brodkey, r. s. and hershey, h. c. - transport phenomena - a unified approach

Bailey and Ouip: Biochemical Engineering FundamentalsBennett and Myers: Momentum, Heat, and Mass Transfer Beveridge and Schechter: Optimization: Theory and Practice Brodkey and Hershey:

TRANSPORT PHENOMENA

A Unified Approach

McGraw-Hill Chemical Engineering Series

Editorial Advisory Board

James J Carbeny, Professor of Chemical Engineering, University of Notre Dame James R Fair, Professor of Chemical Engineering, University of Texas, Austin William P Schowalter, Professor of Chemical Engineering, Princeton University

Matthew ‘IkreU, Professor of Chemical Engineering, University of Minnesota

James Wei, Professor of Chemical Engineering, Massachusetts Institute of Technology

I&xx S Peters, Emeritus Professor of Chemical Engineering, University of Colorado

BUILDING THE LITERATURE OF A PROFESSION

Fifteen prominent chemical engineers first met in New York more than 60 years ago toplan a continuing literature for their rapidly growing profession From industry camesuch pioneer practitioners as Leo H Baekeland, Arthur D Little, Charles L Reese,John V N Dorr, M C Whitaker, and R S McBride From the universities came sucheminent educators as William H Walker, Alfred H White, D D Jackson, J H.James, Warren K Lewis, and Harry A Curtis H C Parmelee, then editor of

Chemical and Metallurgical Engineering, served as chairman and was joined

sub-sequently by S D Kirkpatrick as consulting editor

After several meetings, this committee submitted its report to the McGraw-Hill BookCompany in September 1925 In the report were detailed specifications for a correlatedseries of more than a dozen texts and reference books which have since become theMcGraw-Hill Series in Chemical Engineering and which became the cornerstone of thechemical engineering curriculum

From this beginning there has evolved a series of texts surpassing by far the scope andlongevity envisioned by the founding Editorial Board The McGraw-Hill Series inChemical Engineering stands as a unique historical record of the development ofchemical engineering education and practice In the series one finds the milestones ofthe subject’s evolution: industrial chemistry, stoichiometry, unit operations andprocesses, thermodynamics, kinetics, and transfer operations

Chemical engineering is a dynamic profession, and its literature continues to evolve.McGraw-Hill and its consulting editors remain committed to a publishing policy thatwill serve, and indeed lead, the needs of the chemical engineering profession during thears to come

Bailey and Ouip: Biochemical Engineering Fundamentals

Bennett and Myers: Momentum, Heat, and Mass Transfer

Beveridge and Schechter: Optimization: Theory and Practice

Brodkey and Hershey: Transport Phenomena: 4

iif

fied Approach cprberry: Chemical and Catalytic Reaction En& &ig 1 ) _,

Coughanowr and Koppel: Process Systems Analysis and Control

Edgar and Himmelbhm: Optimization of Chemical Processes

Fabien: Fundamentals of Transport Phenomena

FInlayson: Nonlinear Analysis in Chemical Engineering

Gates, Katzer, and !3chuit: Chemistry of Catalytic Processes

Holland: Fundamentals of Multicomponent Distillation

Holland and Liipis: Computer Methods for Solving Dynamic Separation Problems

Katz, Cornell, Kobayashi, Poettmann, Vary, Elenbaas, and Weinang:

Handbook of Natural Gas Engineering

King: Separation Processes

Loyben: Process Modeling, Simulation, and Control for Chemical Engineers

McCabe, Smith, J C., and Harriott: Unit Operations of Chemical Engineering Mickley, Sherwood, and Reed: Applied Mathematics in Chemical Engineering

Nelson: Petroleum Refinery Engineering

Perry and Cbilton (Editors): Chemical Engineers’ Handbook

Peters: Elementary Chemical Engineering

Peters and Timmerhaus: Plant Design and Economics for Chemical Engineers

Probstein and Hicks: Synthetic Fuels

Reid, Prausnitz, and Shenvood: The Properties of Gases and Liquids

Resnick Process Analysis and Design for Chemical Engineers

Sattertield: Heterogeneous C?talysis in Practice

Sherwood, Pigford, ind Wiie: Mass Transfer

Smith, B D.: Design of Equilibrium Stage Processes

Smith, J M.: Chemical Engineering Kinetics

Smith, J M., and Van Ness: Introduction to Chemical Engineering Thermodynamics Treybal: Mass Transfer Operations

Valle-Riestraz Project Evolution in the Chemical Process Industries

Van Ness and Abbott: Classical Thermodynamics of Nonelectrolyte Solutions: With Applications to Phase Equilibria

Van Wile: Distillation

Volk: Applied Statistics for Engineers

Wdas: Reaction Kinetics for Chemical Engineers

Wei, Russell, and Swartzlander: The Structure of the Chemical Processing Industries Whitwell and Toner: Conservation of Mass and Energy

TRANSPORT PHENOMENA

A Unified Approach

Robert S Brodkey

The Ohio State Universi@

Harry C Hershey

The Ohio State University

McGraw-Hill Book Company

New York St Louis San Francisco Auckland BogotP Hamburg London Madrid Mexico Milan Montreal New Delhi Panama

Paris SHo Paula Singapore Sydney Tokyo -Toronto

A Unified Approach

INTERNATIONAL EDITION \

Copyright @ 1988

Exclusive rights by McGraw-Hill Book Co T Singapore

for manufacture and export This book cannot be

re-exported from the country to which it is consigned by

McGraw-Hill.

34567CM0943210

Copyright 0 1988 by McGraw-Hill Inc All rights reserved.

Except as permitted under the United States Copyright Act of 1976,

no part of this publication may be reproduced or distributed in any form or by any means, or stored in a data base or retrieval system, without the prior written permission of the publisher.

This book was set in Times Roman.

The editor was B.J Clark;

the production supervisor was Denise L Puryear; ’

Project supervision was done by Universities Press, Belfast.

