ Prl ≤ 13. Shah [11.14] later improved and extended this correlation methodology to the following form. For J v ≥ 0.98( Z + 0.263)−0.62: n



0.04 h  µl   3.8 x 0.76 (1 − x )  0.8 − + 1 x = ( )   (11.43) hlo  14µ v   Pr0.38 

where

k   GD  hlo = 0.023  l    D   µ l 

0.8

Prl0.4 (11.44)

523

Internal Convective Condensation

And for J v < 0.98( Z + 0.263)−0.62: 1/3

n

0.04 3 h  µl   3.8 x 0.76 (1 − x )  0.8 −1/3  ρl (ρl − ρv ) gkl  = (1 − x ) +  + 1.32 Rel   (11.45) 0.38 2  hlo  14µ v   Pr µl   



where J v =



 1 − 1 0.5 , Z =   x  [ gDh ρv (ρl − ρv )] xG

0.8

Pr0.4 , n = 0.0058 + 0.557 Pr (11.46)

and Pr is the reduced pressure Pr = P / Pcrit . Example 11.3 Steam condenses at 247 kPa as it flows inside a horizontal tube with an inside diameter of 2.0 cm. For G = 200 kg/m2s, x = 0.7, and a wall temperature of 100°C, compute the local heat transfer coefficient using the correlations of (a) Traviss et al. [11.5], (b) Shah [11.14], and (c) Boyko and Kruzhilin [11.7]. For saturated steam at 247 kPa, Tsat = 400 K. = 127°C, ρl = 938 kg/m3, ρv = 1.37 kg/m3, hlv = 2183 kJ/kg, cpl = 4.24 kJ/kg K, μl = 219 × 10 –6 Ns/m2, μv = 13.6 × 10 –6 Ns/m2, kl = 0.686 W/mK, and Prl = 1.35. At 100°C, Prl = 1.72. a. For the specified conditions,

Rev =

GxD (200)(0.7)(0.02) = 2.06 × 105 = 13.6 × 10 −6 µv



Re l =

G(1− x )D (200)(0.3)(0.02) = = 5480 µl 219 × 10 −6

and for turbulent-turbulent flow,

ρ  Xtt =  v   ρl 

0.5

 µl   µ  v

0.1

 1− x    x 

0.9

 1.37  =  938 

0.5

 219    13.6 

0.1

 0.3    0.7 

0.9

= 0.0235

Substituting into Eq. (11.42a),

{

FT = 5Prl + 5ln {1+ 5Prl } + 2.5ln 0.0031Re 0.812 l

}

= 5(1.35) + 5ln{1+ 5(1.35)} + 2.5ln{0.0031(5480)0.812 } = 20.0 Rearranging Eq. (11.41) and substituting,



0.9 2.85   k  0.15Prl Re l  1 hT =  l   X + X 0.476   D FT tt  tt  0.9 2.85   0.686  0.15(1.35)(5480)  1 2 =    0.0235 + (0.0235)0.476  = 47,900 W / m K  0.02  20.0  

Note also that

FTD

  G2 x 2 =  Dg ( ) ρ ρ − ρ v  v l 

0.5

  (200)2 (0.7)2 =  1.37(938 1.4)(0.02)9.8 −  

0.5

= 8.83

524

Liquid-Vapor Phase-Change Phenomena For the values of X = Xtt = 0.0235 and FTD = 8.83, Fig. 10.5 indicates that the flow is in the annular flow regime. b. For the Shah [11.14] correlation, substituting into Eq. (11.44),



0.8

 k   GD  hlo = 0.023  l    D   µ l 

Prl0.4

 0.686   200(0.02)  = 0.023   0.02   219 × 10 −6 

0.8

(1.35)0.4 = 2280 W / m 2K

and noting that Pc for water is 22,124 kPa, xG

0.7(200)



Jv =



1  Z =  − 1 x 



n = 0.0058 + 0.557Pr = 0.0058 + 0.557(247 / 22,124) = 0.012

 gDρv (ρl − ρv ) 0.8

0.5

=

[9.8(0.02)1.37(938 − 1.37)]0.5

 1  −1 Pr0.4 =   0.7 

= 8.83,

0.8

(247 / 22,124)0.4 = 0.0841,

Direct substitution indicates that the inequality Jv ≥ 0.98( Z + 0.263)−0.62 is satisfied and therefore Eq. (11.43) applies

h  µl  = hlo  14µv 

n

0.04  3.8 x 0.76 (1− x )  0.8  (1− x ) + Pr0.38  

Substituting, we find that

 219  ×  10 −6    hS = (2280)   (14)13.6  ×  10 −6 

0.012

 3.8(0.7)0.76 (0.3)0.04  0.8 2 (0.3) + (247 / 22,124)0.38  = 35,600 W / m K  

c. The heat transfer coefficient is determined using the Boyko and Kruzhilin [11.7] correlation as follows



 k   GD  h0 = 0.021 l    D   µ l 

0.8

 Pr  Prb0.43  b   Prw 

0.25

Noting that at the wall and bulk flow temperatures, Prl = Prw = 1.72 and Prl = Prb = 1.35, respectively, substitution yields

 0.686   200(0.02)  h0 = 0.021  0.02   219 × 10 −6 

0.8

 1.35  (1.35)0.43   1.75 

0.25

= 1970 W / m 2K

Equations (11.26) and (11.27) are combined to obtain 1/ 2



 ρ   hBK = h0 (1− x ) +  l  x   ρv   

1/ 2

   938  0.7  = 1970 (0.3) +   1.37   

= 43,100 W / m 2K

Thus the predicted h values for these three correlations vary from 35.6 to 47.9 kW/m2K.

525

Internal Convective Condensation

The Correlation of Chen et al. [11.15] In a later study, Chen et al. [11.15] proposed a comprehensive film condensation heat transfer correlation based on analytical and theoretical results from the literature. For co-current annular film flow they argued that for falling film condensation in the absence of interfacial shear, the method of Churchill and Usagi [11.16] can be used to determine the local Nusselt number for an arbitrary condition from separate correlations for the laminar-wavy and turbulent film-flow regimes. Based on such arguments, they postulated the following relation for the zero-shear Nusselt number Nu0 at an arbitrary location

Nu 0 = [(Nu lw )n1 + (Nu t )n1 ]1/ n1 (11.47)

where Nulw and Nut are the Nusselt numbers for laminar wavy and turbulent film flow, respectively. The Nusselt number in the correlations developed by these investigators is defined as Nu = hvl2/3 / ( kl g1/3 ) . In a similar fashion, Chen et al. [11.15] proposed the following relation for the local Nusselt number Nux in the presence of vapor shear

Nu x = [(Nu 0 )n2 + (Nu sd )n2 ]1/ n2 (11.48)

where Nu0 is the zero shear value as determined using Eq. (11.47), and Nu sd is the value of the Nusselt number obtained from a prediction for shear-dominated film flow. For laminarwavy flow with zero shear, Chen et al. [11.15] used the correlation recommended by Chun and Seban [11.17]

Nu lw = 0.823 Re −x 0.22 (11.49)

where

Re x =

G (1 − x ) D (11.50) µl

For the turbulent regime, Chen et al. [11.15] used the following relation to predict Nut which is a curve-fit to the theoretical results of Blangetti and Schlunder [11.18].

0.65 Nu t = 0.0040 Re 0.4 (11.51) x Prl

For the high interfacial shear stress regime, a modified form of the relation proposed by Soliman et al. [11.12] was used to predict Nusd.

Nu sd = 0.036 Prl0.65 (τ∗i )1/2 (11.52)

where τ∗i is a dimensionless interfacial shear stress defined as

τ∗i =

τ (11.53) ρl ( gvl )2/3

In obtaining Eq. (11.52) from the original relation of Soliman et al. [11.12], the gravitational and acceleration effects have been neglected, as would be appropriate for the interfacial-sheardominated case.

526

Liquid-Vapor Phase-Change Phenomena

For co-current flow, Chen et al. [11.15] used the following expression to determine τ∗i , which was derived from the empirical two-phase pressure drop correlation developed by Dukler [11.19] τ∗i = AD (Re ter − Re x )1.4 Re 0.4 x (11.54)

where

AD =



0.252 µ1.117 µ 0.156 l v (11.55) 2 2/3 0.553 0.78 D g ρl ρv Re ter =



GD (11.56) µl

Using Eqs. (11.49)–(11.56) to evaluate the terms in Eqs. (11.47) and (11.48), Chen et al. [11.15] determined the values of n1 and n2 that provided a best fit to the measured local heat transfer coefficients obtained by Blangetti and Schlunder [11.18]. They found that n1 = 6 and n2 = 2 provided a very good fit to these data. Combining the above equations for these values of n1 and n2, the following correlation equation is obtained Nux =

3.9  hvl2/3  Re 2.4 x Prl −1.32 = 0.31Re +  x 1/3 14   kl g 2.37 × 10   1/2

1/3

(11.57)

 A Pr            + D (Re ter − Re x )1.4 Re 0.4 x  771.6  1.3 l

This equation applies to annular flow condensation in vertical tubes, and includes the effects of gravity, interfacial waves and interfacial shear. As illustrated in Fig. 11.7, Eq. (11.57) agrees fairly well with the local condensation heat transfer data obtained by Ueda et al. [11.20] for steam.

FIGURE 11.7  Comparison of the data of Ueda et al. [11.20] with the Nusselt number variation with Reynolds number predicted by the correlation of Chen et al. [11.15] for internal flow condensation of steam in a tube.

527

Internal Convective Condensation

If the gravity force terms are neglected, the above equation reduces to

Nu x =

hvl2/3 = 0.036 AD0.5 Prl0.65 (Re ter − Re x )0.7 Re 0.2 x (11.58) kl g1/3

When shear forces are large compared to body forces, the tube orientation is unimportant. Equation (11.58) is therefore expected to apply to shear-dominated annular film condensation in either horizontal or vertical tubes. Although the gravitational acceleration g appears in the definitions of Nux and AD, it cancels out of Eq. (11.58), leaving the heat transfer coefficient independent of g. Chen et al. [11.15] also used Eq. (11.57) to derive a relation for the average heat transfer coefficient for annular film condensation in vertical tubes, and they extended their analysis and developed a correlation for counterflow reflux condensation. For co-current flow, Chen et al. [11.15] noted that their correlation equation (11.57) may be inaccurate near the inlet of the tube and/or at low quality near the end of the condensation process. Near the inlet, high vapor shear may result in high liquid entrainment, resulting in a mist-annular flow or breakdown of a continuous liquid film. Condensation heat transfer for such conditions would be significantly different from the annular flow postulated in their analysis. At the end of the condensation process, the flow pattern will change to slug flow for vertical tubes, or to stratified or intermittent (slug/plug) flow for horizontal tubes. For these non-annular regimes, the heat transfer mechanisms are expected to differ and the correlation for annular flow is not expected to apply. A comparison of the heat transfer coefficient variations along the passage predicted by the above correlations for co-current annular flow is shown in Fig. 11.8. The variations have been computed for R-134a condensing at conditions typical of a vapor compression refrigeration system. Other efforts to develop improved methods to predict the convective condensation heat transfer coefficient for nominally annular flow include:

The Correlation of Moser et al. [11.21] Moser et al. [11.21] proposed a predictive model for condensation heat transfer in the annular regime that uses an effective Reynolds number defined as

Ree = Rel 0 φ8/7 lo (11.59)

FIGURE 11.8  Comparison of the variation of h with x predicted by five correlation methods.

528

Liquid-Vapor Phase-Change Phenomena

where φlo is the two-phase multiplier determined form the correlation of Friedel [11.22] defined by Eqs. (10.85)–(10.90) in Chapter 10. The heat transfer coefficient h is computed from the relation h = hlo F (11.60)

where F is defined

F = (Tl − Tw ) / (Tδ F − Tw ) (11.61)



and hlo is calculated for the entire flow as liquid using an appropriate single-phase correlation with the effective Reynolds number Ree. An example would be the Dittus Boelter equation for turbulent flow in a round tube hlo =



kl 0.023 Ree0.2 Prl0.4 (11.62) D

A dimensionless temperature is defined

T + ( y) =

Tw − T ( y) (11.63) qw′′ / ρl c pl τ w / ρl

(

)

and Tδ+ at the film interface must be computed from eddy diffusivity models of heat in the liquid film. The dimensionless film thickness δ +F = δ F τ w / ρl / νl is first computed from relations based on the similar model of Traviss et al. [11.5]:



 0.7071Rel0.5 for Rel < 50   δ +F =  0.4818Rel0.585 for 50 ≤ Rel > 1125 (11.64)  0.812 for Rel > 1125  0.0095Rel

Once δ +F is determined, R + is computed as R + = 0.0994 Ree7/8 (11.65)



and Tδ+ (δ +F ) is then determined from the integral relations



 δ +F Prl for δ +F ≤ 5   5 {Prl + ln[1 + Prl (δ +F / 5 − 1)]} for 5 < δ +F < 30  + Tδ =  (11.66) δ +F  (1 − y + / R + )dy + for δ +F ≥ 30  5[Prl + ln(1 + 5Prl )] + (1 / Prl ) − 1 + ( y + / 2.5)(1 − y + / R + )  30 

∫

They demonstrated that F can then be computed as

F=

1.07 2 / f + 12.7(Prl2/3 − 1) (11.67) Tδ+

529

Internal Convective Condensation

where f is the single-phase friction factor computed for the flow at the equivalent Reynolds number Ree. To simplify calculations, Moser et al. [11.20] developed the following correlation to the integrated results of the relation for Tδ+ (δ +F ) for δ +F > 30: F = 1.31( R + )C1 ReCl 2 Prl−0.185 (11.68)

where

C1 = 0.126 Prl−0.185 , C2 = −0.113 Prl−0.563 (11.69)



With F determined and Eq. (11.62) used to predict hlo, the heat transfer coefficient is determined as h = hlo F . Alternatively, it is noteworthy that for turbulent flow, Ree is expected to be large, resulting in large R +, often on the order of 103 to 104 or higher. Since the dimensionless film thickness δ +F is typically no more than a few hundred, and the integrals consider y + < δ +F , it follows that y + / R + may often be much smaller than 1. If this is true, and y + / R + is neglected in the integral for δ +F ≥ 30 , it simplifies to a form that is easily integrated to obtain

 Prl δ +F − 2.5 Prl + 2.5  + + + Tδ+ = 5[Prl + ln(1 + 5 Prl )] + 2.5 ln   for δ F ≥ 30, δ F / R 2.5, where jg * =

 Gx  ρv (11.77)  (ρl − ρv ) gD  ρv 

For annular flow, Cavallini et al. [11.25] recommended using a modified version of the heat transfer model proposed by Kosky and Staub [11.29]. The wall shear stress τ w is computed from the twophase frictional pressure gradient (dP / dz ) f : τ w = ( D / 4)(dP / dz ) f , where (dP / dz ) f is computed using a suitable two-phase friction factor model. In the modified Kosky and Staub [11.29] correlation employed in the model of Cavallini et al. [11.25–11.28], the dimensionless film thickness δ +F = δ F τ w / ρl / νl is computed from



 Rel / 2 for Rel ≤ 1145  δ +F = δ F τ w / ρl / νl =  (11.78) 7/8 for Rel > 1145  0.0504 Rel

where Rel = G (1 − x ) D / µ l , and the dimensionless temperature Tδ+ is computed as



 δ +F Prl for δ +F ≤ 5   Tδ+ =  (11.79) 5 {Prl + ln[1 + Prl (δ +F / 5 − 1)]} for 5 < δ +F < 30  + +  5[Prl + ln(1 + 5 Prl ) + 0.495 ln(δ F / 30)] for δ F ≥ 30

The heat transfer coefficient for annular flow hann = (Tw − Tl ,δ F ) / qw′′ is then calculated from the definition of Tδ+:

Tδ+ = ρl c pl τ w / ρl / hann , where hann = (Tw − Tl ,δ F ) / qw′′ (11.80)



⇒ hann = ρl c pl τ w / ρl / Tδ+ (11.81)

In the stratified regime, the heat transfer coefficient in the correlation of Cavallini et al. [11.25–11.28], hstr , is computed as

 k 3ρ (ρ − ρv ) ghlv  hstr = 0.725  l l l   µ l D(Tsat − Tw ) 

0.25

−1

0.268   0.8  1− x   1 + 0.82   + hlo (1 − x ) (1 − θl / π) (11.82) x  

where

θl = π − cos −1 (2α − 1) (11.83)

α is the void fraction, and hlo is calculated using a single-phase correlation with the entire flow as liquid in the passage.

531

Internal Convective Condensation

For the transition and wavy-stratified flow region, where jg * < 2.5 and X tt , defined as ρ  X tt =  v   ρl 



0.5

 µl   µ  v

0.125

 1− x    x 

0.875

(11.84)

is in the range X tt < 1.6, the heat transfer coefficients computed for annular and stratified flow are averaged: htws = hstr + ([hann ] jg∗ = 2.5 − hstr )( jg∗ / 2.5) (11.85)



Here [hann ] jg∗ = 2.5 is the heat transfer coefficient computed with the method for the annular regime at conditions corresponding to jg* = 2.5. For the stratified-slug transition and slug regimes, jg * < 2.5 and X tt > 1.6, and the heat transfer coefficient hstsl is computed as  x  hstsl = hlo +  ([htws ]Xtt =1.6 − hlo ) (11.86) [ x ]  Xtt =1.6 

where

[ x ]Xtt =1.6 =



(µ l / µ v )1/9 (ρv / ρl )5/9 (11.87) 1.686 + (µ l / µ v )1/9 (ρv / ρl )5/9

This methodology for predicting condensation heat transfer coefficients in different regimes is based on data for a variety of refrigerants for reduced pressures Pr < 0.75, density ratio values ρl / ρv > 4 , and channel hydraulic diameters greater than about 2 mm.

The Correlation of Park et al. [11.30] hDh  φv  = 0.0055Rel0.7 Prl1.37  0.89  (11.88) kl  X tt 

where:

ρ  φ = 1 + 13.17  v   ρl 



1− x  X tt =   x 



0.17

0.9

 ρv   ρl 

(

)

1 − exp −0.6 Bo  X tt + X tt2 (11.89)  

2 v

0.5

Rel =

 µl   µ  v

0.1

, Bo =

g(ρl − ρv ) Dh2 (11.90) σ

G (1 − x ) Dh (11.91) µl

The Correlation of Kim and Mudawar [11.31] Kim and Mudawar [11.31] proposed the following predictive relation for the heat transfer coefficient for condensing mini- and microchannel flows.

φ  hDH = 0.048 Rel0.69 Prl0.34  v  (11.92)  X tt  kf

532

Liquid-Vapor Phase-Change Phenomena

where φ v and X tt conform to the Martinelli correlation definitions (see Chapter 10),

φ2v = 1 + CX tt + X tt2 (11.93) X tt2 =

(dP / dz )l (11.94) (dP / dz ) v

with different values for the constant C for different laminar and turbulent liquid and vapor flow states. Although developed for mini- and microcondensing flows, this relation has been shown to agree will experimental data for tubes as large as 7 mm in diameter.

The Correlation of Rosson and Meyers [11.32] For the slug flow often encountered at the end of the condensation process, the intermittent nature of the flow makes analysis of the condensation heat transfer difficult at best. Based on point measurements of the heat transfer coefficient around the perimeter during slug, plug, and wavy (stratified) flow condensation in a horizontal tube, Rosson and Meyers [11.32] recommended that the upper and lower portions of the tube be treated separately. Over the upper portion of the tube, the condensation process is basically falling-film condensation with superimposed effects of vapor shear. They proposed to correlate the heat transfer coefficient over the upper portion of the tube with the relation

 GxD  htop = 0.31   µ v 

0.12

1/4

 gρl (ρl − ρv ) kl3 hlv′   µ (T − T ) D  (11.95) w  l sat 

where hlv′ is given by Eq. (11.30). Based on arguments regarding the analogy between heat and mass transfer, Rosson and Meyers [11.32] suggested the following relation for the heat transfer coefficient over the bottom portion of the tube

hbot D φl ,vt [8G (1 − x ) D / µ l ]1/2 = (11.96) kl 5[1 + Prl−1 ln(1 + 5 Prl )]

where ϕ l,vt is the two-phase multiplier for viscous (laminar) liquid flow and turbulent vapor flow, as predicted by the Martinelli correlation (Eq. (10.58a)) with C = 12. The angular position θm at which the h value undergoes a transition from the higher value on the top to the lower value on the bottom must be determined to obtain an overall mean value for the condensing heat transfer coefficient at a given downstream location. Based on their data, they concluded that the angular position at which this transition occurs is dependent on the vapor and liquid Reynolds numbers Rev and Rel, respectively, and the Galileo number Ga defined as





Re v =

Rel =

Ga =

GxD (11.97) µv

G (1 − x ) D (11.98) µl

D 3ρl (ρl − ρv ) g (11.99) µ l2

Internal Convective Condensation

533

FIGURE 11.9  The graphical correlation for θm proposed as part of the correlation of Rosson and Meyers [11.32] for plug, slug or wavy flow condensation heat transfer in a horizontal tube. (Adapted from reference [11.32] with permission, copyright © 1965, American Institute of Chemical Engineers.)

From their data, Rosson and Meyers [11.32] developed a correlation for θm as a function of the above parameters. This correlation was presented graphically as shown in Fig. 11.9. With the value of θm determined from this plot, the streamwise local condensation heat transfer coefficient, averaged over the tube perimeter can be approximately computed as θ   π − θm  h =  m  htop +  h (11.100)  π   π  bot



A similar but more detailed model of slug-flow condensation has also been developed by Tien et al. [11.33]. Example 11.4 Use the correlation of Chen et al. [11.12] to predict the local heat transfer coefficient for condensation of R-134a at 1604 kPa in a horizontal tube with an inside diameter of 1.0 cm. The local flow conditions are such that G = 500 kg/m2s and x = 0.6. Compare the prediction with that obtained using the Shah correlation. For saturated R-134a at 1604 kPa, Tsat = 331 K = 58°C, ρl = 1063 kg/m3, ρv = 82.7 kg/m3 μl = 128 × 10 –6 Ns/m2, μν = 13.7 × 10 –6 Ns/m2, kl = 0.0670 W/mK, Prl = 3.13, and σ = 0.0040 N/m. To determine the flow regime, we compute Rel and Rev:

Re l =

G(1− x )D 500(0.4) ( 0.01) = = 15,600 µl 128 × 10 −6

534

Liquid-Vapor Phase-Change Phenomena





Rev = ρ  Xtt =  v   ρl 

0.5

 µl   µ  v

0.125

GxD 500(0.6)(0.01) = 2.18 × 105 = 13.7 × 10 −6 µv

 1− x    x 

0.875

 82.7  =  1063 

0.5

 128    13.7 

0.125

 0.4    0.6 

0.875

= 0.259

Evaluating FTD using Eqs. (10.7a) and (10.15b), we find

  G2 x 2 FTD =    ρv (ρl − ρv )Dg 

0.5

  (500)2 (0.6)2 =   82.7(1063 − 82.7)(0.01)9.8 

0.5

= 3.37

For these values of Xtt and FTD, Fig. 10.5 indicates that the flow is well into the annular flow regime. This implies that gravity effects on the annular film are small and may be neglected. Evaluating AD and Reter using Eqs. (11.55) and (11.56), we obtain



AD = =

µv0.156 0.252µ1.177 l 2 2/ 3 0.553 0.78 D g ρl ρv 0.252(128 × 10 −6 )1.177 (13.7 × 10 −6 )0.156 = 1.69 × 10 −6 (0.01)2 (9.8)2/ 3 (1063)0.553 (82.7)0.78



Re ter =

GD 500(0.01) = = 39,100 128 × 10 −6 µl

Neglecting the effects of gravity, the heat transfer coefficient is given by Eq. (11.58):  kl g1/ 3  0.5 0.65 A h = 0.036  (Re ter − Re x )0.7 Re 0.2 x 2/ 3  D Prl  (µ l / ρl ) 



Using the fact that Rex = Rel and substituting yields  (0.0670)(9.8)1/ 3  (1.69 × 10 −6 )0.5 h = 0.036  2/ 3  −6  (128 × 10 / 1063) 

× (3.13)0.65 (39,100 − 15,600)0.7 (15,600)0.2 = 4572 W / m 2K

Using the Shah correlation (noting that Pc = 4059 kPa for R-134a), it follows from Eqs. (11.44) and (11.46) that



 k   GD  hlo = 0.023  l    D   µ l 

0.8

Prl0.4

 0.0670   500(0.01)  = 0.023   0.01   128 × 10 −6  xG

0.8

(3.13)0.4 = 1146 W / m 2K 0.6(500)



Jv =



1  Z =  − 1 x 



n = 0.0058 + 0.557Pr = 0.0058 + 0.557 (1604 / 4059) = 0.226

 gDρv (ρl − ρv ) 0.8

0.5

=

[9.8(0.01)82.7(1063 − 82.7)]0.5

 1  −1 Pr0.4 =   0.6 

0.8

= 3.37

(1604 / 4059)0.4 = 0.499,

535

Internal Convective Condensation

Substituting indicates that the inequality Jv ≥ 0.98( Z + 0.263)−0.62 is satisfied, and therefore Eq. (11.43) applies



n 0.04 3.8 x 0.76 (1− x )   µ   0.8 h = hlo  l  (1− x ) +  0.38 Pr  14µv   

 128 × 10 −6  h = (1146)   (14)13.7 × 10 −6 

0.226

0.76  (0.4)0.04  0.8 3.8(0.6) 4200 W / m 2K (0.4) 0.38  = (1604 / 4059)  

This result is within 9% of the value obtained with the correlation of Chen et al. [11.15].

Since the beginning of research on flow condensation, it has been recognized that methods to predict flow condensation heat transfer should account for any changes in the two-phase flow regime, and its effects on heat transfer mechanisms. Early correlation methods have typically used one or two relations since the annular flow morphology exists during much of the process, and a second method can be invoked when that gives way to wavy stratified or slug/bubbly flow at the end of the process. Over time, improved predictive models have been developed that account for the effect of flow regime transitions in more detailed ways. The improved Shah [11.14] correlation reflects this trend, as do the correlations of Breber et al. [11.34] and Chen et al. [11.15]. The models developed by Cavallini et al. [11.25–11.28] and Thome et al. [11.35] embody this more detailed approach, accounting for several possible regime transitions and associated changes in heat transfer mechanisms. The trends in the heat transfer coefficient variation with quality and mass flux predicted by this type of more detailed model are generally consistent with those in simpler correlation methods. However, the more detailed multiregime methods are expected to provide more accurate predictions of heat transfer for systems within the ranges of conditions for which they were developed.

11.4 CONVECTIVE CONDENSATION IN MICROCHANNELS, ADVANCED MODELING, AND SPECIAL TOPICS Recent research efforts on convective condensation have yielded noteworthy results in three areas: microchannel convective condensation, advanced modeling strategies, and special case studies that relate to enhancement strategies or specific applications. Developments in each of these areas are discussed below.

Microchannel Convective Condensation In the previous sections of this chapter, the discussion of convective condensation in tubes has focused on flow passages with round smooth cross sections and diameters larger than about 3 mm. This geometry and size range are typical of larger scale condensers in many conventional technologies, including commercial refrigeration systems, early building and first-generation automotive air-conditioners, and Rankine power systems for electricity generation and propulsion. However, recent advances in manufacturing technologies have made it possible to fabricate condensers with passages that have very small cross-sectional dimensions. Compact condensers with small hydraulic diameter passages are an attractive option for residential and automotive air conditioning systems because their compactness requires less material and less space in the application system, and, as will be discussed below, the smaller hydraulic diameter tends to enhance heat heat transfer. MEMS lithographic fabrication methodologies and micromachining now make it possible to create condensers with passage dimensions as small as a few hundred nanometers. This provides the possibility of producing ultra-miniature condensers that could be

536

Liquid-Vapor Phase-Change Phenomena

used in thermal control systems for small electronic components or systems. In this section, we will first examine the effect of passage size reduction and passage cross-section geometry on flow and heat transfer at these small scales. Effects of Reducing Flow Passage Size We can gain some insight into the effect of reducing passage diameter by considering the results of the simplistic separate-cylinders model discussed in Section 10.3. Equation (10.69) implies the following link between the two-phase multiplier φl2 and the void fraction  1 (1 − α) =  2   φl 



2/(5− n )

(11.101)

As passage diameter is reduced, the Reynolds number associated with the flow becomes progressively smaller and the flow would be expected to be in the laminar-laminar regime. As noted in earlier sections of this chapter, annular flow is expected to exist during much of the internal flow condensation processes that occur in conventional applications. This is also true for microcondensers. We will therefore focus specifically on how annular flow transport changes with passage size. For annular flow in a round tube with an ultra-thin film of thickness δ with a smooth interface, it follows that 1− α =



2πDδ (11.102) πD 2 / 4

For transport across the film by conduction h=



kl (11.103) δ

As shown in Section 10.3, for laminar-laminar flow, n = 1 and the equation for φl2 becomes 2

  1  φl2 = 1 +    (10.74)   X vv  



Combining the above equations for n = 1 yields the following relations for the film thickness and heat transfer coefficient:



δ=

h=

D D = (11.104) 8φl 8[1 + (1 / X vv )]

8 k l φl 8 k l   1   =  (11.105) 1 + D D   X vv  

As noted in Section 10.3, as quality approaches 1, Xvv → 0 and the factor in square brackets above −1 approaches ( X vv ) . In this limit, the above equations are well approximated as



δ=

h=

D X vv (11.106) 8

8 kl ( X vv )−1 (11.107) D

537

Internal Convective Condensation

For laminar-laminar flow (n = 1), it is easily shown using the separate cylinder model relations from Section 10.3 that

µ  X vv =  l   µv 

1/2

 1− x    x 

1/2

1/2

 ρv   ρ  (11.108) l

Using this to replace Xvv in Eqs. (11.106) and (11.107), they become 1/2



δ=

D  µl  8  µ v 



h=

8 kl  µ v  D  µ l 

 1− x    x 

1/2

1/2

 x    1− x 

1/2

 ρv   ρ  (11.109) l

1/2

1/2

 ρl   ρ  (11.110) v

Note that for a given saturation pressure, the properties of the saturated liquid and vapor in Eqs. (11.109) and (11.110) are specified. Thus, this crude annular flow model indicates that for a given fluid at a specified quality and saturation pressure, the film thickness decreases and the heat transfer coefficient increases as the tube diameter is reduced. These general trends are, in fact, observed in the performance of high-performance compact condensers used in automotive air-conditioning applications. Early condenser designs for this type of application typically used round tubes with inside diameters of about 10 mm. More modern designs use flow passages with much smaller hydraulic diameters. The flow passages in an automotive condenser design developed by Modine Manufacturing Company [11.36] is an example of this strategy. The cross section of this passage is shown schematically in Fig. 11.10. The patent for this design specifies optimal mean hydraulic diameters for the passage in the range of 380–1000 μm. The approximate analysis described above indicates trends that motivate the use of ultrasmall flow passages for high-performance condensers. As passage size is reduced, the flow exhibits specific features that are different from those for comparable macroscale condensing flows. As noted above, the resulting small Reynolds numbers imply laminar flow. In general, it is expected that the two-phase flow in micro- and nanochannels will be dominated by the interplay among flow inertia, viscous forces (shear), and surface tension forces. Gravity induced buoyancy is expected to play a much weaker role. The relative magnitude of these different

FIGURE 11.10  Cross section of the flow passages design used in the ultra-compact Modine automotive condenser [11.36].

538

Liquid-Vapor Phase-Change Phenomena

mechanisms can be quantified in terms of the Bond number Bo D and the Weber numbers Wel and Wev, defined as G 2 (1 − x ) D (11.111) σρl 2

Wel =



We v =



Bo D =



G2x2D (11.112) σρv

g ( ρl − ρv ) D (11.113) σ

The Bond number Bo D characterizes the ratio of buoyancy to surface tension forces. In microchannels this ratio is typically much lower than in macrochannel condensation processes in conventional applications, which is indicative of the weak impact of gravitational buoyancy compared to surface tension forces in microchannel condensation. The Weber numbers Wel and Wev characterize the ratio of liquid inertia to surface tension and vapor inertia to surface tension, respectively. These numbers are generally lower for flow condensation in microchannels, relative to typical condensation processes in macrochannels, reflecting the stronger influence of surface tension effects in microchannel convective condensation. Note that the capillary number Ca l , representing the ratio of liquid viscous forces to surface tension forces, is essentially the ratio of liquid Webber number Wel to Reynolds number for the liquid flowing alone, Rel.

Ca l =

 G 2 (1 − x )2 D  Wel µG (1 − x ) µ =  = Re (11.114) G (1 − x ) D  σρl σρl  l

Investigations of condensation in small channels prior to 2002 have been reviewed by Cavallini et al. [11.37]. Heat transfer coefficients for convective condensation in micro- and minichannels have been obtained experimentally by a number of investigators [11.38–11.52]. Accurate determination of local heat transfer coefficient data for convective condensation in microchannels is a challenging task. Wang and Rose [11.53] have argued that the uncertainty in the data from many previous studies is large. Chowdhury et al. [11.54] and Cavillini et al. [11.55] have developed improved test section designs and experimental procedures in efforts to obtain more accurate data. Wu and Cheng [11.52] carried out experiments in which they simultaneously visualized the flow and measured wall temperature variations during convective condensation of steam in microchannels. When saturated steam enters the channel and condensation continues almost to completion, Wu and Cheng [11.52] observed the sequence of flow regimes indicated in Fig. 11.11. At high quality, a droplet flow was observed in which the condensed liquid formed droplets that were carried downstream on or adjacent to the wall by vapor drag. At slightly lower quality, liquid covered the wall in a thin film and annular flow was observed. With decreasing quality, the thickness of the film increased until the vapor core became unstable, pinching off bubbles. This condition was termed

FIGURE 11.11  Two-phase flow regimes observed during convective condensation in a microchannel.

539

Internal Convective Condensation

an injection flow regime. At low qualities, large bubbles produced by the injection flow regime were carried downstream in a slug-bubbly flow. This sequence was observed in the trapezoidal channel with a hydraulic diameter of 82.6 μm tested by Wu and Cheng [11.52] at a steam/water mass flux of 304 kg/m2. If complete condensation did not occur, then only the droplet or the droplet and annular regimes were observed. Chen and Peterson [11.56] observed a similar sequence of flow regimes for convective condensation in triangular microchannels. As noted by Wang and Rose [11.53], the data from several experimental investigations (see, e.g., reference [11.57]) indicate that annular flow spans an increasing portion of the condensation process as the hydraulic diameter of the passage decreases. As a consequence, annular flow has been a main focus of efforts to empirically and theoretically predict convective condensation heat transfer in microchannels. For annular flow condensation in rectangular microchannels, Wang et al. [11.44] found that the following predictive correlation provided a best fit to 700 annular flow data for convective condensation heat transfer: 0.5



 1.376 + 8 X tt1.655  hDh = 0.0274 Prl Rel0.6792 x 0.2208   (11.115) kl X tt2 

In the above equation, Dh is the hydraulic diameter of the rectangular microchannel, and the Reynolds number Rel = G(1 – x)Dh /μl for the liquid flowing alone is based on Dh. At specified local conditions, this empirical correlation generally predicts a local heat transfer coefficient value that is lower than that predicted by the large round tube correlations discussed in an earlier section of this chapter. As noted by Wang et al. [11.44], this suggests that the annular film in a small rectangular tube has a different configuration than that in round tubes for the same conditions. In Fig. 11.12, the correlation of Wang et al. [11.44] is compared to convective condensation heat transfer data for a round minichannel obtained by Cavallini et al. [11.55]. Also shown are the predictions of two of the large round-tube correlations discussed in Section 11.3. The correlation prediction using Eq. (11.75) was determined taking the hydraulic diameter to be equal to the round-tube diameter for the data (800 μm). It can be seen that the round minichannel data are generally higher than the prediction of the rectangular minichannel correlation of Wang et al. [11.44],

FIGURE 11.12  Comparison of correlation predictions with measured convective condensation heat transfer data for a round minichannel with an inside diameter of 800 μm.

540

Liquid-Vapor Phase-Change Phenomena

but lower than the prediction of correlations developed for larger round tubes. The correlation of Ananiev et al. [11.6] is only slightly higher than the microchannel data. Subsequent investigations by, for example, Baird et al. [11.58], Cavallini et al. [11.28], Kim et al. [11.59], Wu and Cheng [11.60], Agarwal [11.61], and Garamella, et al. [11.62], have experimentally measured local condensation heat transfer coefficients for refrigerant and water flows in tube sizes ranging from 83 µm to 1.4 mm. Data from these studies generally reflect the trends of increasing heat transfer coefficient as mass flux and quality increase, consistent with the trend indicated in Fig 11.12, and predicted by the correlations of Wang and Rose [11.53], and Kim and Mudawar [11.31]. In comparing their data to correlations developed for macro scale channels, Baird et al. [11.58] found that measured heat transfer data for flow condensation in microchannels can differ substantially from the predictions of correlations developed for larger tubes. These observations suggest that correlations and models developed specifically for microchannels will likely be better predictors of flow condensation heat transfer performance in microchannels than predictive models develop for channels with larger hydraulic diameter. Although numerous experimental studies have obtained local heat transfer coefficient data for flow condensation in microchannels, a smaller number of studies have developed correlations or models of heat transfer during flow condensation in microchannels. Convective condensation in microchannels has also been theoretically analyzed in studies by Begg et al. [11.63], Zhao and Liao [11.64], Li and Wang [11.65], and Wang and Rose [11.53]. These studies largely focus on annular flow transport because, as noted above, this regime is most frequently encountered during the condensation process. The correlation methodology of Kim and Mudawar [11.31], Eq. (11.92), is an example of a correlation developed specifically for annular flow dominated condensation in microchannels. Effects of Micro- and Minichannel Flow Passage Cross Section Microchannels can be round in cross section or can have cross sections of other shapes. Etching, micromachining or lithographic processes can be used to produce microchannels in a substrate with rectangular, trapezoidal, or triangular cross sections. A key aspect of the noncircular shapes is the corner region. As discussed in Chapter 9, when a surface is covered with a liquid film, the liquid-vapor interface curves in the corner region, which reduces the liquid pressure in the film in the corner region. This tends to drive liquid into the corners, thickening the film there. This is the Gregorig effect discussed in Chapter 9. When annular flow is first established, liquid condensing on the film on flat surfaces of a passage will tend to be drawn to the corner regions, as depicted in Fig. 11.13a. As the amount of liquid on the walls increases, the wall regions accumulate more liquid and surface tension tends to pull the interface into a more cylindrical surface, as shown in Fig. 11.13b. As condensation continues, eventually the interface becomes cylindrical (Fig. 11.13c), leading to the transition from annular flow to slugbubbly flow. In non-circular microchannels, the very small radius of curvature of the interface in the corner strongly reduces the liquid pressure there, strongly drawing liquid to the corners and significantly

FIGURE 11.13  Cross section of two-phase flow during condensation in a square microchannel.

541

Internal Convective Condensation

thinning the liquid film on the flat wall surfaces. This substantially enhances the condensation heat transfer in annular flow. This strong enhancement of condensation due to the Gregorig effect (see Chapter 9) in micropassages makes their use attractive for high performance compact condensers. Tubes or channels with ridges or grooves in the wall surface or protruding microfins may also benefit from this effect during annular flow. Microchannels and minichannels can also be fabricated with more complex passage geometries, such as offset strip fins, or cross-ribbed channels. Discontinuous fin or rib surfaces are key elements in passages of this sort. In annular flow, liquid may be shed from these types of discontinuous surfaces. The resulting entrainment of liquid thins the liquid film on the passage walls, which generally enhances heat transfer. Mandrusiak and Carey [11.66] have argued that this type of entrainment will only occur if vapor inertia is large compared to surface tension forces. This condition can be quantified in terms of a Weber number We L f defined as

We L f =

G2x2L f (11.116) σρv

where Lf is a length scale associated with the lateral extent of the fin or rib. Entrainment due to shedding off fins or ribs in the channel is expected for We L f >> l. This is consistent with the findings of Mandrusiak and Carey [11.66], which indicated that the threshold Weber number for onset of shedding was about 20 for their channel with offset strip fins. In most microchannels, the characteristic length of a fin or rib would likely be so small that the Weber number would be much smaller than the threshold for onset of shedding entrainment. It is therefore unlikely that this mechanism would be important in microchannels with complex ribbed or finned geometries unless the vapor mass flux was very high and the surface tension was very low. Further discussion of convective condensation in channels with non-circular cross sections may be found in references [11.67–11.71].

Advanced Models and More Complex Operating Conditions The models and correlation described in the previous section are generally applicable to steady flows at fixed flow rate and a steady wall boundary condition with a fixed temperature or heat flux. Some recent studies have explored more advanced modeling or special-case flow condensation conditions that exhibit somewhat different behavior. Some of these studies have explored the use of advanced, multiphase flow volume of fluid (VOF) methodologies. Examples include Chen et al. [11.72] who used the VOF method to simulate condensing flow in a rectangular microchannel; Bortolin et al. [11.73], who used the VOF method to model condensation in a square minichannel; and Yin et al. [11.74], who use the VOF method to model annular film condensation in a horizontal minitube. Other numerical approaches to modeling internal flow condensation are described by Lee et al. [11.75], Kharangat and Mudawar [11.76], and Toninelli et al. [11.77]. It should be noted that the predictive correlations and modeling efforts described above most often focus on steady flow conditions under earth normal gravity, with steady boundary conditions. There are reasons to consider circumstances that deviate from these conditions, however, even under steady inlet and boundary conditions, instabilities in the two-phase flow can produce waves or oscillatory flow behavior. In addition, in applications, a pump or compressor driving the flow may introduce fluctuations in pressure and or flow, which give the flow and transport a non-steady character. These issues have been explored by, for example, Kivisalu et al. [11.78], Naik et al. [11.79], and Narain et al. [11.80]. The interested reader can get more detailed information regarding these issues from these references. Deviation from earth normal gravity also can alter the two-phase flow and transport during internal flow condensation. As expected, in horizontal tubes, reducing gravitational acceleration tends to reduce stratification of the flow. The resulting flow may resemble a shear-dominated annular flow

542

Liquid-Vapor Phase-Change Phenomena

with little or no stratification. Flow condensation under microgravity conditions has been explored experimentally in recent investigations by Lee et al. [11.81] and May et al. [11.82].

11.5  INTERNAL CONVECTIVE CONDENSATION OF BINARY MIXTURES While convective condensation of a pure vapor is perhaps more commonly encountered in refrigeration and powers systems, there are an increasing number of important circumstances in which convective condensation of a multicomponent mixture occurs. In the petrochemical processing industries, processes of this type are, in fact, common. Refrigeration systems that contain a pure refrigerant and a soluble lubricating oil are essentially a binary mixture system. Binary working fluids are also being considered for use in heat pump systems and in heat pipes, as a means of improving the thermal performance of the system. Convective condensation of the mixture plays an important role in the operation of these types of systems. The general treatment of condensation of multicomponent mixtures is beyond the scope of this text. However, most of the important features of the general multicomponent case can be seen in the simpler circumstance of convective condensation of a binary mixture. With some extension, many of the concepts developed for convective condensation of pure vapors can be used for binary systems. To gain some insight into the physics of multicomponent convective condensation, in this section we will therefore examine convective condensation in a binary system in some detail. To limit the complexity of the problem, we will confine our attention to simple binary mixtures without azeotropic points. The added complexity associated with condensation in a binary mixture system results primarily from (1) the more complicated thermodynamics associated with the binary system, and (2) the added effect of mass (species) transport during the condensation process. In modeling internal flow condensation of a binary mixture, we will adopt several idealizations that will simplify the analysis somewhat, but retain most of the essential physics of the process. The thermodynamic characteristics of convective condensation of a binary vapor mixture can best be understood by simultaneously considering the binary phase diagram in Fig. 11.14 and the schematic

FIGURE 11.14  Variation of saturation conditions during convective condensation of a binary mixture.

Internal Convective Condensation

543

FIGURE 11.15  System model for annular downflow film condensation for a binary mixture in a vertical tube.

of co-current downward flow condensation in a round tube depicted in Fig. 11.15. (See Section 8.5 for a discussion of the basic thermodynamic considerations.) Note that species 1 is the more volatile component (the component with the lowest pure fluid boiling point) and xˆ l and yˆl represent the liquid and vapor mole fractions of species 1, respectively. If the vapor entering the tube has a bulk concentration yˆ1,0 the equilibrium saturation temperature at which condensation occurs is Tin = Tdp ( yˆl,0 , P), as indicated in Fig. 11.14. The concentration of liquid first formed is xˆ l,0, as dictated by the bubble point curve in Fig. 11.14. For the purposes of this discussion of the condensation process shown in Fig. 11.15, we will adopt the following idealizations that are similar to those used in the analysis of internal flow condensation of a pure vapor in Section 11.2: 1. The co-current downward flow in the tube is in the annular regime with a smooth interface and no entrainment. 2. Saturated vapor enters the tube. 3. The thickness of the film on the wall of the tube is thin compared to the tube diameter. 4. The liquid film flow and the vapor core flow are turbulent and the transport in the film is shear-dominated. 5. Sensible cooling of the vapor is negligible compared to the latent heat energy associated with the condensation process. 6. Effects of liquid mass transport on the interface condition are negligible. These idealizations simplify analysis of the transport somewhat. They can be relaxed, when necessary to provide a more accurate treatment. However, the main effects of the binary system thermodynamics and species transport are often only slightly different from those for these more idealized circumstances. Because the concentration of the more volatile component 1 in the condensate is lower than in the vapor, the excess of this component left behind in the vapor raises its bulk concentration yˆl,b as

544

Liquid-Vapor Phase-Change Phenomena

the flow proceeds down the tube. This has the effect of lowering the dew point as the flow proceeds along the tube. If complete condensation is achieved at some point along the tube, the liquid bulk concentration ( xˆ l,b )cc (on the bubble point curve) must equal the inlet concentration in the vapor yˆl,0 . These considerations limit the range of saturation temperature exhibited by the system during the condensation process. From Fig. 11.14, it can be seen that the saturation temperature is expected to be between T0 and (Tl)cc. If the binary phase diagram is available, the temperature and concentration limits for the process can be determined from the specified inlet pressure and concentration. For the heat transport in the liquid film, we will adopt the model developed in Example 11.2 for shear-dominated turbulent heat transfer in the liquid film of an annular flow with a smooth interface and no entrainment. That model was developed for upward flow, but since the flow is shear dominated, it works equally well for a downward shear dominated flow. In that model the film was shown to be a constant shear layer. This constant shear in the film must equal the interfacial shear τi . The relation between the heat transfer coefficient and the shear in the film that was obtained from the model can therefore be expressed as

h = 0.134

kl Prl vl

τi (11.117) ρl

In evaluating the interfacial shear stress τi , the analysis here treats the vapor core flow as singlephase flow in a round tube, with the velocity of the vapor taken to be zero at the interface. As noted in Section 11.2, this is a very simplistic treatment, but it provides a reasonable first approximation for τi .

   ρ u2  G2 x2 (11.118) τi = fl  v v  = fi   2   2ρv (1 − 8δ / D) 

For the interfacial friction factor fi, we use the empirical relation suggested for annular flow in Chapter 10:

 300δ  fi = 0.005  1 +  (11.119)  D 

For a general analysis of condensation of a binary mixture under these circumstances, a detailed treatment of conservation of species and energy at the interface would be included to predict the interface conditions for steady transport. In such a general treatment, the local concentration varies with cross-stream position in the vapor and liquid portions of the flow, and the equilibrium conditions specified by the binary phase diagram would be assumed to exist at the liquid-vapor interface. This implicitly invokes local thermodynamic equilibrium at the interface consistent with the phase diagram for the mixture. Satisfying this diagram at the interface becomes a boundary condition, with the concentration fields in the two-phases then being dictated by species transport due to convection and diffusion. Here, however, we will invoke some idealizations to construct a simpler example analysis. In the model adopted here, the treatment of the interface mass transport assumes that turbulent mass transfer in the gas is so effective that the concentration of species 1 at the interface yˆl,i equals the bulk value yˆl,b . The concentration of liquid condensed at the interface xˆ l.i is taken to be equal to the equilibrium liquid concentration (on the bubble point curve) for the system pressure and dew point temperature at the local vapor concentration xˆ l,i = xˆ l,bp (Tdp ( yˆl,b , P), P) . The relationship between the dew point temperature of the vapor at yˆl,i = yˆl,b and xˆ l,i can be seen in Fig. 11.14. The species 1 concentration in the bulk liquid xˆ l,b in the film will generally be slightly lower than ˆx l,i because liquid produced upstream condensed at lower concentration. Mass diffusion of species 1 into the bulk liquid will tend to lower xˆ l in the liquid at the interface below xˆ l,i . In this analysis we

545

Internal Convective Condensation

will assume that the concentration of species 1 in the liquid at the interface varies only slightly from xˆ l,i = xˆ l,bp (Tdp ( yˆl,b , P), P) , and this variance due to liquid mass transfer has a negligible effect on the interface temperature. Consistent with these idealizations, we take the interface temperature Ti to be the dew point temperature at the bulk vapor concentration Tdp ( yˆl,b , P) . With the idealizations described above, the heat transfer rate is dictated by the heat transfer coefficient for turbulent transport across the liquid film h, and the wall-interface temperature difference Tdp ( yˆl,b , P) − Tw :

q ′′ = − h Tdp ( yˆl,b , P) − Tw  (11.120)

Implicit here is the idealization that the species mass transport in the liquid film and vapor adjust so as to accommodate conservation of species, but that transport will not affect h or the interface temperature. It follows from an energy balance on the overall flow that

dx 4 q ′′ 4 h(Tsat − Tw ) = =− (11.121) dz DGhlv DGhlv

Because this is a binary system, x in Eq. (11.121) is not the quality in the usual thermodynamic sense for a pure fluid. Consistent with its interpretation for two-phase flow, x is interpreted here as simply being the ratio of the vapor mass flow rate to the total mass flow rate.

x=

m tot,v m tot,v = (11.122)  mtot G (π D 2 / 4)

where m tot,v is the total vapor mass flow rate (the sum for species 1 and 2 in the vapor):

m tot,v = m l,v + m 2,v (11.123)

and consistent with previous analysis, the mass flux G is the total mass flow rate m tot divided by the tube cross-sectional area G=



m tot (11.124) (πD 2 / 4)

In Eq. (11.123), m l,v and m 2,v are the mass flow rates of species 1 and 2 in the vapor, respectively. The model in Example 11.2 also predicts that the film thickness δ depends on the local shear stress, and quality as

δ = 0.134

 GD(1 − x )  vl  µl 2τi / ρl 

1/2

(11.125)

The following additional relations account for conservation of species, overall mass, and energy in the vapor. First, by definition of the vapor mole fraction of species 1

yˆl,b =

m l,v / M l (11.126)  mlv / M l + m 2,v / M 2 )

In the above equation Ml and M2 are the molecular masses of species 1 and 2, respectively. Rearranging this relation yields the following equation that links the mass flow of the two species:

 1 − yˆl,b   M 2  m 2,v =    m l,v (11.127)  yˆl,b   M l 

546

Liquid-Vapor Phase-Change Phenomena

Combining Eqs. (11.123) and (11.127) yields m l. v =



m tot,v ( M l / M 2 ) yˆl,b (11.128) 1 − yˆl,b + yˆl,b ( M l / M 2 )

The heat flux across the liquid film must equal the rate of decrease of enthalpy of the vapor core. Neglecting sensible cooling compared to latent heat effects, this implies that πDh(Tdp − Tw ) dm tot,v =− (11.129) dz hlv



Combining Eqs. (11.128) and (11.129), we obtain πDh(Tdp − Tw )  M l yˆl,b + M 2 (1 − yˆl,b )  dm l,v =−   (11.130) dz hlv M l yˆl,b  



The relations obtained above are sufficient to solve for the flow and heat transfer variation along the passage using a numerical integration over finite segments of the tube having length Δz. The numerical integration is facilitated by first dividing the length of the tube into N segments. The segments are sequentially assigned a number n. The scheme uses Eqs. (11.121) and (11.130), in tandem with other relations, to compute the variations of h, δ, x, yˆl,b , xˆ l,i , Tdp, and m l,v along each segment. The numerical scheme treats the parameters on the right side of these relations as constant over the segment at their values at the segment inlet. Unless otherwise indicated, parameter values are based on inlet properties for segment n. The computational algorithm proceeds as follows: 1. At the inlet to the first segment (z = 0), set n = 1, δ = 0, x = 1.0, and yˆl,b = yˆl,0. From the specified m tot and D, compute G using Eq. (11.124). Also determine m l,v and m 2,v at the inlet to the first section using Eqs. (11.127) and (11.128) and the fact that m tot ,v = m tot at the inlet.

For yˆl,b = yˆl,0 :

m l,v =

m tot ( M l / M 2 ) yˆl,b , 1 − yˆl,b + yˆl,b ( M l / M 2 )

 1 − yˆl,b   M 2  m 2,v =    m l,v  yˆl,b   M1 

2. If not at the outlet, set the inlet values of δ, x, yˆl,b , m l,v, and m 2,v for the next segment n to the respective exit values of these variables computed for the previous segment. For yˆl,b at the inlet of segment n, use thermodynamic information to determine the equilibrium values of bubble point liquid concentration xˆ l,i = xˆ l,bp (Tdp ( yˆl,b , P), P) and dew point temperature Tdp ( yˆl,b , P) . 3. Use Eqs. (11.118) and (11.119) to calculate τi for segment n using the segment inlet film thickness δ and the value of x at the inlet of the segment. 4. Use the computed τi for the interfacial shear to compute the film heat transfer coefficient h for segment n using Eq. (11.117), evaluating properties for the liquid at the interface temperature Tdp ( yˆl,b , P) . 5. Determine δout,n using Eq. (11.125) with τi , and the value of x at the inlet of the segment.

