Convection Heat Transfer E-Book

Adrian Bejan

• Herausgeber: John Wiley & Sons
• Kategorie: Fachliteratur
• Sprache: Englisch

Seitenzahl: 998

Veröffentlichungsjahr: 2013

Beschreibung

A new edition of the bestseller on convection heat transfer

A revised edition of the industry classic, Convection Heat Transfer, Fourth Edition, chronicles how the field of heat transfer has grown and prospered over the last two decades. This new edition is more accessible, while not sacrificing its thorough treatment of the most up-to-date information on current research and applications in the field.

One of the foremost leaders in the field, Adrian Bejan has pioneered and taught many of the methods and practices commonly used in the industry today. He continues this book's long-standing role as an inspiring, optimal study tool by providing:

• Coverage of how convection affects performance, and how convective flows can be configured so that performance is enhanced
• How convective configurations have been evolving, from the flat plates, smooth pipes, and single-dimension fins of the earlier editions to new populations of configurations: tapered ducts, plates with multiscale features, dendritic fins, duct and plate assemblies (packages) for heat transfer density and compactness, etc.
• New, updated, and enhanced examples and problems that reflect the author's research and advances in the field since the last edition
• A solutions manual

Complete with hundreds of informative and original illustrations, Convection Heat Transfer, Fourth Edition is the most comprehensive and approachable text for students in schools of mechanical engineering.