Library of Congress Cataloging-in-Publication Data

1 Transport theory I Hershey Harry C.

II Title III Series

TPI 56.T7B76 1988 660.2’842 86-34414

ISBN 0-07-007963-3

When ordering this title use ISBN 0-07-100152-2

Printed in Singapore

CONTENTS

I

i Preface

2

2.1

2.22.32.42.5

Introduction to Transport PhenomenaTransport Phenomena and Unit OperationsEquilibrium and Rate Processes

Fundamental Variables and UnitsThe Role of Intermolecular ForcesSimple Balances

ProblemsReferences

Molecular Transport MechanismsThe Analogy

2.1.1 The Case for Heat Transfer2.1.2 The Case for Mass Transfer2.1.3 The Case for Momentum Transfer2.1.4 The Analogous Forms

Heat TransferMass TransferMomentum TransferHeat, Mass and Momentum Diffusivities2.51 Thermal Conductivity

2.5.2 Diffusion Coefficient2.5.3 Viscosity

344599

1 1

1 3

1418182122253032

4 046475051vii

2.6 A Comparison of the Transports 53

3.4 The Continuity Equation

3.5 The General Property Balance for an Incompressible Fluid

3.6 Summary

Problems

4 pI$ek&ar Transport and the General Property

4.1 Steady Transport in One Direction Involving Input-Output

with no Generation

4.1.1 Constant-area Transport

4.1.2 Variable-area Transport

4.2 Steady State Transport With Generation

4.2.1 Heat or Mass Transport with Constant Generation

4.2.2 Momentum Transfer with Generation at Steady-State

4.2.3 Laminar Flow in a Tube

4.2.4 Laminar Flow Between Parallel Plates

4.2.5 Variable Generation

4.3 Concluding Remarks

Problems

References

5 Transport with a Net Convective Flux

5.1 Convective Flux Caused by Forced Convection

5.1.1 The Balance Equation

51.2 Coordinate Systems

51.3 Relationship Between Shear Stress and Shear Rate

51.4 The Continuity Equation

51.5 The Energy Balance

5.1.6 The Navier-Stokes Equation

5.1.7 The Boundary Layer

5.2 Convected Coordinates

5.3 Mass Diffusion Phenomena

5.3.1 Mass Flwes in Stationary and Convected Coordinates

6062

6 4656667

7272727782858787

90

939595103104108113119124125126128

129132134134135138142146157160161161

5.3.2 Total Flux and Fick’s Law

5.3.3 Binary Mass Diffusion in Gases

5.3.4 Binary Mass Diffusion in Liquids

5.3.5 Diffusion in Solids

53.6 Diffusion due to a Pressure Gradient

53.7 Diffusion with Three or More Components

Less Common Types of Mass and Thermal Transport

Transitional and Turbulent Flow

6.1.1 The Reynolds Experiment

6.1.2 Transitional Flow

6.1.3 Fully Developed Turbulent Flow

The Equations for Transport under Turbulent Conditions

6.2.1 Reynolds Rules of Averaging

6.2.2 Reynolds Equation for Incompressible Turbulent Flow

6.2.3 Reynolds Stresses

6.2.4 Turbulent Flow in Channels and Pipes

Turbulence Models

6.3.1 The Boussinesq Theory

6.3.2 The Prandtl Mixing Length Theory

6.3.3 Analogies

6.3.4 Film and Penetration Theories

The Velocity Distribution

ral Methods of Analysis

The eneral Integral Balance

7.1.1 The Integral Mass Balance

7.1.2 The Integral Balance on an Individual Species

7.1.3 The Integral Momentum Balance

7.1.4 The Integral Energy Balance

7.1.5 The Mechanical Energy Equation and the Engineering

195198198201206210214220223225227227229234236240257260261263

265268270273275286

295305305316319321

9.6.1 Scale-up Procedures for Turbulent Flow with Three or

More Test Volumes9.6.2 Scale-up Procedures for Turbulent Flow with Two Test

Volumes9.6.3 Scale-up Procedures for Turbulent Flow with a Single

Test Volume9.6.4 Scale-up Procedure for Laminar Flow

9.6.5 Scale-up Without Geometric Similarity

10.1.1 Laminar Pipe Flow

10.1.2 Turbulent Pipe Flow

Piping Systems

10.2.1 Roughness

10.2.2 Pressure Drop in Rough Pipes

10.2.3 von Karman Number

10.2.4 Solutions of Large Molecules

10.2.5 The Velocity Head Concept

10.2.6 Curved Tubes

10.2.7 Expansion and Contraction Losses

10.2.8 Pipe Fittings and Valves

359362364371372374383

384

385

386395396396397398

400403403406409409413417420421422424430

Heat and Mass Transfer in Duct Flow

Review and Extensions

11.1.1 Radiation

11.1.2 Convection

11.1.3 Conduction

11.1.4 The Resistance Concept

11.15 Slope at the Wall

11.1.6 Bulk and Film Temperatures

Laminar Pipe Flow

11.2.1 Fully Developed Transfer

11.2.2 Entry Region

Heat and Mass Transfer During Turbulent Flow

11.3.1 Review of Turbulence Models

11.3.2 Correlations for Fully Developed HOW

11.3.3 The Analogies

11.3.4 Other Methods

Double-pipe Heat Exchangers

11.4.1 The Overall Heat Transfer Coefficient

11.4.2 Contact Resistance and Fouling Factors

Transport Past Immersed Bodies

The Boundary Layer and the Entry Region

12.1.1 The Laminar Boundary Layer

12.1.2 The Turbulent Boundary Layer

12.1.3 Heat and Mass Transfer During Boundary Layer Flow

Past a Flat Plate

442443455459460469471476479481482484487

489493493493494494504505506506510512512512516520526528530532

5 3 5 539539541546547549

551556557566

571

12.2 Flow Over Cylinders and Spheres 578

13.1.4 Heat Transfer with Negligible Internal Resistance

13.2 Finite Slab and Cylinder

13.2.1 Fourier Series Solution

13.2.2 Lapiace Transform Solution

13.2.3 Generalized Chart Solution

Part III Transport Property

Agitation of Non-Newtonian Fluids

Heat Transfer in Pipe Flow

A.1 Properties of Water and Air

Table A.1 Thermophysical Properties of Saturated

WaterTable A.2 Thermophysical Properties of Dry AirA.2 Prediction of Transport Properties

Table A.3 Constants in the Lennard-Jones 12-6

Potential as Determined from Viscosity DataTable A.4 Le Bas Atomic and Molar Volumes at the

Normal Boiling Point

B Mechanical Characteristics of Pipe and Tubing

Table B.l Standard Steel Pipe Dimensions, Capacities,

and WeightsTable B.2 Condenser and Heat-Exchanger Tube Data

C Physical Constants, Units, and Conversion

Tables

Table C.l Physical ConstantsTable C.2 SI Base and Supplementary Quantities and

U n i t sTable C.3 Derived Units of SI Which Have Special

Names

xlu

752755756761762770771777778783784786786788

791791

792794796

807

808

Table C.4 SI Prefixes 808

Table C 15 Pressure or Momentum Flux or Shear Stress 811

After publication of the pioneering book Transport Phenomena by Bird,

Stewart, and Lightfoot in 1960, educators everywhere recognized that theprevious “unit operations-unit processes” organization of material for thecurricula of chemical engineers was inadequate for modern engineeringeducation Many schools found that the 1960 book was suitable for graduatecourses and an excellent reference, but too difficult for most undergraduates,especially if the course was offered early in the curriculum Others followedthis pioneering effort by writing simpler versions

