Krantz W.B. Scaling Analysis in Modeling Transport and Reaction Processes: A Systematic Approach to Model Building and the Art of Approximation
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SCALING ANALYSIS IN
MODELING TRANSPORT
AND REACTION PROCESSES
A Systematic Approach to Model
Building and the Art of Approximation
William B. Krantz
Isaac M. Meyer Chair Professor
National University of Singapore
Rieveschl Ohio Eminent Scholar and Professor Emeritus
University of Cincinnati
President’s Teaching Scholar and Professor Emeritus
University of Colorado
A JOHN WILEY & SONS, INC., PUBLICATION
SCALING ANALYSIS IN
MODELING TRANSPORT
AND REACTION PROCESSES
SCALING ANALYSIS IN
MODELING TRANSPORT
AND REACTION PROCESSES
A Systematic Approach to Model
Building and the Art of Approximation
William B. Krantz
Isaac M. Meyer Chair Professor
National University of Singapore
Rieveschl Ohio Eminent Scholar and Professor Emeritus
University of Cincinnati
President’s Teaching Scholar and Professor Emeritus
University of Colorado
A JOHN WILEY & SONS, INC., PUBLICATION
Cover: Smaller and Smaller,a38cm× 38 cm wood engraving by Maurits Cornelis Escher
(1898–1972), is a beautiful marriage between art and mathematics that employs hyperbolic tiling of
two-dimensional space. It provides an appropriate introduction to the subject of this book— whereas
the reptilian creatures become progressively smaller toward the center of the figure, they are the same
size in scaled space based on a characteristic length for their geometry.
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Wiley Bicentennial Logo: Richard J. Pacifico
Library of Congress Cataloging-in-Publication Data:
Krantz, William B., 1939–
Scaling analysis in modeling transport and reaction processes : a systematic
approach to model building and the art of approximation / William B. Krantz.
p. cm.
Includes bibliographical references and index.
ISBN 978-0-471-77261-3 (cloth)
1. Transport theory—Mathematics. 2. Chemical reactions—Mathematical
models. 3. Scaling laws (Statistical physics) 4. Approximation theory. 5.
Mensuration. I. Title.
QC175.25.S8K73 2007
530.13
8–dc22
2006030750
Printed in the United States of America.
10987654321
To my wife, June, for her continued love, support, patience, and understanding
throughout the years of our marriage and particularly
during the time that I was writing this book!
CONTENTS
Preface xi
Acknowledgments xv
1 Introduction 1
1.1 Motivation for Using Scaling Analysis 1
1.2 Organization of the Book 5
2 Systematic Method for Scaling Analysis 7
2.1 Introduction 7
2.2 Mathematical Basis for Scaling Analysis 7
2.3 Order-of-One Scaling Analysis 8
2.4 Scaling Alternative for Dimensional Analysis 13
2.5 Summary 18
3 Applications in Fluid Dynamics 19
3.1 Introduction 19
3.2 Fully Developed Laminar Flow 20
3.3 Creeping- and Lubrication-Flow Approximations 26
3.4 Boundary-Layer-Flow Approximation 32
3.5 Quasi-Steady-State-Flow Approximation 38
3.6 Flows with End and Sidewall Effects 43
3.7 Free Surface Flow 45
3.8 Porous Media Flow 52
3.9 Compressible Fluid Flow 56
3.10 Dimensional Analysis Correlation for the Terminal Velocity 62
3.11 Summary 67
3.E Example Problems 70
3.P Practice Problems 110
4 Applications in Heat Transfer 145
4.1 Introduction 145
4.2 Steady-State Heat Transfer with End Effects 146
4.3 Film and Penetration Theory Approximations 153
vii
viii CONTENTS
4.4 Small Biot Number Approximation 159
4.5 Small Peclet Number Approximation 163
4.6 Boundary-Layer or Large Peclet Number Approximation 167
4.7 Heat Transfer with Phase Change 173
4.8 Temperature-Dependent Physical Properties 180
4.9 Thermally Driven Free Convection: Boussinesq Approximation 183
4.10 Dimensional Analysis Correlation for Cooking a Turkey 187
4.11 Summary 193
4.E Example Problems 196
4.P Practice Problems 224
5 Applications in Mass Transfer 252
5.1 Introduction 252
5.2 Film Theory Approximation 253
5.3 Penetration Theory Approximation 259
5.4 Small Peclet Number Approximation 261
5.5 Small Damk
¨
ohler Number Approximation 266
5.6 Large Peclet Number Approximation 269
5.7 Quasi-Steady-State Approximation 273
5.8 Membrane Permeation with Nonconstant Diffusivity 277
5.9 Solutally Driven Free Convection Due to Evapotranspiration 281
5.10 Dimensional Analysis for a Membrane-Lung Oxygenator 287
5.11 Summary 293
5.E Example Problems 297
5.P Practice Problems 336
6 Applications in Mass Transfer with Chemical Reaction 360
6.1 Introduction 360
6.2 Concept of the Microscale Element 362
6.3 Scaling the Microscale Element 364
6.4 Slow Reaction Regime 371
6.5 Intermediate Reaction Regime 371
6.6 Fast Reaction Regime 372
6.7 Instantaneous Reaction Regime 373
6.8 Scaling the Macroscale Element 377
6.9 Kinetic Domain of the Slow Reaction Regime 380
6.10 Diffusional Domain of the Slow Reaction Regime 381
6.11 Implications of Scaling Analysis for Reactor Design 381
6.12 Mass-Transfer Coefficients for Reacting Systems 387
6.13 Design of a Continuous Stirred Tank Reactor 390
6.14 Design of a Packed Column Absorber 394
6.15 Summary 397
6.P Practice Problems 399