Leseprobe

Table of Contents

Other books by Adrian Bejan

Title Page

Copyright

Preface

References

Preface to the Third Edition

Preface to the Second Edition

Preface to the First Edition

List of Symbols

Chapter 1: Fundamental Principles

1.1 Mass Conservation

1.2 Force Balances (Momentum Equations)

1.3 First Law of Thermodynamics

1.4 Second Law of Thermodynamics

1.5 Rules of Scale Analysis

1.6 Heatlines for Visualizing Convection

References

Problems

Chapter 2: Laminar Boundary Layer Flow

2.1 Fundamental Problem in Convective Heat Transfer

2.2 Concept of Boundary Layer

2.3 Scale Analysis

2.4 Integral Solutions

2.5 Similarity Solutions

2.6 Other Wall Heating Conditions

2.7 Longitudinal Pressure Gradient: Flow Past a Wedge and Stagnation Flow

2.8 Flow Through the Wall: Blowing and Suction

2.9 Conduction Across a Solid Coating Deposited on a Wall

2.10 Entropy Generation Minimization in Laminar Boundary Layer Flow

2.11 Heatlines in Laminar Boundary Layer Flow

2.12 Distribution of Heat Sources on a Wall Cooled by Forced Convection

2.13 The Flow of Stresses

References

Problems

Chapter 3: Laminar Duct Flow

3.1 Hydrodynamic Entrance Length

3.2 Fully Developed Flow

3.3 Hydraulic Diameter and Pressure Drop

3.4 Heat Transfer To Fully Developed Duct Flow

3.5 Heat Transfer to Developing Flow

3.6 Stack of Heat-Generating Plates

3.7 Heatlines in Fully Developed Duct Flow

3.8 Duct Shape for Minimum Flow Resistance

3.9 Tree-Shaped Flow

References

Problems

Chapter 4: External Natural Convection

4.1 Natural Convection as a Heat Engine in Motion

4.2 Laminar Boundary Layer Equations

4.3 Scale Analysis

4.4 Integral Solution

4.5 Similarity Solution

4.6 Uniform Wall Heat Flux

4.7 Effect of Thermal Stratification

4.8 Conjugate Boundary Layers

4.9 Vertical Channel Flow

4.10 Combined Natural and Forced Convection (Mixed Convection)

4.11 Heat Transfer Results Including the Effect of Turbulence

4.12 Stack of Vertical Heat-Generating Plates

4.13 Distribution of Heat Sources on a Vertical Wall

References

Problems

Chapter 5: Internal Natural Convection

5.1 Transient Heating from the Side

5.2 Boundary Layer Regime

5.3 Shallow Enclosure Limit

5.4 Summary of Results for Heating from the Side

5.5 Enclosures Heated from Below

5.6 Inclined Enclosures

5.7 Annular Space Between Horizontal Cylinders

5.8 Annular Space Between Concentric Spheres

5.9 Enclosures for Thermal Insulation and Mechanical Strength

References

Problems

Chapter 6: Transition to Turbulence

6.1 Empirical Transition Data

6.2 Scaling Laws of Transition

6.3 Buckling of Inviscid Streams

6.4 Local Reynolds Number Criterion for Transition

6.5 Instability of Inviscid Flow

6.6 Transition in Natural Convection on a Vertical Wall

References

Problems

Chapter 7: Turbulent Boundary Layer Flow

7.1 Large-Scale Structure

7.2 Time-Averaged Equations

7.3 Boundary Layer Equations

7.4 Mixing Length Model

7.5 Velocity Distribution

7.6 Wall Friction in Boundary Layer Flow

7.7 Heat Transfer in Boundary Layer Flow

7.8 Theory of Heat Transfer in Turbulent Boundary Layer Flow

7.9 Other External Flows

7.10 Natural Convection Along Vertical Walls

References

Problems

Chapter 8: Turbulent Duct Flow

8.1 Velocity Distribution

8.2 Friction Factor and Pressure Drop

8.3 Heat Transfer Coefficient

8.4 Total Heat Transfer Rate

8.5 More Refined Turbulence Models

8.6 Heatlines in Turbulent Flow Near a Wall

8.7 Channel Spacings for Turbulent Flow

References

Problems

Chapter 9: Free Turbulent Flows

9.1 Free Shear Layers

9.2 Jets

9.3 Plumes

9.4 Thermal Wakes Behind Concentrated Sources

References

Problems

Chapter 10: Convection with Change of Phase

10.1 Condensation

10.2 Boiling

10.3 Contact Melting and Lubrication

10.4 Melting By Natural Convection

References

Problems

Chapter 11: Mass Transfer

11.1 Properties of Mixtures

11.2 Mass Conservation

11.3 Mass Diffusivities

11.4 Boundary Conditions

11.5 Laminar Forced Convection

11.6 Impermeable Surface Model

11.7 Other External Forced Convection Configurations

11.8 Internal Forced Convection

11.9 Natural Convection

11.10 Turbulent Flow

11.11 Massfunction and Masslines

11.12 Effect of Chemical Reaction

References

Problems

Chapter 12: Convection in Porous Media

12.1 Mass Conservation

12.2 Darcy Flow Model and the Forchheimer Modification

12.3 First Law of Thermodynamics

12.4 Second Law of Thermodynamics

12.5 Forced Convection

12.6 Natural Convection Boundary Layers

12.7 Enclosed Porous Media Heated from the Side

12.8 Penetrative Convection

12.9 Enclosed Porous Media Heated from Below

12.10 Multiple Flow Scales Distributed Nonuniformly

12.11 Natural Porous Media: Alternating Trees

References

Problems

Appendixes

A: Constants and Conversion Factors

Constants

Conversion Factors

Dimensionless Groups Used in This Booka

B: Properties of Solids

References

C: Properties of Liquids

References

D: Properties of Gases

References

E: Mathematical Formulas

Error Function

Leibniz's Formula for Differentiating an Integral

Reference

Author Index

Subject Index

Other books by Adrian Bejan:

Entropy Generation Through Heat and Fluid Flow, Wiley, 1982.

Advanced Engineering Thermodynamics}, Third Edition, Wiley, 2006.

Thermal Design and Optimization, with G. Tsatsaronis and M. Moran, Wiley, 1996.

Entropy Generation Minimization, CRC Press, 1996.

Shape and Structure, from Engineering to Nature, Cambridge, 2000.

Heat Transfer Handbook, with A. D. Kraus, eds., Wiley, 2003.

