CONVECTIVE HEAT AND MASS TRANSFER This book was developed by Professor S. Mostafa Ghiaasiaan during 10 years of teaching a graduate-level course on convection heat and mass transfer. The book is ideal for a graduate course dealing with the- ory and practice of convection heat and mass transfer. The book treats well-established theory and practice on the one hand; on the other hand, it is enriched by modern areas such as flow in microchannels and computational fluid dynamics–based design and analysis methods. The book is primarily concerned with convective heat transfer. Essentials of mass transfer are also covered. The mass transfer material and prob- lems are presented such that they can be easily skipped, should that be preferred. The book is richly enhanced by exercises and end-of-chapter problems. Solutions are available for qualified instructors. The book includes 17 appendices providing compilations of most essential prop- erties and mathematical information for analysis of convective heat and mass transfer processes. Professor S. Mostafa Ghiaasiaan has been a member of the Woodruff School of Mechanical Engineering at Georgia Institute of Technology since 1991 after receiving a Ph.D. in Thermal Science from the Univer- sity of California, Los Angeles, in 1983 and working in the aerospace and nuclear power industry for eight years. His industrial research and development activity was on modeling and simulation of transport processes, multiphase flow, and nuclear reactor thermal hydraulics and safety. His current research areas include nuclear reactor thermal hydraulics, particle transport, cryogenics and cryocoolers, and multi- phase flow and change-of-phase heat transfer in microchannels. He has more than 150 academic publications, including 90 journal arti- cles, on transport phenomena and multiphase flow. Among the hon- ors he has received for his publications are the Chemical Engineering Science’s Most Cited Paper for 2003–2006 Award, the National Heat Transfer Conference Best Paper Award (1999), and the Science Appli- cations International Corporation Best Paper Award (1990 and 1988). He has been a member of American Society of Mechanical Engineers (ASME) and the American Nuclear Society for more than 20 years and was elected an ASME Fellow in 2004. Currently he is the Exec- utive Editor of Annals of Nuclear Energy for Asia, Africa, and Aus- tralia. This is his second book with Cambridge University Press—the first was Two-Phase Flow, Boiling, and Condensation, In Conventional and Miniature Systems (2007). Convective Heat and Mass Transfer S. Mostafa Ghiaasiaan Georgia Institute of Technology cambridge university press Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, S ˜ ao Paulo, Delhi, Tokyo, Mexico City Cambridge University Press 32 Avenue of the Americas, New York, NY 10013-2473, USA www.cambridge.org Information on this title: www.cambridge.org/9781107003507 c S. Mostafa Ghiaasiaan 2011 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2011 Printed in the United States of America A catalog record for this publication is available from the British Library. Library of Congress Cataloging in Publication data Ghiaasiaan, Seyed Mostafa, 1953– Convective heat and mass transfer / Mostafa Ghiaasiaan. p. cm. Includes bibliographical references and index. ISBN 978-1-107-00350-7 (hardback) 1. Heat – Convection. I. Title. QC327.G48 2011 536  .25 – dc22 2011001977 ISBN 978-1-107-00350-7 Hardback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party Internet Web sites referred to in this publication and does not guarantee that any content on such Web sites is, or will remain, accurate or appropriate. To my wife Pari Fatemeh