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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CONTENTS ix

7 Applications in Process Design 414

7.1 Introduction 414

7.2 Design of a Membrane Lung Oxygenator 415

7.3 Pulsed Single-Bed Pressure-Swing Adsorption 424

7.4 Thermally Induced Phase-Separation Process 438

7.5 Fluid-Wall Aerosol Flow Reactor for Hydrogen Production 448

7.6 Summary 464

7.P Practice Problems 467

Appendix A Sign Convention for the Force on a Fluid Particle 480

Appendix B Generalized Form of the Transport Equations 482

B.1 Continuity Equation 482

B.2 Equations of Motion 482

B.3 Equations of Motion for Porous Media 483

B.4 Thermal Energy Equation 483

B.5 Equation of Continuity for a Binary Mixture 484

Appendix C Continuity Equation 486

C.1 Rectangular Coordinates 486

C.2 Cylindrical Coordinates 487

C.3 Spherical Coordinates 487

Appendix D Equations of Motion 489

D.1 Rectangular Coordinates 489

D.2 Cylindrical Coordinates 490

D.3 Spherical Coordinates 492

Appendix E Equations of Motion for Porous Media 494

E.1 Rectangular Coordinates 494

E.2 Cylindrical Coordinates 494

E.3 Spherical Coordinates 495

Appendix F Thermal Energy Equation 496

F.1 Rectangular Coordinates 496

F.2 Cylindrical Coordinates 497

F.3 Spherical Coordinates 497

Appendix G Equation of Continuity for a Binary Mixture 499

G.1 Rectangular Coordinates 499

x CONTENTS

G.2 Cylindrical Coordinates 500

G.3 Spherical Coordinates 502

Appendix H Integral Relationships 504

H.1 Leibnitz Formula for Differentiating an Integral 504

H.2 Gauss Ostrogradskii Divergence Theorem 504

Notation 506

Index 515

PREFACE

How big is big? How small is small? How wide is wide? How tall is tall?

Scaling analysis as deﬁned in this book involves a systematic method for nondimen-

sionalizing the dependent and independent variables as well as their derivatives in a

set of describing equations for a physical process. The unique aspect of this nondi-

mensionalization is that it is done to ensure that the variables and their derivatives

are bounded of order one; this implies that the magnitude of the dimensionless vari-

ables and their derivatives can range between zero and more-or-less 1. When this

order-of-one scaling is done, the magnitude of the resulting dimensionless groups

permits assessing the relative importance of the various terms in the describing

equations; this in turn has many applications. The magnitude of the dimensionless

groups appearing in the resulting dimensionless describing equations can be used to

assess possible simplifying approximations. Order-of-one scaling analysis results

in the minimum parametric representation of the describing equations. As such,

this systematic method of scaling offers many advantages relative to dimensional

analysis using the Pi theorem, which does not necessarily result in the minimum

number of dimensionless groups. A particular advantage of scaling analysis is that

it permits assessing the usefulness of a process or technology without the need

for prior bench- or pilot-scale data. It also provides a template for the design of

experiments to explore a new process or to validate a mathematical model.

The motivation for developing the approach to scaling analysis presented in this

book extends back over 40 years, to when the author was a graduate student in the

Department of Chemical Engineering at the University of California at Berkeley.

The author had difﬁculty in grasping constructs such as hydrodynamic boundary-

layer theory that were introduced using rather intuitive arguments such as those

found in Schlichting’s classic book.

The author strongly believed that boundary-

layer theory as well as approximations such as creeping and lubrication ﬂows, ﬁlm

theory, penetration theory, quasi-steady-state, and so on, which were introduced

using intuitive arguments, could in fact be developed systematically. During this

time the author rather serendipitously became aware of the work of Hellums and

Churchill, who used a type of scaling analysis for systematically arriving at the

Anonymous—attributed to an inquisitive young child!

H. Schlichting, Boundary Layer Theory, McGraw-Hill, New York, 1960.

xii PREFACE

form of dimensionless variables that permitted similarity solutions to partial dif-

ferential equations.

