ii TRANSPORT PHENOMENA i ii TRANSPORT PHENOMENA An Introduction to Advanced Topics LARRY A. GLASGOW Professor of Chemical Engineering Kansas State University Manhattan, Kansas iii Copyright © 2010 by John Wiley & Sons, Inc. All rights reserved. Published by John Wiley & Sons, Inc., Hoboken, New Jersey. Published simultaneously in Canada. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978) 750-4470, or on the web at www.copyright.com. 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TP156.T7G55 2010 530.4’75–dc22 2009052127 Printed in the United States of America 10 9 8 7 6 5 4 3 2 1 iv CONTENTS Preface ix 3.13 Flows in Open Channels, 41 3.14 Pulsatile Flows in Cylindrical Ducts, 42 1. Introduction and Some Useful Review 1 3.15 Some Concluding Remarks for Incompressible 1.1 A Message for the Student, 1 Viscous Flows, 43 1.2 Differential Equations, 3 References, 44 1.3 Classification of Partial Differential Equations and Boundary Conditions, 7 4. External Laminar Flows and Boundary-Layer 1.4 Numerical Solutions for Partial Differential Theory 46 Equations, 8 4.1 Introduction, 46 1.5 Vectors, Tensors, and the Equation of Motion, 8 4.2 The Flat Plate, 47 1.6 The Men for Whom the Navier-Stokes Equations 4.3 Flow Separation Phenomena About Bluff are Named, 12 Bodies, 50 1.7 Sir Isaac Newton, 13 4.4 Boundary Layer on a Wedge: The Falkner–Skan References, 14 Problem, 52 4.5 The Free Jet, 53 2. Inviscid Flow: Simplified Fluid Motion 15 4.6 Integral Momentum Equations, 54 2.1 Introduction, 15 4.7 Hiemenz Stagnation Flow, 55 2.2 Two-Dimensional Potential Flow, 16 4.8 Flow in the Wake of a Flat Plate at Zero 2.3 Numerical Solution of Potential Flow Problems, 20 Incidence, 56 2.4 Conclusion, 22 4.9 Conclusion, 57 References, 23 References, 58 3. Laminar Flows in Ducts and Enclosures 24 5. Instability, Transition, and Turbulence 59 3.1 Introduction, 24 5.1 Introduction, 59 3.2 Hagen–Poiseuille Flow, 24 5.2 Linearized Hydrodynamic Stability Theory, 60 3.3 Transient Hagen–Poiseuille Flow, 25 5.3 Inviscid Stability: The Rayleigh Equation, 63 3.4 Poiseuille Flow in an Annulus, 26 5.4 Stability of Flow Between Concentric 3.5 Ducts with Other Cross Sections, 27 Cylinders, 64 3.6 Combined Couette and Poiseuille Flows, 28 5.5 Transition, 66 3.7 Couette Flows in Enclosures, 29 5.5.1 Transition in Hagen–Poiseuille 3.8 Generalized Two-Dimensional Fluid Motion in Flow, 66 Ducts, 32 5.5.2 Transition for the Blasius Case, 67 3.9 Some Concerns in Computational Fluid 5.6 Turbulence, 67 Mechanics, 35 5.7 Higher Order Closure Schemes, 71 3.10 Flow in the Entrance of Ducts, 36 5.7.1 Variations, 74 3.11 Creeping Fluid Motions in Ducts and Cavities, 38 5.8 Introduction to the Statistical Theory of 3.12 Microfluidics: Flow in Very Small Channels, 38 Turbulence, 74 3.12.1 Electrokinetic Phenomena, 39 5.9 Conclusion, 79 3.12.2 Gases in Microfluidics, 40 References, 81 v vi CONTENTS 6. Heat Transfer by Conduction 83 8.2 Unsteady Evaporation of Volatile Liquids: The 6.1 Introduction, 83 Arnold Problem, 120 6.2 Steady-State Conduction Problems in 8.3 Diffusion in Rectangular Geometries, 122 Rectangular Coordinates, 84 8.3.1 Diffusion into Quiescent Liquids: 6.3 Transient Conduction Problems in Rectangular Absorption, 122 Coordinates, 86 8.3.2 Absorption with Chemical Reaction, 123 6.4 Steady-State Conduction Problems in Cylindrical 8.3.3 Concentration-Dependent Diffusivity, 124 Coordinates, 88 8.3.4 Diffusion Through a Membrane, 125 6.5 Transient Conduction Problems in Cylindrical 8.3.5 Diffusion Through a Membrane with Coordinates, 89 Variable D, 125 6.6 Steady-State Conduction Problems in Spherical 8.4 Diffusion in Cylindrical Systems, 126 Coordinates, 92 8.4.1 The Porous Cylinder in Solution, 126 6.7 Transient Conduction Problems in Spherical 8.4.2 The Isothermal Cylindrical Catalyst Coordinates, 93 Pellet, 127 6.8 Kelvin’s Estimate of the Age of the Earth, 95 8.4.3 Diffusion in Squat (Small L/d) 6.9 Some Specialized Topics in Conduction, 95 Cylinders, 128 6.9.1 Conduction in Extended Surface Heat 8.4.4 Diffusion Through a Membrane with Edge Transfer, 95 Effects, 128 6.9.2 Anisotropic Materials, 97 8.4.5 Diffusion with Autocatalytic Reaction in