Transport Phenomena [1 ed.] 9780471073925, 047107392X

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NUNC COGNOSCO EX PARTE

TRANSPORT PHENOMENA

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TRANSPORT PHENOMENA WAEEEW ■TEWABT EDWil W. LIGHTFOOT Department of Chemical Engineering University of Wisconsin Madison, Wisconsin John Wiley & Sons, Inc. New York • London • Sydney

COPYRIGHT © 1960 BY JOHN WHEY & SONS, INC.

All rights reserved. This book or any part thereof must not be reproduced in any form without the written permission of the publisher.

-SIXTH PRINTING, OCTOBER,

1965

LIBRARY OF CONGRESS CATALOG CARD NUMBER: 60-11717 PRINTED IN THE UNITED STATES OF AMERICA

ONUIB

Preface

This book is intended to be an introduction to the field of transport phenomena for students of engineering and applied science. Herein we present the subjects of momentum transport (viscous flow), energy transport (heat conduction, convection, and radiation), and mass trans¬ port (diffusion). In this treatment the media in which the transport phenomena are occurring are regarded as continua, and very little is said about the molecular explanation of these processes. Surely the continuum approach is of more immediate interest to engineering stu¬ dents, although it should be emphasized that both approaches are needed for complete mastery of the subject. Because of the current demand in engineering education to put more emphasis on understanding basic physical principles than on the blind use of empiricism, we feel there is a very definite need for a book of this kind. Obviously the subject matter is sufficiently basic that it cuts across traditional departmental lines. Our thought has been that the subject of transport phenomena should rank along with thermodynamics, mechanics, and electromagnetism as one of the key “engineering sci¬ ences.” Knowledge of the basic laws of mass, momentum, and energy transport has certainly become important, if not indispensable, in en¬ gineering analysis. In addition, the material in this text may be of interest to some who are working in physical chemistry, soil physics, meteorology, and biology.

AflWId

Preface

vi

Since the field of transport phenomena has not heretofore been recog¬ nized as a distinct engineering subject, it seems worthwhile for us to tell the reader how we have organized the material.

Diverse methods

of organization were studied, and, with the help of our departmental colleagues, we settled on the outline shown in Table I. been

assigned

a

Each topic has

pigeonhole in a two-dimensional array in order to

emphasize the relation of each subject to other subjects in the same row or column.

Division of the material into columns labeled mass,

momentum, and energy transport allows for one method of classification, based on the entity being transported.

In the various rows another mode

of classification, based on the type of transport, is indicated.

Clearly, on

the basis of this chart, one can organize a course on transport phenomena in one of two ways: by working down the columns (Chapters 1, 2, 3, 4, 5, etc.) or by working across the rows (Chapters 1, 8, 16, 2, 9, 17, 3, etc.). Actually, the text material is arranged in such a way that either method may be used.

The “column” approach is probably better for beginners,

whereas the “row” approach may be more suited to advanced students. Each chapter is provided with illustrative examples which show how to use various techniques or which give further elaboration on the text. Discussion questions at the end of the chapter are included in an effort to catalyze thinking about the material from several different viewpoints. The problems at the end of each chapter have been grouped into four classes (designated by a subscript after the problem number): Class 1: Problems that illustrate direct numerical applications of the formulas in the text. Class 2: Problems that require elementary analysis of physical situa¬ tions, based on the subject material in the chapter. Class 3: Problems that require somewhat more mature analysis, some¬ times involving information from several chapters or material not specifi¬ cally covered in the text. Class 4: Problems that require mathematical analysis involving Bessel functions, partial differential equations, Laplace transforms, complex variable, and tensor analysis. Of these four classes of problems the first three should be appropriate for junior and senior courses in transport phenomena; none of the prob¬ lems in these classes involves mathematics beyond ordinary differential equations. Obviously there is more material in this book than can be conveniently used in an introductory course.

As a guide to prospective teachers of

transport phenomena, we have indicated with an asterisk (*) those sec¬ tions that we feel are suitable for a well-balanced three- or four-credit undergraduate course.

Having some additional material in the book

TABLE I.

SCHEMATIC DIAGRAM OF THE ORGANIZATION OF TRANSPORT PHENOMENA

Entity Being Transported -> Type of Transport -^

TRANSPORT BY MOLECU¬ LAR MOTION

Momentum

1

VISCOSITY

n

Newton’s law of vis¬ cosity Temperature, pressure, and composition dependence of m Kinetic theory of m TRANSPORT IN LAMINAR FLOW OR IN SOLIDS, IN ONE DIMENSION

TRANSPORT IN AN ARBITRARY CONTINUUM

2

SHELL MOMENTUM BALANCES

Velocity profiles Average velocity Momentum flux at surfaces 3

EQUATIONS OF CHANGE (isothermal)

4

MOMENTUM TRANSPORT WITH TWO INDEPENDENT VARIABLES

5

TURBULENT MOMENTUM TRANSPORT

Time-smoothing of equations of change Eddy viscosity Turbulent velocity profiles 6

PHASES

INTERPHASE MOMENTUM TRANSPORT

Friction factor/ Dimensionless correla¬ tions

TRANSPORT IN LARGE FLOW SYSTEMS

9

SHELL ENERGY BALANCES

Temperature profiles Average temperature Energy flux at surfaces 10

EQUATIONS OF CHANGE (nonisothermal)

Equation of motion for forced and free convection Equation of energy (nonisothermal)

FLOW

TRANSPORT BY RADIATION

k Fourier's law of heat conduction Temperature, pressure, and composition dependence of k Kinetic theory of k

Equation of motion

Two-dimensional viscous flow Ideal two-dimensional flow Boundary-layer mo¬ mentum transport

TRANSPORT BETWEEN TWO

thermal CONDUCTIVITY

Equation of continuity

Unsteady viscous flow

TRANSPORT IN TURBULENT

8

Equation of continuity

Equation of energy (isothermal) TRANSPORT IN LAMINAR FLOW OR IN SOLIDS, WITH TWO INDEPENDENT VARIABLES

Energy

11

ENERGY TRANSPORT WITH TWO INDEPENDENT VARIABLES

Unsteady heat con¬ duction Heat conduction in viscous flow Two-dimensional heat conduction in solids Boundary-layer energy transport 12

turbulent ENERGY TRANSPORT

Time-smoothing of equations of change Eddy thermal con¬ ductivity Turbulent temperature profiles 13