This book was designed to provide an integrated treatment of the threeareas of transport: momentum, heat, and mass The similarities and thedifferences of the three transports are clearly stated at a level suitable forsecond-semester sophomores and first-semester juniors in engineering or theother sciences where the mathematics requirement is similar Many of thebasic equations are mathematically identical, when expressed in terms of thegeneralized flux and property variables This identity helps the studentunderstand transport phenomena and forms the basis for the organization ofthe material here A typical curriculum teaches momentum transfer beforeheat and mass because a complete treatment of these latter two is not possiblewithout a prior discussion of fluid dynamics This text allows heat transfer,which is encountered daily by everyone and easily visualized, to explain byanalogy momentum transfer, which is not easily visualized or understood byneophytes Transport is rapidly becoming more widely used in most branches

of engineering, and this text provides all engineering disciplines with areadable and otherwise useful treatment of this difficult subject In most of theother books on this subject, these topics are covered separately

We believe that this text provides a solid foundation for engineeringdesign and research At the same time, some interesting and importantproblems are solved A study of transport phenomena does not replace unitoperations, but understanding of transport phenomena provides deeper insight

xv

into the fundamental processes occurring in the unit operations The engineerwho masters the material in this text will be better able to analyze the unitoperations he or she encounters.

McGraw-Hill and the authors would like to express their thanks for themany useful comments and suggestions provided by colleagues who reviewedthis text during the course of its development, especially Charles E Hamrin,Jr., University of Kentucky; Richard W Mead, University of New Mexico;Robert Powell, University of California-Davis; and James Wei, MassachusettsInstitute of Technology

Finally, the authors owe much thanks to many who have helped over theyears with this project A partial list (in alphabetical order) includes F.Bavarian, A M Cameron, J F Davis, L Economikos, L S Fan, L Fishler,

K S Knaebel, S G Nychas, J Y Oldshue, C E Patch, A Syverson, G B.Tatterson, J L Zakin, and the many typists who have helped with this effort,especially Pat Osborn

Robert S Brodkey Harry C Hershey

TO THE INSTRUCTOR

This text covers transport phenomena in an integrated manner It is ouropinion that a solid understanding of fluid mechanics is essential tounderstanding and solving problems in heat and mass transport Hence, alltopics suitable for a course in undergraduate fluid mechanics are covered indetail This text introduces the basic equations of heat and mass transfer aswell This text also covers heat and mass transport applications that are in thetransport phenomena area It does not cover topics that are traditionallytaught as unit operations It is expected that the students will purchase a unitoperations book for course work and reference beyond our coverage

After the introductory chapter, the basic equations of molecular port are covered first, then the general property balance, followed by thecombination of the balance and molecular transport The topic of convection is

trans-in Chapter 5 Our treatment is especially strong trans-in discerntrans-ing the differencesbetween transport problems with flow but no net convective flux, and those with a net convective flux Chapter 5 also contains a lengthy section on thefundamentals of mass transport phenomena Chapter 6 on turbulent flowprovides a thorough discussion of modern turbulence theory There are alsotwo chapters (7 and 8) on methods of analysis-integral methods anddimensional and modeling approaches Dimensional analysis is applied toagitation in Chapter 9 The remaining chapters contain advanced applications.Chapters 10 and 11 cover transport in ducts Flow past immersed bodies andfluidization are discussed in Chapter 12 Chapter 13 covers unsteady-statetransport phenomena Chapter 14 covers the estimation of the transport

properties p, k, and DAB; this chapter can be covered in conjunction with

Chapter 2, or anything thereafter, as the instructor wishes Chapter 15 onnon-Newtonian phenomena is unique in that this important topic is largelyignored in other texts This chapter also can be covered whenever theinstructor wishes; at Ohio State, we have found that our students cannot reallyappreciate non-Newtonian flow until they understand Newtonian flow Hence,

we teach Chapter 15 in conjunction with Chapter 10 If the instructor desires

to cover only the laminar aspects of non-Newtonianism, the appropriatematerial from Chapter 15 could be taught much earlier in the course The

$ appendix is in five parts: properties of materials, mechanical characteristics ofpiping and tubing, conversion tables, vector mathematics, and a list ofcomputer programs.

Taken as a whole, this text covers the area of fluid mechanics thoroughly.The basic equations of change are in the early chapters Laminar flow solutionsare found in both Chapters 4 and 5 on molecular and convective transport Thesections on agitation, turbulence, and fluidization contain the most modemconcepts and procedures Fluid statics is covered in Chapter 7 on integralmethods, where it arises naturally from the general integral balance equations.Advanced topics include discussions of design of complex piping systems, theboundary layer, ideal flow, flow past immersed bodies such as spheres andcylinders, fluidization, packed beds, banks of tubes, and non-Newtonian fluids.The inclusion of an entire chapter on non-Newtonian transport phenomena is

an indication of the importance of this subject to chemical engineers;non-Newtonian fluids are encountered daily in our lives, as well as beingcommon in industry The engineer needs some familiarity with this area.The topics of heat and mass transfer are covered only to the extent thattransport phenomena can be applied Excellent books exist for these topics,especially for heat transfer, for which our colleagues in mechanical engineeringhave written well-conceived textbooks In 1984 and 1985, three major books

on mass transfer were published Because heat and momentum transport are

so closely linked, a weakness of many heat transfer texts lies in their limitedtreatment of the fluid mechanics topics needed for heat-exchanger design Ourintegrated approach is intended to explain fully the coupled nature of heat andmomentum transport The heat transport and momentum transport equationsare presented together for laminar applications, turbulent flow, flow pastimmersed bodies, fluidized beds, etc Similarly, the basic equations for masstransport are integrated with those for heat and momentum Chapter 5discusses mass transport phenomena in detail, including the additionalcomplexities inherent in mass diffusion The presentation is in a clear fashionthat undergraduates can understand, especially the reasons why the massdiffusion equations as simplified for gases do not strictly apply to liquidsystems The basic principle of diffusion in solids is also covered; this topic isimportant in catalysis, and other areas as well The unsteady-state chaptercombines heat and mass transfer and includes the modern numerical methods

as the Crank-Nicolson

Our text is expected to serve widely as a reference Hence, more materialand more detail have been included than undergraduates can usually assimi-late At the Ohio State University, this text is used for a 4-credit-hour,one-quarter course, in transport phenomena which is offered to sophomoreswho have completed differential equations, freshman chemistry, stoichiometry,and two quarters of physics In our course, the material in Chapters 1 to 8 iscovered in detail The topics of agitation (Chapter 9) is covered rapidly Thedesign material in Chapters 10 and 12 occupy the last part of the course.Chapter 15 on non-Newtonian phenomena is covered briefly after Chapter 10

on fluid flow in ducts Our thermodynamics course, usually taken later or

concurrently, emphasizes the applications of the first law to flow problems.Note that this material is in Chapter 7 in our text and in Chapter 2 in