δ out ,n = 0.134

 GD(1 − x )  vl  µl 2τi / ρl 

1/2

547

Internal Convective Condensation

6. Using the results of previous steps, determine (dx / dz )n from Eq. (11.121) and determine (dm l,v / dz )n from Eq. (11.130). Then compute segment outlet values of x , m l,v , m 2,v , and yˆl,b as follows:

4 hn Tdp ( yˆl,b , P) − Tw   dx    = − dz n DGhlv



πDhn Tdp ( yˆl,b , P) − Tw   M l yˆl,b + M 2 (1 − yˆl,b )   dm l,v    = −   dz n hlv M l yˆl,b  



 dx  xout,n = x +   ∆z  dz  n



 dm  (m l,v )out ,n = m l,v +  l,v  ∆z  dz  n from Eq. (11.122): (m tot,v )out,n = m tot xout,n

from Eq. (11.123):

(m 2,v )out,n = (m tot,v )out,n − (m l,v )out,n

from Eq. (11.126):

( yˆ1 )out,n =

(m 1,v )out,n / M1 (m 1,v )out,n / M1 + (m 2,v )out,n / M 2

7. Increment n. If for the new n, n ≥ N exit the algorithm, the computation is complete. If the new n is less than N, go to step 2. Inspection of the above algorithm reveals that even for the simplified model considered here, the level of computational effort required to predict the heat transfer performance is significantly higher than that for a comparable pure fluid condensation process. Nevertheless, this type of algorithm can be easily programmed on a computer, if the thermophysical property data are available. It cannot be overemphasized that the idealized model analysis described in this section for annular film condensation in a round tube is only approximate in many respects. It is presented here to illustrate the nature of this type of computation. Before attempting to use this scheme, the reader would be wise to carefully examine the idealizations in the analysis to determine the appropriateness of each to the application of interest. The determination of the interfacial shear and the treatment of mass transfer effects are particularly crude in this model, but they could be upgraded by using more detailed models of these effects. Incorporating the effects of possible entrainment would also improve the model (see Chapter 10). In the above analysis, the fully mixed turbulent core flow and the assumption of negligible liquid mass transfer effects on the interface temperature allows specification of the local interface temperature in a relatively simple manner. If these idealizations are inaccurate, then the species transport equation must be solved in the film and the core flow to determine the interface condition at each downstream location. The species transport equation must then be numerically solved

548

Liquid-Vapor Phase-Change Phenomena

downstream from the inlet condition in both the core vapor flow and the liquid film. This makes determination of the interface condition in each segment more complicated. As noted in the review article by Koyama and Yu [11.83], accurate treatment of the mass transfer effects are among the most important and most challenging aspects of modeling transport in internal flow condensation of binary mixtures. Additional information about internal convective condensation of binary mixtures is available in the open literature. In an early study, van Es and Heertjes [11.49] theoretically and experimentally investigated the natural convection condensation of non-azeotropic vapor mixtures of benzene and toluene in a vertical tube. In their experiments, the vapor was only partially condensed in the tube. The theory developed by these investigators was applied to flows driven by gravity and by the interfacial shear associated with the turbulent downward flow of the vapor mixture. Somewhat later, experimental and theoretical investigations of convective film-wise condensation of non-azeotropic mixtures of R-11 and R-114 inside a vertical tube were conducted by Mochizuki et al. [11.85]. Experiments were conducted in which local and overall heat transfer coefficients were determined for co-current downflow condensation of the mixture. A theoretical model of the condensation process, similar to that outlined above, was developed and its predictions were compared to the experimentally determined variation of the heat transfer coefficient with downstream distance along the tube. The predictions of the model for conditions resulting in laminar film flow agreed reasonably well with measured data. However, for turbulent flow, the predictions of the model were substantially different from the measured h profiles along the tube. For the overall mean heat transfer coefficient for the entire condensation process, these investigators developed the following correlation as a best fit to their data

hL L   = 0.38  Ja      kl D 

−0.3

0.8

 Re * Prl   (11.131)  R* 

where c pl ∆T (11.132) hlv



Ja =



Re* =

U in L (11.133) vl 1/2



 ρµ  R* =  l l  (11.134)  ρv µ v 

In the above relations, Uin is the inlet vapor mean velocity and ∆T is the mean difference between the interface and wall temperatures. This relation was reported to be accurate to ±10% for 5.8 × 106 < Re* < 7.5 × 107 and 60 < L/D < 222. While this correlation scheme seems to work well for this specific binary mixture, further testing of its predictions against data for other fluids is necessary before it can be widely used with confidence. Non-equilibrium models of condensation of binary mixtures inside horizontal round tubes have been developed by Zhang et al. [11.86] and Koyama et al. [11.87]. Zhang et al. [11.86] found that their model heat transfer predictions agreed well with data for an R-32/134a refrigerant binary mixture. The model of Koyama et al. [11.87] incorporated mass, species, and energy conservation treatments similar to those in the idealized model described earlier in this section. However, the model of Koyama et al. [11.87] used the correlation of Shah [11.13, 11.14], developed for pure fluids, to

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predict the heat transfer across the liquid film. Their model also incorporated a more detailed treatment of mass transfer effects, with phase equilibrium at the interface and the bulk liquid and vapor phases in non-equilibrium states. Koyama et al. [11.87] found that their model predictions of heat transfer and wall temperatures along the tube agreed well with data for an R-134a/123 binary mixture. Recent investigations by Koyama et al. [11.88], Kedzierski and Kim [11.89], and Kedzierski and Goncalves [11.90] have also explored internal forced flow condensation of binary mixtures in enhanced tubes with internal fins or twisted tape inserts. While the investigations described in this section have provided important insight into the heat transfer mechanisms in internal flow condensation of binary mixtures, they have been tested against very limited data, and further study of their performance for wider conditions and other mixtures is needed to define their range of applicability and fully assess how well they model the transport mechanisms in binary mixture internal flow condensation.

REFERENCES 11.1 Rykaczewski, K., Paxson, A. T., Staymates, M., Walker, M. L., Sun, X., Anand, S., Srinivasan, S., McKinley, G. H., Chinn, J., Scott, J. H. J., and Varanasi, K. K., Dropwise condensation of low surface tension fluids on omniphobic surfaces, Sci. Rep., vol. 4, pp. 4158-1 to 4158-8, doi:10.1038/srep04158, 2014. 11.2 Sett, S., Yan, X., Barac, G., Bolton, L.W., and Miljkovic, N., Lubricant-infused surfaces for low-surface-tension fluids: Promise versus reality, ACS Appl. Mater. Interfaces, vol. 9, pp. 36400–36408, 2017. 11.3 Taitel, Y., and Dukler, A. E., A model for predicting flow regime transitions in horizontal and near horizontal gas-liquid flow, AIChE J., vol. 22, pp. 47–55, 1976. 11.4 Coleman, J. W., and Garimella, S., Two-phase flow regimes in round, square and rectangular tubes during condensation of refrigerant R134a, Int. J. Refrig., vol. 26, pp. 117–128, 2003. 11.5 Traviss, D. P., Rohsenow, W. M., and Baron, A. B., Forced convection condensation in tubes: A heat transfer correlation for condenser design, ASHRAE Trans., vol. 79, Part I, pp. 157–165, 1972. 11.6 Annaiev, E. P., Boyko, L. D., and Kruzhilin, G. N., Heat transfer in the presence of steam condensation in a horizontal tube, Proc. 1st Int. Heat Transfer Conf., part II, p. 290, 1961. 11.7 Boyko, L. D., and Kruzhilin, G. N., Heat transfer and hydraulic resistance during condensation of steam in a horizontal tube and in a bundle of tubes, Int. J. Heat Mass Transf., vol. 10, pp. 361–373, 1967. 11.8 Miropolsky, Z. L., Heat transfer during condensation of high-pressure steam inside a tube, Teploenergetika, vol. 3, pp. 79–83, 1962. 11.9 Chato, J., Laminar condensation inside horizontal and inclined tubes ASHRAE J., pp. 52–60, 1962. 11.10 Jaster, H., and Kosky, P. G., Condensation in a mixed flow regime, Int. J. Heat Mass Transf., vol. 19, pp. 95–99, 1976. 11.11 Rufer, C. E., and Kezios, S. P., Analysis of stratified flow with condensation, J. Heat Transf., vol. 88, pp. 265–275, 1966. 11.12 Soliman, M., Schuster, J. R., and Berenson, P. J., A general heat transfer correlation for annular flow condensation, J. Heat Transf., vol. 90, pp. 267–276, 1968. 11.13 Shah, M. M., A general correlation for heat transfer during film condensation inside pipes, Int. J. Heat Mass Transf., vol. 22, pp. 547–556, 1979. 11.14 Shah, M. M., An improved and extended general correlation for heat transfer during condensation in plain tubes, HVAC&R Res., vol. 15, pp. 889–913, 2009. 11.15 Chen, S. L., Gerner, F. M., and Tien, C. L., General film condensation correlations, Exp. Heat Transf., vol. 1, pp. 93–107, 1987. 11.16 Churchill, S. W., and Usagi, R., A general expression for the correlation of rates of transfer and other phenoemena, AIChE J., vol. 18, pp. 1121–1128, 1972. 11.17 Chun, K. R., and Seban, R. A., Heat transfer to evaporating liquid films, ASME J. Heat Transf., vol. 93, pp. 391–396, 1971. 11.18 Blangetti, F., and Schlunder, E. O., Local heat transfer coefficients of condensation in a vertical tube, Proc. 6th Int. Heat Transfer Conf., vol. 2, pp. 437–442, 1978. 11.19 Dukler, A. E., Fluid mechanics and heat transfer in vertical falling-film systems, Chem. Eng. Prog. Symp. Ser., vol. 56, no. 30, pp. 1–10, 1960.

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11.20 Ueda, T., Kubo, T., and Inoue, M., Heat transfer for steam condensing inside a vertical tube, Proc. 5th Int. Heat Transfer Conf., vol. 3, pp. 304–308, 1976. 11.21 Moser, K. W., Webb, R. L., and Na, B., A new equivalent Reynolds number model for condensation in smooth tubes, J. Heat Transf., vol. 120, 410–417, 1998. 11.22 Friedel, L., Improved friction pressure drop correlations for horizontal and vertical two-phase pipe flow, paper E2, Proc. European Two Phase Flow Group Meeting, Ispra, Italy, 1979. 11.23 Dobson, M. K., and Chato, J. C., Condensation in smooth horizontal tubes, J. Heat Transf., vol. 120, pp. 193–213, 1998. 11.24 Koyama, S., Kuwahara, K., Nakashita, K., and Yamamoto, K., An experimental study on condensation of refrigerant R134a in a multi-port extruded tube, Int. J. Refrig., vol. 24, pp. 425–432, 2003. 11.25 Cavallini, A., Censi, G., Del Col, D., Doretti, L., Longo, G. A., and Rossetto, L., Experimental investigation on condensation of new HFC refrigerants (R134a, R125, R32, R410a, R236ea) in a horizontal smooth tube, Int. J. Refrig., vol. 24, pp. 73–87, 2001. 11.26 Cavallini, A., Censi, G., Del Col, D., Doretti, L., Longo, G.A., and Rossetto, L., Condensation of halogenated refrigerants inside smooth tubes, HVAC&R Res., vol. 8, pp. 429–451, 2002. 11.27 Cavallini, A., Censi, G., Del Col, D., Doretti, L., Longo, G. A., Rossetto, L. and Zilio, C., Condensation inside and outside smooth and enhanced tubes – A review of recent research, Int. J. Refrig., vol. 26, 373–392, 2003. 11.28 Cavallini, A., Doretti, L., Matkovic, M., and Rossetto, L., Update on condensation heat transfer and pressure drop inside minichannels, Heat Transf. Eng., vol. 27, pp. 74–78, 2006. 11.29 Kosky, P. G. and Staub, F. W., Local condensation heat transfer coefficients in the annular flow regime, AIChE J., vol. 17, pp. 1037–1043, 1971. 11.30 Park, J. E., Vakili-Farahani, F., Consolini, L., Thome, J. R., Experimental study on condensation heat transfer in vertical minichannels for new refrigerant R1234ze(E) versus R134a and R236fa, Exp. Therm. Fluid Sci., vol. 35, pp. 442–454, 2011. 11.31 Kim, S. M., Mudawar, I., Universal approach to predicting heat transfer coefficient for condensing mini/micro-channel flow, Int. J. Heat Mass Transf., vol. 56 pp. 238–250, 2013. 11.32 Rosson, H. F., and Meyers, J. A. Point values of condensing film coefficients inside a horizontal tube, Chem. Eng. Prog. Symp. Ser., vol. 61, no. 59, pp. 190–199, 1965. 11.33 Tien, C. L., Chen, S. L., and Peterson, P. F., Condensation inside tubes, Electric Power Research Institute, report no. EPRI NP-5700, 1988. 11.34 Breber, G., Palen, J., and Taborek, J., Prediction of horizontal tube-size condensation of pure components using flow regime criteria, J. Heat Transf. – Trans. ASME, vol. 102, pp. 471–476, 1980. 11.35 Thome, J. R., El Hajal, J., and Cavallini, A., Condensation in horizontal tubes, Part 2: New heat transfer model based on flow regimes, Int. J. Heat Mass Transf., vol. 46, pp. 3365–3387, 2003. 11.36 Guntly, L. A., and Costello, N. F., U.S. Patent No. 4998580, Condenser with small hydraulic diameter flow path, 1991. 11.37 Cavallini, A., Censi, G., Del Col, D., Doretti, L., Longo, A. G., and Rossetto, L., Condensation heat transfer and pressure drop inside channels for AC/HP application, Proc. 12th Int. Heat Transfer Conf., Grenoble, France, August 18–23, 2002 vol. 1, pp. 171–186, 2002. 11.38 Yang, M., and Webb, R. L., Condensation of R-12 in small hydraulic diameter extruded aluminum tubes with and without micro-fins, Int. J. Heat Mass Transf., vol. 39, pp. 791–800, 1996. 11.39 Vardhan, A., and Dunn, E. E., Heat Transfer and Pressure Drop Characteristics of R-22, R-134a and R-407C in Microchannel Tubes, ACRC TR-133, University of Illinois at Urbana-Champaign, 1997. 11.40 Yan, Y. Y., and Lin, T. F., Condensation heat transfer and pressure drop of refrigerant R134a in a small pipe, Int. J. Heat Mass Transf., vol. 42, pp. 697–708, 1999. 11.41 Kim, N. H., Cho, J. P., and Kim, J. O., R-22 condensation in flat aluminum multi-channel tubes, J. Enhanc. Heat Transf., vol. 7, pp. 427–438, 2000. 11.42 Zhang, M., and Webb, R. L., Correlation of two-phase friction for refrigerants in small-diameter tubes, Exp. Therm. Fluid Sci., vol. 25, pp. 131–139, 2001. 11.43 Webb, R. L., and Ermis, K., Effect of hydraulic diameter on condensation of R-134a in flat, extruded aluminum tubes, J. Enhanc. Heat Transf., vol. 8, pp. 77–90, 2001. 11.44 Wang, W. W. W., Radcliff, T. D., and Christensen, R. N., A condensation heat transfer correlation for millimeter-scale tubing with flow regime transition, Exp. Therm. Fluid Sci., vol. 26, pp. 473–485, 2002. 11.45 Garimella, S., and Bandhauer, T. M., Measurement of condensation heat transfer coefficients in microchannel tubes, Proc. 2001 Int. Mechanical Engineering Congress and Exposition, NY, IMECE 2001/ HTD-24221, November, pp. 1–7, 2001.

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11.46 Ermis, K., and Ekmekci, I., Condensation heat transfer in a small hydraulic diameter extruded aluminum tubes, Proc. 1st Int. Exergy, Energy and Environment Symp., Izmir, Turkey, pp. 201–206, July 13–17, 2003. 11.47 Kim, N. H., Cho, J. P., Kim, J. O., and Youn, B., Condensation heat transfer of R-22 and R-410A in flat aluminum multi-channel tubes with and without micro-fins, Int. J. Refrig., vol. 26, pp. 830–839, 2003. 11.48 Kim, M. H., Shin, J. S., Huh, C., Kim, T. J., and Seo, K. W., A study of condensation heat transfer in a single mini-tube and a review of Korean micro- and mini-channel studies, Proc. 1st Int. Conf. on Microchannels and Minichannels, Rochester, NY, pp. 47–58, April 24–25, 2003. 11.49 Baird, J. R., Fletcher, D. F., and Haynes, B. S., Local condensation heat transfer rates in fine passages, Int. J. Heat Mass Transf., vol. 46, pp. 4453–4466, 2003. 11.50 Mederic, B., Miscevic, M., Platel, V., Lavieille, P., and Joly, J. J., Complete convective condensation inside small diameter horizontal tubes, Proc. 1st Int. Conf. on Microchannels and Minichannels, Rochester, New York, NY, pp. 707–712, April 21–23, 2003. 11.51 Chen, Y. P., and Cheng, P., Condensation of steam in silicon microchannels, Int. Commun. Heat Mass Transf., vol. 32, pp. 175–183, 2005. 11.52 Wu, H. Y., and Cheng, P., Condensation flow patterns in silicon microchannels, Int. J. Heat Mass Transf., vol. 48, pp. 2186–2197, 2005. 11.53 Wang. H. S., and Rose, J. W., A theory of film condensation in horizontal noncircular section microchannels, J. Heat Transf., vol. 127, pp. 1096–1105, 2005. 11.54 Chowdhury, S., Al-hajri, E., Dessiatoun, S., Shooshtari, A., and Ohadi, M., An experimental study of condensation heat transfer and pressure drop in a single high aspect ratio microchannel for refrigerant R134a, paper ICNMM206–96211, Proc. 4th Int. Conf. on Nanochannels, Microchannels and Minichannels, Limerick, Ireland, June 19–21, 2006. 11.55 Cavallini, A., Del Col, D., Matkovic, M., and Rossetto, L., Local heat transfer coefficient during condensation in a 0.8 mm diameter pipe, paper ICNMM2006–96137, Proc. 4th Int. Conf. on Nanochannels, Microchannels and Minichannels, Limerick, Ireland, June 19–21, 2006. 11.56 Chen, Y., Li, J., and Peterson, G. P., Influence of hydraulic diameter on flow condensation in silicon microchannels, paper CSN-01, Proc. 13th Int. Heat Transfer Conf., Sydney, Australia, August 13–18, 2006. 11.57 Garimella, S., Condensation flow mechanisms in microchannels: Basis for pressure drop and heat transfer models, Proc. 1st Int. Conf. on Microchannels and Minichannels, ASME, Rochester, New York, NY, pp. 181–192, April 24–25, 2003. 11.58 Baird, J. R., Fletcher, D. F., and Haynes, B. S., Local condensation heat transfer in fine passages, Int. J. Heat Mass Transf., vol. 46, pp. 4453–4466, 2003. 11.59 Kim, M. H., Shin, J. S., Kim, T. J., and Seo, K. W., A study of condensation heat transfer in a single mini tube and review of Korean micro and mini channel studies, Proc. 1st Int. Conf. on Microchannels and Minichannels, Rochester, NY, pp. 47–58, 2003. 11.60 Wu, H.Y., and Cheng, P., Condensation flow patterns in silicon microchannels, Int. J. Heat Mass Transf., vol. 48, pp. 2186–2197, 2005. 11.61 Agarwal, A., Heat transfer and pressure drop during condensation of refrigerants in microchannels, PhD dissertation, Georgia Institute of Technology, Atlanta, GA, 2006. 11.62 Garamella, S., Argarwal, A., and Fronk, B. M., Condensation heat transfer in rectangular microscale geometries, Int. J. Heat Mass Transf., vol. 100, pp. 98–110, 2016, 11.63 Begg, E., Khrustalev, D., and Faghri, A., Complete condensation of forced convection two-phase flow in a miniature tube, J. Heat Transf., vol. 121, pp. 904–915, 1999. 11.64 Zhao, T. S., and Liao, Q., Theoretical analysis of film condensation heat transfer inside vertical mini triangular channels, Int. J. Heat Mass Transf., vol. 45, pp. 2829–2842, 2002. 11.65 Li, J.-M., and Wang, B.-X., Size effect on two-phase regime for condensation in micro/mini tubes, Heat Transf. – Asian Res., vol. 32, pp. 65–71, 2003. 11.66 Mandrusiak, G., and Carey, V. P., Pressure drop characteristics of two-phase flow in a vertical channel with offset strip fins, Exp. Therm. Fluid Sci., vol. 1, pp. 41–50, 1988. 11.67 Webb, R. L., Principles of Enhanced Heat Transfer, John Wiley & Sons, New York, NY, Chapter 14, 1994. 11.68 Kim, M. H., Srinivasan, V., and Shah, R. K., Augmentation techniques and condensation inside advanced geometries, Handbook of Phase Change: Boiling and Condensation, S. Kandlikar, M. Shoji, and V. K. Dhir (editors), Taylor & Francis, Philadelphia, PA, Chapter 24, pp. 639–678, 1999. 11.69 Wang, H. S., and Rose, J. W., film condensation in horizontal microchannels: Effect of channel shape, Int. J. Therm. Sci., vol. 45, pp. 1205–1212, 2006.

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11.70 Agarwal, A., Bandhauer, T. M., and Garimella, S., Measurement and modeling of condensation heat transfer in non-circular microchannels, Int. J. Refrig., vol. 33, pp. 1169–1179, 2010 11.71 Garimella, S., Agarwal, A., and Fronk, B. M., Condensation heat transfer in rectangular microscale geometries, Int. J. Heat Mass Transf., vol. 100, pp. 98–110, 2016. 11.72 Chen, S., Yang, Z., Duan, Y., Chen, Y., and Wu, D., Simulation of condensation flow in a rectangular microchannel, Chem. Eng. Process., vol. 76, pp. 60–69, 2014. 11.73 Bortolin, S., Da Riva, E., and Del Col, D., Condensation in a square minichannel: Application of the VOF method, Heat Transf. Eng., vol. 35, pp. 193–203, 2014. 11.74 Yin, Z., Guo, Y., Sunden, B., Wang, Q., and Zeng, M., Numerical simulation of laminar film condensation in a horizontal minitube with and without non-condensable gas by the VOF method, Numer. Heat Transf. A, vol. 68, pp. 958–977, 2015. 11.75 Lee, Y., Kharangate, C. R., Mascarenhas, N., Park, I., and Mudawar, I., Experimental and computational investigation of vertical downflow condensation, Int. J. Heat Mass Transf., vol. 85, pp. 865–879, 2015. 11.76 Kharangate, C. R., and Mudawar, I., Review of computational studies on boiling and condensation, Int. J Heat Mass Transf., vol. 108, pp. 1164–1196, 2017. 11.77 Toninelli, P., Bortolin, S., Azzolin, M., and Del Col, D., Visualization and numerical simulations of condensing flow in small diameter channels, Heat Transf. Eng., vol. 40, pp. 802–817, 2019, DOI: 10.1080/01457632.2018.1443255. 11.78 Kivisalu, M. T., Gorgitrattanagul, P., Mitra, S., Naik, R. R., and Narain, A., Prediction and control of internal condensing flows in the experimental context of their inlet condition sensitivities, Microgravity Sci. Technol., vol. 24, pp. 147–155, 2012. 11.79 Naik, R., Narain, A., and Mitra, S., Steady and unsteady simulations for annular internal condensing flows, part I: Algorithm and its accuracy, Numer. Heat Transf. B, vol. 69, pp. 473–494, 2016. 11.80 Narain, A., Ranga Prasad, H. P., and Koca, A., Internal annular flow condensation and flow boiling: Context, results, and recommendations, (Chapter 51) in Handbook of Thermal Science and Engineering, F.A. Kulacki (editor), Springer International Publishing AG, part of Springer Nature, Cham, Switzerland, 2018, https://doi.org/10.1007/978-3-319-26695-4_51 11.81 Lee, H., Mudawar, I., and Hasan, M. M., Experimental and theoretical investigation of annular flow condensation in microgravity, Int. J. Heat Mass Transf., vol. 61, pp. 293–309, 2013. 11.82 Lee, H., Park, I., Konishi, C. Mudawar, I, May, R. I., Juergens, J. R., Wagner, J. D., Hall, N. R., Nahra, H. K., Hasan, M. M., and Mackey, J. R., Experimental investigation of flow condensation in microgravity, ASME J. Heat Transf., vol. 136, pp. 021502-1–021502-11, 2013. 11.83 Koyama, S., and Yu, J., Heat Transfer and pressure drop in internal flow condensation, in Handbook of Phase Change: Boiling and Condensation, S. Kandlikar, M. Shoji, and V. K. Dhir (editors), Taylor & Francis, Philadelphia, PA, chapter 23, pp. 621–637, 1999. 11.84 Van Es, J. P., and Heertjes, P. M., On the condensation of a vapor of a binary mixture in a vertical tube, Chem. Eng. Sci., vol. 5, pp. 217–225, 1956. 11.85 Mochizuki, S., Yagi, Y., Tadano, R., and Yang, W.-J., Convective filmwise condensation of nonazeotropic binary mixtures in a vertical tube, J. Heat Transf., vol. 106, pp. 531–538, 1984. 11.86 Zhang, L., Hihara, E., Saito, T., Oh, J. T., and Ijima, H., A theoretical model for predicting the boiling and condensation heat transfer for a ternary mixture inside a horizontal smooth tube, Proc. of the Thermal Engineering Conf., JSME, vol. 95, pp. 94–96 (in Japanese), 1995. 11.87 Koyama, S., Yu, J., and Ishibashi, A., Condensation of binary refrigerant mixtures in a horizontal smooth tube, Therm. Sci. Eng., vol. 6, pp. 123–129, 1998. 11.88 Koyama, S., Yu, J., and Ishibashi, A., Heat and mass transfer of binary refrigerant mixtures condensing in a horizontal microfin tube, Proc. 5th ASME/JSME Thermal Engineering Joint Conf., San Diego, CA, paper AJTE99–6358, 1999. 11.89 Kedzierski, M. A., and Kim, M, S., Convective boiling and condensation heat transfer with a twisted tape insert for R12, R22, R152a, R134a, R290, R32/R134a, R32/R152a, R290/R13a, and R134a/ R600a, Therm. Sci. Eng., vol. 6, pp. 113–122, 1998. 11.90 Kedzierski, M. A., and Goncalves, J. M., Horizontal convective condensation of alternative refrigerants within a micro-fin tube, J. Enhanc. Heat Transf., vol. 6, pp. 161–178, 1999.

PROBLEMS 11.1 Oxygen condenses at a nominal pressure of 196 kPa inside a horizontal tube. The tube inside diameter is 8 mm and the mass flux is 250 kg/m2s. Determine the condensation regime at qualities of 0.9, 0.5, and 0.05.

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11.2 Mercury vapor condenses inside a horizontal tube with an inside diameter of 1.7 cm. The pressure is essentially constant along the tube at 145 kPa. For a mass flux of 2000 kg/m2s, determine the condensation regime at locations where the quality is 0.95, 0.4, and 0.01. 11.3 R-134a condenses in the horizontal tubes of an air-conditioning condenser at a pressure of 815 kPa. The tubes of the condenser are 0.9 cm inside diameter and the mass flux through each tube is 500 kg/m2s. Determine the condensation regime at locations where the quality is equal to 0.95, 0.4, and 0.02. 11.4 R-410A condenses at 2000 kPa in a vertical tube having an inside diameter of 5 mm. At a quality of 0.8, use the correlations of Traviss et al. [11.5] and Chen et al. [11.15] to predict the heat transfer coefficient for downflow mass flux values of 100, 200, 500, and 1000 kg/m2s. Plot both sets of results on a log-log plot. How does the dependence of h on G for these correlations compare? Do they obey a power-law variation? 11.5 R-134a condenses at 1604 kPa in a vertical tube having an inside diameter of 8 mm. For downward flow at a quality of 0.95 and a mass flux of 400 kg/m2s, use the correlation of Travis et al. [11.5] and Chen et al. [11.15] to predict the heat transfer coefficient. Also use one of the correlations from Table 10.2 to predict the void fraction. From the heat transfer results, estimate the liquid film thickness as δ = kl   /h. Compute the film thickness from the void fraction as δ = D(1 − α) / 4 and compare the results to the values of δ computed from the heat transfer coefficients. How would you expect entrainment to affect the validity of these estimates for δ? 11.6 Ammonia condenses in the horizontal tubes of a refrigeration system condenser. The tubes have an inside diameter of 1.3 cm, and the mass flux through each tube is 600 kg/m2s. At a particular location where the pressure is 1425 kPa and the quality is 0.8, determine the void fraction and condensation regime. Also determine the condensation heat transfer coefficient using (a) the heat transfer relation obtained from the analysis in Example 11.2, (b) the correlation of Boyko and Kruzhilin [11.7], and (c) the correlation of Traviss et al. [11.5]. 11.7 Nitrogen condenses in downward co-current flow in a vertical tube. The tube has an inside diameter of 8 mm and a mass flux of 200 kg/m2s. At a location where the heat flux is 500 kW/m2, the quality is 0.6, and the pressure is 540 kPa, determine the condensing heat transfer coefficient using the correlation of Soliman et al. [11.12]. Compare your result with the prediction of the correlation of Boyko and Kruzhilin [11.7]. 11.8 In a miniature condenser unit, it is proposed to condense ammonia at a mass flux of 100 kg/m2s in a round tube with an inside diameter of 0.2 mm. The tube wall temperature is 50°C. For a pressure of 2442 kPa and a quality of 0.8, estimate the heat transfer coefficient using Shah’s [11.14] correlation. Estimate how much the saturation temperature shifts because of the curvature of the liquid film on the wall of the tube (use results given in Chapter 5). If the h value is based on the normal (flat interface) saturation temperature h = q″/(Tsat – Tw), how much (by how many percent) does this effectively alter the h value? 11.9 R-134a condenses inside a horizontal tube with an inside diameter of 1.2 cm. The mass flux is 280 kg/m2s, and the pressure is virtually constant at 1604 kPa along the tube. Determine the approximate value of quality at which the flow is expected to be in the slug, plug or bubbly regime. For a quality of 0.02 and a wall subcooling Tsat – Tw of 10°C, use the correlation of Rosson and Meyers [11.32] to estimate the heat transfer coefficient. How does this value compare with that predicted by the DittusBoelter equation (11.44) for the entire flow as liquid? 11.10 Steam condenses at 247 kPa as it flows inside a horizontal tube with an inside diameter of 2.0 cm. For G = 200 kg/m2s and x = 0.9, 0.7, and 0.5, compute the local heat transfer coefficient using the correlation of Chen et al. [11.15]. Compare your results for x = 0.7 to the values obtained with other correlations in Example 11.3. 11.11 Ammonia condenses inside a horizontal tube with an inside diameter of 9 mm. The mass flux is 400 kg/m2s. For a location near the end of the condensation process, the quality is 0.01, the wall temperature is 65°C, and the pressure is 3870 kPa. (a) What are the void fraction and flow regime for these circumstances? (b) Use the correlation of Rosson and Meyers [11.32] to estimate the heat transfer coefficient at this location. 11.12 Nitrogen condenses during downflow in a round tube with an inside diameter of 1.5 cm. The pressure is nominally constant along the tube at 229 kPa, and the mass flux is 100 kg/m2s. Use the correlation of Chen et al. [11.15] to determine the heat transfer coefficient at x = 0.9, 0.65, and 0.4. Are gravity effects important for these conditions? Explain briefly. 11.13 Steam condenses in a horizontal tube with an inside diameter of 25 mm. The mass flow rate through the tube is 5.9 g/s, the wall temperature is held constant at 112°C, and the pressure is essentially constant along the tube at 247 kPa. Consider a location near the inlet where the quality is 0.9 and one near the

554

Liquid-Vapor Phase-Change Phenomena

outlet where the quality is 0.1. Determine (a) the void fraction for each location, (b) the two-phase flow regime for each location, and (c) the local heat transfer coefficient for each location. Repeat parts (a)–(c) for a flow rate of 25 g/s. 11.14 A downflow heat exchanger for condensing R-410A has a tube configuration like that shown in Fig. 11.2. Each tube in the condenser is 1.2 m long and has an inside diameter of 9 mm. The flow rate through the unit is such that the mass flux through each tube is 250 kg/m2s. The R-410A enters as saturated vapor at 2800 kPa. If the walls of the tube are essentially at a constant and uniform temperature of 40°C, estimate the exit quality of the flow by dividing the tube into four segments and computing the heat transfer for each section assuming that the heat transfer coefficient is constant over the segment. Begin at the inlet segment and work progressively downstream, using the exit condition from the previous segment as the inlet condition to the next. (You may compute the heat transfer coefficient for each segment based on its inlet condition as a first approximation.) 11.15 A cross-flow condenser like that shown in Fig. 11.1 is to be used for condensing ammonia at 1425 kPa. The tubes in the exchanger have an inside diameter of 8 mm, and the temperature of the tube wall is held essentially constant at 20°C. The ammonia enters as saturated vapor, and the mass flow rate is such that the mass flux through each tube is 300 kg/m2s. Estimate the tube length required in the exchanger to completely condense the vapor for these conditions. Do so by considering segments of the tube 20 cm long and assuming that the heat transfer coefficient is constant over each segment. Begin at the inlet and work progressively downstream until the exit quality from a segment equals zero or would be negative based on an energy balance. 11.16 Steam condenses at 247 kPa as it flows through a rectangular minichannel with a hydraulic diameter of 600 μm. The wall temperature is 100°C and the mass flux G is 200 kg/m2s. Using the correlation of Wang et al. [11.44], determine the heat transfer coefficient at a location in the channel where x = 0.7. Compare your result to the large tube predictions of h for these conditions in Example 11.3. Are your results consistent with the trends in the predictions of minichannel and macrochannel correlations in Fig. 11.12? 11.17 R-134a condenses in a horizontal minichannel with a square cross section and a hydraulic diameter of 800 μm. The mass flux is 150kg/m2s. At a particular location along the channel, the pressure is 815 kPa, the quality is 0.6 and the wall temperature is 15°C. (a) Use the correlation of Wang et al. [11.44] to compute the heat transfer coefficient at this location and use its value to determine the local heat flux, (b) Use the heat flux from part (a) to compute dx/dz. (c) Use the Martinelli correlations for ϕ l and α and the separated flow model to determine the friction and acceleration components of dP/dz at this location.

12

Convective Boiling in Tubes and Channels

12.1 REGIMES OF CONVECTIVE BOILING IN CONVENTIONAL (MACRO) TUBES Flow boiling in tubes and channels is perhaps the most complex convective phase-change process encountered in applications. In most evaporator and boiler applications, the flow is either vertically upward or horizontal. To make this discussion more concrete, we will therefore focus on these two specific flow circumstances. Figure 12.1 depicts schematically a typical low-heat-flux vaporization process in a horizontal round tube. The widest possible range of flow conditions is encountered if the liquid enters as subcooled liquid and leaves as superheated vapor. As the vaporization process proceeds, the vapor content of the flow increases with distance along the tube. To maintain the specified mass flow rate as the mean density of the flow decreases, the mean flow velocity must increase substantially. As noted in Chapter 10, the two-phase flow regime generally is strongly dependent on the relative velocities of the two phases. The acceleration of the flow results in an increasing difference between the mean liquid and vapor velocities, which produces a sequence of changes in the flow regime. When boiling is first initiated, bubbly flow exists in the tube. Increasing quality typically produces transitions from bubbly to plug, plug to annular flow, and annular to mist flow, as indicated in Fig. 12.1. Regimes of slug flow, stratified flow, or wavy flow may also be observed at intermediate qualities, depending on the flow conditions. Flow boiling generally is further complicated by the fact that in addition to flow regime changes, different vaporization mechanisms may be encountered at different locations along the tube. As indicated in Fig. 12.1, nucleate boiling is usually the dominant vaporization mechanism near the onset of boiling. As more vapor is generated in the flow and the void fraction increases, the flow undergoes a transition to an annular or nearly annular configuration, whereupon evaporation from the liquid-vapor interface becomes increasingly important. At low to moderate qualities, both mechanisms may be important. As the liquid film on the wall thins, however, film evaporation may become so effective that it becomes the dominant mechanism. In some cases nucleate boiling may be completely suppressed, leaving film evaporation as the only active vaporization mechanism. Acceleration of the vapor core during the latter annular flow stages of the vaporization process very often produces entrainment of liquid droplets. This effect, together with direct vaporization of the film, tends to reduce the film thickness as it flows downstream. Eventually, the film may disappear completely from portions of the tube wall. This is usually referred to as dryout, or in this case, partial dryout of the tube wall. For horizontal tubes, because of the tendency of gravity to thin the liquid film on the top of the tube, partial dryout of the tube usually is first observed along the top portion of the tube wall, as indicated in Fig. 12.1. The lower portion of the tube wall may remain wetted with the liquid film for a substantial distance beyond the first partial dryout of the top portion. The portion of the tube wall covered by the liquid film generally decreases with downstream distance until the wall is completely dry around the entire perimeter of the tube. Prior to the onset of partial dryout, transfer of heat across the liquid film becomes more efficient as the film becomes progressively thinner. As a result, the heat transfer coefficient associated with 555

556

Liquid-Vapor Phase-Change Phenomena

FIGURE 12.1  Qualitative variation of the heat transfer coefficient and flow regime with quality for internal convective boiling in a horizontal tube.

the combined nucleate boiling and film evaporation mechanisms often (but not always) increases with downstream distance prior to the onset of dryout, as indicated in Fig. 12.1. When the tube wall is partially dry, heat transfer from dry portions of the wall surface is negligible compared to that at locations wetted by liquid where film evaporation is occurring. Because these dry locations are relatively inactive, the local heat transfer coefficient, averaged around the perimeter of the tube, is lower than if the entire perimeter were covered by the liquid film. Furthermore, as the wetted fraction of the wall decreases with downstream distance, more of wall becomes inactive and the local heat transfer coefficient, averaged over the perimeter, also progressively decreases, as indicated in Fig. 12.1. At some point along the tube, the film may completely disappear from the wall of the tube. Liquid may still be present in the flow, however, as entrained droplets. Transport of heat from the tube wall to the droplets is necessary if the vaporization process is to continue. In general, the continued vaporization of the droplets may be accomplished by a combination of mechanisms, including convection through the gas, radiation, and collisions or near collisions of droplets with the wall. These mechanisms are not very effective, and the associated heat transfer coefficient is usually significantly lower than the values associated with nucleate boiling and/or liquid film evaporation. In the mist evaporation process, the heat transfer coefficient usually continues to decrease as the quality increases, until ultimately the single-phase vapor value is attained. For convective boiling in vertical tubes at moderate heat flux levels, the typical sequence of flow regimes observed is shown in Fig. 12.2. Near the onset of boiling, bubbly flow exists. As the quality increases, progressive transitions from bubbly to slug, slug to churn, churn to annular, and annular to mist flow are generally observed. The progression from one regime to the next can be traced on a flow regime map such as that of Hewitt and Roberts [12.1] for vertical upward co-current flow. As liquid is converted to vapor, jv = Gx/ρv increases and jl = G(1 − x)/ρl decreases. On the Hewitt and Roberts map, the sequence of flow regimes would trace a curve similar to that shown in Fig. 12.3. This specific curve represents the locus of state points for vaporization of R-134a at a pressure of 338 kPa and a mass flux of 285 kg/m2s.

Convective Boiling in Tubes and Channels

557

FIGURE 12.2  Flow regimes and boiling mechanisms for upflow convective boiling in a vertical tube.

FIGURE 12.3  Representation of a convective boiling process on the flow pattern map for vertical co-current flow.

558

Liquid-Vapor Phase-Change Phenomena

Since the total mass flux is the same everywhere along the tube, the path of the system on this map is defined by the relation ρv jv + ρl jl = G (12.1)

which can be rearranged to the form

(ρv )1/ 2 (ρv jv2 )1/ 2 + (ρl )1/ 2 (ρl jl2 )1/ 2 = G (12.2)

From the latter form of this relationship, it can be easily shown that for a specified value of G, the curve representing the system state points on the flow regime map will have a shape like that of the curve shown on the log-log plot in Fig. 12.3. While different values of G or fluid densities shift the curve somewhat, the overall shape will remain similar to the curve in Fig. 12.3. Upward flow boiling in a vertical tube generally exhibits a sequence of vaporization mechanisms similar to those in the horizontal flow case described above. As indicated schematically in Fig. 12.2, nucleate boiling is a major factor in the early stages of the vaporization process where bubbly and slug flow occurs. Once churn or annular flow is attained, film evaporation usually becomes important. If the vaporization process continues, dryout of the liquid film will eventually occur, leaving vaporization of entrained droplets of liquid as the final stage of the boiling process. As in the horizontal case, convection, radiation, and droplet collisions with the wall may all play a role in the mist evaporation that occurs during the final stage of vaporization in a vertical tube. The sequences of flow regimes and boiling mechanisms described above correspond to low wall heat flux conditions and/or low wall superheat levels. The effect of varying heat flux or wall superheat on the flow and boiling mechanisms can be better understood by considering the boiling regime maps shown in Figs. 12.4 and 12.5. Figure 12.4 is a modified version of a boiling regime

FIGURE 12.4  Boiling regimes for a constant wall heat flux condition.

Convective Boiling in Tubes and Channels

559

FIGURE 12.5  Boiling regimes for an isothermal tube wall condition.

map presented by Collier [12.2]. This type of map applies specifically to flow boiling with a constant applied heat flux condition at the tube wall. The horizontal coordinate is the local bulk enthalpy of the boiling fluid. The vertical coordinate is the applied heat flux. For a specified heat flux, the system state traverses a horizontal line on this map. The line labeled q1′′ in Fig. 12.4 is typical of the path followed by a system at a low applied heat flux. The sequence of boiling regimes is exactly that described above for the horizontal and vertical flows shown in Figs. 12.1 and 12.2. If, however, a higher heat flux is applied, the sequence of boiling regimes may be quite different. At the higher heat flux of q2′′ indicated in Fig. 12.4, nucleate boiling is initiated while the bulk fluid is still subcooled. Subcooled nucleate boiling gives way to saturated nucleate boiling when the bulk enthalpy reaches the value for saturated liquid. As indicated in this diagram, at this higher value of heat flux, as the quality of the flow increases, a departure from nucleate boiling (DNB) eventually occurs, resulting in a transition to saturated film boiling. For a constant heat flux wall condition, this transition is usually accompanied by a substantial rise in wall temperature. If the vaporization process continues in the film boiling mode, the quantity of liquid in the core eventually becomes so small that it breaks up into droplets, thus resulting in a transition to the mist evaporation mode (liquid deficient region). As in the low heat flux case, when droplet vaporization is complete, further heat transfer occurs as single-phase vapor convection. As illustrated by the q3′′ line in Fig. 12.4, at very high heat flux levels, subcooled film boiling may exist immediately at the entrance of the tube. Subsequent transitions to saturated film boiling

560

Liquid-Vapor Phase-Change Phenomena

and the mist evaporation region occur as the flow proceeds downstream. Because all three boiling regimes are characterized by inefficient heat transfer mechanisms, the wall superheat is expected to be very high all along the tube. Because the heat transfer coefficient associated with these regimes is typically so low, heat transfer equipment is usually designed to operate at relatively low heat flux levels. Vaporization will then occur mostly in the nucleate boiling and two-phase forced-convection regimes, in which the heat transfer coefficient is usually high. An alternative view of boiling regime transitions can be seen in Fig. 12.5. In this figure, the horizontal coordinate again is the bulk enthalpy of the fluid and the vertical coordinate is the imposed wall superheat. For a fixed wall superheat along the tube, the flow will exhibit the sequence of flow regimes that lie along the horizontal constant superheat line. The bottom of this diagram reflects the fact that if the wall superheat is less than that required for the onset of boiling, no vaporization will take place, and only superheated liquid will leave the tube. At low wall superheats above the onset threshold, regimes of nucleate boiling, two-phase forcedconvective boiling and mist evaporation will be encountered, in that order, as the flow proceeds downstream. Because the wall superheat is controlled, at high wall superheats, beyond the departure from nucleate boiling, transition boiling may occur. At even higher wall superheat levels, subcooled and saturated film boiling will occur in the early stages of the vaporization process, followed again by mist evaporation. Heat transfer equipment is generally designed to operate at low wall superheat levels so as to avoid transition and film boiling. The high wall temperatures required to sustain film boiling can result in thermal stresses that may lead to failure of the equipment. In most real evaporators, an isothermal wall condition is not exactly maintained. Because the wall has finite conductivity and thermal capacity, its temperature can be perturbed by changes in the boiling side heat transfer coefficient. Intermittent contact of the liquid with the tube wall during transition boiling may produce large, rapid wall temperature fluctuations in such cases. These fluctuations may, in turn, produce severe thermal stress fluctuations that may cause failure of the tube wall. Because of these effects, conditions that result in transition boiling are also usually avoided when designing evaporators or boilers. One final aspect of flow boiling processes worth noting is that non-equilibrium conditions can exist in several of the regimes shown in Figs. 12.4 and 12.5. At low wall superheats or low wall heat flux levels, a region of superheated liquid may exist near the wall before the onset and/or during nucleate boiling. During film boiling, the vapor near the wall in the film may be superheated while the liquid is subcooled or saturated. Also, in the mist evaporation region, although the liquid droplets are at the saturation temperature, the vapor may be superheated, particularly near the wall. It is possible, therefore, for the bulk enthalpy to be equal to or greater than that for saturated vapor, while the system exists as a mixture of superheated vapor and saturated liquid droplets. For this reason, the mist evaporation region is shown to extend beyond the hˆv line in Figs. 12.4 and 12.5. It is important to realize that the two-phase flow regime maps described in Chapter 10 apply to equilibrium, adiabatic two-phase flows. The existence of non-equilibrium conditions may cause the two-phase flow behavior to deviate from the predictions of these maps. In general, the adiabatic two-phase flow regime maps will provide a reasonably good prediction of the twophase flow characteristics during convective boiling at low to moderate heat flux levels. They may not be accurate at high heat flux levels, however. For example, the inverted annular flow that would result during convective film boiling is not represented on the adiabatic flow regime map (Fig. 12.3). In attempting to predict the flow conditions for convective boiling processes, the effects of possible departures from equilibrium and possible effects of the phase-change process on the two-phase morphology must be taken into account if an accurate prediction is to be obtained. These effects are discussed further in the following sections of this chapter.

Convective Boiling in Tubes and Channels

561

12.2  ONSET OF BOILING IN INTERNAL FLOWS For convective flows in tubes and channels, the onset of nucleate boiling (ONB) marks the beginning of the transition from single-phase liquid convection to combined convection and nucleate boiling. In a convective flow of this type, the onset is usually defined as occurring at the location where active nucleation sites are first observed. The conditions at which potential nucleation sites may become active, and methods for predicting such conditions, are discussed at length in Chapter 6. The reader who is unfamiliar with these concepts is urged to review the relevant material in that chapter prior to reading further in this section. This section will focus on extension of those fundamental concepts to internal convective flows. As discussed in Section 6.3, Hsu’s [12.3] model analysis predicts that the range of active nucleation sites on a heated surface depends directly on the thermal boundary layer thickness, subcooling, wall superheat, and the thermophysical properties of the liquid. It further predicts that a threshold value of wall superheat must be attained before any sites will become active. While this model analysis is approximate in many respects, qualitatively it agrees with trends observed in many boiling systems. The trends predicted by this analysis are particularly relevant to the internal flows considered here. Hsu’s analysis implies that for a given wall temperature, boundary layer thickness, subcooling, and fluid properties, the range of active cavity sizes is determined. Because the temperature profile is assumed to be linear across the thermal boundary layer in that model, for a given wall temperature and set of fluid properties, specifying the bulk temperature and the boundary layer thickness is equivalent to specifying the heat flux. This suggests that the onset conditions for a coolant with given properties corresponds to a specific combination of wall temperature (or equivalently, wall superheat) and heat flux. This is, in fact, the way in which several commonly used onset correlations are cast. For a constant heat flux boundary condition, if the single-phase heat transfer coefficient hle associated with the pure liquid flow is constant along the tube, it follows from the definition of hle and conservation of energy that

q ′′ = hle [ Tw ( z ) − Tl ( z )] (12.3)



d  q ′′ =  h  Gc pl [Tl ( z ) − Tl ,in ] (12.4)  4z 

where dh is the hydraulic diameter, Tl(z) is the local bulk liquid temperature, Tw(z) is the local wall temperature, and Tl,in the bulk temperature of the liquid at the inlet of the channel. Combining these relations to eliminate Tl(z), it can be shown that

Tw ( z ) − Tsat =

 hle   z   q ′′  1 + 4      − (Tsat − Tl ,in ) (12.5) hle   G c pl   d h  

Thus for a constant heat flux wall boundary condition, the wall superheat increases linearly with distance along the tube z, and at each location is proportional to the heat flux q″. Equation (12.3) applies also to the case of an isothermal wall boundary condition, except that Tw is constant. For this circumstance, the heat flux will now vary along the tube, since as the bulk temperature rises, the driving temperature difference will vary. The energy balance in this case is given by

 G c pl d h  dTl q ′′ =  (12.6)  4  dz

562

Liquid-Vapor Phase-Change Phenomena

Combining Eqs. (12.3) and (12.6) yields the differential equation

dTl  4 hle  = (Tw − Tl ) (12.7) dz  G c pl d h 

which can be solved using the boundary condition Tl = Tl,in at z = 0 to obtain

 4 hle z  Tw − Tl ( z ) = (Tw − Tl ,in ) exp  −  (12.8)  G c pl d h 

It follows directly from Eqs. (12.3) and (12.8) that for the isothermal wall condition

 4 hle z  q ′′( z ) = hle [(Tw − Tsat ) + (Tsat − Tl ,in )]exp  − (12.9)  G   c pl d h 

To further explore the necessary conditions for the onset of boiling, we now will consider a fixed z location along the tube. It can be seen from Eq. (12.5) that for a uniform applied heat flux, at a fixed value of z the wall superheat increases linearly with applied heat flux. Similarly, Eq. (12.9) indicates that for a constant wall temperature condition, at a given location, the resulting heat flux increases linearly with the specified wall superheat. For either the isothermal or uniform flux wall conditions, as the wall superheat or heat flux is increased, the onset of boiling will eventually occur, initiating a transition from single-phase liquid convection to fully developed nucleate boiling. The nature of this transition can be better understood by considering Fig. 12.6, which shows possible variations of the heat flux with wall superheat near the onset condition. For the isothermal wall condition, increasing the wall temperature causes the operating point at the specified z location to move up the single-phase operating curve shown

FIGURE 12.6  The variation of wall heat flux with wall superheat near the onset of boiling.

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Convective Boiling in Tubes and Channels

in this diagram. If this process is continued, eventually the point representing the system operating condition approaches the intersection of the single-phase curve and the fully developed nucleate boiling curve. As discussed in Chapter 8, the nucleate boiling curve is usually only weakly dependent on flow velocity and subcooling. Consequently, the fully developed nucleate boiling curve in the presence of forced convection and subcooling is generally taken to be identical to the ordinary saturated nucleate pool boiling curve at the same pressure. Correlating equations that relate the heat flux to the wall superheat for saturated nucleate boiling are generally of the form q ′′ ( z )  = γ [ Tw − Tsat ( P )] (12.10) m



where γ is a factor that depends on surface and fluid properties, and the exponent m is typically between 2 and 4. A necessary (but not sufficient) condition for the onset of boiling is that the wall temperature must be above Tsat at the local system pressure (i.e., the wall superheat must be greater than zero). If the wall superheat is gradually increased at a given downstream location, the onset of boiling generally occurs in the vicinity of the intersection of the single-phase convection and nucleate boiling curves. The onset may occur before the intersection is reached at, for example, point 2 in Fig. 12.6, or it may occur at a point 2′ beyond the intersection. If the former is true, and the wall is held isothermal, the system operating point jumps vertically to a curve that marks a smooth transition between the single-phase curve and the fully developed boiling curve. If the latter is true, the system operating point will similarly jump vertically from point 2′ to point 3b′ on the fully developed boiling curve. The smooth transition sometimes observed between the single-phase and fully developed boiling curves is usually referred to as the partial boiling regime. As the superheat is increased in this regime, the active nucleation site density increases and the nucleate boiling contribution to the total heat transfer increases until it essentially equals that for nucleate pool boiling at the same wall superheat and pressure conditions. Similar arguments apply for a uniform heat flux wall condition. At a given location, increasing the heat flux would cause the wall superheat to rise, moving the system operating point up the single-phase curve. The onset may occur either before or after the intersection of the single-phase curve with the fully developed boiling curve. If the onset occurs at point 2, and the heat flux is held constant, the operating point may jump horizontally to point 3a′ in Fig. 12.6. If the transition is delayed to point 2′, the operating point may jump horizontally to point 3a. The transition to fully developed boiling once the onset condition is achieved will be discussed further in the next section. The main point here is that, regardless of whether the wall boundary condition is isothermal or uniform heat flux, the onset condition may occur before or after the intersection of the single-phase and fully developed boiling curves depending on the fluid properties, wall cavity size distribution, and imposed flow conditions. Because the onset location represents the location where a new heat transfer mechanism “turns on,” predicting this location is essential to any effort to predict the heat transfer performance of an evaporator tube. Several semi-empirical methods have been proposed for predicting the onset condition. As a first approximation, the onset condition could be taken to be the point of intersection of the single-phase convection curve and the fully developed boiling curve. For the uniform heat flux wall condition, combining Eqs. (12.5) and (12.10) yields the following relation for the intersection condition:

 q ′′   γ 

1/ m

−

 hle   z   q ′′  1 + 4      + [Tsat − Tl ,in ] = 0 (12.11) hle   G c pl   d h  

564

Liquid-Vapor Phase-Change Phenomena

Alternatively, for the isothermal wall condition, combining Eqs. (12.8) and (12.10), the following relation is obtained

 4 hle z  γ (Tw − Tsat )m − hle [(Tw − Tsat ) + (Tsat − Tl ,in )]exp  −  = 0 (12.12)  G c pl d h 

Solving Eq. (12.11) for q″ or Eq. (12.12) for Tw – Tsat yields the values of these parameters for which the intercept condition is just met at the specified z location. Alternatively, if q″ or Tw – Tsat is specified, the appropriate equation can be solved for the value of z at which the intercept condition is met. If the intercept is interpreted as the onset condition, these equations thus predict the onset conditions for these boundary conditions. Collier [12.2] reports that this approach was, in fact, proposed in an early study by Bowring [12.4]. As noted in the discussion above, the actual onset condition is a function of a number of system parameters, and this idealization is, at best, a very crude approximation to the actual system behavior. Several investigators have developed methods for predicting the onset conditions based on semitheoretical arguments similar to those used in the model analysis of Hsu [12.3] (see Chapter 6). Bergles and Rohsenow [12.5] developed a model analysis similar to Hsu’s and used a graphical technique to solve the governing equations. Based on their computed results, they recommended the following relation as a means of predicting the onset of boiling for flow of water in a heated tube

qONB ′′ = 5.30 P1.156 [1.80(Tw − Tsat )ONB ]2.41/ P

0.0234

(12.13)

This is a dimensional relation in which the pressure, temperature, and heat flux are in units of kPa, °C, and W/m2, respectively. This relation was reported to match the values calculated using the model analysis for 103 ≤ P ≤ 13,700 kPa. As noted by Bergles and Rohsenow [12.5], their model analysis, like Hsu’s [12.3], is plausible only for surfaces that have cavity sizes distributed over a wide range. They argue, however, that commercially produced surfaces usually do have cavities over a wide range of sizes, and therefore the results of this model should be applicable to many real systems. Sato and Matsumura [12.6] developed an analytical treatment of the onset problem similar to that proposed by Bergles and Rohsenow [12.5]. They proposed the following relation as a means of predicting the onset condition:

qONB ′′ =

kl hlv ρv [(Tw − Tsat )ONB ]2 (12.14) 8σTsat

Davis and Anderson [12.7] modified and extended the analytical treatment of Bergles and Rohsenow [12.5]. Their detailed analysis yielded the following expression for the onset condition

(Tw − Tsat )ONB =

2 q ′′ y ′ ( RTsat /hlv ) ln(1 + ξ ′) + ONB (12.15) kl 1 − ( RTsat /hlv ) ln(1 + ξ ′)

where

y′ =

Cθ σ C σ  2C k σT +  θ  + θ l sat (12.16)  P P  qONB ′′ hlv ρv ξ′ =

2Cθ σ (12.17) Py ′

Cθ = 1 + cos θ (12.18)

Convective Boiling in Tubes and Channels

565

In the above expressions, θ is the contact angle of the liquid-vapor interface for the model bubble considered in the analysis. Davis and Anderson [12.7] also argued that for systems at higher pressures or for low surface tension, the relation for the onset condition could be simplified to

qONB ′′ =

kl hlv ρv [(Tw − Tsat )ONB ]2 (12.19) 8Cθ σTsat

It can readily be seen that for a hemispherical model bubble θ = 90°, C θ = 1 and Eq. (12.19) becomes identical to the correlation of Sato and Matsumura [12.6]. Davis and Anderson [12.7] found that Eq. (12.19) with C θ = 1 (the Sato and Matsumura correlation) agreed well with experimentally reported onset conditions for water reported by Sato and Matsumura [12.6] and Rohsenow [12.8]. This equation was also shown to agree well with the correlation of Bergles and Rohsenow [12.5] over wide ranges of pressure and wall heat flux. Somewhat later, Frost and Dzakowic [12.9] explored the applicability of the analytical treatment used by Davis and Anderson [12.7] to other liquids. Based on additional arguments regarding the effect of liquid Prandtl number on the onset condition, they recommended the following relation for the onset condition

qONB ′′ =

kl hlv ρv [(Tw − Tsat )ONB ]2 Prl−2 (12.20) 8σTsat

This correlation, which is identical to the Sato and Matsumura [12.6] correlation except for the inclusion of the Prl−2 multiplier, was found to agree well with data for a wide variety of fluids, including water, various hydrocarbons, mercury, and cryogenic liquids. It is important to note that the onset correlations described above are expected to accurately predict the onset conditions only if there exists a sufficiently wide range of potential nucleation site sizes on the wall surface. In some instances, larger active sites may not be present in large numbers due to the surface being polished, or due to the highly wetting nature of the liquid which may cause the liquid to displace vapor or air in all but the smallest cavities. If such conditions exist in the system of interest, the above relations should be used with caution. The above predictive relations for the onset condition indicate the threshold condition at which nucleation is initiated or turned off. When analyzing internal convective boiling in a tube, the first step is usually to determine the portion of the tube over which boiling occurs. While the correlations described above indicate the boundary between the boiling and non-boiling regions, determining the region in which boiling occurs can be a little tricky. The variation of the wall superheat with heat flux at the onset condition, as predicted by the Sato and Matsumura [12.6] correlation, is plotted in Fig. 12.7 for water at atmospheric pressure. The curve in this plot represents combinations of wall superheat and heat flux at which nucleation will just begin. The question to be answered, then, is: “Where along the tube will the actual operating conditions first cross this curve?” For a tube with a constant heat flux applied along the wall, the wall superheat increases linearly with downstream distance z, as indicated by Eq. (12.5). As the flow proceeds downstream, the system state point thus moves upward along a vertical (constant q″) line on Fig. 12.7. This has the effect of raising Tw while keeping the slope of the temperature profile near the wall constant. At a given distance y away from the tube wall, the temperature will be higher, and nucleation conditions will be more favorable at locations that are farther downstream. Moving upward across the onset curve thus carries the system from a region with no boiling present to one where nucleate boiling is more favored. State points above the onset curve in Fig. 12.7 are therefore in the nucleate boiling region, and points below the curve are in the single-phase region. Equation (12.5) can be combined with one of the onset correlations described above to eliminate the wall superheat and solve for the z location of the onset for the specified q″ value.