The book by Hansen, who used the Lie group of uniform

magniﬁcations and contractions to explore the spectrum of differential equations

and associated boundary and initial conditions that would permit similarity solu-

tions, was invaluable for establishing the mathematical basis for scaling analysis.

The pioneering books of van Dyke

and Nayfeh,

which focused on the use of

perturbation expansions for solving differential equations, were particularly helpful

in developing the concepts of ordering and multiple scales. Development of the

microscale–macroscale modeling concept for modeling heterogeneous systems was

strongly inﬂuenced by the book Mass Transfer with Chemical Reaction by Astarita.

Since the author was a product of the “transport phenomena” generation of engi-

neering students, many of the examples and problems in this book were inspired

by his perceived need to justify the assumptions underlying material presented in

the classic textbook Transport Phenomena by Bird, Stewart, and Lightfoot.

The

conﬂuence of these inﬂuences led the author to develop the approach to scaling

analysis presented in this book. Whereas this systematic method for scaling analysis

borrows liberally from these prior mathematical developments, the author believes

this to be an original contribution that he proffers to the community of scholars in

science and engineering as both a teaching and a research tool.

When the author began his academic career in the Department of Chemical

Engineering at the University of Colorado in 1968, he introduced the use of scal-

ing analysis in the courses he taught involving ﬂuid dynamics, heat transfer, mass

transfer, and reactor design. The use of scaling analysis was very well received by

the many engineering students who passed through the courses the author taught

during his 38 years in academia. In particular, it helped his students by provid-

ing a systematic method for understanding subtle concepts such as creeping and

boundary-layer ﬂows in ﬂuid dynamics, the Boussinesq approximation for ther-

mally driven free convection, Taylor dispersion in mass transfer, and the various

reaction regimes in mass transfer with chemical reaction. Scaling analysis was also

helpful to students because it provided a uniﬁed approach to teaching transport and

reaction processes. For example, scaling analysis provides a systematic method of

illustrating the analogous roles played by the Reynolds number in ﬂuid dynamics

and the Peclet number in heat or mass transfer, or the Biot number in heat transfer

and the Damk

ohler number in mass transfer with chemical reaction. The effec-

tive use of scaling analysis as a pedagogical tool in his courses contributed in no

small way to the author being recognized by many teaching awards, including life-

time designation as a President’s Teaching Scholar of the University of Colorado

and several awards from the American Society for Engineering Education. The

J. D. Hellums and S. W. Churchill, A.I.Ch.E.J., 10, 110 (1964).

A. G. Hansen, Similarity Analysis of Boundary Value Problems in Engineering, Prentice-Hall, Engle-

wood Cliffs, NJ 1964.

M. Van Dyke, Perturbation Methods in Fluid Mechanics, Parabolic Press, Stanford, CA, 1975.

A. H. Nayfeh, Perturbation Methods, Wiley, New York, 1973.

G. Astarita, Mass Transfer with Chemical Reaction, Elsevier, New York, 1967.

R. B. Bird, W. E. Stewart, and E. N. Lightfoot, Transport Phenomena, Wiley, New York, 1960.

PREFACE xiii

favorable response the author received from his students to the use of scaling anal-

ysis in his courses led him to present and publish several papers on this subject. This

national as well as international exposure for this systematic approach to scaling

analysis catalyzed a response from the academic community whose encouragement

motivated the author to write this book.

Whereas scaling analysis clearly is invaluable as a pedagogical tool, it also has

application in research and development. For example, scaling analysis allows one

to assess the value of a new process by providing order-of-magnitude estimates of

the anticipated performance. It also can be used to establish the process parameters

in the design of both numerical and laboratory experiments to explore new tech-

nologies. For this reason, timely examples drawn from the author’s experience are

included that effectively illustrate how scaling analysis was used to design a novel

membrane–lung oxygenator,

to assess the use of pulsed pressure-swing adsorption

in producing oxygen from air,

to develop a model for polymeric membrane fab-

rication,

and to explore the potential of a novel process for producing hydrogen

from methane using solar energy.