Introduction Co Chemical Engineering Thermodynamics by Smith and VanNess Fluid statics is also covered at the start of the thermodynamics course

At Ohio State, a second course in transport focuses entirely on heat transfer;that course covers Chapter 11 in detail plus, of course, much more Thatsecond course requires the students to purchase a specialized heat transfer text(usually a mechanical engineering series) in order to cover the specializedtopics in heat transfer, such as radiation, boiling, and condensation

Further topics in mass transfer can be taught in an additional course orcourses that combine discussions of mass transfer with unit operations notpreviously covered, such as absorption, distillation, drying, evaporation, andfiltration The basic material for mass diffusion is in Section 5.3; most of thematerial presented is not covered in the traditional unit operations texts.Again, the analogy with heat and momentum assists the student in under-standing the difficult concepts in mass diffusion

In solving the example problems, we used a computer or a hand-heldcalculator Calculations with these retain many digits in order to reducetruncation errors The example problems in this text make no serious effort toascertain the correct number of significant digits for every final answer,inasmuch as the purpose of the examples is to illustrate the method ofcalculation The instructor should point out from time to time the probableaccuracy of final answers, especially when the physical constants (such as mass.diffusivity) are not usually known accurately or approximate methods ofsolution are used Also, many of the example problems and homeworkproblems in this text have been in use at Ohio State for more than 20 years.Hence, their origins are obscure: We sincerely apologize if we have inadver-tently used problems that originated with someone else

A book on transport phenomena always encounters nomenclatureproblems, because the three areas of transport developed independently in theearly days A problem of more recent vintage is the decision of the AmericanInstitute of Physics to switch the viscosity notation from ~1 to q ChemicalEngineers have used p from the beginning; this text will also use cc, althoughthe instructor may wish to point out that the other symbol is also recom-mended by some

Finally, there are some excellent films available, which illustrate most of

available for purchase or loan twenty-two 16-mm films and a hundredthirty-three &mm film loops as a part of their fluid mechanics program At theOhio State University, we use the &mm loops, which are shown with a smallportable projector and lend themselves easily to informal discussion Thesefihn loops are referenced by number at the appropriate-locations in the text

Robert S Brodkey Harry C Hershey

TRANSPORT PHENOMENA

A Unified Approach

I

BASIC CONCEPTS

IN TRANSPORT PHENOMENA

INTRODUCTION

TO TRANSPORT PHENOMENA

Number of moles of gas (kmol, lb mol)

Pressure (kPa, atm, lb, in.-*)

Gas constant, see Appendix, Table C.l

Temperature (K, “R, “C, “F)

Volume (m3, ft’)

Unknown in algebraic equation

Unknown in algebraic equation

Unknown in algebraic equation

Shear stress (N m-*, lbf ft-‘)

This chapter provides a brief introduction to the material to be covered indetail in subsequent chapters First, a brief historical perspective of the role oftransport phenomena in the solution of engineering problems is discussed

3

? Then some fundamental concepts from physics, chemistry, and mathematicsare presented.

1.1 TRANSPORT PHENOMENA AND

UNIT OPERATIONS

The pioneering work Principles of Chemical Engineering was published in 1923

under the authorship of Walker, Lewis, and McAdams [Wl] This book wasthe first to emphasize the concept of unit operations as a fundamentalapproach to physical separations such as distillation, evaporation, drying, etc.This was the era when the profession of chemical engineering matured into aseparate area, no longer the province of the industrial chemist The study ofunit operations such as distillation is predicated on the idea that similarities inequipment and fundamentals exist regardless of the process In other words,the principles of distillation apply equally to the separation of liquid oxygenfrom liquid nitrogen as well as to the thousands of other distillations routinelycarried out in industries around the world The study of transport phenomena

is undertaken because this topic is the basis for most of the unit operations Insimple terms, transport phenomena comprise three topics: heat transfer, masstransfer, and momentum transfer (fluid flow) In many of the unit operations(such as distillation), all three transport phenomena (i.e., fluid flow, heattransfer, and mass transfer) occur, otten simultaneously The conceptspresented in transport phenomena underly the empirical procedures that areused in the design of unit operations Empiricism is required because the exactequations cannot be solved

1.2 EQUILIBRIUM AND RATE

PROCESSES

Many problems can conveniently be divided into two classifications: librium and nonequilibrium Under conditions of nonequilibrium, one or morevariables change with time The rates of these changes are of much interest,naturally A typical engineer-scientist reading this book will be involved withfour types of rate processes: rate of heat transfer, rate of mass transfer, rate ofmomentum transfer, and rate of reaction The first three of these are thesubject of this text The fourth, rate of reaction, will not be covered in anydetail, except for the inclusion of the appropriate terms in the generalequations and in a few elementary examples

equi-Equilibrium processes The science of thermodynamics deals mainly with

systems in equilibrium Consider Fig 1.1 which shows a gas composed of 50mole percent nitrogen and 50 mole percent oxygen enclosed in a tank at apressure of 2 atm at 300 K Let this gas be surrounded by ambient air at thesame temperature After an appropriately long time, the gas inside the tank is

at physical equilibrium Its temperature is the same as that of the surrounding

MTRODUCnON T O T R A N S P O R T PHENOMENA 5

gas, 3OOK Inside the tank, there will be no concentration gradients Thescience of chemistry tells us that this gas will not be in chemical equilibrium.Although oxygen and nitrogen can form a series of compounds, such as NzO,

NO, NO,, and N,O,, in actuality none of these is formed in the present systembecause the rate of reaction is essentially zero If the gas in the tank were purenitrogen, then there would be complete equilibrium inside the tank, i.e.,

