multicomponent mass transfer

We are thinking, in particular, of the materials in Section 2.1 theMaxwell-Stefan relations for ideal gases, Section 2.2 the Maxwell-Stefan equations fornonideal systems, Section 3.2 the

Wiley Series in Chemical Engineering

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Bird, Stewart and Lightfoot: TRANSPORT PHENOMENA

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Gates: CATALYTIC CHEMISTRY

Henley and Seader: EQUILIBRIUM-STAGE SEPARATION

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Taylor and Krishna: MULTICOMPONENT MASS TRANSFER

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AND MASS TRANSFER, 3rd Edition

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Library of Congress Cataloging in Publication Data:

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1954-Multicomponent mass transfer/Ross Taylor and R Krishna.

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Includes bibliographical references and indexes.

ISBN 0-471-57417-1 (acid-free)

1 Mass transfer I Krishna, R II Title III Series TP156.M3T39 1993

660'.28423—dc20 92-40667 Printed in the United States of America

10 9 8 7 6 5 4 3 2

Chemical engineers frequently have to deal with multicomponent mixtures; that is, systemscontaining three or more species Conventional approaches to mass transfer in multicompo-nent mixtures are based on an assumption that the transfer flux of each component isproportional to its own driving force Such approaches are valid for certain special cases

• Diffusion in a two component (i.e., binary) mixture

• Diffusion of dilute species in a large excess of one of the components

• The case in which all of the components in a mixture are of a similar size and nature.The following questions arise

• Does the presence of three or more components in the system introduce additionalcomplications unpredicted by binary mass transfer theory alone?

• If the answer to the above question is in the affirmative, how can the problem ofmulticomponent mass transport be tackled systematically?

• Do the transport processes of mass and heat interact with each other in normalchemical engineering operations?

Though the first question has been in the minds of chemical engineers for a long time(Walter and Sherwood in 1941 raised doubts about the equalities of the componentefficiencies in multicomponent distillation), it has been established beyond doubt in the lasttwo decades that multicomponent systems exhibit transport characteristics completelydifferent from those of a simple binary system Furthermore, procedures have beendeveloped to extend the theory of binary mass transfer to multicomponent systems in aconsistent and elegant way using matrix formulations; such formulations have also beenincorporated into powerful computational algorithms for equipment design taking intoaccount simultaneous heat transfer effects These advanced models have been incorporatedinto design software for distillation, absorption, extraction, and condensation equipment.This is one example where commercial application has apparently preceded a formalacademic training in this subject even at the graduate level

This textbook is our attempt to address two needs:

1 The needs of the academic community for a reference text on which to base advancedlectures at the graduate level in transport phenomena or separation processes

2 The requirements of a process design or research engineer who wishes to use rigorousmulticomponent mass transfer models for the simulation and design of processequipment

This textbook has grown out of our research and teaching efforts carried out separatelyand collaboratively at The University of Manchester in England, Clarkson University in theUnited States, Delft University of Technology, and The Universities of Groningen andAmsterdam in the Netherlands, The Royal Dutch Shell Laboratory in Amsterdam, and TheIndian Institute of Petroleum

vi PREFACE

This textbook is not designed as a first primer in mass transfer theory; rather, it is meant

to follow an undergraduate program of lectures wherein the theory of mass transfer andfundamentals of transport phenomena have already been covered

The 15 chapters fall into three parts Part I (Chapters 1-6) deals with the basic equations

of diffusion in multicomponent systems Chapters 7-11 (Part II) describe various models ofmass and energy transfer Part III (Chapters 12-15) covers applications of multicomponentmass transfer models to process design

Chapter 1 serves to remind readers of the basic continuity relations for mass, tum, and energy Mass transfer fluxes and reference velocity frames are discussed here.Chapter 2 introduces the Maxwell-Stefan relations and, in many ways, is the cornerstone ofthe theoretical developments in this book Chapter 2 includes (in Section 2.4) an introduc-tory treatment of diffusion in electrolyte systems The reader is referred to a dedicated text(e.g., Newman, 1991) for further reading Chapter 3 introduces the familiar Fick's law forbinary mixtures and generalizes it for multicomponent systems The short section ontransformations between fluxes in Section 1.2.1 is needed only to accompany the material inSection 3.2.2 Chapter 2 (The Maxwell-Stefan relations) and Chapter 3 (Fick's laws) can bepresented in reverse order if this suits the tastes of the instructor The material onirreversible thermodynamics in Section 2.3 could be omitted from a short introductorycourse or postponed until it is required for the treatment of diffusion in electrolyte systems(Section 2.4) and for the development of constitutive relations for simultaneous heat andmass transfer (Section 11.2) The section on irreversible thermodynamics in Chapter 3should be studied in conjunction with the application of multicomponent diffusion theory inSection 5.6

momen-Chapter 4 suggests usable procedures for estimating diffusion coefficients in nent mixtures Chapters 5 and 6 discuss general methods for solution of multicomponentdiffusion problems Chapter 5 develops the linearized theory taking account of multicompo-nent interaction effects, whereas Chapter 6 uses the conventional effective diffusivityformulations We considered it appropriate to describe both of these approaches and to givethe readers a flavor of the important differences in their predictions We stress theinadequacy of the effective diffusivity approach in several cases of practical importance It is

multicompo-a mmulticompo-atter of continuing surprise to us thmulticompo-at the effective diffusivity multicompo-appromulticompo-ach is still being used

in the published literature in situations where it is clearly inapplicable By delineating theregion of applicability of the effective diffusivity model for multicomponent mixtures andpointing to the likely pitfalls in misapplying it, we hope that we will be able to warnpotential users

In the five chapters that make up Part II (Chapters 7-11) we consider the estimation ofrates of mass and energy transport in multicomponent systems Multicomponent masstransfer coefficients are defined in Chapter 7 Chapter 8 develops the multicomponent filmmodel, Chapter 9 describes unsteady-state diffusion models, and Chapter 10 considersmodels based on turbulent eddy diffusion Chapter 11 shows how the additional complica-tion of simultaneous mass and energy transfer may be handled

Chapter 12 presents models of mass transfer on distillation trays This material is used todevelop procedures for the estimation of point and tray efficiencies in multicomponentdistillation in Chapter 13 Chapter 14 uses the material of Chapter 12 in quite a differentway; in an alternative approach to the simulation and design of distillation and absorptioncolumns that has been termed the nonequilibrium stage model This model is applicable toliquid-liquid extraction with very little modification Chapter 15 considers the design ofmixed vapor condensers

A substantial portion of the material in this text has been used in advanced levelgraduate courses at The University of Manchester, Clarkson University, The Universities ofAmsterdam, Delft, Groningen and Twente in the Netherlands, and The University ofBombay in India For a one semester course at the graduate level it should be possible to

PREFACE viicover all of the material in this book In our experience the sequence of presentation of thechapters is also well suited to lecture courses.