a 6.9.3 Composite Spheres, 99 Cylinder, 129 6.10 Conclusion, 100 8.5 Diffusion in Spherical Systems, 130 References, 100 8.5.1 The Spherical Catalyst Pellet with Exothermic Reaction, 132 8.5.2 Sorption into a Sphere from a Solution of 7. Heat Transfer with Laminar Fluid Motion 101 Limited Volume, 133 7.1 Introduction, 101 8.6 Some Specialized Topics in Diffusion, 133 7.2 Problems in Rectangular Coordinates, 102 8.6.1 Diffusion with Moving Boundaries, 133 7.2.1 Couette Flow with Thermal Energy 8.6.2 Diffusion with Impermeable Production, 103 Obstructions, 135 7.2.2 Viscous Heating with 8.6.3 Diffusion in Biological Systems, 135 Temperature-Dependent Viscosity, 104 8.6.4 Controlled Release, 136 7.2.3 The Thermal Entrance Region in Rectangular 8.7 Conclusion, 137 Coordinates, 104 References, 137 7.2.4 Heat Transfer to Fluid Moving Past a Flat Plate, 106 7.3 Problems in Cylindrical Coordinates, 107 9. Mass Transfer in Well-Characterized Flows 139 7.3.1 Thermal Entrance Length in a Tube: The 9.1 Introduction, 139 Graetz Problem, 108 9.2 Convective Mass Transfer in Rectangular 7.4 Natural Convection: Buoyancy-Induced Fluid Coordinates, 140 Motion, 110 9.2.1 Thin Film on a Vertical Wall, 140 7.4.1 Vertical Heated Plate: The Pohlhausen 9.2.2 Convective Transport with Reaction at the Problem, 110 Wall, 141 7.4.2 The Heated Horizontal Cylinder, 111 9.2.3 Mass Transfer Between a Flowing Fluid and 7.4.3 Natural Convection in Enclosures, 112 a Flat Plate, 142 7.4.4 Two-Dimensional Rayleigh–Benard 9.3 Mass Transfer with Laminar Flow in Cylindrical Problem, 114 Systems, 143 7.5 Conclusion, 115 9.3.1 Fully Developed Flow in a Tube, 143 References, 116 9.3.2 Variations for Mass Transfer in a Cylindrical Tube, 144 8. Diffusional Mass Transfer 117 9.3.3 Mass Transfer in an Annulus with Laminar 8.1 Introduction, 117 Flow, 145 8.1.1 Diffusivities in Gases, 118 9.3.4 Homogeneous Reaction in Fully-Developed 8.1.2 Diffusivities in Liquids, 119 Laminar Flow, 146 CONTENTS vii 9.4 Mass Transfer Between a Sphere and a Moving 11.2 Liquid–Liquid Systems, 180 Fluid, 146 11.2.1 Droplet Breakage, 180 9.5 Some Specialized Topics in Convective Mass 11.3 Particle–Fluid Systems, 183 Transfer, 147 11.3.1 Introduction to Coagulation, 183 9.5.1 Using Oscillatory Flows to Enhance 11.3.2 Collision Mechanisms, 183 Interphase Transport, 147 11.3.3 Self-Preserving Size Distributions, 186 9.5.2 Chemical Vapor Deposition in Horizontal 11.3.4 Dynamic Behavior of the Particle Size Reactors, 149 Distribution, 186 9.5.3 Dispersion Effects in Chemical 11.3.5 Other Aspects of Particle Size Distribution Reactors, 150 Modeling, 187 9.5.4 Transient Operation of a Tubular 11.3.6 A Highly Simplified Example, 188 Reactor, 151 11.4 Multicomponent Diffusion in Gases, 189 9.6 Conclusion, 153 11.4.1 The Stefan–Maxwell Equations, 189 References, 153 11.5 Conclusion, 191 References, 192 10. Heat and Mass Transfer in Turbulence 155 Problems to Accompany Transport Phenomena: An 10.1 Introduction, 155 Introduction to Advanced Topics 195 10.2 Solution Through Analogy, 156 10.3 Elementary Closure Processes, 158 Appendix A: Finite Difference Approximations for 10.4 Scalar Transport with Two-Equation Models of Derivatives 238 Turbulence, 161 10.5 Turbulent Flows with Chemical Reactions, 162 Appendix B: Additional Notes on Bessel’s Equation and 10.5.1 Simple Closure Schemes, 164 Bessel Functions 241 10.6 An Introduction to pdf Modeling, 165 10.6.1 The Fokker–Planck Equation and pdf Appendix C: Solving Laplace and Poisson (Elliptic) Modeling of Turbulent Reactive Partial Differential Equations 245 Flows, 165 10.6.2 Transported pdf Modeling, 167 Appendix D: Solving Elementary Parabolic Partial 10.7 The Lagrangian View of Turbulent Differential Equations 249 Transport, 168 10.8 Conclusions, 171 Appendix E: Error Function 253 References, 172 Appendix F: Gamma Function 255 11. Topics in Multiphase and Multicomponent Systems 174 Appendix G: Regular Perturbation 257 11.1 Gas–Liquid Systems, 174 11.1.1 Gas Bubbles in Liquids, 174 Appendix H: Solution of Differential Equations by 11.1.2 Bubble Formation at Orifices, 176 Collocation 260 11.1.3 Bubble Oscillations and Mass Transfer, 177 Index 265 viii