566

Liquid-Vapor Phase-Change Phenomena

FIGURE 12.7  Onset of boiling conditions predicted for water at atmospheric pressure.

For a constant and uniform wall temperature condition, the wall heat flux decreases exponentially with downstream distance z, as indicated by Eq. (12.9). The system point thus moves horizontally to the left in Fig. 12.7 along a constant superheat line as the fluid proceeds downstream. Decreasing q″ has the effect of decreasing the temperature gradient at the wall with downstream distance. Hence, at a given y location (away from the wall), the fluid will be hotter, and nucleation will therefore be favored, at a z location that is further downstream. Thus the decreasing heat flux with downstream distance will carry the system from a region of no boiling into the boiling region as its operating point crosses the onset curve. Combining Eq. (12.9) with one of the onset relations to eliminate q″, the z location at which the onset occurs can be determined for the specified wall superheat. In applications involving convective boiling, the combined variations of heat flux and wall superheat with downstream distance may be more complicated than the variations that result for the constant q″ and Tw circumstances described above. However, once the variation of q″ with superheat along the tube has been determined (using energy balance requirements and the imposed boundary condition), the intersection of the resulting q″ versus superheat relation with the onset curve will define onset and/or suppression locations. The most noteworthy trends are that increasing superheat and decreasing heat flux favor nucleation. The direction in which the onset curve is crossed, and the associated changes in heat flux and superheat determine whether nucleation is initiated or suppressed. Example 12.1 Water flows upward in a vertical round tube with an inside diameter of 1.0 cm. The pressure along the tube is virtually constant at 6124 kPa. The water enters as subcooled liquid at a flow rate that corresponds to a mass flux of 9000 kg/m2s. The wall is held at a uniform temperature of 281°C. Estimate the inlet subcooling that will suppress nucleate boiling over the first 10 cm of the tube. For water at 6124 kPa, Tsat = 550 K. = 276.8°C, ρl = 756 kg/m3, ρv = 31.5 kg/m3, hlv = 1563 kJ/kg, cpl = 5.07 kJ/kgK, μl = 99.2 × 10−6 Ns/m2, kl = 0.581 W/mK, Prl = 0.87, and σ = 0.0197 N/m. Using the correlation of Sato and Matsumura [12.6], the onset heat flux is predicted by Eq. (12.14): kl hlv ρv [(Tw − Tsat )ONB ]2 8σTsat (0.581)(1563 × 1000)(31.5) [281.0 − 276.8]2 = 8(0.0197)(550)

qONB ′′ =

= 5.82 × 106 W/m 2

Convective Boiling in Tubes and Channels

567

The Reynolds number for the entire flow as liquid is given by

Re le =

GD (9000)(0.01) = = 9.07 × 105 µl 99.2 × 10 −6

Assuming that fully developed turbulent flow is established just downstream of the tube inlet, the heat transfer coefficient for pure liquid flow is computed using the Dittus-Boelter equation:



k  hle = 0.023  l  Re 0.8 Pr 0.4  D  le l  0.581 = 0.023  (9.07 × 105 )0.8 (0.87)0.4 = 7.38 × 10 4 W/m 2K  0.01 

Up to the onset of boiling, the heat flux variation is given by Eq. (12.9):  4hle z  q′′( z ) = hle[(Tw − Tsat ) + (Tsat − Tl ,in )]exp  −   Gc pl D 



We want to find the value of Tsat − Tl,in that just makes q′′ = qONB ′′ at z = 0.10 m. Rearranging this equation to solve for Tsat − Tl,in yields

Tsat − Tl ,in =

 4hle z  q′′ exp   − (Tw − Tsat ) hle  Gc pl D 

Substituting q′′ = qONB ′′ , z = 0.10 m, and the respective values of the other parameters, we obtain



Tsat − Tl ,in =

 4(7.38 × 10 4 )(0.10)  5.82 × 106 − (281.0 − 276.8) exp  4 7.38 × 10  9000(5.07 × 1000)(0.01) 

= 79.9°C Note that for z < 10 cm, Eq. (12.9) predicts that q″ will be greater than q′′ONB for this subcooling and from Fig. 12.7 it can be seen that no boiling will occur. Thus if the inlet subcooling is 80°C, the onset is predicted not to occur over the first 10 cm downstream of the inlet.

12.3  SUBCOOLED FLOW BOILING Regimes of Subcooled Flow Boiling Subcooled internal flow boiling can arise in a number of applications. Boilers and vapor generators are occasionally fed liquid that is somewhat subcooled. Subcooled flow boiling has also been of particular interest as a means of providing high-heat-flux cooling in some specialized thermal control applications. In electronic systems, for example, cooling of microelectronic chips may require removal of heat at flux levels exceeding 200 W/cm2. At the present time, one of the simplest ways to achieve flux levels of this magnitude is with a subcooled flow boiling process. As noted in Chapter 8, for external flow over a surface, the critical heat flux increases with subcooling and flow velocity over the surface. Similar trends are observed for internal flow in tubes, which makes subcooled flow boiling an attractive prospect for removing heat at high heat flux levels from coolant passage walls in special applications that require it. Figure 12.8 indicates schematically the sequence of regimes associated with subcooled boiling. Immediately downstream of the onset of boiling, a region of partial subcooled boiling exists.

568

Liquid-Vapor Phase-Change Phenomena

FIGURE 12.8  Transition of flow through regimes of subcooled boiling.

This is a transition region in which both forced-convective effects and nucleate boiling effects are important, with nucleate boiling effects increasing in strength as the flow proceeds downstream. Near the onset, active nucleation sites are few and widely spaced. The effect of bubble growth and release on the overall heat transfer from the wall is small in this region. For either the uniform wall temperature or uniform wall heat flux conditions, as the flow proceeds downstream, the conditions become progressively more favorable to nucleation (see the discussion in the previous section). This generally results in an increase in the nucleation site density with increasing downstream distance in the partial boiling region. Eventually the nucleation site density becomes so high that the nucleate boiling contribution to the heat transfer is essentially equivalent to that for saturated nucleate pool boiling. The flow is then said to have entered the fully developed nucleate boiling regime. In the fully developed nucleate boiling regime, the nucleate boiling mechanism generally is so strong that it completely dominates the heat transfer process. At extremely high flow velocities and subcooling levels, convective effects may be important well beyond the point where the nucleation site density has attained a level equal to that for pool boiling at the same superheat. However, for most cases of practical interest, nucleate boiling completely dominates in the fully developed nucleate boiling regime. Once fully developed nucleate boiling dominates the transport, the heat transfer rate becomes virtually independent of the flow rate and subcooling. This is a direct consequence of the observation (discussed in Chapter 8) that nucleate pool boiling at moderate to high wall superheats is affected only slightly by ambient motion and subcooling of the surrounding liquid. The heat flux versus wall superheat curve will vary with flow velocity or subcooling in the single-phase liquid convection regime and in the partial subcooled boiling regime. However, the above observations imply that the different curves will all eventually merge into the fully developed boiling curve, as indicated in Fig. 12.9.

Methods of Predicting Partial Subcooled Boiling Heat Transfer As noted in Chapter 8, the mechanisms of subcooled nucleate boiling have been the subject of active research over the past fifty years. For subcooled flow boiling in particular, a number of

Convective Boiling in Tubes and Channels

569

FIGURE 12.9  The partial boiling transition from single-phase forced convection to fully developed nucleate boiling.

investigators have examined the nucleate boiling process experimentally and/or analytically (see, e.g., references [12.10–12.14]). In the partial subcooled boiling regime, methods used to predict the heat transfer rate have typically been based on the premise that the forced convection and nucleate boiling mechanisms act in parallel and independently. To account for the contributions of these two mechanisms, Kutateladze [12.15] proposed the following interpolation formula for the heat transfer coefficient associated with partial subcooled flow boiling

2 2 1/ 2 h = (hspl + hsnb ) (12.21)

As the wall temperature increases, the subcooled nucleate boiling contribution hsnb generally becomes large compared to the single-phase liquid convection contribution hspl, and this relation predicts that h approaches hsnb. Similarly, as the wall temperature decreases toward Tsat, hsnb goes to zero, and consequently, h approaches hspl. This type of interpolation relation thus correctly predicts the limiting behavior of h as one mechanism turns on and the other turns off. In subsequent investigations, the total heat flux has more often been postulated to be the sum of contributions due to single-phase liquid qspl ′′ convection and nucleate boiling qsnb ′′ :

qtotal ′′ = qspl ′′ + qsnb ′′ (12.22)

One possible superposition scheme of this type for predicting partial subcooled boiling heat transfer can be understood by considering Fig. 12.10. This figure is a plot of surface heat flux as a function of wall temperature Tw. In this plot, the region of partial subcooled boiling is idealized as beginning when the wall temperature just exceeds the saturation temperature. The objective is to predict the portion of the curve BCD in the partial boiling regime. The broken curve in this figure represents the difference between the actual system curve BCD and the fully developed nucleate boiling curve FGD. This difference curve can be interpreted as the convective contribution to the total heat flux

570

Liquid-Vapor Phase-Change Phenomena

FIGURE 12.10  Reference points used in analysis of the partial boiling regime.

in the partial boiling regime qspl ′′ . If this curve could be predicted somehow, adding it to the fully developed boiling curve would yield the desired curve for the total heat flux. One method of predicting the total heat transfer in this manner would be to approximate the broken curve as the step function represented by BC′D′ in Fig. 12.10. This is specified mathematically as

Tsat ≤ Tw ≤ Tw , fdb :

qspl ′′ = hle [Tsat − Tl ( z )] (12.23a)

Tw > Tw , fdb :

qspl ′′ = 0 (12.23b)

To use this method, the condition at point D where fully developed boiling begins must be specified. Based on experimental data, Engelberg-Forester and Greif [12.16] concluded that the heat flux at the point where fully developed boiling begins is given approximately by qspl ′′ = 1.4 qI′′ (12.24)



where qI′′ is the heat flux at the intersection of the single-phase liquid convection curve and the fully developed nucleate boiling curve. If the nucleate boiling curve is given by Eq. (12.10), then for the uniform heat flux wall condition, the heat flux at the intercept point qI′′ can be determined from Eq. (12.11), as described in Section 12.2. If, on the other hand, the wall is held at a uniform temperature, qI′′ can be determined from Eqs. (12.10) and (12.12). Then, using Eq. (12.24), the wall temperature corresponding to the beginning of fully developed boiling can be determined as m



Tw , fdb

 1.4 qI′′  + Tsat (12.25) =  γ 

571

Convective Boiling in Tubes and Channels

With Tw,fdb specified, the relations for qspl ′′ can be evaluated and added to the fully developed nucleate boiling curve, as specified by Eq. (12.22), to determine the total heat flux in the partial subcooled regime. This scheme for predicting the heat transfer during partial subcooled boiling was proposed by Bowring [12.4]. Rohsenow [12.17] developed a slightly different method for predicting partial subcooled boiling heat transfer. He proposed to use a conventional single-phase correlation to predict the single-phase contribution to the total heat flux

qspl ′′ = hle [Tw − Tl ( z )] (12.26)

and he further postulated that the remaining additional nucleate boiling contribution to the total heat flux could be computed using the Rohsenow correlation for nucleate pool boiling  g ( ρl − ρv )  qsmb ′′ = µ l hlv   σ  

1/ r

1/ 2



−s /r l

Pr

 c pl Tw − Tsat ( Pl )    (12.27) Csf hlv  

As described in Chapter 7, for this correlation, r = 0.33, s = 1.0 for water and 1.7 for other fluids, and the constant Csf may vary from one fluid-surface combination to another. To correlate partial subcooled boiling data with Rohsenow’s method, the convective contribution to the total heat flux is first computed using Eq. (12.26) and subtracted from the total measured heat flux at each data point. The value of Csf is then determined that best fits the data representing the implied nucleate boiling contributions. This method has been used to correlate partial subcooled boiling data obtained by Rohsenow and Clarke [12.18], Kreith and Sommerfield [12.19], Piret and Isbin [12.20], and Bergles and Rohsenow [12.5]. Values of Csf obtained in this manner are listed in Table 7.2. Yet another method of predicting the heat transfer during partial subcooled boiling was proposed by Bergles and Rohsenow [12.5]. This method can be best understood by considering the heat flux versus wall temperature plot shown in Fig. 12.11. For the partial subcooled boiling regime, the following relation was proposed to predict the heat transfer:



  q ′′ For q ′′ > qONB ′′ : q ′′ = qspl ′′ 1 +  snb q ′′   spl

 qD′′    1 − q ′′   snb 

2

1/ 2

  (12.28) 

In this relation, the value of qspl ′′ is determined using an appropriate single-phase liquid convection correlation and qsnb ′′ was determined using the fully developed subcooled boiling correlation at the local conditions. The value of qD′′ is determined using the fully developed boiling correlation at the wall temperature corresponding to the onset of boiling as determined using the Bergles and Rohsenow onset correlation (12.13) described in the previous section. As the wall temperature increases, qsnb ′′ becomes large compared to qspl ′′ and qD, ′′ and q″ approaches qsnb ′′ . Similarly, as the wall temperature decreases toward Tsat, qsnb ′′ goes to zero, and q″ approaches qspl ′′ . In evaluating the fully developed nucleate boiling contribution in the above schemes, it has often been assumed that the heat flux contribution due to subcooled nucleate boiling in the presence of liquid convection is identical to that for saturated pool boiling. Bergles and Rohsenow [12.5] specifically explored this issue by testing a heated cylinder in forced convective boiling (as the inside wall of an annulus) and in pool boiling in a nearly saturated liquid pool. The forced-convection q″ versus superheat curves for different velocities did approach a common limiting curve at high superheat levels. However, this limiting curve was not exactly an extension of the curve indicated by the pool boiling data.

572

Liquid-Vapor Phase-Change Phenomena

FIGURE 12.11  Reference points used in the method of Bergles and Rohsenow [12.5] for prediction of partial subcooled boiling heat transfer.

Their results indicate that using a correlation based on pool boiling data to predict the fully developed nucleate boiling curve during subcooled convective boiling can result in significant errors in some cases. Better results can be obtained if the fully developed nucleate boiling curve is determined by correlation of the implied nucleate boiling effect back-calculated from measured data using one the correlation schemes described above. It may be necessary to use an extension of the saturated pool boiling curve to predict qsnb ′′ , if no better information is available. The predictions may be quite good in some cases. In general, however, the results of such a calculation should be used with caution. Somewhat later, Kandlikar [12.21] proposed a non-linear interpolation scheme for predicting the heat flux variation with superheat in the partial flow boiling regime. This approach provides a smooth transition from the onset condition to fully developed nucleate boiling, but requires knowledge of the conditions at which fully developed boiling is achieved. Example 12.2 Subcooled liquid ethanol at 226 kPa flows through a vertical round tube having walls held at 140°C. The liquid enters the tube at 50°C and the onset of boiling occurs immediately at the entrance of the tube. The tube diameter is 1.2 cm and the mass flux is 600 kg/m2s. Determine the partial boiling heat transfer coefficient predicted by Rohsenow’s method at a location downstream of the inlet where the bulk fluid temperature is 90°C. For saturated ethanol at 226 kPa, Tsat = 373 K = 99.8°C, ρl = 734 kg/m3, ρv = 3.18 kg/m3, hlv = 927 kJ/kg, cpl = 3.30 kJ/kgK, μl = 314 × 10 −6 Ns/m2, kl = 0.151 W/mK, Prl = 6.88, and σ = 0.0157 N/m. Rohsenow’s [12.17] method postulates that the total partial subcooled boiling heat flux is given by

q′′ = qspl ′′ + qsnb ′′

573

Convective Boiling in Tubes and Channels where, for ethanol qspl ′′ = hle Tw − Tl ( z ) 



 c pl Tw − Tsat ( Pl )    g ( ρl − ρv )  −5.15 qsnb ′′ = µ l hlv      Prl σ Csf hlv     1/ 2



3.0

For the entire flow as liquid

Re le =

GD 600 ( 0.012) = = 2.24 × 10 4 µl 314 × 10 −6

Assuming fully developed turbulent flow k  hle = 0.023  l  Re 0.8 Pr 0.4  D  le l



(

 0.151 = 0.023  2.24 × 10 4  0.012 

)

0.8

(6.88)0.4 = 1930 W/m2K

It follows that qspl ′′ = 1930 (140 − 90 ) = 9.65 × 10 4 W/m 2



Using Csf = 0.013 and substituting respective values of the fluid properties in the above equation for qsnb ′′ yields



 9.8 (734 − 3)  qsnb ′′ = 314 × 10 −6 ( 927 × 1000 )    0.0157 

(

)

1/ 2

(6.88)

−5.15

 3.33[140 − 99.8]     0.013 ( 927) 

3.0

= 1.44 × 10 4 W/m 2 The total heat flux and overall heat transfer coefficient are then given by q′′ = qspl ′′ + qsnb ′′ = 9.65 × 10 4 + 1.44 × 10 4 = 1.11× 105 W/m 2

h =

1.11 × 105 q′′ = = 2220 W/m 2K Tw − Tl ( z ) 140 − 90

Pressure Drop and Void Fraction Determination of the two-phase pressure drop during subcooled flow boiling is an important aspect of the design of some power systems and heat exchangers. To determine the two-phase pressure drop during the subcooled boiling process a means of predicting the void fraction is also needed. Predicting the void fraction and two-phase pressure drop for these conditions is a challenging problem, requiring methods that account for the fact that the two-phase flow is not one-dimensional. Fortunately, the simple methods described above can be used to predict the heat transfer for subcooled flow boiling in spite of the complexity of the two-phase flow. A full discussion of the twophase flow characteristics for these circumstances is beyond the scope of this book. However, because two-phase pressure drop is important to heat exchanger design, a brief overview of the void fraction and pressure drop characteristics of subcooled boiling will be presented in the remaining portion of this section.

574

Liquid-Vapor Phase-Change Phenomena

Prediction of the void fraction and two-phase pressure drop is generally even more complicated for convective subcooled boiling than for saturated convective boiling. The added complexity arises from the non-uniformity of the flow across the tube. As indicated schematically in Fig. 12.8, during partial subcooled boiling in a highly subcooled flow, vapor bubbles generated at the wall of the tube generally condense rapidly when they depart from the wall region and enter the subcooled bulk liquid. As a result, in a highly subcooled turbulent flow, vapor bubbles are confined to the thin thermal boundary layer, which is often limited to the viscous sublayer near the wall. The average void fraction in this regime is extremely small, but the impact of the bubbles on the wall region transport can be quite significant. An approximate model of the flow in this regime near the wall for these conditions was developed by Griffith et al. [12.22]. From an analysis based on this model, the following relation for the void fraction was obtained:

α = 3.73

 kl  qsnb ′′ Prl (12.29) hle [ Tsat − Tl ( z )]  hle d h 

This relation is limited to highly subcooled flow boiling at moderate pressures. In the partial subcooled boiling regime, as heat is added and the flow proceeds downstream, the bulk subcooling decreases. At lower subcooling levels, bubbles that leave the wall region and enter the bulk flow condense more slowly. As a result, vapor bubbles are present in a progressively larger portion of the bulk flow as the subcooling continues to decrease with downstream distance. The point in the flow at which the subcooling is low enough that significant amounts of vapor begin to enter the bulk flow is of particular importance, since the pressure drop characteristics of the two-phase flow upstream and downstream of this point are quite different. This location is designated as the point where α = ε+ in Fig. 12.8. Upstream of this point (or at high subcooling), vapor is present only very near the wall. Downstream of this location, vapor exists near the wall and in a significant portion of the bulk flow near the wall. From a simple model, Levy [12.23] has developed a relation that can be used to predict the void fraction at the α = ε+ point between these two regimes. It has also been argued by Griffith et al. [12.22] that the α = ε+ point corresponds to the onset of fully developed boiling (Fig. 12.8). Based on this premise, it is possible to directly estimate the α = ε+ location from the boiling heat transfer relations described previously in this section. Collier [12.2] reports that an alternative methodology for predicting the α = ε+ location has also been developed by Bowring [12.4]. Levy [12.23] interpreted the α = ε+ point as being the location where bubbles are first able to detach from the surface. By considering the conditions necessary for growth and the force balance on an attached bubble, Levy [12.23] developed an analytical method for predicting the conditions at the α = ε+ point. The predictions of the Levy [12.23] model agree fairly well with the data of Egen et al. [12.24] and Maurer [12.25]. An analysis similar to Levy’s has also been presented by Staub [12.26]. An alternative approach to predicting the α = ε+ point, interpreted as the point of net vapor generation, was also developed by Saha and Zuber [12.27]. As noted above, in the low subcooling regime (downstream of the α = ε + point), the vapor void fraction increases progressively with downstream distance (and decreasing subcooling). Methods that can be used to predict the void fraction in the flow for conditions in this regime have been developed by Griffith et al. [12.22], Bowring [12.4], Kroeger and Zuber [12.28], and Levy [12.23]. In the low subcooling region, vapor bubbles are present over much of the passage cross section. Prediction of the two-phase pressure gradient using the separated flow model would therefore seem appropriate, since the flow deviates only slightly from saturated flow boiling. The main difficulty in applying this approach is determination of the vapor void fraction and the local mass fraction of vapor in the flow. Note that because the flow is far from equilibrium, the local vapor mass fraction is not equal to the quality, and thus cannot be calculated from simple thermodynamic considerations. Collier [12.2] describes a methodology proposed by Sher [12.29] that handles this difficulty.

Convective Boiling in Tubes and Channels

575

For the highly subcooled region, the presence of vapor in the near-wall region may have two possible effects: (1) the bubbles may act like surface roughness elements acting to increase the pressure gradient, or (2) the vapor in the layer near the wall may act to reduce the effective viscosity of the layer, thereby resulting in a lower wall shear stress and lower pressure gradient for a given bulk velocity. Because these are opposite effects, the magnitude of the pressure gradient relative to that for single-phase liquid at the same flow rate is not clear. In an attempt to examine some of the mechanisms associated with momentum transfer during highly subcooled flow boiling, Hirata and Nishikawa [12.30] developed an analysis of the analogous circumstance of liquid flow over a porous plate through which gas was injected into the boundary layer. Their results suggest that generation of the gas at the wall can produce an increase in the wall shear stress. The increase in the shear stress was found to be strongly affected by the bubble (nucleation) site density and the size of the bubbles at departure. Results of experimental investigations indicate that the pressure gradient associated with highly subcooled flow boiling may be either larger or smaller than that for pure liquid at the same flow rate. The lack of a clear trend in the data is not surprising given the complexity of the mechanisms described above. Measured values of the total pressure gradient for subcooled flow boiling obtained by Reynolds [12.31] imply that just downstream of the onset of subcooled boiling, the pressure gradient may be lower than that for single-phase liquid flow. Further downstream, however, the pressure gradient was observed to increase rapidly, becoming much larger that that for single-phase liquid at the same flow rate. Data obtained by Dormer and Bergles [12.32] exhibit a similar trend. On the other hand, the pressure gradients for subcooled flow boiling measured by Buchberg et al. [12.33] were virtually always higher than that for the entire flow as liquid. Additional experimental studies of the pressure gradient associated with subcooled flow boiling have been conducted by Sher [12.29], Owens and Schrock [12.34], Jicha and Frank [12.35], and Jordan and Leppert [12.36]. Based on the data obtained in his experiments, Reynolds [12.30] developed an empirical correlation for the total two-phase pressure gradient. Owens and Schrock [12.34] also developed a pressure gradient correlation based on a fit to their data. Additional information regarding empirical correlations for the two-phase pressure gradient for subcooled flow boiling can be found in Collier [12.2] and Tong [12.37].

12.4  SATURATED FLOW BOILING Saturated internal flow boiling is most often encountered in applications where complete or nearly complete vaporization of the coolant is desired. Perhaps the most frequently encountered example is the evaporator in a refrigeration or air-conditioning system. Other examples include cryogenic processing applications, boilers in nuclear and conventional power plant systems, and chemical processing involving pure hydrocarbons. As seen in Figs. 12.4 and 12.5, to avoid the high wall temperatures and/or the poor heat transfer associated with the saturated film boiling regime, the vaporization must be accomplished at low superheat or low heat flux levels. For this reason, evaporators and boilers are usually designed to avoid the high heat flux and high wall superheat levels that may produce film boiling at some point during the process. Because most equipment operates in this range, this section will focus on flow boiling processes at low to moderate superheat and heat flux conditions. Before discussing the boiling process itself, it is worth noting that at low wall superheat conditions, it is possible for the onset of nucleate boiling to be delayed until the mean coolant enthalpy is higher than that for saturated liquid hˆl . The required wall superheat for the onset of boiling for saturated or superheated bulk liquid can be predicted by the correlations of Bergles and Rohsenow [12.5], Sato and Matsumura [12.6], Davis and Anderson [12.7], or Frost and Dzakowic [12.9], as described in Section 12.2. In most (but not all) systems of practical interest, the onset of nucleate boiling is achieved at or just beyond the point where the bulk flow reaches the saturated liquid condition.

576

Liquid-Vapor Phase-Change Phenomena

When boiling is initiated, both nucleate boiling and liquid convection may be active heat transfer mechanisms. Usually the walls of the passage have an abundance of active nucleation sites, and at low quality, the vapor void fraction is relatively low, and the nucleate boiling mechanism is much stronger than the forced convective effect. In general, however, the relative importance of these two mechanisms varies over the length of the passage. As the flow proceeds downstream and vaporization occurs, the void fraction rapidly increases at low to moderate pressures. As a result, the flow must accelerate, which tends to enhance the convective transport from the heated wall of the tube. As described in Section 12.1, the increasing void fraction and acceleration of the flow also produces changes in the flow regime with downstream location. For vertical upward flow, bubbly flow at the onset location subsequently changes to slug, churn, and then annular flow. When there is a large difference in the liquid and vapor densities, the transition from the bubbly to the annular configuration associated with churn or annular flow can occur over a very short portion of the tube length. Once such an annular configuration is achieved, convective transport of heat across the liquid film on the wall can directly vaporize liquid at the liquid-vapor interface of the film. Further, it is clear that as vaporization continues, the thickness of the liquid film on the tube wall will decrease, reducing its thermal resistance and thereby enhancing the effectiveness of this mechanism. In the case of a uniform applied heat flux, it can be seen that as annular film evaporation increases in effectiveness, if the nucleate boiling contribution is unchanged, the wall-to-interface temperature difference needed to drive the heat flux is reduced. However, the decrease in wall superheat resulting from this effect tends to extinguish the smaller active nucleation sites, reducing the effectiveness of the nucleate boiling mechanism. As the liquid film becomes very thin, near the latter stages of the vaporization process, the required superheat to transport all the surface heat flux across the liquid film may become so low that nucleation is completely suppressed. This is indicated schematically in Fig. 12.12.

FIGURE 12.12  Boiling regime transitions at moderate quality.

577

Convective Boiling in Tubes and Channels

For a tube with a constant wall superheat condition, the variation of the transport mechanisms is similar to that for the uniform flux wall condition. As in the uniform heat flux case described above, the transition from bubbly to annular flow is generally expected to strengthen the convective transport mechanism. Although the wall superheat is fixed at a level adequate to result in the onset of nucleation initially, the enhancement of the convective effect increases the heat flux from the wall as the flow proceeds downstream. As indicated in Fig. 12.7, increasing the heat flux from the surface at a fixed wall superheat level may ultimately cause nucleate boiling to be suppressed. Thus for flow in a round tube with an isothermal wall condition or a uniform applied heat flux, the forced-convective effect generally becomes stronger and the nucleate boiling effect tends to become weaker as the flow proceeds downstream. Because of the trends described above, prediction of the convective boiling heat transfer coefficient requires an approach that accommodates a transition from a nucleate-pool-boiling-like condition at low qualities to a nearly pure film evaporation condition at higher qualities. Heat transfer in the latter case can be modeled in the manner described in Section 11.2 for annular film-flow condensation. The only differences here are that the direction of the heat flux is reversed and the driving temperature difference Tsat − Twall used in the Section 11.2 analysis is replaced with Twall − Tsat. In fact, with these changes, the integral analysis and resulting heat transfer relation presented in Example 11.2 apply equally well to annular film flow evaporation in a vertical round tube. The approximate analysis presented in Example 11.2 indicates that, for convective transport across the liquid film, the resulting heat transfer coefficient can be correlated in terms of the turbulent-turbulent Martinelli parameter Xtt, the Reynolds number for the liquid flowing alone Rel, and the liquid Prandtl number Prl:

hD = f ( X tt ,Rel ,Prl ) (12.30) kl

Variations of the above form that have been used to correlate boiling heat transfer data have included relations of the form

 1  h = f  X tt  hle

and

 1  h = f (12.31)  X tt  hl

In these relations hle is the single-phase convection coefficient for the entire flow as liquid and hl is the single-phase coefficient for the liquid phase flowing alone. Xtt is the turbulent-turbulent Martinelli parameter. If the single-phase turbulent-flow correlation

f =  0.046Re −0.2

is used to evaluate the friction factors in the definition of the Martinelli parameter (Eqs. (10.11b) and (10.12)), it can be shown that Xtt is given by

1− x  X tt =   x 

0.9

 ρv   ρ  l

0.5

0.1

 µl   µ  (12.32) v

If the Dittus-Boelter correlation is used to evaluate hle and hl, relations of the type indicated in Eqs. (12.31) can be rearranged to show that h is a function of Rel, Prl, and Xtt, making the form of these relations similar to that indicated by Eq. (12.30). Several investigators have, in fact, correlated convective boiling data in the absence of strong nucleate boiling effects, using relations similar to the forms indicated above. Dengler and Addoms [12.38]

578

Liquid-Vapor Phase-Change Phenomena

proposed the following relation as a fit to heat transfer data for convective vaporization of water in a vertical tube: 0.5

 1  h = 3.5  (12.33)  X tt  hle



Based on a fit to data for convective boiling of organic liquids, Guerrieri and Talty [12.39] similarly proposed the correlation  1  h = 3.4   X tt  hle



0.45

(12.34)

Equations (12.33) and (12.34) strictly apply only to conditions where nucleate boiling has been completely suppressed. Hence, their usefulness is very limited. However, if one of these relations is assumed to hold at the location where complete suppression is just achieved, it can be combined with one of the onset correlations described in Section 12.2 to obtain a relation for the conditions at which nucleation will be suppressed. Combining Eq. (12.33) with the Sato and Matsumura [12.6] correlation (12.14), for example, it can easily be shown that the quality at which nucleation is suppressed is given by the relation x SUP =



γ (12.35) 1+ γ

where for an isothermal wall boundary condition ρ  γ = γT =  v   ρl 



0.56

 µl   µ  v

0.11

 kl hlv ρv ( Tw − Tsat )    28hle σTsat  

2.22

(12.36a)

and for a uniform heat flux applied to the wall ρ  γ = γH =  v   ρl 



0.56

 µl   µ  v

0.11

1.11

 q ′′kl hlv ρv   98σT h 2  sat le  

(12.36b)

The variation of xSUP with γ is shown in Fig. 12.13. Thus if the coolant properties and the wall boundary condition are specified, the quality at which nucleate boiling is virtually completely suppressed can be estimated using these relations. Example 12.3 Saturated flow boiling of R-410A at 220 kPa occurs in a small vertical round tube with a diameter of 0.7 cm. The mass flux is 200 kg/m2s, and a uniform heat flux of 6000 W/m2 is applied to the tube wall. Estimate the quality at which nucleation is expected to be completely suppressed. For saturated R-410A at 220 kPa, Tsat = 238 K = −34.9°C, ρl = 1296 kg/m3, ρv = 8.70 kg/m2, hlv = 258.2 kJ/kg, cpl = 1.40 kJ/kgK, μl = 272 × 10−6 Ns/m2, μv = 10.6 × 10−6 Ns/m2, kl = 0.140 W/mK, Prl = 2.72, and σ = 0.0149 N/m.

Re le =

GD ( 200 )( 0.007) = = 5147 µl 272 × 10 −6

579

Convective Boiling in Tubes and Channels

FIGURE 12.13  Variation of the quality at which nucleation is suppressed with the parameters defined by Eqs. (12.36).

So, for fully turbulent flow,



k  hle = 0.023  l  Re 0.8 Pr 0.4  D  le l  0.140  = 0.023  (5147)0.8 ( 2.72)0.4 = 639 W/m2K  0.007 

Using Eq. (12.36b), γ is determined as ρ  γ = v  ρl 

0.56

 8.70  =  1296 

 µl   µ  v

0.56

0.11

1.11

 q′′kl hlv ρv   98σT h2  sat le  

 272    10.6 

0.11

1.11

 6000 ( 0.140 )( 258.2 × 1000 )(8.70 )    2 98 ( 0.0149)( 238)(639)  

= 1.52 Substituting this value of γ into Eq. (12.35) yields

xSUP =

1.52 γ = = 0.603 1+ γ 1+ 1.52

Thus, in this system, nucleation is expected to exist to about 60% quality.

The above discussion indicates that nucleate boiling may dominate at low qualities while at moderate to high qualities nucleate boiling may be completely suppressed and film evaporation may dominate. At intermediate qualities, both nucleate boiling and film evaporation effects may be important. A number of correlations have been proposed that attempt to use a superposition technique to account for the gradual transition between these limiting circumstances. Several correlations of this type are described and briefly discussed below.

580

Liquid-Vapor Phase-Change Phenomena

The Chen Correlation Chen [12.40] argued that the heat transfer coefficient h for saturated convective boiling is equal to the sum of a microscopic (nucleate boiling) contribution hmic and a macroscopic (bulk convective) contribution hmac.

h = hmic + hmac (12.37)

Chen [12.40] proposed to evaluate the macroscopic contribution using a correlation similar in form to the Dengler and Addoms [12.38] correlation.

hmac = hl F ( X tt ) (12.38)

where the single-phase coefficient for the liquid alone hl is evaluated using the Dittus-Boelter equation,

k  hl = 0.023  l  Rel0.8 Prl0.4 (12.39a)  D Rel =

G (1 − x ) D (12.39b) µl

The microscopic contribution to the overall heat transfer coefficient was determined by applying a correction to the Forster-Zuber [12.41] relation for the heat transfer coefficient for nucleate pool boiling:

0.49  kl0.79 c 0.45  0.24 0.75 pl ρl hmic = 0.00122  0.5 0.29 0.24 0.24  Tw − Tsat ( Pl )  Psat ( Tw ) − Pl  S (12.40) σ µ ρ h l lv v  

The so-called suppression factor S corrects the fully developed nucleate boiling prediction of hmic to account for the fact that as the macroscopic convective effect increases in strength, nucleation is more strongly suppressed. Chen argued that this suppression factor ought to be a function of an appropriately defined two-phase Reynolds number Retp. He further conjectured that the macroscopic contribution hmac should also be related to this two-phase Reynolds number via an extension of the Dittus-Boelter equation

 ktp  hmac = 0.023   Retp0.8 Prtp0.4 (12.41)  D

Taking ktp = kl and Prtp = Prl and combining Eq. (12.41) with Eq. (12.39a), it was concluded that

Retp =   Rel  F ( X tt )

1.25

(12.42)

From a regression analysis of available data, Chen obtained F(Xtt) and S(Retp) curves that provided a best fit for this correlation technique. The correlation curves for F(Xtt) and S(Retp) were originally presented only in graphical form. Somewhat later, Collier [12.42] proposed the following empirical relations as fits to Chen’s original F(Xtt) and S(Retp) curves:

F ( X tt ) = 1

for X tt−1 ≤ 0.1 (12.43a)

 1  F ( X tt ) = 2.35  0.213 +  X tt 

(

0.736

for X tt−1 > 0.1 (12.43b)

S ( Retp ) = 1 + 2.56 × 10 −6 Re1.17 tp

)

−1

(12.44)

581

Convective Boiling in Tubes and Channels

If the wall superheat, mass flux, fluid properties and quality are specified, Chen’s [12.40] correlation can then be used to calculate the heat transfer coefficient as follows: 1. For the specified G, x, and fluid properties, Eqs. (12.32), (12.39a), and (12.39b) can be used to calculate Xtt, Rel and hl. 2. Equation (12.43) can be used to compute F(Xtt). 3. Equation (12.42) can be used to calculate Retp, which can then be inserted into Eq. (12.44) to determine S. 4. With the results of steps 1–3, Eqs. (12.38) and (12.40) can be used to determine hmac and hmic. The overall heat transfer coefficient is then computed using Eq. (12.37): h = hmac + hmic. If a uniform heat flux is applied to the tube walls, the above calculation process must be iterated to determine the wall superheat that, when multiplied by the resulting h value, yields the specified heat flux. Modifications to the original Chen correlation have also been proposed in more recent publications. Based on a model analysis of the thermal region near the wall and its effect on nucleation, Bennett et al. [12.43] proposed that the suppression factor S in the Chen correlation be computed using the relation

1 − exp {− F ( X tt ) hl X 0 / kl }  S=  (12.45) F ( X tt ) hl X 0 / kl

where 0.5



  σ X 0 = 0.041   (12.46)  g ( ρl − ρv ) 

In most cases, use of this correlation for S yields values of h that are comparable to those obtained using the empirical relation (12.44). Noting that the original Chen correlation was developed mainly to fit flow boiling data for water, Bennett and Chen [12.44] modified the correlation to account for the effect of the liquid Prandtl number being significantly different from one. To generalize the Chen correlation for all non-metallic liquids they recommended replacing Eq. (12.36) with the relation

hmac = hl F ( X tt ) Prl0.296 (12.47)

The variation of h with x predicted using the Bennett and Chen [12.44] correlation for upward vertical flow boiling of R-134a at a saturation pressure of 337.8 kPa is shown in Fig. 12.14. Computed results are shown in this figure for G = 300 kg/m2s and a wall superheat of 10°C. In addition to the overall h variation, the variations of hmic and hmac are also plotted in this figure. As the quality increases, trends of increasing hmac and decreasing hmic are clearly discernible in Fig. 12.14. Combining the two values, the net trend in this case is a slight but steady increase in h as x increases.

The Shah Correlation For saturated flow boiling in vertical and horizontal tubes, Shah [12.45] proposed a correlation for the heat transfer coefficient in the form

ψS =

h = f ( Co, Bo,Frle ) (12.48) hl

582

Liquid-Vapor Phase-Change Phenomena

FIGURE 12.14  Convective boiling heat transfer coefficient variation with quality as predicted for R-134a using the Bennett and Chen correlation [12.44].

where Co is the convection number, Bo is the boiling number, and Frle is the Froude number defined as

1− x  Co =   x 

0.8

0.5

 ρv   ρ  (12.49) l



Bo =

q ′′ (12.50) Ghlv



Frle =

G2 (12.51) ρl2 gD

Here, hl is the single-phase coefficient for the liquid phase flowing alone in the tube, as predicted by Eqs. (12.39). Note that the convection number is similar to the turbulent-turbulent Martinelli parameter defined in Eq. (12.32). The correlation in terms of these variables was specified in graphical form. The graphical representation of Shah’s correlation is shown in Fig. 12.15. For specified quality, mass flux and heat flux values, the dimensionless parameters Co, Bo, and Frle can be computed and the ratio ψS = h/hl can be determined from the plot in Fig. 12.15. Now, however, this plot is more useful for understanding trends, rather than as a computational tool. For vertical tubes, the value of the Froude number Frle is ignored. The user reads up along the vertical line corresponding to the computed Co value to its intersection with the appropriate Bo line and then horizontally across to the vertical axis to determine the h/hl value. As Bo decreases, the constant Bo lines merge into line AB. For horizontal tubes, the use of the plot is somewhat different. In this case, the user reads vertically up along the appropriate constant Co line to its intersection with the Frle line corresponding to the computed value. One then proceeds horizontally to line AB, vertically to the appropriate Bo line, and then horizontally to the left to read the h/h l value from the vertical axis.

583

Convective Boiling in Tubes and Channels

FIGURE 12.15  Graphical representation of Shah’s correlation. (Adapted from [12.45] with permission, copyright © 1976, American Society of Heating Refrigerating and Air-Conditioning Engineers.)

Shah [12.46] later recommended the following computational representation of his correlation:

N S = Co

for Frle ≥ 0.04 (12.52a)



N S = 0.38 Frle−0.3 Co for Frle < 0.04 (12.52b)



FS = 14.7 for Bo ≥ 11 × 10 −4 (12.53a)



FS = 15.4 for Bo < 11 × 10 −4 (12.53b)



ψ cb = 1.8 N S−0.8 (12.54)

For Ns > 1.0:

ψ nb = 230Bo0.5

for Bo > 0.3 × 10 −4 (12.55a)



ψ nb = 1 + 46Bo0.5

for Bo ≤ 0.3 × 10 −4 (12.55b)



ψS =

 ψ nb h = the maximum of  (12.56) hl  ψ cb

For Ns ≤ 1.0:

(

ψ bs = FS Bo0.5 exp 2.74 N S−0.1

(

)

ψ bs = FS Bo0.5 exp 2.47 N S−0.15 ψS =

for 0.1 < N S ≤ 1.0 (12.57a)

)

for N S ≤ 0.1 (12.57b)

 ψ bs h = the maximum of  (12.58) ψ hl  cb

584

Liquid-Vapor Phase-Change Phenomena

Shah [12.46] indicates that the equations given above agree with the curves in the graphical representation within ±6% over most of the chart except for two regions: 1. Near Co = 0.004 and Bo = 50 × 10−4. 2. For horizontal tubes at Frle < 0.04 and Bo < 1 × 10−4. Shah [12.46] notes that these equations overpredict h by about 11% in the first region, but this is not expected to be a problem since these conditions usually fall into the post-dryout region. Inaccuracy in the second region can be as much as 20%. However, Shah [12.46] elected not to refine the correlating equations to achieve a better fit in this range because values of Bo below 1 × 10−4 are rarely encountered.

The Correlation of Schrock and Grossman Schrock and Grossman [12.47] recommended the following correlation based on a fit to vertical upward flow boiling heat transfer data for water:

0.66   1   h = C1  Bo + C2   (12.59)  X tt   hl 

where Bo and Xtt are the boiling number and Martinelli parameter defined above. These investigators recommended values for C1 and C2 of 7.39 × 103 and 1.5 × 10−4, respectively. Wright [12.48] later recommended C1 = 6.70 × 103 and C2 = 3.5 × 10−4 for this correlation.

The Gungor and Winterton Correlation Gungor and Winterton [12.49] proposed the following correlation for heat transfer during convective flow boiling in vertical tubes

0.41 0.75  x   ρl   h = hl 1 + 3000Bo0.86 +   (12.60)   1 − x   ρv    

Here again hl is the heat transfer coefficient for the liquid phase flowing alone, and Bo is the boiling number defined by Eq. (12.48).

The Correlation of Bjorge, Hall, and Rohsenow For saturated upward flow boiling in vertical tubes at qualities above 0.05, Bjorge et al. [12.50] recommended the following correlation technique for predicting h:

h=

qtot ′′ (12.61) Tw − Tsat

where

  ( Tw − Tsat )  3  i qtot ′ = q ′′fc + q ′′fdb 1 −    (12.62)   Tw − Tsat   q ′′fc = FB Prl ( kl /D )( Tw − Tsat )

Rel0.9 (12.63) C2

585

Convective Boiling in Tubes and Channels

 1 2  FB = 0.15  + (12.64)  X tt X tt0.32 



(

C2 = 5Prl + 5Prl ln (1 + 5Prl ) + 2.5ln 0.0031Rel0.812

(

for Rel > 1125 (12.65a)

)

C2 = 5Prl + 5ln 1 + Prl 0.0964 Rel0.585 − 1  for 50 < Rel ≤ 1125 (12.65b) C2 = 0.0707 Prl Rel0.5



 g ( ρl − ρv )  q ′fdb = BM µ l hlv   σ  

1/ 2



)

( Tw − Tsat )i =



for Rel ≤ 50 (12.65c)

3  1/8  kl1/ 2 ρ17/8 c19/8 l pl ρv ( Tw − Tsat )  9/8 5/8 1/8  (12.66) 7/8  µ l hlv ( ρl − ρv ) σ Tsat 

8σTsat h fc  1 1 −  (12.67)  kl hlv  ρv ρl 

This correlation technique postulates a superposition of heat fluxes rather than a superposition of heat transfer coefficients. Correlation of the forced-convective (macroscopic) contribution to the heat flux again presumes that the corresponding heat transfer coefficient is primarily a function of Rel, Xtt, and Prl. The nucleate boiling (microscopic) contribution is predicted using the MikicRohsenow [12.51] correlation (12.66) for fully developed nucleate boiling. BM in this relation is a dimensional constant that depends on the solid-surface cavity size distribution and the fluid properties. For forced convection of water, they recommend BM = 1.89 × 10−14 for properties in SI units. (Tw – Tsat)i in Eq. (12.62) is the wall superheat at the incipient boiling (onset) condition.

K andlikar’s Correlation Kandlikar [12.52] proposed a saturated flow boiling correlation for the heat transfer coefficient h that was developed to fit a very broad spectrum of data for flow boiling heat transfer in vertical and horizontal tubes. Initially [12.52], this correlation was formulated with different constants for specified ranges of convection number Co defined by Eq. (12.49). This provided different treatments of the convective boiling dominant and nucleate boiling dominant regimes. Kandlikar [12.53] subsequently proposed the following slightly modified form of the correlation:

 hNBD h = the maximum of  (12.68)  hCBD

where for the nucleate boiling dominant regime



ρ  hNBD = 0.6683  l   ρv 

0.1

x 0.16 (1 − x )

0.64

f2 ( Frle ) hle

(12.69)

+1058.0 Bo0.7 FK (1 − x ) hle 0.8

and for the convective boiling dominant regime



ρ  hCBD = 1.1360  l   ρv 

0.45

x

0.72

(1 − x )0.08 f2 ( Frle ) hle

+667.2 Bo0.7 FK (1 − x ) hle 0.8

(12.70)

586

Liquid-Vapor Phase-Change Phenomena

Note that the terms proportional to Bo account for the variation of nucleate boiling effects, similar to the Shah correlation described above. In this correlation, hle is the single-phase heat transfer coefficient for the liquid phase flowing alone. In the initial formulation of this correlation [12.52], the DittusBoelter correlation was used to predict hle. Kandlikar later [12.53] proposed use of the PetukhovPopov [12.54] and Gnielinski [12.55] correlations for hle because these appeared to better account for the Prandtl number effect for different fluids. For 0.5 ≤ Prl ≤ 2000 and 2300 ≤ Rele < 104, hle, is therefore computed using the correlation of Gnielinski [12.55]  k  ( Rele − 1000 ) Prl ( f /2 ) (12.71) hle =  l   D  1 + 12.7 Prl2/3 − 1 ( f /2 )0.5



(

)

and for 0.5 ≤ Prl ≤ 2000 and 104 ≤ Rele ≤ 5 × 106, hle is computed using the correlation of Petukhov and Popov [12.54]: Rele Prl ( f /2 ) k  hle =  l  (12.72)  D  1.07 + 12.7 Prl2/3 − 1 ( f /2 )0.5



(

)

In the above relations for hle, f is the friction factor computed as −2

f =   1.58 ln ( Rele ) − 3.28  (12.73a)



Rele =



GD (12.73b) µl

and the function f2(Frle) is defined as

 ( 25Fr )0.3 for Fr < 0.04 for horizontal tubes le le f2 ( Frle ) =  (12.74) 1 for Fr le > 0.04 for horizontal tubes and for vertical tubes 

Note that on the right side of Eqs. (12.69) and (12.70), the first term accounts for the convective effect and the second term accounts for the nucleate boiling effect. The factor FK is a fluid-dependent parameter, values of which are listed for various fluids in Table 12.1. For fluids other than those listed

TABLE 12.1 Fluid Constant Values for the Kandlikar [12.53] Correlation Fluid Water R-11 R-12 R-13B1 R-22 R-113 R-114 R-134a R-152a Nitrogen Neon

FK 1.00 1.30 1.50 1.31 2.20 1.30 1.24 1.63 1.10 4.70 3.50

Convective Boiling in Tubes and Channels

587

in Table 12.1, Kandlikar [12.52] recommends that FK be estimated as the multiplier that must be applied to the Forster-Zuber [12.41] correlation to correlate pool boiling data for the fluid of interest. In addition, a non-linear superposition scheme for flow boiling heat transfer has been proposed by Steiner and Taborek [12.56] which requires knowledge of a characteristic roughness for the surface. Generalized correlations have also been developed by Kilmenko [12.57, 12.58]. Note that the flow boiling heat transfer methodologies described above that, in some way, superimpose convective and nucleate boiling effects, generally are presumed to apply from the onset of nucleate boiling to the onset of CHF or dryout. As in the case of convective condensation (see Chapter 11), the two-phase flow regime is expected to affect the heat transfer mechanisms during flow boiling. This suggests that heat transfer prediction methods that have different treatments for different flow regimes may offer the prospect of a more accurate predictive tool. This is particularly relevant to horizontal evaporator passages in HVAC and refrigeration systems. In such cases, gravity may stratify the flow, producing partial dryout at higher qualities, which alters the mean heat transfer coefficient from that predicted for vertical or low-gravity-effects flow boiling. Examples of flow regime based correlations for saturated flow boiling include those developed by Kattan et al. [12.59–12.61], Zurcher et al. [12.62], and Wojtan et al. [12.63, 12.64]. Use of this type of framework requires first predicting the two-phase flow regime for the local conditions, and then using void fraction and heat transfer predictive tools for that regime to determine the local heat transfer coefficient. This results in a more complex computation for the local heat transfer coefficient compared to the simpler combined-convection-and-nucleate-boiling methods described above, but implementation of flow regime based predictive models in a computer program is generally straightforward. The interested reader can find more detailed information on flow-regime modeling in the references by Kattan et al. [12.59–12.61], Zurcher et al. [12.62], and Wojtan et al. [12.63, 12.64].

Comparison of Correlations Like most of the other correlations described above, Kandlikar’s [12.53] correlation sums the contributions of terms representing nucleate boiling and forced convective effects. The variation of these contributions and that of the overall heat transfer coefficient are shown in Fig. 12.16 for flow boiling

FIGURE 12.16  Convective boiling heat transfer coefficient variation with quality as predicted for R-134a using Kandlikar’s [12.53] correlation.