Shafiei, and my son Saam CONTENTS Preface page xv Frequently Used Notation xvii 1 Thermophysical and Transport Fundamentals 1 1.1 Conservation Principles 1 1.1.1 Lagrangian and Eulerian Frames 1 1.1.2 Mass Conservation 2 1.1.3 Conservation of Momentum 3 1.1.4 Conservation of Energy 6 1.2 Multicomponent Mixtures 11 1.2.1 Basic Definitions and Relations 11 1.2.2 Thermodynamic Properties 15 1.3 Fundamentals of Diffusive Mass Transfer 17 1.3.1 Species Mass Conservation 17 1.3.2 Diffusive Mass Flux and Fick’s Law 18 1.3.3 Species Mass Conservation When Fick’s Law Applies 19 1.3.4 Other Types of Diffusion 20 1.3.5 Diffusion in Multicomponent Mixtures 20 1.4 Boundary and Interfacial Conditions 22 1.4.1 General Discussion 22 1.4.2 Gas–Liquid Interphase 24 1.4.3 Interfacial Temperature 24 1.4.4 Sparingly Soluble Gases 27 1.4.5 Convention for Thermal and Mass Transfer Boundary Conditions 30 1.5 Transport Properties 31 1.5.1 Mixture Rules 31 1.5.2 Transport Properties of Gases and the Gas-Kinetic Theory 32 1.5.3 Diffusion of Mass in Liquids 37 1.6 The Continuum Flow Regime and Size Convention for Flow Passages 38 Problems 39 vii viii Contents 2 Boundary Layers 44 2.1 Boundary Layer on a Flat Plate 44 2.2 Laminar Boundary-Layer Conservation Equations 48 2.3 Laminar Boundary-Layer Thicknesses 51 2.4 Boundary-Layer Separation 53 2.5 Nondimensionalization of Conservation Equations and Similitude 54 Problems 58 3 External Laminar Flow: Similarity Solutions for Forced Laminar Boundary Layers 61 3.1 Hydrodynamics of Flow Parallel to a Flat Plate 61 3.2 Heat and Mass Transfer During Low-Velocity Laminar Flow Parallel to a Flat Plate 65 3.3 Heat Transfer During Laminar Parallel Flow Over a Flat Plate With Viscous Dissipation 71 3.4 Hydrodynamics of Laminar Flow Past a Wedge 73 3.5 Heat Transfer During Laminar Flow Past a Wedge 78 3.6 Effects of Compressibility and Property Variations 80 Problems 85 4 Internal Laminar Flow 90 4.1 Couette and Poiseuille Flows 90 4.2 The Development of Velocity, Temperature, and Concentration Profiles 94 4.2.1 The Development of Boundary Layers 94 4.2.2 Hydrodynamic Parameters of Developing Flow 97 4.2.3 The Development of Temperature and Concentration Profiles 100 4.3 Hydrodynamics of Fully Developed Flow 103 4.4 Fully Developed Hydrodynamics and Developed Temperature or Concentration Distributions 107 4.4.1 Circular Tube 107 4.4.2 Flat Channel 110 4.4.3 Rectangular Channel 113 4.4.4 Triangular Channel 113 4.4.5 Concentric Annular Duct 114 4.5 Fully Developed Hydrodynamics, Thermal or Concentration Entrance Regions 117 4.5.1 Circular Duct With Uniform Wall Temperature Boundary Conditions 117 4.5.2 Circular Duct With Arbitrary Wall Temperature Distribution in the Axial Direction 124 4.5.3 Circular Duct With Uniform Wall Heat Flux 126 4.5.4 Circular Duct With Arbitrary Wall Heat Flux Distribution in the Axial Coordinate 129 Contents ix 4.5.5 Flat Channel With Uniform Heat Flux Boundary Conditions 130 4.5.6 Flat Channel With Uniform Wall Temperature Boundary Conditions 132 4.5.7 Rectangular Channel 135 4.6 Combined Entrance Region 135 4.7 Effect of Fluid Property Variations 137 Appendix 4A: The Sturm–Liouville Boundary-Value Problems 141 Problems 141 5 Integral Methods 151 5.1 Integral Momentum Equations 151 5.2 Solutions to the Integral Momentum Equation 153 5.2.1 Laminar Flow of an Incompressible Fluid Parallel to a Flat Plate Without Wall Injection 153 5.2.2 Turbulent Flow of an Incompressible Fluid Parallel to a Flat Plate Without Wall Injection 156 5.2.3 Turbulent Flow of an Incompressible Fluid Over a Body of Revolution 158 5.3 Energy Integral Equation 159 5.4 Solutions to the Energy Integral Equation 161 5.4.1 Parallel Flow Past a