INTERPHASE ENERGY TRANSPORT

Heat-transfer coef¬ ficient h Dimensionless correla¬ tions (forced and free convection) 14

MACROSCOPIC BALANCES (isothermal)

IS

16

DIFFUSIVITY

2)AS

Fick's law of diffusion Temperature, pressure, and composition dependence of 33ab Kinetic theory of ‘Qab 17

SHELL MASS BALANCES

Concentration profiles Average concentration Mass flux at surfaces

18

EQUATIONS OF CHANGE (multicomponent)

Equations of continuity for each species Equation of motion for forced and free convection Equation of energy (multicomponent) 19

MASS TRANSPORT WITH TWO INDEPENDENT VARIABLES

Uns/^ady diffusion Diffusion in viscous flow Two-dimensional diffu¬ sion in solids Boundary-layer mass transport 20

TURBULENT MASS TRANSPORT

Time-smoothing of equations of change Eddy diffusivity Turbulent concentration profiles 21

INTERPHASE MASS TRANSPORT

Mass-transfer coefficient kx Dimensionless correla¬ tions (forced and free convection)

RADIANT ENERGY TRANSPORT

Planck's radiation law ’> Numbers refer to the XvXvX Stefan-Bo'tzmann law ‘///.chapters in this book///: Geometrical problems Radiation through ab¬ sorbing media 7

Mass

MACROSCOPIC BALANCES (NONISOTHERMAL)

Mass balance

Mass balance

Momentum balance Mechanical energy balance (Bernoulli equation)

Momentum balance Mechanical and total energy balance

>This book may be studied'. •Xv either by "col urn ns" vXv X*X or by "rows' ’ v!y!v’.y!;.v.;.v.

22

MACROSCOPIC BALANCES (multicomponent)

Mass balances for each species Momentum balance Mechanical and total energy balance

viii

Preface

will be helpful to instructors and advanced students and will, in addition, serve as a warning to the undergraduate that the “boundaries of the course” do not coincide with the “boundaries of the subject.” Our notation is uniform throughout the text, and a table of notation has been appended for the readers’ convenience. Unfortunately, it is not possible to adopt notation in agreement with that used by all our readers, inasmuch as the subject material includes several fields that have developed independently. Generally, our notation represents a compromise between that used by physicists and that used by engineers. Early in 1957 the Chemical Engineering Department of the University of Wisconsin decided, after considerable deliberation, to inaugurate a required one-semester junior course in transport phenomena. No text¬ book was available; hence mimeographed notes were prepared for the students’ use and in the fall of 1958 were published as Notes on Transport Phenomena. These notes have also been used at several other univer¬ sities, and we have benefited immensely from the comments sent to us by both students and teachers. This book represents the result of an exhaustive revision of the Notes on Transport Phenomena. The text has been completely rewritten, several chapters have been entirely reorganized, and numerous problems and examples have been added. Most of the changes were made in an effort to provide a better text for beginning students. R. Byron Bird Warren E. Stewart Edwin N. Lightfoot Madison, Wisconsin June 1960

Acknowledgments

Many persons have contributed either directly or indirectly to this book; we should like to mention some of them by name: Professor O. A. Hougen and Dean W. R. Marshall, Jr., of the Uni¬ versity of Wisconsin deserve many thanks for their continued interest in the field of transport phenomena and their enthusiasm for promoting increased instruction in this area. Professor R. A. Ragatz, Chairman, of the Chemical Engineering De¬ partment, University of Wisconsin, assisted us by handling the adminis¬ trative problems associated with the introduction of this course into the new chemical engineering curriculum and by providing us with some additional time for preparation of the manuscript. Our colleagues, Professors R. J. Altpeter, C. C. Watson, W. K. Neill, and E. J. Crosby, who have worked with us in developing the under¬ graduate course in transport phenomena, have given us many useful suggestions. Professor J. E. Powers and his undergraduate class at the University of Oklahoma and Professor J. Dranoff and his graduate class at North¬ western University supplied us with detailed reviews of the Notes on

Transport Phenomena. Professor Eric Weger (Johns Hopkins University) and Professor K. M. Watson (Illinois Institute of Technology) also offered their comments after experiences in teaching with Notes on Transport Phenomena.

Acknowledgments

X

A number of our students have read certain chapters of the manuscript of this book and have contributed materially to the accuracy of the final text: Donald

R. Woods, Allyn J.

Ziegenhagen,

David 0.

Edwards,

Paul F. Korbach, Donald W. McEachern, Rosendo J. Sanchez Palma, James P. Hutchins, Rafifi M. Turian, Davis W. Hubbard, Boudewijn van Nederveen, Wiliam A. Hunt, John P. Lawler.

In addition the follow¬

ing students have checked the statements and solutions to all of the Class 1 and Class 2 problems: Vipin D. Shah, Thomas J. Sadowski, Richard H. Weaver, Gary F. Kuether. Professors J. O. Hirschfelder and C. F. Curtiss of the University of Wisconsin, with whom we have had many years of pleasant associa¬ tion, first introduced our chemical engineering department to the subject of transport phenomena some ten years ago via a graduate course; our present course is in a sense a direct descendent of theirs. Professor H. Kramers (Technische Hogeschool, Delft, Holland) in 1956 prepared a set of lecture notes entitled Physische Transportverschijnselen, which represented the first attempt that we know of to teach transport phenomena to engineering students; one of us (R.B.B.) had the pleasure of spending a semester at Professor Kramers’ laboratory as a Fulbright Lecturer and Guggenheim Fellow, during which period he profited very much from discussions related to the teaching of transport phenomena. Miss Jeanne O. Lippert deserves our warmest thanks for typing the bulk of the manuscript and some parts of it several times.

We are deeply

indebted to Mr. Stuart E. Schreiber for his tireless efforts in mimeo¬ graphing and assembling the original set of notes.

Also we wish to thank

Miss Ellen Gunderson for her part in assisting us with the preparation of the manuscript. R. B. B. W. E. S. E. N. L.