Design with Constructal Theory, with S. Lorente, Wiley, 2008.

Design in Nature, with J. P. Zane, Doubleday, 2012.

Convection in Porous Media, with D. A. Nield, Fourth Edition, Springer, 2013.

Cover image: Courtesy of Adrian Bejan

Cover design: John Wiley & Sons, Inc.

This book is printed on acid-free paper.

Copyright © 2013 by John Wiley & Sons, Inc. All rights reserved

Published by John Wiley & Sons, Inc., Hoboken, New Jersey

Published simultaneously in Canada

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ISBN 978-0-470-90037-6; ISBN 978-1-118-33008-1 (ebk); ISBN 978-1-118-33282-5 (ebk); ISBN 978-1-118-33448-5 (ebk); ISBN 978-1-118-51975-2 (ebk); ISBN 978-1-118-51976-9 (ebk)

The entrepreneur, as a creator of the new and a destroyer of the old, is constantly in conflict with convention. He inhabits a world where belief precedes results, and where the best possibilities are usually invisible to others. His world is dominated by denial, rejection, difficulty, and doubt. And although as an innovator, he is unceasingly imitated when successful, he always remains an outsider to the “establishment.”

Theodore Forstmann, 2003.

In science, the “entrepreneur” is the one who gets the unusual idea, climbs out on a limb, jumps, and runs with it on the landscape. His fate at the feet of the establishment is the same.

Preface

An author is fortunate if his book is popular enough to merit a second edition somewhere down the line, yet the flow of ideas that grew around this book since the first edition (1988) has been beyond expectations. I will let others comment on this flow. In this brief Preface, I comment on just one feature of the flow of ideas and one bit of history.

The flow of ideas is illustrated by the changes made in this new edition. Good ideas (in this or any other field) attract interesting minds—researchers, educators, and authors with ideas. These minds grow the field the way that the yeast grows the cake. While revising this edition, it was not possible to keep up with this growth, but I tried, even though this meant abandoning some of the material from earlier editions. The new growth is represented by the impact of the science of discovering effective flow configurations (constructal theory and design), the streamlining of the discipline along methods that are direct, muscular, and at the same time lean (scale analysis, intersection of asymptotes, heatlines), the oneness with thermodynamics through the irreversibility (entropy generation) phenomenon, and new references and problems at the end of chapters.

Because we know where convection and thermodynamics come from, this growth illustrates that science (education, knowledge, information) is an evolutionary design [1-4], a flow system that constantly morphs and improves so that our own movement and life are facilitated and extended on the landscape. This is nature, the animate and the inanimate alike.

Because research is autobiographical, good research is a book of wonderful memories. I close this preface with the story of how the first edition of this book was born. It was an accident, literally. At age 33, I was behaving as if I was meant to play basketball forever, and I was wrong. During a game in January 1982, one of my Achilles' tendons was severed, and I ended up in a wheelchair for the entire semester. I had to teach my convection course, for which I had written notes, but this time I was forced to write each lecture on transparencies, for the screen. My first graduate student, Shigeo Kimura, now professor at Kanazawa University, Japan, was my teaching assistant. He would wheel me into the classroom every morning, and my convection book would come to life, one original drawing at a time, one original (solved) problem after another. One such problem was the method of intersecting the asymptotes and the back-of-the-envelope prediction of optimal spacings (Problem 11, Chapter 4, p. 157, in the first edition).

There was so much richness during the spring of 1982 that the accident was a blessing.

Adrian Bejan Duke University

References

1. A. Bejan and J. P. Zane, Design in Nature, Doubleday, New York, 2012.

2. A. Bejan, Science and technology as evolving flow architectures, Int. J. Energy Res., Vol. 33, 2009, pp. 112-125.

3. A. Bejan and S. Lorente. The physics of spreading of ideas, Int. J. Heat Mass Transfer, Vol. 55, 2012, pp. 802-807.

4. A. Bejan, Two hierarchies in science: The free flow of ideas and the academy, Int. J. Design Nature Ecodynam., Vol. 4, No. 4, 2009, pp. 386−394.