Flat Surface 161 5.4.2 Parallel Flow Past a Flat Surface With an Adiabatic Segment 163 5.4.3 Parallel Flow Past a Flat Surface With Arbitrary Wall Surface Temperature or Heat Flux 165 5.5 Approximate Solutions for Flow Over Axisymmetric Bodies 167 Problems 173 6 Fundamentals of Turbulence and External Turbulent Flow 177 6.1 Laminar–Turbulent Transition and the Phenomenology of Turbulence 177 6.2 Fluctuations and Time (Ensemble) Averaging 180 6.3 Reynolds Averaging of Conservation Equations 181 6.4 Eddy Viscosity and Eddy Diffusivity 183 6.5 Universal Velocity Profiles 185 6.6 The Mixing-Length Hypothesis and Eddy Diffusivity Models 188 6.7 Temperature and Concentration Laws of the Wall 192 6.8 Kolmogorov Theory of the Small Turbulence Scales 196 6.9 Flow Past Blunt Bodies 200 Problems 205 7 Internal Turbulent Flow 208 7.1 General Remarks 208 7.2 Hydrodynamics of Turbulent Duct Flow 211 7.2.1 Circular Duct 211 7.2.2 Noncircular Ducts 217 x Contents 7.3 Heat Transfer: Fully Developed Flow 218 7.3.1 Universal Temperature Profile i n a Circular Duct 218 7.3.2 Application of Eddy Diffusivity Models for Circular Ducts 221 7.3.3 Noncircular Ducts 224 7.4 Heat Transfer: Fully Developed Hydrodynamics, Thermal Entrance Region 224 7.4.1 Circular Duct With Uniform Wall Temperature or Concentration 224 7.4.2 Circular Duct With Uniform Wall Heat Flux 226 7.4.3 Some Useful Correlations for Circular Ducts 229 7.4.4 Noncircular Ducts 231 7.5 Combined Entrance Region 231 Problems 238 8 Effect of Transpiration on Friction, Heat, and Mass Transfer 243 8.1 Couette Flow Film Model 243 8.2 Gas–Liquid Interphase 248 Problems 256 9 Analogy Among Momentum, Heat, and Mass Transfer 258 9.1 General Remarks 258 9.2 Reynolds Analogy 259 9.3 Prandtl–Taylor Analogy 261 9.4 Von Karman Analogy 263 9.5 The Martinelli Analogy 265 9.6 The Analogy of Yu et al. 265 9.7 Chilton–Colburn Analogy 267 Problems 272 10 Natural Convection 275 10.1 Natural-Convection Boundary Layers on Flat Surfaces 275 10.2 Phenomenology 278 10.3 Scaling Analysis of Laminar Boundary Layers 280 10.4 Similarity Solutions for a Semi-Infinite Vertical Surface 285 10.5 Integral Analysis 289 10.6 Some Widely Used Empirical Correlations for Flat Vertical Surfaces 294 10.7 Natural Convection on Horizontal Flat Surfaces 295 10.8 Natural Convection on Inclined Surfaces 297 10.9 Natural Convection on Submerged Bodies 298 10.10 Natural Convection in Vertical Flow Passages 300 10.11 Natural Convection in Enclosures 304 10.12 Natural Convection in a Two-Dimensional Rectangle With Heated Vertical Sides 305 10.13 Natural Convection in Horizontal Rectangles 307 10.14 Natural Convection in Inclined Rectangular Enclosures 309 [...]... length (m) Mass transfer entrance length (m) Thermal (heat transfer) entrance length (m) Turbulence mixing length (m) Turbulence mixing length for mass transfer (m) Heated length (m) Turbulence mixing length for heat transfer (m) Molar mass (kg/kmol) Mach number Mass fraction; dimensionless constant Mass (kg); mass of a single molecule (kg) Mass flux (kg/m2 s) Ratio between concentration-based and thermal-based... in science and technology education caused by the proliferation of computing power and information Like most other science and technology fields, convective heat and mass transfer is already too vast to be covered in a semester-level course even at an outline level and is yet undergoing exponential expansion The expansion is both quantitative and qualitative On the quantitative side, novel and hitherto... of these topics should not be at the expense of basics and classical methods This book is the outcome of more than 10 years of teaching a graduate-level course on convective heat