588

Liquid-Vapor Phase-Change Phenomena

FIGURE 12.17  Comparison of the predictions of six different correlations for convective boiling heat transfer.

of R-134a at 338 kPa in a vertical tube. The tube inside diameter is 1.0 cm, G = 300 kg/m2s, and the wall superheat is 10°C. The conditions are identical to those for Fig. 12.14, which shows similar variations as predicted by the Bennett and Chen [12.44] correlation. Comparing Figs. 12.14 and 12.16, it can be seen that both correlations predict that, as the quality increases, the nucleate boiling contribution diminishes while the forced-convective effect increases. However, the net effect is different in the two cases. As quality increases, the Kandlikar correlation predicts a steadily decreasing heat transfer coefficient whereas the Bennett and Chen correlation predicts a gradual increase in overall h. Kandlikar [12.52, 12.53] points out that data for refrigerants often exhibit a decrease in h with quality at a specified mass flux. Figure 12.17 shows the variation of overall h predicted by all of the correlations described above for the same conditions as Figs. 12.14 and 12.16. While the Bennett and Chen [12.44] correlation and the correlation of Bjorge et al. [12.50] show a slight monotonic increase with quality, the other correlations exhibit different trends. The Shah [12.46] and Schrock-Grossman [12.47] correlations generally indicate that h first increases and then decreases slightly with quality. The Gungor and Winterton [12.49] and Kandlikar [12.52, 12.53] correlations indicate a steady drop in heat transfer coefficient with quality. While there seems to be a consensus regarding the behavior of the individual mechanisms as quality increases, the net combined effect of these mechanisms varies depending on which of these correlations is used. The agreement of refrigerant data with the low-quality predictions of Kandlikar’s correlation implies that the other correlations may underpredict the strength of the nucleate boiling effect in hydrocarbon fluids at low quality. The relatively good agreement with data for a broad spectrum of fluids over a wide range of conditions suggests that Kandlikar’s [12.53] may be a reliable general correlation. The original version of this correlation was found to agree with water data to a mean deviation of 15.9% and to a mean deviation of 18.8% for all refrigerant data. Agreement with data is slightly better when the

589

Convective Boiling in Tubes and Channels

alternate single-phase correlations (12.71) and (12.72) are used. It should be noted that the correlations of Shah [12.46] and Kandlikar [12.53] can be applied to horizontal tubes or vertical tubes with upflow. The other correlations described above were developed to match data for vertical upflow, and they can be expected to yield reasonable results for other orientations only if the Froude number Frle = G 2 /(ρl2 gD) is large (> 0.05). All of the correlations described above have been used with some success to correlate flow boiling heat transfer data over a finite range of flow conditions. In selecting one of the above correlations to predict the flow boiling heat transfer coefficient for a specific set of circumstances, perhaps the best advice is to select one that has been verified against data for fluid and flow conditions that are as similar as possible to those under consideration. It should be noted that the correlations described above generally will not be accurate at qualities beyond that at which dryout of the liquid film on the wall occurs. For round tubes, this is usually estimated to occur at a quality somewhere above x = 0.7. Determination of the exact condition at which dryout occurs will be discussed further in the next section. Example 12.4 Use Kandlikar’s [12.53] correlation to predict the heat transfer coefficient for flow boiling of nitrogen in a vertical tube at a pressure of 778 kPa and qualities of 0.2 and 0.6. The tube diameter is 0.9 cm, the flow rate is such that the mass flux is 200 kg/m2s, and a uniform heat flux of 20 kW/m2 is applied to the tube wall. Compare the results with the predictions of the Gungor and Winterton [12.49] correlation under the same conditions. For saturated nitrogen at 778 kPa, Tsat = 100K = – 173.2°C, ρl = 691 kg/m3, ρv = 32.0 kg/m3, hlv = 162.2 kJ/kg, cpl = 2.31 kJ/kgK, μl = 86.9 × 10−6 Ns/m2, μv = 7.28 × 10 –6 Ns/m2, kl = 0.0955 W/mK, Prl = 2.10, and σ = 0.00367 N/m. At x = 0.2,

Re le =

GD 200(0.009) = = 20,800 µl 86.9 × 10 −6

Because 104 ≤ Rele ≤ 5 × 106, hle is computed using the correlation of Petukhov and Popov [12.54]:



f =   1.58 ln (Rele ) − 3.28 

−2

=   1.58 ln ( 20,800 ) − 3.28 

−2

=  0.00647

Re le Prl (f / 2) k  hle =  l   D  1.07 + 12.7 Prl2/ 3 − 1 (f / 2)0.5

(

)

(20,800)(2.10)(0.00647/ 2)  0.0955  = 978 W/m 2K =  0.009  1.07 + 12.7 (2.1)2/ 3 − 1 (0.00647/ 2)0.5

(

)

The boiling number is computed as

q′′ 20 = = 6.17 × 10 −4 Ghlv (200)(162.2)

Bo =

For vertical tubes f2 = 1 and FK is 4.70 for nitrogen. The relations for hNBD and hCBD then simplify to

ρ  hNBD = 0.6683  l   ρv 



ρ  hCBD = 1.1360  l   ρv 

0.1

x 0.16 (1− x )0.64 hle + 1058.0Bo0.7 (4.70)(1− x )0.8 hle

0.45

x 0.72 (1− x )

0.08

hle + 667.2Bo0.7 (4.70)(1− x )0.8 hle

590

Liquid-Vapor Phase-Change Phenomena

Substituting yields



 691  hNBD = 0.6683   32.0 

0.1

(0.2)0.16 (0.8)0.64 (978)

+1058.0(6.17 × 10 −4 )0.7 (4.70)(0.8)0.8 (978) = 2.36 × 10 4 W/m 2K

 691  hCBD = 1.1360   32.0 

0.45

(0.2)0.72(0.8)0.08 (978)

+667.2(6.17 × 10 −4 )0.7 (4.70)(0.8)0.8 (978) = 1.59 × 10 4 W/m 2K h is taken to be the larger of the two results above, and hence it follows that h is predicted to be 2.36 × 104 W/m2K. Repeating the calculations above for x = 0.6 yields h = 1.37 × 104 W/m2K. The correlation of Gungor and Winterton [12.49] is given by Eq. (12.60):

0.41 0.75   x   ρl    h = hl 1+ 3000Bo0.86 +   1− x   ρl    

For the liquid phase flowing alone at x = 0.2,



Re l =

G(1− x )D 200(0.8)(0.009) = = 16,600 µl 86.9 × 10 −6

k  hl = 0.023  l  Rel0.8 Prl0.4  D  0.0955  = 0.023  (16,600)0.8 (2.10)0.4 = 780 W/m 2K  0.009 

A similar calculation for x = 0.6 indicates that hl = 448 W/m2K. Substituting the appropriate hl and x values for x = 0.2 and x = 0.6 into the Gungor and Winterton [12.49] correlation yields



0.75 0.41   0.2   691   h = 780 1+ 3000(0.000617)0.86 +       0.8   32.0    

= 5.82 × 103 W/m 2 K



for x = 0.2

0.75 0.41   0.6   691   h = 448 1+ 3000(0.000617)0.86 +       0.4   32.0    

= 4.92 × 103 W/m 2 K

for x = 0.6

Thus, it can be seen that for these circumstances, the values of h predicted by these schemes differ by a factor of 4.1 at x = 0.2 and a factor of 2.8 at x = 0.6.

12.5  CRITICAL HEAT FLUX CONDITIONS FOR INTERNAL FLOW BOILING Mechanisms The terms critical heat flux condition (CHF) and boiling burnout are used to describe the conditions at which the wall temperature rises and/or the heat transfer coefficient decreases sharply due to a change in the heat transfer mechanism. The term “burnout” is used even when failure of the

Convective Boiling in Tubes and Channels

591

FIGURE 12.18  Film dryout and the transition from annular to mist flow.

passage wall does not occur due to overheating and melting, although clearly its origins can be traced back to circumstances where the tube wall does fail in this manner. The critical heat flux condition is indicated schematically in Figs. 12.4 and 12.5 as diagonal (or in the limit, vertical) lines. The nature of the transition indicated by these lines varies with the enthalpy of the flow. At subcooled (bulk flow) conditions and low qualities, this transition corresponds to a change in the boiling mechanism from nucleate to film boiling. For this reason, the critical heat flux condition for these circumstances is often referred to as the departure from nucleate boiling (DNB). In the European literature, this transition is sometimes referred to as burnout of the first kind. At moderate to high qualities, the flow is almost invariably in an annular configuration, and the transition corresponds to dryout of the liquid film on the tube wall (Fig. 12.18). For this range of conditions, this transition is usually referred to simply as dryout. Some European investigators have used the terminology burnout of the second kind for this transition. As indicated in Figs. 12.4 and 12.5, once dryout occurs, the flow enters the so-called liquid-deficient regime, in which the remaining liquid exists as entrained droplets. The heat transfer associated with this process will be discussed in more detail in the Section 12.6. However, it is worth noting at this stage that because of the high vapor velocity typical of these conditions, the convection of heat from the tube wall to the vapor is generally strong. As a result, the drop in the heat transfer coefficient for dryout is usually not as severe as that which accompanies departure from nucleate boiling at the same total mass flux. Before discussing measurements of the critical heat flux conditions and methods for predicting them, it is useful to first consider the mechanisms responsible for part or all of the tube wall becoming dry. Although numerous investigations of critical heat flux conditions have been conducted, at the present time, some aspects of the mechanisms responsible for the critical heat flux transitions are not well understood. Some light has been shed on the nature of these mechanisms, however. At subcooled or very low quality conditions, bubbly or slug flow is typically encountered. For such conditions, three potential mechanisms have been discussed by Tong and Hewitt [12.65]. These mechanisms are illustrated in Fig. 12.19. The first of these mechanisms is associated with the evaporation of the liquid microlayer under a growing vapor bubble on the heated tube wall (Fig. 12.19a). Just prior to release of the bubble, evaporation of the microlayer may leave a portion of the wall under the bubble completely dry. If a constant heat flux condition is applied to the wall, the temperature of the dry patch may rise above the Leidenfrost temperature, preventing the patch, or a portion of it, from rewetting. Continued evaporation of the microfilm at the perimeter of the dry patch may cause it to grow. This may eventually lead to the entire wall becoming dry, whereupon a transition to convective film boiling occurs. The second mechanism identified by Tong and Hewitt [12.65] may be encountered at moderate bulk subcooling levels. For such conditions, bubbles may be concentrated in a boundary layer near the wall (Fig. 12.19b). If the number density of the bubbles and size of the bubble boundary layer

592

Liquid-Vapor Phase-Change Phenomena

FIGURE 12.19  Schematic representations of postulated CHF mechanisms at low quality.

become large enough, liquid flow to the surface may be impeded. Liquid under and between bubbles at the surface may then be evaporated away, producing dry patches at the wall. Dry patches produced in this fashion may continue to grow as liquid near the surface is evaporated away, ultimately producing a transition to convective film boiling. A third mechanism described by Tong and Hewitt [12.65] is associated with slug flow encountered for low-quality saturated flows. As a slug flow bubble moves downstream, if the heat flux from the wall is high enough, the liquid film between the bubble interface and the wall may completely evaporate at a particular location, forming a dry patch, as indicated in Fig. 12.19c. For a sufficiently high applied heat flux, the surface temperature may exceed the Leidenfrost temperature and prevent the patch from rewetting. The dry patch may subsequently grow in size, leading to a transition to convective film boiling. Despite numerous studies of critical heat flux (CHF) conditions, how the mechanisms described above vary with flow conditions is not well understood. The complexity of these mechanisms has also made analytical modeling generally difficult. As a result, most proposed methods for predicting CHF conditions have been correlations based on experimental data.

CHF Data for Water The body of CHF data generated in prior experimental studies is extremely large. Bergles [12.66] has estimated that several hundred thousand CHF data points have been obtained in such studies, and that over 200 correlations have been developed in attempts to correlate the data. Most of the available data have been obtained for upward flow boiling of water in a uniformly heated tube. A review of all the available CHF data is beyond the scope of this text. The interested reader may wish to consult the summary of CHF data for water in vertical tubes compiled by Thompson and Macbeth [12.67] for further information. With the large available database, evaluation of the data to eliminate questionable points is a formidable task. In the late 1960s, a group in the Soviet Union at the All Union Heat Engineering Institute (VTI) addressed this issue by reviewing available data. They subsequently tabulated recommended values of CHF conditions for upflow of water in a vertical tube with an inside diameter of 8 mm. These tabulated values were listed in a paper by Doroshchuk and Lantsman [12.68]. This paper also described methods for extrapolating the CHF conditions to round tubes with different diameters. The CHF values summarized in this paper for the most part are departure from nucleate boiling (DNB) transitions at low-quality or subcooled conditions.

593

Convective Boiling in Tubes and Channels

FIGURE 12.20  Predicted CHF conditions for water at 29.5 bar, based on data collected by the USSR Academy of Sciences.

A more extensive tabulation, including DNB as well as dryout transitions, was published somewhat later by the Heat and Mass Transfer Section of the Scientific Council, USSR Academy of Sciences [12.69]. The tabulated CHF conditions were for water in a vertical round tube with an inside diameter of 8 mm, and the following method was recommended for predicting the critical heat flux for other tube diameters: 1/ 2

 8 qcrit ′′ = (qcrit ′′ )8 mm   (12.75)  D



where D is the tube inside diameter in millimeters. Values of the CHF presented in tabulated form in the USSR Academy of Sciences report [12.69] are plotted for two pressure levels in Figs. 12.20 and 12.21. These figures clearly indicate the effects of varying system conditions on the critical heat flux.

Parametric Effects in DNB Data In general, for upward flow of water in a round tube, the parameters that may affect the critical heat flux include the mass flux G, the pressure P, the tube diameter D, the location z downstream of the tube inlet, and either the quality x or subcooling Tsat – Tbulk. The bulk of the available data indicates that for z/D> 40, the downstream location z has no significant effect on the CHF condition, implying that the critical heat flux is dictated by local conditions alone. For sufficiently long tubes, it follows that the critical heat flux depends mainly on the four remaining variables.

qcrit ′′ = f ( D, P, G , xcrit ) for saturated flow (12.76a)

or

qcrit ′′ = f ( D, P, G , Tsat − Tbulk ) for subcooled flow (12.76b)

594

Liquid-Vapor Phase-Change Phenomena

FIGURE 12.21  Predicted CHF conditions for water at 69 bar, based on data collected by the USSR Academy of Sciences.

Figures 12.20 and 12.21 provide some indication of the effects of each of these parameters on the critical heat flux. Figure 12.20 shows the variation of the critical heat flux qcrit ′′ with subcooling and quality for various mass flux levels at a pressure of 29.5 bar. Figure 12.21 shows similar variations of the critical heat flux at a pressure of 69 bar. The effects of the different parameters in Eqs. (12.76) on the CHF conditions are discussed below. Note that the parametric trends described here are for DNB transition. (Wall dryout will be considered later in this section.) Mass Flux Figures 12.20 and 12.21 indicate that the effects of increasing mass flux are clearly different for subcooled and saturated flows. For strongly subcooled conditions, increasing the flow rate increases the critical heat flux, apparently because doing so strengthens convective transport, and enhances delivery of liquid to the wall to replace vapor generated there. The increase in qcrit ′′ is generally somewhat weaker than a linear dependence. These trends were also exhibited in data obtained in the studies of subcooled flow boiling of water by Boyd [12.70–12.72] and Celata et al. [12.73]. For saturated flow, Figs. 12.20 and 12.21 indicate that increasing the mass flux decreases the critical heat flux at a given quality. When the flow is in the churn or annular regimes, this trend is apparently a consequence of the increase in entrainment that generally accompanies increasing mass flux. Since a greater portion of the liquid inventory flows as entrained droplets on the core, less flows in the liquid film, making it possible to blanket the wall with vapor at lower heat flux levels. Celata and Mariani [12.74] note, however, that this trend is characteristic of changes in mass flux with the exit condition fixed, and that when the inlet condition is fixed, the trend is reversed: the critical heat flux then increases with mass flux for the same pressure and quality. Pressure Comparison of these two figures suggests that, other parameters being the same, qcrit ′′ generally increases as the pressure decreases over this pressure range. For subcooled conditions, this trend persists throughout the tabulated conditions. For subcooled flow boiling, Boyd [12.75] noted that,

Convective Boiling in Tubes and Channels

595

for water, some researchers found a maximum in the CHF versus pressure trend near P/Pc = 0.75, with the exact value being weakly dependent on mass flux. However, results of CHF experiments reported by Celata et al. [12.73], Nariai and Inasaka [12.76], and Vandervort et al. [12.77] indicated that the pressure dependence of CHF for subcooled flow boiling of water is weak over the range 100 < P < 5000 kPa. For saturated conditions, Figs. 12.20 and 12.21 also suggest that, for fixed mass flux and quality, qcrit ′′ also generally tends to increase as the pressure decreases. However, as noted by Celata and Mariani [12.74] and Collier and Thome [12.78], the available data for saturated flow boiling of water indicate that the variation of CHF with pressure is very complex, with some data indicating that the variation with pressure exhibits a maximum at a particular pressure under some flow conditions. Subcooling and Quality It is clear from the plots that the critical heat flux decreases as the bulk enthalpy of the coolant increases: qcrit ′′ decreases as subcooling decreases or quality increases. For subcooled boiling at moderate to high subcooling, Figs. 12.20 and 12.21 suggest that qcrit ′′ increases about linearly with subcooling. This nearly linear trend is consistent with data obtained by Celata et al. [12.73], Vandervort et al. [12.77], Bergles [12.79], and Nariai et al. [12.80]. At very low subcooling the dependence becomes non-linear. For saturated conditions, Figs. 12.20 and 12.21 indicate that at fixed P and G, the DNB qcrit ′′ decreases with increasing quality. Increasing quality accelerates the core flow, which can enhance entrainment, making it more difficult to sustain a continuous liquid film flow on the tube walls. This tends to make it easier to blanket the wall with vapor at a lower heat flux level. Effects of Tube Length and Diameter Data from a number of studies (including Bergles [12.79], Nariai and Inasaka [12.76], and Vandervort et al. [12.77]) imply that the CHF for subcooled flow boiling of water increases as diameter decreases. The effect of tube diameter on critical heat flux can also be surmised from inspection of the recommended correction relation (12.75). It also implies that tubes of larger diameter will have lower critical heat flux values. Since this relation applies to subcooled as well as saturated flow boiling, it implies that this trend applies in both cases. Subcooled flow boiling data from several studies (e.g., those by Narai et al. [12.80] and Vandervort et al. [12.77]) indicate that CHF decreases as the heated passage length to diameter ratio L/D increases. This variation is strongest for L/D below about 20, and the variation is negligible above a threshold that is typically in the range 20 < L/D < 40. For saturated flow boiling, some experimental data suggest that for fixed inlet conditions, CHF tends to decrease with L/D, while for lengths above a certain threshold the effect disappears (see e.g., references [12.78, 12.81]). For saturated boiling DNB, the threshold at which this occurs apparently varies with system parameters. Other Factors Affecting DNB Some studies indicate that other factors such as heat flux distribution, tube orientation, and tube wall properties and surface geometry may affect the DNB condition in subcooled or saturated flow boiling. Further information on these effects can be found in references [12.74, 12.76–12.78, 12.82–12.86]. Parametric Trends in Dryout Data for Water The CHF conditions indicated in Figs. 12.20 and 12.21 are all for relatively high heat flux levels. Figure 12.22 shows a sample of CHF data reported by Levitan and Lantsman [12.87] for lower heat flux levels. At these lower heat flux levels, their data indicate that below a certain heat flux, the critical condition occurs at a quality that is a function only of the mass flux and pressure (and is independent of heat flux). In considering this type of behavior in low-heat-flux CHF data, Doroshchuk et al. [12.88] interpreted this range as corresponding to dryout of the liquid film on the wall of the

596

Liquid-Vapor Phase-Change Phenomena

FIGURE 12.22  Low-heat-flux CHF data for water. (Adapted from [12.87] with permission, copyright 1975, British Library Document Supply Centre.)

tube (burnout of the second kind). At heat flux levels above this threshold, the CHF transition was interpreted to be a departure from nucleate boiling. Based on an assessment of the available data base, values of the limiting dryout quality conditions for low heat flux values were recommended in the report by the members of the USSR Academy of Sciences [12.69]. These recommended values are presented graphically in Fig. 12.23. This plot clearly indicates that increasing mass flux decreases the dryout quality. This apparently is a consequence of increased entrainment, which leaves less liquid in the liquid film on the walls. The lack of heat flux dependence on the measured dryout quality values for these conditions was attributed by Doroshchuk et al. [12.88] to the non-precipitation (non-deposition) of entrained droplets on the wall at low mass flux levels and moderate heat flux levels. Doroshchuk et al. [12.88] also presented CHF data at high mass flux levels that suggest that the dryout quality may exhibit some renewed heat flux dependence at very low heat flux levels and high mass flux levels. These investigators attributed this dependence on heat flux to the increased precipitation (deposition) of droplets on the tube wall at higher mass flux levels. The model of the entrainment and deposition mechanisms discussed in Chapter 10 does imply that the deposition rate increases as the mass flux increases, which tends to support this conjecture. This matter apparently was explored in only a limited fashion because the heat flux levels at which it occurs are low compared to those of interest

597

Convective Boiling in Tubes and Channels

FIGURE 12.23  Predicted dryout conditions for low-heat-flux flow boiling of water based on data assembled by the USSR Academy of Sciences.

in power system boilers. These low heat flux trends in the dryout quality may be of importance, however, to refrigeration system evaporators and other applications where low heat flux evaporation processes are encountered. All values shown in Fig. 12.23 are for a tube diameter of 8 mm. For other tube sizes, the following correction was recommended:

8 x crit = ( x crit )8mm    D

0.15

(where D  is in mm) (12.77)

The parametric effects of pressure and mass flux can be discerned in Fig. 12.23. For fixed mass flux, decreasing pressure generally increases the quality xcrit at which incipient dryout occurs. For fixed pressure, at low G values, increasing G initially decreases xcrit. This is likely due to enhanced entrainment as the mass flux increases. However, the effect diminishes with increasing mass flux, suggesting that as mass flux increases, eventually a maximum entrainment effect is achieved, and increasing the mass flux further produced little change in the entrainment and deposition effect at the film interface. The effect of tube diameter on dryout quality is indicated by the recommended correction relation (12.77). It implies that xcrit is generally expected to decrease weakly with increasing tube diameter.

CHF Correlations and Models for Subcooled Flow Boiling A number of correlation methods have been proposed to predict the CHF condition for subcooled flow boiling (e.g., Gunther [12.89], Tong [12.90, 12.91], and Celata et al. [12.92]). An example is

598

Liquid-Vapor Phase-Change Phenomena

the following correlation proposed by Celata et al. [12.92], which is a modification of a correlation proposed earlier by Tong [12.90]:

qcrit C ′′ = c (12.78) Ghlv Re 0.5 Cc =  ( 0.216  +  0.0474 P ) ψ



 1 ψ=  0.825 + 0.986 xout



Re =



xout =

( P in MPa)

(12.79)

for xout < −0.1 for 0 > xout > −0.1

(12.80)

GD (12.81) µl

c pl (Tl ,out − Tsat ) (12.82) hlv

This correlation is recommended for P ≤ 5.5 MPa, a mean velocity (G/ρl) of 2.2–40m/s, outlet subcooling (Tsat – T l,out) of 15–190 K, and tube diameters in the range 0.3 ≤ D ≤ 15 mm. Note that the critical heat flux depends on the exit quality, which depends on the flow and rate of heat input. Thus, an iterative approach may be needed to apply the correlation method. Celata and Mariani [12.74] indicate that use of a heat balance method to link heat input to exit conditions is preferred. The correlation methods discussed above are generally empirical relations based on fits to CHF data. An alternate approach, adopted by a number of researchers, is to develop a model of the CHF mechanism and use the model as a tool to predict the CHF condition. Models of the subcooled flow boiling CHF have been based on several postulated mechanisms. Although these models all generally consider the effect of vapor generation on transport in the near-wall region, each type has a slightly different interpretation of the critical mechanism. A summary description of these types of models is provided below: The liquid layer superheat model. This model, which has been developed and explored by Tong et al. [12.93], treats transport across the bubbly layer near the wall (Fig. 12.19b) as the key CHF mechanism. When transport decreases to a critical level, superheat builds up in the layer, resulting in rapid vaporization and leading to vapor blanketing of the surface. Boundary layer separation. In this model, the generation of vapor at the wall is modeled as fluid injection into the wall boundary layer. The CHF condition is postulated to correspond to the injection rate that results in separation of the wall boundary layer. Several investigators have explored this type of model, including Kutateladze and Leontiev [12.94], Tong [12.95], Fiori and Bergles [12.96], Hino and Ueda [12.97], and van der Molen and Galjee [12.98]. The liquid flow blockage model. This model attributes the CHF condition to the inability of the flow to transport liquid from the bulk to the wall. The CHF condition is taken to correspond to a critical flow resistance associated with either the incipient instability of the liquid-vapor interface (Smogalev [12.99]) or the kinetic energy of vapor motion being insufficient to depart the wall region and allow liquid to reach the wall (Bergel’son [12.100]). The near-wall bubble crowding model. This model attributes the CHF to bubble crowding near the wall which prevents cooler bulk liquid from reaching the wall, leading to the onset of vapor blanketing. In this type of model it is often postulated that this CHF condition corresponds to a critical void fraction in the bubbly layer near the wall, at which liquid flow to the wall diminishes to a level that leads to vapor blanketing. The critical void fraction is typically taken to be about 0.82. This type of model has been explored by Hebel et al. [12.101] and Weisman and co-workers [12.102, 12.103]. Celata and Mariani [12.74] note, however, that

Convective Boiling in Tubes and Channels

599

recent experimental results suggest that the wall region void fraction at CHF varies from as low as 0.3 to as high as 0.95, which makes the assumption of a single critical value suspect. The liquid sublayer dryout model. This type of model considers the vaporization of a thin liquid layer under an elongated bubble (Fig. 12.19c) or a vapor mass formed by merging bubbles near the wall. The CHF condition is taken to correspond to conditions that lead to complete evaporation of the liquid film in the time it takes for the bubble to pass by the wall location of interest. A number of recent modeling efforts have focused on the liquid sublayer dryout mechanism as the key CHF mechanism (Lee and Mudawar [12.104], Katto [12.105–12.107], and Celata et al. [12.108]. Variations of this type of model have been shown to agree well with CHF data for water at high and low pressure, and nitrogen, R-11, R-12, and R113.

CHF Correlations and Models for Saturated Flow Boiling For saturated flow boiling, the CHF transition at low quality may be a departure from nucleate boiling, whereas for moderate to high quality, where annular flow is often encountered, the CHF condition is more likely to be due to dryout of the liquid film. In general, correlation methods applicable to saturated flow boiling usually are developed to account for either type of CHF mechanism. As noted above, a very large number of correlation methods for predicting the CHF condition have been described in the literature. Discussion of all such correlations here is clearly impossible. However, a representative sample of these correlations will be described in this sub-section. Building on earlier investigations by Thompson and Macbeth [12.67], Bowring [12.109] proposed the following empirical correlation for the CHF condition:

A′ + 0.25GD(hˆl ,sat − hˆin ) (12.83) C′ + z

qcrit ′′ =

where

0.579 FB1GDhlv (12.84a) 1.0 + 0.0143FB 2 D1/ 2 G 0.077 FB 3GD C′ = (12.84b) 1.0 + 0.347 FB 4 (G /1356)n A′ =

n = 2.0 − 0.00725P (12.84c)



In these relations, qcrit ′′ is in W/cm2, D is the internal tube diameter in meters, z is the distance downstream of the inlet in meters, G is the mass flux in kg/m2s, hlv is the latent heat of vaporization in J/kg, and the pressure P is in bar. In Eq. (12.83), hˆl ,sat is the enthalpy of saturated liquid at the local pressure P and hˆin is the inlet enthalpy. The correlation parameters FB1, FB2, FB3, and FB4, (for water) were taken to be functions of pressure. Variations of these parameters recommended for use with this correlation scheme are shown graphically in Fig. 12.24. Values of these correlation parameters can be computed using the following empirical relations: p = P /69 ( P in bar) (12.85)

for p < 1:

FB1 =

[ p 18.942 exp {20.8(1 − p )}] + 0.917 (12.86a) 1.917

600

Liquid-Vapor Phase-Change Phenomena

FIGURE 12.24  Pressure dependence of constants in Bowring’s [12.109] CHF correlation for water.



FB 2 = FB 3 =

1.309 FB1 (12.86b) 1.316  [p exp {2.444(1 − p )}] + 0.309

[ p 17.023 exp {16.658(1 − p )}] + 0.667 (12.86c) 1.667 FB 4 = FB 3 p 1.649 (12.86d)

for p > 1:

FB1 = p −0.368 exp {0.648(1 − p )} (12.87a)



FB 2 = p −0.448 exp {0.245(1 − p )} (12.87b)



FB3 = p 0.219 (12.87c)



FB 4 = FB 3 p 1.649 (12.87d)

This correlation is based on a fit to data in the ranges 136 ≤ G ≤ 18,600 kg/m2s, 2 ≤ P ≤ 190 bar, 2 ≤ D ≤ 45 mm, 0.15 ≤ z ≤ 3.7 m. Collier [12.2] reports that the RMS error for this correlation is 7%. He further notes that extrapolation outside the range of parameters indicated above is not recommended. Biasi et al. [12.110] proposed the following CHF correlation scheme for vertical upflow boiling of water in uniformly heated tubes:



qcrit ′′ = qcrit ′′ =

1883  FBi ( P) − x  m 1/6  1/6 D G  G 

for low qualities (12.88a)

3780 H Bi ( P) [1 − x ] D m G 0.6

for high qualities (12.88b)

601

Convective Boiling in Tubes and Channels

where

m = 0.4

for D ≥ 1cm (12.89a)



m = 0.6

for D < 1cm (12.89b)



FBi ( P )  =  0.7249  +  0.099 P exp ( −0.032 P ) (12.90a) H Bi ( P) = −1.159 + 0.149 P exp(−0.019 P) +

8.99 P (12.90b) 10 + P 2

In these dimensional relations, qcrit ′′ is in W/cm 2 , D is in cm, P is in bar, and G is in g/cm 2s. The correlation is reported to be valid over the ranges 10 < G < 600 g/cm 2s, 2.7 < P < 140 bar, 0.3 < D < 3.75 cm, 20 < z < 600 cm, ρv/(ρl – ρv) < x < 1. Operationally, one calculates the critical heat flux using both Eqs. (12.88a) and (12.88b) and the larger of the two is taken as the correct value. The RMS error of this correlation against a database of over 4500 points was reported to be 7.3%. For upflow boiling of saturated water in a vertical tube with constant applied heat flux, Levitan and Lantsman [12.87] recommended the following correlation for burnout of the first kind (DNB) in an 8 mm diameter tube:

1.2{[0.25( P − 98)/98]− x} 2   P  P   G  e −1.5 x (12.91) qcrit ′′ = 10.3 − 7.8   + 1.6       98   98    1000  

For burnout of the second kind (dryout) they recommend the following relation for predicting the dryout quality for an 8 mm tube:

−0.5 2 3  P P P  G  x crit =  0.39 + 1.57   − 2.04   + 0.68     (12.92)   98   98   98    1000  

In these relations, qcrit ′′ is in MW/m2, P is the pressure in bar, and G is in kg/m2s. Equation (12.91) may be extended into the subcooled range by using negative qualities defined as x = (hˆ − hˆl ,sat )/hlv Equations (12.75) and (12.77) are used to obtain critical values for other tube diameters. Equation (12.91) was reported to be accurate to ± 15% for 29.4 ≤ P ≤ 196 bar and 750 ≤ G ≤ 5000 kg/m2s. Equation (12.92) was stated to predict xcrit values accurate to within ±0.05 for 9.8 ≤ P ≤ 166.6 bar and 750 ≤ G ≤ 3000 kg/m2s. The correlations discussed above are all for flow boiling of water. There is a need, however, for CHF prediction methods for other fluids as well. This need stems from (1) a desire to accurately design evaporators for boiling applications involving other coolants such as fluorocarbons and hydrocarbons, and (2) a desire to model large-scale boiling heat transfer equipment using water with smaller experimental simulations using other fluids. Because of the complexity of most flow boiling systems, there are a number of dimensionless groups that potentially may affect the critical heat flux conditions. The interested reader is referred to the early report by Barnett [12.111], the papers by Ahmad [12.112] and Ishii and Jones [12.113], and the summary of Tong and Tang [12.114] for a more in-depth discussion of dimensional analysis and modeling of critical heat flux phenomena. The Katto and Ohno [12.115] CHF correlation for vertical uniformly heated tubes is design to be applicable to a wide variety of fluids and conditions. This correlation is an updated version of a correlation proposed earlier by Katto [12.116, 12.117]. The updated version attempts to correct some

602

Liquid-Vapor Phase-Change Phenomena

inconsistencies and improve the accuracy of the earlier version. This correlation takes the form of the following algorithm:

γ=

ρv (12.93) ρl



We k =

G2 L (12.94) ρl σ



for L /D < 50 : CK = 0.25 (12.95a)



 L  for 50 ≤ L /D ≤ 150 : CK = 0.25 + 0.0009   − 50  (12.95b)   D  



for L /D > 150 : CK = 0.34 (12.95c)



L qcol ′′ = CK Ghlv We −K0.043   (12.96a)  D



1   qco2 ′′ = 0.10Ghlv γ 0.133 We −K1/3   (12.96b) 1 + 0.0031( L / D )  



  ( L /D)0.27 qco3 ′′ = 0.098Ghlv γ 0.133 We −K0.433   (12.96c) 1 + 0.0031( L / D )  



1   qco4 ′′ = 0.0384Ghlv γ 0.60 We −K0.173  −0.233  (12.96d)  1 + 0.280We K ( L /D) 



 ( L /D)0.27  qco5 ′′ = 0.234Ghlv γ 0.513 We −K0.433   (12.96e)  1 + 0.003( L /D) 

−1



K Kl =

1.043 (12.97a) 4CK We −K0.043

5 0.0124 + D /L K K 2 =   0.133 −1/3 (12.97b)  6 γ We K K K 3 = (1.12)

1.52We −K0.233 + D /L (12.97c) γ 0.6 We −K0.173

For γ K K 2 : K K = K Kl (12.101a)



for K K 1 ≤ K K 2 : K K = the smaller of {K K 2 , K K 3 } (12.101b)

603

Convective Boiling in Tubes and Channels

After executing the above calculations, the critical heat flux qcrit ′′ is computed as  hˆ − hˆin  qcrit ′′ = qco ′′  1 + K K l ,sat  (12.102) hlv  



The recommended parameter ranges for this correlation are: 0.01 m ≤ L ≤ 8.8 m, 0.001 m ≤ D ≤ 0.038 m, 5 ≤ L/D ≤ 880, 0.00003 ≤ ρv/ρl ≤ 0.41 and 3 × 10 –9 ≤ σ ρl/ (G2L) ≤ 2 × 10 –2. This correlation method has been found to agree reasonably well with available data for flow boiling of water, R-12, R-22, and liquid helium for tube diameters near 1 cm, heated lengths near 1 m, and mass flux levels from 120 to 2100 kg/m2s. Additional CHF prediction methods have also been proposed by Shah [12.118, 12.119], Levy et al. [12.120], and Groeneveld et al. [12.121]. The correlation of Groeneveld et al. [12.121] is a simple tabulated correlation cast in terms of the following dimensionless variables:

Bo =

qcrit ′′ , Ghlv

1/ 2

 G2 D  ρl , ψG =  (12.103) ρv  ρl σ 

This relatively simple correlation provided a reasonably good fit to CHF data for water, R-11, R-12, R-21, R-113, R-l 14, CO2, and N2. Based on extensive experimental data, Shah [12.119] developed a CHF correlation that relates a boiling number at CHF, BoCHF , to a dimensionless parameter Y defined as

Y=

GDc pl  ρl2 gD  kl  G 2 

−0.4

0.6

 µl   µ  (12.104) v

The correlation is organized into two versions: an upstream conditions correlation and a local conditions correlation. Upstream condition version:

BoCHF =

 D qCHF ′′ = 0.124    Le  Ghlv

−.89

n

 10 4   Y  (1 − xin ) (12.105)

For subcooled inlet conditions, the negative equivalent inlet quality xin = xin ,eq is computed as

xin ,eq = −

c pl (Tsat − T f ,in ) (12.106) hlv

and Le equals the distance L downstream from inlet. If the inlet quality is greater than zero, xin is set to zero and Le is equal to the boiling length computed as

Le = L +

Dxin ,eq (12.107) 4BoCHF

The exponent n is determined as follows:

For all fluids when Y ≤ 10 4 , n =  0 (12.108a)



For helium, with Y > 10 4 , n = ( D /Le )0.33 (12.108b)

604



Liquid-Vapor Phase-Change Phenomena

 ( D / L )0.54 for Y ≤ 10 6 e  For other fluids, when with Y > 10 4 , n =  (12.108c) 0.12 6  (1 − x )0.5 for Y > 10 in 

Local condition version: In this version, Lc is the downstream distance from the inlet and

BoCHF =

qCHF ′′ = Fe Fx Bo0 (12.109) Ghlv

where

Fe = 1.54 − 0.32( Lc /D) (12.110)



  Bo0 = 15Y −0.612   Bo0 = max  Bo0 = 0.082Y −0.3 (1 + 1.45Pr4.03 )  (12.111)   −0.105 (1 + 1.15Pr3.39 )   Bo0 = 0.0024Y 

and Fx is determined differently in two ranges of local quality xeq : for xeq < 0: b



(1 − F2 )( Pr − 0.6)  Fx = F1 1 − (12.112) 0.35  

where

F1 = 1 + 0.0052(− xeq )0.88 Y 0.41 (12.113)

and Y is taken to be 1.4 × 10 7 when the computed value for Y exceeds this value, and



 F −0.42 for F ≤ 4  0 for Pr ≤ 0.6 1  1 ,b= F2 =  (12.114)  0.55 for F1 > 4  1for Pr > 0.6

For xeq ≥ 0: c



 ( F −0.29 − 1)( Pr − 0.6)  Fx = F3 1 + 3  (12.115) 0.35  

where

 1.25 × 10 5  F3 =   Y  

0.833 xeq

 0 for Pr ≤ 0.6 ,c= (12.116)  1for Pr > 0.6

Shah [12.119] recommends that the upstream condition correlation always be used for helium, and be used for other fluids when Y ≤ 10 6 , or the downstream distance L (in meters) is greater than 160 / Pr1.14 . For all other conditions, the lower value of Bo predicted by the two models should be

Convective Boiling in Tubes and Channels

605

used. Shah’s [12.119] CHF correlation is recommended for water, potassium, cryogens, refrigerants, and hydrocarbons over the ranges

0.32 < D < 37.8 mm



0.0014 < Pr < 0.961



4 < G < 2.9 × 10 5 kg/m 2 s



0.11 < qw′′ < 4.5 × 10 4 kW/m 2



1.3 < L /D < 940



−4.0 < xeq ,in < 0.81



−2.6 < xeq ,CHF < 1.0

Although based on a wide collection of data, Katto [12.107] has noted that the predictions of this exhibit strong gravity dependence at high mass flux levels, which is unexpected on physical grounds.

Modeling of DNB and Dryout For saturated flow boiling, the CHF transition at low quality may be a departure from nucleate boiling, whereas for moderate to high quality, where annular flow is often encountered, the CHF condition most likely is due to dryout of the liquid film. CHF models for DNB in saturated flow boiling, are generally similar to those described above for subcooled flow boiling. They include models of dryout under bubbles or vapor slugs, near-wall bubble crowding, or liquid sublayer dryout theory. The key difference is that at saturation, vapor is not confined to the near-wall region. Explorations of models of these types of mechanisms include studies by Weisman and Pei [12.102], Weisman and Ying [12.103], Lin et al. [12.122], Lee and Mudawar [12.104], Katto [12.107], and Celata et al. [12.123]. In addition, empirical CHF correlations for flow circumstances relevant to nuclear reactor design have been proposed by, for example, Tong [12.91, 12.124, 12.125]. As mentioned above, for saturated conditions at moderate to high qualities, the critical heat flux condition most often corresponds to dryout of the liquid film on the tube wall. As suggested by Figs. 12.4 and 12.5, this variation of the critical heat flux condition is encountered at low heat flux and/or low wall superheat conditions. Perhaps the most common application where such conditions are encountered is in the tubes of evaporators used in air-conditioning and refrigeration systems. In such systems, the two-phase flow leaving the expansion valve generally enters the evaporator at about 20–30% quality and the flow leaving the unit is usually near 100% quality or slightly superheated. Dryout is generally encountered somewhere along the refrigerant flow path in the evaporator. To predict the conditions at which dryout occurs, one approach would be to model the annular film flow evaporation process in the manner described in Sections 10.4 and 12.4 and computationally predict the conditions at which the film thickness or the film flow rate goes to zero. As indicated in previous sections, this would require analysis of the mass and momentum transport in the core flow and the liquid film on the tube wall and transport of heat across the liquid film. As noted in Section 10.4, this usually requires including the effect of entrainment of liquid droplets in the vapor core. The condition at which the film dries out is largely dictated by the transport in the core and the film regions and the interaction of the mechanisms of deposition, entrainment, and vaporization. This approach to predicting the dryout condition is plausible because, at low to moderate heat flux levels, the entrainment and deposition mechanisms apparently are largely unaffected by the presence of the applied heat flux. This makes it possible to apply the entrainment/deposition models developed for adiabatic equilibrium annular flow to annular flow with evaporation of the liquid film. An analysis of this type proceeds by integrating the transport equations in the core flow and

606

Liquid-Vapor Phase-Change Phenomena

in the film along the channel from known or assumed boundary conditions at the starting point, to the point where the film thickness goes to zero. Dryout is thus determined as the point where the processes of evaporation, deposition, and entrainment lead to a condition in which the film flow rate becomes zero. Analytical treatments of this type have been explored in depth by Whalley, Hewitt, and coworkers [12.126–12.129]. The results of these investigations indicate that this type of method can be used successfully to predict dryout conditions for forced convective boiling in tubes, annuli, and rod bundles. For a further discussion of this method of predicting dryout conditions, the interested reader may wish to consult references [12.114, 12.130–12.132].

Other Factors The discussion of CHF conditions in this section has focused on the relatively idealized circumstances of vertical upward flow boiling in a round tube with a uniform applied heat flux. Obviously, the conditions of interest in applications involving flow boiling may deviate in a number of ways from these conditions. Deviations from the uniformly heated, round tube, upward flow circumstances are expected to alter the CHF conditions somewhat. Some of the previous investigations of how these deviations affect the CHF were mentioned earlier in this section. In addition, other investigations have examined the effects of a non-uniform heat flux profile (axially and circumferentially) [12.133–12.138], step changes in heat flux [12.139], noncircular geometries [12.140–12.147], pulsating or transient flow [12.148–12.151], and channel orientation [12.151]. Further information on these specialized aspects of CHF phenomena may be obtained in the indicated references or in the summary articles by Bergles [12.66], Hewitt [12.131], Katto [12.153], and Tong and Tang [12.114]. While the studies cited above provide some insight into the nature of CHF transitions, many aspects of CHF phenomena are still not well understood, particularly in circumstances that deviate from the upward flows in large round tubes that have been the focus of most previous investigations in this area. Continued research in this area is clearly needed to fill the gaps in the current knowledge of this aspect of flow boiling processes. Example 12.5 Saturated flow boiling of water at low applied heat flux occurs in a vertical round tube with an inside diameter of 1.2 cm. The flow rate is such that the mass flux is 2000 kg/m2s, and the pressure along the tube is virtually uniform at 3773 kPa. Use the correlation of Levitan and Lantsman [12.77] to determine the dryout quality for these conditions. For an 8 mm tube, the correlation of Levitan and Lantsman [12.77] is given by Eq. (12.92):

2 3 −0.5   P   G   P   P  ( xcrit )8mm = 0.39 + 1.57   − 2.04   + 0.68       98    1000   98   98   

where P and G must be in bar and kg/m2s, respectively. Here

 1.0  = 37.4bar P = 3773   101

Substituting into the above equation yields



2 3 −0.5   37.4    2000   37.4   37.4  + 0.68  − 2.04  ( xcrit )8mm = 0.39 + 1.57        98    1000   98   98    = 0.516

607

Convective Boiling in Tubes and Channels Using Eq. (12.77) to correct for the diameter effect yields

 8 xcrit = ( xcrit )8mm    D



 8 xcrit = 0.516    12 

0.15

(D in mm)

0.15

= 0.486

Thus dryout is predicted to occur at a quality of about 49%.

Example 12.6 Flow boiling of R-134a at low applied heat flux occurs in a vertical round tube with a diameter and length of 1.1 cm and 30 cm, respectively. The pressure is virtually uniform along the tube at 338 kPa. The mass flow rate is such that the mass flux is 500 kg/m2s. Use the correlation of Katto and Ohno [12.115] to predict the critical heat flux for these circumstances if (a) the inlet condition is saturated liquid and (b) the inlet flow is subcooled by 20°C. For saturated R-134a at 338 kPa, Tsat = 277.2 K = 4.0°C, ρl = 1281 kg/m3, ρv = 16.6 kg/m3, hlv = 195.5 kJ/kg, cpl = 1.352 kJ/kgK, and σ = 0.0110 N/m. a. Using the embodiment of the Katto and Ohno correlation in Eqs. (12.93)–(12.102), we proceed as follows:

γ=



WeK =

ρv 16.6 = = 0.0130 ρl 1281

G 2L (500)2(0.30) = = 5322 ρl σ 1281(0.0110)

L/D = 30/1.1 = 27.3, which implies that CK = 0.25.



 L qco1 ′′ = CK Ghlv WeK−0.043    D

−1

= (0.25)(500)(195.5)(5322)−0.043(27.3)−1 = 619 kW/m 2 1   qco2 ′′ = 0.10Ghlv γ 0.133WeK−1/ 3    1+ 0.0031(L /D) 

= 0.10(500)(195.5)(0.0130)0.133 (5322)−1/ 3[1+ 0.0031(27.3)]−1 = 290 kW/m 2   (L /D)0.27 qco3 ′′ = 0.098Ghlv γ 0.133WeK−0.433    1+ 0.0031(L /D) 



  (27.3)0.27 = 0.098(500)(195.5)(0.0130)0.133 (5322)−0.433   + 1 0.0031 (27.3)   = 295 kW/m 2 Because qco1 ′′ > qco2 ′′ and qco2 ′′ < qco3 ′′ , qco ′′ = qco2 ′′ = 290 kW/m 2.



KK 1 =

1.043 1.043 = = 1.51 4CK WeK−0.043 4(0.25)(5322)−0.043

0.0124 + 1/ 27.3  5  0.0124 + D /L  5  KK 2 =   0.133 =  = 1.22  6 γ  6  (0.0130)0.133(5322)−1/ 3 WeK−1/ 3

608

Liquid-Vapor Phase-Change Phenomena Because KK1 > KK2, KK = KK1 = 1.51. For saturated liquid at the inlet, Eq. (12.102) reduces to qcrit ′′ = qco ′′



And it follows from the above results that for these conditions, q′′crit = 290 kW/m 2



b. Evaluation of q′′co for the subcooled inlet condition is identical to the saturated case. However, for the subcooled case,  hˆ − hˆin  qcrit ′′ = qco ′′  1+ KK l ,sat  hlv  



Computing hˆl ,sat − hˆin as cpl(Tsat – Tin), this becomes c pl (Tsat − Tin )   qcrit ′′ = qco ′′  1+ KK   hlv

Substituting yields

(1.352)(20)   2 q′′crit = 290  1+ (1.51)  = 351 kW/m  195.5 



Thus the 20°C subcooling is estimated to increase the critical heat flux by about 18%.

12.6  POST-CHF INTERNAL FLOW BOILING As indicated in Figs. 12.4 and 12.5, if the flow boiling process exceeds the CHF condition, the boiling process may enter the transition boiling, film boiling, or mist evaporation regime. Usually, heat transfer equipment is designed so that transition boiling and film boiling are avoided because they result in lower heat transfer performance, and because in some cases the associated high wall temperatures may damage the walls of the flow passages in the unit. Mist flow evaporation is often encountered in the latter stages of vaporization processes at low to moderate heat flux or wall superheat levels. It may be particularly important in refrigeration and air-conditioning evaporators where complete vaporization of the working fluid is usually desired. Each of the three possible regimes of vaporization are discussed in this section.

Transition Flow Boiling Of the three regimes considered in this section, transition boiling is perhaps the least understood. In order to encounter this regime, the wall temperature of the passage must be controlled in the physical system so that it remains in the transition boiling range. While this regime is rarely encountered under the normal operating conditions, it may potentially arise during loss-of-coolant accident scenarios for nuclear power plants. Reflooding the core of the reactor with coolant may produce the forced-flow version of the quenching process described in Chapter 7 (for pool boiling). Because of the need to understand how transition flow boiling may affect emergency core cooling, many of the studies of flow transition boiling have been conducted in connection with nuclear power applications. Experimental investigations of forced convection transition boiling have been conducted by McDonough et al. [12.154], Iloeje et al. [12.155], and Cheng and Ng [12.156]. Early work on forcedflow transition boiling up to 1976 has been summarized by Groeneveld and Fung [12.157]. Attempts

Convective Boiling in Tubes and Channels

609

to develop heat transfer correlations for forced-flow transition boiling were made by McDonough et al. [12.154], Tong [12.124], and Ramu and Weisman [12.158]. Tong and Young [12.159] proposed the following correlation for transition boiling of water

1+ 0.00288( Tw − Tsat )   x 2/3  Tw − Tsat  qtb′′ = q ′′fb + qnb ′′ exp −0.0394 e  (12.117)   dxe / dz 55.6  

where qtb′′ is the total transition boiling heat flux, qnb ′′ is the nucleate boiling heat flux based on the instantaneous local conditions, q ′′fb is the film boiling heat flux based on the instantaneous local conditions, xe is the equilibrium quality, Tw – Tsat is the wall superheat in °C, and dxe /dz is the quality gradient in m−1. This is a dimensional correlation, and terms must be evaluated in the proper units. Collier [12.160] questioned the manner in which this correlation was derived, but nevertheless acknowledged its usefulness as a predictive tool.

Convective Film Boiling As seen in Figs. 12.4 and 12.5, at low qualities and subcooled conditions, the CHF transition may lead to film boiling at the walls of the tube. Knowledge of the transport for this type of flow condition is relatively limited. For flow boiling of water, which is of central interest to the power industry, to achieve sustained film boiling at high mass flux levels requires wall temperatures so high that most conventional materials would melt. Thus conducting controlled experiments is extremely difficult for such circumstances. It is not surprising, therefore, that few extensive experimental investigations of convective film boiling have been conducted. Experimental investigations by Dougall and Rohsenow [12.161] and others indicate that for convective film boiling at low to moderate flow rates, the flow takes on the so-called inverted annular flow configuration shown in Fig. 12.25 (regular annular flow corresponds to a liquid film on the wall with vapor flowing in the core). The vapor film along the tube wall is generally not smooth but exhibits irregularities at random locations. This behavior is similar to the irregular film behavior for buoyancy-driven film boiling on a vertical surface described in Chapter 7. These irregular bulges

FIGURE 12.25  Transition to internal flow film boiling.

610

Liquid-Vapor Phase-Change Phenomena

FIGURE 12.26  Simplistic model of inverted annular flow internal film boiling.

or bubbles in the film tend to retain their identity, moving downstream at a velocity slightly slower than the peak velocity in the vapor film. A simplistic model of inverted annular upflow film boiling can be constructed using an approach similar to the analysis described in Section 11.2 for film condensation. To do so, the flow would be idealized as having a smooth interface and a uniform film thickness as shown in Fig. 12.26. The film boiling analysis follows the same steps as the condensation analysis in Section 11.2, except for changes to invert the role of liquid and vapor, reverse the direction of heat flow, and modify the model to upward flow. The force momentum balance on the differential element of the film shown in Fig. 12.26 is embodied in the equation −ρv g(δ − y)π( D − 2δ)dz − (δ − y)π( D − 2δ)

dP dz + τi π( D − 2δ)dz dz

 du                                       − µ v   π( D − 2 y)dz = 0                 dy 

(12.118)

Because the vapor film is thin (δ 1 (12.153)

2

∑b x

i i e

for 0 ≤ ψ ≤

i=0

π (12.154) 2

(if computed ψ < 0, set ψ = 0; if computed ψ > π/2, set ψ = π/2)

hˆv ,a − hˆv ,e = exp(− tan ψ ) (12.155) hlv



hˆv ,e = hˆv ,sat  for 0 ≤ xe ≤ 1, hˆv ,e = hˆv ,sat + ( xe − 1)hlv  for  xe > 1 (12.156)

Equations (12.150) and (12.151) are consequences of conservation of energy and the definitions of the actual and equilibrium qualities. Equations (12.152)–(12.155) embody an empirical correlation for the degree of non-equilibrium in the flow. In Eq. (12.155), hˆv ,a is the actual vapor enthalpy and hˆv ,e is the equilibrium vapor enthalpy. Values of the constants ai and bi were determined to provide a best fit to a database of over 1400 data points for water. The recommended values of these constants are given below



a1 = 0.13864 a2 = 0.2031 a3 = 0.20006 a4 = −0.09232

b0 = 1.3072 b1 = −1.0833 b2 = 0.8455

To compute the local wall temperature for a specified wall heat flux, the Groeneveld and Delorme [12.176] correlation is used in the following manner: 1. Assuming the local equilibrium quality xe and mass flux are given, a value of the film temperature Tv,f is guessed so that properties may be evaluated. 2. Equations (12.152)–(12.154) are used to determine Rehom and ψ, which are then substituted into Eqs. (12.155) and (12.156) to determine hˆv ,a . 3. Using the value of hˆv ,a determined in step 2, Eq. (12.150) is used to determine xa, and Eq. (12.151) is iteratively solved to determine Tv,a. 4. Using the values of hˆv ,a and xa determined in step 3, Eq. (12.149) can be solved for the wall temperature Tw for the specified heat flux. 5. For the computed values of Tv,a and Tw, a new value of the film temperature is computed as the arithmetic average of the two. If this value agrees with the previous guess of Tv,f, the computation is complete. If not, a new value of Tv,f is guessed and the process is repeated beginning with step 2. Although iteration is required, this correlation procedure is straightforward and can be easily programmed on a computer. While this correlation accounts for non-equilibrium conditions, it does not account for direct transfer of heat from the wall to liquid droplets or for radiation heat transfer. These investigators argue, however, that these effects are small for many circumstances of practical interest. They specifically cite the visual observations of the post-dryout region made by Cumo et al. [12.184], which suggest that droplet collisions with the wall were infrequent, and contributed little to

Convective Boiling in Tubes and Channels

621

FIGURE 12.28  Comparison of the post-dryout wall temperature data of Era [12.185] with wall temperature variations predicted by two correlation techniques. (Adapted from [12.176] with permission, copyright © 1976, Elsevier Science Publishers.)

the overall heat transfer. They also note that the droplet velocity to vapor velocity ratios determined experimentally by Cumo et al. [12.184] were typically close to one, supporting the homogeneous flow idealization implicit in this model. Cumo et al. [12.184] also argued that calculations indicate that the radiative heat flux component is very small compared to the convective component except for low flow rates and very high surface temperatures. The generally good agreement of the predictions of this correlation with available data supports arguments regarding the validity of the idealizations in this model. In using it, however, one must be sure that the conditions of interest conform to the assumptions in this model. Figure 12.28 shows a comparison of the predicted wall temperature variation with equilibrium quality and measured data obtained by Era [12.185]. Also shown is the variation predicted by Groeneveld’s [12.170] empirical correlation described by Eqs. (12.145) and (12.146). It can be seen that the non-equilibrium correlation exhibits trends similar to the data, whereas the empirical correlation generally does not, particularly at equilibrium qualities greater than 1. Empirical correlations that account for thermodynamic non-equilibrium have also been developed by Chen et al. [12.186], Jones and Zuber [12.187], and Plummer et al. [12.188]. Yoo and France [12.189] have proposed updated constants for the correlation of Plummer et al. [12.188] that fit data for water and refrigerants.