We have included three appendices to provide the necessary mathematical background.Appendix A reviews matrix algebra Appendix B deals with solution of coupled lineardifferential equations; this material is essential for the solution of multicomponent diffusionproblems Appendix C presents two numerical methods for solving systems of nonlinearalgebraic equations; these algorithms are used to compute rates of mass transfer inmulticomponent systems and in the solution of the design equations for separation equip-ment We have usually found it necessary to include almost all of this material in ouradvanced level courses; either by setting aside time at the start of the course or byintroducing the necessary mathematics as it is needed

We also feel that portions of the material in this book ought to be taught at theundergraduate level We are thinking, in particular, of the materials in Section 2.1 (theMaxwell-Stefan relations for ideal gases), Section 2.2 (the Maxwell-Stefan equations fornonideal systems), Section 3.2 (the generalized Fick's law), Section 4.2 (estimation ofmulticomponent diffusion coefficients), Section 5.2 (multicomponent interaction effects),and Section 7.1 (definition of mass transfer coefficients) in addition to the theory of masstransfer in binary mixtures that is normally included in undergraduate courses

A special feature of this book is the large number of numerical examples that have beenworked out in detail With very few exceptions these examples have been based on actualphysicochemical data and many have direct relevance in equipment design The workedexamples can be used by the students for self-study and also to help digest the theoreticalmaterial

To gain a more complete understanding of the models and procedures discussed it isvery important for students to undertake homework assignments We strongly encouragestudents to solve at least some of the exercises by hand, although we recognize that acomputer is essential for any serious work in multicomponent mass transfer We have foundequation solving packages to be useful for solving most of the simpler mass transferproblems For some problems these packages are not yet sufficiently powerful and it isnecessary to write special purpose software (e.g., for distillation column simulation or forcondenser design)

Our research and teaching efforts in multicomponent mass transfer have been stronglyinfluenced by two people The late Professor George Standart of the University ofManchester who impressed upon us the importance of rigor and elegance Professor HansWesselingh of the University of Groningen motivated us to present the material in a formmore easily understandable to the beginner in this area It is left to our readers to judgehow well we have succeeded in achieving both rigor and simplicity

R TAYLOR

R KRISHNA

Potsdam, New York

Amsterdam, The Netherlands

June 1993

computa-A library of Fortran 77 routines for performing multicomponent mass transfer calculations

is available from R Taylor These routines can be made to work with any number ofcomponents and are easily incorporated into other programs We have checked all of ouroriginal calculations by repeating the examples using software that has been designed formathematical work We have used several such packages in the course of our work Withthe exception of the design examples in Chapters 14 and 15, all of the examples have beensolved using Mathcad for DOS (Version 2.5) from MathSoft A disk containing our Mathcadfiles is provided with this book

The distillation design examples in Chapter 14 were solved using a software package

called ChemSep (Kooijman and Taylor, 1992) ChemSep (or an equivalent software package) will be needed for solving some the exercises Information on the availability of ChemSep

can be obtained from R Taylor

The authors would like to express their appreciation to: Gulf Publishing Company, ton, TX for permission to base portions of this textbook on the authors contribution entitled

Hous-Multicomponent Mass Transfer: Theory and Applications, which we published in Handbook

of Heat and Mass Transfer, edited by N P Cheremisinoff, 1986; H L Toor, E U.

Schllinder, and A Gorak kindly provided copies of experimental data (some of it lished) that we have used in creating a number of examples, figures, and exercises; H A.Kooijman for creating the software that allowed us to prepare several of the illustrationsshown in this book (including the three dimensional plots of diffusion coefficients inChapter 4); Norton Chemical Process Products Corporation of Stow, Ohio for supplying thephotographs of packing elements in Chapter 12; and BP Engineering for permission toinclude several industrial applications of the nonequilibrium model in Chapter 14

unpub-R.T.R.K

1.2.1 Transformations Between Fluxes, 6

1.3 Balance Relations for a Two-Phase System Including a Surface

of Discontinuity, 9

1.4 Summary, 12

2 The Maxwell-Stefan Relations 13

2.1 Diffusion in Ideal Gas Mixtures, 13

2.1.1 The Mechanics of Molecular Collisions, 13

2.1.2 Derivation of the Maxwell-Stefan Equation for Binary

Diffusion, 142.1.3 The Maxwell-Stefan Equations for Ternary Systems, 17

2.1.4 The Maxwell-Stefan Equations for Multicomponent

Systems, 192.1.5 Matrix Formulation of the Maxwell-Stefan

Equations, 19

Example 2.1.1 Multicomponent Diffusion

in a Stefan Tube: An Experimental Test

of the Maxwell-Stefan Equations, 212.2 Diffusion in Nonideal Fluids, 23

2.2.1 Matrix Formulation of the Maxwell-Stefan Equations

for Nonideal Fluids, 252.2.2 Limiting Cases of the Maxwell-Stefan Equations, 25

Example 2.2.1 Diffusion of Toluene

in a Binary Mixture, 262.3 The Generalized Maxwell-Stefan Formulation of Irreversible

Thermodynamics, 28

2.3.1 The Generalized Driving Force, 28

2.3.2 The Generalized Maxwell-Stefan Equations, 30

xiii

Example 23.2 Separation of Uranium Isotopes

With a Gaseous Ultracentrifuge, 362.4 Diffusion in Electrolyte Systems, 37

2.4.1 The Nernst-Planck Equation, 40

Example 2.4.1 Diffusion in the System KC1 - H2Oat25°C, 41

2.4.2 Conductivity, Transference Numbers, and the Diffusion

Potential, 432.4.3 Effective Ionic Diffusivities, 45

Example 2.4.2 Diffusion in an Aqueous Solution of HC1

and BaCl2, 46

3 Fick's Law 503.1 Diffusion in Binary Mixtures: Fick's First Law, 50

3.1.1 Fick Diffusion Coefficients, 50

3.1.2 Alternative Forms of Fick's Law, 51

3.2 The Generalized Fick's Law, 52

3.2.1 Matrix Representation of the Generalized Fick's Law, 53

3.2.2 Alternative Forms of the Generalized Fick's Law, 54

3.2.3 Multicomponent Fick Diffusion Coefficients, 54

3.2.4 Transformation of Multicomponent Diffusion Coefficients

From One Reference Velocity Frame to Another, 56

Example 3.2.1 Fick Diffusion Coefficients for the System

Acetone-Benzene-Methanol, 573.3 Irreversible Thermodynamics and the Generalized Fick's Law, 59

Example 33.1 Calculation of the Onsager Coefficients, 61

3.3.1 Diffusion in the Region of a Critical Point, 62

4 Estimation of Diffusion Coefficients 67

4.1 Diffusion Coefficients in Binary Mixtures, 67

4.1.1 Relationship Between Fick and Maxwell-Stefan Diffusion

Coefficients, 674.1.2 Estimation of Diffusion Coefficients in Gas Mixtures, 68

4.1.3 Diffusion Coefficients in Binary Liquid Mixtures, 69

4.1.4 Estimation of Diffusion Coefficients in Dilute Liquid

Mixtures, 73

Example 4.1.1 Diffusion of Alcohols Infinitely Diluted

in Water, 754.1.5 Estimation of Diffusion Coefficients in Concentrated

Liquid Mixtures, 76

4.2.1 Estimation of Multicomponent Diffusion Coefficients

for Gas Mixtures, 80

Example 4.2.1 The Structure of the Fick Matrix [D]