Preface to the Third Edition

Research is autobiographical. I often say this when I lecture, and I find it true as I look at this new edition of Convection Heat Transfer. It is even more true as I look at all three editions together. This book is a chronicle of the heat transfer side of my career, the methods I developed and taught along the way, and the great fortune I had to work with extremely gifted colleagues. The three editions are also a story of how the field has grown and prospered. It has done so based on new challenges and especially, new ideas.

One trend that is made visible (and useful, I hope) in this edition is the new emphasis on design as science—the generation of flow configuration based on principle. For many years, the field of convection was preoccupied with documenting the transport characteristics of various but simple flow configurations—relationships between temperature differences and heat transfer rates. This information is essential in the modeling and simulations that are necessary in design. The reality, however, is harsh: Constraints exist, and one overriding constraint is space (size, volume, weight). Putting more and more heat transfer into a given volume has been the objective, from the compact heat exchangers of my MIT years to the heat transfer augmentation techniques and the cooling of electronics packages of today. Doing more with limited resources has been the driving force.

Miniaturization marches forward, but this is not even half of the story. The reason is that the devices we touch must be made at our scale—they must be macroscopic, no matter how small the smallest components. The more successful we are in making smaller components, the greater the challenge to install larger numbers of such components and to connect them with currents (heat, fluid, electricity), to keep them alive. The challenge is to “construct,” to assemble and design while assembling (i.e., to design complexity and to deduce the flow configuration of the macroscopic device).

Construction must be shouted from the rooftops, especially today as the crowd marches toward smaller scales. To construct is to proceed in the opposite direction, from small to large, because only in this direction can the small scales be made useful. Only after the achievement of constructal assembly can small-scale components deliver high densities of heat transfer.

In this new edition, the first steps toward constructs with high heat transfer density are used as an introduction to constructal theory and design1

: the generation of flow architecture in the pursuit of maximal global performance subject to global constraints, when the flow architecture is free to morph. The focus is on method, on design as science, on the generation of optimal and complex architectures based on the constructal law. To emphasize this facet of the third edition is appropriate not only because of its importance today, but also because it had its start in the 1984 edition [see the optimization of spacings with natural convection (p. 157, Problem 11, Chapter 4).

The focus on methodology is why in this new edition I chart the progress made by three other methods that were pioneered in the 1984 edition. These methods have become recognized and now occupy growing sections of the literature:

The intersection of asymptotes method, which delivered in amazingly direct fashion the optimal spacing for natural convection (see above), has since been extended to spacings for forced convection and the constructal theory prediction of all the basic features of Bénard convection. The intersection of asymptotes is also useful pedagogically, in the teaching of the concept of transition (e.g., laminar-turbulent flow, natural-forced convection).

Heatlines are now being used to visualize the true paths followed by convection: the paths of energy flow, not fluid flow. They were introduced in the 1984 edition, with an example of natural convection in an enclosure. The concept has since been extended to mass transfer and a variety of basic and applied configurations with natural and forced convection in fluids and fluid-saturated porous media. This method of visualization is particularly well suited for computational heat transfer and should be included in commercial computational packages.

Scale analysis continues to be the main method for teaching the basics of convection in this new edition. The rules and promise of scale analysis as a problem-solving method were first formulated in the 1984 edition. Today the method is used widely, and this makes it even more essential in a basic course of convection. The increased importance of scale analysis is also due to the proliferation of computational heat transfer. If done correctly, scale analysis can shed light on what the deluge of numerical results is trying to tell us. Even more, to teach scale analysis is to remind the student not to give up on pencil and paper. Not everything must be done on the computer.

Porous media were brought into a heat transfer course for the first time by the 1984 edition of this book. Since then, convection in porous media has developed into a field of its own. In this edition we continue to emphasize the basic method and the most basic results. A connection is also made between porous media and designed complex flow structures,2 and this serves as one more bridge to the constructal design method.