622

Liquid-Vapor Phase-Change Phenomena

The Model of Ganic and Rohsenow In contrast to the model of Groeneveld and Delorme [12.176], the model proposed by Ganic and Rohsenow [12.178] attempts to account for more of the mechanisms. The total heat flux was postulated to consist of contributions due to vapor convection qv′′ , droplet heat transfer qd′′ , and radiation qr′′ :

q ′′ = qv′′ + qd′′ + qr′′ (12.157)

To predict the convective contribution, the McAdams correlation was specified

 k   GxD  qv′′ = 0.023   v    D   αµ v 

0.8

Prv0.4 (Tw − Tsat ) (12.158)

in which properties were evaluated at saturation conditions. Note that in the above definition of the vapor Reynolds number, α is the void fraction. The radiation contribution was postulated to be given by the sum of radiation from the surface to the liquid drops and the vapor:

4 4 qr′′ = Fwl σ SB (Tw4 − Tsat ) + Fwv σ SB (Tw4 − Tsat ) (12.159)

Implicit in the above relation is the assumption that the vapor and the liquid are at the saturation temperature. To evaluate the view factors Fwl and Fwv, the method described by Sun et al. [12.190] for a dispersed system was recommended. For the specific circumstances of convective boiling of nitrogen, Ganic and Rohsenow [12.166] concluded that the radiation flux contribution was negligible. When comparing the predictions of their model with experimental data for nitrogen, qr′′ was therefore taken to be zero. A key element of this model is its very detailed analytical treatment of the wall to droplet heat transfer process. The Groeneveld and Delorme [12.176] model neglects this mechanism, which is appropriate for high-quality conditions where the liquid inventory is low and droplet collisions with the wall are infrequent. At lower qualities, where more liquid is present in the flow, liquid is likely to contact the wall more frequently, and the importance of this mechanism to the overall heat transfer is expected to be greater. Also, since more liquid and less vapor is present at lower qualities, there is a reduced tendency for the vapor to superheat, since it contacts liquid more frequently. Ganic and Rohsenow’s model, which neglects non-equilibrium effects but includes the effects of heat transfer to the droplets, is therefore most directly applicable to lower quality flows. Based on a detailed model of the interaction between liquid drops and the wall, Ganic and Rohsenow [12.178] proposed the following relation for its contribution to the total heat flux:



  T  qr′′ = v0 (1 − α)ρl hlv fcd exp 1 −  w    Tsat 

2

  (12.160) 

where v0 is the droplet deposition velocity, and fcd is the cumulative deposition factor. Based on an analysis of the deposition process, the following relation was developed to predict v0:

v0 = 0.15

Gx ρl α

fv (12.161) 2

623

Convective Boiling in Tubes and Channels

where f v is the friction factor computed from a correlation for single-phase flow at the vapor Reynolds number GxD/αμv. The cumulative deposition factor fcd was shown in the analysis to be a function only of the ratios ac /a and am /a , where ac is the critical droplet radius, am is the maximum droplet radius, and a is the mean droplet radius. In any dispersed flow, a range of droplet sizes may exist. The maximum size am is dictated by the critical Weber number, at which dynamic pressure forces in the vapor flow overcome surface tension forces, leading to the break up of liquid droplets. Based on arguments of this type, it was recommended that am be computed as am =



7.5σ (12.162) ρv (uv − ul )2

where G (1 − x ) (12.163) ρl (1 − α)



uv =

Gx ρv α



α=

x (12.164) x + (ρv / ρl ) S (1 − x )

ul =

In Eq. (12.164), S is the slip ratio uv /ul. Although Ganic and Rohsenow developed a specific correlation for S, they suggested that other correlations, such as those described by Tong and Young [12.159], may also be used. To predict the mean droplet size a , the following empirical correlation was proposed:

a=

0.732 uv − ul

1/ 2

 σ  µv (12.165) ρl  (uv − ul ) Dρv 

This is a dimensional relation in which a and D are in m, uv and ul are in m/s, σ in N/m, ρl and ρv in kg/m3 and μv in Ns/m2. The critical droplet radius ac is a parameter that emerges from analysis of the motion of droplets near the wall. Ganic and Rohsenow [12.178] solved the differential equations governing the motion of droplets near the wall, including the effects of lift and drag forces, buoyancy, and the repelling effect of rapid evaporation on the wall side of the drop. They found that for a given set of flow and wall temperature conditions, droplets below a critical size were returned to the main flow without striking the wall. Above this critical size, droplets did strike the wall. For a typical case in their study, Ganic and Rohsenow [12.178] found that ac was between 20 and 50 μm. Their computed results indicated that, as expected, for a given drop size the tendency to hit the wall was increased as the wall temperature was decreased and as the deposition velocity v0 increased. The computational scheme used to determine ac is rather lengthy and will not be described in detail here. A detailed description of this scheme can be found in reference [12.178]. Once ac is obtained for the circumstances of interest, and am and a are computed using the equations given above, the ratios ac / a and am / a can be computed, and fcd can be computed from an explicit relation given by Ganic and Rohsenow [12.178]. Equation (12.160) can then be evaluated, using Eqs. (12.161) and (12.164), to determine qd′′ . While programming this model into a computer is a straightforward task, it makes prediction of the heat transfer for these circumstances less than convenient. Perhaps the main significance of this model is that it illuminates the physics of the droplet-wall interaction and demonstrates the manner in which these physical mechanisms affect the overall heat transfer. Further discussion of dropletwall interactions may be found in references [12.191–12.194].

624

Liquid-Vapor Phase-Change Phenomena

FIGURE 12.29  Comparison of wall temperature variations in the post-dryout region predicted by three correlation methods.

Comparison of Correlation Methods A comparison of the tube wall temperatures predicted by the correlation methods described in this section is shown in Fig. 12.29 for a typical set of flow conditions. The results shown in this figure were computed for water at 6124 kPa in a 1 cm diameter tube for G = 1200 kg/m2s and q″ = 600 kW/m2. It can be seen that the Groenveld [12.170] and Dougall-Rohsenow [12.161] correlations predict almost identical tube wall variations. In the past, these two correlations have been recommended for use in safety analysis codes by the US Nuclear Regulatory Commission. The temperature variation predicted by the non-equilibrium correlation of Groeneveld and Delorme [12.176] is substantially higher than those for the other equilibrium correlations. Plots of the actual vapor temperature and quality as predicted by the non-equilibrium correlation are shown in Fig. 12.30. It is clear from these plots that this correlation predicts a higher wall temperature because it predicts a significant departure from equilibrium, with the vapor being superheated by as much as 100°C and the actual quality being lower than the equilibrium value. This model further predicts that all droplets will not be evaporated until the (effective) equilibrium quality has reached 1.11. As noted earlier in this section, additional semi-theoretical models of post-dryout heat transfer have been proposed by other investigators. While the models discussed in this section are useful, further experimental investigation of the mechanisms and verification of the predictions of these models are needed before the best of them can be used with confidence over a wide range of conditions. In particular, it is worth noting that most of the experimental work in this area has been done with water. Much less information on post-dryout heat transfer exists for hydrocarbons, refrigerants, and cryogenic fluids. Further general information on post-dryout heat transfer can be obtained from the review articles by Collier [12.160], Mayinger and Langner [12.169], and Chen and Costigan [12.195].

Convective Boiling in Tubes and Channels

625

FIGURE 12.30  Actual quality and vapor temperature variations in the post-dryout regime predicted by the Groeneveld-Delorme correlation [12.176].

Over the past decade, there has been an extensive effort to better understand flow boiling mechanisms in microchannels. This has been stimulated by an interest in enhancing flow boiling heat transfer, and using flow boiling to remove heat at high flux levels to thermally manage microelectronic components. New developments in this area are discussed in detail in the following section.

12.7 INTERNAL FLOW BOILING IN MICROCHANNELS AND COMPLEX ENHANCED FLOW PASSAGES The discussion of internal flow boiling processes in the previous sections of this chapter has focused on flows in macroscale round tubes (i.e., those with hydraulic diameters greater than 3 mm), which are typical of vaporization processes in conventional power systems, air-conditioning, refrigeration, and chemical processing applications. In recent years, there has been increasing interest in convective boiling in ultra-small passages of highly compact evaporators. This type of highly compact design is attractive in a number of applications in which small size and small weight are important advantages. Minimum size and weight are important, for example, in automotive air-conditioning systems, thermal management systems, and fuel supply systems for spacecraft, and electronics cooling. Advances in metal forming and bonding processes, micromachining, and widespread access to MEMS lithographic techniques have made it possible to create microevaporators with flow passage hydraulic diameters that are less than 100 μm. In flow passages that small, the physics of the convective vaporization change significantly from that observed in larger macroscopic tubes. Complex non-circular flow passages also change features of the convective boiling process from those observed in simple round tubes. In this section, we will take an introductory look at how the boiling process is affected by small channel size and the presence of fins or ribs in complex channel geometries.

626

Liquid-Vapor Phase-Change Phenomena

Effect of Reducing Flow Passage Size to the Microchannel Range In earlier sections of this chapter, convective flow boiling at low to moderate wall superheat levels was typically found to consist of a nucleate-boiling dominated regime at low quality and a regime dominated by annular flow evaporation at moderate to high qualities. Another important feature of internal convective boiling is that often the flow is rapidly accelerating, which induces significant pressure drop along the flow passage. Reducing the hydraulic diameter of the flow passage impacts all of these important mechanisms: nucleation and bubble growth, annular flow vaporization, and pressure loss. As discussed in Section 11.4, for condensation, in two-phase flow with phase change in a microchannel, the dimensionless groups that define the flow features are in different ranges than macroscopic flows in more conventional applications. For convective boiling, there are several dimensionless parameters that have been used to characterize the physics of the process, each of which reflects a ratio of mechanisms. Here we will examine the relevant dimensionless parameters to assess how they are affected by reducing the passage hydraulic diameter. Kandlikar [12.196] has argued that surface tension, momentum change during evaporation, viscous shear, and flow inertia are the primary physical effects that define the nature of the convective boiling process. This suggests that dimensionless groups that involve these mechanisms will be of particular interest. The flow rate in microchannel boiling circumstances is generally low by design because pressure loss scales with flow dynamic pressure and keeping the flow inertia down is a compromise to keep the pressure drop in a reasonable range. As a result, any of the following Reynolds numbers Rele =



Rel =





Retp =

GD µl

(for the entire flowas liquid) (12.166)

G (1 − x ) D µl

Re v =



GD µl

GxD µv

(for liquid flowingalone) (12.167) (for vapor flowingalone) (12.168)

(with the two-phase flow µ l a weighted average of µ l and µ v ) (12.169)

are usually small because flow inertia is low and hydraulic diameter is very small. This generally puts the flow in the laminar regime. In flow boiling in conventional macroscopic tubes, the Reynolds number is typically large enough that turbulent flow is likely. Bond and Weber numbers are also generally much lower than macroscopic flows. The Bond number Bo D =



g(ρl − ρv ) D 2 (12.170) σ

quantifies the ratio of gravitational body forces to surface tension forces for a system with dimensions in the size range of the channel cross section. As noted by Kandlikar [12.184], the hydrostatic forces generated over the small cross-sectional dimensions of a microchannel are generally small compared to surface tension, inertia, or viscous shear. As a result, the Bond number in microchannel boiling is much smaller than for boiling in conventional larger tube, and gravitational forces are usually sufficiently small that they can be neglected. This implies that tube orientation is not likely to significantly affect microchannel boiling processes. The capillary number −1



Ca =

 x 1− x  µl G , ρ= + (12.171) ρσ ρl   ρv

627

Convective Boiling in Tubes and Channels

represents the ratio of viscous to surface tension forces. Because both mechanisms are expected to be important in microchannel boiling, this number is expected to a useful system parameter. In recent experimental studies of flow boiling in microchannels Kandlikar [12.196] found that Ca is typically between 10−3 and 10−2, which is slightly higher than the range typical of boiling in larger tubes. The Weber number

We =

G2 D (12.172) ρl σ

can be viewed as an indicator of the ratio of dynamic pressure of the flow (or flow inertia) to surface tension forces (capillary pressure) across interfaces that span the channel. This can be seen more clearly by rearranging the Weber definition to

We = 2

G 2 /ρl (12.173) σ /( D /2)

The Weber number is most often between 1 and 104 in microchannel boiling experiments. This is lower than typical values for boiling in larger channels, but it is a useful parameter because the flow inertia and surface tension forces are expected to be important in microchannel boiling, and they are typically comparable in strength. The Martinelli parameter X, the convection number Co, and the boiling number Bo defined in Section 12.4 also are useful for characterizing flow boiling in microchannels. In addition, Kandlikar [12.196] has proposed two additional dimensionless groups that he recommends as useful parameters for characterizing flow boiling in microchannels. He argued that the evaporating momentum force will be an important mechanism in microchannel boiling. He therefore recommended the following dimensionless group that represents the ratio of evaporation momentum force to flow inertia: 2



 q ′′  ρl K1 =  (12.174)  Ghlv  ρv

This contains the boiling number Bo = q ′′ /Ghlv and density ratio, but the combination of factors in the definition is indicative of the ratio of evaporation momentum force to inertia. Previous experimental studies of flow boiling in microchannels typically had values of this parameter ranging from 10−4 to 0.1. The high end of this range is significantly above that typically found in flow boiling in larger tubes. Kandlikar [12.196] also recommended the following dimensionless group that represents the ratio of evaporation momentum force to the surface tension force: 2



 q ′′  D K2 =   (12.175)  hlv  ρv σ

For previous experimental studies of flow boiling in microchannels, Kandlikar [12.196] reports that K2 typically ranges from about 10−4 to 10−2, which is comparable to the range typical for flow boiling in larger tubes. Since evaporation momentum forces and surface tension are expected to be important mechanisms, it appears that K2 should be a useful additional parameter that can be used to characterize these mechanisms for flow boiling processes in microchannels.

Flow Boiling in Microchannels Over the past three decades, there has been a progressively increasing level of research on flow boiling in microchannels, motivated by interest in enhancing flow boiling heat transfer and potentially

628

Liquid-Vapor Phase-Change Phenomena

using microchannel evaporator heat sinks to cool electronics at high heat flux levels. Early studies [e.g., 12.197–12.218] focused on experimental exploration of boiling heat transfer in microchannels. Subsequent studies have explored the two-phase flow behavior and the differences between adiabatic two-phase flow and two-phase flow with vaporization in microchannels. More recent studies have worked to more fully define the physics, understand the mechanisms, and develop models that can predict heat transfer performance. In general, investigators have found that in some cases, flow through the channel during flow boiling in microchannels was a conventional, nominally steady annular flow, while in others the flow exhibited fluctuations or oscillations. The oscillatory features of convective boiling in microchannels have been linked to nucleation in the channel. The results of recent studies suggest that the onset of nucleation in micropassages may be substantially different from that in larger flow channels. In general, the roughness of the wall in micropassages is expected to be smaller scale than the channel hydraulic diameter, suggesting that the passage walls are likely to be smooth compared to typical macroscale passages. As discussed in Sections 5.7 and 8.3, some earlier studies have suggested that wall-fluid attractive forces may affect the onset of nucleation in some cases, and it has been argued that reflection of pressure waves off opposing walls of microchannels may tend to suppress the onset of nucleation (see Section 5.7). The analysis of the onset of nucleation process described in Chapter 6 implies that the relatively smooth walls of microchannels will require a high wall superheat to initiate nucleation. In some microchannels the walls may be so smooth and so effectively wetted that the onset of boiling is achieve at wall superheats close to that for homogeneous nucleation. Early experimental data obtained by Moriyama et al. [12.216] indicated that for flow boiling of R113 in microchannels, the heat transfer coefficient increases rapidly with quality, at very low quality, leveling off at mid-range qualities and then dropping rapidly with quality at high qualities values beyond the onset of dryout. The data of Moriyama et al. suggest that nucleate boiling is the dominant heat transfer mechanism at low-quality levels. Subsequent studies by Peng and Wang [12.217], Bowers and Mudawar [12.218], and others have documented the strong effect of nucleate boiling on the flow boiling process and defined effects of channel hydraulic diameter, heat flux, and mass flux level. A later study of differences between adiabatic two-phase flow and flow boiling in microchannels by Hetsroni et al. [12.219] documented the effects of nucleation and bubble growth on the flow regime. A key outcome of this research was the finding that vaporizing flows in parallel channels may commonly exhibit instabilities and/or oscillations. In fact, numerous researchers have observed flow boiling instabilities in their microchannel experiments [12.207,12.219–12.226]. As noted above, if only very small cavities exist in the wall surfaces, onset of heterogeneous nucleation will require a high level of wall superheat. In such cases, once bubble growth starts, the high superheat will drive rapid bubble growth with high internal bubble pressure levels. When a bubble first nucleates and starts to grow at the wall of the passage, the pressure inside the bubble is near Psat(Tw). The saturation pressure generally increases rapidly with temperature. Accumulated high superheat thus may produce a high vapor pressure inside the bubble initially, causing the bubble to grow rapidly to a diameter nearly equal to that of the channel. This is depicted in Figs. 12.31a and 12.31b. Experimental observations indicate that the resulting evaporating momentum force may sometimes be large enough to overcome the relatively low dynamic pressure of the flow, and/or the manifold pressure at the channel inlet. When such conditions exist, the bubble may continue growing as a slug bubble upstream and downstream of the nucleation site, stopping, and then reversing, flow in the channel. This is shown schematically in Fig. 12.31c. This type of reversal has, in fact, been observed in microchannels in flow boiling experiments (see, e.g., Steinke and Kandlikar [12.207]). The bidirectional growth of the slug depicted in Fig. 12.31c may slow as growth proceeds and eventually stop as superheat in the liquid is consumed (Fig. 12.31d). The pressure gradient along the channel may then act to move the slug bubble toward the outlet, as shown in Fig. 12.31e. This may effectively create an annular flow along the channel, with the liquid film between the slug bubble interface and the wall evaporating as the slug and liquid film move downstream. Evaporation of the

Convective Boiling in Tubes and Channels

629

FIGURE 12.31  Schematic of bubble growth and motion in a microchannel.

liquid film as the slug moves downstream may lead to dryout of portions of the wall (Fig. 12.31f). This may lead to the onset of sustained dryout, or the wall may be rewet as the slug passes by. If the channel fills with liquid as the slug move downstream, nucleation and growth of a bubble may initiate another cycle of this process. Under the right conditions, the cyclic sequence described above may sustain an oscillatory flow in the channel. On the other hand, Cheng and Wu [12.227] concluded that annular two-phase flow with minimal pressure and temperature fluctuations could be obtained when the inlet subcooling is low. These conditions apparently lead to onset of nucleation a short distance downstream from the entrance to the channel, with the flow sustaining a stable flow of vapor downstream from the onset location. Some investigators have developed models of the instability mechanisms [12.228–12.232], which have enhanced understanding of the causes. Other studies have proposed using artificial nucleation sites, adding inlet restrictors, or adding localized heaters to reduce oscillatory behavior. Figure 12.31 depicts the behavior of a single bubble as it might behave during the onset of boiling when the growth and motion are not affected by other adjacent bubbles. If the frequency of repeated nucleation at a single site is high, or nucleation occurs at multiple sites, the result may be the production of successive slug bubbles that flow in tandem down the passage. They may eventually merge into longer slugs that subsequently evolve into annular flow as they move downstream and the quality increases. Figure 12.32 shows photographs of slug bubble flow and annular flow observed in the study of Warrier and Dhir [12.233] during flow boiling of water in a minichannel. Consistent with the discussion above, the slug bubbles fill the channel except for a thin film on the walls. The flow of successive bubbles is virtually like an annular flow, and, as described above, the successive slug flow can evolve into a conventional annular flow. Experimental studies have now generated a large body of heat transfer data for flow boiling in microchannels [12.218–12.224, 12.234, 12.235] that indicate a strong decreasing trend with increasing quality at low qualities, and mixed parametric trends for mass flux and heat flux at

630

Liquid-Vapor Phase-Change Phenomena

FIGURE 12.32  Photos of [a] slug bubble flow and [b] annular flow during flow boiling of water in a 325 mm long minichannel with a 1.0 mm by 0.5 mm cross section (hydraulic diameter of 750 μm). The exit pressure was fixed at 1.05 bar, the mass flow was 2.15 g/s, and the inlet subcooled was 10°C. The heat flux was varied from 2.1 to 8.86 W/cm2. (From G. R. Warrier and V. K. Dhir, J. Heat Transfer, vol. 126, p. 495, 2004, reproduced with permission, copyright © 2004 American Society of Mechanical Engineers.)

mid-range qualities. Many researchers have noted that their data indicate that flow boiling in microchannels is strongly impacted by heat flux level, suggesting that the process is dominated by nucleate boiling effects. Kandlikar [12.236, 12.237] has pointed out that a moving slug bubble in a microchannel has advancing and receding interfaces that behave similar to interface motion at the base of bubbles during nucleate boiling. This suggests that this may result in heat transfer trends similar to those in nucleate pool boiling. For saturated flow boiling in microchannels, the heat transfer correlation methods described in Section 12.4 generally do not yield accurate predictions of the heat transfer coefficient. There appear to be at least two plausible explanations for this. First, the correlations are constructed for conditions in larger tubes that are likely to produce turbulent flow, whereas convective boiling in microchannels often results in laminar flow. In addition, in microchannel flow boiling, the evaporating momentum force is often strong compared to flow inertia, and convective transport is often weaker than nucleate boiling effects, even at moderate to high quality. As a result, the transport in flow boiling in a microchannel is likely to have a stronger contribution due to nucleate boiling effects than comparable flow in larger tubes. To accurately predict the heat transfer in saturated flow boiling in microchannels, a correlation method must account for the differences described above. A number of microchannel flow boiling heat transfer correlations have been developed to match limited data. An early example is the model of Moriyama et al. [12.216] who combined a slug flow model for low quality with a film flow model for higher quality conditions.

631

Convective Boiling in Tubes and Channels

Based on the models, a correlation methodology was constructed in terms of dimensionless parameters, with the capillary number Ca playing a central role. This modeling agreed with trends for flow boiling of R113 in high aspect ratio channels with minimum dimensions ranging between 35 and 110 µm. Another example is the correlation of Kandlikar [12.196], which extends the larger tube correlation of Kandlikar [12.53] to channels with smaller hydraulic diameters. The Kandlikar correlation [12.53] for macro scale tubes is defined by the relations  hNBD h = the maximum of  (12.176a)  hCBD



where for the nucleate boiling dominant regime ρ  hNBD = 0.6683  l   ρv 



0.1

x 0.16 (1 − x )

0.64

f2 ( Frle )hle

(12.176b)

+1058.0Bo FK (1 − x ) hle 0.7

0.8

and for the convective boiling dominant regime ρ  hCBD = 1.1360  l   ρv 



0.45

x 0.72 (1 − x )

0.08

f2 ( Frle )hle

(12.176c)

+ 667.2Bo0.7 FK (1 − x ) hle 0.8

In this correlation, hle is the single-phase heat transfer coefficient for the liquid phase flowing alone, computed using one of the turbulent correlations defined by Eq. (12.71) or Eq. (12.72). This correlation was developed for larger conventional tubes for flows with moderate to large Reynolds numbers. For flow boiling in microchannels, Kandlikar and Balasubramanian [12.238] recommended the following modifications to this correlation. 1. The function f 2(Frle) is taken to be 1.0 because in small channels, the effect of Froude number is expected to be negligible. The original turbulent-flow correlation proposed by Kandlikar [12.53] for conventional tubes is appropriate in microchannel boiling for Rele > 3000. 2. In microchannels, the Reynolds number for the entire flow as liquid Rele is often low enough that the flow is expected to be laminar. For Rele < 1600, Kandlikar and Balasubramanian [12.238] recommend using the correlation equations (12.68)–(12.70) with f 2(Frle) = 1 and hle computed using the laminar relation

h D  Nu le =  le h  = Cl (12.177a)  kl 

where Cl is a constant that varies with the channel cross section for laminar fully developed flow (Cl = 4.36 for a round tube). For 1600 < Rele < 3000 they recommend interpolating between the computed values for Rele = 1600 and Rele = 3000. 3. For Rele < 100, Kandlikar and Balasubramanian [12.238] argued that the flow boiling mechanism is dominated by nucleate boiling. The forced convection mechanism never

632

Liquid-Vapor Phase-Change Phenomena

contributes significantly to the heat transfer mechanism, even at moderate to high quality. For Rele < 100, they recommend that h be set equal to h NBD for the entire range of quality.



ρ  h = hNBD = 0.6683  l   ρv 

0.1

x 0.16 (1 − x )0.64 hle

(12.177b)

+1058.0Bo FK (1 − x ) hle 0.7

0.8

In this relation, hle is computed using the laminar relation defined by Eq. (12.177a). In Fig. 12.33, the variation of the heat transfer coefficient with quality predicted by this modified correlation is compared with data obtained by Yen et al. [12.239] for flow boiling of R-123 in a stainless tube with an inside diameter of 191 μm. Use of conventional macrochannel correlations predicts heat transfer coefficients that do not agree well with these data. For example, the Kandlikar [12.53] large tube correlation and the Gungor and Winterton [12.49] correlation predict heat transfer coefficient values that are generally higher than those for the microtube data, and they exhibit the wrong trend. These macrochannel correlations predict increasing h with increasing quality, which is contrary to the trend in the data. The modified Kandlikar correlation described above agrees fairly well with these data over most of the quality range. The correlation developed by Liu and Garimella [12.240] is a modification of the macro-channel Chen-type correlation [12.40, 12.44] to account for microchannel effects on flow boiling. As discussed in Section 12.4, this correlation computes the heat transfer coefficient as the sum of a singlephase convective contribution and a nucleate boiling contribution:

h = hsp F + hnb S (12.178)

FIGURE 12.33  Variation of the heat transfer coefficient with quality predicted by the microchannel correlation of Kandlikar and Balasubramanian [12.238]. Also shown is experimental heat transfer data obtained by Yen et al. [12.239].

633

Convective Boiling in Tubes and Channels

Adjusting the model development to account for expected flow conditions in the microchannel, Liu and Garimella [12.240] derived the following relations to compute the single-phase convection and nucleate boiling contributions:



k  Re Pr D hsp = 1.86  l   l l   D  L 



 µl   µ  s

0.14

,

Rel =

G (1 − x ) D (12.179) µl

1 − x   ρv   µ l   for laminar flow (12.180a) X 2 =   x   ρl   µ v  φl2 = 1 +



1/3

( )

F = 2 φl2

1/ 4

 µ tp   µ  l

 q ′′   Rp  hnb = h0     q0′′   Rp 0 

{

0.105

0.133

C 1 + 2 (12.180b) X X

 c p,tp   c  p ,l

1/ 4

 ktp   k  l

3/ 4

Prl0.167 (12.181)

  0.68  2  0.27 pr  (12.182) 1.73 pr +  6.1 + 1 − pr    

}

S = exp 36.57 − 55746/(Rel F 3 ) − 3.4 ln(Rel F 3 ) (12.183)

In the hsp relation, µ s is the viscosity at the surface temperature. The relation for hnb used here is based on a nucleate boiling heat transfer correlation proposed by Gorenflo [12.241]. Here, two-phase equivalent properties (with the “tp” subscript) are computed as

ψ tp = xψ v + (1 − x )ψ l (12.184)

where ψ can be specific heat c p , viscosity µ, or thermal conductivity k, and pr = P / Pc is the reduced pressure. The exponent n varies with pressure and is computed from the relation

n = 0.9 − 0.3 pr (12.185)

Constants in this model were chosen to provide a best fit to heat transfer data for flow boiling of water at nominally atmospheric pressure in microchannels. Values chosen for reference parameters values are q0′′ = 20,000 W/m 2, h0 = 5600 W/m 2 K , and Rp 0 = 0.4  µm For their study, Liu and Garimella [12.240] used a surface roughness value of 1.0 µm for Rp. Also, for the two-phase multiplier φl2, the value of C in the Martinelli correlation is expected to be 5, the value for viscous-viscous flow for the low Reynolds number flows in microchannels (Table 10.1). In this correlation, X is the Martinelli parameter defined on Chapter 10, with (dP /dz )l and (dP /dz ) v being the frictional pressure gradients for the liquid and vapor phases flowing alone in the passage, respectively. The relation for X2 in terms of property ratios indicated above applies for viscous-viscous flow. With these parameter choices, the predictions of this model were found to agree well with experimental heat transfer coefficient measurements for flow boiling of water in microchannels. In contrast, Jacobi and Thome [12.242] developed a three-zone model for flow boiling in microchannels than combined treatments for single-phase liquid heat transfer in a slug region, thin film evaporation in a liquid film region, and gas transport in the dryout region. In addition, Zhang et al. [12.243] combined information from available correlation methods to construct a model to predict the local heat transfer coefficient along a channel as local quality and other parameters change. The resulting model predictions were found to agree well with measured data for flow boiling in a silicon parallel silicon microchannels.

634

Liquid-Vapor Phase-Change Phenomena

Model concepts for flow boiling in microchannels developed to this point have helped to illuminate important mechanisms. However, as noted by Kandlikar [12.244], correlations developed to date are often found to agree with data for limited ranges of channel size, length and type, fluid type, mass flux, pressure, and wall boundary condition (heat flux or superheat), while not agreeing well with some others. Kandlikar [12.244] further argues that shortcomings in available experimental data may be clouding efforts to converge on widely applicable models of flow boiling in microchannels. He suggest that some of the data may have large uncertainty in measurements of channel dimensions, local wall temperatures and wall heat flux, and that different sets of data may correspond to different wall boundary conditions. Variations in nucleation cavity size distributions among surfaces tested, and variable levels of instability in the flow may also make it difficult to compare models with data in a consistent way. These observations lead to the conclusion that there is a need to generate experimental data in microchannel flow boiling systems with a higher level of rigor to facilitate accurate validation of heat transfer performance models. This is especially challenging given the tendency for microchannel boiling processes to exhibit instability or oscillations. Critical Heat Flux in Small Channels Numerous studies have explored CHF during flow boiling in macrochannels. A summary of CHF experimental data sources can be found in papers documenting efforts to develop CHF model using large databases. The mechanisms of CHF in microchannel flow are complicated by the potential for flow instabilities and oscillations, which undoubtedly contribute to scatter in experimental CHF data. As discussed by Bergles and Kandlikar [12.245], the instabilities in these flows tend to result in lower CHF values. CHF correlations for flow boiling in microchannels have often been developed with a small data set. The resulting correlations can provide insight into the mechanisms and parametric trends affecting CHF. But often they are only limitedly successful in matching CHF data over broad ranges of fluids and conditions. Kandlikar [12.244] notes that the Katto and Ohno [12.115] and Shah [12.119] correlations developed for larger channels predicted CHF values generally within 20–40% mean absolute error over widely varying conditions. Alternatively, correlations developed to match specific types of conditions have been proposed [12.245–12.250]. If the application of interest falls with the conditions spanned by these studies, the correlation may be a valuable CHF prediction tool. Based on scale analysis, Kandlikar [12.250] developed a CHF model for flow boiling in microchannels that is framed in terms of the quality, Weber number We, capillary number Ca, and the non-dimensional number K 2 defined above in Eq. (12.175), with K 2 evaluated at the critical heat flux qCHF ′′ . The effect of channel L/D ratio, which appears to be linked to flow instability effects, is also included. The resulting correlation takes the form for the low inertia region We < 900, 2



L /D ≤  140 : K 2,CHF

 q ′′  D = a1 (1 + cosθ) + a2 We(1 − x ) + a3 Ca(1 − x ) (12.186) =  CHF   hlv  ρv σ 2



 q ′′  D = a4 [ a1 (1 + cosθ) + a2 We(1 − x ) + a3 Ca(1 − x ) ] (12.187) L /D ≥  230 :  CHF   hlv  ρv σ

for the high inertia region We ≥ 900, 2



 q ′′  D = a1 (1 + cosθ) + a2 We(1 − x ) + a3 Ca(1 − x ) (12.188) L /D ≤  60 :  CHF   hlv  ρv σ



 q ′′  D = a4 [ a1 (1 + cosθ) + a2 We(1 − x ) + a3 Ca(1 − x ) ] (12.189) L /D ≥  100 :  CHF   hlv  ρv σ

2

635

Convective Boiling in Tubes and Channels

The different regime specifications reflect the fact that the transition between the high and low inertia behavior shifts as L/D varies. Also, Kandlikar [12.250] found that for We > 900, a better fit to data was achieved if a4 was replaced with a factor that varied weakly with We and Ca:

n

1  a4 = a5  for We ≥ 900  We Ca 

The numerical constants a1 − a5 and n were determined to provide a best fit to a very large CHF data base for flow boiling in microchannels. The resulting recommended values are

a1 = 1.03 × 10 −4 , a2 = 5.78 × 10 −5 , a3 = 0.783



a4 = 0.125, a5 = 0.14, n =  0.07

Kandlikar [12.250] noted that although a large data base was used to determine these constants, refinement of the constant values should be expected as the available database is extended to broader ranges of conditions. Further Information There clearly remain challenges to developing useful microchannel flow boiling systems for thermal management applications. Reliable heat transfer and CHF data for stable operating conditions and well defined and controlled boundary and operating conditions are needed to validate models. Understanding the physics and modeling multiple stability mechanisms in flow boiling in microchannels remains a challenge. Work is needed to sort out the effects of upstream compressibility, explosive boiling, parallel channel instabilities, and density wave instabilities to develop the capability to model strategies to provide stable operation. In systems of this type, steady, fully developed flow models are inadequate, and modeling is needed that can account for the time-varying characteristics of flow boiling processes in microchannels. In addition, strategies to enhance flow boiling heat transfer in microchannels is centrally important to applications. Some explorations of using fins structures and nanostructured walls [12.251–12.253] to enhance flow boiling in microchannels have shown promising results, but exploration of these issues has been limited in extent to this point. The interested reader can obtain more information regarding mechanisms and modeling of flow boiling in microchannels in the review references [12.196, 12.202, 12.227, 12.244, 12.254–12.258].

Non-Circular Enhanced Channel Geometries Numerous investigators have explored the use of non-circular enhanced flow channel geometries for flow boiling applications. Investigations of this sort have examined flow boiling in tubes with microfins, twisted-tape inserts, offset or perforated fins, and cross-ribbed passages. A full discussion of the findings of these investigations is beyond the scope of this text. However, it is useful to note some of the key physical differences between flow boiling in simple passage geometries and enhanced passages. Some of the different enhanced flow channel geometries considered for use in high performance evaporators are shown in Fig. 12.34. The features of these channel geometries can have an important impact on the mechanisms of the boiling process. They can, for example, affect the distribution of active nucleation sites in at least two ways. As discussed earlier in this chapter, in annular flow evaporation, thinning the liquid film tends to suppress nucleation. In channels with a plate-fin surface, like that in Fig. 12.34a, liquid shed from the trailing edge of finite fins enhances entrainment, leaving less liquid in the film on the wall. This tends to enhance film evaporation and suppress nucleation. For annular flow in channels containing crossed rib structures on the walls, the wake region downstream of rib structures are locations of recirculation in the vapor flow and thickening

636

Liquid-Vapor Phase-Change Phenomena

FIGURE 12.34  Enhanced passage geometries that are sometimes used for applications involving flow boiling.

of the liquid film on the wall. The thickening of the liquid film at these locations weakens convective evaporation and enhances nucleation. The nucleation site density can also be affected by the temperature variation along fin surfaces resulting from less than perfect fin efficiency. This can occur in channels with plate-fin surfaces (Fig. 12.34a) or in internally finned tubes (Fig. 12.34c) or in tubes with a twisted tape insert attached to the wall. In enhanced flow channel geometries, during saturated flow boiling at moderate pressure, the flow rapidly undergoes a transition to annular flow, just as in simple round tubes. For such circumstances in some complex enhanced channel geometries, the core vapor flow twists and turns as it flows through the passage. As a result, the vapor shear on the liquid film on the wall surfaces varies substantially over the wall surfaces, which would generally produce variations in the liquid film thickness and the local heat transfer and evaporation rates. The non-uniform film thickness may also be a result of geometry effects on entrainment and deposition in the channel. This implies that the complexity of the channel geometry may enhance heat transfer over some portions of the wall surface, and reduce it at others. Whether the net result is an overall enhancement may depend on the channel geometry and flow circumstances. The non-uniformity of the liquid film thickness and evaporation rate generally results in dryout of the liquid film occurring first at wall locations where the film tends to be thinner. In annular flow, dryout may then occur over a finite length of the channel, with more of the wall perimeter becoming dry with increasing downstream distance. An example of this can be seen in Fig. 12.35, which shows a photograph of convective boiling of water at atmospheric pressure in a vertical channel with offset strip fins [12.259]. The mass flux is

Convective Boiling in Tubes and Channels

637

FIGURE 12.35  Photograph of annular film flow boiling of water in a channel with offset strip fins at atmospheric pressure. The flow is vertically upward at a mass flux of 12.5 kg/m2s. The quality at the arrow location is 0.38. The back wall is heated to a superheat of about 5°C and the front transparent wall is adiabatic. The channel cross section is 1.9 cm wide by 3.8 mm high and the channel is 46 cm long. The fins are 1.59 mm thick and 12.7 mm long. The hydraulic diameters based on wetted and heat perimeters are 7.78 mm and 5.15 mm, respectively. (From V. P. Carey and G. D. Mandrusiak [12.258], reproduced with permission, copyright © 1986 Pergamon Press.)

12.5 kg/m2s and the quality in the channel is about 0.38. The channel wall nearest to the camera is glass and is essentially adiabatic. The base wall and fins are heated. The wall superheat in this case is about 5°C. On the back wall of the channel, the light regions of the surface are dry areas and the darker regions are covered with liquid. The core flow field variation as it weaves its way through the fins results in a variation of interfacial shear on the film, which produces in variations in the film thickness. Dryout is seen to occur first at locations where the core flow shear tends to thin the film. More of the wall becomes drier with increasing downstream distance. Thus, dryout occurs progressively over a finite distance in this flow. The non-uniform transport that results during flow boiling in complex enhanced channels such as those shown in Fig. 12.34 makes modeling of the transport a challenging task. Some correlation methods developed for flow boiling heat transfer in enhanced channel geometries have been based on extensions of correlations initially developed for simple round tubes. In a few instances, more detailed models have been constructed using computational methods to solve the governing equations for two-phase flow and convective boiling in complex flow channels (see, e.g., reference [12.260]). However, detailed models of this type require considerable effort to construct and computationally execute, and the accuracy of the submodels of interfacial shear, turbulent transport, entrainment, deposition, and fin efficiency effects in complex flow passages geometries is often unclear. In at least some circumstances of practical interest, use of enhanced flow channels, with fins, rib structures, or twisted tape inserts, appears to offer performance advantages. Further information

638

Liquid-Vapor Phase-Change Phenomena

on research investigations exploring the transport mechanisms in flow boiling in complex enhanced channels of these types can be found in references [12.260–12.277]. Example 12.8 Saturated boiling of water occurs in a microchannel with hydraulic diameter D = 588 µm, length L = 25.4 mm, at P = 101 kPa and mass flux G = 221 kg/m2s. For these conditions and an applied heat flux of 40 W/cm2, use the correlation of Liu and Garimella [12.240] to estimate the heat transfer coefficient for x = 0.1. For saturated water at 101 kPa, Tsat = 373 K. = 100°C, ρl = 958 kg/m3, ρv = 0.597 kg/m3, μv = 12.6 × 10 –6 Ns/m2, μl = 278 × 10 –6 Ns/m2, kv = 0.0250 W/mK, kl = 0.679 W/mK, cpv = 2.03 kJ/kgK, cpl = 4.22 kJ/kgK, and Prl = 1.72. Substituting in Eqs. (12.179–12.183),



Re l =

G(1− x )D 221(1− 0.1)588 × 10 −6 = = 420 µl 278 × 10 −6

 k   Re Pr D  hsp = 1.86  l   l l   D  L 

1/ 3

 µl   µ  s

0.14

−6  0.679   420(1.72)588 × 10  = 1.86      588 × 10 −6   0.0254 

1/ 3

(1)0.14 = 5490

W/m 2K



−6  1− x   ρv   µ l   1− 0.1  0.597   278 × 10  = X2 =  = 0.0739           x   ρl   µv   0.1   958   12.6 × 10 −6 



φl2 = 1+

C 1 5 1 + = 1+ + = 32.9 X X2 (0.0739)1/ 2 0.0739

Using the relation ψ tp = xψ v + (1− x )ψ l to compute the equivalent two-phase properties yields c ptp = 4.00 kJ/kgK, µtp = 251× 10 −6 Ns/m2, and ktp = 0.614. F is then computed as

( )

F=2φ

 µtp   µ  l

2 1/ 4 l

= 2 ( 32.9)

1/ 4

0.105

 c p ,tp  c   p ,l 

 251× 10 −6   278 × 10 −6 

1/ 4

0.105

 ktp   k  l

3/ 4

 4.00    4.22 

Prl0.167 1/ 4

 0.614    0.679 

3/ 4

(1.72)0.167 = 4.76

Next the nucleate boiling contribution is computed as

pr = P /Pc = 101/ 22129 = 0.00456



n = 0.9 − 0.3pr = 0.9 − 0.32(0.00456) = 0.766  q′′   Rp  hnb = h0      q0′′   Rp0 



0.133

  0.68  2  0.27 pr  1.73pr +  6.1+ 1− pr    

 400000   1  = 5600   20000   0.4 

0.133

 0.68   0.27 2 1.73(0.00456) +  6.1+ 1− 0.00456  (0.00456)   

= 25,400 W/m 2K



{ } = exp {36.57 − 55746/(420[4.76] ) − 3.4ln(420[4.76] )} = 0.330

S = exp 36.57 − 55746/(Re l F 3 ) − 3.4ln(Re l F 3 ) 3

3

639

Convective Boiling in Tubes and Channels The overall heat transfer coefficient is calculated as

h = hsp F + hnb S = 5,490(4.76) + 25,400(0.330) = 34,500 W/m 2 K

The heat transfer coefficient for flow boiling in a microchannel at about these conditions was determined by Liu and Garimella [12.240] to be about 32,000 W/m2K, which agrees fairly well with this prediction.

12.8  INTERNAL FLOW BOILING OF BINARY MIXTURES While preceding sections of this chapter have focused on flow boiling of pure fluids, there are, in fact, numerous applications in which flow boiling of multicomponent fluid mixtures is important. Vaporization of hydrocarbon mixtures in the petrochemical industry is perhaps the most conspicuous example. Binary working fluids have also been considered for use in refrigeration and air-conditioning systems in which boiling of the binary fluid is the mechanism that provides the cooling effect in the evaporator. In addition to the complex transport mechanisms found in pure fluid vaporization processes, vaporization of multicomponent fluid mixtures exhibits added complexities due to its more complicated thermodynamics, and the additional effects of species mass transfer. In general, the complexity of the transport rapidly increases as the number of mixture components increases. To examine multicomponent convective boiling processes with minimal added complexity (beyond that for pure fluids), this section will focus on flow boiling of binary mixtures. The twophase flow characteristics for convective boiling of binary mixtures are generally similar to those for flow boiling of pure fluids. In considering the flow, it is useful to extend the definition of the quality x to be the ratio of the vapor mass flow rate m v to the total mass flow rate, which is the sum of the vapor and liquid flow rates x=



m v (12.190) m v + m l

It should also be noted that in binary mixture systems, the mass fraction of the more volatile species (1) is related to the mole fraction by the relations

xˆ1 =

x1 /M1 , [ x1 /M1 ] + [(1 − x1 ) /M 2 ]

yˆ1 =

y1 /M1 (12.191) [ y1 /M1 ] + [(1 − y1 ) /M 2 ]

where M1 and M2 are the molecular masses of species 1 and 2, respectively. As a result, mass fraction and mole fraction information can be used interchangeably in the analysis of binary mixture systems. The boiling mechanisms that arise during flow boiling of binary mixtures are similar to those for flow boiling of pure fluids. At low-quality conditions, where the vapor void fraction is low, nucleate boiling at the wall of the tube is the dominant mechanism of vaporization. As the flow proceeds downstream and the quality and void fraction increases, a transition to annular flow is expected to occur, which facilitates the increasing importance of film flow evaporation. As in the case of pure fluid convective boiling, nucleate boiling may be largely suppressed at high qualities, and dryout of the liquid film on the tube wall can result in a transition into the mist flow evaporation regime.

Prediction of Convective Boiling Heat Transfer Prediction of the flow boiling heat transfer performance for a binary mixture vaporizing in a vertical tube requires an analysis that spans two tasks: (1) at a number of locations along the passage, we must compute the heat and mass transfer coefficients from properties at that location,

640

Liquid-Vapor Phase-Change Phenomena

and (2) we must account for downstream variation of properties that result from cross-stream transport and downstream convection. At low qualities, where nucleate boiling is the dominant vaporization mechanism, methodologies for predicting heat transfer for nucleate pool boiling of binary mixtures can be adapted to flow boiling circumstances. As described in Section 8.5, there are a number of proposed methods for predicting the heat transfer coefficient for nucleate boiling of binary mixtures, ranging from the simpler methods proposed by Stephan and Körner [12.278] and Calus and Leonidopoulos [12.279], to the more detailed methods proposed by Fujita et al. [12.280] and Kandlikar [12.281]. The key point is that these methods are constructed so that the boiling heat transfer coefficient is effectively a function of the pure fluid nucleate boiling heat transfer coefficient for each species at the local pressure and wall heat flux or superheat, the bulk liquid mass or mole fraction of the more volatile component (1), the bulk vapor concentration of the more volatile species, and the properties of the fluids at the local conditions. Mathematically this can be expressed as

hbl = f (hb1 ( P, q ′′), hb 2 ( P, q ′′), x1 , y1 ,properties) (12.192)

As a simple example, we consider here the correlation of Calus and Leonidopoulos [12.279], which can be stated as −1



−1

 α T ,l   c pl   dTbp   (1 − x1 )   x hbl =  1 + 1 + ( y1 − x1 )   (12.193)  hb 2   D12l   hlv   dx1    hb1

where dT bp /dx1 is the slope of the bubble point curve in the equilibrium phase diagram (see Section 8.5), αT,l is the thermal diffusivity of the liquid, and D12l is the mass diffusivity for species 1 in the liquid. As discussed in Section 12.3, the increasing void fraction usually results in acceleration of the core flow and a transition to an annular configuration, both of which tend to enhance the forced convective effect and suppress nucleate boiling. To account for this transition, a superposition correlation method, similar to the Chen [12.40] correlation, could also be used for convective vaporization of binary mixtures. Some modification of the nucleate boiling contribution predicted, for example, by one of the binary mixture pool boiling correlations described above would be necessary to account for the suppression of nucleation with increasing forced-convection effect. One approach would be to modify the nucleate boiling heat transfer coefficient predicted by Eq. (12.193) with a suppression factor S. If the correlation of Calus and Leonidopoulos [12.279] is used, the resulting relation for the nucleate boiling contribution would be −1

−1



 α T ,l   c pl   dTbp   (1 − x1 )   x hbl = S  1 + 1 + ( y1 − x1 )   (12.194)  hb 2   D12l   hlv   dx1    hb1

Following the usual reasoning for a superposition model, we add a convective contribution to the nucleate boiling contribution to get the overall flow boiling heat transfer coefficient: −1



−1

 α T ,l   c pl   dTbp   (1 − x1 )   x hbl = S  1 +  1 + ( y1 − x1 )  D   h   dx   + hc (12.195) h h 12 l 1 b2 lv  b1   

One logical way to proceed would be to adapt relations for the suppression factor and forced convection component from a correlation developed for pure fluids. A simplistic approach would be to adopt the relations proposed by Bennett and Chen [12.43, 12.44] for pure fluids (see Section 12.4):

hc = hmac = hl F ( X tt ) Prl0.296 (12.196)

641

Convective Boiling in Tubes and Channels

S=



[1 − exp{− F ( X tt )hl X 0 / kl }] (12.197) F ( X tt ) hl X 0 / kl

where F ( X tt ) = 1



for X tt−1 ≤ 0.1 (12.198a)

 1  F ( X tt ) = 2.35  0.213 +  X tt 



0.736

for X tt−1 > 0.1 (12.198b) 0.5



  σ X 0 = 0.041   (12.199)  g ( ρl − ρv ) 



k hl = 0.023  l  Rel0.8 Prl0.4 (12.200a)  D Rel =



G (1 − x ) D (12.200b) µl

Although the above analysis selects relations for the components of the heat transfer coefficient relation in an ad hoc way, the logic is plausible, given what is known about the contributions of nucleate boiling and convective effects. Similar lines of reasoning have been used by several investigators to develop correlation methods for predicting the heat transfer coefficient for internal flow boiling of binary mixtures. The above treatment is lacking in one important respect, however. In binary mixture flow boiling, the suppression factor may, in general, be a function of concentrations in the liquid and vapor and other concentration-related parameters. In correlations developed using this type of reasoning, the effect of concentration on suppression has usually been taken into account. Stephan and Körner [12.278] developed a flow boiling correlation based on this type of reasoning in their early investigation of binary mixture boiling. Bennett and Chen [12.44] also adopted this strategy to extend the original Chen correlation to convective evaporation of a binary mixture. The modified Chen correlation developed by Bennett and Chen [12.44] retains the postulated form h = hmic + hmac (12.201)



As in the original correlation, the microscopic contribution hmic is again given by



0.49  kl0.79 c 0.45  pl ρl hmic = 0.00122  0.5 0.29 0.24 0.24  [Tw − Tbp ( x1b , Pl )]0.24 σ µ h ρ (12.202) l lv v  

×[Psat ( x1b ,Tw ) − Pl ]0.75 Sbin where −1



Sbin

 c pl ( y1b − x1b )  ∂Tbp   ρl c pl  1/ 2  = [ Spure (Retp )] 1 −  ∂ x   k D∗   (12.203) hlv l 1l 1   p

Here, x1 is the mass fraction of the more volatile component (species 1) in the liquid, x1b is the bulk liquid mass fraction of species 1, and ylb is the bulk vapor mass fraction of species 1. In the above

642

Liquid-Vapor Phase-Change Phenomena

relations, Spure is the suppression factor for convective boiling of a pure fluid at the same two-phase Reynolds number, D1∗l is the binary diffusion coefficient for species 1 in the liquid, and Tbp is the bubble point of the binary mixture, which is a function of pressure and liquid concentration. The correction factor Sbin accounts for suppression of nucleate boiling and corrects the driving potential for bubble growth in a manner suggested by Florshuetz and Kahn [12.282] based on their study of bubble growth in binary liquid mixtures. Bennett and Chen [12.44] further postulated that mass transfer in the flow does not affect the heat transfer coefficient, but that it does affect the driving potential for heat transfer. Based on this hypothesis and a simple model analysis, they recommended the following relations for the macroscopic contribution to the heat transfer coefficient

 k   m l D  hmac = 0.023  l    D   (πD 2 /4)µ l 

0.8

 Pr + 1  Prl0.4 FBC  l   2 

0.444

Γ ∆T (12.204)

where  (dP /dz ) Ftp  FBC =    (dP /dz ) Fl 



Γ ∆T = 1 +

0.444

(12.205)

y1b q ′′(∂Tbp / ∂ x1 ) P (12.206) ρl hlv h* [Tw − Tbp ( x1b , P)]



 D*  h∗ = 0.023  1l  Retp0.8 Scl0.4 (12.207)  D 



0.444 1.25  m l D    Prl + 1   Retp =    FBC    (12.208) 2  2  πD /4 µ l   

(

)

Note that Fbc is basically the Chen F parameter, except that rather than explicitly correlating it with the Martinelli parameter, they have related it directly to the friction pressure gradient ratio in Eq. (12.205). The subscripts “Ftp” and “Fl” indicate the frictional pressure gradients for the two-phase flow and the liquid flowing alone, respectively. The Martinelli correlation or any other suitable correlation method can be used to determine the two-phase pressure gradient in Eq. (12.205). The factor ΓΔT corrects for mass transfer effects on the interface temperature. The overall heat transfer coefficient h must be computed iteratively using this correlation because the relation for ΓΔT contains the total heat flux q ′′ = h[Tw − Tbp ( x1b , P)]. This correlation method was found to agree well with data for flow boiling of water and ethylene glycol mixtures to a mean deviation of 14.9%. It has the further advantage of reducing to a form that is virtually identical to the original Chen correlation when the concentration of the more volatile component approaches zero. Kandlikar [12.283] also used the line of reasoning described above to develop a correlation for flow boiling of binary mixtures. More recently, however, Kandlikar [12.284] developed a more detailed correlation methodology for predicting flow boiling heat transfer for binary mixtures that accounted for different regimes of mass transfer effects. The regimes are categorized in terms of a volatility parameter V1 defined as

 c pl   α  V1 =    T ,l   hlv   D12l 

0.5

 dTbp   dx  ( x1 − y1 ) (12.209) 1

643

Convective Boiling in Tubes and Channels

FIGURE 12.36  Regimes of transport in flow boiling of binary mixtures.

The regimes indicated by the Kandlikar [12.284] correlation are shown in the regime diagram in Fig. 12.36. Kandlikar [12.284] defines Region I to correspond to V1 < 0.03, where mixture effects are negligible and a pure-fluid correlation is a good predictor of the heat transfer. Because this would apply in mixtures that form an azeotrope when the concentration is close to the azeotropic condition, this is termed the Near-Azeotropic Region. The prediction method for this region is:  hNBD h = the maximum of  (12.210) h  CBD

where



hNBD

ρ  = 0.6683  l   ρv 

0.1

x 0.16 (1 − x )0.64 f2 (Frle )hle

(12.211)

+1058.0Bo FK ,m (1 − x ) hle 0.7



ρ  hCBD = 1.1360  l   ρv 

0.8

0.45

x 0.72 (1 − x )0.08 f2 (Frle )hle

(12.212)

+667.2Bo FK ,m (1 − x ) hle 0.7

0.8

and

FK ,m = x1 FK ,1 = (1 − x1 ) FK ,2 (12.213)

In the above equation, Fk,1 and FK,2 are pure values of the FK constant for species 1 and 2 listed in Table 12.1. Liquid and vapor properties are calculated at the equilibrium phase concentrations, Bo and Frle are defined by Eqs. (12.50) and (12.51), and hle is the single-phase entire-flow-as-liquid heat transfer coefficient computed using Eq. (12.71) or Eq. (12.72) (see Section 12.4).