When All of the Binary Diffusion Coefficientsare Nearly Equal, 81

Example 4.2.2 [D] for Dilute Gas Mixtures, 82

Example 4.2.3 Composition Dependence of the Fick

Matrix [D], 82 Example 4.2.4 Effect of Component Numbering

on the Fick Matrix, 84

Example 4.2.5 Prediction of Multicomponent Diffusion

Coefficients in the Mass Average Reference VelocityFrame, 86

4.2.2 Estimation of Multicomponent Fick Diffusion Coefficients

for Liquid Mixtures, 88

4.2.3 Estimation of Maxwell-Stefan Diffusion Coefficients

for Multicomponent Liquid Mixtures, 89

Example 4.2.6 Prediction of [D] in the System

Acetone-Benzene-Carbon Tetrachloride, 914.3 Maxwell-Stefan, Fick, and Onsager Irreversible

Thermodynamics Formulations: A Summary Comparison, 93

5 Solution of Multicomponent Diffusion Problems: The Linearized Theory 95

5.1 Mathematical Preliminaries, 95

5.1.1 The Binary Diffusion Equations, 95

5.1.2 The Multicomponent Diffusion Equations, 96

5.1.3 Solving the Multicomponent Equations, 97

5.1.4 Special Relations for Ternary Systems, 99

5.2 Interaction Effects, 100

5.3 Steady State Diffusion, 102

Example 5.3.1 Steady-State Diffusion in a Ternary

System, 1035.4 Diffusion in a Two Bulb Diffusion Cell, 105

5.4.1 Binary Diffusion in a Two Bulb Diffusion Cell, 106

5.4.2 Multicomponent Diffusion in a Two Bulb

Diffusion Cell, 106

Example 5.4.1 Multicomponent Diffusion in a Two Bulb

Diffusion Cell: An Experimental Test of the LinearizedTheory, 107

xvi CONTENTS

5.5 The Loschmidt Tube, 110

Example 5.5.1 Multicomponent Diffusion in the Loschmidt

Tube: Another Test for the Linearized Theory, 1125.6 Multicomponent Diffusion in a Batch Extraction Cell, 115

5.6.1 Equilibration Paths, 115

Example 5.6.1 Equilibration Paths in a Batch Extraction

Cell, 1185.6.2 Equilibration Paths in the Vicinity of the Plait Point, 121

5.7 The Linearized Theory: An Appraisal, 122

6 Solution of Multicomponent Diffusion Problems: Effective Diffusivity

Methods 124

6.1 The Effective Diffusivity, 124

6.1.1 Definitions, 124

6.1.2 Relationship Between Effective, Maxwell-Stefan, and

Multicomponent Fick Diffusion Coefficients, 1256.1.3 Limiting Cases, 126

Example 6.1.1 Computation of the Effective

Diffusivity, 1276.2 Solution of Multicomponent Diffusion Problems Using

an Effective Diffusivity Model, 129

6.3 Steady-State Diffusion, 129

Example 6.3.1 Computation of the Fluxes with

an Effective Diffusivity Model, 1306.4 The Two Bulb Diffusion Cell, 131

Example 6.4.1 Diffusion in a Two Bulb Diffusion Cell:

A Test of the Effective Diffusivity, 1316.5 The Loschmidt Tube, 133

Example 6.5.1 Multicomponent Diffusion in the Loschmidt

Tube: Another Test of the Effective Diffusivity, 1346.6 Diffusion in a Batch Extraction Cell, 136

Example 6.6.1 Multicomponent Diffusion in a Batch

Extraction Cell, 1366.7 The Effective Diffusivity—Closing Remarks, 138

PART II INTERPHASE TRANSFER 139

7 Mass Transfer Coefficients 141

7.1 Definition of Mass Transfer Coefficients, 141

7.1.1 Binary Mass Transfer Coefficients, 141

7.1.2 Multicomponent Mass Transfer Coefficients, 143

7.1.3 Interaction Effects (Again), 144

7.2.4 Flux Ratios Specified, 146

7.2.5 The Generalized Bootstrap Problem, 147

7.2.6 The Bootstrap Matrix, 148

7.3 Interphase Mass Transfer, 149

7.3.1 Overall Mass Transfer Coefficients, 150

8 Film Theory 152

8.1 The Film Model, 152

8.2 Film Model for Binary Mass Transfer, 153

8.2.1 Equimolar Counterdiffusion, 156

8.2.2 Stefan Diffusion, 156

8.2.3 Flux Ratios Fixed, 156

8.2.4 Generalization to Other Geometries, 156

Example 8.2.1 Equimolar Distillation of a Binary

Mixture, 157

Example 8.2.2 Production of Nickel Carbonyl, 158 Example 8.2.3 Condensation of a Binary Vapor

Mixture, 1608.3 Exact Solutions of the Maxwell-Stefan Equations

for Multicomponent Mass Transfer in Ideal Gases, 162

8.3.1 Formulation in Terms of Binary Mass Transfer

Coefficients, 1658.3.2 Limiting Cases of the General Solution, 167

8.3.3 Computation of the Fluxes, 168

Example 8.3.1 Equimolar Counterdiffusion in a Ternary

Mixture, 170

Example 8.3.2 Diffusional Distillation, 174

8.3.4 Advanced Computational Strategies, 179

8.3.5 An Alternative Formulation, 182

8.4 Multicomponent Film Model Based on the Assumption

of Constant [D] Matrix: The Linearized Theory of Toor,

Stewart, and Prober, 184

8.4.1 Comparison with Exact Method, 185

8.4.2 The Toor-Stewart-Prober Formulation, 187

8.4.3 Computation of the Fluxes, 189

Example 8.4.1 Vapor-Phase Dehydrogenation

of Ethanol, 1918.5 Simplified Explicit Methods, 196

8.5.1 Method of Krishna, 197

8.5.2 Method of Burghardt and Krupiczka, 197

8.5.3 Method of Taylor and Smith, 199

xviii CONTENTS

8.5.4 Computation of the Fluxes, 200

Example 8.5.1 Evaporation into Two Inert Gases, 201

8.5.5 Comparison With Exact and Linearized Solutions, 2038.6 Effective Diffusivity Methods, 204

Example 8.6.1 Diffusion in a Stefan Tube, 206

8.6.1 Comparison With the Matrix Methods, 208

8.7 Multicomponent Film Models for Mass Transfer in NonidealFluid Systems, 209

8.7.1 Exact Solutions, 209

8.7.2 Approximate Methods, 209

Example 8.7.1 Mass Transfer in a Nonideal Fluid

Mixture, 2118.8 Estimation of Mass Transfer Coefficients from Empirical

Correlations, 212

8.8.1 Estimation of Binary Mass Transfer Coefficients, 2138.8.2 Estimation of Multicomponent Mass Transfer Coefficients:The Method of Toor, Stewart, and Prober, 214

8.8.3 Estimation of Multicomponent Mass Transfer

Coefficients for Gas Mixtures from Binary MassTransfer Coefficients, 215

8.8.4 Estimation of Mass Transfer Coefficients for NonidealMulticomponent Mixtures, 216

Example 8.8.1 Ternary Distillation in a Wetted Wall

Column, 2168.8.5 Estimation of Overall Mass Transfer Coefficients:

A Simplified Result, 219

9 Unsteady-State Mass Transfer Models

9.1 Surface Renewal Models, 220

9.2 Unsteady-State Diffusion in Binary Systems, 222

Example 9.2.1 Regeneration of Triethylene Glycol, 225

9.3 Unsteady-State Diffusion in Multicomponent Systems, 2289.3.1 An Exact Solution of the Multicomponent PenetrationModel, 228

9.3.2 Multicomponent Penetration Model Based

on the Assumption of Constant [D] Matrix, 230

9.3.3 Toor-Stewart-Prober Formulation, 232

Example 9.3.1 Mass Transfer in a Stirred Cell, 233

9.4 Diffusion in Bubbles, Drops, and Jets, 235

9.4.1 Binary Mass Transfer in Spherical and Cylindrical

Geometries, 2359.4.2 Transport in Multicomponent Drops and Bubbles, 238

Example 9.4.1 Diffusion in a Multicomponent Drop, 239

CONTENTS xix

10 Mass Transfer in Turbulent Flow 242

10.1 Balance and Constitutive Relations for Turbulent Mass

Transport, 242

10.2 Turbulent Eddy Diffusivity Models, 244

10.2.1 Estimation of the Turbulent Eddy Viscosity, 246

10.3 Turbulent Mass Transfer in a Binary Fluid, 248

10.3.1 Solution of the Diffusion Equations, 248

10.3.2 Mass Transfer Coefficients, 250

Example 10.3.1 Thin Film Sulfonation of Dodecyl

Benzene, 25210.4 Turbulent Eddy Transport in Multicomponent Mixtures, 255

10.4.1 Solution of the Multicomponent Diffusion

Equations, 25510.4.2 Multicomponent Mass Transfer Coefficients, 257

10.4.3 Computational Issues, 258

Example 10.4.1 Methanation in a Tube

Wall Reactor, 25910.4.4 Comparison of the Chilton-Colburn Analogy

with Turbulent Eddy Diffusivity Based Models, 264

11 Simultaneous Mass and Energy Transfer 266

11.1 Balance Equations for Simultaneous Heat and

Mass Transfer, 266

11.2 Constitutive Relations for Simultaneous Heat and

Mass Transfer, 267

11.3 Definition of Heat Transfer Coefficients, 269

11.4 Models for Simultaneous Heat and Mass Transfer, 270

11.4.1 The Film Model, 270

Example 11.4.1 Heat Transfer in Diffusional

Distillation, 27311.4.2 The Penetration Model, 274

11.4.3 Turbulent Eddy Diffusivity Model, 274

Example 11.4.2 Estimation of the Heat Transfer

Coefficient in a Thin-Film Sulfonator, 27711.4.4 Empirical Methods, 278

11.5 Interphase Mass and Energy Transfer, 279

11.5.1 The Bootstrap Problem Revisited, 281

11.5.2 Nonequimolar Effects in Multicomponent Distillation, 282

Example 11.5.1 Nonequimolar Effects in Ternary

Distillation, 28311.5.3 Computation of the Fluxes, 285

Example 11.5.2 Distillation of a Binary Mixture, 287 Example 11.5.3 Interphase Mass Transfer in the Presence

of an Inert Gas, 292

xx CONTENTS

PART III DESIGN 305

12 Multicomponent Distillation: Mass Transfer Models 307

12.1 Binary Distillation in Tray Columns, 307

12.1.1 Material Balance Relations, 309

12.1.2 Composition Profiles, 310

12.1.3 Mass Transfer Rates, 310

12.1.4 Numbers of Transfer Units, 311

12.1.5 Numbers of Transfer Units from Empirical

Correlations, 312

Example 12.1.1 Distillation of

Toluene-Methylcyclohexane, 31412.1.6 Numbers of Transfer Units—A Simplified Approach, 317

12.1.7 A Fundamental Model of Tray Performance, 318

Example 12.1.2 Regeneration of Triethylene Glycol, 324

12.2 Multicomponent Distillation in Tray Columns, 330

12.2.1 Composition Profiles, 330

12.2.2 Mass Transfer Rates, 332

12.2.3 Numbers of Transfer Units for Multicomponent

Systems, 333

Example 12.2.1 Numbers of Transfer Units for the

Methanol-1-Propanol-Water Systems, 33412.2.4 A Fundamental Model of Mass Transfer

in Multicomponent Distillation, 336

Example 12.2.2 Distillation of Ethanol-tert-Butyl

Alcohol-Water in a Sieve Tray Column, 33912.3 Distillation in Packed Columns, 348

12.3.1 Material and Energy Balance Relations, 350

12.3.2 Transfer Units for Binary Systems, 353

12.3.3 Mass Transfer Coefficients for Packed Columns, 355

Example 12.3.1 Distillation of Acetone-Water

in a Packed Column, 358

Example 12.3.2 Mass Transfer Coefficients in a Column

with Structured Packing, 36212.3.4 Transfer Units for Multicomponent Systems, 364