Rate proeemes When nonequilibrium processes are considered, the systemunder consideration progresses in a manner such as to approach equilibrium.All such rate processes are characterized by a driving force The rate oftransport is proportional to the driving force The topic will be discussedthoroughly in Chapter 2

U N I T S

Temperature Interestingly, temperature (T) can be defined only in theempirical sense as a relative measure of “hotness” [Dl, Ml] The temperaturescales in present use are defined with only one fixed point, the triple point ofwater, 273.16 K Temperature scales are based on changes in properties ofmaterials with temperature The change of resistivity of a solid such asplatinum or the change of volume of a liquid such as mercury is easilymeasured as a function of temperature, and therefore can be used as anindication of temperature Temperature units are Kelvin (K), Celsius (“C),Fahrenheit (“F), and Rankine (“R) The reader already knows how to convertfrom one to the other Temperature is one of the most important quantities in

a system Temperature manifests itself in the motions of the molecules: thehigher the temperature, the higher are the velocities of the molecules Almostall properties are strongly dependent on temperature The rate processes arelikewise functions of temperature

Pressare The pressure in the tank in Fig 1.1 has units of force (F) per unit

? area (A); The force is a result of the collisions of molecules with the wall of thetank The pressure acts equally in all directions At equilibrium, the pressureinside the tank is uniform It is important to distinguish between pressure andforce Force equals pressure times area Since the tank in Fig 1.1 is arectangularly shaped box, with width 4 m and height and depth both 2 m, thetotal force on the front face is twice as great as that on either end, since thefront area is twice the area of an end.

Volume The volume (V) is the easiest variable to understand An equation of

state expresses the volume of a material in terms of temperature, pressure (p),and composition or total number of moles (n) For an ideal gas, the equation

of state is

where R is the gas constant (see Appendix, Table C.l).

Coneentralion The concentration of species A (CA) has units of moles (ormass) per volume The following example illustrates concentration

Example 1.1 Calculate the concentration of nitrogen in the tank in Fig 1.1, assuming that the ideal gas law holds The answer is to be in lb mol tY3.

Answer The total volume of the tank is given as ‘0.01283 m3, which in English

units is 0.4531 ft? (using the conversion 35.316 ft’ = 1 m3 from the Appendix, Table C.18) The total number of pound moles is found from Eq (1.1):

Nz in the tank is 1.149 x low3 lb mol Therefore the concentration is

6, = (1.149 x lo-‘)/(0.4531) (lb mol)/(ft’) = 2.54 x lo-’ lb mol fte3 (ii)

Shear stress The shear stress (t) is also a force per area, as is pressure The

shear stress may have components in any or all directions in contrast topressure which acts in a direction normal to a surface For now, let us consider

a simple view of shear stress, Fig 1.2 A block with an area of 2m2 isimbedded in a concrete floor A force of 5 newtons is impressed against theside of a second block that is glued to the bottom block The shear stress onthe glue is 5 N per 2 m*, or 2.5 N m-* The pressure on the glue is atmosphericpressure plus the weight of the block (due to gravity) divided by the area

Flux A flux is a certain quantity per unit area per unit time Answers to

INTRODUCTION TO TRANSPORT PHENOMENA 7as

Force=SN-Area = 2 m2 FIGURE 1.2

Shear stress and pressure

problems in ‘heat transfer are often expressed ,in units of Btu, cal, or J.Therefore, a heat flux may have units of J me2 s-i Similarly, a mass flux mayhave units of kg m-* s-l

Phases In a simple view of matter, there are three states: solid, liquid, andgas Any given system contains only one gas phase, but may possess severalsolid or liquid phases For example, if a small amount of crude oil is added to acontainer half filled with water, then at equilibrium there are two liquidphases, water on the bottom and hydrocarbon floating on top, plus a vaporphase consisting of air, water molecules, and hydrocarbon molecules Trans-port phenomena often occur in systems where several phases are present.Naturally, solutions of such problems are more complicated than solutions ofsingle-phase problems

Units Engineers must be familiar with all systems of units The abundance oftables in the literature that use English or CGS units requires all of us to bereasonably familiar with all systems of units This text will use SI units(Systeme International dUnit&), as well as the traditional English system withunits such as Btu, pound force (lbt), and pound mass (lb,,,)

In the SI system, mass in kilograms (kg), length in meters (m), time inseconds (s), and temperature in kelvins (K) are taken as basic units Manyother units are derived from these In practice, some of the SI units are quitecumbersome For example, the SI unit for pressure is the Pascal (Pa), which isdefined as N me2 One atmosphere is 101325 Pa; obviously the use of the SIsystem is clumsy here Some authors use the bar, which is 100 kPa or0.986 923 atm But one can argue that the bar is no more fundamental a unitthan the atmosphere

Another point of confusion in the use of SI units is in the definition of themole The basic SI unit is mol, which is defined as the amount of a specieswhose mass in grams is numerically equal to its molecular weight (also calledits molar mass, symbol M) Thus 1 mol of a molecular substance alwayscontains Avogadro’s Number of molecules Since most tables present molecu-lar weights in units of g mol-‘, this SI unit is the same as the old “g-mole”unit Since the SI unit of mass is the kilogram, many authors prefer the kmol

;sp unit (1600 mol) as the most practical for tables of properties, etc This text will

use kmol, as well as lb mol For example, the molecular weight of CO2 isapproximately 44 kg kmol-‘ Therefore, 1 kmol of CO? contains 44 000 g, and

1 lb mol of CO1 contains 44 lb,

Force The relationship between force and mass may be expressed by

Newton’s second law of motion:

In the SI system, two viewpoints of the role of g, are prevalent The unit

of force is a derived unit from Eq (1.2), i.e., kg m s-*, which has been namedthe newton (N) in honor of Sir Isaac Newton One view is that g, equals1.0 kg m NW1 s * A second interpretation is that g, is really unity anddimensionless, since 1 N is identical to 1 kg m s-* This text takes the latterview, and therefore g, will be omitted from equations in the future

Numerous example problems with English units have been included, inorder to illustrate the proper use of g= Note that in the CGS system, thegram-force unit (analogous to the lbr) is almost never used and, since the dyne

is defined as gems-*, when F is expressed in dynes, g, is again unity and

dimensionless

Many equations in transport phenomena involve both lb,,, and lbr ifEnglish units are used The reader must always be very careful to useconsistent and correct units in all problems Also, there are some unusual massand force units in the English system, such as the slug (lbrs2 ft-‘), and thepoundal (lb,,, ft s-*) Each of these is defined via Eq (1.2) These will beignored in this text

’ The acceleration due to gravity is approximately 32.174 ft SK’ here on earth It is this value that changes from location to location Its equivalent is 9M665ms-* in the SI system or 980.665 cm s-’ in the CGS system.