Contents

PART I

MOMENTUM TRANSPORT

Chapter 1 *§1.1

Viscosity and the Mechanism of Momentum Transport

3

Newton’s Law of Viscosity

3

* Example 1.1-1. Calculation of Momentum. Flux, 7 *§1.2 *§1.3

Non-Newtonian Fluids

10

Pressure and Temperature Dependence of Viscosity

15

* Example 1.3-1. Estimatio?i of Viscosity from Critical Properties, 18 *Example 1.3-2. Effect of Pressure on Gas Viscosity, 19 §1.4

Theory of Viscosity of Gases at Low Density

19

Example 1.4-1. Computation of the Viscosity of a Gas at Low Density, 25 Example 1.4-2. Prediction of the Viscosity of a Gas Mixture at Low Density, 25 §1.5

Theory of Viscosity of Liquids

26

Example 1.5-1. Estimation of the Viscosity of a Pure Liquid, 29

Chapter 2

Velocity Distributions in Laminar Flow

34

*§2.1

Shell Momentum Balances: Boundary Conditions

35

*§2.2

Flow of a Falling Film

37

* Example 2.2-1. Calculation of Film Velocity, 41 Example 2.2-2. Falling Film with Variable Viscosity, 41 xi

Contents

xii *§2.3

Flow through a Circular Tube

42

* Example 2.3-1. Determination of Viscosity from Capillary

Flow Data, 48 Example 2.3-2. Bingham Flow in a Circular Tube, 48 *§2.4 §2.5 *§2.6

Flow through an Annulus

51

Adjacent Flow of Two Immiscible Fluids

54

Creeping Flow Around a Solid Sphere

56

* Example 2.6-1. Determination of Viscosity from Terminal Velocity of a Falling Sphere, 60 Chapter 3

The Equations of Change for Isothermal Systems

71

*§3.1

The Equation of Continuity

74

*§3.2

The Equation of Motion

76

The Equation of Mechanical Energy

81

*§3.4

The Equations of Change in Curvilinear Coordinates

82

*§3.5

Use of the Equations of Change to Set Up Steady

§3.3

Flow Problems

92

*Example 3.5-1. Tangential Annular Flow of a Newtonian Fluid, 94 *Example 3.5-2. Shape of the Surface of a Rotating Liquid, 96 Example 3.5-3. Torque Relationships and Velocity Distribu¬ tion in the Cone-and-Plate Viscometer, 98 §3.6

The Equations of Change for Incompressible NonNewtonian Flow

101

Example 3.6-1. Tangential Annular Flow of a Bingham Plastic, 104 Example 3.6-2. Components of the Momentum Flux Tensor For Non-Newtonian Radial Flow between Two Parallel Disks, 106 |c§3.7

Dimensional Analysis of the Equations of Change

107

* Example 3.7-1. Prediction of Vortex Depth in an Agitated Tank, 108 Chapter 4

*§4.1

Velocity Distributions with More Than One Independent Variable

123

Unsteady Viscous Flow

123

*Example 4.1—1. Flow Near a Wall Suddenly Set in Motion, 124 Example 4.1-2. Unsteady Laminar Flow in a Circular Tube, 126 §4.2

Steady Viscous Flow With Two Nonvanishing Velocity Components: The Stream Function

Example 4.2-1. “Creeping Flow” Around a Sphere, 132

130

Contents

XII

>4.3

Steady Two-Dimensional Potential Flow

133

Example 4.3-1. Ideal Flow Around a Cylinder, 136 Example 4.3-2. Flow into a Rectangular Channel, 138 >4.4

Boundary-Layer Theory

140

Example 4.4-1. Flow Near a Wall Suddenly Set in Motion, 140 Example 4.4-2. Flow Near the Leading Edge of a Flat Plate, 142 Chapter 5

Velocity Distributions in Turbulent Flow

153

*§5.1

Fluctuations and Time-Smoothed Quantities

154

*§5.2

Time-Smoothing of the Equations of Change for an Incompressible Fluid

*§5.3

Semiempirical Expressions for the Reynolds Stresses

158 160

* Example 5.3-1. Derivation of the Logarithmic Distribution

Law for Tube Flow (Far from Wall), 161 * Example 5.3-2. Velocity Distribution for Tube Flow

(Near Wall), 163 *Example 5.3-3. Relative Magnitude of Molecular and Eddy Viscosity, 165 §5.4

The Second-Order Correlation Tensor and Its Propagation (the von Karman-Howarth Equation)

166

Example 5.4-1. Decay of Turbulence Behind a Grid, 173 Interphase Transport in Isothermal Systems

180

*§6.1

Definition of Friction Factors

181

*§6.2

Friction Factors for Flow in Tubes

183

Chapter 6

* Example 6.2-1. Pressure Drop Required for a Given

Flow Rate, 188 * Example 6.2-2. Flow Rate for a Given Pressure Drop, 189 *§6.3

Friction Factors for Flow Around Spheres

190

* Example 6.3-1. Determination of Diameter of a Falling

Sphere, 194 §6.4

Chapter 7

Friction Factors for Packed Columns

Macroscopic Balances for Isothermal Systems

196

208

*§7.1

The Macroscopic Mass Balance

209

*§7.2

The Macroscopic Momentum Balance

210

*§7.3

The Macroscopic Mechanical Energy Balances (Bernoulli equation)

211

Example 7.3-1. Derivation of Mechanical Energy Balance for Steady Incompressible Flow, 213 *§7.4

Estimation of the Friction Loss * Example 7.4-1. Power Requirements for Pipe-Line Flow, 217

214

Contents

xiv *§7.5

Use of the Macroscopic Balances to Set Up Steady Flow Problems

219

* Example 7.5-1. Pressure Rise and Friction Loss in a

Sudden Expansion, 219 *Example 7.5-2. Performance of a Liquid-Liquid Ejector, 220 *Example 7.5-3. Thrust on a Pipe Bend, 222 *Example 7.5-4. Isothermal Flow of a Liquid through an Orifice, 224 §7.6

Use of the Macroscopic Balances to Set Up Unsteady Flow Problems

226

Example 7.6-1. Efflux Time for Flow from a Funnel, 226 Example 7.6-2. Oscillation of a Damped Manometer, 229

PART II

ENERGY TRANSPORT

Chapter 8

*§8.1

Thermal Conductivity and the Mechanism of Energy Transport

243

Fourier’s Law of Heat Conduction

244

*Example 8.1-1. Measurement of Thermal Conductivity, 247 *§8.2

Temperature and Pressure Dependence of Thermal Conductivity in Gases and Liquids

249

* Example 8.2-1. Effect of Pressure on Thermal

Conductivity, 251 §8.3

Theory of Thermal Conductivity of Gases at Low Density

253

Example 8.3-1. Computation of the Thermal Conductivity of a Monatomic Gas at Low Density, 258 Example 8.3-2. Estimation of the Thermal Conductivity of a Polyatomic Gas at Low Density, 258 Example 8.3-3. Prediction of the Thermal Conductivity of a Gas Mixture at Low Density, 259 §8.4