Hence, the book has been written to serve as

both a textbook and as a reference book for researchers.

The book includes 62 examples that are worked in some detail to illustrate the

scaling method as well as 165 unworked problems that can be assigned when the

book is used as a textbook. Many of these problems are open ended; as such, they

provide excellent material to stimulate creative thinking for students. The author

has used selected chapters of this book complemented either with an appropriate

textbook or his lecture notes to teach courses in transport phenomena, ﬂuid dynam-

ics, heat transfer, mass transfer, and reactor design. Whether one intends to use this

as a reference book or a textbook, it is necessary to read Chapters 1 and 2, which

provide an overview of the systematic approach to scaling analysis. A course in

ﬂuid dynamics can easily cover Chapters 1 through 3, including working nearly all

the problems at the end of the Chapter 3. A course in heat transfer would cover

Chapters 1, 2, and 4, and a course in mass transfer would cover Chapters 1, 2,

and 5. For both courses, parts of Chapter 3 would be required to consider con-

vective heat or mass transfer. A course in mass transfer with chemical reaction

would cover Chapters 1, 2, and 6 as well as the parts of Chapters 3 and 5 needed

to consider convective mass transfer. A course in modeling transport and reac-

tion processes necessarily would involve all the chapters. The author used a draft

version of this book to teach a course on process modeling to graduate students

whose pre- or corequisites included at least a graduate-level course in transport

phenomena; this one-semester course covered all the chapters, at least in part.

The author conscientiously tried to ferret out the errors in the book. but, a

few more were found with each rereading. Unfortunately, perfection is a quality

R. R. Bilodeau, R. J. Elgas, W. B. Krantz, and M. E. Voorhees, U.S. patent 5,626,759, issued

May 6, 1997.

E. M. Kopaygorodsky, V. V. Guliants, and W. B. Krantz, A.I.Ch.E. J., 50(5), 953 (2004).

D.Li,A.R.Greenberg,W.B.Krantz,andR.L.Sani,J. Membrane Sci., 279, 50 (2006).

J.K.Dahl,A.W.Weimer,andW.B.Krantz,Int. J. Hydrogen Energy, 29, 57 (2004).

xiv PREFACE

accorded only to the gods! The author would greatly appreciate receiving correc-

tions and suggestions for improving the book. In particular, he welcomes contribu-

tions of new examples and problems that will possibly be included in subsequent

editions of the book, with due credit given to contributors.

ACKNOWLEDGMENTS

Preparing these acknowledgments made the author reﬂect on the wonderful people

who made this book possible. We are inﬂuenced by many people in our lives.

However, a few people really make a difference in what we become and what we

accomplish. Educators should take some encouragement from this ﬁrst acknowl-

edgment since it indicates that students remember at least some of what you say to

them—in this case the student carried the sage advice of his teacher given to him

over 50 years ago! The late Dom Wulstan Mork, the author’s high school English

teacher, conveyed to his class that every person can write one good book. Although

the author has edited research monographs, this is his ﬁrst solo effort to write a

book that more or less represents his own work. Hence, the author sincerely hopes

that this effort will be his “good book” and hopefully, not his last!

The author ﬁrst pursued a degree in chemistry at Saint Joseph’s College in

Indiana. The small liberal arts college environment helped to develop the author

as a “whole man” by cultivating his interests in music, philosophy, languages,

mathematics, and science. His life would have taken a much different path had

it not been for the guidance of the late Brother John Marling at Saint Joe’s, who

encouraged him to pursue engineering studies at the University of Illinois.