Example 12.3.3 Distillation of a Quaternary System

in a Sulzer Packed Column, 365

13 Multicomponent Distillation: Efficiency Models 371

13.1 Introduction, 371

13.1.1 Definitions of Efficiency, 371

13.2 Efficiencies of Binary Systems, 373

13.2.1 Point Efficiency for Binary Systems, 373

13.3 Efficiencies of Multicomponent Systems, 375

13.3.1 Point Efficiency of Multicomponent Systems, 375

Example 13.3.1 Point Efficiency in the Distillation

of the Methanol-1-Propanol-Water System, 376

Example 13.3.2 Point Efficiencies of Ethanol-tert-Butyl

Alcohol-Water System, 37813.3.3 Tray Efficiency for Multicomponent Systems, 379

Example 13.3.3 Tray Efficiency in the Distillation

of the Methanol-1-Propanol-Water System, 38213.4 Column Simulation, 384

13.4.1 The Equilibrium Stage Model, 384

13.4.2 Solving the Model Equations, 387

13.5 Simulation and Experimental Results, 388

13.5.1 Two Nonideal Systems at Total Reflux, 389

13.5.2 Industrial Scale Columns, 390

14.1.1 The Conservation Equations, 399

14.1.2 The Rate Equations, 401

14.1.3 The Interface Model, 402

14.1.4 The Hydraulic Equations, 402

14.1.5 Specifications for Nonequilibrium Simulation, 403

14.2 Solving the Model Equations, 403

14.2.1 Variables and Equations for a Nonequilibrium

Stage, 40414.2.2 Condensers and Reboilers, 405

14.2.3 Equations and Variables for a Multistage Column, 405

14.2.4 Solution of the Model Equations, 406

15 Condensation of Vapor Mixtures 435

15.1 Mass and Energy Transfer in Condensation, 435

15.1.1 Condensation Flow Patterns, 435

15.1.2 Mass and Energy Transfer, 437

15.1.3 Computation of the Fluxes in Multicomponent

Systems, 440

Example 15.1.1 Condensation of a Methanol-Water

Mixture, 442

Example 15.1.2 Condensation of a Binary Vapor in the

Presence of an Inert Gas, 44815.1.4 Condensation of a Binary Vapor Mixture, 457

15.1.5 Condensation of a Single Vapor in the Presence

of an Inert Gas, 458

Example 15.1.3 Condensation of a Methanol in the

Presence of Nitrogen, 45815.2 Condenser Design, 461

15.2.1 Material Balance Relations, 462

15.2.2 Energy Balance Relations, 463

15.2.3 Solving the Model Equations, 464

15.4.2 Ternary Mass Transfer in a Wetted Wall Column, 473

15.5 Conclusions and Recommendations, 476

Postface 478

Exercises 480

A.5.1 The Cayley-Hamilton Theorem, 518A.5.2 Functions of a Matrix, 520

A.5.3 Functions of Diagonalizable Matrices, 520A.6 Matrix Computations, 522

A.6.1 Arithmetic Operations, 522A.6.2 Matrix Functions, 522

Appendix B Solution of Systems of Differential Equations

B.I Generalization of the Solutions of Scalar Differential

Equations, 524B.2 The Method of Successive Substitution, 525

B.3 Solution of Coupled Differential Equations

Using Similarity Transformations, 529

524

Appendix C Solution of Systems of Algebraic Equations

C.I Solutions of Systems of Linear Equations, 531

C.2 Solutions of Systems of Nonlinear Equations, 532

C.2.1 Repeated Substitution, 532C.2.2 Newton's Method, 532

xxiv CONTENTS

D.2 Thermodynamic Factors for Multicomponent

Systems, 542D.3 Thermodynamic Fluid Stability and the Gibbs Free

Energy, 548

Appendix E About the Software 549

E.I What is on This Disk, 549

E.2 Hardware Requirements, 549

E.3 What is Mathcad?, 550

E.4 Making a Backup Copy, 550

E.5 Installing the Disk, 551

E.6 How to Use the Files on this Disk, 552

References 555 Author Index 571 Subject Index 575

a t Weighting factor (Chapter 1) [various]

a t Activity of component / in solution [ —]

a 1 Interfacial area per unit volume of vapor [m 2 /m 3 ]

a Interfacial area per unit volume of liquid [m2 /m 3 ]

a Interfacial area per unit volume of dispersion [m2 /m 3 ]

a p Specific surface area of packing [m 2 /m 3 ]

a'j Interfacial area per unit volume of vapor in bubble formation zone (Sections

12.1 and 12.2) [m 2 /m 3 ]

a'jj k Interfacial area per unit volume of vapor in A:th bubble population (Sections

12.1 and 12.2) [m 2 /m 3 ]

A n Eigenvalue in Kronig-Brink model

A + Damping constant in van Driest mixing length model [ — ]

A b Active bubbling area on tray [m 2 ]

A h Hole area of sieve tray [m 2 ]

A c Cross-sectional area [m 2 ]

A Interfacial area in batch extraction cell [m2 ]

A(y + ) Quantity defined by Eqs 10.3.4

[A] Matrix defined by Eqs 8.5.21 and 8.5.22 [s/m2 ]

[A(0] Matrix defined by Eq 9.3.8

b Weir length per unit bubbling area (Section 12.1) [m"1 ]

B Channel base (Section 12.3) [m]

B Inverse of binary diffusion coefficient [s/m2 ]

[B] Matrix function of inverted binary diffusion coefficients defined by Eqs 2.1.21

and 2.1.22 [s/m 2 ]

[B n ] Matrix function of inverted binary diffusion coefficients defined by Eqs 2.4.10

and 2.4.11 [s/m 2 ]

[B uV ] Transformation matrix defined by Eqs 1.2.21 [ - ]

[B Vu ] Transformation matrix defined by Eqs 1.2.23 [ - ]

[B uo ] Transformation matrix defined by Eqs 1.2.25 [ - ]

[B ou ] Transformation matrix defined by Eqs 1.2.27 [ - ]

[B(y + )] Matrix defined by Eq 10.4.7

c, Molar density of component / [mol/m 3 ]

c t Mixture molar density [mol/m 3 ]

C pi Specific heat of component / [J/kg] Also, molar heat capacity of component i

[J/mol]

C p Specific heat of mixture [J/kg] Also, molar heat capacity of mixture [J/mol]

Ca Capillary number (Section 12.3) [ - ]

XXV

xxvi NOMENCLATURE

d Characteristic length of contacting device [m]

d Diameter [m]

d t Driving force for mass diffusion [ m " 1 ]

dj Diameter of jet in bubble formation zone (Sections 12.1 and 12.2) [m] djj k Diameter of bubble in kth bubble population (Sections 12.1 and 12.2) [m]

d eq Equivalent diameter [m]

d p Nominal packing size [m]

D Maxwell-Stefan diffusivity for pair i-j [ m2 / s ]

D ijXk ^ l Limiting value of Maxwell-Stefan diffusivity for pair i-j when x k tends to

Dj Thermal diffusion coefficient [kg/m s]

D rei Reference value for diffusion coefficient [ m 2 / s ]

[D] Matrix of Fick diffusion coefficients [m2 /s]

[D°] Matrix of Fick diffusion coefficients in mass average velocity reference frame

[m 2 /s]

[D v ] Matrix of Fick diffusion coefficients in volume average velocity reference frame

[ m 2 / s ]

[D f ] Matrix of Fick diffusion coefficients relative to a reference diffusivity [ —]

[Z) turb ] Matrix of turbulent diffusion coefficients [m 2 /s]

D t /th eigenvalue of [D] [ m2 / s ]

(e t ) /th eigenvector of [D] [ — ]

E Energy flux in stationary coordinate frame of reference [ W / m2 ]

W Energy transfer rate (Chapter 14) [W]

E Energy balance equation (Chapters 11, 13, 14, and 15) [ W / m2 , W]

E o Overall efficiency (Eq 13.1.1) [ - ]

E MV Murphree tray efficiency (Eq 13.1.2) [ - ]

E ov Murphree point efficiency (Eq 13.1.3) [ - ]

/ Fanning friction factor [ — ]

fi Fugacity of component / [Pa]

F Discrepancy functions [various]

Fj Flow rate of feed stream [mol/s]

f IIk Fraction of vapor in A:th bubble population (Sections 12.1 and 12.2) [ —]

& Faraday's constant [9.65 X 104 C / m o l ]

[/] Matrix function [various] (Chapter 5)

F s F-factor based on superficial velocity [ k g 1 / 2 / m 1 / 2 s]

F(N t ) Function of total molar flux (Section 8.4) [ - ]

Fo Fourier number [ - ]

Fr Froude number [ — ]

g Acceleration due to gravity [9.81 m / s2 ]

G tj Chemical potential—composition derivative (Eq 3.3.9) [ J / m o l ]

G Parameter in N R T L model (Appendix D)

NOMENCLATURE xxvii

h Heat transfer coefficient (Eq 11.3.2) [ W / m2 K]

A Heat transfer coefficient (Eq 11.3.1) [m/s]

h f Froth or dispersion height [m]

hj Height of bubble formation zone (Sections 12.1 and 12.2) [m]

h n k Height of bubbling zone (Sections 12.1 and 12.2) [m]

h w Weir height [m]

H Height of packing (Section 12.3) [m]

H K Height of a transfer unit for the vapor (Section 12.3) [m]

H L Height of a transfer unit for the liquid (Section 12.3) [m]

H ov Overall height of a transfer unit (Section 12.3) [m]

