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Chemical Reactor Design,Optimization, and ScaleupSecond Edition

E. Bruce NaumanRensselaer Polytechnic Institute

A John Wiley & Sons, Inc., Publication

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Chemical Reactor Design,Optimization, and Scaleup

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ii


Chemical Reactor Design,Optimization, and ScaleupSecond Edition

E. Bruce NaumanRensselaer Polytechnic Institute

A John Wiley & Sons, Inc., Publication

iii


Copyright C© 2008 by John Wiley & Sons, Inc. All rights reserved

Published by John Wiley & Sons, Inc., Hoboken, New JerseyPublished simultaneously in Canada

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Library of Congress Cataloging-in-Publication Data:

Library of Congress Cataloging-in-Publication DataNauman, E. B.Chemical reactor design, optimization, and scaleup / E. Bruce Nauman. – 2nd ed.

p. cm.Includes index.ISBN 978-0-470-10525-2 (cloth)

1. Chemical reactors. I. Title.TP157.N393 2008660’.2832–dc22

2007051403

Printed in the United States of America

10 9 8 7 6 5 4 3 2 1

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http://www.copyright.comhttp://www.wiley.com/go/permissionhttp://www.wiley.com


Contents

Preface to the Second Edition xiiiSymbols xv

1 Elementary Reactions in Ideal Reactors 1

1.1 Material Balances 11.1.1 Measures of Composition 41.1.2 Measures of Reaction Rate 5

1.2 Elementary Reactions 51.2.1 Kinetic Theory of Gases 61.2.2 Rate of Formation 61.2.3 First-Order Reactions 81.2.4 Second-Order Reactions with One Reactant 81.2.5 Second-Order Reactions with Two Reactants 91.2.6 Third-Order Reactions 9

1.3 Reaction Order and Mechanism 91.4 Ideal, Isothermal Reactors 12

1.4.1 Ideal Batch Reactors 121.4.2 Reactor Performance Measures 171.4.3 Piston Flow Reactors 191.4.4 Continuous Flow Stirred Tanks 24

1.5 Mixing Times and Scaleup 261.6 Dimensionless Variables and Numbers 311.7 Batch Versus Flow and Tank Versus Tube 33Suggested Further Readings 36Problems 37

2 Multiple Reactions in Batch Reactors 41

2.1 Multiple and Nonelementary Reactions 412.1.1 Reaction Mechanisms 422.1.2 Byproducts 43

2.2 Component Reaction Rates for Multiple Reactions 432.3 Multiple Reactions in Batch Reactors 442.4 Numerical Solutions to Sets of First-Order ODEs 462.5 Analytically Tractable Examples 52

2.5.1 The nth-Order Reaction 522.5.2 Consecutive First-Order Reactions, A→B→C→ · · · 53

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vi Contents

2.5.3 Quasi-Steady Hypothesis 562.5.4 Autocatalytic Reactions 62

2.6 Variable-Volume Batch Reactors 652.6.1 Systems with Constant Mass 652.6.2 Fed-Batch Reactors 71

2.7 Scaleup of Batch Reactions 732.8 Stoichiometry and Reaction Coordinates 74

2.8.1 Matrix Formulation of Reaction Rates 742.8.2 Stoichiometry of Single Reactions 762.8.3 Stoichiometry of Multiple Reactions 77

Suggested Further Readings 78Problems 79Appendix 2.1 Numerical Solution of Ordinary DifferentialEquations 84

3 Isothermal Piston Flow Reactors 89

3.1 Piston Flow with Constant Mass Flow 903.1.1 Gas Phase Reactions 943.1.2 Liquid Phase Reactions 104

3.2 Scaleup Relationships for Tubular Reactors 1073.2.1 Scaling Factors 1073.2.2 Scaling Factors for Tubular Reactors 112

3.3 Scaleup Strategies for Tubular Reactors 1133.3.1 Scaling in Parallel and Partial Parallel 1133.3.2 Scaling in Series for Constant-Density Fluids 1143.3.3 Scaling in Series for Gas Flows 1163.3.4 Scaling with Geometric Similarity 1173.3.5 Scaling with Constant Pressure Drop 119

3.4 Scaling Down 1203.5 Transpired-Wall Reactors 122Suggested Further Readings 124Problems 124

4 Stirred Tanks and Reactor Combinations 129

4.1 Continuous Flow Stirred Tank Reactors 1294.2 Method of False Transients 1314.3 CSTRs with Variable Density 135

4.3.1 Liquid Phase CSTRs 1364.3.2 Computational Scheme for Variable-Density CSTRs 1374.3.3 Gas Phase CSTRs 138

4.4 Scaling Factors for Liquid Phase Stirred Tanks 1434.5 Combinations of Reactors 145

4.5.1 Series and Parallel Connections 1454.5.2 Tanks in Series 148


Contents vii

4.5.3 Recycle Loops 1504.5.4 Maximum Production Rate 153

4.6 Imperfect Mixing 154Suggested Further Readings 154Problems 155Appendix 4.1 Solution of Nonlinear Algebraic Equations 158