Theory of Thermal Conductivity of Liquids

260

coq 0° Cn

Example 8.4-1. Prediction of the Thermal Conductivity of a Liquid, 261

Chapter 9

Thermal Conductivity of Solids

262

Temperature Distributions in Solids and in Laminar Flow

265

*§9.1

Shell Energy Balances; Boundary Conditions

266

*§9.2

Heat Conduction with an Electrical Heat Source

267

* Example 9.2-1. Voltage Required for a Given Temperature

Rise in a Wire Heated by an Electric Current, 271 Example 9.2-2. Heating of an Electric Wire with TemperatureDependent Electrical and Thermal Conductivity, 272

Contents

xv §9.3

Heat Conduction with a Nuclear Heat Source

274

*§9.4

Heat Conduction with a Viscous Heat Source

276

Heat Conduction with a Chemical Heat Source

279

§9.5 *§9.6

Heat Conduction through Composite Walls: Addition of Resistances

283

* Example 9.6-1. Composite Cylindrical Walls, 286 §9.7

Heat Conduction in a Cooling Fin

288

Example 9.7-1. Error in Thermocouple Measurement, 290 *§9.8

Forced Convection

291

*§9.9

Free Convection

297

Chapter 10

The Equations of Change for Nonisothermal Systems

310

*§10.1

The Equations of Energy

311

*§10.2

The Energy Equation in Curvilinear Coordinates

317

*§10.3

The Equations of Motion for Forced and Free Convection in Nonisothermal Flow

*§10.4

Summary of the Equations of Change

*§10.5

Use of the Equations of Change to Set Up Steady-State Heat Transfer Problems

*Example 10.5-1. Tangential Flow in an Annulus with Viscous Heat Generation, 325 *Example 10.5-2. Steady Flow of a Nonisothermal Film, 326 *Example 10.5-3. Transpiration Cooling, 328 Example 10.5-4. Free-Convection Heat Transfer from a Vertical Plate, 330 Example 10.5-5. One-Dimensional Compressible Flow: Velocity, Temperature, and Pressure Gradients in a Stationary Shock Wave, 333 *Example 10.5-6. Adiabatic Frictionless Processes in an Ideal Gas, 337

*§10.6

Dimensional Analysis of the Equations of Change

*Example 10.6-1. Forced-Convection Heat Transfer in an Agitated Tank, 339 *Example 10.6-2. Surface Temperature of an Electric Heating Coil, 340 Chapter 11

Temperature Distributions with More Than One Independent Variable

*§11.1

Unsteady Heat Conduction in Solids

*Example 11.1-1. Heating of a Semi-Infinite Slab, 353 *Example 11.1-2. Heating of a Finite Slab, 354 Example 11.1-3. Cooling of a Sphere in Contact with a Well-Stirred Fluid, 357

317 321

321

Contents

xvi §11.2

Steady Heat Conduction in Laminar Flow of a Viscous Fluid

Example Flux at Example Flux at §11.3

361

11.2-1: Laminar Tube Flow with Constant Heat Wall, 362 11.2-2. Laminar Tube Flow with Constant Heat Wall: Asymptotic Solution for Small Distances, 363

Steady Two-Dimensional Potential Flow of Heat in Solids

364

Example 11.3—1. Temperature Distribution in a Wall, 365 §11.4

Boundary-Layer Theory

366

Example 11.4-1. Heat Transfer in Forced-Convection Laminar Flow along a Heated Flat Plate, 367

Chapter 12 *§12.1

Temperature Distributions in Turbulent Flow

375

Temperature Fluctuations and the Time-Smoothed Temperature

375

*§12.2

Time-Smoothing the Energy Equation

377

*§12.3

Semiempirical Expressions for the Turbulent Energy Flux

379

* Example 12.3-1. Temperature Profiles in Steady Turbulent

Flow in Smooth Circular Tubes, 380 §12.4

The Double Temperature Correlation and Its Propagation: The Corrsin Equation

384

Example 12.4-1. Decay Equation for the Double Temperature Correlation, 386

Chapter 13 *§13.1

Interphase Transport in Nonisothermal Systems Definition of the Heat-Transfer Coefficient

389 390

*Example 13.1-1. Calculation of Heat-Transfer Coefficients from Experimental Data, 394 *§13.2

Heat-Transfer Coefficients for Forced Convection in Tubes

396

* Example 13.2-1. Design of a Tubular Heater, 405 *§13.3

Heat-Transfer Coefficients for Forced Convection around Submerged Objects

§13.4

Heat-Transfer Coefficients for Forced Convection through Packed Beds

*§13.5

407

Heat-Transfer Coefficients for Free Convection

411 412

* Example 13.5-1. Heat Loss by Free Convection from a

Horizontal Pipe, 414 §13.6

Heat-Transfer Coefficients for Condensation of Pure Vapors on Solid Surfaces

Example 13.6-1. Condensation of Steam on a Vertical Surface, 418

415

Contents Chapter 14

xvii Energy Transport by Radiation

426

*§14.1

The Spectrum of Electromagnetic Radiation

427

*§14.2

Absorption and Emission at Solid Surfaces

429

*§14.3

Planck’s Distribution Law, Wien’s Displacement Law, and the Stefan-Boltzmann Law

433

* Example 14.3-1. Temperature and Radiant-Energy Emission

of the Sun, 437 *§14.4

Direct Radiation between Black Bodies in Vacuo at Different Temperatures

437

* Example 14.4-1. Estimation of the Solar Constant, 443

*Example 14.4-2. Radiant Heat Transfer between Disks, 444 *§14.5

Radiation between Nonblack Bodies at Different Temperatures

445

* Example 14.5-1. Radiation Shields, 446

*Example 14.5-2. Radiation and Free-Convection Heat Losses from a Horizontal Pipe, 448 Example 14.5-3. Combined Radiation and Convection, 448 §14.6

Radiant Energy Transport in Absorbing Media

449

Example 14.6-1. Absorption of a Monochromatic Radiant Beam, 451 Chapter 15

Macroscopic Balances for Nonisothermal Systems

456

*§15.1

The Macroscopic Energy Balance

456

*§15.2

The Macroscopic Mechanical Energy Balance (Bernoulli Equation)

*§15.3

Summary of the Macroscopic Balances for Pure Fluids

*§15.4

Use of the Macroscopic Balances for Solving Steady-State Problems * Example 15.4-1. The Cooling of an Ideal Gas, 463