and mass transfer It also benefits from my more than 20 years of experience of teaching undergraduate heat transfer and other thermal fluid science courses to mechanical and nuclear engineering students The book is designed to... as the basis for a semester-level graduate course dealing with theory and practice of convection heat and mass transfer My incentive in writing the book is to strike a balance between well-established theory and practice on the one hand, and modern areas such as flow in microchannels and computational fluid dynamics (CFD)–based design and analysis methods on the other I have had much difficulty finding such... existing textbooks while teaching convection to graduate students and had to rely on my own class notes and recent issues of journals for much of the syllabi of my classes The book is primarily concerned with convective heat transfer Essentials of mass transfer are also covered, although only briefly The mass transfer material xv xvi Preface and problems are presented such that they can be easily skipped,... cross-sectional dimension (m) Interfacial surface area concentration (surface area per unit) mixture volume (m−1 ) Blowing parameter Mass- flux-based heat transfer driving force Molar-flux-based heat transfer driving force Mass- flux-based mass transfer driving force Molar-flux-based mass transfer driving force Biot number = hl/k μU 2 Brinkman number = k| T| Buoyancy number = Gr/Rem One-half of the shorter cross-sectional... Henry number Specific enthalpy (J/kg) Heat transfer coefficient (W/m2 K); height (m) Radiative heat transfer coefficient (W/m2 K) Latent heats of vaporization, fusion, and sublimation (J/kg) Molar-based latent heats of vaporization, fusion, and sublimation (J/kmol) Modified Bessel’s function of the first kind and mth order Diffusive molar flux (k mol/m2 s) Diffusive mass flux (kg/m2 s); molecular flux (m−2... the Combined Thermal and Mass Diffusion Effects 10.15.1 Conservation Equations and Scaling Analysis 10.15.2 Heat and Mass Transfer Analogy 10.16 Solutions for Natural Convection Caused by Combined Thermal and Mass Diffusion Effects Problems 311 311 316 317 327 11 Mixed Convection 332 11.1 11.2 11.3 11.4 11.5 11.6 Laminar Boundary-Layer Equations and Scaling Analysis... turbulent transport models in current convective heat transfer research and analysis The discussions are meant to show the students where these models have come from, with an emphasis on how they treat not just the fluid mechanics aspects of turbulent flow but also the transport of heat and mass Although access to and practice with CFD tools are helpful for understanding these turbulence models, the chapter... Near-Wall Turbulence Modeling and Wall Functions 12.4 The K–ε Model 12.4.1 General Formulation 12.4.2 Near-Wall Treatment 12.4.3 Turbulent Heat and Mass Fluxes 12.5 Other Two-Equation Turbulence Models 12.6 The Reynolds Stress Transport Models 12.6.1 General Formulation 12.6.2 Simplification for Heat and Mass Transfer 12.6.3 Near-Wall Treatment of Turbulence 12.6.4 Summary of Equations and Unknowns 12.7 Algebraic . parameter B h Mass- flux-based heat transfer driving force ˜ B h Molar-flux-based heat transfer driving force B m Mass- flux-based mass transfer driving force ˜ B m Molar-flux-based mass transfer driving. microchannels and computational fluid dynamics–based design and analysis methods. The book is primarily concerned with convective heat transfer. Essentials of mass transfer are also covered. The mass transfer. on convective heat and mass transfer. It also benefits from my more than 20 years of experience of teaching undergraduate heat transfer and other thermal fluid science courses to mechanical and