INTRODUCTION To TRANSPORT PHENOMENA 9

;rpt 1.4 TI-iE ROLE OF INTERMOLECULAR

FORCES

Intermolecular forces are responsible for the behavior of matter in the worldaround us The balance between attractive (long-range) forces and repulsive(short-range) forces is responsible for the existence of the gas, the liquid, andthe solid phases Liquids and solids exist at the lower temperatures, at whichthe kinetic energy of the molecules is less than at higher temperatures at whichonly the gas phase can be present

Intermolecular forces are also responsible for the transport phenomena.The basic equations for momentum, heat, and mass can be derived directlyfrom the Boltzmann equation of statistical mechanics This derivation isextremely complex [Hl] Any effort at an exact solution gives results sosimplistic as to have no direct use in the solution of engineering problems.Instead, the following chapters will introduce transport phenomena via bothempirical laws such as Newton’s law of viscosity, and fundamental laws such asconservation of mass, momentum, and energy The empirical nature of theselaws obscures their molecular origin The reader should always keep in mindthat intermolecular forces are responsible for the phenomena at hand but thatthe exact equations are too difficult to solve

1.5 SIMPLE BALANCES

Material balances Perhaps most fundamental of the physical laws is the

conservation of mass The classic reference is in Chapter 7 of the text byHougen, Watson, and Ragatz [H2], although more modem books are usedtoday The idea of conservation of mass is simple: the total mass entering (IN)must equal the total mass leaving (OUT)-unless there is generation,depletion, or accumulation Generation or depletion might come from anuclear reaction For example, uranium-238 can decay upon emission of an LYparticle into thorium-234 An example of accumulation is the simple filling of atank The reader of this text is expected to be familiar with these types ofbalances, at least in a general manner

Example 1.2 The waste acid from a nitrating process contains 15 percent HN03,

45 percent H2S04, and 40 percent H,O by weight This acid must be concentrated

to 25 percent HN03, 50 percent H2S04, and 25 percent H,O Available are concentrated solutions of acid in water, one of 95 percent H2S04 and the other of

85 percent I-INO, If 1500 kg of final product is required, find the mass of each concentrated solution to be added.

Answer In this problem, there is no accumulation or generation The solution requires an overall mass balance, plus balances of two of the specie-water, HN03, or H2S04 All concentrations are known All weight percentages are converted to weight fractions by dividing by 100 The convenient basis is to

consider 1500 kg of product Convenient variables are as follows:

Let x = kg, concentrated H$O.,, 5 percent H,O

Let y = kg concentrated HNO,, 15 percent Hz0

Let z = kg of waste acid before concentration

Overall balance (IN = OUT)

x+y+.2=1500Water balance (Hz0 IN = Hz0 OUT)

0.05~ + 0.15~ + 0.40~ = (1500)(0.25)H,SO, balance (H,SO, IN = H&SO, OUT)

0.95x + o.ooy + 0.452 = (1500)(0.50)HNO, balance (I-INO, IN = I-IN03 OUT)

(9

(ii)

(iii)

Since the summation of Eqs (ii), (iii), and (iv) results in Eq (i), only three

of Eqs (i) through (iv) are independent Arbitrarily, Eq (iv) will not be used.Now Eqs (i) through (iii) constitute three equations in three unknowns Solution

is by elimination of variables First, Eq (i) is solved for z and the results aresubstituted into Eqs (ii) and (iii):

0.05x + 0.15y + (0.40)(1500 - x - y) = 375 6.9

Simplifying these:

Equation (viii) is multiplied by 2 and rearranged to

Energy balances The principle of the energy balance is similar to the massbalance: IN equals OUT, if there is no accumulation or generation However,

in practice energy balances involve several more concepts not yet introduced

These will be explained thoroughly in Chapter 7 on Integral Methods Briefly,

either the first law of thermodynamics or the mechanical energy balance is needed in order to provide the correct relationship of the many terms in the

INTRODUCTION TO TRANSPORT PHENOMENA 11

energy balance: molecular (internal) energy, potential energy, kinetic energy, radiant energy, electrical energy, magnetic energy, chemical reaction effects, heat supplied from external sources, and work done.

A further consideration of Example 1.2 illustrates some of the plexities of the energy balance Suppose in that example all three streams (waste, concentrated HN03, and concentrated H2S04) were at 25°C What would be the final temperature upon mixing? The answer requires knowledge

com-of the “heat com-of mixing” When the concentrated acids are added to the waste stream, there will be a substantial temperature rise, owing to the large heats of mixing in this system.

PROBLEMS

1.2 The heat capacity of carbon dioxide gas at very low pressure is expressed by the equation

cp = 10.57 + 0.0021T - (2.06 x l@)V where cp has units of calmol-‘K-l and T is in K What are the units of the constant of value 2.06 X laS?

1.3 Pure sulfur is burned with air at the rate of 400 lb,,, h-r of s,uIfur The outlet gas is

10 percent 9

(a) Find the number of lb mol of SO, produced per hour.

(b) Find the number of tih-’ of air required if the entering air is at 1 atm pressureand 100°F

1.4 A waste acid stream (to be designated as stream W) contains 4 percent HN4, 20 percent HzS04, and the r&t water, by weight A ton of acid of concentration 21

percent HNOs and 35 percent H2S04 is required Available are concentrated acidsolutions as follows? stream A (92 percent H2S04, 8 percent H,O); stream B (81percent HNO,, 2 percent H,SO,, and 17 percent H,O) Find the number ofpounds of each concentrated solution to be added to make 1 ton of product.1.!5 Wet green lumber containing 12.5 percent moisture is fed continuously to a dryingoven at a rate of 10 ton h-‘ The drying oven consists of two kilns, operated inseries From the first kiln the “partially dried” lumber is fed to a second Tests onthe second show that the final dried lumber leaving contains 4.0 percent moisture.Also, 650 lb, h-’ of moisture are removed from the entering lumber in the secondkiln

(a) Find the lb, h-’ of “dried” lumber exiting from the second kiln

(b) Find the lb, h-’ of water removed in the first kiln.

(c) What is the percentage moisture in the “partially dried” lumber exiting Eromthe first kiln?