At the University of Illinois the author was extremely fortunate to take an

undergraduate course from Professor John Quinn, who used the ﬁrst printing of

the ﬁrst edition of Transport Phenomena by Bird, Stewart, and Lightfoot as the

textbook. This superbly written book stimulated the author to begin asking the

questions regarding how the underlying assumptions in the many models used to

describe transport phenomena might be rigorously justiﬁed—the seed for writing

this book had been planted! The author would be remiss if he did not acknowledge

the profound inﬂuence that Professor Max Peters had on his career. Max had an

inimitable way of giving advice that he must have picked up while serving in

the 10

Mountain Division of the U.S. Army during World War II! Despite the

author’s intention of taking a job in industry after receiving his B.S. degree in

chemical engineering, Max “issued orders” for him to apply to graduate school,

which he dutifully obeyed!

The University of California at Berkeley provided an unbelievably fertile envi-

ronment for graduate studies in the early 1960s. The undercurrent of thought at

Berkeley was to question and challenge irrespective of whether the subject involved

science or societal issues. Professor John Prausnitz, who was the author’s aca-

demic advisor, was particularly inﬂuential in stimulating the author to think more

creatively. Indeed, “thinking outside the box” did not come naturally to a young

man who had spent four years in the highly disciplined environment of a military

xvi ACKNOWLEDGMENTS

high school and an all-men’s Catholic college isolated deep in the cornﬁelds of

Indiana—you did what you were told to do and did not question the reason for

doing it! Serving as a teaching assistant for several courses at Berkeley gave the

author the opportunity to discover something that he might not otherwise have

known: namely that his vocation in life was to be an educator.

Once having decided to enter academia, the decision as to where to apply for

a faculty position was easy. Professor Max Peters had moved on from Illinois to

become Dean of Engineering at the University of Colorado. The author wanted to

work for the man who was so inﬂuential in charting the course of his professional

life. The author’s academic career began with some very good advice from the late

Professor R. Curtis (Curt) Johnson, then chairman of the Department of Chemical

Engineering at Colorado, who told him to ﬁrst become a good teacher and then

focus on building a credible research program—uncommon advice for young edu-

cators today. The author’s senior faculty mentor at Colorado was Professor Klaus

Timmerhaus, who provided invaluable guidance for his career development. With-

out Klaus’s continued encouragement and support, it is doubtful that this book

would ever have been written.

The author would be an ingrate if he did not acknowledge the many students

at the University of Colorado, the University of Cincinnati, and most recently at

the National University of Singapore who provided useful feedback on the scaling

analysis method that he inculcated into his courses. The author sincerely wishes

that he could acknowledge all these students by name. If they had not responded

favorably to the use of scaling analysis in his courses, there would have been no

impetus to write the book.

This book is signiﬁcantly improved as a result of the opportunity provided to

the author by Professor Raj Rajagopalan, head of the Department of Chemical

and Biomolecular Engineering at the National University of Singapore. Raj offered

a visiting faculty appointment to the author and also encouraged him to teach a

graduate-level course on scaling analysis using a draft copy of the book. This

convinced the author that the book could be covered in sufﬁcient depth in a one-

semester course.

Finally, and most appropriately, the author would like to acknowledge the staff

of his publisher, John Wiley & Sons. The directives provided by the Wiley staff

made preparing this book much easier. In particular, the author acknowledges Bob

Esposito, Executive Editor for Wiley’s Scientiﬁc, Technical, and Medical Division,

with whom he worked most closely in preparing the manuscript for publication.

1 Introduction

Through and through the world is infested with quantity: To talk sense is to talk

quantities. It is no use saying the nation is large ... How large?

It is no use saying the radium is scarce ... How scarce?

You cannot evade quantity. You may ﬂy to poetry and music, and quantity and

number will face you in your rhythms and your octaves.

1.1 MOTIVATION FOR USING SCALING ANALYSIS

This book is directed to a broad spectrum of readers since modeling transport and

reaction processes is common to many ﬁelds of pure and applied science. The

book should be useful to educators who are seeking effective pedagogical tools for

introducing their students to an ever-expanding body of knowledge in the ﬁeld of

transport phenomena and reactor design. It should also be of value to engineers and

scientists who need to apply and develop mathematical models for transport and

reaction processes. It will be helpful to students who are seeking ways to better

understand the broad range of subjects encompassed by transport and reaction

processes.