5 Thermal Effects and Energy Balances 163

5.1 Temperature Dependence of Reaction Rates 1635.1.1 Arrhenius Temperature Dependence 1635.1.2 Optimal Temperatures for Isothermal Reactors 166

5.2 Energy Balance 1705.2.1 Nonisothermal Batch Reactors 1725.2.2 Nonisothermal Piston Flow 1755.2.3 Heat Balances for CSTRs 178

5.3 Scaleup of Nonisothermal Reactors 1855.3.1 Avoiding Scaleup Problems 1855.3.2 Heat Transfer to Jacketed Stirred Tanks 1875.3.3 Scaling Up Stirred Tanks with Boiling 1905.3.4 Scaling Up Tubular Reactors 191

Suggested Further Readings 194Problems 195

6 Design and Optimization Studies 199

6.1 Consecutive Reaction Sequence 1996.2 Competitive Reaction Sequence 216Suggested Further Readings 218Problems 218Appendix 6.1 Numerical Optimization Techniques 220

7 Fitting Rate Data and Using Thermodynamics 225

7.1 Fitting Data to Models 2257.1.1 Suggested Forms for Kinetic Models 2267.1.2 Fitting CSTR Data 2287.1.3 Fitting Batch and PFR Data 2337.1.4 Design of Experiments and Model Discrimination 2387.1.5 Material Balance Closure 2397.1.6 Confounded Reactors 241

7.2 Thermodynamics of Chemical Reactions 2447.2.1 Terms in the Energy Balance 2447.2.2 Reaction Equilibria 252

Suggested Further Readings 269Problems 269Appendix 7.1 Linear Regression Analysis 274


viii Contents

8 Real Tubular Reactors in Laminar Flow 279

8.1 Flow in Tubes with Negligible Diffusion 2808.1.1 Criterion for Neglecting Radial Diffusion 2818.1.2 Mixing-Cup Averages 2828.1.3 Trapezoidal Rule 2848.1.4 Preview of Residence Time Theory 287

8.2 Tube Flows with Diffusion 2888.2.1 Convective Diffusion of Mass 2888.2.2 Convective Diffusion of Heat 2908.2.3 Use of Dimensionless Variables 2908.2.4 Criterion for Neglecting Axial Diffusion 291

8.3 Method of Lines 2928.3.1 Governing Equations for Cylindrical Coordinates 2928.3.2 Solution by Euler’s Method 2948.3.3 Accuracy and Stability 2958.3.4 Example Solutions 296

8.4 Effects of Variable Viscosity 3018.4.1 Governing Equations for Axial Velocity 3028.4.2 Calculation of Axial Velocities 3038.4.3 Calculation of Radial Velocities 304

8.5 Comprehensive Models 3078.6 Performance Optimization 307

8.6.1 Optimal Wall Temperatures 3088.6.2 Static Mixers 3088.6.3 Small Effective Diameters 310

8.7 Scaleup of Laminar Flow Reactors 3118.7.1 Isothermal Laminar Flow 3118.7.2 Nonisothermal Laminar Flow 312

Suggested Further Readings 312Problems 313Appendix 8.1 Convective Diffusion Equation 316Appendix 8.2 External Resistance to Heat Transfer 317Appendix 8.3 Finite-Difference Approximations 319

9 Packed Beds and Turbulent Tubes 323

9.1 Packed-Bed Reactors 3249.1.1 Incompressible Fluids 3249.1.2 Compressible Fluids in Packed Beds 333

9.2 Turbulence 3349.2.1 Turbulence Models 3359.2.2 Computational Fluid Dynamics 336

9.3 Axial Dispersion Model 3369.3.1 Danckwerts Boundary Conditions 3399.3.2 First-Order Reactions 340


Contents ix

9.3.3 Utility of the Axial Dispersion Model 3429.3.4 Nonisothermal Axial Dispersion 3449.3.5 Shooting Solutions to Two-Point Boundary Value

Problems 3449.3.6 Axial Dispersion with Variable Density 352

9.4 Scaleup and Modeling Considerations 352Suggested Further Readings 352Problems 353

10 Heterogeneous Catalysis 355

10.1 Overview of Transport and Reaction Steps 35710.2 Governing Equations for Transport and Reaction 35810.3 Intrinsic Kinetics 360