[H v ] Matrix of heights of transfer units for the vapor (Section 12.3) [m]

[H L ] Matrix of heights of transfer units for the liquid (Section 12.3) [m]

[H ov ] Matrix of overall heights of transfer units (Section 12.3) [m]

HETP Height equivalent to a theoretical plate [m]

H t Partial specific enthalpy [J/kg] Also, partial molar enthalpy of component i

[J/mol]

A// v a p i Latent heat of vaporization of component / [J/mol]

i Current [amps]

/ Referring to interphase or interface

[H] Matrix of transport coefficients (Chapter 3) [J/mol m2 s]

[ / ] Identity matrix [ — ]

/ Unit tensor [ — ]

j D Chilton-Colburn j factor for mass transfer [ - ]

j H Chilton-Colburn j factor for heat transfer [ - ]

j m Roots of zero-order Bessel function (Chapter 9) [ — ]

/ Mass diffusion flux relative to the mass average velocity [kg/m 2 s]

j u Mass diffusion flux relative to the molar average velocity [kg/m 2 s]

j v Mass diffusion flux relative to the volume average velocity [kg/m 2 s]

j r Mass diffusion flux relative to velocity of component r [kg/m2 s]

Ji, turb Turbulent diffusion flux of component / [kg/m2 s]

/ Molar diffusion flux relative to the molar average velocity [mol/m 2 s]

J v Molar diffusion flux relative to the volume average reference velocity [mol/

m2 s]

J r Molar diffusion flux relative to the velocity of component r [mol/m2 s]

J u Molar diffusion flux relative to the mass average reference velocity [mol/m2 s]

Ji, turb Molar turbulent diffusion flux of component i [mol/m2 s]

J t Pseudodiffusion flux (Chapter 5) [mol/m 2 s]

J o Zero-order Bessel function [ - ]

[J] Jacobian matrix [various]

k B Boltzmann constant [1.38048 J / K ]

k Mass transfer coefficient in a binary mixture [m/s]

K t Equilibrium ratio (K value) for component / [ —]

[K] Diagonal matrix of the first n — 1 K values [ - ]

K t eff "Effective" volumetric mass transfer coefficient [s" 1 ] or [h" 1 ]

xxviii NOMENCLATURE

[K] Matrix of volumetric mass transfer coefficients (Section 5.6) [s"1 ]

K ov Overall mass transfer coefficient in a binary mixture [m/s]

[k] Matrix of multicomponent mass transfer coefficients [m/s]

^/,eff Pseudobinary (effective) mass transfer coefficient of component / in a mixture

[m/s]

[K ov ] Matrix of multicomponent overall mass transfer coefficients [m/s]

Jf t Equivalent conductivity of component / (Section 2.4)

X Equivalent conductivity of mixture (Section 2.4)

i Generalized characteristic length [m]

( Mixing length describing turbulent transport (Chapter 10) [m]

i + Reduced mixing length [ - ]

6 t Component flow of a liquid [mol/s]

Le Lewis number [ — ]

L Liquid flow rate [mol/s]

m Mass of molecule [kg]

M Molar mass of mixture [kg/mol]

M t Molar mass of component / [kg/mol]

M t Moles of i in batch extraction cell [mol]

M t Total moles of mixture in batch extraction cell [mol]

[M] Matrix of equilibrium constants (Eq 7.3.5) [ - ]

Mfj Component material balance equation (Chapters 13 and 14) [mol/s]

M /; Component material balance equation (Chapters 13 and 14) [mol/s]

M v Mass flow of vapor [kg/s]

M L Mass flow of liquid [kg/s]

n Number of components in the mixture [ — ]

n t Mass flux component / referred to a stationary coordinate reference frame

M v Number of transfer units for the vapor phase in binary system [ —]

N' L Number of transfer units for the liquid phase in a binary system [ — ]

[N v ] Matrix of numbers of transfer units for the vapor phase [ - ]

[N L ] Matrix of numbers of transfer units for the liquid phase [ —]

[N ov ] Matrix of overall number of transfer units [ - ]

JV V Number of transfer units for a vapor defined by Eq 12.1.42 [ - ]

JV' L Number of transfer units for a liquid defined by Eq 12.1.42 [ —]

JV Mass transfer rate (Chapter 14) [mol/s]

P Pressure [Pa]

P Perimeter in structured packing (Section 12.3) [m]

Pj Pressure drop equation (Chapter 14)

NOMENCLATURE xxix

p Sieve tray hole pitch (Section 12.1) [m]

p Pressure tensor (Chapter 1) [Pa]

[P] Modal matrix [-]

Pf Vapor pressure of component / [Pa]

Pi Partial pressure of component i [Pa]

P i Parachor (Section 4.2) [g 1 / 4 cm 3 /niol s1/2 ]

Pr Prandtl number [ - ]

Pr t u r b Turbulent Prandtl number [ - ]

q Conductive heat flux [ W / m2 ] Also, integer parameter [ - ]

<3Wb Turbulent contribution to the conductive heat flux [ W / m 2 ]

Qj Unaccomplished equilibrium in bubble formation zone (Section 12.1) [ - ] Qn,k Unaccomplished equilibrium in kth bubble population (Section 12.1) [ - ] [Q] Matrix describing unaccomplished equilibrium (Section 12) [ - ]

[Qj] Matrix of unaccomplished equilibrium in bubble formation zone (Section 12.2)

[-]

[Q//.J Matrix describing unaccomplished equilibrium in kth bubble population

(Section 12.2) [ - ]

Q t Equilibrium equation (Chapters 11-15) [ - ]

Qj Heat duty (Chapters 13-14) [W]

Q L Volumetric liquid flow rate [m 3 /s]

Q v Volumetric vapor flow rate [m 3 /s]

r Coordinate direction or position [m]

r 0 Inner edge of film [m]

r 0 Radius of spherical particle [m]

r 8 Outer edge of film [m]

r t Radius of molecule in Eq 4.1.7 [m]

R Gas constant [8.314 J / m o l K]

R t Radius of gyration (Section 4.2 only) [nm]

(R) Vector of rate equations [mol/m2 s or mol/s]

[R] Matrix function of inverted binary mass transfer coefficients defined by Eqs.