*Example 15.4-2. Parallel- or Counter-Flow Heat Exchangers, 465 * Example 15.4-3. Power Requirements for Pumping a

Compressible Fluid through a Long Pipe, 467 Example 15.4-4. Mixing of Two Ideal-Gas Streams, 470 *Example 15.4-5. Flow of Compressible Fluids through Head Meters, 471 §15.5

Use of the Macroscopic Balances for Solving Unsteady-State Problems

Example 15.5-1. Heating of a Liquid in an Agitated Tank, 473 Example 15.5-2. Operation of a Simple Temperature Controller, 476 Example 15.5-3. Free Batch Expansion of a Compressible Fluid, 480

460 462

463

Contents

xviii

PART III

MASS TRANSPORT

Chapter 16 *§16.1

Diffusivity and the Mechanisms of Mass Transport Definitions of Concentrations, Velocities, and Mass Fluxes

495 496

Example 16.1-1. Relations among the Molar Fluxes, 501 *§16.2

Fick’s Law of Diffusion

502

*§16.3

Temperature and Pressure Dependence of Mass Diffusivity

504

*Example 16.3-1. Estimation of Mass Diffusivity at Low Density, 507 * Example 16.3-2. Estimation of Mass Diffusivity at High Density, 507 §16.4

Theory of Ordinary Diffusion in Gases at Low Density

508

Example 16.4-1. Computation of Mass Diffusivity at Low Density, 512 §16.5

Theories of Ordinary Diffusion in Liquids

513

Example 16.5-1. Estimation of Mass Diffusivity for a Binary Liquid Mixture, 515 Chapter 17

Concentration Distributions in Solids and in Laminar Flow

*§17.1 *§17.2

519

Shell Mass Balances: Boundary Conditions

521

Diffusion Through a Stagnant Gas Film

522

*Example 17.2-1. Determination of Diffusivity, 526 Example 17.2-2. Diffusion Through a Nonisothermal Spherical Film, 527 *§17.3

Diffusion with Heterogeneous Chemical Reaction

529

*Example 17.3-1. Diffusion with Slow Heterogeneous Reaction, 531 *§17.4

Diffusion with Homogeneous Chemical Reaction

532

*Example 17.4-1. Gas Absorption with Chemical Reaction in an Agitated Tank, 534 *§17.5

Diffusion into a Falling Liquid Film: Forced-Convection Mass Transfer

537

*Example 17.5-1. Gas Absorption f rom Rising Bubbles, 541 §17.6

Diffusion and Chemical Reaction Inside a Porous Catalyst: the "Effectiveness Factor"

Chapter 18

542

The Equations of Change for Multicomponent Systems

554

*§18.1

The Equations of Continuity for a Binary Mixture

555

*§18.2

1 he Equation of Continuity of A in Curvilinear Coordinates

558

§18.3

The Multicomponent Equations of Change in Terms of the Fluxes

560

Contents §18.4

xix The Multicomponent Fluxes in Terms of the Transport Properties

§18.5

Use of the Equations of Change to Set Up Diffusion Problems

553

572

Example 18.5-1. Simultaneous Heat and Mass Transfer, 572 Example 18.5-2. Thermal Diffusion, 574 Example 18.5-3. Pressure Diffusion, 575 Example 18.5-4. Forced. Diffusion, 577 Example 18.5-5. Three-Component Ordinary Diffusion with Heterogeneous Chemical Reaction, 578 *§18.6

Dimensional Analysis of the Equations of Change for a Binary Isothermal Fluid Mixture

580

* Example 18.6-1. Blending of Miscible Fluids, 582 Chapter 19

Concentration Distributions with More Than One Independent Variable

§19.1

Unsteady Diffusion

592 594

Example 19.1-1. Unsteady-State Evaporation, 594 Example 19.1-2. Unsteady Diffusion with First-Order Reaction, 598 Example 19.1-3. Gas Absorption with Rapid Chemical Reaction, 599 §19.2

Boundary-Layer Theory: von Karman Approximate Method

601

Example 19.2-1. Unsteady Evaporation into a Multicomponent Mixture, 602 Example 19.2-2. Diffusion and Chemical Reaction in Iso¬ thermal Laminar Flow Along a Soluble Flat Plate, 605 §19.3

Boundary-Layer Theory: Exact Solutions for Simultaneous Heat, Mass, and Momentum Transfer

608

Example 19.3-1. Calculation of Mass-Transfer Rate, 619 Chapter 20 *§20.1

Concentration Distributions in Turbulent Flow Concentration Fluctuations and the Time-Smoothed Concentration

*§20.2 §20.3

626

626

Time-Smoothing of the Equation of Continuity of A

627

Semiempirical Expressions for the Turbulent Mass Flux

629

Example 20.3-1. Concentration Profiles in Turbident Flow in Smooth Circular Tubes, 630 Example 20.3-2. Evaporation of Ammonia in a Wetted Wall Column, 630 §20.4

The Double Concentration Correlation and Its Propagation: the Corrsin Equation

633

Contents

XX

Chapter 21

Interphase Transport in Multicomponent Systems

*§21.1

Definition of Binary Mass-Transfer Coefficients in One Phase

*§21.2

Correlations of Binary Mass-Transfer Coefficients in One Phase at Low Mass-Transfer Rates

636 637

642

* Example 21.2-1. Evaporation of a Freely Falling Drop, 648

*Example 21.2-2. The Wet-and-Dry-Bulb Psychrometer, 649 *§21.3

Definition of Binary Mass-Transfer Coefficients in Two Phases at Low Mass-Transfer Rates

*§21.4

Definition of the Transfer Coefficients for High MassTransfer Rates

§21.5

652

656

Transfer Coefficients at High Mass-Transfer Rates: Film Theory

658

Example 21.5—1. Rapid Evaporation of a Pure Liquid, 666 Example 21.5-2. Use of Correction Factors in Droplet Evaporation, 667 Example 21.5-3. Wet-Bulb Performance at High MassTransfer Rates, 667 §21.6