10.3.1 Intrinsic Rate Expressions from Equality of Rates 36110.3.2 Models Based on a Rate-Controlling Step 36310.3.3 Recommended Models 367

10.4 Effectiveness Factors 36810.4.1 Pore Diffusion 36810.4.2 Film Mass Transfer 37110.4.3 Nonisothermal Effectiveness 37210.4.4 Deactivation 374

10.5 Experimental Determination of Intrinsic Kinetics 37610.6 Unsteady Operation and Surface Inventories 380Suggested Further Readings 381Problems 382

11 Multiphase Reactors 385

11.1 Gas–Liquid and Liquid–Liquid Reactors 38511.1.1 Two-Phase Stirred Tank Reactors 38611.1.2 Measurement of Mass Transfer Coefficients 40111.1.3 Fluid–Fluid Contacting in Piston Flow 40411.1.4 Other Mixing Combinations 41011.1.5 Prediction of Mass Transfer Coefficients 412

11.2 Three-Phase Reactors 41511.3 Moving-Solids Reactors 417

11.3.1 Bubbling Fluidization 41911.3.2 Fast Fluidization 42011.3.3 Spouted Beds 42011.3.4 Liquid-Fluidized Beds 421

11.4 Noncatalytic Fluid–Solid Reactions 42111.5 Scaleup of Multiphase Reactors 427

11.5.1 Gas–Liquid Reactors 42711.5.2 Gas-Moving Solids Reactors 429


x Contents


12 Biochemical Reaction Engineering 433

12.1 Enzyme Catalysis 43412.1.1 Michaelis–Menten Kinetics 43412.1.2 Inhibition, Activation, and Deactivation 43812.1.3 Immobilized Enzymes 43912.1.4 Reactor Design for Enzyme Catalysis 440

12.2 Cell Culture 44412.2.1 Growth Dynamics 44612.2.2 Reactors for Freely Suspended Cells 45012.2.3 Immobilized Cells 45712.2.4 Tissue Culture 458

12.3 Combinatorial Chemistry 458Suggested Further Readings 459Problems 459

13 Polymer Reaction Engineering 461

13.1 Polymerization Reactions 46113.1.1 Step Growth Polymerizations 46213.1.2 Chain Growth Polymerizations 466

13.2 Molecular Weight Distributions 46813.2.1 Distribution Functions and Moments 46913.2.2 Addition Rules for Molecular Weight 47013.2.3 Molecular Weight Measurements 470

13.3 Kinetics of Condensation Polymerizations 47113.3.1 Conversion 47113.3.2 Number- and Weight-Average Chain Lengths 47213.3.3 Molecular Weight Distribution Functions 473

13.4 Kinetics of Addition Polymerizations 47813.4.1 Living Polymers 47913.4.2 Free-Radical Polymerizations 48113.4.3 Transition Metal Catalysis 48613.4.4 Vinyl Copolymerizations 486

13.5 Polymerization Reactors 49013.5.1 Stirred Tanks with a Continuous Polymer Phase 49213.5.2 Tubular Reactors with a Continuous Polymer Phase 49513.5.3 Suspending-Phase Polymerizations 507

13.6 Scaleup Considerations 50913.6.1 Binary Polycondensations 50913.6.2 Self-Condensing Polycondensations 50913.6.3 Living Addition Polymerizations 51013.6.4 Vinyl Addition Polymerizations 510


Contents xi


14 Unsteady Reactors 513

14.1 Unsteady Stirred Tanks 51314.1.1 Transients in Isothermal CSTRs 51514.1.2 Nonisothermal Stirred Tank Reactors 523

14.2 Unsteady Piston Flow 52614.3 Unsteady Convective Diffusion 529Suggested Further Readings 530Problems 530

15 Residence Time Distributions 535

15.1 Residence Time Theory 53515.1.1 Inert Tracer Experiments 53615.1.2 Means and Moments 539

15.2 Residence Time Models 54015.2.1 Ideal Reactors and Reactor Combinations 54015.2.2 Hydrodynamic Models 552

15.3 Reaction Yields 55715.3.1 First-Order Reactions 55715.3.2 Other Reactions 560

15.4 Extensions of Residence Time Theory 56915.4.1 Unsteady Flow Systems 57015.4.2 Contact Times 57015.4.3 Thermal Times 571

15.5 Scaleup Considerations 571Suggested Further Readings 572Problems 572

16 Reactor Design at Meso-, Micro-, and Nanoscales 575

16.1 Mesoscale Reactors 57716.1.1 Flow in Rectangular Geometries 57816.1.2 False Transients Applied to PDEs 58016.1.3 Jet Impingement Mixers 584