Interdisciplinary teaching and research is one of the missions of this course, but with this warning: Learn your disciplines first; only then you will be strong on the interdisciplinary frontiers. The teaching of convection in porous media is a good example. This is presented not as a self-standing subject but as an interaction between principles of convection in pure fluids, which we all learn, and newly emerging technological applications that employ porous flow structures.

In my work on this new edition I benefited from the help and ideas offered by Professors C. Biserni, J. Bonjour, I. Dincer, M. Feidt, D. Gobin, Y. Fautrelle, S. J. Kim, A. D. Kraus, S. Lorente, E. Lorenzini, G. Lorenzini, N. Mazet, F. Meunier, A. F. Miguel, W. J. Minkowycz, P. Neveu, D. A. Nield, A. H. Reis, E. Sciubba, B. Spinner, F. B. Tehrani, J. V. C. Vargas, M. E. Weber, and C. Zamfirescu. In particular, I wish to thank my doctoral students Y. Azoumah, T. Bello-Ochende, A. K. da Silva, L. Gosselin, J. C. Ordonez, Luiz A. O. Rocha, and W. Wechsatol.

Adrian Bejan Durham, North CarolinaApril 2004

1 A. Bejan, Shape and Structure, from Engineering to Nature, Cambridge University Press, Cambridge, 2000.

2 A. Bejan, I. Dincer, S. Lorente, A. F. Miguel, and A. H. Reis, Porous and Complex Flow Structures in Modern Technologies, Springer-Verlag, New York, 2004.

Preface to the Second Edition

I want to thank John Wiley & Sons, Inc. and the users of my Convection Heat Transfer for giving me this opportunity to prepare a second edition. The changes and additions that I made are due to the suggestions received from many colleagues and students, and to the evolution of my own research activity.

I made changes in both format and content. The format is now based on numbered sections and equations, to make it easier for the first-time user to use this book as a reference. I assembled all the symbols in a list that precedes the text. The Author Index acknowledges one more time the individuals whose work is quoted in the text. The Solutions Manual is now produced on the word processor, and has the appearance of a companion book.

The changes in content are more significant and at more than one level. New topics covered in the second edition are convection with change of phase (condensation, boiling, melting), the cooling of electronic packages by forced and natural convection, lubrication by contact melting, and several examples of conjugate heat transfer, i.e., convection coupled with conduction or radiation. I augmented most chapters with results, namely, formulas, tables, charts, and appendixes that are recommended for use in engineering design work. And, speaking of design, many of the new problems at the end of chapters refer to basic principles of thermal design.

Relative to the first edition, the chapters dealing with laminar and, especially, turbulent forced convection have been expanded. To make room for the new material and still respect the prescribed space limits, I had to eliminate the chapter on numerical methods, and to condense the treatment of convection in porous media. Numerical methods are now covered in courses devoted entirely to computational fluid dynamics and heat transfer. For porous media, I recently completed with Professor D. A. Nield a separate textbook, Convection in Porous Media (Springer, 1992; now in 4th edition, 2013).

As in the first edition, the most important feature of this book is that many of the topics and problems came from my own research. These problems recommended themselves as interesting and beautiful, i.e., worthy of study. They represent my argument in favor of practicing laissez faire in engineering research, and against the dirigiste policy advocated by others.

Adrian Bejan Durham, North CarolinaJune 1994

Preface to the First Edition

My main reason for writing a convection textbook is to place the field's past 100 years of growth in perspective. This book is intended for the educator who wants to present his students with more than a review of the generally accepted “classical” methods and conclusions. Through this book I hope to encourage the convection student to question what is known and to think freely and creatively about what is unknown.

There is no such thing as “unanimous agreement” on any topic. The history of scientific progress shows clearly that our present knowledge and understanding—contents of today's textbooks—are the direct result of conflict and controversy. By encouraging our students to question authority, we encourage them to make discoveries on their own. We can all only benefit from the scientific progress that results.