644

Liquid-Vapor Phase-Change Phenomena

Region II for the Kandlikar correlation [12.284] is the Moderate Diffusion-Induced Suppression Region, corresponding to 0.03 < V1 < 0.2 and Bo > 10 −4. For this regime, convection is expected to dominate the boiling heat transfer and the heat transfer coefficient is therefore computed as



hCBD

ρ  = 1.1360  l   ρv 

0.45

x 0.72 (1 − x )0.08 f2 (Frle )hle

(12.214)

+667.2Bo FK ,m (1 − x ) hle 0.7

0.8

In using this relation, the properties and parameters are determined in the same way as for Region I. Region III in this correlation scheme is the Severe Diffusion-Induced Suppression Region. This region corresponds to 0.03 < V1 < 0.2 with Bo < 10−4 or V1 > 0.2 for any Bo value. This region is characterized by strong mixture effects. In this region, Kandlikar proposed use of the convectiondominated relation with an added mixture-induced suppression factor multiplying the nucleate boiling term:



ρ  hCBD = 1.1360  l   ρv 

0.45

x 0.72 (1 − x )0.08 f2 (Frle )hle

(12.215)

+667.2Bo FK ,m (1 − x ) hle FD 0.7

0.8

where −1



  c pl   α  0.5  dTbp   FD = 0.678 1 +    T ,l   ( x1 − y1 )  (12.216)    hlv   D12l   dx1  

In Eq. (12.216), x1 and y1 are equilibrium concentrations calculated at the local system pressure, and the diffusivity D12l is computed from the relation

0 xˆ1 D12l = ( D120 )1− xˆ1 ( D21 ) (12.217)

0 where D120 and D21 are low concentration diffusion coefficients computed using the Wilke-Chang correlation:



D120 = 1.173 × 10 −16

(φM 2 )1/ 2 T (12.218) µ l ,2 ( vˆ1 )0.6

and xˆ1 is the molar concentration of species 1. In this relation, D120 is the mutual diffusion coefficient of solute 1 at very low concentration in solvent 2 in m2/s, M2 is the molecular mass of species 2 in kg/kmol, T is the absolute temperature in Kelvin, μl,2 is the viscosity of solvent 2 in Ns/m2, and vˆl is the molar volume in m3/kmol for pure solute 1 at its normal boiling point. The parameter ϕ is an association factor for solvent 2, which is 1.0 for unassociated solvents, but is higher for associated solvents (ϕ = 2.26 for water, 1.9 for methanol, 1.5 for ethanol). Methane and most refrigerants 0 are unassociated. Note that to compute D21 , the same relation is used with 1 in the place of 2 and 2 in the place of 1 in Eq. (12.218). Kandlikar [12.284] found that predictions of this correlation agreed well with flow boiling heat transfer data for a variety of refrigerants spanning all three regions shown in Fig. 12.36.

Convective Boiling in Tubes and Channels

645

Given local property information, one of the correlation methods described above can be used to compute the heat transfer coefficient for flow boiling of a binary mixture. As noted above, to predict the flow boiling heat transfer for the vaporization process in the tube, computations of the heat transfer coefficient must be combined with a calculation scheme that accounts for downstream variation of properties that result from cross-stream transport and downstream convection. To illustrate how this works, we will consider implementation of this type of model analysis for annular flow film evaporation of a binary mixture flowing upward in a vertical tube. As noted above, at moderate to high qualities, for vertical upward flow (and horizontal flow at high flow rates) the flow is expected to take on an annular morphology. For heat pump systems using a nonazeotropic refrigerant blend, the throttling process in the expansion valve may result in 20–30% quality at the inlet of the evaporator. The vaporization process may be annular film flow evaporation at the inlet and over most of the refrigerant flow passage in such circumstances. Because this type of vaporization process may be of major importance in applications, we will consider it in this example analysis. To facilitate an analytical examination of annular flow evaporation of a binary mixture, the following idealizations will be adopted: 1. The two components of the binary mixture are miscible at all concentrations. 2. The sensible heat gain (or loss) of the liquid film and the vapor core are negligible compared to the latent heat of vaporization. 3. Steady-state conditions exist throughout the flow. 4. The interface is smooth and the flow field is axisymmetric. 5. All liquid flows in the annular film on the tube wall (no entrainment). 6. Thermo-diffusion and diffusion-thermo effects are negligible. 7. Rates of axial diffusion and conduction are negligible. 8. Turbulent flow in the liquid film and in the vapor result in rapid transport in the flow in each of these regions, with the effect that the interface concentration of the more volatile component in the liquid is equal to the bulk concentration xl. 9. At the interface, vapor is generated at a rate dictated by heat transfer across the liquid film. 10. The concentration of the vapor generated at the interface is the dew point concentration at the interface temperature, which equals the bubble point temperature at the interface liquid concentration of xl. The interface property relationships can be seen in the phase diagram shown in Fig. 12.37. The liquid enters the tube at xl,in, and vapor initially generated has a concentration equal to y1,dp(Tbp(x1,in, P), P), which is greater than xl,in. As the boiling process proceeds and the liquid film flows downstream, the liquid concentration decreases because vaporization produces vapor with a higher concentration of species 1 and depletes the liquid film of species 1 more rapidly than species 2. As a result, the liquid concentration state point moves to the left in Fig. 12.37 along the bubble point curve as the liquid flows downstream. At any point along the passage where the film liquid concentration is xl, vapor generated has a concentration equal to y1,dp(Tbp(xl, P), P). The idealized physical system to be modeled is shown schematically in Fig. 12.38. In this model, heat transferred across the liquid from the wall to the interface is absorbed by the latent heat for the vaporizing liquid. It follows from conservation of energy that

dm v πD = h Tw − Tbp ( x l , P)  (12.219) dz hlv 

Conservation of overall mass also requires that

dm l dm = − v (12.220) dz dz

646

Liquid-Vapor Phase-Change Phenomena

FIGURE 12.37  Equilibrium properties in a binary mixture during flow boiling.

Invoking conservation of the more volatile species (1) in the liquid and vapor portions of the differential control volume in Fig. 12.38 and using Eqs. (12.219) and (12.220) results in the relations

 yl − y1,dp  dyl πD = h[Tw − Tbp ( x l , P)]  (12.221)  m v  dz hlv



 y1,dp − x l  dx l πD =− h[Tw − Tbp ( x l , P)]  (12.222)  m l  dz hlv

FIGURE 12.38  System model for analysis of convective vaporization of a binary mixture inside a tube.

647

Convective Boiling in Tubes and Channels

The system of equations described above can be used to predict the heat transfer performance for annular film-flow boiling in a round tube if the operating conditions, fluid properties and boundary conditions are specified. Specifically, the performance calculation may proceed subject to the following conditions: 1. The tube diameter D, wall temperature Tw, inlet vapor and liquid flow rates, (m v and m l ), and bulk concentrations (xl and yl), and the system pressure P are all specified. 2. All necessary thermodynamic property information is available, including the variation of xl,bp along the bubble point curve, and y1,dp along the dew point curve, with interface temperature. 3. A correlation is available to compute the heat transfer across the liquid film from local flow conditions. If these conditions are satisfied, the tube may be analytically divided into segments and the performance calculation proceeds as follows: 1. Using the inlet liquid concentration xl and pressure P for a given segment, thermodynamic relations representing the bubble point curve and the dew point curve are used to determine the bubble point temperature Tbp(xl, P) and the concentration of the vapor at the interface yl,dp (Tbp(xl, P) P). 2. The heat transfer coefficient is determined from the local flow conditions (at the inlet of the segment) and the interface conditions obtained in step 1. Any of the flow boiling correlations discussed earlier in this section could be used to determine the heat transfer coefficient h. If the correlation depends on Bo = q ′′ /Ghlv , the calculation may require an iterative solution scheme. 3. Using the results of steps 1 and 2, Eqs. (12.219) and (12.220) are used to determine dm v /dz  and  dm l /dz . 4. Equations (12.221) and (12.222) are used to determine dx1/dz and dy1/dz. 5. Values of m v, m l , x1, and y1 at the inlet to the next (n + 1) segment, are computed as



 dm  (m v )n +1 = (m v )n +  v  ∆z  dz 



 dm  (m l )n +1 = (m l )n +  l  ∆z  dz 



x n +1 =

(m v )n +1  (mv )n +1 + (m l )n +1



 dx  ( x l )n +1 = ( x l )n +  1  ∆z  dz 



 dy  ( y1 )n +1 = ( y)n +  1  ∆z  dz  where Δz is the length of the tube segment. 6. If not computed as part of the determination of h, the heat flux from the wall is computed as q ′′ = h[Tw − Tbp ( x1 , P)].

648

Liquid-Vapor Phase-Change Phenomena

7. The quality gradient is computed as dx /dz = (dm v /dz )/(m v + m l ) and the separated flow model with a suitable correlation for the void fraction and two-phase multiplier is use to determine dP/dz. The pressure at the inlet of the next tube segment is then computed as  dP  ∆z Pn +1 = Pn +   dz 



8. If this is the last segment of the tube, the exit condition is known, as is the heat flux at each segment, and the computation is complete. If this is not the last segment, the algorithm is repeated, beginning with step 1. The computational scheme described above will thus predict the variation of the heat flux along the flow passage. As specified, the integration of the equations for the bulk concentrations is obviously very simplistic, and the algorithm could be improved by the use of a more sophisticated numerical scheme. However, the simplistic scheme described above does indicate the nature of the algorithm required for this type of computation. The analysis of annular film-flow evaporation of a binary mixture described above considers only the film evaporation process. In an actual flow boiling circumstance, it is possible for both forced convective evaporation and nucleate boiling effects to be present. This effect can be treated by selecting a flow boiling heat transfer correlation that includes both convective and nucleate boiling contributions, such as the Bennett and Chen [12.44] or Kandlikar [12.284] correlation. The model described above incorporates the idealization that turbulent transport in the liquid and vapor phases results in rapid transport. This is equivalent to assuming high heat and mass transfer coefficients between the bulk fluid and the interface in the liquid film and core vapor flow. In real systems, the heat and mass transfer coefficients will be finite, and accurate calculation of the transport in each segment of the tube would require a treatment that accounts for mass transfer effects. Further discussion of methods for predicting the heat and mass transfer coefficients in models of annular flow boiling of binary mixtures can be found in the paper by Shock [12.285].

CHF Conditions The complicated variation of properties with concentration, temperature, and pressure in binary mixture flow boiling in tubes makes development of methods for predicting CHF conditions in such circumstances a challenging task. Collier [12.2] recommends the method proposed by Hewitt and co-workers [12.129, 12.130] for pure fluid convective vaporization processes. In binary systems, the interfacial tension may vary strongly with concentration. Because entrainment is directly dependent on interfacial tension (see Section 10.4), the entrainment characteristics in a binary system might be expected to differ from those in a pure fluid system under comparable conditions. Shock [12.285] presented evidence that suggests that entrainment characteristics in convective evaporation of binary mixtures are not strongly affected by mass transfer effects. Based on this evidence Collier concludes that the model analysis of Hewitt and co-workers [12.129, 12.130] should yield reasonably good results. It should be noted, however, that during convective evaporation of a binary mixture, if the more volatile component has a higher surface tension, Marangoni effects at the interface of the liquid film may cause the film to become unstable and break down into rivulets (see Section 2.5). This may lead to the onset of dryout prior to the point where dryout is predicted by models that ignore such effects. A number of experimental investigations of the departure from nucleate boiling (DNB) during flow boiling of a binary mixture have been conducted (see, e.g., references [12.286–12.290]). In general, the DNB heat flux for flow boiling of a binary mixture increases with increasing subcooling and flow velocity in much the same way as for a pure fluid. As a simplistic model, one might expect

Convective Boiling in Tubes and Channels

649

FIGURE 12.39  Qualitative variation of the flow boiling critical heat flux observed in a binary mixture that forms an azeotrope at one concentration.

that the DNB heat flux would vary linearly with liquid concentration between the pure fluid values (at the same subcooling and flow velocity) for x1 = 0 and x1 = 1.

(qcr′′ )i = (qcr′′ ) x1 = 0 (1 − x1 ) + (qcr′′ ) x1 =1 x1 (12.223)

This variation is indicated by the broken line in Fig. 12.39. The solid curve in this figure indicates the trend typically observed in experimental data for a binary mixture that forms an azeotrope at one concentration. As in the case of pool boiling, maxima in the critical heat flux values are observed at or near locations where y1 − x1 attains a maximum. For a binary mixture that does not form an azeotrope, y1 − x1 attains a maximum value at one concentration, and only one maximum in the critical heat flux is expected. To obtain a better fit to available data, a correlation having the following form has often been postulated qcr′′ = (qcr′′ )i )(1 + χ) (12.224)



where ((qcr′′ )i , is given by Eq. (12.223). Note that χ is a correction term that accounts for the deviation from the idealized linear relation (12.223). Tolubinsky and Matorin [12.287] proposed the following relation for χ.

χ =  1.5 y1 − x1

1.8

 Tbp ( x1 , P) − Tsat (1, P)  + 6.8 y1 − x1     (12.225) Tsat (1, P)  

where Tbp(xl, P) is the bubble point temperature of the mixture and Tsat(1, P) is the pure component saturation temperature of the more volatile component, y1 and x1 are the bulk mole fractions in the vapor and liquid phases, respectively. This relation fit DNB data for flow boiling of ethanol-water, acetone-water, ethanol-benzene and, water-ethylene glycol mixtures to within ±20%. Note that

650

Liquid-Vapor Phase-Change Phenomena

this relation reflects the fact that the maxima in the critical heat flux variation with concentration roughly corresponds to the maxima in y1 − x1 . The trends described above suggest that addition of small amounts of a more volatile liquid to another pure fluid may increase the critical heat flux above the pure fluid value for the same conditions. However, for subcooled flow boiling of mixtures of water with small amounts of 1-pentanol, Bergles and Scarola [12.290] observed a trend quite different from that suggested above. For added amounts of 1-pentanol (about 2% by weight), they found that the critical heat flux actually decreased. They attributed the decrease to the formation of smaller bubbles during the boiling process in the mixture, producing local liquid velocities that were smaller than those associated with the pure fluid boiling process where larger bubbles are produced. The reduction in the local liquid velocities apparently allows vapor blanketing of the surface to more easily occur at a lower heat flux level. This implies that the effect of adding small amounts of a more volatile fluid on the DNB condition may vary, depending in the fluids involved. Additional information on the effects of composition on CHF for flow boiling of binary mixtures has been reported by Mori et al. [12.291], Celata et al. [12.292], and Auracher and Marroquin [12.293]. Further discussion of the CHF condition for binary mixture boiling in tubes can also be found in Collier and Thome [12.78].

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12.19 Kreith, F., and Sommerfield, M., Heat transfer to water at high flux densities with and without surface boiling, Trans. ASME, vol. 71, pp. 805–815, 1949. 12.20 Piret, E. L., and Isbin, H. S., Two-phase heat transfer in natural circulation evaporators, Chem. Eng. Prog. Symp. Ser., vol. 50, no. 6, p. 305, 1953. 12.21 Kandlikar, S.G., Heat transfer and flow characteristics in partial boiling, fully developed boiling and significant void flow regions of subcooled flow boiling, J. Heat Transf., vol. 120, pp. 395–401, 1998. 12.22 Griffith, P., Clark, J. A., and Rohsenow, W. M., Void volumes in subcooled boiling systems, Paper 58-HT-19, presented at the 1958 National Heat Transfer Conference, Chicago, IL, 1958. 12.23 Levy, S., Forced convection subcooled boiling prediction of vapor volumetric fraction, Int. J. Heat Mass Transf., vol. 10, pp. 951–965, 1967. 12.24 Egen, R. A., Dingee, D. A., and Chastain, J. W., Vapor formation and behavior in boiling heat transfer, ASME Paper No. 57-A-74, 1957. 12.25 Maurer, G. W., A method of predicting steady state boiling vapor fraction in reactor coolant channels, Bettis Technical Review, WAPD-BT-19, 1960. 12.26 Staub, F. W., The void fraction in subcooled boiling – Prediction of the point of net vapor generation, ASME paper 67-HT-36 presented at the 1967 National Heat Transfer Conference, 1967. 12.27 Saha, P., and Zuber, N., Point of net vapor generation and vapor void fraction in subcooled boiling, Proc. 5th Int. Heat Transfer Conf., paper B4.7, Tokyo, Japan, 1974. 12.28 Kroeger, P. G., and Zuber, N., An analysis of the effects of various parameters on the average void fractions in subcooled boiling, Int. J. Heat Mass Transf., vol. 11, pp. 211–233, 1968. 12.29 Sher, N. C., Estimation of boiling and non-boiling pressure drop in rectangular channels at 2000 psia, USAEC Report WAPD-TH-300, 1957. 12.30 Hirata, M., and Nishiwaka, N., Skin friction and heat transfer for liquid flow over a porous wall with gas injection, Int. J. Heat Mass Transf., vol. 6, pp. 941–949, 1963. 12.31 Reynolds, J. B., Local boiling pressure drop, USAEC Report ANL-5178, 1954. 12.32 Dormer, J., Jr., and Bergles, A. E., Pressure drop with surface boiling in small diameter tubes, Report No. 8767–31, Department of Mechanical Engineering, MIT, 1964. 12.33 Buchberg, H., Romie, F., Lipkis, R., and Greenfield, M., Heat transfer, pressure drop and burnout studies with and without surface boiling for de-aerated and gassed water at elevated pressures in a forced flow system, Proc. 1951 Heat Transfer and Fluid Mechanics Institute, Stanford University, pp. 171–191, 1951. 12.34 Owens, W. L., and Schrock, V. E., Local pressure gradients for subcooled boiling of water in vertical tubes, ASME Paper No. 60-WA-249, 1960. 12.35 Jicha, J. J., and Frank, S., An experimental local boiling heat transfer and pressure drop study of a round tube, Paper 62-HT-48 presented at the 1962 National Heat Transfer Conference, Houston, TX, 1962. 12.36 Jordan, D. P., and Leppert, G., Pressure drop and vapor volume with subcooled nucleate boiling, Int. J. Heat Mass Transf., vol. 5, pp. 751–761, 1962. 12.37 Tong, L. S., Boiling Heat Transfer and Two-Phase Flow, Robert Krieger Publishing Company, Malabar, FL, 1975. 12.38 Dengler, C. E., and Addoms, J. N., Heat transfer mechanism for vaporization of water in a vertical tube, Chem. Eng. Prog. Symp. Ser., vol. 52, no. 18, pp. 95–103, 1956. 12.39 Guerrieri, S. A., and Talty, R. D., A study of heat transfer to organic liquids in single-tube natural circulation vertical tube boilers, Chem. Eng. Prog. Symp. Ser., vol. 52, no. 18, pp. 69–77, 1956. 12.40 Chen, J. C., Correlation for boiling heat transfer to saturated fluids in convective flow, Ind. Eng. Chem. Proc. Des. Dev., vol. 5, no. 3, pp. 322–339, 1966. 12.41 Forster, H. K., and Zuber, N., Dynamics of vapor bubbles and boiling heat transfer, AIChE J., vol. 1, pp. 531–535, 1955. 12.42 Collier, J. G., Forced convective boiling, in Two-Phase Flow and Heat Transfer in the Power and Process Industries, A. E. Bergles, J. G. Collier, J. M. Delhaye, G. F. Hewitt, and F. Mayinger (editors), Hemisphere Publishing, New York, NY, chapter 8, 1981. 12.43 Bennett, D. L., Davis, M. W., and Hertzler, B. L., The suppression of saturated nucleate boiling by forced convective flow, AIChE Symp. Ser., vol. 76, no. 199, pp. 91–103, 1980. 12.44 Bennett, D. L., and Chen, J. C., Forced convective boiling in vertical tubes for saturated pure components and binary mixtures, AIChE J., vol. 26, pp. 454–461, 1980. 12.45 Shah, M. M., A new correlation for heat transfer during boiling flow through pipes, ASHRAE Trans., vol. 82, part 2, pp. 66–86, 1976. 12.46 Shah, M. M., Chart correlation for saturated boiling heat transfer: Equations and further study, ASHRAE Trans., vol. 88, part 1, pp. 185–196, 1982.

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12.206 Wu, H. Y., and Cheng, P., Liquid/two-phase/vapor alternating flow during boiling in microchannels at high heat flux, Int. Commun. Heat Mass Transf., vol. 30, pp. 295–302, 2003. 12.207 Steinke, M. E., and Kandlikar, S. G., Flow boiling and pressure drop in parallel microchannels, paper no. ICMM2003-1070, Proc. 1st Int. Conf. Microchannels and Minnichannels, ASME, New York, NY, pp. 567–579, 2004. 12.208 Wu, H. Y., and Cheng, P., Boiling instability in parallel silicon microchannels at different heat flux, Int. J. Heat Mass Transf., vol. 47, pp. 3631–3641, 2004. 12.209 Lee, P. C., Tseng, F. G., and Pan, C., Bubble dynamics in microchannels. Part I: single microchannel, Int. J. Heat Mass Transf., vol. 47, pp. 5575–5589, 2004. 12.210 Li, H. Y., Tseng, F. G., and Pan C., Bubble dynamics in microchannels. Part II: Two parallel microchannels,. Int. J. Heat Mass Transf., vol. 47, pp. 5591–5601, 2004. 12.211 Steinke, M. E., and Kandlikar, S. G., Control and effect of dissolved air in water during flow boiling in microchannels, Int. J. Heat Mass Transf., vol. 47, pp. 1925–1935, 2004. 12.212 Balasubramanian, P., and Kandlikar, S. G., Experimental study of flow patterns, pressure drop and flow instabilities in a parallel rectangular minichannels. Proc. 2nd Int. Conf. Microchannels and Minichannels, ICMM2004-2371, pp. 475–481, Rochester, NY, 2004. 12.213 Xu, J., Zhou, J., and Gang, Y., Static and dynamic flow instability of a parallel microchannels heat sink at high heat fluxes, Energy Convers. Manag., vol. 46, pp. 313–334, 2005. 12.214 Kandlikar, S. G., Willistein, D. A., and Borrelli, J., Experimental evaluation of pressure drop elements and fabricated nucleation sites for stabilizing flow boiling in minichannels and micro channels, Proc. 3rd Int. Conf. Microchannels and Minichannels, paper ICMM2005-75197, Toronto, Ontario, Canada, 2005. 12.215 Hetsroni, G., Mosyak, A., Pogrebnyak, E., and Segal, Z., Periodic boiling in parallel microchannels at low vapor quality, Int. J. Multiphase Flow, vol. 32, pp. 1141–1159, 2006. 12.216 Moriyama, K., Inoue, A., and Ohira, H., Thermohydraulic characteristics of two-phase flow in extremely narrow channels (the frictional pressure drop and void fraction of adiabatic two-component two-phase flow), Heat Transf. Jpn. Res., vol. 21, pp. 823–837, 1992. 12.217 Peng, X. F., and Wang, B. X., Forced convection and flow boiling heat transfer for liquid flowing through microchannels, Int. J. Heat Mass Transf., vol. 36, pp. 3421–3427, 1993. 12.218 Bowers, M. B., and Mudawar, I., High flux boiling in low flow rate, low pressure drop mini-channel and micro-channel heat sinks, Int. J. Heat Mass Transf., 37(2), pp. 321–332, 1994. 12.219 Hetsroni, G., Mosyak, A., Segal, Z., and Pogrebnyak, E., Two-phase flow patterns in parallel microchannels, Int. J. Multiphase Flow, vol. 29, pp. 341–360, 2003. 12.220 Yen, T.-H., Kasagi, N., and Suzuki, Y., Forced convective boiling heat transfer in microtubes at low mass and heat fluxes, Int. J. Multiphase Flow, vol. 29, pp. 1771–1792, 2003. 12.221 Steinke, M. E., and Kandlikar, S. G., An experimental investigation of flow boiling characteristics of water in parallel microchannels ASME J. Heat Transf., vol. 126, pp. 518–526, 2004. 12.222 Kosar, A., Kuo, C.-J., and Peles, Y., Boiling heat transfer in rectangular microchannels with eeentrant cavities, Int. J. Heat Mass Transf., vol. 48, pp. 4867–4886, 2005. 12.223 Hetsroni, G., Klein, D., Mosyak, A., Segal, Z., and Pogrebnyak, E., Convective boiling in parallel microchannels, Microscale Thermophys. Eng., vol. 8, pp. 403–421, 2004. 12.224 Qu, W., and Mudawar, I., Transport phenomena in two-phase micro-channel heat sinks, J. Electron. Packag., vol. 126, pp. 213–224, 2004. 12.225 Wu, H. Y., and Cheng, P., Visualization and measurements of periodic boiling in silicon microchannels, Int. J. Heat Mass Transfer, vol. 46, pp. 2603–2614, 2003. 12.226 Hetsroni, G., Mosyak, A., Pogrebnyak, E., and Segal, Z., Explosive boiling of water in parallel microchannels, Int. J. Multiphase Flow, vol. 31, pp. 371–392, 2005. 12.227 Cheng, P., and Wu, H. Y., Phase change heat transfer in microsystems, Keynote Paper KN-02, Proc. 13th Int. Heat Transf. Conf., Sydney, 2006. 12.228 Kandlikar, S. G., Methods for stabilizing flow in channels and systems thereof, US Patent Application No. 20090266436, 2003. 12.229 Mukherjee, A., and Kandlikar, S. G., Numerical simulation of growth of a vapor bubble during flow boiling of water in a microchannel, Microfluid. Nanofluid., vol. 1, pp. 137–145, 2005. 12.230 Kandlikar, S. G., Kuan, W. K., Willistein, D. A., and Borrelli, J., Stabilization of flow boiling in microchannels using pressure drop elements and fabricated nucleation sites, ASME J. Heat Transf., vol. 128, pp. 389–396, 2006. 12.231 Kosar, A., Kuo, C. J., and Peles, Y. P., Suppression of boiling flow oscillations in parallel microchannels by inlet restrictors, ASME J. Heat Transf., vol. 128, pp. 251–260, 2006.

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12.232 Zhang, T., Tong, T., Chang, J.-Y., Peles, Y., Prasher, R., Jensen, M., Wen, J., and Phelan, P., Ledinegg instability in microchannels, Int. J. Heat Mass Transf., 52(25–26), pp. 5661–5674, 2009. 12.233 Warrier, G. R., and Dhir, V. K., Visualization of flow boiling in narrow rectangular channels, J. Heat Transf., vol. 126, p. 495, 2004. 12.234 Jiang, L., Wong, M., and Zohar, Y., Phase change in microchannel heat sinks with integrated temperature sensors, J. Microelectromech. Syst., vol. 8, pp. 358–365, 1999. 12.235 Hetsroni, G., Mosyak, A., and Segal, Z., Nonuniform temperature distribution in electronic devices cooled by flow in parallel microchannels, IEEE Trans. Compon. Packag. Technol., vol. 24, pp. 16–23, 2000. 12.236 Kandlikar, S. G., Similarities and differences between flow boiling in microchannels and pool boiling, Heat Transf. Eng., vol. 31, pp. 159–167, 2010. 12.237 Kandlikar, S. G., Kuan, W. K., and Mukherjee, A., Experimental study of heat transfer in an evaporating meniscus on a moving heated surface, ASME J. Heat Transf., vol. 127, pp. 244–252, 2005. 12.238 Kandlikar, S. G., and Balasubramanian, P., An extension of the flow boiling correlation to transition, laminar and deep laminar flow in minichannels and microchannels, Heat Transf. Eng., vol. 25, pp. 86–93, 2004. 12.239 Yen, T.-H., Kasagi, N., and Suzuki, Y., Forced convective boiling heat transfer in microtubes at low mass and heat fluxes, Compact Heat Exchangers, a Festschrift on the 60th Birthday of Ramesh K. Shah, G. P. Celata, B. Thonon, A. Bontemps, and S. G. Kandlikar (editors), Edizioni ETS, Pisa, Italy, 2002. 12.240 Liu, D. and Garimella, S.G., Flow boiling heat transfer in microchannels, J. Heat Transf., vol 129, pp. 1321–1332, 2007. 12.241 Gorenflo, D., 1993, Pool Boiling: VDI-Heat Atlas, VDI, Dusseldorf. 12.242 Jacobi, A. M., and Thome, J. R., Heat transfer model for evaporation of elongated bubble flows in microchannels, ASME J. Heat Transf., vol. 124, pp. 1131–1136, 2002. 12.243 Zhang, L., Koo, J.-M., Jiang, L., Asheghi, M., Goodson, K. E., Santiago, J. G., and Kenny, T. W., Measurements and modeling of two-phase flow in microchannels with nearly constant heat flux boundary conditions, J. Microelectromech. Syst., vol. 11, pp. 12–19, 2002. 12.244 Kandlikar, S.G., History, advances, and challenges in liquid flow and flow boiling heat transfer in microchannels: A critical review, J. Heat Transf., vol. 134, pp. 034001-1 to 034001-15, 2012. 12.245 Qu, W., and Mudawar, I., Measurement and correlation of critical heat flux in two-phase micro-channel heat sinks, Int. J. Heat Mass Transf., vol. 47, pp. 2045–2059 2004. 12.246 Kosar, A., and Peles, Y., Critical heat flux of R-123 in silicon-based microchannels, ASME J. Heat Transf., vol. 129, pp. 844–851, 2007. 12.247 Park, J. E., and Thome, J. R., Critical heat flux in multi-microchannel copper elements with low pressure refrigerants, Int. J. Heat Mass Transf., vol. 53, pp. 110–122, 2010. 12.248 Zhang, W., Hibiki, T., Mishima, K., and Mi, Y., Correlation of critical heat flux for flow boiling of water in mini-channels, Int. J. Heat Mass Transf., vol. 49, pp. 1058–1072, 2006. 12.249 Kosar, A., Peles, Y., Bergles, A. E., and Cole, G. S., Experimental investigation of critical heat flux in microchannels for flow-field probes, Paper No. ICNMM2009-82214, Proc. ASME Seventh Int. Conference on Nanochannels, Microchannels, and Minichannels, June 22–24, Pohang, South Korea, 2009. 12.250 Kandlikar, S. G., A scale analysis based theoretical force balance model for critical heat flux (CHF) during saturated flow boiling in microchannels and minichannels, ASME J. Heat Transf., vol. 132, p. 081501, 2010. 12.251 Krishnamurthy, S., and Peles, Y., Flow boiling heat transfer on micro pin fins entrenched in a microchannel, ASME J. Heat Transf., vol. 132, p. 041002, 2010. 12.252 Khanikar, V., Mudawar, I., and Fisher, T., Effects of carbon nanotube coating on flow boiling in a micro-channel, Int. J. Heat Mass Transf., vol. 52, pp. 3805–3817, 2009. 12.253 Gao, L. and Bhavnani, S.H., Enhanced boiling in microchannels due to recirculation induced by repeated saw- toothed cross-sectional geometry, Appl. Phys. Lett., vol. 111, pp. 184105-1 to 184105-5, 2017. 12.254 Kandlikar, S. G., Fundamental issues related to flow boiling in minichannels and microchannels, Exp. Therm. Fluid Sci., vol. 26, pp. 389–407, 2002. 12.255 Mudawar, I., and Bowers, M. B., Ultra-high critical heat flux (CHF) for subcooled water flow boiling – I: CHF data an parametric effects for small diameter tubes, Int. J. Heat Mass Transf., vol. 42, pp. 1405–1428, 1999. 12.256 Watel, B., Review of standard flow boiling in small passages of compact heat-exchangers, Int. J. Therm. Sci., vol. 42, pp. 107–140, 2003.

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12.257 Bergles, A. E., Lienhard, J. H. V., Kendall, G. E., and Griffith, P., Boiling and evaporation in small diameter channels, Heat Transf. Eng., vol. 24, pp. 18–40, 2003. 12.258 Thome, J. R., Boiling in microchannels: A review of experiment and theory, Int. J. Heat Fluid Flow, vol. 25, pp. 128–139, 2004. 12.259 Carey, V. P., and Mandrusiak, G. D., Annular film-flow boiling of liquids in a partially-heated vertical channel with offset strip fins, Int. J. Heat Mass Transf., vol. 29, pp. 927–939, 1986. 12.260 Mandrusiak, G. D., and Carey, V. P., A finite difference model of annular film-flow evaporation in a vertical channel with offset strip fins, Int. J. Heat Mass Transf., vol. 16, pp. 1071–1098, 1990. 12.261 Panitsidis, H., Gresham, R. D., and Westwater, J. W., Boiling of liquids in a compact plate-fin heat exchanger, Int. J. Heat Mass Transf., vol. 18, pp. 37–42, 1975. 12.262 Withers, J. G., and Habdas, E. P., Heat transfer characteristics of helical-corfregated tubes for in tube boiling of refrigerant R-12, AIChE Symp. Ser., vol. 70, no. 138, pp. 98–106, 1974. 12.263 Cumo, M., Farello, G. E., Ferrari, G., and Palazzi, G., The influence of twisted tapes in subcritical, once-through vapor generators in counterflow, J. Heat Transf., vol. 96, pp. 365–370, 1974. 12.264 Robertson, J. M., The correlation of boiling coefficients in plate-fin heat exchanger passages with a filmflow model, Heat Transfer 1982 – Proc. 7th Int. Heat Transfer Conf, Munich, vol. 6, pp. 341–345, 1982. 12.265 Panchal, C. B., Hillis, D. L., and Thomas, A., Convective boiling of ammonia and freon 22 in plate heat exchangers, Proc. ASME/JSME Therm. Eng. Joint Conf., ASME, New York, NY, vol. 2, pp. 261–268, 1983. 12.266 Marseille, T. J., Carey, V. P., and Estergreen, S. L., Full-core test methods for experimental determination of convective boiling heat transfer coefficients in tubes of cross-flow compact evaporators, Exp. 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G., Celata, G. P., and Mariani, A., Flow boiling augmentation, in Handbook of Phase Change: Boiling and Condensation, S. Kandlikar, M. Shoji, and V. K. Dhir (editors), Taylor & Francis, Philadelphia, PA, chapter 18, pp. 495–522, 1999. 12.278 Stephan, K., and Kömer, M., Calculation of heat transfer in evaporating binary liquid mixtures, Chem. Ing. Tech., vol. 41, pp. 409–417, 1969. 12.279 Calus, W. F., and Leonidopoulos, D. J., Pool boiling – binary liquid mixtures, Int. J. Heat Mass Transf., vol. 17, pp. 249–256, 1974. 12.280 Fujita, Y., Bai, Q., and Tsutsui, M., Heat transfer of binary mixtures in nucleate boiling, 2nd Eur. Thermal Sci. and 14th UIT Nat. Heat Trans. Conf, G. P. Celata, P. DiMarco, and A. Mariani (editors), Edizioni ETS, Pisa, Italy, pp. 1639–1646, 1996. 12.281 Kandlikar, S. G., Boiling heat transfer with binary mixtures: Part I – a theoretical model for pool boiling, J. Heat Transf., vol. 120, pp. 380–387, 1998. 12.282 Florshuetz, L. W., and Khan, A. 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PROBLEMS 12.1 Liquid nitrogen flows upward in a vertical round tube with an inside diameter of 0.7 cm. The pressure along the tube is virtually constant at 360 kPa. The nitrogen enters subcooled at 80 K and a flow rate corresponding to a mass flux of 800 kg/m2s. Assuming the flow of liquid becomes fully developed immediately at the entrance, estimate the location (distance downstream of the inlet) at which the onset of boiling occurs. 12.2 Subcooled liquid ammonia at a temperature of 250 K and a pressure of 382 kPa enters a horizontal tube with an inside diameter of 1.2 cm. The mass flux is 1000 kg/m2s. Estimate the wall superheat necessary to initiate nucleate boiling at a location 0.5 m downstream of the inlet. 12.3 Subcooled water at 9460 kPa and 550 K enters a vertical evaporator tube with an inside diameter of 1.3 cm. The tube wall temperature is constant along the length of the tube at 600 K. Estimate the mass flux at which the onset of boiling just occurs at a distance 0.3 meters downstream of the inlet. If the mass flux is decreased below the onset level, does nucleate boiling persist, or is it suppressed? 12.4 Subcooled liquid water at 571 kPa flows through a vertical round tube having walls held at 460 K. The liquid enters the tube at 400 K, and the onset of boiling occurs immediately at the entrance of the tube. The tube diameter is 0.9 cm and the mass flux is 1200 kg/m2s. Determine the partial-boiling heat transfer coefficient predicted by Rohsenow’s method [12.17] at a location downstream of the inlet where the bulk fluid temperature is 420 Κ. 12.5 Subcooled liquid nitrogen at 229 kPa and 77.4 K enters a round tube with an inside diameter of 1.0 cm. The wall of the tube is held at a constant temperature of 90 K. The mass flux through the tube is 300 kg/m2s. At a location along the tube where the bulk fluid temperature is 81 K, estimate the partial-boiling heat transfer coefficient using (a) Rohsenow’s method [12.17] and (b) the method of Bergles and Rohsenow [12.5]. 12.6 Saturated flow boiling of R-134a at 537 kPa occurs in a vertical round tube with a diameter of 0.9 cm. The mass flux is 1000 kg/m2s, and a uniform heat flux of 7000 W/m 2 is applied to the wall. Estimate the quality at which nucleation is expected to be completely suppressed. 12.7 Saturated flow boiling of ammonia at 775 kPa occurs in a vertical round tube with a diameter of 1.3 cm. The mass flux is 200 kg/m2s, and a uniform heat flux of 50,000 W/m2 is applied to the wall. Estimate the quality at which nucleation is expected to be completely suppressed. 12.8 Use Kandlikar’s [12.53] correlation to predict the heat transfer coefficient for flow boiling of R-134a in a vertical tube at a pressure of 338 kPa and qualities of 0.3 and 0.7. The tube diameter is 1.2 cm, the mass flux is 250 kg/m2s and a uniform heat flux of 15 kW/m2 is applied to the tube wall. Compare the results to the predictions of the Shah [12.44] correlation for the same conditions. 12.9 Use the correlation of Bjorge et al. [12.50] to predict the heat transfer coefficient for flow boiling of water in a vertical tube at a pressure of 6124 kPa and qualities of 0.3 and 0.7. The tube diameter is 1.4 cm, the

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Liquid-Vapor Phase-Change Phenomena mass flux is 900 kg/m2s and the wall superheat is 25°C. Compare the results to the predictions of the Bennett and Chen [12.44] correlation for the same conditions. Saturated flow boiling of water at low wall superheat occurs in a vertical round tube with an inside diameter of 1.8 cm. The mass flux through the tube is 3000 kg/m 2s and the pressure along the tube is essentially constant at 6124 kPa. Estimate the dryout quality for these conditions. Saturated flow boiling of water occurs in a vertical round tube with an inside diameter of 0.9 cm. The mass flux through the tube is 4000 kg/m2s and the pressure along the tube is essentially constant at 9460 kPa. Estimate the critical heat flux at qualities of 0.1, 0.3, and 0.5 using the Levitan and Lantsman [12.87] correlation. Compare the results with the predictions of the correlation of Biasi et al. [12.110] for the same conditions. Flow boiling of nitrogen occurs in a vertical tube with a diameter of 0.8 cm and a length of 50 cm. The flow enters the tube as saturated liquid. The pressure is virtually uniform along the tube at 540 kPa. The mass flux through the tube is 300 kg/m2s. Use the correlation of Katto and Ohno [12.115] to predict the critical heat flux for these conditions. Convective vaporization of R-134a occurs at low wall superheat in a vertical tube with a diameter of 0.6 cm. The mass flux is 300 kg/m2s and the pressure is essentially constant at 338 kPa. For these conditions and an applied heat flux of 100 kW/m2, use the correlation of Groeneveld [12.170] to estimate the heat transfer coefficient for x = 0.7 and x = 0.9. Compare the results with those predicted by the correlation of Dougall and Rohsenow [12.161]. Write a computer program to compute the heat transfer coefficient for mist flow evaporation using the correlation of Groeneveld and Delorme [12.176], Use the program to compute the heat transfer coefficient value for the conditions specified in problem 12.13, and compare the computed values of the heat transfer coefficient and wall temperature with those predicted by the Groeneveld [12.170] correlation. Saturated liquid water at 9460 kPa enters a vertical tube 2.4 m long and having an inside diameter of 1.27 cm. The mass flow rate through the tube is 0.130 kg/s. A constant heat flux of 1.27 MW/m2 is applied all along the tube. The onset of boiling occurs immediately at the entrance of the tube (z = 0). The pressure is essentially constant along the tube, (a) Determine the regimes of convective boiling that exist along the tube and the portion of the tube (range of z) that corresponds to each regime, (b) Also estimate the wall temperature at z = 1.2 m. Water at 1172 kPa and 100°C enters (at z = 0) a horizontal copper tube that is 25 mm in inside diameter and 5.0 m long (the exit corresponds to z = 5.0 m). A uniform heat flux of 200 kW/m2 is applied to the tube. The water mass flow rate is 0.25 kg/s. The last 2 m of the tube contains a twisted-tape insert that for single-phase flow, produces a heat transfer coefficient of 6000 W/m2K. The first 2 m of the tube contains no insert and has a single-phase h value of 3000 W/m2K. Does nucleate boiling occur on the wall of the tube? If so, determine the portion, or portions of the tube (range of z) over which nucleate boiling is expected to occur. Saturated liquid water at 6124 kPa enters a vertical tube 1.5 m long and having an inside diameter of 1.27 cm. The mass flow rate through the tube is 0.191 kg/s. The applied heat flux to the tube is 3.65 MW/m2 over the first 0.5 m of the tube (0 < z ≤ 0.5 m) and 2.40 MW/m2 over the last 1.0 m of the tube (0.5 < z ≤ 1.5 m). The onset of boiling occurs immediately at the entrance of the tube (z = 0). The pressure is essentially constant along the tube, (a) Determine the regimes of convective boiling that exist along the tube and the portion of the tube (range of z) that corresponds to each regime, (b) Estimate the wall temperature at z = 0.8 m. (c) At z = 1.45 m, will the Hewitt and Roberts flow regime map shown in Fig. 12.3 predict the correct flow regime? Briefly explain your answer. Saturated flow boiling of R-134a occurs in a horizontal microchannel with an inside hydraulic diameter of 230 μm. The flow conditions are such that the mass flux through the channel is 200 kg/m 2s and the wall heat flux is 4.0 W/cm2. At a location along the channel, the pressure is 201 kPa and the quality is 0.5. Use the microchannel correlation of Kandlikar and Balasubramanian [12.238] to estimate the wall temperature at this location. Water flows at a mass flux of 2000 kg/m2s in a heated microchannel with a hydraulic diameter of 700 μm. Saturated liquid enters the tube and boiling occurs immediately at the inlet. The wall heat flux for the microchannel is 15.0 W/cm2. At a location downstream, the quality is 0.4 and the pressure is 247 kPa. Use the microchannel correlation of Kandlikar and Balasubramanian [12.238] to determine the wall temperature at this location.

Appendix I: Basic Elements of the Kinetic Theory of Gases The success of the kinetic theory of gases is a direct consequence of the relatively simple nature of most gases. The kinetic theory of gases is based on the following idealizations of the structure of gases: 1. The gas is composed of a very large number of moving particles (atoms or molecules). 2. The particles collide with one another infrequently. 3. Collisions between particles are elastic, that is, the total kinetic energy of the particles before and after the collision is constant. 4. Between collisions, the particles move in straight lines, in the absence of any force field, and hence obey Newton’s first law of motion. 5. The motion of the particles is completely random or chaotic. The relationship between the motion of molecules in the gas and the pressure P can be understood by considering the rectangular box of length L and cross-sectional area A shown in Fig. AI. 1. If a single molecule with a velocity u in the x direction hits the end of the box, it is reflected and travels in the opposite direction with velocity –u. Newton’s second law of motion requires that the force F exerted on the wall during the collision is equal to the rate of change of momentum:

F=

d (mu) (I.1) dt

where m is the mass of the molecule. The momentum of the molecule before collision is mu, whereas after the collision it is –mu. The change in the momentum is –2mu. The total change in momentum per unit time due to collisions of this molecule with the wall is equal to the change in momentum per collision multiplied by the number of collisions per unit time that the molecule makes with the wall. The time between collisions is equal to the time it takes the molecule to travel to the other end of the box and back again, 2L/u. Thus the total change in momentum per unit time is –2mu(u/2L), which must equal the force F on the molecule. The force on the wall, Fw = –F, must therefore equal mu2/L. Since the pressure (normal stress) P′ exerted by the molecule must equal Fw/A,

P′ =

mu 2 mu 2 = (I.2) AL V

where V is the volume of the box. If we have a collection of particles all moving in the x direction with various speeds u1, u2, . . ., un then the total force, and consequently the total pressure, will equal the sum of the contributions from all particles:

P=

m V

n

∑(u ) (I.3) i

2

i =1

663

664

Appendix I: Basic Elements of the Kinetic Theory of Gases

FIGURE AI.1  A single particle in a box moving between two parallel walls.

Defining the average of the squares of the velocities, 〈u2〉, as

〈u 2 〉 =

1 n

n

∑(u ) (I.4) i

2

i =1

where n is the total number of particles in the box, the expression for the pressure becomes

P=

nm 2 〈u 〉 (I.5) V

Because the molecules in this model can move in only one dimension, this can be thought of as being the equation for the pressure of a one-dimensional gas. Note that, so far, we have ignored the effect of molecular collisions. Since the molecules can move only in the x direction, collisions in this one-dimensional gas can occur only if two molecules travel in exactly the same linear path. If we consider two such molecules, as shown in Fig. AI.2, it is clear that the leftmost molecule will never hit the right wall and the rightmost molecule will never hit the left wall. However, because of collisions in the center of the box, the leftmost molecule will strike the left wall twice as often as it would if the two molecules traveled parallel paths and did not collide. Similarly, the rightmost molecule in Fig. AI.2 will strike the right wall twice as often as it would if the molecules traveled parallel paths. On the average, then, the net effect of the molecules traveling the same or parallel paths is identical. This implies that Eq. (I.5) is valid for a one-dimensional gas even if some of the molecules collide. In a real gas, of course, molecules can move in three dimensions, and the velocity vector of a molecule has three components as shown in Fig. AI.3. The square of the velocity vector is related to the square of its components as

c 2 = u 2 + v 2 + w 2 (I.6)

For any given molecule, the velocity components u, v, and w may generally all be different. However, averaging both sides of Eq. (I.6) over all the molecules yields

〈c 2 〉 = 〈u 2 〉 + 〈 v 2 〉 + 〈 w 2 〉 (I.7)

FIGURE AI.2  Two particles moving along a collinear path between two parallel walls.

Appendix I: Basic Elements of the Kinetic Theory of Gases

665

FIGURE AI.3  Graphical representation of a differential element in Cartesian velocity space.

Since there is no reason to believe that any of the directions x, y, or z is preferred, it is expected that 〈u2〉 = 〈v2〉 = 〈w2〉. This conclusion, together with Eq. (I.7), implies that

〈u 2 〉 = 〈 v 2 〉 = 〈 w 2 〉 =

1 2 〈c 〉 (I.8) 3

If we now consider a molecule that strikes a wall in a plane normal to the x axis, as shown in Fig. AI.4, it can be seen that only the normal velocity u is reversed in the collision. The tangential components v and w (not shown in Fig. AI.4) have the same direction and magnitude before and after the collision. Hence, only reversal of the normal velocity component is relevant to the momentum interaction with the wall. The frequency with which molecules strike the wall also depends on the normal velocity component only. The change in momentum per collision with the wall is –2mu, and the number of impacts per second is u/2L. By the same reasoning used for the one-dimensional gas, the pressure on the wall must equal

P=

nm 2 〈u 〉 (I.9) V

FIGURE AI.4  Particle velocity components before and after a perfectly elastic collision with a wall.

666

Appendix I: Basic Elements of the Kinetic Theory of Gases

Using the expression (I.8) for 〈u2〉, Eq. (I.9) becomes 1  nm  2   〈c 〉 (I.10) 3 V 

P=



If we denote the kinetic energy of a molecule as ε = 12 mc 2, Eq. (I.10) can be written as PV =



2 n 〈ε〉 (I.11) 3

Equation (I.11) resembles the ideal gas relation, ˆ (I.12) PV = NRT



Eliminating PV by combining Eqs. (I.11) and (I.12) yields 2 ˆ NRT = n 〈ε〉 (I.13) 3



Noting that n / Nˆ is equal to Avogadro’s number NA, Eq. (I.13) can be written as RT =



2 N A 〈ε〉 (I.14) 3

If we denote the total energy associated with the random motion of the molecules in one mole of gas as û, then û must be given by

uˆ =

1 N A m 〈c 2 〉 = N A 〈ε〉 (I.15) 2

Combining Eqs. (I.14) and (I.15), we obtain

uˆ =

3 RT (I.16) 2

Equation (I.16) is one of the most interesting results of the kinetic theory. This relation links the temperature of the gas directly to the kinetic energy associated with the random motion of the molecules. For this reason, the random or chaotic motion of the molecules is sometimes referred to as the thermal motion of the molecules. This also implies that at absolute zero this thermal motion ceases completely. If we define a root-mean-square (RMS) speed of the molecules, cRMS, as

cRMS = 〈c 2 〉1/2 (I.17)

then Eqs. (I.15) and (I.16) can be combined to obtain 1/2



cRMS

 3 RT  = (I.18)  M 

where M = N A m is the molecular mass of the gas. Thus kinetic theory also provides a link between the mean speed of the gas molecules and the temperature of the gas. To develop the kinetic theory of gases further, we must first consider the concept of distribution functions. So far, kinetic theory has provided a relation between the RMS speed of the molecules and the temperature of the system. However, individual molecules in the gas are traveling in various

Appendix I: Basic Elements of the Kinetic Theory of Gases

667

directions at different speeds. In general, each molecule will have a different combination of u, v, and w velocity components. Let dnu be the number of molecules having a component of velocity in the x direction in the range between u and u + du. The probability of finding such a molecule in a box containing n molecules is dnu/n. If the width of the interval is du is small, it is reasonable to expect that the number of molecules in the interval will be proportional to its size du (i.e., doubling its size would double the number of molecules in the interval). It is therefore expected that dnu/n is proportional to du. Because the probability dnu/n will depend on the velocity component u itself, the variation of dnu/n with u and du is postulated as being of the form

dnu = f (u 2 )du (I.19) n

where f(u2) is an as-yet-undetermined function. Note that f is postulated to be a function of u2 rather than u. Because the molecular motion is completely random, motion in all directions is equally probable. Consequently, the probability of finding a molecule with velocity u between u and u + du is equal to that of finding one with a velocity between –u and –u –du. By assuming that f is a function of u2, this symmetry condition is automatically satisfied. Note also that it is implicitly assumed that the probability dnu/n is independent of the velocity components v and w in the y and z directions, respectively. Although arguments can be presented to justify this assumption, here we will simply adopt it as a plausible idealization that is consistent with the expected symmetry of the probability distribution. Because the arguments described above apply equally well to the velocity components in the y and z directions, we must also have

dnv = f ( v 2 ) dv (I.20) n



dnw = f ( w 2 )dw (I.21) n

From basic probability theory, it is known that the probability of three independent things occurring simultaneously must equal the product of the probabilities of each occurring individually. It follows, therefore, that the probability of a molecule simultaneously having velocity components in the intervals u to u + du, v to v + dv, and w to w + dw is equal to the product (dnu/n)(dnv/n)(dnw/n). Denoting this probability as dnuvw/n, we thus obtain

dnuvw = f (u 2 ) f ( v 2 ) f ( w 2 )  dudvdw (I.22) n

Noting that dnuvw /(dudvdw) is the number density of molecules at a particular location in velocity space coordinates, u, v, w, shown in Fig. AI.3, the above equation can be written as

dnuvw = nf (u 2 ) f ( v 2 ) f ( w 2 ) (I.23) dudvdw

Suppose now that the above analysis was repeated with a different set of coordinate axes that had the same origin as our original set but were rotated relative to the original ones. If we denote these new coordinates as x′, y′, and z′ and the respective velocities in each direction as u′, v′, and w′, the line of reason previously described implies that the number density of molecules must be given by

dnu′v ′w′ = nf (u ′ 2 ) f ( v ′ 2 ) f ( w′ 2 ) (I.24) du ′dv ′dw ′

668

Appendix I: Basic Elements of the Kinetic Theory of Gases

Because the number density at a given location in velocity space must be independent of the coordinate system used, we conclude from Eqs. (I.23) and (I.24) that

f (u 2 ) f ( v 2 ) f ( w 2 ) = f (u ′ 2 ) f ( v ′ 2 ) f ( w′ 2 ) (I.25)

Also, since (u, v, w) and (u′, v′ w′) represent the same point,

c 2 = u 2 + v 2 + w 2 = u ′ 2 + v ′ 2 + w′ 2 (I.26)

Furthermore, because the rotation of the primed axes is arbitrary, for convenience, we establish them by rotating the original axes in Fig. AI.3 (about the original v and w axes) so that in the primed coordinate system the point (u, v, w) corresponds to (u′, 0, 0). Using the fact that v’ = w’ = 0 and Eqs. (I.25) and (I.26), it is easily shown that

f (u 2 ) f ( v 2 ) f ( w 2 ) = A2 f (u 2 + v 2 + w 2 ) (I.27)

where

A = f (0) (I.28)

If both sides of Eq. (I.27) are differentiated in turn with respect to u2, v2, and w2, three relations are obtained, which indicate that

f ′(u 2 ) f ( v 2 ) f ( w 2 ) = f (u 2 ) f ′( v 2 ) f ( w 2 ) = f (u 2 ) f ( v 2 ) f ′( w 2 ) (I.29)

where f ′ denotes the derivative of f with respect to its argument. These relations can be rearranged to show that

f ′(u 2 ) f ′( v 2 ) f ′( w 2 ) = = (I.30) f (u 2 ) f (v 2 ) f (w2 )

Because the expressions given above are each functions of a different independent variable, the equalities can hold only if they all equal a constant, which will designate as –β. If each one of these expressions is set equal to β and the differential equations are solved, it is found that 2



f (u 2 ) = Ae −βu (I.31a)



f ( v 2 ) = Ae −βv (I.31b)



f ( w 2 ) = Ae −βw (I.31c)

2

2

Note that either positive or negative values of β would satisfy Eq. (I.30). However, if β were negative, f would become infinite as u2 approaches infinity. This would mean that the probability of finding molecules with infinite kinetic energy is infinite, which is impossible for a gas having finite internal energy. We therefore conclude that only positive values of β are physically realistic. The variation of f for positive β values is the well-known Gaussian distribution associated with many random processes. Substituting Eqs. (I.31a)–(I.31c) into Eq. (I.23) yields the following relation for the number density of molecules in velocity space:

2 2 2 dnuvw = nA3e −β (u + v + w ) (I.32) dudvdw

669

Appendix I: Basic Elements of the Kinetic Theory of Gases

FIGURE AI.5  Velocity vector in Cartesian and spherical coordinates.