Transfer Coefficients at High Mass-Transfer Rates: Penetration Theory

§21.7

668

Transfer Coefficients at High Mass-Transfer Rates: Boundary-Layer Theory

672

Example 21.7—1. Rapid Evaporation from a Plane Surface, 676 §21.8

Transfer Coefficients in Multicomponent Systems

676

Example 21.8-1. Mass Transfer in a Fixed-Bed Catalytic Reactor, 678 Chapter 22

Macroscopic Balances for Multicomponent Systems

685

*§22.1

The Macroscopic Mass Balances

686

*§22.2

The Macroscopic Momentum Balance

688

*§22.3

The Macroscopic Energy Balance

689

*§22.4

The Macroscopic Mechanical Energy Balance

689

*§22.5

Use of the Macroscopic Balances to Solve Steady-State Problems

690

*Example 22.5-1. Energy Balance for a Sulfur Dioxide Converter, 690 *Example 22.5-2. Height of a Packed-Tower Absorber, 692 Example 22.5-3. Expansion of a Reactive Gas Mixture through a Frictionless Adiabatic Nozzle, 697 §22.6

Use of the Macroscopic Balances for Solving UnsteadyState Problems

Example 22.6-1. Start-Up of a Chemical Reactor, 700 Example 22.6-2. Unsteady Operation of a Packed Column, 702

700

Contents

XXI

Postface

712

Appendix A

Summary of Vector and Tensor Notation

715

§A.l

Vector Operations from a Geometrical Viewpoint

716

§A.2

Vector Operations from an Analytical Viewpoint Example A.2-1. Proof of a Vector Identity, 722

719

§A.3

The Vector Differential Operations

723

§A.4

Second Order Tensors Example A.4-1. Proof of a Tensor Identity, 731

726

§A.5

Integral Operations for Vectors and Tensors

731

§A.6

Vector and Tensor Components in Curvilinear Coordinates Example A.6-1. Transformation Characteristics of Vector and Tensor Products, 736

733

§A.7

Differential Operations in Curvilinear Coordinates Example A.7-1. Derivation of Several Differential Operations in Cylindrical Coordinates, 737

736

Appendix B §B. 1 §B.2

Appendix C

Tables for Prediction of Transport Properties Intermolecular Force Parameters and Critical Properties Functions for Prediction of Transport Properties of Gases at Low Densities Constants and Conversion Factors

743 744 746 747

§C.l

Mathematical Constants

747

§C.2 §C.3

Physical Constants Conversion Factors

747 748

Notation

757

Author Index

765

Subject Index

769

'

MOMENTUM TRANSPORT

CHAPTER

I

Viscosity and the Mechanism of Momentum Transport

The first part of this book deals with the flow of viscous fluids. The physical property that characterizes the flow resistance of simple fluids is the viscosity. Anyone who has purchased motor oil for his automobile is aware of the fact that some oils are more “viscous” than others and that viscosity is a function of the temperature. It is the purpose of this chapter to discuss the viscosities of gases and liquids in a quantitative way. This information will be needed immediately in Chapter 2 for the solution of viscous flow problems. We begin in §1.1 by stating Newton’s law of viscosity and then present a few numerical values to show how the viscosity varies with the conditions and the nature of the fluid. In §1.2 we touch briefly on the subject of nonNewtonian fluids, for which Newton’s law of viscosity is inadequate. The effects of temperature and pressure on viscosities of gases and liquids are summarized in §1.3. Finally, in §§1.4 and 1.5, viscosity is discussed in terms of molecular processes, and the mechanisms of momentum transport in gases and liquids are contrasted. §1.1

NEWTON’S LAW OF VISCOSITY

Consider a fluid—either a gas or a liquid—contained between two large parallel plates of area A, which are everywhere separated by a very small 3

Viscosity and the Mechanism of Momentum Transport

4

t< o

Fluid initially at rest

c-tII o

distance Y. (See Fig. 1.1—1.) We imagine that the system is initially at rest but that at time t = 0 the lower plate is set in motion in the x-direction at a constant velocity V. As time proceeds, the fluid gains momentum, and finally the steady-state velocity profile shown in Fig. 1.1-1 is established. When this final state of steady motion has been attained, a constant force

Lower plate set in motion

V 1 1 . , L /vx(y,t) 1 V/ I -1 V

Small t

^(y> -1

L X

Fig. I.l-I. plates.

—

Large t

Velocity buildup in unsteady flow

Final velocity distribution in steady flow

Xj v

Buildup to steady laminar velocity profile for fluid contained between two

F is required to maintain the motion of the lower plate. This force may be expressed as follows (provided the flow is laminar): (1.1-1)

That is, the force per unit area is proportional to the velocity decrease in the distance Y; the constant of proportionality fi is called the viscosity of the fluid. We shall find it useful in the treatment that follows to rewrite Eq. 1.1-1 in a somewhat more explicit form. The shear stress exerted in the x-direction on a fluid surface of constant y by the fluid in the region of lesser y is designated as tvx, and the x-component of the fluid velocity vector is desig¬ nated as vx. Note that vx is not equal to dv/dx; in this text we do not use

Newton’s Law of Viscosity

5

Subscripts to indicate differentiation of velocity components. Then, in terms of these symbols, Eq. 1.1-1 is rewritten as1

’V.= -/*TS ay

(11-2)

This states that the shear force per unit area is proportional to the negative of the local velocity gradient; this is known as Newton’s law of viscosity, and fluids that behave in this fashion are termed Newtonian fluids. All gases and most simple liquids are described by Eq. 1.1-2; fluids that do not obey this simple law (primarily pastes, slurries, and high polymers) are discussed in §1.2. Equation 1.1-2 may be interpreted usefully in another fashion. In the very neighborhood of the moving surface at y = 0 the fluid acquires a certain amount of x-momentum. This fluid, in turn, imparts some of its momentum to the adjacent “layer” of liquid causing it to remain in motion in the x-direction. Hence x-momentum is transmitted through the fluid in the ^-direction. Consequently, tvx may also be interpreted as the viscous flux2 of x-momentum in the ^-direction. This interpretation ties in better with the molecular nature of the momentum transport process and corre¬ sponds to the treatment given later for energy and mass transport. Further¬ more, the convention of sign for rvx seems easier to visualize in terms of momentum flux. It may be seen from Eq. 1.1-2 that the viscous momentum flux is in the direction of the negative velocity gradient;3 * that is, the momentum tends to go in the direction of decreasing velocity. In other words, the momentum goes “downhill” in the sense that it “coasts” from a region of high velocity to a region of low velocity—just as a sled goes downhill from a region of high elevation to a region of low elevation or heat flows from a hot region toward a colder one. A velocity gradient can thus be thought of as a “driving force” for momentum transport. We shall in the ensuing paragraphs refer to Newton’s law in Eq. 1.1-2 sometimes in terms of forces (this brings out the essentially mechanical nature of the subject with which we are dealing) and sometimes in terms of momentum transport (this brings out the analogies with energy and mass 1 The correspondence between these two equations is clearer if we note that Eq. 1.1-1 is equivalent to

2 By flux is meant “rate of flow per unit area.” Momentum flux then has units of momen¬ tum per unit area per unit time. The student should verify that this is equivalent to force per unit area. 3 A glance at §8.1 will show that this is the same behavior that one experiences with heat conduction, in which the heat flux is proportional to the negative temperature gradient.