In writing this book, I sought to make available a textbook alternative that offers something new on two other fronts: (1) content, or the selection of topics, and (2) method, or the approach to solving problems in convection heat transfer.

Regarding content, this textbook reflects the relative change in the priorities set by our technological society over the past two decades. Historically, the field of convective heat transfer grew out of great engineering pursuits such as energy conversion (power plant technology), the aircraft, and the exploration of extraterrestrial space. Today, we are forced to face additional challenges, primarily in the areas of “energy” and “ecology.” Briefly stated, engineering education today places a strong emphasis on man's need to coexist with the environment. This new emphasis is reflected in the topics assembled in this book. Important areas covered for the first time in a convection textbook are: (1) natural convection on an equal footing with forced convection, with application to energy conservation in buildings and to geophysical dynamics, (2) convection through porous media saturated with fluid, with application to geothermal and thermal insulation engineering, and (3) turbulent mixing in free-stream flow, with application to the dispersion of pollutants in the atmosphere and the hydrosphere.

Regarding method, in this book I made a consistent effort to teach problem solving (a Solutions Manual is available from the publisher or from me). This book is a textbook to be used for teaching a course, not a handbook. Of course, important engineering results are listed; however, the emphasis is placed on the thinking that leads to these results. A unique feature of this book is that it stresses the importance of correct scale analysis as an eligible and cost-effective method of solution, and as a precondition for more refined methods of solution. It also stresses the need for correct scaling in the graphic reporting of more refined analytical results and of experimental and numerical data. The cost and the “return on investment” associated with a possible method of solution are issues that each student-researcher should examine critically: these issues are stressed throughout the text.

I wrote this book during the academic year 1982-1983, in our mountain-side house on the greenbelt of North Boulder. This project turned out to be a highly rewarding intellectual experience for me, because it forced upon me the rare opportunity to think about an entire field, while continuing my own research on special topics in convection and other areas (specialization usually inhibits the ability to enjoy a bird's-eye-view of anything). It is a cliché in education and research for the author of a new book to end the preface by thanking his family for the “sacrifice” that allowed completion of the work. My experience with writing Convection Heat Transfer has been totally different (i.e., much more enjoyable!), to the point that I must thank this book for making me work at home and for triggering so many inspiring conversations with Mary. Convection can be entertaining.

Adrian Bejan Boulder, ColoradoJuly 1984

List of Symbols

a, b

dimensions of rectangular duct cross section (

Fig. 3.5

)

A

area

A

c

cross-sectional area

A, B

constants in the logarithmic law of the wall [eqs. (

7.41

) and (

7.42

)]

Ar

Archimedes number [eq. (

10.80

)]

b

empirical constant, Forchheimer flow [eq. (

12.15

)]

b

natural convection parameter [eq. (

5.117

)]

b

radial length scale of round velocity jet [eq. (

9.40

)]

b

stratification parameter [eq. (

12.116

)]

b

taper parameter [eq. (

2.140

)]

b

thermal stratification number [eq. (

4.81

)]

b

T

radial length scale of round thermal jet [eq. (

9.43

)]

empirical factors (

Table 11.6

)

B

condensation driving parameter [eq. (

10.26

)]

B

cross-sectional shape number (

Fig. 3.7

)

B

dimensionless group [eq. (

2.147

)]

B

dimensionless group [eq. (

12.107

)]

Be

L

Bejan number, pressure drop number [eq. (

3.120

)]

Be

p

Bejan number for a porous medium [eq. (

12.113

)]

Bo

H

Boussinesq number [eq. (

4.35

)]

c

specific heat of incompressible substance

specific heat at constant volume

c

P

specific heat at constant pressure

c

1, 2

constants

C

compressive impulse or reaction [eq. (

6.7

)]

C

concentration [eq. (

11.1

)]

C

constant

C

f

,

x

local skin friction coefficient [eqs. (

2.57

) and (

7.52

)]

C

n

factor (

Fig. 7.11

)