It is also useful to determine the number density of molecules with speed c in the interval c to c + dc, regardless of direction. In the velocity space shown in Fig. AI.3, this corresponds to the number of points inside a spherical shell of radius c and thickness dc. If the velocity vector is transformed into the spherical coordinates shown in Fig. AI.5 using the relations

u = c sin θ cos φ

v = c sin θ sin φ

w = c cos θ (I.33a)

c = (u 2 + v 2 + w 2 )1/2 (I.33b)



the velocity distribution can be written as

2

dncθφ = nA3c 2e −βc sin θ dθ dφ dc (I.34)

The left side of Eq. (I.34) is the number of molecules with speed c and directional angles in the ranges c to c + dc, θ to θ + dθ, ϕ to ϕ + dϕ. The number of molecules with speeds in the range c to c + dc, regardless of direction, can be found by integrating Eq. (I.34) over all values of ϕ and θ: φ= 2 π θ=π



dnc =

∫ ∫ dn

2

cθφ

= 4 π nA3c 2 e −βc dc (I.35)

φ= 0 θ= 0

Although this analysis has suggested the form of the relation for the velocity distribution, the constants A and β are, as yet, undetermined. These constants can be determined, however, by requiring that the distribution be consistent with the total number of molecules and the total energy of the system. The total number of molecules n must equal the integral of dnc over all possible speeds c from zero to infinity: c =∞

n=



∫ dn (I.36) c

c= 0

Substituting Eq. (I.35) for dnc into Eq. (I.36) yields c =∞



n=

∫ 4π nA c e

3 2 −βc 2

c= 0

dc (I.37)

670

Appendix I: Basic Elements of the Kinetic Theory of Gases

It is a straightforward manipulation to evaluate this integral. Upon doing so and solving for A, the following relation is obtained:  β A=   π



1/2

(I.38)

The average kinetic energy 〈ε〉 per molecule is obtained by multiplying 12 mc 2 by the number dnc of molecules (in each differential speed range), integrating over all values of c, and dividing by n: ε =



1 n

c =∞

1

∫ 2 mc dn (I.39) 2

c

c= 0

Substituting Eq. (I.35) for dnc into Eq. (I.39) yields

1 ε = n

c =∞

1

∫ 2 mc (4π n) A c e

3 2 −βc 2

2

dc (I.40)

c= 0

Canceling n on the right-hand side, substituting Eq. (I.38) to eliminate A, and evaluating the integral yields the following expression for 〈ε〉: 〈ε〉 =



3m (I.41) 4β

It has been shown above, using the kinetic theory, that the average energy per molecule 〈ε〉 is related to the temperature of the gas as

〈ε〉 =

3 R  3 T = k B T (I.42) 2  N A  2

where kB is the Boltzmann constant, defined as

kB =

R (I.43) NA

Combining Eqs. (I.41) and (I.42) yields the following relation for β:

β=

m (I.44) 2k BT

It follows directly using Eqs. (I.38) and (I.44) that 1/2



 m  A=  (I.45)  2π k B T 

Replacing A and β in Eq. (I.35) with the relations given in Eqs. (I.44) and (I.45) yields

 m  dnc = 4 π n    2π k B T 

3/2

c 2e − mc

2

/2 k B T

dc (I.46)

The number distribution of molecules given by the above equation is the Maxwell distribution.

671

Appendix I: Basic Elements of the Kinetic Theory of Gases

If the relations (I.44) and (I.45) are substituted into Eqs. (I.31a)–(I.31c), it can be shown using Eqs. (I.19)–(I.21) that the number distributions for the individual velocity components are given by dnu  m  =  n  2π k B T 

1/2



dnv  m  =  n  2π k B T 

1/2



1/2



dnw  m  =  n  2π k B T 

2

/2 k B T

du (I.47a)

2

/2 k B T

dv (I.47b)

2

/2 k B T

dw (I.47c)

e − mu

e − mv

e − mw

Substitution of Eqs. (I.44) and (I.45) into Eq. (I.32) also yields the following form of the Maxwell distribution:

 m  dnuvw = n    2π k B T 

3/2

e − m (u

2

+ v 2 + w 2 )/2 k B T

du dv dw (I.48)

Associated with the Maxwell distribution of molecular velocities, one can consider the most probable speed cmp, the root-mean-square speed 〈c2〉1/2, and the mean speed 〈c〉. These quantities can be determined directly from Eq. (I.46). The most probable speed is the value of c that maximizes dnc/dc. Solving Eq. (I.46) for this quantity, differentiating, and setting the result to zero yields 1/2

 2 k BT  cmp =  (I.49)  m 



The mean speed 〈c〉 and the root-mean-square speed 〈c2〉1/2, each of which is weighted with the fraction of the molecules in each speed interval, are calculated using Eq. (I.46) as ∞

c =



∫ 0

dn  c  c  dc  ndc 

 8 kBT  =  m  12



c

2 12

∞ dn   =  c 2  c  dc   ndc    0 

∫

12

(I.50) 12

 3 kBT  =  m  (I.51)

Thus the three representative molecular velocities defined above are all proportional to (kBT/m)1/2, with the proportionality constant ranging from 21/2 to 31/2. The kinetic theory of gases provides an important conceptual foundation for understanding the behavior of real gases. Further discussion of the kinetic theory and its implications can be found in references [4.10–4.12].

Appendix II: Saturation Properties of Selected Fluids* ACETONE Critical temperature: 508.15 K Critical pressure: 4761 kPa Critical density: 273 kg/m3

Chemical formula: CH3COCH3 Molecular weight: 58.1 Tsat (K) Psat (kPa) ρl (kg/m3) ρv (kg/m3) hlv (kJ/kg) cpl (kJ/kgK) cpv (kJ/kgK) μι (μNs/m2) μv (μNs/m2) kl (mW/mK) kv (mW/mK) Prl Prv σ (mN/m)

329.25 101.3 750 2.23 506 2.28 1.41 235 9.4 142 12.7 3.77 1.04 18.4

340 152 736 3.11 494 2.32 1.46 213 9.8 137 14.1 3.61 1.01 17.0

360 274 710 5.49 465 2.42 1.55 188 10.4 129 16.1 3.53 1.00 14.5

380 452 683 9.13 439 2.53 1.66 165 11.1 121 18.5 3.49 1.00 12.1

400 731 655 14.5 414 2.65 1.79 141 11.8 112 21.2 3.34 1.00 9.6

420 1082 625 22.3 382 2.83 1.95 119 12.6 104 24.2 3.24 1.02 7.1

440 1637 590 33.6 344 3.03 2.18 99 13.5 96 27.2 3.12 1.08 4.6

460 2279 553 50.3 300 3.29 2.54 80 14.4 87 31.0 3.03 1.18 3.1

480 3252 504 77.2 242 3.76 3.38 64 15.8 77 36.0 3.13 1.48 1.6

508.15 4761 273 273

49 49 58 58

AMMONIA Critical temperature: 405.55 K Critical pressure: 11290 kPa Critical density: 235 kg/m3

Chemical formula: NH3 Molecular weight: 17.032 Tsat (K) Psat (kPa) ρl (kg/m3) ρv (kg/m3) hlv (kJ/kg) cpl (kJ/kgK) cpv (kJ/kgK) μι (μNs/m2) μv (μNs/m2) kl (mW/mK) kv (mW/mK) Prl Prv σ(mN/m)

*

239.75 101.3 682 0.86 1368 4.472 2.12 285 9.25 614 18.8 2.06 1.04 33.9

250 165.4 669 1.41 1338 4.513 2.32 246 9.59 592 19.8 1.88 1.11 31.5

270 381.9 643 3.09 1273 4.585 2.69 190 10.30 569 22.7 1.58 1.17 26.9

290 775.3 615 6.08 1200 4.649 3.04 152 11.05 501 25.2 1.39 1.25 22.4

310 1424.9 584 11.0 1115 4.857 3.44 125 11.86 456 28.9 1.36 1.31 18.0

330 2422 551 18.9 1019 5.066 3.90 105 12.74 411 34.3 1.32 1.34 13.7

350 3870 512 31.5 899 5.401 4.62 88.5 13.75 365 39.5 1.34 1.49 9.60

370 5891 466 52.6 744 5.861 6.21 70.2 15.06 320 50.4 1.41 1.70 5.74

390 8606 400 93.3 508 7.74 8.07 50.7 17.15 275 69.2 1.43 1.86 2.21

400 10280 344 137 307

39.5 19.5 252 79.4

0.68

Adapted from the Heat Exchanger Design Handbook, with permission, copyright © Hemisphere, New York, 1983. (Some refrigerant properties are included from the ASHRAE Handbook – Fundamentals, ASHRAE, Atlanta, GA, 2009.)

673

674

Appendix II: Saturation Properties of Selected Fluids

n-BUTANOL Critical temperature: 561.15 K Critical pressure: 4960 kPa Critical density: 270.5 kg/m3

Chemical formula: C2H5CH2CH2OH Molecular weight: 74.12 Tsat (K) Psat (kPa) ρl (kg/m3) ρv (kg/m3) hlv (kJ/kg) cpl (kJ/kgK) cpv (kJ/kgK) μι (μNs/m2) μv(μNs/m2) kl (mW/mK) kv(mW/mK) Prl Prv σ (mN/m)

390.65 101.3 712 2.30 591.3 3.20 1.87 403.8 9.29 127.1 21.7 10.3 0.81 17.1

410.2 182 688 4.10 565.0 3.54 1.95 346.1 10.3 122.3 24.2 9.86 0.83 15.6

429.2 327 664 7.9 537.3 3.95 2.03 278.8 10.7 117.5 26.7 9.17 0.81 13.9

446.5 482 640 12.5 509.7 4.42 2.14 230.8 11.4 112.6 28.2 8.64 0.86 12.3

469.5 759 606 23.8 468.8 5.15 2.24 188.5 12.1 105.4 31.3 10.2 0.87 10.2

485.2 1190 581 27.8 437.2 5.74 2.37 144.2 12.7 101.4 33.1 8.10 0.91 7.50

508.3 1830 538 48.2 382.5 6.71 2.69 130.8 13.9 91.7 36.9 8.67 1.01 6.44

530.2 2530 487 74.0 315.1 7.76 3.05 115.4 15.4 82.9 40.2 9.08 1.17 4.23

545.5 3210 440 102.3 248.4

558.9 4030 364 240.2 143.0

3.97 111.5 17.1 74.0 43.6

105.8 28.3 62.8 51.5

1.56 2.11

0.96

CARBON TETRACHLORIDE Critical temperature: 556.35 K Critical pressure: 4560 kPa Critical density: 588 kg/m3

Chemical formula: CCl4 Molecular weight: 153.8 Tsat (K) Psat (kPa) ρl(kg/m3) ρv(kg/m3) hlv(kJ/kg) cpl(kJ/kgK) cpv(kJ/kgK) μl(μNs/m2) μv(μNs/m2) kl(mW/mK) kv(mW/mK) Prl Prv σ (mN/m)

349.95 101.3 1484 5.44 195 0.92 0.58 494 11.9 92 8.6 4.94 0.80 20.2

370 184 1442 9.40 188 0.94 0.60 407 12.5 87 9.3 4.40 0.81 17.6

390 307 1397 15.2 180 0.97 0.62 352 13.3 83 10.0 4.16 0.82 15.4

410 473 1351 23.4 172 1.01 0.65 309 14.1 78 10.7 4.08 0.85 13.1

430 701 1303 34.8 159 1.06 0.68 274 14.9 74 11.5 3.93 0.88 10.9

450 1020 1250 50.3 152 1.14 0.73 241 15.7 70 12.3 3.92 0.93 8.8

470 1390 1199 71.2 140 1.24 0.80 205 16.7 65 13.2 3.91 1.01 6.9

495 2020 1107 108.5 126 1.36 0.91 154 18.9 57 14.3 3.67 1.20 4.4

525 3160 989 184.5 98 1.57 1.30 98 21.0 45 16.3 3.42 1.67 2.0

556.35 4560 588 588

63 63 25 25

675

Appendix II: Saturation Properties of Selected Fluids

ETHANOL Critical temperature: 516.25 K Critical pressure: 6390 kPa Critical density: 280 kg/m3

Chemical formula: CH3CH2OH Molecular weight: 46.1 Tsat (K) Psat (kPa) ρl (kg/m3) ρv (kg/m3) hlv (kJ/kg) cpl (kJ/kgK) cpv (kJ/kgK) μι (μNs/m2) μv (μNs/m2) kl (mW/mK) kv (mW/mK) Prl Prv σ (mN/m)

351.45 101.3 757.0 1.435 963.0 3.00 1.83 428.7 10.4 153.6 19.9 8.37 0.96 17.7

373 226 733.7 3.175 927.0 3.30 1.92 314.3 11.1 150.7 22.4 6.88 0.95 15.7

393 429 709.0 5.841 885.5 3.61 2.02 240.0 11.7 146.5 24.5 5.91 0.96 13.6

413 753 680.3 10.25 834.0 3.96 2.11 185.5 12.3 141.9 26.8 5.18 0.97 11.5

433 1256 648.5 17.15 772.9 4.65 2.31 144.6 12.9 137.2 29.3 4.90 1.02 9.3

453 1960 610.5 27.65 698.9 5.51 2.80 113.6 13.7 134.8 32.1 4.64 1.20 6.9

473 2940 564.0 44.40 598.3 6.16 3.18 89.6 14.5 129.1 35.3 4.28 1.31 4.5

483 3560 537.6 56.85 536.7 6.61 3.78 79.7 15.1 125.6 37.8 4.19 1.51 3.3

503 5100 466.2 101.1 387.3

513 6020 420.3 160.2 280.5

6.55 63.2 16.7 108.0 43.9

56.3 18.5 79.11 50.7

2.49 0.9

0.34

MERCURY Critical temperature: 1763.2 K Critical pressure: 151,000 kPa Critical density: 5500 kg/m3

Chemical formula: Hg Molecular weight: 200.51 Tsat (K) Psat (kPa) ρl (kg/m3) ρv (kg/m3) hlv (kJ/kg) cpl (kJ/kgK) cpv (kJ/kgK) μl (μNs/m2) μv (μNs/m2) kl (mW/mK) kv (mW/mK) Prl Prv σ (mN/m)

630.1 101.3 12737 3.91 294.9 0.136 0.104 884 61.7 121.9 10.4 0.987 0.617 417

650 145 12688 5.37 294.2 0.136 0.104 870 63.5 123.6 10.8 0.957 0.612 413

700 316 12567 10.9 292.3 0.137 0.105 841 68.6 128.0 11.7 0.900 0.616 403

750 620 12444 20.1 290.2 0.138 0.106 816 73.5 131.9 12.6 0.854 0.618 393

800 1120 12318 34.2 287.8 0.140 0.107 794 78.4 135.1 13.5 0.823 0.621 383

850 1880 12190 54.6 285.1 0.142 0.108 776 83.5 137.8 14.4 0.800 0.626 372

900 2990 12059 82.7 282.1 0.144 0.109 760 88.4 141.8 15.3 0.772 0.630 362

950 4530 11927 119.9 278.6 0.146 0.111 746 93.2 144.5 16.2 0.754 0.637 352

1000 6580 11791 167.7 274.7 0.149 0.113 736 98.0 146.9 17.2 0.744 0.644 341

1050 9230 11650 227.3 269.2 0.153 0.116 723 103.0 147.9 18.1 0.748 0.660 331

676

Appendix II: Saturation Properties of Selected Fluids

METHANOL Critical temperature: 513.15 K Critical pressure: 7950 kPa Critical density: 275 kg/m3

Chemical formula: CH3OH Molecular weight: 32.00 Tsat (K) Psat (kPa) ρl (kg/m3) ρv (kg/m3) hlv (kJ/kg) cpl (kJ/kgK) cpv (kJ/kgK) μl (μNs/m2) μv (μNs/m2) kl (mW/mK) kv (mW/mK) Prl Prv σ (mN/m)

337.85 101.3 751.0 1.222 1101 2.88 1.55 326 11.1 191.4 18.3 5.13 0.94 18.75

353.2 178.4 735.5 2.084 1070 3.03 1.61 271 11.6 187.0 20.6 4.67 0.91 17.5

373.2 349.4 714.0 3.984 1022 3.26 1.69 214 12.4 181.3 23.2 4.15 0.90 15.7

393.2 633.3 690.0 7.142 968 3.52 1.83 170 13.1 178.5 26.2 3.61 0.92 13.6

413.2 1076 664.0 12.16 922 3.80 1.99 136 14.0 170.0 29.7 3.34 0.94 11.5

433.2 1736 634.0 19.94 843 4.11 2.20 109 14.9 164.0 33.8 2.82 0.97 9.3

453.2 2678 598.0 31.86 756 4.45 2.56 88.3 16.0 158.7 39.4 2.56 1.04 6.9

473.2 3970 553.0 50.75 645 4.81 3.65 71.6 17.4 153.0 46.9 2.42 1.35 4.5

493.2 5675 490.0 86.35 482

511.7 7775 363.5 178.9

5.40 58.3 20.1 147.3 60.0

41.6 26.0 142.0 98.7

1.81 2.1

0.09

NITROGEN Critical temperature: 126.25 K Critical pressure: 3396 kPa Critical density: 304 kg/m3

Chemical formula: N2 Molecular weight: 28.016 Tsat (K) Psat (kPa) ρl (kg/m3) ρv (kg/m3) hlv (kJ/kg) cpl (kJ/kgK) cpv (kJ/kgK) μl (μNs/m2) μv (μNs/m2) kl (mW/mK) kv (mW/mK) Prl Prv σ (mN/m)

77.35 101.3 807.10 4.621 197.6 2.064 1.123 163 5.41 136.7 7.54 2.46 0.81 8.85

85 229 771.01 9.833 188.0 2.096 1.192 127 5.60 122.9 8.18 2.17 0.82 7.20

90 360 746.27 15.087 180.5 2.140 1.258 110 6.36 112.0 9.04 2.10 0.89 6.16

95 540 719.42 22.286 172.2 2.211 1.350 97.2 6.80 104.0 9.77 2.07 0.94 4.59

100 778 691.08 31.989 162.2 2.311 1.474 86.9 7.28 95.5 10.60 2.10 1.01 3.67

105 1083 660.5 44.984 150.7 2.467 1.666 78.5 7.82 88.0 11.69 2.20 1.11 2.79

110 1467 626.17 62.578 137.0 2.711 1.975 70.8 8.42 80.2 14.50 2.39 1.15 1.98

115 1940 583.43 87.184 119.9 3.180 2.586 59.9 9.25 70.4 20.76 2.71 1.16 1.18

120 2515 528.54 124.517 95.7 4.347 4.136 48.4 10.68 62.8 30.91 3.35 1.43 0.52

126 3357 379.22 237.925 32.1

19.1 19.1 52.8 51.11

0.01

677

Appendix II: Saturation Properties of Selected Fluids

OXYGEN Critical temperature: 154.77 K Critical pressure: 5090 kPa Critical density: 405 kg/m3

Chemical formula: O2 Molecular weight: 32.00 Tsat (K) Psat (kPa) ρl (kg/m3) ρv (kg/m3) hlv (kJ/kg) cpl (kJ/kgK) cpv (kJ/kgK) μl (μNs/m2) μv (μNs/m2) kl (mW/mK) kv (mW/mK) Prl Prv σ (mN/m)

90.18 101.3 1135.72 4.48 212.3 1.63 0.96 195.83 6.85 148 8.5 2.16 0.77 13.19

97 196 1102.05 8.23 205.9 1.66 1.00 161.75 7.50 139 9.5 1.93 0.79 11.53

104 352 1065.07 14.14 198.3 1.70 1.05 136.55 8.35 130 10.5 1.79 0.84 9.88

111 583 1025.64 22.79 189.4 1.76 1.12 116.80 9.36 121 11.7 1.70 0.90 8.27

118 908 982.32 35.03 178.7 1.86 1.23 101.20 10.6 111 13.4 1.70 0.97 6.71

125 1348 934.58 52.05 165.7 2.00 1.36 89.00 11.24 102 14.8 1.75 1.03 5.20

132 1924 880.28 75.81 150.1 2.22 1.68 80.15 13.35 92.5 16.9 1.92 1.33 3.77

140 2782 808.41 116.12 127.3 2.63 2.27 69.66 15.8 82.0 20.1 2.23 1.78 2.23

146 3591 737.56 163.34 104.6 3.28 3.63 60.65 18.5 71.2 23.6 2.79 2.85 1.18

154 3939 557.10 304.41 46.1

42.48 26.9 35.2 19.93 0.40

1-PROPANOL Critical temperature: 536.85 K Critical pressure: 5050 kPa Critical density: 273 kg/m3

Chemical formula: CH3CH2CH2OH Molecular weight: 60.1 Tsat (K) Psat (kPa) ρl (kg/m3) ρv (kg/m3) hlv (kJ/kg) cpl (kJ/kgK) cpv (kJ/kgK) μl (μNs/m2) μv (μNs/m2) kl (mW/mK) kv (mW/mK) Prl Prv σ (mN/m)

373.2 109.4 732.5 2.26 687 3.21 1.65 447 9.61 142.4 20.9 10.1 0.76 17.6

393.2 218.5 711 4.43 645 3.47 1.82 337 10.3 139.2 23.0 8.40 0.82 16.15

413.2 399.2 687.5 8.05 594 3.86 1.93 250 10.9 138.4 26.2 6.97 0.80 14.42

433.2 683.6 660 13.8 544 4.36 2.05 188 11.5 133.5 28.9 5.14 0.82 12.7

453.2 1089 628.5 22.5 486 5.02 2.20 148 12.2 127.9 31.4 5.81 0.85 10.77

473.2 1662 592.0 35.3 427 5.90 2.36 119 12.9 120.7 34.7 5.82 0.88 8.85

493.2 2426 548.5 55.6 356 6.78 2.97 90.6 14.2 111.8 38.0 5.50 1.11 6.35

513.2 3402 492.0 90.4 264 7.79 3.94 70.0 15.7 100.6 43.9 5.42 1.41 4.04

523.2 3998 452.5 118.0 209

533.1 4689 390.5 161.0 138

61.4 17.0 94.1 47.5

53.9 19.3 89.3 53.5

2.6

0.96

678

Appendix II: Saturation Properties of Selected Fluids

REFRIGERANT-12 Critical temperature: 348.8 K Critical pressure: 4132 kPa Critical density: 561.8 kg/m3

Chemical formula: CCl2F2 Molecular weight: 120.92 Tsat (K) Psat (kPa) ρl (kg/m3) ρv (kg/m3) hlv (kJ/kg) cpl (kJ/kgK) cpv (kJ/kgK) μl (μNs/m2) μv (μNs/m2) kl (mW/mK) kv (mW/mK) Prl Prv σ (mN/m)

243.2 101.3 1486 6.33 168.3 0.896 0.569 373 10.3 95.1 6.9 3.51 0.85 15.5

260 200 1436 11.8 161.5 0.911 0.614 303 11.0 87.4 7.7 3.16 0.88 13.5

275 333 1388 19.2 154.7 0.932 0.646 262 11.7 80.5 8.4 3.03 0.90 11.4

290 528 1338 29.9 146.6 0.957 0.689 231 12.5 73.3 9.2 3.02 0.94 9.4

305 793 1284 44.8 137.7 0.990 0.746 208 13.3 66.8 10.0 3.14 0.99 7.7

320 1145 1225 65.4 127.2 1.03 0.825 187 14.2 59.8 10.8 3.22 1.08 5.9

335 1602 1157 94.6 114.0 1.08 0.920 167 15.2 53.0 11.6 3.40 1.21 4.2

350 2183 1075 136.4 97.6 1.13 1.22 144 16.5 46.2 12.3 3.52 1.64 2.8

365 2907 969.7 203.2 75.8 1.22 1.68 119 18.1 39.2 13.4 3.70 2.27 1.3

384.8 4132 561.8 561.8

15.4 15.4

REFRIGERANT-22 Critical temperature: 369.3 K Critical pressure: 4986 kPa Critical density: 513 kg/m3

Chemical formula: CHClF2 Molecular weight: 86.48 Tsat (K) Psat (kPa) ρl (kg/m3) ρv (kg/m3) hlv (kJ/kg) cpl (kJ/kgK) cpv (kJ/kgK) μl (μNs/m2) μv (μNs/m2) kl (mW/mK) kv (mW/mK) Prl Prv σ (mN/m)

242.4 101.3 1413 4.70 233.4 1.10 0.599 332 10.1 119 7.15 3.07 0.85 18.3

250 218 1360 9.59 225.6 1.13 0.646 282 10.9 109 8.22 2.92 0.86 15.5

265 376 1313 16.1 210.8 1.16 0.691 251 11.7 101 9.10 2.88 0.89 13.0

280 619 1260 26.3 198.6 1.19 0.747 225 12.3 94.2 10.1 2.84 0.91 10.6

295 958 1206 40.6 185.2 1.24 0.820 204 13.2 86.6 11.2 2.92 0.97 8.4

310 1420 1146 60.9 169.8 1.30 0.930 187 14.2 78.8 12.4 3.09 1.07 6.2

325 2020 1076 90.2 151.6 1.41 1.09 172 15.7 70.2 14.0 3.45 1.22 4.3

340 2800 991 134 128.7 1.65 1.40 150 16.4 59.2 16.0 4.18 1.69 2.5

355 3800 877 208 95.7 2.43 2.31 119 18.8 44.0 18.8 6.89 2.31 1.0

369.3 4986 513 513

31.9 31.9

679

Appendix II: Saturation Properties of Selected Fluids

REFRIGERANT 134A Critical temperature: 374.2 K Critical pressure: 4059.3 kPa Critical density: 511.9 kg/m3

Chemical formula: CF3CH2F Molecular weight: 102.03 Tsat (K) Psat (kPa) ρl (kg/m3) ρv (kg/m3) hlv (kJ/kg) cpl (kJ/kgK) cpv (kJ/kgK) μl (μNs/m2) μv (μNs/m2) kl (mW/mK) kv (mW/mK) Prl Prv σ (mN/m)

247.1 101.3 1377 5.26 217.0 1.281 0.794 384.2 9.68 103.9 9.31 4.74 0.83 15.4

263.2 200.6 1327 10.0 206.0 1.316 0.854 308.6 10.3 96.5 10.7 4.21 0.83 13.0

277.2 337.8 1281 16.6 195.5 1.352 0.916 257.6 10.9 90.2 11.9 3.86 0.84 11.0

291.2 537.2 1233 26.1 184.0 1.397 0.989 216.0 11.5 84.1 13.4 3.59 0.87 9.0

305.2 815.4 1180 39.8 171.2 1.456 1.080 181.1 12.1 78.1 14.5 3.38 0.90 7.2

319.2 1190.3 1121 59.3 156.5 1.537 1.202 151.0 12.9 72.1 16.2 3.22 0.96 5.4

331.2 1603.6 1063 82.7 141.8 1.638 1.354 127.9 13.7 67.0 18.0 3.13 1.03 4.0

345.2 2123.2 983.8 122.4 121.1 1.843 1.665 102.9 14.9 60.8 21.0 3.12 1.18 2.4

358.2 2925.8 887.2 181.8 95.5 2.306 2.397 80.2 16.7 54.9 26.2 3.37 1.52 1.1

374.3 4059.3 511.9 511.9

REFRIGERANT-410A Composition: R-32/125 (CH2F2/CHF2CF3) 50/50% by mass (50/50 difluoromethane/pentafluoroethane) Molecular weight: 72.6 Tsat (K) Psat (kPa) ρl (kg/m3) ρv (kg/m3) hlv (kJ/kg) cpl (kJ/kgK) cpv (kJ/kgK) μl (μNs/m2) μv (μNs/m2) kl (mW/mK) kv (mW/mK) Prl Prv σ (mN/m)

221.7 101.3 1350 4.17 273.0 1.37 0.807 346 9.90 151 8.94 3.14 0.89 17.8

238.3 220 1296 8.70 258.2 1.40 0.893 272 10.6 140 9.77 2.72 0.97 14.9

251.8 380 1250 14.7 245.1 1.43 0.975 226 11.2 130 10.6 2.49 1.03 12.6

264.5 600 1204 23.0 231.6 1.48 1.064 191 11.8 121 11.5 2.34 1.09 10.5

Critical temperature: 344.5 K Critical pressure: 4903 kPa Critical density: 459.5 kg/m3 278.7 950 1148 36.5 214.5 1.55 1.188 159 12.4 112 12.8 2.20 1.15 8.2

292.0 1400 1089 55.0 196.0 1.65 1.35 133 13.1 103 14.6 2.13 1.21 6.2

305.4 2000 1021 82.1 173.9 1.80 1.60 110 14.0 94.6 17.4 2.09 1.29 4.3

319.2 2800 936 125 145.7 2.10 2.14 88.6 15.2 86.0 22.3 2.16 1.46 2.5

332.7 3800 821 198 106.8 3.07 3.82 67.7 17.4 77.9 33.0 2.67 2.01 1.0

344.5 4903 459.5 459.5

680

Appendix II: Saturation Properties of Selected Fluids

WATER Critical temperature: 647.3 K Critical pressure: 22,129 kPa Critical density: 351 kg/m3

Chemical formula: H2O Molecular weight: 18.0156 Tsat (K) Psat (kPa) ρl (kg/m3) ρv (kg/m3) hlv (kJ/kg) cpl (kJ/kgK) cpv (kJ/kgK) μl (μNs/m2) μv (μNs/m2) kl (mW/mK) kv (mW/mK) Prl Prv σ (mN/m)

373.15 101.3 958.3 0.597 2256.7 4.22 2.03 277.53 12.55 679.0 25.0 1.72 1.02 58.91

400 247 937.5 1.370 2183 4.24 2.16 218.9 13.57 685.7 28.1 1.35 1.04 53.50

430 571 910.3 3.020 2092.8 4.28 2.35 175.73 14.716 683.3 31.6 1.10 1.09 47.16

460 1172 879.4 5.975 1990.4 4.45 2.70 147.24 15.86 671.3 36.6 0.98 1.17 40.66

490 2185 844.3 10.95 1871.5 4.60 3.17 126.6 17.00 646.0 42.3 0.90 1.27 33.90

520 3773 803.8 18.90 1731.0 4.84 3.84 111.05 18.14 618.3 50.1 0.87 1.39 26.96

550 6124 756.1 31.52 1562.6 5.07 4.87 99.21 19.33 580.9 60.2 0.87 1.56 19.66

WATER – LOW PRESSURE Critical temperature: 647.3 K Critical pressure: 22,129 kPa Critical density: 351 kg/m3

Chemical formula: H2O Molecular weight: 18.0156 Tsat (K) Psat (kPa) ρl (kg/m3) ρv (kg/m3) hlv (kJ/kg) cpl (kJ/kgK) cpv (kJ/kgK) μl (μNs/m2) μv (μNs/m2) kl (mW/mK) kv (mW/mK) Prl Prv σ (mN/m)

303.2 4.25 995.6 0.0387 2429.9 4.18 1.92 797.4 10.0 615.5 18.9 5.41 1.02 71.19

323.2 12.35 988.0 0.0831 2382.0 4.18 1.95 546.8 10.6 643.6 20.4 3.55 1.02 67.94

343.2 31.20 977.7 0.1984 2332.9 4.19 1.99 403.9 11.3 663.1 22.1 2.55 1.01 64.48

363.2 70.2 965.3 0.423 2282.5 4.21 2.04 314.4 11.9 675.3 24.0 1.96 1.01 60.82

373.15 101.3 958.3 0.597 2256.7 4.22 2.03 277.53 12.55 679.0 25.0 1.72 1.02 58.91

580 9460 697.2 51.85 1350.3 5.70 6.71 89.40 20.51 536.6 77.3 0.950 1.78 12.71

610 14044 619.5 87.5 1064.2 8.12 11.2 78.60 21.68 464.0 111.4 1.38 2.17 6.26

647.3 22129 315 315 0.0

23.1 23.1 914 914

0.0

Appendix III: Analysis Details for the Molecular Theory of Capillarity In the van der Waals theory of capillarity discussed in Section 1.2, the z = 0 location in the interfacial region (Fig. III.1) is chosen so that ∞

0

∫



∫

(ρˆ – ρˆ v ) dz + (ρˆ – ρˆ 1 ) dz = 0 (III.1)

–∞

0

and the interfacial excess free energy σ (per unit area of interface) is computed as ∞

0



σ=

∫

∫

[ψ – ψ (ρˆ v )] dz + [ψ – ψ (ρˆ 1 )] dz (III.2)

–∞

0

where ψ = F/V is the volumetric free energy. The van der Waals model of capillarity is postulated to apply to a system held at constant temperature with a volume that encompasses the interfacial region over a unit area of the interface. The second law of thermodynamics dictates that for such a system (with fixed V and T), equilibrium corresponds to a minimum in the Helmholtz free energy. Since volume is fixed, this also corresponds to a minimum in volumetric free energy ψ = F/V. This equilibrium free energy per unit of interface area is the property commonly referred to as interfacial tension or surface tension. Determination of the equilibrium interfacial tension using the van der Waals molecular theory of capillarity thus requires solution of a constrained minimization problem. We must determine the mean density variation ρˆ ( z ) that minimizes the right side of Eq. (III.2). The interfacial tension or interfacial free energy σlv is the value of the integral on the right side of Eq. (III.2) for ρˆ ( z ) that satisfies Eq. (III.1) and minimizes the integral. To execute this scheme requires a means of predicting ψ (ρˆ ,  z ) in the interfacial region that accounts for the effects of the density gradient there. To evaluate ψ and other properties in the interfacial region, the statistical thermodynamics model for properties of a van der Waals fluid described in Section 1.1 can be extended using a methodology known as Rayleigh’s model. For a molecule at an arbitrary z location, the mean energy per molecule is computed by summing the potential energy interaction with all the surrounding molecules and dividing by 2 to assign half the energy to each molecule in the pair. This is represented by the following relation

Φ( z ) 1 = ρn ( z )φdV (III.3) N 2

∫

where ρn is the number density of molecules, ϕ is the radially symmetric molecular interaction potential, and Φ/N is the mean energy per molecule at a location z in the interfacial region. Rayleigh’s model treats the interaction of two molecules in the interfacial region using the coordinates shown in Fig. III.2. Considering a donut-shaped differential volume dV around the z axis

dV = 2πr 2 sin θ dθdr (III.4)

681

682

Appendix III: Analysis Details for the Molecular Theory of Capillarity

FIGURE III.1  Variation of mean molar density and volumetric free energy ψ = F/V across the interfacial region.

we integrate over the entire surrounding space: over 0 ≤ θ ≤ π and from an average minimum separation of closest molecules rmin up to a maximum separation distance rmax representing the effective range of the long-range attraction forces between molecules. The resulting volume integral can be cast in the form

Φ(z ) =π N

π

rmax

∫

∫

φ(r )r 2 ρn (r cos θ + z )sin θ d θ dr (III.5) 0

rmin

In the van der Waals mean field theory, ρn is expanded about a specific z value for small r cos θ

ρn (r cos θ + z ) = ρn ( z ) + ρ′n ( z )r cos θ +

1 ρ′′n ( z )r 2 cos2 θ (III.6) 2

where primes denote derivatives with respect to z. Substituting the right side of the above equation into Eq. (III.5) and integrating with respect to θ yields

Φ( z ) 2πN = N V

rmax

∫

rmin

π φ(r )r dr + ρ′′n ( z ) 3 2

FIGURE III.2  Pair relative position in interfacial region.

rmax

∫ φ(r)r

rmin

4

dr (III.7)

Appendix III: Analysis Details for the Molecular Theory of Capillarity

683

Defining rmax

a v 0 = –2 π



∫ φ(r )r

2

dr (III.8)

rmin



κv =

– 23π ∫ rrmax φ(r )r 4 dr (III.9) min –2π ∫ rrmax φ(r )r 2 dr min

and substituting yields

Φ( z ) = − av 0

N2 1 − avo κ v Nρ′′n ( z ) (III.10) V 2

When the density is uniform, ρ′′n = 0 and use of Eq. (III.10) for the potential energy in standard statistical thermodynamic analysis leads to the relation (1.4) for the log of the partition function for a van der Waals fluid described in Section 1.1. Using Eq. (III.10) in its entirety, the same analysis leads to the following relation for ln Q in which the last term accounts for the effect of non-uniform density.



3 N   2πMk B T (V − Nbv′ )2/3    (ξ − 5) ln Q = N +  ln  + N  2 ln π − ln σ s   2   N 2/3 h 2    (ξ − 5) N  T  avθ N 2 avθ N 2  ρ′′  + + κv  n             + ln  2  θrot ,m  Vk B T 2Vk B T  N /V 

(III.11)

Equation (III.11) for the partition function applies to a system in which the density varies with z location, as is the case in the interfacial region. Using this result in the thermodynamic relation F = − k B T ln Q (III.12)



together with the definitions of the volumetric free energy and the molar density

ψ = F /V (III.13a)



ρˆ = ρn /N A = ( N /N A )/V (III.13b)

the following relation is obtained for ψ:

1 ψ = ψ 0 (ρˆ , T ) −   aˆ v 0 κ v ρˆ ρˆ ′′ (III.14)  2

where ψ0 is the value of ψ at local conditions for uniform density given by



 T    ξ − 3 ˆ  ξ−5 ρRT ln  ψ 0 (ρˆ , T ) = −ρˆ RT 1 + ln π − ln σ s  −   2  θrot ,m    2    1 − ρˆ bˆv   2πMk B T  3/2  2                  −ρˆ RT ln    h 2   − aˆ v 0 ρˆ ˆ ρ N A   

(III.15)

684

Appendix III: Analysis Details for the Molecular Theory of Capillarity

In Eq. (III.14), aˆ v 0 κ v ρˆ ρˆ ′′ provides the correction for non-uniform density, where

aˆ v 0 = av 0 N A2 (III.16)



ρˆ ′′ = d 2 ρˆ /dz 2 (III.17)

and in Eq. (III.15), bˆv is defined by Eqs. (1.6) and (1.14b) in Section 1.1. One approach to computing the equilibrium interfacial free energy with the above results is to postulate an interfacial region density variation such as

(ρˆ − ρˆ v ) ρˆ = ρˆ v + 1 [1 + tanh(z /δzi )] (III.18) 2

Using this relation for the density variation with z and the extended van der Waals relation (III.14) for ψ, the integrals on the right side of Eq. (III.2) can be numerically evaluated at a specified temperature for chosen δzi values to iteratively determine the value of δzi that minimizes the integral. The value of σ so determined is the equilibrium interfacial tension σlv. An alternative approach is to use the calculus of variations to determine a relation for the minimized interfacial tension integral. Application of the calculus of variations to this problem (see reference [III.1]) leads to the conclusion (see reference [III.2]) that the minimized integral can be computed as ∞

σ lv =



∫ 2ψ (ρˆ , T )dz (III.19) e

−∞

where ψe is the excess volumetric free energy defined as

ψ e = ψ 0 (ρˆ , T ) − ψ 0 (ρˆ v , T ) + µˆ v (ρˆ − ρˆ v ) (III.20)

and µˆ v is determined from Eq. (1.15) in Section 1.1 with ρˆ set to ρˆ v. The results of the calculus of variations analysis further indicate that the integral (III.19) can be converted to the following integral with respect to density ρˆ l



σ lv =

∫ 4aˆ

1/ 2

v0

κ v ψ e (ρˆ , T ) 

d ρˆ (III.21)

ρˆ v

and it also yields a relation between z position and local density in the layer ρˆ

z – zv =

∫

ρˆ v +δρˆ

 ψ e (ρˆ , T )   aˆ κ  v u

–1/ 2

d ρˆ (III.22)

Numerical evaluation of the integrals using the property relations derived above yields predictions of the surface tension and the density profile across the interfacial region for a specified temperature. Note that the lower limit on the integral (III.22) is chosen to exceed ρˆ v by a small amount δρˆ because ψe is zero and the integrand is singular at ρˆ = ρˆ v. Likewise, the upper limit of the integral is chosen to be no more than ρˆ l − δρˆ because the integrand is singular at ρˆ = ρˆ l. The integral approaches a well-defined limit as δρˆ → 0 because of the asymptotic behavior of the integrand. Computationally, choosing δρˆ to be very small compared to ρˆ v provides an accurate prediction of the asymptotic value of the integral.

Appendix III: Analysis Details for the Molecular Theory of Capillarity

685

With the van der Waals property relations derived in this section, Eqs. (III.21) and (III.22) can be used to compute the surface tension and the density profile across the interfacial region if a value can be specified for κ v. A prediction of κ v can be obtained using Eq. (III.9) with an appropriate force interaction potential model. For example, using such an approach with the Lennard-Jones potential (Eq. (1.3) in Section 1.1) and the limits set to rmin = r0 and rmax = ∞ yields κ v = (9 / 7)r02 . This model thus makes it possible to compute predictions of the surface tension and density profile in the interfacial region for a van der Waals fluid using Eqs. (III.21) and (III.22) together with the Lennard-Jones model constants and other molecular parameters for the fluid. Alternative fluid property models can also be used, as illustrated by Carey [III.2] who used the Redlich-Kwong fluid property model with Eqs. (III.21) and (III.22) to construct relations for the surface tension and density profile.

REFERENCES III.1 Sokolnikoff, I. S., and Redheffer, R. M., Mathematics of Physics and Modern Engineering, 2nd ed., Chapter 5, McGraw-Hill, New York, NY, 1966. III.2 Carey, V. P., Thermodynamic properties and structure of the liquid-vapor interface – A neoclassical Redlich-Kwong model, J. Chem. Phys., vol. 118, pp. 5053–5064, 2003.

Index Note: Locators in italics represent figures and bold indicate tables in the text. 1-propanol, 677

A Abdel-Khalik, S. I., 498–499 Abdelsalam, M., 265, 268, 270, 271, 272–273, 275, 340 Abu-Orabi, M., 401 Accommodation coefficient, 137, 399 Acetone, 673 Active nucleation sites, 561 Actual flow fields, 261 Adams, J. C., 47 Additional flow regimes, 460, 460 Addoms, J. N., 259, 260, 577, 580 Adhesion, 72–76, 80–81, 90 Adiabatic equilibrium, annular flow, 605 Adiabatic flow, 510 Adsorption, 79–80, 87 Advancing contact angle, 85, 85–86, 87–88, 92 Ahmad, S. Y., 601 Ahmed, S., 341, 380 Air-conditioning, 507 Air-cooled condenser, 508 Aksan, S. N., 402 Alad’yev, I. T., 265 Alcohol mixtures, 54–55 Alder, B. J., 21 Al-Diwany, H. K., 438 All Union Heat Engineering Institute (VTI), 592 Ammonia, 673 Analytical models of annular flow, 487–498 Ananiev et al., 519–521; see also Correlation Anderson, G. H., 564–565, 575 Andrews, J. R., 360 Annular downflow film condensation, 543 Annular flow, 455, 458, 509, 511, 537 adiabatic equilibrium, 605 advanced analysis, 495–498 analytical models of, 487–498 computational fluid dynamics (CFD), 495–498 condensation heat transfer, 527 convective condensation, 522 during flow boiling of water, 630, 637 heat transfer coefficient, 531 modeling with entrainment, 491–495 with no entrainment, 488, 488–491 in rectangular microchannels, 539 transition between wispy-annular flow and, 457 transition from churn flow, 456

Ansari, M., 500 Apfel, R. E., 178–179 Apparent contact angle, 86, 87, 93–94, 96, 98, 101–103, 102 Aqueous mixtures, 55 Archimedes number, 239, 427 Arzandi, B., 500 Asai, A., 189 ASHRAE Fundamentals Handbook, 16, 16, 20, 20, 23, 24, 56 Atmospheric pressure benzene-toluene mixture, 374 polished aluminum surface at, 351 pool boiling, 255 saturated liquid water at, 222 saturated nitrogen at, 336, 494 vapor at, 462 water at, 215, 227–228, 232, 236, 242, 273, 313–314, 332, 408, 418 water/methanol mixture, 376 Attard, P., 217 Auracher, H., 650 Avedisian, C. T., 179, 196 Avogadro’s number, 6, 134, 154, 172, 184 Axisymmetric bodies, film condensation, 419–423 Azeotrope, 368, 368–369

B Bae, S., 241 Baer, E., 396 Bailey, N. A., 613 Baird, J. R., 540 Baker, O., 458 Bakhru, N., 338, 339 Balasubramanian, P., 631, 632 Balss, K. M., 361 Banerjee, S., 619 Bankoff, S. G., 210–211, 230, 249 Barnett, P. G., 601 Baroczy, C. J., 479, 479–481, 481–482 Baroczy correlation, 481, 482, 486 Bartell, L. S., 30 Bashforth, F., 47 Baumeister, K. J., 346 Baxter, S., 98–99, 101 Bedeaux, D., 30 Begg, E., 540 Benard circulation cells, 57, 62 Bennett, D. L., 581, 582, 588, 640–642, 648 Benzene-toluene mixture, 374 Berenson, P. J., 283, 292, 310, 312, 318, 320, 348, 351

687

688 Berenson’s correlation, 313 Bergles, A. E., 268, 333, 367, 441, 564–565, 571, 572, 575, 592, 595, 598, 606, 634, 650 Bernardin, J. D., 349–351 Bernath, L., 280 Berttrand, E., 95–96 Betz, A. R., 366 Beysens, D., 56, 57 Bhattacharya, A., 266 Biasi, L., 600 Binary mixtures, 53–55, 63 annular downflow film condensation, 543 convective condensation of, 542, 542–549 convective vaporization of, 646 critical heat flux (CHF), 377–380 equilibrium phase diagram, 368–369 equilibrium properties, 646 evaporation, 645 features, 380 flow boiling, 643, 649 internal flow boiling of, 639–648 non-azeotropic, 371 nucleate boiling heat transfer, 371–377, 372 pool boiling in, 367–380 thermodynamics of, 368–371 vertical tube, 543 Biot number, 59–61 Bird, R. B., 51 Bjorge, R. W., 584, 588 Bjorge correlation, 584–585 Blake, T. D., 95–96 Blander, M., 175, 178 Blanghetti, F., 490, 549 Blasius correlation, 468 Blokhuis, E. M., 30 Boiling, 460; see also Film boiling; Pool boiling burnout, 143, 590 convective, see Convective boiling heat transfer correlations, 271 number, 582, 584, 589 regimes, 558–559, 560, 576 Boiling curve classic, 337 for decreasing controlled heat flux, 254 effects of subcooling on, 332 forced convection on, 334 for increasing controlled heat flux, 253 non-wetting liquid, 338 Boltzmann constant, 5, 134, 154, 174 Bond number, 47, 48, 59–61, 238–239, 498, 538, 626 Borishansky, V. M., 238, 268, 312 Borishansky-Mostinski correlation, 265 Bortolin, S., 541 Boundary conditions, 344, 436, 491, 562 interface, 146–149 thermal, 113, 147 transport, 109–113

Index Boundary-layer analysis, 293–299, 294 equation, 129 film condensation, 408–412 separation, 598 Boure, J. A., 463, 496 Bousman, W. S., 499 Bowers, M. B., 628 Bowring, R. W., 564, 571, 574, 599, 600 Boyd, R. D., 594 Boyko, L. D., 519, 523–524 Brauer, H., 426, 428 Breber, G., 535 Brock, J. R., 51 Brodskaya, E. N., 25 Bromley, L. A., 336 Bromley’s equation, 613 Bruin, C., 30 Bubble blowing, 207 buoyancy, 379 computational model, 236 departure diameter, 235–242, 237–238 dispersed, 511 formation, 164, 168, 175–176, 184–185, 189, 193 frequency, 262–263 Bubble growth, 629 in extensive liquid pool, 222–228 heat-transfer-controlled, 224, 226, 228–230, 234, 256 inertia-controlled growth, 223, 225–230, 234 near heated surfaces, 228–235 within and out of idealized surface cavity, 212 process, 223 from re-entrant cavity, 211 in superheated liquid, 223 transient temperature profiles in, 231 Bubbly flow, 458, 509, 555–556, 576; see also Flow Buchberg, H., 575 Bui, T. D., 320–321 Buie, C. R., 101 Bulk fluid enthalpy, 558–559, 558–559 Bumping, 274 Buoyancy, 379, 612, 623 Burnout, 601 of first kind, 591 heat flux, 253 of second kind, 591 Busse, C. A., 359 Butterworth, D., A., 474, 481

C Cabrera, S., 366, 367 Calus, W. F., 374, 376, 380, 640 Capillarity, 51 defined, 50 molecular theories of, 11–17, 21, 51, 681–685 van der Waals theory of, 12

689

Index Capillary bubble flow, 460, 460 flow, 460, 460 number, 538 pressure difference, 110 tube, 50 Carbon tetrachloride, 357–358, 380, 674 Carey, V. P., 14–17, 19–21, 24, 24–25, 31–32, 31–33, 56, 93, 101, 145, 189–192, 283, 341, 359, 361, 366–367, 367, 378, 378–380, 541 Carpenter, F. S., 425 Carslaw, H. S., 219 Cartesian velocity, 134, 665 Cassie, A. D. B., 98–101, 103 Cassie-Baxter relation, 99, 101 Cassie penetrating (impregnating) state, 100 Catton, I., 359 Cavallini, A., 529–531, 535–540 Cavicchi, R. E., 196 Cavitation processes, 170, 176 Cavities embryo vapor bubble in, 213 number distribution of, 215 range of active, 221 re-entrant, see Re-entrant cavity Celata, G. P., 594–595, 597–599, 605, 650 Cess, R. D., 332, 336 Chandrasekhar, S., 613 Chang, Y. P., 278–280, 291, 310 Chao, David F., 241 Charles, M. E., 459 Chato, J. C., 519–520, 529; see also Correlation Chemical potential, 153, 156, 162, 162–165, 170, 180–181, 190 Chen, J. C., 580–581, 582, 588, 608, 621, 624, 632, 640–642, 648 Chen, S. L., 525–527, 535, 541; see also Correlation Chen correlation, 580–581, 642; see also Correlation Cheng, L., 460 Cheng, P., 538–540, 629 Cheng, S. C., 608 Chenoweth, J. M., 481, 482 Chen-type correlation, 632; see also Correlation Cherniakov, P. S., 283 Chisholm, D., 470, 483 Chowdhury, S. K. R., 320 Chu, K.-H., 366 Chuah, Y. K., 367 Chun, K. R., 417, 525 Churchill, S. W., 525 Churn flow, 455 Cicchitti, A., 468 Clapeyron equation, 227, 397 Clark, H. B., 222 Clarke, J. A., 268, 274, 571 Clark’s correlation, 275; see also Correlation Classical nucleate boiling, 338; see also Nucleate boiling Classical pool boiling, 338; see also Pool boiling

Clausius-Clapeyron equation, 139, 212, 219 Co-current annular film flow, 525 gas-liquid flow, 457–458 Cognata, T. J., 500 Cohesion, 72–76, 80–81 Colburn, A. P., 417, 425 Cole, R., 206, 230, 238, 240 Coleman, J. W., 507, 509, 510–511 Collier, J. G., 402, 559, 564, 574–575, 580, 595, 600, 609, 613, 616, 624, 648, 650 Complex finned passage, 500 Complex operating conditions, convective condensation, 541–542 Compressibility effects, 483 Computational algorithm, 546–547 Computational domain, 31 Computational fluid dynamics (CFD), 495–498 Concentration variations, 379 Condensation, 428–438; see also Convective condensation; External condensation dropwise, 395–404 falling-film, 511, 513 film, see Film condensation heat transfer, see Condensation heat transfer horizontal co-current flow with, 462, 508 processes, 134–142 shock, 159, 159 steam, 395 Condensation heat transfer annular flow, 527 dropwise condensation, 438–440 enhancement of, 438–442 film condensation, 440–442 Condensers, refrigeration, 507 Condensing refrigerant flow, 509 Conduction resistance, 399 Conservation of energy, 645 of mass, 226 of thermal energy, 226 Constant applied flux, 360 Constriction resistance, 398 Contact angle advancing, 85, 85–86, 87–88, 92 apparent, 86, 87, 93–94, 96, 98, 101–103, 102 defined, 47, 69, 71 effect on capillary rise, 91 equilibrium, 69–72, 85 finite, 83 hysteresis, 85–89, 88, 210, 318 of interface, 47, 49–50 intrinsic, 79, 94, 102, 102, see also Young’s angle liquid surface tension effect on, 76–79 liquid-vapor, 78, 565 measurements, 87, 91 receding, 85, 85–86, 88, 90, 91, 92 on smooth surfaces, 69–72

690 temperature effects on, 347 time variation of, 93, 93–94 wettability, 72 Young’s equation, 71–72 Contact line, 69, 70, 87 defined, 69 displacement of, 96, 97–98 heat flux, 144 interfacial tensions at, 70, 71, 96 macroscopic, 100–101 motion, 85, 87, 90, 283–284 over dry solid, 85 pinning of droplet, 88 Controlled heat flux, 253–254 Controlled surface temperature, 250 Convection analogy models, 256–257 Convection number, 582, 585 Convective boiling of binary mixtures, 639–648 conventional (macro) tubes, 555–560 critical heat flux condition (CHF), 590–608, 648–650 in horizontal tube, 556 in microchannels, 625–639 onset of nucleate boiling (ONB), 561–567 post-critical heat flux condition, 608–625 process for vertical co-current flow, 557 regimes of, 555–560 saturated flow, 575–590 subcooled flow, 567–575 upflow, 557 Convective boiling heat transfer, prediction of, 639–648 Convective condensation, 507–549 advanced models, 541–542 analytical modeling of downflow internal, 511–518 annular flow, 522 of binary mixtures, 542, 542–549 complex operating conditions, 541–542 in conventional (macro) tubes, 507–511 correlation methods for heat transfer, 519–535 heat transfer, 539 microchannel flow passage cross section, 540–541 in microchannels, 535–536 minichannel, 539, 540–541 reducing flow passage size, 536–540 regimes of, 507–511 Convective effects, see Sensible heat and convective effects Convective film boiling, 609–613 Conventional (macro) tubes, 507–511 Cooper, J. R., 441 Cooper, M. G., 232, 380 Copper oxide (CuO), 439 Cornwell, K., 217 Corrections for sensible heat and convective effects, 314–316 Correlation of Ananiev et al., 519–521 of Cavallini et al., 529–531

Index of Chen et al., 525–527, 535 comparison of, 624–625 critical heat flux (CHF), 599–605 developed from integral analysis, 305–309 of Dobson and Chato, 529 of Dougall and Rohsenow, 617–619 of Groeneveld, 616–617, 617 of Groeneveld and Delorme, 619–621 of Kim and Mudawar, 531–532 of Koyama et al., 529 methods for heat transfer, 519–535 of Moser et al., 527–529 of Park et al., 531 of Rosson and Meyers, 532–535 of Shah, 522–524 of Soliman et al., 521–522 of Traviss et al., 522 Costello, C. P., 280 Costigan, G., 624 Crispation number, 59 Critical angle, 100 Critical bubble packing, 278 Critical heat flux (CHF), 250, 279, 282, 285, 340, 362 2-propanol in water on, 378 augmentation due to liquid thermal conductivity, 285 binary mixtures, 377–380 conditions, 591, 599, 648–650 convective boiling, 590–608, 648–650 correlations, 597–605, 600 data for water, 592–593, 593, 597 effects of tube length and diameter, 595 factors, 595, 606–608 flow boiling, 649 internal flow boiling, 590–625 low-heat-flux data for water, 596 mass flux, 594 mechanisms, 590–592, 592 parametric effects in DNB data, 593–594 parametric trends in dryout data for water, 595–597, 597 prediction, 601 pressure, 594–595 quality, 595 saturated flow boiling, 599–605 in small channels, 634–635 subcooled flow boiling, 597–599 subcooling, 595 Critical radius, 173, 183, 192, 195–196 Critical surface tension, 77–79 Critical wavenumber, 118–119 Cumo, M., 620–621 Cylinders, film condensation, 419–423 Czikk, A. M., 364