Viscosity and the Mechanism of Momentum Transport

6

transport). This dual viewpoint should cause no special difficulty, and indeed it should actually prove helpful in some instances. In some formulas that appear in later chapters it will be useful to have a symbol to represent the viscosity divided by the mass density (mass per unit volume) of the fluid. Hence at this point we define the quantity v by

v=nlp

(1-1-3)

which is called the kinematic viscosity. A few words deserve to be said about the units of some of the quantities we have already defined. The situation is the simplest in the cgs system for which4 Tvx

[=] dyne cm-2

vx [=] cm sec-1

(1.1-4)

y[=]cm Since the two sides of Eq. 1.1-2 must agree in units as well as in numerical value, we may solve for the units of pi in the cgs system as follows:

V = ~rvx

[=] (^ cm'1 sec~2)(cm sec-1 cm'1)'1

[=] g cm-1 sec-1

(1.1-5)

Correspondingly, v = /i/p [=] cm2 sec'1

(1.1-6)

The cgs unit of g cm-1 sec-1 is called the poise; most viscosity data are reported either in this unit or in centipoises (1 cp = 0.01 poise). The analogous set of units in the English system is rvx [=] poundals ft 2 vx [=] ft sec-1 y [=] ft

(i.i-7)

k- [=] lbm ft-1 sec-1 v [=] ft2 sec-1 These units are consistent with Eq. 1.1-2. Because it is not common to work in terms of poundals of force, many people prefer to rewrite Eq. 1.1-2 thus: Sc TVX

1 Read [=] as “has units of.”

dv~ ft — dy

(1.1—8)

Newton’s Law of Viscosity

7

in which Tvx

[=] lb, ft-2

vx [=] ft sec-1 (1.1-9)

y[=]ft P [ = ] lbTO ft-1 sec-1 gc [=] (lbTO/lb,)(ft sec-2)

or

poundals/lb,

In these units the numerical value of gc, the “gravitational conversion factor,” is 32.174. Note that gctvx in Eq. 1.1-8 has units of poundals ft-2 and that division by gc gives tvx in lb, ft-2. In this book we shall consistently employ Eq. 1.1-2 and understand that the units given in Eqs. 1.1-4 or 1.1-7 are used. The student should, however, be able to use formulas in either system because both systems are in current use in the technical literature. Careful checks for dimensional consistency are needed in all practical calculations. Example l.l-l.

Calculation of Momentum Flux

Referring to Fig. 1.1-1, compute the steady-state momentum flux ryx in lb, ft-2 when the lower plate velocity V is 1 ft/sec in the positive ^-direction, the plate separation Y is 0.001 ft, and the fluid viscosity y is 0.7 cp. Solution. Since tvx is desired in lb, ft-2, we first convert all data to lb,-ft-sec units. Thus, making use of Appendix C, Table C.3-4, we find y = (0.7 cp)(2.0886 x 10“5)

= 1.46 x 10~5 lb, sec ft-2 The velocity profile is linear; hence dvx

Avx

— 1.0 ft sec 1

dy

Ay

0.001 ft

= —1000 sec 1

Substitution in Eq. 1.1-2 then gives T

= -y^ = -(1.46 x 10-5)( —1000) dy = 1.46 x 10~2 lb, fr2

If Eq. 1.1-8 were used, one would first convert y as follows: y = (0.7 cp)(6.7197

X

10~4) = 4.70 x 10“4 lbTO ft”1 sec"1

Then substitution into Eq. 1.1-8 gives

= 1.46 x 10-2 lb, fr2 This agrees with the result obtained from Eq. 1.1-2.

Viscosity and the Mechanism of Momentum Transport

8

In Tables 1.1-1, 1.1-2, and 1.1-3 some experimental viscosity data are given for pure fluids at 1 atm pressure.5 Note that at room temperature ^ is about 1 cp for water and about 0.02 cp for air. Note also that for gases at low density the viscosity increases with increasing temperature, whereas for

TABLE

l.l-l

VISCOSITY OF WATER AND AIR AT

I

ATM

Water (liq.)a Temperature Viscosity T (°C) /x (cp) 0 20 40 60 80 100

1.787 1.0019 0.6530 0.4665 0.3548 0.2821

PRESSURE

Airb

Kinematic Viscosity v x 102 (cm2 sec-1)

Viscosity /i (cp)

Kinematic Viscosity v x 102 (cm2 sec-1)

1.787 1.0037 0.6581 0.4744 0.3651 0.2944

0.01716 0.01813 0.01908 0.01999 0.02087 0.02173

13.27 15.05 16.92 18.86 20.88 22.98

a Calculated from the results of R. C. Hardy and R. L. Cottington, J. Research Nat. Bur. Standards, 42, 573--578 (1949), and J. F. Swindells, J. R. Coe, Jr., and T. B. Godfrey, J. Research Nat. Bur. Standards, 48, 1-31 (1952). b Calculated from “Tables of Thermal Properties of Gases,” Nat. Bur. Standards Circ. 464 (1955), Chapter 2. TABLE

1.1-2

VISCOSITIES OF SOME GASES AND LIQUIDS AT ATMOSPHERIC

Substance

Temperature Viscosity T (°C) A (cp)

Gases i-C4H10 ch4 h2o co2 n2. o2 Hg

Substance

PRESSURE11

Temperature Viscosity T (°C) A (cp)

Liquids 23 20 100 20 20 20 380

0.0076 0.0109b 0.0127 0.0146b 0.0175b 0.0203b 0.0654

(c2h5)2o C6H6 Br2 c2h5oh Hg h2so4 Glycerol

20 20 26 20 20 25 20

0.245 0.647 0.946 1.194 1.547 19.15 1069.

a Values taken from N. A. Lange, Handbook of Chemistry, McGraw-Hill, New York (1956), Ninth Edition, pp. 1658-1664. b H. L. Johnston and K. E. McKloskey, J. Phys. Chem., 44, 1038 (1940). 6 Very complete data may be found in the Landolt-Bornstein Physikochemische Tabellen.