C

1

,

C

2

,

C

constants [eq. (

8.61

)]

C

D

drag coefficient [eq. (

7.103

)]

C

sf

constant (

Table 10.1

)

d, D

diameter

D

mass diffusivity [eq. (

11.24

),

Tables 11.1

and

11.2

]

D

plate-to-plate spacing (

Fig. 3.1

)

D

stream transversal length scale

D

h

hydraulic diameter [eq. (

3.26

)]

D

k-k

knee-to-knee thickness of time-averaged turbulent shear layer (

Fig. 9.3

)

D

T

distance of maximum thermal penetration in the

y

direction, in the vicinity of a direct contact spot [eq. (

7.94

)]

e

specific energy (labeled

u

in

Table 1.1

)

f

Blasius streamfunction similarity profile [eq. (

2.80

)]

f

factor [eq. (

7.113

)]

f

friction factor [eq. (

3.24

)]

f

porous medium friction factor [eq. (

12.12

)]

f

roll thickness [eq. (

5.92

)]

f

u

curve fit for the velocity profile [eq. (

7.53

)]

frequency of vortex shedding [eq. (

7.102

)]

F

force

F

streamfunction similarity profile [eqs. (

4.60

) and (

12.139

)]

Fo

Fourier number [eq. (

10.104

)]

F

D

drag force

F

n

normal force

F

t

tangential force

g

gravitational acceleration

Gr

H

Grashof number [eq. (

4.38

)]

Gr

*

Grashof number based on heat flux (

Table 6.1

)

Gz

Graetz number [eq. (

3.107

)]

G

constant (

Table 4.3

)

h

heat transfer coefficient [eq. (

2.4

)]; local heat transfer coefficient [eq. (

2.100

)]

h

specific enthalpy

h

fg

latent heat of condensation or evaporation (

Table 10.2

)

augmented latent heat [eq. (

10.10

)]

augmented latent heat [eq. (

10.41

)]

h

m

mass transfer coefficient [eq. (

11.46

)]

h

sf

latent heat of melting

H

enthalpy flow rate [eq. (

10.5

)]

H

heatfunction [defined via eqs. (

1.68

) and (

1.69

)]

H

height

H

Henry's constant [eq. (

11.35

) and

Table 11.3

]

I

area moment of inertia

I

integral [eq. (

3.135

)]

j

diffusion flux [eq. (

11.20

)]

j

app

apparent mass flux [eq. (

11.102

)]

J

dimensionless thickness parameter [eq. (

2.139

)]

Ja

Jakob number [eq. (

10.19

)]

k

thermal conductivity

k

wave number

reaction rates [eqs. (

11.135

) and (

11.136

)]

k

s

sand grain size [eq. (

8.16

)]

K

jet strength [eq. (

9.33

)]

K

permeability [eq. (

12.9

)]

K

1, 2

constants

l

effective length [eq. (

4.127

)]

l

mixing length [eq. (

7.27

)]

L

length

L

length of direct viscous contact [eq. (

7.92

)]

L

c

characteristic length

equivalent length [eq. (

10.86

)]

L

m

length of direct thermal contact [eq. (

7.95

)]

effective length [eq. (

4.128

)]

Le

Lewis number [eq. (

11.93

)]

m

exponent in flow over a wedge [eq. (

2.124

)]

m

function [eq. (

6.27

)]

m

profile shape function for integral analysis [eq. (

2.54

)]

mass flow rate

mass transfer rate per unit length [eq. (

11.52

)]

volumetric mass generation rate [eq. (

11.15

)]

M

bending moment [eq. (

6.8

)]

M

function [eq. (

8.22

)]

M

impulse or reaction force due to fluid flow into or out of a control volume (

Fig. 2.3

)

M

mass

M

massfunction [eqs. (

11.133

)-(

11.134

)]

M

material constraint [eq. (

3.132

)]

M

molar mass [eq. (

11.4

)]

n

dimensionless coordinate across the velocity boundary layer (/) [eq. ()]