1.6 Zinc is to be extracted from a roasted ore All zinc is present as ZnSO, Roastedore of the following composition is obtained: ZnSO, 18 percent; gangue 75percent; moisture 7 percent The roasted ore at a rate of 25 ton h-’ is extractedwith pure water The resulting solution containing 12 percent ZnSO, represents a

100 percent recovery of the ZnSO., One ton of inert gangue will carry with it 2tons of solution Calculate the lb, h-’ of water required

FIGURE 1.3

Residual liquor

Process flow diagram for production of borax (Problem 1.8).

1.7 Peat as dug from the ground contains 88 percent moisture, 8.05 percent volatilecombustible matter, 3.18 percent fixed carbon, and 0.77 percent ash For use as adomestic fuel, the peat is dried until it contains 10 percent moisture The cost ofdrying is 9 cents per 100 lb, water removed, and the cost of mining the peat is

$7.58 per ton Find the cost of processing 1 ton of dried peat

1.8 Borax (NazB40,-10HrO) is produced from a mineral containing 85 percentNa2B40,-4Hz0 by dissolving in water under pressure at lOOT, filtering, andcrystallixing at 20°C The flow sheet for the process is shown in Fig 1.3 Themineral is fed along with the water into a vessel at 100°C When all theNarB,O,*4H,O is dissolved, the solution passes to a second vessel where thetemperature is lowered to 20°C The borax is then filtered, leaving a residualliquor The solubility of anhydrous sodium tetraborate is 525 parts per 100 partswater (by weight) at lOO“C, and 3.9 parts per 100 at 20°C For 1000 kg of boraxproduced, calculate the following:

(a) number of kg of mineral required

(b) number of kg of water used

’ (c) number of kg of residual liquor produced

1.9 Methanol is’ synthesized from carbon monoxide and hydrogen according to thereaction

CO + 2Hr-t CH,OHThe feed stream to a methanol plant consists of 250 kmol h-’ CO, 625 kmol h-’

Hz, and 50 kmol h-r Nr The process flow diagram for this process is shown in Fig.1.4 The gross feed enters a reactor where the conversion per pass is 50 percent.The gross product stream enters a condenser where all the methanol is removed

The unreacted gases are recycled to the gas mixer located just before the reactor.

From the condenser, there is a purge stream which removes the inert nitrogen, plus some valuable CO and HP The concentrations in the purge stream are identical to those in the recycle line If the ratio of moles of feed gas to moles of purge gas is 3.5, find the flow rate (km01 h-‘) of all components in all streams.

REFERENCES

Dl Denbigh, K.: The Principles of Chemicul Equilibrium, 3d ed., Cambridge University Press,Cambridge, 1971

Hl Hirsehfelder, J O., C F Cnrtiss, and R B Bird: Molecular Theory of Gases and Liquids,

April 1%7 printing, Wiley, New York, 1954

H2 Hougen, 0 A., K M Watson, and R A Ragatz: Chemical Process Principles Part 1 Material and Energy Balances, 2d ed., Wiley, New York, 1954

Ml Moore, W J.: Physical Chemistry, 4th ed., Prentice-Hall, Englewood Cliffs, NJ, 1972

Wl Walker, W H., W K Lewis, and W H McAdams: Principles of Chemical Engineering,

McGraw-Hill, New York, 1923

MOLECULAR TRANSPORT MECHANISMS

Species A: Al and A2 are species A at locations 1 and 2

Species B; subscripts 1 and 2 represent locations

Empirical constant in viscosity correlation, Eq (2.52) (units same

Gravitational conversion constant (32.174 lb,,, lb;’ ft s-~)

Unit vector in x direction

Molar flux vector in Fick’s law, Eq (2.4), defined with respect to

a plane of no net volume flow (kmolm-2s-‘, lb mol ftT2 s-l);

Chapter 5

Mass flux of species A in the x direction, defined with respect to aplane of no net volume flow (kg m-’ s-‘, lb, ftT2 s-‘)

Unit vector in y direction

Unit vector in z direction

Thermal conductivity (W m-l K-’ or J m-l K-l s-l, Btu ft-’OR-’ s-‘1

Mass flow vector, equal to molar flow N times molecular weight

Number of moles of gas (kmol, lb mol)

partial pressure of species A, Eq (2.38)

Energy (heat) flow vector (J s-l, Btu s-l)

Gas constant; see Appendix, Table C.l, for values

Superscript meaning transpose of a tensor or matrix

Velocity vector (m s-l, ft s-l); U is magnitude of U; U,, U,, U,are components in directions x, y, z

Volume (m3, ft’)

Rectangular (Cartesian) coordinate

Mole fraction of species A in liquid in Problem 2.17

Rectangular (Cartesian) coordinate

Mole fraction of species A in gas, dimensionless

Rectangular (Cartesian) coordinate

Thermal dSusivity (m2 s-‘, ft2 s-l), defined by Eq (2.10)

Angle in Problem 2.21

value of the temperature at location 2 minus the value at location1

W m-*, Btu ft-* s-‘; see Table 2.1 for more details); YX, YY, Yzare components in directions X, y, z

Generalized concentration of property (e.g., units for tion of heat are J me3 or Btu ftT3; see Table 2.1 for more details)Momentum flux (or shear stress) tensor (N m-*, lbf ft-‘); rXY, rYX,etc are components of the momentum flux tensor, wheresubscripts refer to direction of momentum transfer and direction

concentra-of velocityVector operator del, defined by Eq (2.16) (m-l, ft-‘)Shear rate tensor, defined by Eq (2.41) (s-l)

Transpose of shear rate tensor, defined by Eq (2.42)

In Chapter 1, the role of intermolecular forces was briefly introduced Thediscussion concluded that the exact equations for the rate processes couldnot be solved for most engineering problems The empirical approach usuallyseparates transport into two major divisions: transport by c turbulent mechan-isms and transport by molecular means, with or without convection Turbulentflow will be introduced in Chapter 6 This chapter treats the equations andmechanisms of molecular transport Molecular transport may occur in solids,liquids, gases, or mixtures thereof The simplest example of moleculartransport is the conduction of heat from a high-temperature region to alow-temperature region through a rod, as shown in Fig 2.l(a) If one end of arod at ambient temperature is held firmly while the other end is thrust into aroaring fire, heat is transferred to the hand-held end of the rod from the end inthe fire by molecular transport, The hot molecules in the fire have more energythan the adjacent cooler molecules of the rod As the molecules collide, energy

is transferred from the hotter molecules to the cooler molecules The process isrepeated millions of times until the rod is too hot to hold The difference intemperature (temperature of the hot fire minus hand temperature) is thedriving force for the heat transfer For mass transport, the situation is morecomplicated because there must be at least two species present Consider twoidentical flasks joined through a valve as shown in Fig 2.1@) Let one flask befilled with pure nitrogen, the other with pure oxygen, both at the samepressure and temperature If the valve in the middle is opened, oxygen willdiffuse into the nitrogen srhleand nitrogen into the oxygen side until each flaskcontains 50 percent nitrogen and 50 percent oxygen Concentration is thedriving force