16.2 Microscale Reactors 58416.2.1 Mixing Times 58516.2.2 Radial or Cross-Channel Diffusion 58616.2.3 False Transients Versus Method of Lines 58716.2.4 Axial Diffusion in Microscale Ducts 58716.2.5 Second-Order Reactions with Unmixed Feed 59116.2.6 Microelectronics 59416.2.7 Chemical Vapor Deposition 595


xii Contents

16.3 Nanoscale Reactors 59616.3.1 Self-Assembly 59716.3.2 Molecular Dynamics 598

16.4 Scaling, Up or Down 599Suggested Further Readings 599Problems 599

References 601Index 603


Preface to the Second Edition

When I told a friend of mine who is not a chemical engineer that I was writinga new edition of my book, she said that I should include a murder mystery, as thatwould make the book more enjoyable. Now that was a challenge. How can a bookcalled Chemical Reactor Design, Optimization, and Scaleup, second edition, includea murder mystery? Well, it doesn’t, but it does have an evil assistant professor and abeautiful princess who is also an assistant professor. Their rather sophomoric adven-tures begin at Problem 1.13 and wander through Chapter 6.

This book can be considered a third edition since there was an earlier book, Chem-ical Reactor Design, John Wiley & Sons, 1987, that was followed by the first editionbearing the current title. The new title reflected an emphasis on optimization andparticularly on scaleup, a topic rarely covered in detail in undergraduate or graduateeducation but of paramount importance to many practicing engineers. The treatmentof biochemical and polymer reaction engineering is also more extensive than normal.There is a completely new chapter on meso-, micro-, and nanoreactors that includessuch topics as axial diffusion in microreactors and self-assembly of nanostructures.

Practitioners are a major audience for the new book. Here, in one spot, youwill find a reasonably comprehensive treatment of reactor design, optimization, andscaleup. Spend a few minutes becoming comfortable with the notation (anyone both-ering to read a Preface obviously has the inclination), and you will find practicalanswers to many design problems.

The book is also used for undergraduate and graduate courses in chemical en-gineering. Some faults of the old book were eliminated. One fault was its level ofdifficulty. It was too hard for undergraduates at most U.S. universities. The new bookis better. Known rough spots have been smoothed. However, the new book remainsterse and somewhat more advanced in its level of treatment than is the current U.S.standard. Its goal is less to train students in the qualitative understanding of existingsolutions than to prepare them for the solution of new problems. The reader shouldbe prepared to work out the details of some examples rather than expect a completesolution.

There is a continuing emphasis on numerical solutions. Numerical solutions areneeded for most practical problems in chemical reactor design, but sophisticatednumerical techniques are rarely necessary given the speed of modern computers.Euler’s method is routinely used to integrate sets of ordinary differential equations(ODEs). Random searches are used for optimization and least-squares analyses. Theseare appallingly inefficient but marvelously robust and easy to implement. The methodof lines is used for solving the partial differential equations (PDEs) that govern realtubular reactors and packed beds. This technique is adequate for most problems in

xiii


xiv Preface to the Second Edition

reactor design, but the method of false transients is now introduced as well. The goal isto make the techniques understandable and easily accessible and to allow continuedfocus on the chemistry and physics of the problem. Computational elegance andefficiency are gladly sacrificed for simplicity.

Too many engineers are completely in the dark when faced with variable physicalproperties and tend to assume them away without full knowledge of whether the effectsare important. They are often unimportant, but a real design problem, as opposed to anundergraduate exercise or preliminary process synthesis, deserves careful assemblyof data and a rigorous solution. Thus the book gives simple but general techniquesfor dealing with varying physical properties in reactors of all types.

No CD ROM is supplied with the book. Many of the numerical problems canbe solved with canned ODE and PDE solvers, but most of the solutions are quitesimple to code. Creative engineers must occasionally write their own code to solveengineering problems. Due to their varied nature, the solutions require use of a general-purpose language rather than a specific program. Computational examples in thebook are illustrated using Basic. This choice was made because Basic is indeed basicenough that it can be sight-read by anyone already familiar with another general-purpose language and because the ubiquitous spreadsheet, Excel, uses Basic macros.Excel provides input/output, plotting, and formatting routines as part of its structureso that coding efforts can be concentrated on the actual calculations. This makesit particularly well suited for students who have not yet become comfortable withanother language. Those who prefer another language such as C or Fortran or amathematical programming system such as Mathematica, Maple, Mathcad, or Matlabshould be able to translate quite easily

I continue with a few eccentricities in notation, using a, b, c, . . . to denote molarconcentrations of components A, B, C, . . . . Equations are numbered when the resultsare referenced or important enough to deserve some emphasis. The problems at theback of each chapter are generally arranged to follow the flow of the text rather thanlevel of difficulty. I have tried to avoid acronyms and other abbreviations unless theusage is very common and there is a true economy of syllables. The abbreviationsthat did slip through include CSTR, PFR, ODE, PDE, MWD, PD, RTD, and CPU.