8.3.25 [ s / m ]

[R ov ] Inverse of [K ov ] [ s / m ]

Re Reynolds number [ - ]

R w Inner radius of tube wall [m]

R+ Reduced tube radius [-]

R Mass transfer rate equation

s Surface renewal frequency [s"1 ]

S Quantity defined by Eq 4.2.4

S Structured packing channel side (Section 12.3) [m]

S Summation equation [ —]

Sc Schmidt number [ - ]

[Sc] Matrix of Schmidt numbers [ - ]

Sc turh Turbulent Schmidt number [ - ]

Sh Sherwood number [ - ]

[Sh] Matrix of Sherwood numbers [ - ]

xxx NOMENCLATURE

St Stanton number [ — ]

St H Stanton number for heat transfer [ - ]

[St] Matrix of Stanton numbers [ - ]

t Time [s]

t t Transference number (Section 2.4) [ - ]

t e Exposure time [s]

t v Vapor residence time (Section 12.1) [s]

t L Liquid residence time (Section 12.1) [s]

tj Residence time in bubble formation zone (Section 12.1) [s]

t u k Residence time in A:th bubble population (Section 12.1) [s]

T Temperature [K]

u t Velocity of diffusion of species / [m/s]

u Molar average reference velocity [m/s]

u v Volume average reference velocity [m/s]

u l Velocity of the interface [m/s]

u s f Superficial velocity at flooding [ m / s ]

U Internal energy (Section 1.3)

U Velocity [ m / s ]

v Mass average mixture velocity [ m / s ]

V t Molar volume at normal boiling point (Section 4.1) [m 3 /mol]

V t Partial molar volume [m 3 /mol]

V t Mixture molar volume [m 3 /mol]

v i Molar flow rate of component i [mol/s]

V Molar flow rate of mixture [mol/s]

V o Volume of bulb in two-bulb diffusion cell (Chapter 5) [m 2 ]

V e Volume of bulb in two-bulb diffusion cell (Chapter 5) [m 2 ]

W Weir length (Section 12.1) [m]

[W] Matrix of mass transfer coefficients: [W] = [/3][k] [ m / s ]

Greek Letters

a Relative froth density (Section 12.1) [ - ]

a e Parameter in Bennett method for pressure drop (Eq 12.1.27)

a tj Multicomponent thermal diffusion factors [ - ]

f3 Cell constant in two-bulb diffusion cell (Eq 5.4.6) [ m ~2 ]

[/3] Bootstrap matrix [ — ]

y i Activity coefficient of component / in solution [ — ]

F Liquid flow per unit length of perimeter (Section 12.3) [ k g / m 3 s]

F Thermodynamic factor for binary system (Eq 2.2.12) [ —]

NOMENCLATURE xxxi

[T] Matrix of thermodynamic factors with elements defined by Eqs 2.2.5 [ - ]

8 Distance from interface [m]

8 U Kronecker delta, 1 if i = k, 0 if / # k [-]

s Void fraction [ - ]

£ Rate of production of field quantity in bulk fluid mixture (Section 1.3)

£ l Rate of production of field quantity at the interface (Section 1.3)

£ Combined variable, £ = z/y/4t (Chapter 9)

T] Dimensionless distance [ - ]

K Maxwell-Stefan mass transfer coefficient in a binary mixture (Eqs 8.3.26 and

8.8.16) [ m / s ]

A Dimensionless parameters [ — ]

A Stripping factor (Chapters 12 and 13) [ - ]

A, Difference between component molar enthalpies (Eq 11.5.13) [ J / m o l ]

A Parameter in mixing length models (Section 10.2)

A Molecular thermal conductivity [ W / m K]

A t u r b Turbulent thermal conductivity [ W / m K]

A n Eigenvalue in the Kronig-Brink model [ —]

fx t Molar chemical potential of component / [ J / m o l ]

ju,; Viscosity of component i (Section 4.1) [Pa s]

fi Molecular (dynamic) viscosity of mixture [Pa s]

ju, turb Turbulent eddy viscosity [Pa s]

v Molecular kinematic viscosity of mixture [ m2 / s ]

v tmh Turbulent eddy kinematic viscosity [m 2 /s]

v t Determinacy coefficients for species / [various]

v Mole fraction weighted sum of component determinacy coefficients (Section

8.5) [various]

f Unit normal directed from phase "x" to phase " y " [ —] Also, dimensionless

distance along dispersion or column height [ — ]

£ Ratio of component mass flux to total mass flux (Sections 10.3 and 10.4) [ - ]

S Correction factor for high fluxes in binary mass transfer [ - ] Also, correction,

factor for high fluxes in explicit methods [ — ]

Si eff Correction factor for high fluxes in pseudobinary (effective diffusivity) methods

[-]

B H Correction factor for the effect of high fluxes on the heat transfer coefficient

[-]

[H] Matrix of high flux correction factors [ - ]

p { Mass density of component i [ k g / m3 ]

p t Mixture mass density [ k g / m 3 ]

a R a t e of entropy production (Chapter 2) [ J / m3 s K]

cr diff R a t e of entropy production due to diffusion (Chapter 2) [ J / m 3 s K]

a Characteristic diameter of molecule (Section 4.1) [A]

a Surface tension [N/m]

a c Critical surface tension (Section 12.3) [N/m]

r Shear stress [Pa]

T 0 Shear stress at the wall [Pa]

r Turbulent shear stress [Pa]

xxxii NOMENCLATURE

T Stress tensor [Pa]

r /y Parameter in NRTL activity coefficient model [ — ]

</> Mass transfer parameter defined by Eq 8.2.6 (Section 8.2) [ - ]

<t> t Volume fraction (Section 4.1) [ - ]

<\>i Fugacity coefficient (Chapter 2) [ - ]

(/> Electrical potential (Section 2.4) [V]

</> Association parameter in Eq 4.1.8

(f> fractional free area (Section 12.1) [ - ]

(</>) C o l u m n matrix of dimensionless m a s s transfer p a r a m e t e r s (Section 8.2) [ —] 4> Nonconvective flux of field quantity ( C h a p t e r 1) [various]

O Mass transfer rate factor for binary mass transfer (Eqs 8.2.5, 9.2.3, and

10.3.10) [ - ]

<I> M a s s transfer r a t e factor for explicit m e t h o d s ( E q 8.5.13) [ —]

<£, eff M a s s transfer r a t e factor in p s e u d o b i n a r y (effective diffusivity) m e t h o d s [ — ]

O H H e a t transfer r a t e factors [ — ]

[<l>] Matrix of mass transfer rate factors [ - ]

\fj(t) Surface age distribution [ s "1 ]

ifj Referring to any field variable (Section 1.3) [various]

[W] Matrix of mass transfer rate factors in linearized film model (Eq 8.4.4) [-]

[^] Matrix of mass transfer rate factors in turbulent diffusion model (Eq 10.3.9)

[-]

o) t Mass fraction of component / [ — ]

[co] Diagonal matrix of mass fractions [ —]

[S] Matrix of rate factors for nonideal systems [ —]

X Arbitrary independent variable

(x) Vector defined by Eq 9.3.4 (Section 9.3)

E Quantity entering zone under consideration

eff Pseudobinary or "effective" parameter

H Parameter relevant to heat transfer

/ Referring to the interface

/ Referring to bubble formation zone (Sections 12.1 and 12.2)

//, k Referring to kth bubble population in bubble rise zone (Sections 12.1 and

12.2)

NOMENCLATURE xxxiii

/ Component i property or parameter

/, j , k Component indices, stage or section numbers (; only)

L Quantity leaving zone under consideration

m Mean value Also, refers to the mass average velocity

n nth component

O Overall parameter Also, denotes reduced energy and heat conduction fluxes

OV Overall parameter referred to the vapor phase

ref Denotes reference quantity

t Referring to total mixture

T, P Constant temperature and pressure

x Referring to the "x" phase

y Referring to the "y" phase

8 Quantity evaluated at position 17 = 8

0 Quantity evaluated at position 17 = 0

00 Quantity evaluated at long time or long distance

Superscripts

C Referring to the coolant

F Referring to the feed

/ Referring to the interface

(k) Denotes iteration number

L Referring to the liquid phase

m Referring to the mass average velocity

V Referring to the vapor-gas phase

v Referring to the volume average velocity

W Referring to the wall

x Referring to the ' V phase

y Referring to the "y" phase

' Referring to the ' phase

" Referring to the " phase

Referring to mass average reference velocity frame

Referring to finite transfer rates

Miscellaneous

Overall denotes partial molar property Also, averaged parameter

Eigenvalue of corresponding matrix

Mathematical Symbols

V Gradient

A Difference operator

lim Limit

MULTICOMPONENT MASS TRANSFER

Molecular Diffusion

1 Preliminary Concepts

The reader should not be intimidated by the great generality expressed by the vectorial character

of these equations, because a simple one-dimensional approximation is almost always used in applications (But it is hard to resist the lure of cheap generality when writing down equations.)