Of the three types of molecular transport, momentum transfer is the mostdifficult to explain briefly and concisely First, the basic equation relates shearstress (introduced in Chapter 1) and velocity gradient The velocity of each

ice Direction of

(a) Molecular heat transfer

Valves

X

(b) Molecular mass transfer

Laminar or molecular transfer

Direction of

fluid flow Turbulent or eddy transport

molecule in the fluid changes from point to point in many flow problems.Mathematically the velocity gradient is XJJdy, the rate of change of thevelocity in the x direction (U,) with respect to the y direction In the last twoparagraphs describing heat and mass transfer, the reader easily visualized whatwas being transferred and the nature of the driving force In the case ofmomentum transfer, momentum flux (t) is being transferred, and the velocitygradient (dU,/dy) is the driving force; both of these are difficult to visualizeand will be discussed further

Fluid flow is a simple example of momentum transfer The driving forcefor fluid flow is a pressure difference For example, when the valve in adrinking fountain is opened, the water flows out in a jet because the waterpressure inside the fountain is much higher than the atmospheric pressure intowhich the jet discharges Figure 2.l(c) shows a simple example of the flow of a

? fluid (gas or liquid) in a pipe A pump or fan may force the fluid through thepipe If a very small pump or fan is used (thus creating only a small pressuredrop), the flow in the pipe will be relatively slow and will be laminar If there

is a large pressure drop, the flow in the pipe will be much larger and probablyturbulent Let Fig 2.l(c) represent smoke-filled air being blown through thepipe In the laminar case (molecular transport), the fluid issues from the pipe

in a smooth, ordered fashion In the turbulent case, the fluid motion is chaoticwith blocks of molecules (called eddies) moving in all directions

In summary, the molecular mechanisms involve transport of heat byconduction, of mass by molecular dilfusion, and of momentum as occurs inlaminar fluid flow A limited analogy among these three transport phenomenacan be used to help gain better insight into the processes of the transfer.However, care must be taken not to carry the analogy too far, and itslimitations will be indicated as our development proceeds

2.1 THE ANALOGY

It is common to formulate a general rate equation as

In Eq (2.1), as the driving force increases, the rate increases Also the largerthe resistance, the smaller is the rate Common sense verifies Eq (2.1), and it

is useful to begin discussion of the transport analogy with a simple examplefrom our experience of heat transfer in the world around us

2.1.1 The Case for Heat Transfer

In heat transfer, the driving force is the temperature difference Our intuitionand experience tells us that heat can be transferred from a hot region to acolder area For example, consider a block of copper, in which the sides areinsulated so that heat conduction occurs only in one direction, the x direction

At this point, it may be helpful for the reader to draw a picture of the block on

a piece of scratch paper Let the initial temperature of the block be 273.15 K(OOC) Next to your drawing of the block, plot a temperature profile, i.e., T

versus x Note that initially for all values of , T is constant and equal to

273.15 K Label this curve “t = to” if

Now a temperature difference is established by placing the copper block

on top of a block of ice and by immersing the toR of the block in steam so thatthe top temperature is instantaneously raised to 373.15 K Let us draw asecond picture of the block and a second temperature profile, this time withthe temperature at the hot end equal to 373.15 K Elsewhere, the block is still

at 273.15 K Since this is the very first instant of time for the experiment, therehas been no time allowed for the temperature below the upper surface to beraised However, shortly after commencing the experiment, the temperaturebegins to rise in the areas below the upper surface At this point a third

MOLECULAR TRANSPORT MECHANlShi.9 19

q temperature profile must be drawn-a curved shape from 273.15 to 373.15 K.Label this curve “t = t ”i This curve is shaped like a parabola, but its exactnature is a complex solution of the unsteady-state problem (to be covered inChapter 13)

At some later time tz, the temperature profile will still be curved.However, the profile will be flatter and more nearly a straight line Finally,when time equals infinity (t = t,, steady-state), the profile will become astraight line

These temperature profiles have been plotted together in Fig 2.2 Thelinear temperature gradient in Fig 2.2 is an experimental observation and,provided enough time is allowed, the linear temperature distribution isobserved as long as the temperatures at the bottom and the top are maintained

at the same preset values The observation is attributed to Fourier, and theequation given below is named after him The readers’ attention is drawn tothe fact that once steady-state is achieved (t = r,), the temperature profile inthe block is invariant with further increase in time The system is therefore said

to be at steady-state The heat from the steam is conducted down thetemperature gradient to the bottom where it is absorbed by the ice and causesmelting The heat being transferred per unit time and unit area, or what iscalled the heat flux, is directly proportional to the difference between thetemperatures and inversely proportional to the distance;~ this is the tempera-,ture gradient (dT/dx) The proportionality is shown by the equation in thefigure and is

where q is the amount of heat transferred per unit time, A is the area, and the

subscript x on the flux term denotes that in Eq (2.2) the heat flux is considered

in the x direction only The proportionality constant k is called the thermal

conductivity It varies from material to material over a wide range, as will bediscussed later For now, only this one-dimensional case will be treated.;multidimentional cases will be introduced in Section 2.2 Partial derivativeswere used in Eq (2.2), rather than total derivatives because more than onedirection may be involved

The minus sign in Eq (2.2) is required because the heat flows from hot tocold In Fig 2.2, the plot of T versus x shows that the gradient or derivative (dT/dx) is positive Common sense tells us that the heat will flow from the top

of the block (373.15 K) to the bottom (273.15 K) Hence the heat flux (q/A)x is

in the negative direction, and Eq (2.2) requires the minus sign Note that thedirection of (q/A), is labeled in Fig 2.2.

The quantity q is the rate of heat transfer, and has typical units J s-’ or

Btu h-‘ Therefore, if Eq (2.2) were to be rearranged into the form of thegeneralized rate equation, Eq (2.1), the results would be

rate = q

resistance = ax/(M)driving force = LJT

( 2 3 )

- T

VA/A), I

(aTlax)

Km-’

concentration gradient

(ac,iw

kmol mm4

shear rate w-u+9

m s-’ m-’

FIGURE 2.2

The analogy of the transport phenomena (SI units are shown below the quantities in the equations).