9

Newton’s Law of Viscosity

liquids the viscosity usually decreases with increasing temperature.

This difference in temperature dependence is discussed from a molecular viewpoint in §1.4 and §1.5; we simply mention here that in gases (in which the molecules travel long distances between collisions) the momentum is transported primarily by the molecules in free flight, whereas in liquids (in which the molecules travel only very short distances between collisions) the principal mechanism for momentum transfer is the actual colliding of the molecules. In §§1.3, 1.4, and 1.5 we take up the matter of calculating viscosities of gases and liquids. Before doing so, however, we digress briefly to indicate the kinds of deviations from Eq. 1.1-2 that are known to exist. TABLE

1.1-3

VISCOSITIES OF SOME LIQUID METALSa

Temperature T (°C)

Viscosity A (cp)

Li

183.4 216.0 285.5

0.5918 0.5406 0.4548

Na

103.7 250 700

0.686 0.381 0.182

K

69.6 250 700

0.515 0.258 0.136

103.7 250 700

0.546 0.316 0.161

Metal

Na-K alloy 56% Na by wt. 44% K by wt. Hg

-20 20 100 200

1.85 1.55 1.21 1.01

Pb

441 551 844

2.116 1.700 1.185

a Data taken from The Reactor Handbook, Vol. 2, Atomic Energy Commission AECD-3646, U.S. Government Printing Office, Washington D.C. (May 1955), pp. 258 et seq.

Viscosity and the Mechanism of Momentum Transport

10 §1.2

NON-NEWTONIAN

FLUIDS1

According to Newton’s law of viscosity in Eq. 1.1-2, a plot of ryx versus -(dvjdy) for a given fluid should give a straight line through the origin, and the slope of this line is the viscosity of the fluid at the given temperature and pressure. (See Fig. 1.2-1.) Experiments have shown that ryx is indeed

Two-parameter models Fig. 1.2-1.

Three-parameter models

S ummary of steady-state non-Newtonian models (the Newtonian model is

shown for reference).

proportional to — (dvjdy) for all gases and for homogeneous nonpolymeric liquids. There are, however, quite a few industrially important materials that are not described by Eq. 1.1-2, and they are referred to as non-Newtonian fluids. The subject of non-Newtonian flow is actually a subdivision of the larger science of rheology. This is the “science of deformation and flow” 1M. Reiner, Deformation, Strain, and Flow, Interscience, W. Philippoff, Viskositat der Kolloide, Steinkopff, Leipzig (1942).

New

York,

(1960);

Non-Newtonian Fluids

II

and includes the study of the mechanical properties of gases, liquids, plastics, asphalts, and crystalline materials. Hence rheology includes Newtonian fluid mechanics at one end of the spectrum of subject material, and Hookean elasticity at the other. The region in between concerns the deformation and flow of all sorts of gunky and gooey materials. The steady-state rheological behavior of most fluids in the situation of Fig. 1.1-1 can be expressed by a generalized2 form of Eq. 1.1-2: . ..

dvx

(1-2-1)

Tvx=~V~

dy

where r\ may be expressed as a function of either dvjdy or tvx. In regions in which p decreases with increasing rate of shear {—dvjdy) the behavior is termed pseudoplastic; in regions in which rj increases with increasing rate of shear the behavior is termed dilatant. If rj is independent of the rate of shear, the behavior is Newtonian, with p = p. (See Eq. 1.1-2.) Numerous empirical equations, or “models,” have been proposed to express the steady-state relation between tvx and dvjdy. Five representative models are summarized below. Each of these equations contains empirical positive parameters, which can be evaluated numerically to fit data on ryx versus dvjdy at constant temperature and pressure.

The Bingham Model Tvx = ~Bo ~~ i To dy ~ = 0 dy

if

\Tvx\

if|rj To

(1.2-2a) (1-2-2 b)

The positive sign is used in Eq. 1.2-2a when rvx is positive, and the negative sign is used when tvx is negative. A substance that follows this two-parameter model is called a Bingham plastic, it remains rigid when the shear stress is of smaller magnitude than the yield stress r0 but flows somewhat like a Newtonian fluid when the shear stress exceeds r0. This model has been found reasonably accurate for many fine suspensions and pastes. Bingham parameters for suspensions of nuclear fuel particles in heavy water are given in Table 1.2-1.

The Ostwald-de Waele Model Tyx

m

dvx

dvx

dy

dy

(1.2-3)

This two-parameter equation is also known as the power law. For n — 1, it reduces to Newton’s law of viscosity with m — p; thus the deviation of n 2 For a more complete discussion, see §3.6.

Viscosity and the Mechanism of Momentum Transport

12

TABLE

1.2-1

BINGHAM PLASTIC PARAMETERS FOR AQUEOUS NUCLEAR FUEL SLURRIES1

ln/)

= volume fraction solids in the suspension z,max;

2 Q^o 3

2,F2

fiL

Interrelation of Slit and Annulus Formulas

When an annulus is very thin, it may to a good approximation be considered as a thin slit. Then the results of Problem 2.E can be applied. For example, the volume rate of flow in an annulus with outer wall of radius R and inner wall of radius (1 — e)R, where e is small, may be obtained from Problem 2.E by setting 2B equal to eR and W equal to 2tt( 1 — \c)R and thereby obtaining

(2.F-1) (sf.iL Show that this same result may be obtained from Eq. 2.4-16 by setting

k

equal to 1 —

e

Problems

63

everywhere in the formula and expanding the expression for Q in powers of e. This opera¬ tion involves using the Taylor series In (1 - e) = -e - ie2 - Je3 - ie4-

(2.F-2)

and then performing a long division. The first term in the resulting series will be Eq. 2.F-1. (Hint: In the derivation use the first four terms of the Taylor series in Eq. 2.F-2.)

2.G2

Laminar Flow of Falling Film on Outside of a Circular Tube

In a gas absorption experiment a viscous fluid flows upward through a small circular tube and then downward on the outside. (See Fig. 2.G.) Set up a momentum balance over a shell of thickness Ar in the film, as shown in the figure. Note that the “Momentum in” and “Momentum out” arrows are always taken in the positive /--direction in setting up the balance, even though in this case the momentum turns out to be flowing in the negative /•-direction. a. Show that the velocity distribution in the falling film (neglecting end effects) is

pgR2 Vz

4/