—E A Mason and H K Lonsdale (1990)

1.1 CONCENTRATION MEASURES

In the description of the interphase mass transfer process, a variety of measures for constituent concentrations, mixture reference velocities, and diffusion fluxes (with respect to the arbitrarily defined mixture velocity) are used Table 1.1 summarizes the most commonly used concentration measures together with a number of other quantities that will be needed from time to time.

1.2 FLUXES

If u t denotes the velocity of component / (with respect to a stationary coordinate reference

frame) then the mass flux of that species is defined by

»/ = P,-«,- (1.2.1) and has units of kilograms per meter squared per second (kg/m 2 s) If we sum the component fluxes we obtain

which has units of moles per meter squared per second (mol/m 2 s) The total molar flux is

the sum of these quantities

N,= E A l - c , i i (1.2.5)

Partial molar volume of species i; ^

Mixture molar volume; V t = \/c t Volume fraction of species /; 4>i = c t Fugacity of i

Molar chemical potential of species i

particular, the diffusion flux, which is the flux of species / relative to the flux of the mixture

as a whole The definition of this flux raises the first of our problems, which mixture velocity

are we going to use? We have already introduced two, v and w, and there are others that we

have not discussed yet The literature on diffusion would be a good deal simpler if therewere only one way to define diffusion fluxes For each choice of reference velocity there are

at least two different diffusion fluxes that we could define, mass fluxes and molar fluxes

Perhaps an example will help to clarify the situation If we choose v as the reference

velocity, then the mass diffusion flux with respect to the mass average velocity is

7 , = p , ( « , - " ) (1-2-7)and

E ; , = 0 (1.2.8)

FLUXES 5

The mass flux n i is related to the mass diffusion flux as

On the other hand, we could choose u as the reference velocity and define molar diffusion

fluxes relative to it as

/ , = c,.( « , - « ) (1.2.10) with

L/, = 0 (1.2.11)

/• = 1

The molar flux A^ is related to the molar diffusion flux by

N t = J t + c t u = J; + x t N t (1.2.12) These are the most commonly encountered sets of fluxes; other sets could be defined We could, for example, define a mass diffusion flux relative to the molar average velocity or a molar diffusion flux relative to the mass average velocity Still other choices of reference

velocity are sometimes used; for example, the volume average velocity u v

u v = £ c; > > , = ! > , « , (1.2.13)

1=1 i=\

where <j> i is the volume fraction of species / defined in Table 1.1.

Table 1.2 summarizes the most commonly used reference velocities.

Let us define an arbitrary reference velocity u a

6 PRELIMINARY CONCEPTS

TABLE 1.2 Reference Velocities 0

u a Arbitrary mixture velocity, weighting factor a t

n n

v Mass average mixture velocity, weighting factor o) i

i=\ i=1

u Molar averaged reference velocity, weighting factor x t

u v Volume averaged reference velocity, weighting factor </>,

w r Velocity of species r, weighting factor 8 ir ,

where 8 ir is the Kronecker delta

8 ir = 1 if i == r

8 ir = 0 if i # r

"Units are in meters per second [m/s].

The sum of these fluxes gives

1.2.1 Transformations Between Fluxes

It will sometimes prove necessary to transform fluxes from one reference velocity toanother We give some examples of the required relations here

To relate the molar diffusion flux relative to the volume average velocity to the molardiffusion flux relative to the molar average reference velocity we use the transformation

/, = E B?W (1.2.20)

FLUXES 7

TABLE 1.3 Diffusion Fluxes 0

jf Mass diffusion flux relative to arbitrary reference velocity

ji Mass diffusion flux relative to mass average velocity

jf Mass diffusion flux relative to molar average velocity

Jf Molar diffusion flux relative to arbitrary reference velocity

J t Molar diffusion flux relative to molar average velocity

n

7, = C,(H,-«) £7,-0

1 = 1

JY Molar diffusion flux relative to volume average velocity

Ji Molar diffusion flux relative to mass average velocity

Jf = cf ( B| -v) E ^ = 0

i = l Xi

Jf Molar diffusion flux relative to component r velocity

Jf = c,(«,,- «,) 7rr = 0

Units are kilograms per meter squared per second (kg/m 2 s) for mass diffusion

fluxes and moles per meter squared per second (mol/m 2 s) for molar diffusion

fluxes.

8 PRELIMINARY CONCEPTS

TABLE 1.4 Fluxes With Respect to a Laboratory Fixed Frame

of Reference 0

n t Mass flux relative to stationary coordinates

n t Total mass flux relative to stationary coordinates

"Mass fluxes units are kilograms per meter squares per second

(kg/m 2 s); molar fluxes units are moles per meter squared per second

BALANCE RELATIONS FOR A TWO-PHASE SYSTEM 9

The inverse transformation is

where the Bf£ are

For more on the use of these transformations see Section 3.2.4 and Exercises 1.1, and 1.2

1.3 BALANCE RELATIONS FOR A TWO-PHASE SYSTEM INCLUDING

A SURFACE OF DISCONTINUITY

Let us consider a two-phase system including a surface of discontinuity (phase interface)

Let x and y represent the two phases For example, y may refer to the gas phase and x to the liquid phase in a two-phase system Let the number of components in each phase be n Let / represent the phase interface and the unit normal directed from phase x to y The

system considered is shown pictorially in Figure 1.1 Our immediate task is to develop thebalance relations describing the interphase transport processes taking place is this system.During interphase mass transfer, concentration gradients will be set up across the

interface The concentration variations in the bulk phases x and y will be described by

differential equations; whereas at the interface /, we will have jump conditions or boundaryconditions Standart (1964) and Slattery (1981) give detailed discussions of these relationsfor the transport of mass, momentum, energy, and entropy It will not be possible to givehere the complete derivations and the reader is, therefore, referred to these sources Amasterly treatment of this subject is also available in the article by Truesdell and Toupin(1960), which must be compulsory reading for a serious researcher in transport phenomena.The equations of change for each fluid phase and the "jump" balance conditions thatmust be met at the interface are summarized in Table 1.5 There is an important restriction

on the equations in Table 1.5; the effect of chemical reactions in the bulk fluid phase hasbeen neglected For all of the applications considered in this book this neglect is justified.Our first major task is the description of the interfacial mass transfer process and,therefore, we shall examine further the equations for continuity of species / and theequation for conservation of total mass of mixture

/, surface of discontinuity

Figure 1.1 Pictorial representation of a two-phase

system showing a surface of discontinuity or face.