As deﬁned in this book, the subject of scaling analysis, deals with a system-

atic method for nondimensionalizing a system of describing equations for transport

or reaction processes. The resulting dimensionless system of equations represents

the minimum parametric representation of the process. By this we mean that the

solution for any quantity that can be obtained from these equations will be at

most a function of the dimensionless independent variables and the dimensionless

groups generated by the scaling process. For example, scaling a heat-conduction

process will lead to a set of dimensionless equations whose solution for the dimen-

sionless temperature will be a function of the dimensionless spatial and temporal

Alfred North Whitehead (1861–1947), in The World of Mathematics,J.R.Newman,ed.,Simon&

Schuster, New York, 1956.

Scaling Analysis in Modeling Transport and Reaction Processes: A Systematic Approach

to Model Building and the Art of Approximation, By William B. Krantz

2 INTRODUCTION

independent variables and dimensionless parameters such as the Prandtl number

μ/k, in which μ is the shear viscosity, C

the heat capacity, and k the thermal

conductivity). Quantities that are obtained by evaluating the solution to the

dimensionless equations at ﬁxed values of the spatial and temporal variables or

by integrating a dimensionless dependent variable over the spatial or temporal

domain will be functions of a reduced set of dimensionless spatial or temporal

variables and the relevant dimensionless groups. In some cases the dimension-

less dependent variable of interest might be a function of only the dimensionless

groups. For example, in a steady-state heat-conduction process, the dimension-

less heat-transfer coefﬁcient (Nusselt number) will be a function of the relevant

dimensionless groups, such as the Prandtl number and geometric aspect ratios. This

minimum parametric representation of a transport or reaction process is useful since

it identiﬁes the dimensionless variables and groups that can be used to correlate

data from either laboratory or numerical experiments (i.e., computer simulations).

The resulting dimensionless groups can also be used for scale-up or scale-down

analyses by invoking the principles of geometric and dynamic similarity.

There is no unique set of dimensionless dependent and independent variables and

associated dimensionless groups for a system of equations describing a transport

or reaction process. For any system of describing equations, one set of dimension-

less dependent and independent variables and corresponding dimensionless groups

can always be obtained from any other set. However, one can scale a system of

describing equations in a unique way to ensure that the relevant dependent and

independent variables and their derivatives are bounded of order one. By this we

mean that the magnitude of the particular dimensionless variable or its derivative

is bounded between zero and more-or-less 1. For those familiar with formal order-

ing arguments, we are bounding our variables to be little oh of 1 [i.e.,

◦

(1)] as

opposed to big oh of 1 [i.e.,

◦

(1)], which means that the quantity is essentially 1.

Note that by of order one we do not mean exactly 1. In

◦

(1) scaling, one can say,

for example, that 0.8

∼

1or3

∼

1; that is, the quantity is well within an order of

magnitude of 1. In this book this special application of scaling that leads to unique

dimensionless variables and groups is referred to as

◦

(1) scaling.

The nondimensionalization associated with

◦

(1) scaling is indeed unique. How-

ever, arriving at this unique scaling often involves a process of trial and error. That

is, one has to assume that a particular transport or reaction process is dominated by

some mechanism(s) (e.g., heat conduction in a particular direction for a multidi-

mensional heat-transfer process) and then has to nondimensionalize the describing

equations by comparing the other terms to the one that embodies this mechanism.

After obtaining a system of dimensionless describing equations, one evaluates the

resulting dimensionless groups for the relevant geometric and physical parameters

of interest. If all the dimensionless groups are bounded of order one [i.e.,

◦

(1)],

the original assumption as to the controlling mechanism(s) was correct. However,

if any of the dimensionless groups is much larger than 1, it can indicate that the

scaling was not correct for the geometric and physical parameters of interest or

that there is a region of inﬂuence or boundary layer in which a temporal or spa-

tial derivative becomes very large. In either case one has to repeat the scaling