D Dalton’s law, 368, 370 Damianides, C. A., 498–499

691

Index Danilova, G. N., 364 Davis, E. J., 564–565, 575 Deceleration pressure gradient, 512 De Coninck, J., 96 Deissler, R. G., 425–426 Deissler correlation, 490 Delhaye, J. M., 496 Delorme, G. G. J., 619–622; see also Correlation Demiray, F., 241 Dengler, C. E., 577, 580 Denny, V. E., 426, 437 Density functional calculations, 29 Departure from nucleate boiling (DNB), 559, 591, 593, 601, 605–606, 648 De Ruijter, M. J., 95–96 Deryagin, B. V., 84 De Schepper, S. C. K., 498 Dew-point, 368, 368, 428 Dhir, V. K., 215, 217, 234–235, 236, 264–265, 279–281, 284, 287, 301–303, 320–321, 422, 629 Diesselhorst, T., 283 Differential equation, 47, 124 Diffusion resistance, 372 Diffusion thermo effect, 431 Diffusivity, see Eddy diffusivity DiMarco, P., 340–341 Dimensionless drop radius, 48 temperature, 528 variables, 47 wavenumber, 59, 61 Disjoining pressure, 81–82, 82–85, 352–357 equilibrium vapor pressure with liquid film, 357 on evaporation heat transfer, 357–359 Dispersed bubble, 511 Dispersed flow heat transfer, 614; see also Heat transfer Dispersion forces, 4, 12 Dittus-Boelter correlation, 577, 586; see also Correlation Dittus-Boelter equation, 528, 567, 580, 615 Dobson, M. K., 529; see also Correlation Dong, L., 366 Dormer, J., Jr., 575 Doroshchuk, V. E., 592, 595–596 Dougall, R. S., 609, 617–619; see also Correlation Dougall-Rohsenow correlation, 624; see also Correlation Downflow air-cooled condenser, 508 Downflow internal convective condensation, 511–518, 512; see also Convective condensation Drew, T. B., 249 Drop-flow, 460, 460 Droplet array appearance, 396 embryos, 396 nucleation, 439 simulation model, 26 vapor system, 30–31 wetted footprint area, 93

Dropwise condensation, 395–404; see also Condensation condensation heat transfer, 438–440 defined, 395 droplet array appearance for, 396 heat transfer coefficient, 404 heat transfer data, 403 heat transfer resistance components for, 398 Dryness fraction, 453 Dryout, 285–287, 555, 591, 605–606, 637; see also Partial dryout Dufour effect, 431 Dukler, A. E., 416–417, 419, 425–426, 427, 456, 458–459, 468, 526 Dynamic behavior, 109–149 Dynamic equilibrium, 136 Dzakowic, G. S., 565, 575

E Eberhart, J. G., 178 Ebullition cycle, 218, 218, 337 Eddy diffusivity, 415–416, 491, 497, 515–517 Egen, R. A., 574 Electrostatic forces, 4 Embryo bubbles, 164, 165, 167, 169, 175, 192, 195–196, 204, 207 droplet formation, 395 formation at water-solid interface, 207 liquid droplet, 392 vapor bubble, 204, 213 Energy balance, 512 Engelberg-Forester, K., 570 Enhance film condensation, 441–442; see also Film condensation Enthalpy, 112, 147, 559–560, 575, 591, 595, 599, 620 Entrainment, 488, 488–495, 493, 541 Entrapped gas, heterogeneous nucleation, 209–217 Eötvös number, 498 Equilibrium, 45–51, 56, 136, 157 adiabatic, 605 contact angle, 69–72, 71, 85 dynamic, 136 interface shapes at, 45–51 liquid-vapor interface, 146–147 phase diagram, 368–370 thermodynamic, 615 Era, A., 621, 621 Escobedo, J., 55 Ethanol, 675 Ethylene glycol, 54–55, 55 Eucken, A., 396 Evaporation binary mixtures, 645 horizontal co-current flow with, 461 microlayer, 228, 232, 234 mist flow, 614–616 time of water droplets on aluminum surface, 347

692 Evaporator with thin liquid films, 325 water, 484 Expected variation, 173, 173 Extended meniscus, 83, 83–84, 358 External condensation, 391–442; see also Condensation dropwise condensation, 395–404 enhancement of condensation heat transfer, 438–442 film condensation, 404–423 heterogeneous nucleation in vapors, 391–395 interfacial waves, 423–428 in presence of noncondensable gas, 428–438 vapor motion, 423–428

F Faghri, A., 359 Falling-film condensation, 405, 405, 437, 511, 513; see also Condensation Fanning friction factor, 467 Fictitious vapor density, 512 Field emission scanning electron microscopy (FESEM), 439 Film boiling, 293–316; see also Pool boiling; Transition boiling boundary-layer analysis, 293–299, 294 convective, 609–613 copper surface in water, 315 corrections for sensible heat and convective effects, 314–316 correlations developed from integral analysis, 305–309 for different surface conditions, 318 finite horizontal surfaces, 314 interfacial waves and turbulence, 301–305 internal flow boiling, 609 inverted annular flow internal, 610 large horizontal surfaces, 310–314 Leidenfrost phenomenon, 341–351 radiation effects, 299–301 regime, 251, 251 subcooling, 332 vapor release cycle during, 292 Film condensation, 404–423; see also Condensation; Enhance film condensation annular downflow, 543 on axisymmetric bodies, 419–423 boundary-layer, 408–412 condensation heat transfer, 440–442 on cylinders, 419–423 on flat surface, 404–419 forced-convection, 429, 437 heat transfer coefficient, 407, 427 on horizontal cylinder, 419 on in-line bank of round tubes, 421 laminar, 404–408 noncondensable gas, 428 similarity analysis of, 408

Index turbulent, 412–419 on vertical surface, 404–419, 412 Film dryout, 591; see also Dryout Film thickness, 32, 33, 84 Film-transition, 318, 319 Findlay, J. A., 496 Finite contact angle, 83 Finite horizontal surfaces, 314 Fiori, M. R., 598 Flow actual fields, 261 analytical models of annular, 487–498 annular, see Annular flow capillary, 460, 460 churn, 455 condensing refrigerant, 509 homogeneous, 466–469 horizontal, 457–459 idealized fields, 261 inverted stagnation model, 260–261 one-dimensional two-phase, 463–470 passages, 536–541, 537, 625–627 separated, 469–470 slug, 456, 458 turbulent-turbulent, 484 two-phase, see Two-phase flow upward vertical, 454–457 wispy annular, 455 Flow boiling, 555, 607 binary mixtures, 643, 646 critical heat flux (CHF), 649 in microchannels, 627–634 passage geometries, 636 transition, 608–609 Flow regimes, 455, 458, 460, 460–461, 557 condensing refrigerant flow, 509 horizontal tube, 509 horizontal tubes, 507 Fluid(s), 18 length scale values of, 18 molecules near solid metal wall, 352 regimes, 31 saturation properties, 673–680 Fluid-dependent parameter, 586 Fluid-solid interface, 392 Fluid-wall interactions, 351–361 Fluorocarbon refrigerants, 438 Fluorocarbon surfaces, 79 Flux, see Heat flux Fog flow model, 466 Forced convection, 333–337, 334 Forced-convection film condensation, 429, 437 Forced-convection vs. superheat curves, 571 Forces, 4–5, 10–12 dispersion, 4, 12 electrostatic, 4 induction, 4, 10 interfacial region, 4

693

Index liquid adhesion, 96 London dispersion, 39 momentum, 283 van der Waals, 39 wall-fluid attractive, 189–190 Forster, H. K., 225, 256–258, 265, 580, 587 Forster-Zuber correlation, 587 Forster-Zuber relation, 580 Fourier components, 115, 122, 126, 147 Fourier conduction, 112–113, 147–148, 431 Fourier expressions, 148 Fourier wavelengths, 115 Fowkes, F. M., 39, 75 Fox, H. W., 77 France, D. M., 621 Frank, J. P., 185–186, 188 Frank, S., 575 Frea, W. J., 280 Free energy, 11–14, 13, 27, 42–43, 96; see also Gibbs free energy Gibbs, 96, 163, 168, 170, 170, 182 Helmholtz, 14, 42, 70, 167 interfacial region, 11–14, 13 molar specific, 194 surface excess, 43 volumetric, 682 Free liquid film, 23, 23 Frenkel, D., 22 Frequency of bubble release, 235–242 Frictional gradients, 456 Friction factors, 466, 530, 586 Friedel, L., 480–482, 528 Friedel correlation, 482 Fritz, W., 238, 240, 255 Frost, W., 565, 575 Froude number, 582, 589 Fu, F., 496–497 Fujii, M., 364 Fujita, Y., 374–376, 378 Fully developed nucleate boiling regime, 568–572, 569 Fung, K. K., 608

G Gaertner, R. F., 261, 263 Gaidarov, S. A., 377 Galileo number, 532 Galjee, F. W. B. M., 598 Gambill, W. R., 143–145, 277 Gan, Y., 32, 33 Ganic, E. N., 622–624 Garamella, S., 540 Garber, H. J., 240 Garimella, S., 507, 509, 510–511, 632–633, 638–639 Gases, 8, 73, 663–671 Gas-liquid two-phase flow, 464; see also Flow Gaussian distribution, 668 Gerner, F. M., 359

Gerweck, V., 189, 191, 359 Ghiaasiaan, S. M., 498–499 Ghoshdastidar, P. S., 265 Gibbs-Duhem equation, 160, 162, 165, 169, 180, 355 Gibbs free energy, 96, 163, 168, 170, 170, 182 Gibbs function, 355 Girifalco, L., 75–76 Gnielinski, V., 586 Goglia, G. L., 187 Gogonin, I. I., 238, 239 Goncalves, J. M., 549 Good, R. J., 75–76 Gorenflo, D., 239, 633 Govan, A. H., 493, 493–494 Graham, C., 399–401 Graham, R. W., 218 Gravitational acceleration, 339–341 Gregg, J. L., 409–411 Gregorig, R., 440 Gregorig effect, 441, 540–541 Greif, R., 257–258, 265, 570 Griffith, P., 230, 232, 258, 261, 278, 396–397, 399–402, 574 Groeneveld, D. C., 603, 608, 613 Groeneveld-Delorme correlation, 625 Groeneveld’s correlation, 616–617, 617, 619–621 Grossman, L. M., 584, 588 Grossman correlation, 584 Guerrieri, S. A., 578 Gungor, K. E., 584, 588–590, 632, 632 Gungor correlation, 584; see also Correlation Gunther, F. C., 257

H Hadaller, G., 619 Hahne, E., 283 Haile, J. M., 22 Hall correlation, 584–585; see also Correlation Hallinan, K. P., 359 Hall-Taylor, N. S., 488, 491, 494 Hamaker constants, 190, 353 Hammer, J., 264 Hamzekhani, S., 239 Han, C.-Y., 230, 232, 258, 261 Hannemann, R. J., 402 Happel, O., 374 Haramura, Y., 334–335 Harris, J. G., 94 Hasan, M. M., 282 Haye, M. J., 30 He, Y., 264 Heat and Mass Transfer Section of the Scientific Council, 593 Heated wall, extended meniscus region near, 358 Heat flux, 138, 253 boiling curve for decreasing controlled, 254 boiling curve for increasing controlled, 253

694 burnout, 144, 253 contact line, 144 controlled, see Controlled heat flux due to Fourier conduction, 113 effects, 146–149 function of, 145 higher temperatures and, 141 limitations, 142–146, 143 maximum pool boiling, 281 maximum transition, 275–290 minimum conditions, 291–293 phase change, 138 thermal energy transfer by, 134 transport, 112 variation of wall, 562, 562 wall superheat vs., 317 vs. wall temperature, 571, 572 Zuber critical, 279, 284 Heat transfer, 110 boiling experiments, 143 coefficient, 134, 139, 141, 401, 404, 407, 418, 423, 426–427, 435, 524, 531–532, 556, 567, 580, 582, 618, 632, 642 controlled bubble growth, 224, 226, 228–230, 234, 256 convective condensation, 539 correlation methods for, 519–535 data, 265–275 dispersed flow, 614 enhance film condensation, 441 mechanisms, 614, 614 model, 145 resistance, 145, 398 by vaporization, 144 Hebel, W., 598 Heck, K., 483 Heertjes, P. M., 548 Heine, D. R., 95–96 Heinzinger, K., 95 Heitich, L. V., 366 Heled, Y., 263 Helmholtz free energy, 14, 42, 70, 167; see also Gibbs free energy Helmholtz instability, 278, 278–282, 285, 456 Helmholtz unstable wavelength, 613 Hemi spreading, 100 Henderson, J. R., 30 Henry, R. E., 348 Henry’s law, 368–369 Herdt, G. C., 359 Hertz, H. G., 185–186, 188 Heterogeneous nucleation, 203–242; see also Nucleation bubble departure diameter, 235–242 bubble growth in extensive liquid pool, 222–228 bubble growth near heated surfaces, 228–235 defined, 164, 203 embryo liquid droplet, 392 from entrapped gas or vapor in cavities, 209–217 frequency of bubble release, 235–242

Index nucleate boiling, 217–222 at smooth interface, 203–208 thermodynamic analysis, 204 from vapor in cavities, 209–217 in vapors, 391–395 Heterogeneous surface, 98–103 Heterophase fluctuations, 164 Hetsroni, G., 483, 628 Hewitt, G. F., 455, 455, 488, 491, 494, 556, 591–592, 606, 648 Hibiki, T., 496, 499 High-energy surfaces, 78–80, 86 Hino, R., 598 Hirata, M., 575 Hirt, C. W., 497 Holden, K. M., 403 Homogeneous flow, 466–469 Homogeneous model, 466, 469, 482, 482 Homogeneous nucleation; see also Nucleation defined, 164 in supercooled vapor, 180–184 in superheated liquid, 164–171 surface temperature, 347 wall interaction effects on, 189–192 Hoover, W. G., 22 Horizontal co-current flow, 461–462, 508 Horizontal flow, 457, 457–459 Horizontal surfaces, 310–314 Horizontal tubes, 510 flow regimes, 507, 509 mass flux, 511 Horizontal wire, pool boiling for, 339 Hovestreijdt, J., 378 Hsieh, D. Y., 120 Hsu, Y. Y., 218–220, 221, 222, 267, 362, 561, 564 Huang, C.-K., 346, 347 Hummel, R. L., 362–363 Hung, C.-H., 186 Hutter, C., 241 Hydrocarbon chains, 53, 53 Hydrodynamic instability, 347 Hydrophilic liquid, 72 Hydrophilic surfaces, 98–101, 99 Hydrophobic liquid, 72 Hydrophobic ordered nanostructured surfaces, 440 Hydrophobic stochastic nanostructured surfaces, 439 Hydrophobic surfaces, 101, 101–102 Hydrostatic pressure, 83–84

I Ibele, W. E., 412 Ideal gas law, 165, 169, 171, 174, 181 Idealized flow fields, 261 Ideal mixture, 370 Iloeje, O. C., 608 Iltscheff, S., 438 Impurities on surface, 86

695

Index Inasaka, F., 595 Inclined tube, 464 Induced subcooling, 377; see also Subcooling Induction forces, 4, 10 Inertia-controlled bubble growth, 223, 225–230, 234 Initialization, 21 Instability Helmholtz, 278, 278–282, 285, 456 hydrodynamic, 347 Kelvin-Helmholtz, 113–121 Rayleigh-Taylor, 113–121 Zuber Helmholtz model, 282 Interface boundary conditions, 146–149 Interface height, 48 Interface shapes at equilibrium, 45–51 sessile drop on flat solid surface, 45, 45 Interface stability; see also Instability linear analysis of, 113, 114 of liquid jets, 121–127, 122 of motionless liquid, 119 Interfacial mass flux, 139 Interfacial region, 70, 682 binary mixture, 53 density in, 40 dimensionless thickness, 17 forces, 4 free energy, 11–14, 13 geometry of, 44 macroscopic treatment, 38–63 mean internal energy, 40 molecular density, 12 molecular dynamics simulation studies of, 21–26 molecular perspective on transitions, 3–11 molecular theories of capillarity, 11–17 nanodroplet surrounded by vapor, 26 nanoscale features of, 17–21 nanoscale perspective, 3–33 potential function, 3–4 small system effects, 26–33 structure, 24 temperature, 18 thermodynamic analysis model system, 41 thermodynamic theory, 5 thickness, 18, 56–57, 57 Interfacial resistance, 398 in condensation processes, 134–142 in vaporization, 134–142 Interfacial shear, 423, 514, 526, 637 forces, 465 stress, 490, 511 Interfacial tension, 11, 14–17, 16, 74, 98, 441, 681 acting on contact line, 70 contact angle vs. liquid-vapor, 78 contact line, 71 defined, 42–43 effects of, 57–63 equilibrium properties, 71

experimental data, 51 of simple systems, 75 surface excess free energy, 43 surfactant effects on, 51–54 temperature, 51–54 thermodynamic analysis of, 39–45 between two liquid phases, 75 two-phase flows, 453 Interfacial turbulence, 62, 301–305 Interfacial waves, 301–305, 423–428, 526 Interline region, 359 Internal convective condensation, see Convective condensation Internal film boiling, 610; see also Film boiling Internal flow boiling of binary mixtures, 639–648 critical heat flux (CHF), 608–625 film boiling, 609 flow passages, 625–627 in microchannels, 625–639 Interstitial spaces, 101 Intrinsic meniscus, defined, 83–84 Intrinsic stability, 158, 161 Inverted annular flow, 609, 610; see also Annular flow Inverted stagnation flow, 260–261 Isachenko, V. P., 404 Isbin, H. S., 268, 571 Ishii, M., 463, 496, 601 Isolated bubbles; see also Bubble regime, 249 slugs and columns, 266 Isothermal lines, 160 Isothermal wall condition, 564, 578 Israelachvili, J., 353, 355 Ivey, H. J., 240, 331

J Jacobi, A. M., 633 Jaeger, J. C., 219 Jaikumar, A., 366–367 Jakob, M., 240, 249, 254, 265, 397 Jakob number, 226, 230–231, 233, 238–239, 241, 314 Jarvis, T. J., 207 Jasper, J. J., 51, 52 Jaster, H., 520 Jensen, M. K., 238–239 Jicha, J. J., 575 Jones, O. C., 621 Jones, O. C., Jr., 601 Jordan, D. P., 575 Joung, Y. S., 101 “Jumping droplets,” 440

K Kagen, Y., 170, 175 Kalikmanov, V. I., 30

696 Kandlikar, S. G., 95, 283–284, 284, 290, 359, 366–367, 374–376, 500, 572, 585–589, 626–628, 630–632, 634–635, 640, 642–644, 648 Kandlikar correlation, 585–587, 586; see also Correlation Kapitza effect, 276 Karimi, A., 179 Kast, W., 397 Kattan, N., 587 Katto, Y., 334–335, 601, 605–607, 634 Katz, J. L., 175, 178 Kautzky, D. E., 380 Kawaji, M., 499 Kedzierski, M. A., 549 Keller, J. B., 127 Kelvin-Helmholtz instability, 113–121 Kelvin-Helmholtz stability, 128 Kerosene-air interfaces, 75 Kerosene-water interfaces, 75 Kezios, S. P., 520 Khrustalev, D., 359 Kim, B., 101, 366 Kim, J., 241 Kim, M, S., 549 Kim, S. M., 531–532, 540; see also Correlation Kimura, T., 95 Kinetic energy, 9, 112–113, 147 Kinetic limit of superheat, 171–176 of supersaturation, 184–189 Kinetic theory, 8, 143, 174, 185, 663–671 Kirishenko, Y. A., 283 Klausner, J. F., 215, 217, 496–497 Klett, J. D., 30 Klimenko, V. V., 312–313, 587 Klimenko’s correlation, 318 Koh, J. C. Y, 295, 298, 316, 407, 411 Kömer, M., 640–641 Koplik, J., 95 Körner, M., 374 Kosky, P. G., 520, 530 Koyama, S., 529, 548–549; see also Correlation Kreith, F., 257, 268, 571 Kroeger, P. G., 574 Kruzhilin, G. N., 265, 519, 523–524 Kucherov-Rikenglaz equation, 139 Kuipers, J., 30 Kun, L. C., 364 Kunkelmann, C., 241 Kunkle, C. M., 93 Kunz, H. R., 417 Kutateladze, S. S., 238–239, 261, 265, 277, 280, 331–332, 377–378, 417, 426–427, 569, 598 Kutateladze number, 277

L Labuntsov, D. A., 265 Laird, A. D. K., 470

Index Lamas, M. I., 241 Laminar film condensation, 404–408, 421; see also Condensation; Film condensation Laminar-laminar flow, 536–537 Laminar transport equations, 114 Langer, H., 616, 624 Lantsman, F. P., 592, 595, 601, 606 Laplace equation, 44, 115 Large-diameter tubes, 482 Latent heat, 261–262 Lateral wavy perturbation, 63, 63 Lattice Boltzmann method, 498 Lebouché, M., 500 Lee, C. H., 605 Lee, H. S., 234, 542 Lee, J., 416, 419 Lee, K.-S., 359 Lee, M. S., 402 Lee, Y., 541 LeFevre, E. J., 400–401 Leibniz’s rule, 130 Leidenfrost, J. G., 342 Leidenfrost phenomenon, 341–351 film boiling heat transfer, 342–346 schematic depiction, 342 transition, 346–351 water spheroid on aluminum surface, 342 Leidenfrost point, 342 Leidenfrost temperature, 342, 347, 591, 615 Leidenfrost vaporization, 343 Lekner, J., 30 Lennard-Jones 6–12 potential, 4, 4–5, 9–10, 14, 23, 94–95 Lennard-Jones fluid, 94 Lennard-Jones interaction potential, 353 Leonard, J. E., 613 Leonidopoulos, D. J., 374, 376, 380, 640 Leontiev, A. I., 598 Leppert, G., 575 Level Set method, 497 Levitan, L. L., 595, 601, 606 Levy, S., 265, 574, 603 Li, C., 366 Li, J.-M., 540 Liao, Q., 540 Liaw, S. P., 264, 287, 321 Lienhard, J. H., 120, 143–145, 179, 252, 265–266, 279–282, 284, 293, 313, 316–318, 320, 338, 339, 349, 378, 380, 411, 422 Limitations of nucleate boiling, 275–290 Limit of intrinsic stability, 161 Lin, L., 189 Lin, W. S., 605 Linear stability analysis, see Stability Linear temperature profile, 407 Linetskiy, V. N., 421 Linke, W., 249, 254, 265 Liquefied natural gas (LNG), 180

697

Index Liquid adhesion forces, 96 deficient region, 614 film, see Liquid films flow blockage model, 598 jets, 121–127, 122 layer superheat model, 598 penetration, 364, 364 pool, 222–228 Prandtl number, 565, 577, 581 sublayer dryout model, 599 surface tension, 76–79 transport, 439 viscosity, 468 Liquid droplet, 63, 72, 95, 97 Leidenfrost vaporization, 343 liquid and vapor states for, 182 Liquid films, 423 free, 23, 23 low-Reynolds-number, 132 penetrating, 99, 100 pressure variation, 129 thin, 127–128 waves on, 127–133, 128 Liquid-solid interface, embryo vapor bubble, 204 Liquid-vapor interface, 109, 146–147, 276 contact angle of, 78, 565 energy transport across, 112 equilibrium, 146–147 force-momentum interactions at, 111 hydrodynamic instability of, 347 instability, 113 Kelvin-Helmholtz stability of, 128 mass fluxes across, 110 mass fluxes at, 136 molecular fluxes at, 137 motion of vapor molecules, 135 tangential interactions at, 111 wave motion on, 133 Liquid-vapor interfacial region, see Interfacial region; Interfacial tension Liu, D., 632–633, 638–639 Lloyd, A. J. P., 232 Lockhart, R. W., 470, 474, 482, 483, 485 Lockhart-Martinelli correlation, 473–475, 474, 482, 487, 497, 499 Lockhart-Martinelli-Nelson correlation, 477 London dispersion forces, 39 Lorenz, J. J., 213, 214, 215, 216 Low-density gaseous phase, 73 Low-energy surfaces, 78–79 Low-gravity-effects flow boiling, 587 Low-heat-flux, 555 Lowry, B., 499 Low surface tension refrigerants, 507 Lu, M.-C., 196, 366 Lustrous bare area, 397

M Maa, J. R., 401 Macbeth, R. V., 592, 599 Malenkov, I. G., 240 Mandhane, J. M., 458 Mandrusiak, G., 541 Mansoori, G. A., 55 Marangoni effects, 57, 61–63, 380, 648 Marangoni flow, 62–63 Marangoni number, 59, 61–62 Mariani, A., 594–595, 598 Marroquin, A., 650 Marschall, E., 380 Martin, M. W., 481, 482 Martinelli, R. C., 470, 474–477, 482, 483, 485 Martinelli correlation, 485, 486, 518, 532, 542, 633 Martinelli-Nelson correlation, 475–476 Martinelli parameter, 456, 459, 470–471, 474–475, 479, 484, 487, 522, 584, 633 Marto, P. J., 211, 364, 441 Maruyama, S., 25, 94–96 Mason, B. J., 188 Mass fluxes, 110, 136, 136, 453, 558, 594, 636–637 Mass transfer, 435, 453 Mass velocity, 453 Matorin, A. S., 377 Matorin, P. S., 649 Matsumoto, S., 94 Matsumura, H., 564–566, 575, 578 Maurer, G. W., 574 Maximum flux limitations, 142–146 Maximum heat flux transition, 275–290 Maximum pool boiling heat flux, 281 Maxwell-Boltzmann speed distribution, 8–9 Maxwell distribution, 671 Maxwellian ideal gas, 142 Maxwell relations, 153 Maxwell velocity distribution, 134 May, R. I., 542 Mayinger, F., 616, 624 McAdams, W. H., 412, 468 McAdams correlation, 622; see also Correlation McCormick, J. L., 396 McDonough, J. B., 608–609 McEligot, D. M., 377 McGillis, W. R., 54–55, 341, 378, 378–380 Mean field theory, 12–14 Mean film velocity, 129 Mean internal energy, 40 Mean molar density, 13, 682 Mechanical stability, 159 Mei, M., 402 Mei, R., 234–235 Memmel, G. J., 238–239 MEMS, 535, 625 Meniscus, see Extended meniscus; Intrinsic meniscus Mercury, 675

698 Merte, H., 404, 613 Merte, H., Jr., 234 Metastable regions, 160 Metastable states, 153–164 Methanol, 676 Metropolis, N. A., 21 Meyers, J. A., 532–535; see also Correlation Michel, G., 500 Micro and nano structured surfaces, 362–367 Microchannels annular flow in rectangular, 539 bubble growth and motion in, 629 convective condensation in, 535–536 flow boiling in, 627–634 flow passage, 540–541, 626–627 internal flow boiling in, 625–639 minichannels and, 541 non-circular, 540–541 saturated boiling of water in, 638 two-phase flow, 498–500, 540 Microconvection models, 254–256 Microhydrodynamic region, 286 Microlayer evaporation effects, 261–262 Microstructured surfaces, 96–97 Mikesell, R. D., 230 Mikic, B. B., 216, 226–227, 230–232, 241, 258–259, 261, 263, 402, 585 Mikic-Rohsenow correlation, 585 Miljkovic, N., 440 Miller, C. A., 59 Miller, D. G., 53 Mills, A. F., 417, 426, 437–438 Minichannels boiling of water, 629, 630 convective condensation heat transfer, 539 flow passage cross section, 540–541 microchannels and, 541 Minimum heat flux conditions, 291–293 Minimum wall superheat, 313 Miropolskii, Z. L., 617 Miropolsky, Z. L., 519 Mirzamoghadam, A., 359 Mishima, K., 499 Mist flow evaporation, 614–616 Mixture surface tension, 54 Mochizuki, S., 548 Moderate Diffusion-Induced Suppression Region, 644 Modine Manufacturing Company, 537 Moissis, R, 455–456 Moissis-Berenson transition, 266 Molar specific free energies, 194 Molar specific heat, 154 Molecular dynamic (MD) simulations, 21–24, 24, 29, 94–95 Molecular flux, 135 Molecular force interactions, 352–357 Molecular-kinetic theory (MKT), 96; see also Kinetic theory

Index Molecular theories of capillarity, 11–17 Momentum, 59, 88 balance, 109, 112, 225, 465, 513 equation, 123–125, 129 fluid, 110, 223 forces, 283 liquid and vapor, 110 mechanisms of, 225 transfer, 109, 117, 224 Momentum transport gas-liquid two-phase flow, 464 homogeneous model of, 466 one-dimensional model of, 465 Monte Carlo simulations, 29 Moresco, L. L., 380 Mori, H., 650 Moriyama, K., 628, 630 Morris, D. J., 331 Moser, K. W., 527–529; see also Correlation Mostinski, I. L., 268 Motion, 57, 62, 136–137 Motte, E. I., 336 Mudawar, I., 349–351, 531–532, 540, 605, 628; see also Correlation Mueller, C., 249 Mukheijee, A., 235, 265 Muller-Steinhagen, H., 483 Mulroe, M. D., 440 Myers, J. G., 241

N Nagai, N., 91, 93, 283 Nakayama, W., 364 Nanobubbles, 192–197; see also Bubble Nanochannels, two-phase flow, 498–500 Nanoporous enhanced boiling surface, 367 Nanoscale roughness, wall-affected region, 360 Nanostructured surfaces, 96–97, 287–288, 362–367 Nanostructuring, 438 Nariai, H., 595 NASA, 339 Natural-convection, 251, 259 Navier-Stokes equations, 353 N-butanol, 674 Near-Azeotropic Region, 643 Near critical point behavior, 56–57 Near-wall bubble crowding model, 598–599 nucleation, 359–361 temperature variation, 360 variations, 191 Nelson, D. B., 474–477 Nelson, R., 265, 613 Neogi, P., 59 Neumann’s formula, 69; see also Young’s equation Newton’s laws of motion, 21, 663 Ng, W., 608

699

Index Nichols, B. D., 497 Nijmeijer, M. J. P., 25, 30 Niknejad, J., 401 Nishikawa, K., 265–266, 266 Nishio, S., 349 Nishiwaka, N., 575 Nitrogen, 676 Nitrogen/oxygen mixture, 370 Non-azeotropic binary mixture, 371 Non-circular enhanced channel geometries, 635–639 Non-circular microchannels, 540–541 Noncondensable gas, 428–438 film condensation, 428 forced-convection film condensation, 429 Non-equilibrium effects, 146–149 Non-homogeneous surfaces, 98 Non-linear differential equation, 225 Non-linear interpolation scheme, 572 Non-linear superposition scheme, 587 Non-polar liquids, 39, 53, 77 Non-polar solids, 78 Non-wetting liquid, 72 boiling curve, 338 pool boiling for, 337, 338 N-pentane, pool boiling of, 318 Nuclear Regulatory Commission, 624 Nucleate boiling, 217–222, 555, 580, 638, 640; see also Boiling; Onset of nucleate boiling (ONB) augmentation of CHF due to high liquid thermal conductivity, 285 classical, 338 convection analogy models, 256–257 critical bubble packing, 278 cross sections of three enhanced surfaces for, 365 dryout of the liquid film under larger slugs, 285–287 effects of nanostructured surface morphologies, 287–288 heat transfer, 265–275, 362, 371–377, 372 Helmholtz instability of large vapor columns, 278–282 inverted stagnation flow model, 260–261 latent heat, 261–262 limitations of, 275–290 maximum heat flux transition, 275–290 microconvection models, 254–256 microlayer evaporation effects, 261–262 minimum heat flux condition, 291–293 natural-convection analogy model, 259 observations, 262–263 onset of flooding, 277–278 overview, 289 pool boiling, 254–290, 276 regime, 249 Rohsenow’s model, 254–256 shortcomings of Helmholtz instability CHF models, 282 thermodynamic similitude, 262

vapor-liquid exchange models, 257–259 in water, 363 wetting and vapor recoil on contact line motion, 283–284 Nucleate transition, 317, 319 Nucleation deactivation of, 211 droplet, 439 from entrapped gas, 209–217 heterogeneous, see Heterogeneous nucleation homogeneous, see Homogeneous nucleation Nukiyama, S., 249 Number; see also specific number capillary, 538 crispation, 59 density, 216, 667–668 distribution of cavities, 215 Eötvös, 498 Numerical simulation of pool boiling, 264–265 Nunn, R. H, 441 Nusselt, W., 421–422 Nusselt correlation, 417 Nusselt integral analysis, 404 Nusselt number, 256–257, 406–407, 410, 411, 417, 418, 435, 525, 526 Nusselt-type relation, 612

O O’Bara, J. T., 403 Observations, 262–263 Ochiai, J., 400, 402 Ohno, H., 601, 607, 634 O’Horo, M. P., 360 Ohta, H., 341 Ohtake, H., 499 Oktay, S., 364 One-dimensional gas, 664 One-dimensional two-phase flow, 463–470 general considerations, 463–466 homogeneous flow, 466–469 separated flow, 469–470 O’Neil, P. S., 364, 367 Onset of flooding, 277–278 Onset of nucleate boiling (ONB), 249, 251, 252, 561–567, 568; see also Nucleate boiling at atmospheric pressure, 566 bubbly flow, 556 conditions for, 562–563, 566 curves, 563 for flow of water, 564 heat flux variation, 567 partial subcooled boiling, 567 single-phase convection, 563 wall heat flux, 562 wall superheat, 560 Onset of transition boiling, 317 Orozco, J. A., 336

700 Oshinowo, T., 459 Owens, W. L., 575 Oxygen, 677

P Padday, J. F., 47 Parabolic velocity profile, 344 Parallel walls single particle in box moving between two, 664 two particles moving along collinear path between two, 664 Park, J. E., 531; see also Correlation Park, K., 359 Partial boiling regime, 563, 569–571, 570 Partial boiling transition, 569, 569, 571 Partial dryout, 555, 587; see also Dryout Particle velocity components, 665 Pasamehmetoglu, K. O., 613 Passage size, two-phase flow, 498–500 Pastuszko, R., 366 Patel, P. D., 227 Pearson, J. R. A., 59–60 Peebles, F. N., 240 Pei, B. S., 605 Penetrating liquid film, 99, 100 Peng, X. F., 628 Peng-Robinson equations, 21 Peng-Robinson fluid models, 21 Perturbed interface, 113, 114–115 Peterson, A. C., 403 Peterson, G. P., 359, 366 Petukhov, B. S., 586 Petukhov-Popov and Gnielinski correlations, 586 Phan, H. T., 239 Phase stability, 153–164, 154 Philpott, M. R., 95 Piasecka, M., 366 Pinning, 88 Piret, E. L., 268, 571 Planck’s constant, 5, 191 Plesset, M. S., 225–227 Plug flow, 458, 509, 533 Plummer, D. N., 621 Pohlhausen, E., 232 Poling, B. E., 55 Polished aluminum surface, 351 Polytetrafluoroethylene (Teflon), 72, 77, 79 Pool boiling, 249–321; see also Nucleate boiling additional factors, 341 atmospheric pressure, 255 in binary mixtures, 367–380 classical, 338 curve exhibiting two transition curves, 252 curves compared with plain tube, 365 data obtained for water on heated horizontal wire, 260 defined, 249

Index for different surface conditions, 319 film boiling, 293–316 forced convection, 333–337, 334 gravitational acceleration, 339–341 for horizontal wire, 339 on micro and nano structured surfaces, 362–367 minimum heat flux conditions, 291–293 for non-wetting liquid, 337, 338 of n-pentane, 318 nucleate boiling, 254–290 numerical simulation of, 264–265 parametric effects on, 331–341 regimes of, 249–253, 250–251 size and wettability of surface, 337–338 subcooling, 331–333, 332 surface roughness, 339 transition boiling, 316–321 Popov, V. N., 586, 589 Porteus, A., 456 Post-dryout heat transfer mechanisms, 614 Post-dryout wall temperature, 621, 624; see also Dryout Potash, M. Jr., 84, 357–358 Potential function, 3–4 Power-law, 216, 233 Prandtl number, 256, 316, 404, 411, 414, 417, 418, 515, 517, 565, 577, 581, 586, 617–618 Prediction of convective boiling heat transfer, 639–648 Pressure, 270 drop, 486, 573–575 for water, 269 Preusser, P., 377 Property index, 479, 486 Prosperetti, A., 227 Pruppacher, H. R., 30

Q Qi, Y., 215, 217 Qualitative variation, 173, 173 Quality, see Dryness fraction Quasi-critical value, 132–133

R Radiation effects, 299–301 Radovcich, N. A., 455–456 Rahman, M. M., 366 Raiff, R. J., 367 Ramilison, J. M., 313, 318, 320, 349, 351 Ramu, K., 609 Range of active cavity, 221 Rankine power cycles, 507 Rankine power systems, 535 Raoult’s law, 368–370 Rayleigh, L., 126, 224 Rayleigh equation, 225 Rayleigh’s subsequent analysis, 57 Rayleigh-Taylor instabilities, 113–121

701

Index Rayleigh-Taylor waveform, 120 Receding contact angle, 85, 85–86, 88, 90, 92 Reddy, R. P., 378, 380 Redlich-Kwong constants, 190 Redlich-Kwong model, 14–15, 16–17, 21, 24, 24, 56 capillarity theory, 19, 56 equation of state, 15, 176–178, 185–186 fluid, 21 fluid property model, 190 spinodal curve, 179 thermodynamic property, 16 Redlich-Kwong statistical thermodynamics fluid model, 359 Re-entrant cavity; see also Cavities bubble growth from, 211 liquid penetration, 364, 364 stability of, 211 Refrigerant-12, 678 Refrigerant-22, 678 Refrigerant-134A, 679 Refrigerant-410A, 679 Refrigeration condensers, 507 Regimes boiling, 558–559, 560, 576 for controlled surface temperature, 250 of convective condensation, 507–511 film boiling, 251 fluid, 31 isolated bubble, 249 nucleate boiling, 249 of pool boiling, 249–253, 250–251 of slugs and columns, 250, 251 of subcooled flow boiling, 567–568, 568 transition boiling, 250, 251 Regular interface, 76 Reidel parameter, 51, 52 Relaxation microlayer, 228, 232, 234 Rembe, C., 361 Resistance, 145, 398–399 Reynolds, J. B., 575 Reynolds numbers, 128, 131–132, 255–257, 412, 417, 417, 420, 425–426, 426–427, 467, 472–473, 485, 490, 514, 519–520, 522, 526, 527, 529, 536–539, 567, 577, 580, 622–623, 626, 631, 633, 642 Reynolds stresses, 458 Rezkallah, K. S., 499 Ribbed passage, 500 Rice, P, 374 Robert, M., 56, 57 Roberts, D. N., 455, 455, 556 Rohsenow, W. M., 140, 211, 216, 230–232, 241, 254–256, 258–259, 261, 263, 267–268, 333, 407, 411, 423–425, 564–565, 571–572, 572, 575, 584–585, 609, 617–619, 622–624; see also Correlation Rohsenow’s correlation, 265, 267–268, 274, 339, 363, 571, 584–585, 617–619 Rohsenow’s model, 254–256

Rose, J. W., 400–404, 435, 438, 441, 539–540 Rosson, H. F., 532–535; see also Correlation Rough hydrophilic surface wetting model, 99 Rough surface, 96–103 Round-tube cross-flow, 508 Round tubes, 421 Ruckenstein, E., 238 Rufer, C. E., 520 Rule, T. D., 241 Rusanov, A. I., 25 Ru-Zeng, Z., 30

S Sadasivan, P., 316, 411 Saha, P., 574 Sandall, O. C., 417 Sanna, A., 264 Sato, T., 564–566, 575, 578 Saturated boiling of water, 638 Saturated flow boiling Bjorge, Hall, and Rohsenow correlation, 584–585 Chen correlation, 580–581 comparison of correlations, 587–588, 587–590 convective boiling, 575–590 critical heat flux (CHF), 599–605 Gungor and Winterton correlation, 584 Kandlikar correlation, 585–587, 586 nucleation, 579 Schrock and Grossman correlation, 584 Shah correlation, 581–584 of water, 606 Saturated liquid water, 222 Saturated nitrogen, 336, 494; see also Nitrogen Savic, P., 230 Scanning electron micrographs, 440 Scarola, L. S., 650 Schlunder, E. U., 490, 549 Schmeckenbecher, A. F., 364 Schmidt number, 433 Schnyders, H. C., 178 Schrage, R. W., 137, 137, 142, 145–146 Schrock, V. E., 575, 584, 588 Schrock correlation, 584 Scriven, L. E., 59, 226 Seban, R., 414–417, 425, 438, 525 Second law of thermodynamics, 13–14 Second-order linear differential equation, 124 Second-order ordinary differential equation, 47 Sedighi, N., 95 Segev, A., 349 Selin, G., 420–421 Semi-empirical thermodynamic models, 29 Sensible heat and convective effects, 314–316 Separated flow, 469–470, 471 Sernas, V., 120 Sessile drop, 45, 45, 47 Severe Diffusion-Induced Suppression Region, 644

702 Shah, M. M., 522–524, 533–535, 548, 581–584, 586, 588–589, 603–605, 634; see also Correlation Shah correlation, 533, 581–584, 583 Shakir, S., 374 Shear stress distribution, 415 Shear stress variation, 423 Sher, N. C., 574–575 Sheynkman, A. G., 421 Shimizu, S., 403 Shoji, M., 92, 265 Shortcomings of Helmholtz instability CHF models, 282 Shulman, H. L., 230 Silane deposition, 439 Silicone oils, 80 Silver, R. S., 139, 397 Simple polar groups, 54 Simpson, H. C., 139 Single particle in box moving between two parallel walls, 664 Single-phase convection coefficient, 577, 580 Single-phase forced convection, 569 Single-phase vapor convection, 559 Size and wettability of surface, 337–338 Skripov, V. P., 178–179, 189 Slaughterback, D. C., 617 Slug bubble flow, 630 Slug flow, 456, 458, 509, 533, 555, 558, 591–592, 629–630 Small channels, 634–635 Small-scale roughness elements, 97, 97 Small system effects, 26–33 Smit, B., 22 Smooth heterogeneous surface, 98–103 Smooth interface, heterogeneous nucleation, 203–208 Snyder, N. W., 278 Soave-Redlich-Kwong equation, 21 Sold-liquid interface, 276 Solid metal wall, fluid molecules, 352 Solid surface, ultrathin liquid film on, 355 Soliman, M., 521–522, 525; see also Correlation Sommerfield, M., 268, 571 Son, G., 235, 241 Soret effect, 430 Soviet Union, 592 Sparrow, E. M., 332, 336, 409–411, 432–436 Spencer, D. L., 412 Spheroidal state, 342 Spinodal curve, 161, 177, 177 limit, 161, 164, 177, 177–180, 179, 185–186, 186, 191 lines, 160 temperature, 191 Spohr, E., 95 Spreading coefficient, 73, 75, 80, 81, 83, 90–91 Spreading process, 74 Spread thin films, 80–85 Stability analysis, 58, 58–60, 60 intrinsic, 161

Index mechanical, 159 phase, 153–164 plane, 60 thermal, 159 Statistical thermodynamics fluid model, 359 Staub, F. W., 530, 574 Staub, J., 340 Steam condensation, 395; see also Condensation Steam-water system, 119 Steiner, D., 587 Steinke, M. E., 628 Stemling, C. V., 59 Stenby, E. H., 55 Stephan, K., 238, 271, 340, 374, 640–641 Stephan, P., 241, 264, 359 Stephan-Abdelsalam correlations, 376 Stone, C. R, 380 Stratified flow, 458, 531, 555 Stylianou, S. A., 401, 403 Suarez, J. T., 55 Subbotin, V. I., 274 Subcooled flow boiling, 597–599; see also Flow boiling convective boiling, 567–575 heat transfer, 568–573 partial boiling transition, 569, 569 pressure drop, 573–575 regimes of, 567–568, 568 single-phase forced convection, 569 transition of flow, 568, 568 void fraction, 573–575 Subcooling, 331–333, 595; see also Induced subcooling effects on boiling curve, 332 film boiling, 332 Nusselt number dependence on, 411 Sukhatme, S. P., 140 Sun, K. H., 622 Supercooled vapor, 180–184 Superheated liquid, 223 Superheat limits, 177–179 data, 177, 179 experimental observations, 177–178 for metastable liquid, 176 temperature, 179 theoretical vs. measured, 176–180 Superhydrophilic surface, 72, 144 Superhydrophobic surface, 72 Supersaturated vapor, 159, 164 Supersaturation limit, 185–188, 186, 188 Supersaturation ratios, 187 Surface(s) active materials, 53 inhomogeneity, 86, 86–87, 96 pressure, 80 roughness, 72, 86, 86, 93–94, 96, 101, 339 temperature, 250, 347 Surface excess free energy, 43, 78, 80 of thermodynamic properties, 41

703

Index Surface tension, 14, 16, 18–19, 21, 23–29, 24, 29, 681; see also Interfacial tension cellular convection, 58 critical, 77–78 effect on contact angle, 76–79 linear relation, 51, 52 of liquid metals, 39 in mixtures, 54–55 of non-metallic liquids, 79 of non-polar liquids, 51 of pure water, 51 SI units of, 39 temperature for water, 52 values of, 40 water/2-propanol mixtures, 55 water/ethylene glycol mixtures, 55 Suryanarayana, N. V., 613 Swanson, L., 359

T Taborek, J., 587 Taitel, Y., 456, 458–459 Tajima, K., 265 Takata, Y., 366 Takeyama, T., 403 Talty, R. D., 578 Tanaka, H., 401–403 Tanasawa, I., 400, 402, 438 Tang, J. Z., 94 Tang, Y. S., 601, 606 Tarek, M., 25, 25 Taylor series, 155–156, 169, 183 Teflon, 206, 208, 318, 337, 363, 363 Temperature, 347, 415; see also Leidenfrost temperature Theofanous, T. G., 217, 227, 282 Thermal boundary condition, 113, 147 Thermal diffusion, 430 Thermal energy, 112, 134 Thermal motion, 666 Thermal stability, 159 Thermodynamics analysis, 39–45, 41, 181, 204 equilibrium, 615 Gibbs-Duhem equation from, 160 homogeneous nucleation, 164–171, 180–184 limit of superheat, 164 macroscopic, 6, 11 Maxwell relations from, 153 properties, 5–7, 16, 27 similitude, 262 stability of, 154 temperature and chemical potential, 156 theory, 5, 159 Thermo-Fluid Dynamic Theory of Two-Phase Flow (Ishii and Hibiki), 496 Thin film, 32, 58, 58–59, 83 Thom, J. R. S., 477

Thomas, D. G., 441 Thom correlation, 477–478, 482; see also Correlation Thome, J. R., 367, 374, 500, 535, 595, 633, 650 Thompson, B., 592, 599 Thompson, S. M., 25, 26, 30 Three-dimensional waves, 120 Tien, C.-L., 260–261, 263, 533 Togashi, S., 402 Tolman, R. C., 26 Tolman lenght, 26, 28–30, 29 Tolubinsky, V. I., 649 Tong, L. S., 575, 591–592, 597–598, 601, 605, 606 Transient temperature profiles in bubble growth model, 231 near surface, 219 Transition flow boiling, 608–609 liquid-vapor, 3–11 Transition boiling, 316–321; see also Film boiling; Pool boiling breakdown of the vapor film, 317 for different surface conditions, 318 film boiling, 318 onset of, 317 regime, 250, 251 Transport boundary conditions, 109–113 effects, 109–149 energy, 112 equations, 109, 148 heat, 113, 148 interfacial, 140, 145 interfacial molecular, 140 laminar equations, 114 of mass, 109, 112–113 molecular, 146–148 of momentum, 109, 112 processes in nucleate pool boiling, 276 radiative, 112 thermal, 113 thermal energy, 112 of thermal energy, 515 Transverse wavy perturbation, 63 Traviss, D. P., 521–523, 528; see also Correlation Triplett, K. A., 499 Tsuruta, T., 402 Turbulent falling liquid film, 415 Turbulent film condensation, 412–419, 413, 417 Turbulent-turbulent flow, 484, 510; see also Flow Turbulent-turbulent Martinelli parameter, 577, 582 Turner, R. H., 437 Two-dimensional waves, 120 Two-phase flow, 453–500, 555, 573 adiabatic, 560, 628 analytical models of annular flow, 487–498 annular, 629 behavior, 560, 628 characteristics of, 574

704 in complex finned passage, 500 complexity of, 573 frictional pressure gradients, 642 geometry, 498–500 homogeneous flow model, 615 horizontal flow, 457–459 in microchannels, 498–500 in nanochannels, 498–500 observations, 459–463 one-dimensional, 463–470 passage size, 498–500 with phase change, 626 regimes, 453–463, 499, 587 in ribbed passage, 500 separate cylinders analysis of, 471 two-phase multiplier, 470–487 upward vertical flow, 454–457 void fraction, 470–487 Two-phase mixture of saturated nitrogen liquid, 462 Two-phase multipliers, 465–466 Tyrrell, J. W. G., 217

U Ueda, T., 526, 598 Ultrathin liquid film on solid surface, 355 Umur, A., 396–397 Unperturbed interface, 113 Upward vertical flow, 454–457 Usagi, R., 525 USSR Academy of Sciences, 593, 596 Utaka, Y., 403

V Van der Molen, S. B., 598 Vandervort, C. L., 595 Van der Waals capillarity, 681 constants, 6–7 equation, 6–7, 10, 161–162, 161–163, 176–178, 185–186 fluid, 5–6, 14 forces, 39 isotherms, 7, 8 model, 6–7, 13–14, 16, 56, 153, 681 spinodal curve, 179 theory of capillarity, 12–14, 56 Van Es, J. P., 548 Van Ouwerkerk, S. J., 380 Van Stralen, S. J. D., 230, 232, 234, 261–262, 377, 380 Van Wylen, G. J., 187 Vapor at atmospheric pressure, 462 bubble, 165, 165–166, 167, 169–170, 170, 181, 193, 196–197, 229 in cavities, 209–217, see also Heterogeneous nucleation embryo with cavity cone angle, 214

Index film, 317 heterogeneous nucleation in, 391–395 layer formation, 350 and liquid entrapment in groove, 209 liquid exchange models, 257–259 motion, 423–428 partial pressure, 434 pressure, 166–167, 181, 183, 185–186, 188, 193 recoil, 283 release cycle during film boiling, 292 slugs, 379 temperature variations, 625 trapping process, 213 viscosity, 468 Vaporization, 3, 5, 9–10, 134–142, 371, 453, 615, 639 Varanasi, K. K., 440 Velocities, 114 Velocity vector in Cartesian coordinates, 669 in spherical coordinates, 669, 669 Vertical surface falling-film condensation on, 405 film condensation of steam on, 412 Vertical tube binary mixture, 543 flow regimes, 461 saturated boiling of water, 617 Violation of intrinsic stability, 347 Void fraction, 463, 470–487, 573–575, 576 correlation, 481 defined, 453 relation, 468 tube, 512 Volume of fluid (VOF), 241, 497, 541 Volumetric free energy, 682 Von Karman’s mixing-length model, 517 Von Karman’s relation, 426

W Wainwright, T. E., 21 Waiting period, 218, 228, 229 Wall-affected region, nanoscale roughness, 360 Wall-fluid attractive forces, 189–190 Wall interaction effects, 189–192 Wallis, G. B., 456–457, 471, 482, 483, 490 Wallis separate-cylinders model, 482 Wambsganss, M. W., 499 Wang, B.-X., 540, 628 Wang, C. H., 215, 217 Wang, H. S., 539–540 Wang, P.-I., 196 Wang, W. W. W., 539 Wang, Z., 196 Warrier, G. R., 629 Water, 53, 53–55, 55, 79, 178, 680 2-propanol in, 378 annular film flow boiling of, 637

705

Index at atmospheric pressure, 215, 227–228, 232, 236, 242, 273, 313–314, 332, 408, 418 droplets on aluminum surface, 347 evaporator, 484 exit quality and pressure for, 478 film boiling in copper surface in, 315 methanol mixture, 376 nucleate boiling in, 363 pressure for, 269 saturated boiling in microchannel, 638 saturated boiling in vertical tube, 617 saturated flow boiling of, 606 vaporization of, 578 Water-solid interface, embryo formation at, 207 Wavenumber, 126, 131–132 Waves amplitude, 133 effects of interfacial, 133 on liquid films, 127–133, 128 motion, 129, 133 Wavy film, 63 Wavy flow, 458, 533, 555; see also Flow Wavy surface, 441 Wayner, P. C., Jr., 84, 357–359, 367 Webb, R. L., 364, 367, 441 Weber, M. E., 380 Weber numbers, 538, 541, 623, 626–627, 634 Weisman, J., 598, 605, 609 Welch, J. F., 397 Wemhoff, A. P., 19, 21, 24–25, 31, 31–32, 189–191, 359, 361 Wemp, C. K., 101, 145 Wenzel, U., 374 Wenzel’s model, 97–98, 102 Wenzel state, 97, 99 Westwater, J. W., 261, 263, 336, 364, 380, 395–397, 498–499 Wetness fraction, 453 Wettability, 72–76 contact angle, 72 defined, 69 metrics for, 90–94 of microstructured surfaces, 96–97 molecular-kinetic theory (MKT) of, 96 nanoscale view of, 94–96 of nanostructured surfaces, 96–97 for rough and hydrophilic surfaces, 98–101, 99 for rough and hydrophobic surfaces, 101–102 for rough and non-homogeneous surfaces, 98 for rough surface, 96–97 for smooth heterogeneous surface, 98–103 spreading coefficient, 73, 75, 80, 81, 83, 90–91 of surface, see Size and wettability of surface wickability, 91–92 work of adhesion, 74–75, 80–81, 90 Wetting liquid, 48, 50, 72 number, 93 vapor recoil on contact line motion, 283–284

Whalley, P. B., 606 Wickability, 90–92 Wilke-Chang correlation, 644 Wine tears, 63 Winterton, R. H. S., 320, 584, 588–590, 632, 632 Winterton correlation, 584 Wispy annular flow, 455 Witte, L. C., 252, 317, 336 Wojtan, L., 587 Woodruff, D. M., 396 Work of adhesion, 74–75, 80–81, 90 of cohesion, 74–75, 80–81 defined, 74–75 interaction, 74–75 Wright, R. M., 584 Wu, H. W., 401 Wu, H. Y., 538–540, 629

X Xiao-Song, W., 30 Xu, X., 359

Y Yadigaroglu, G., 189, 191, 359 Yagov, V. V., 380 Yamagata, K., 261 Yanagita, T., 265 Yang, J., 241, 264 Yang, Y. M., 377 Yao, S., 348 Yellot, J. I., 187 Yen, T.-H., 632, 632 Yerazunis, S., 417 Yilmaz, S., 336, 364 Yin, Z., 541 Ying, S. H., 605 Yoo, S. J., 621 Young, J. M., 613 Young, R. K., 362–363 Young-Dupré equation, 69 Young-Laplace equation, 44–46, 48, 50, 83, 89, 110, 117, 125, 129, 165, 168, 174–175, 180, 194, 205, 211–212, 219, 224, 393 Young’s angle, 96, 102 Young’s equation, 69, 71–72, 76, 80, 90, 98, 204, 441 Young’s relation, 100 Yu, J., 548 Yue, P.-L., 380

Z Zatell, V. A., 364 Zeldovich, Y. B., 170

706 Zeng, L. Z., 239 Zhang, L., 548, 633 Zhang, X. Y., 92 Zhao, L., 499 Zhao, T. S., 540 Zhu, S.-B., 95 Zisman, A., 77 Zivi model, 482 Zorin, A. M., 84

Index Zuber, N., 225, 230, 238, 240, 256–257, 259, 263, 265–266, 279, 279, 280–282, 284–287, 291–292, 310, 331, 496, 574, 621 Zuber correlation, 340–341, 379 Zuber critical heat flux, 279, 284 Zuber Helmholtz instability model, 282 Zuo, Y.-X., 55 Zurcher, O., 587 Zwick, S. A., 225–226