Troy, New York E. BRUCE NAUMANMay, 2008


Symbols

BASIC LANGUAGE CODES

Program segments and occasional variables within the text are set in a fixed-widthfont to indicate that they represent computer code.

SI UNITS

Some reaction rates and concentrations for biochemical reactions and polymerizationsare normally in mass units rather than molar units.

Symbol Definition SI Units Where Used∗

A Component A — 1.9A A-type end group in condensation

polymerization— Section 13.1

A Amount of injected tracer kg Example 15.1[A] Concentration of component A mol m−3 1.8A, B, C Various constants Varies VariousAb Cross-sectional area associated with

bubble phasem2 11.46

Ac Cross-sectional area of tubularreactor

m2 Table 1.1

Ae Cross-sectional area of emulsionphase

m2 11.45

Aext External surface area m2 5.13A′ext External surface per unit length of

reactorm 5.22

Ag Cross-sectional area of gas phase m2 11.28Ai Interfacial area per unit volume of

reactorm−1 11.2

A′i Interfacial area per unit height ofreactor

m 11.27

Ainlet Cross-sectional area at reactor inlet m2 Problem 3.6Al Cross-sectional area of liquid phase m2 11.27

* Refers to equation number, except as noted.

xv


xvi Symbols

As Cross-sectional area of solid phase intrickle bed

m2 11.44

As External surface area of catalyst perunit volume of gas phase

m−1 10.2

[AS] Surface concentration of A inadsorbed state

mol m−2 10.5

Av Avogadro’s number Dimensionless 1.11

a Concentration of component A mol m−3 1.6a Vector of component concentrations

(N× 1)mol m−3 2.44

a(0−) Concentration just before inlet toclosed system

mol m−3 Example 9.2

a(0+) Concentration just after inlet to closedsystem


a(L−) Concentration just before outlet ofclosed system


a(L+) Concentration just after outlet ofclosed system


a(t, z) Concentration in unsteady tubularreactor

mol m−3 14.14

a′ Auxiliary variable, da/dz, used toconvert second-order ODEs to setof first-order ODEs


a* Dimensionless concentration Dimensionless 1.64

a*l Liquid phase concentration atgas–liquid interface

mol m−3 11.1

a0 Initial concentration of component A mol m−3 1.24ab Concentration of component A in

bubble phasemol m−3 11.46

abatch(t) Concentration in batch reactor attime t

mol m−3 8.16

ac Catalyst surface area per mass ofcatalyst

m2 kg−1 10.38

ae Gas phase concentration in emulsionphase

mol m−3 11.45

aequil Concentration of component A atequilibrium

mol m−3 Problem 1.15

afull Concentration when reactor becomesfull during startup


ag Concentration of component A in gasphase

mol m−3 11.2

a*g Gas phase concentration atgas–liquid interface

mol m−3 11.1

ain Inlet concentration of component A mol m−3 1.6


Symbols xvii

a j Concentration on j th tray of trayreactor


al Concentration of component A inliquid phase

mol m−3 11.2

al(l) Concentration at position lwithin pore mol m−3 10.3amix Mixing-cup average concentration mol m−3 4.19, 8.5aout Outlet concentration of component A mol m−3 1.6

Example 11.13as Concentration at surface of solid, e.g.,

solid catalystmol m−3 10.2

atrans Concentration of transpiredcomponent

mol m−3 3.48

a(t, z) Concentration in unsteady PFR mol m−3 14.4

B Component B —B B-type end group in condensation

polymerization— Section 13.1

[B] Concentration of component B mol m−3 1.9[BS] Surface concentration of B in

adsorbed statemol m−2 10.5

b Concentration of component B mol m−3 1.9b0 Initial concentration of component B mol m−3 1.32bl Liquid phase concentration of

component Bmol m−3 Example 11.6

C Component C — 1.20C Constant in various equations VariesC Scaling exponent for equipment cost Dimensionless Problem 4.19C Concentration of inert tracer mol m−3 15.1C(t, z) Concentration of inert tracer in

unsteady tubular reactormol m−3 Example 15.4

C0 Initial value for tracer concentration mol m−3 15.1CA Capacity of ion exchange resin for

component Amol m−3 11.49

CAB Collision rate between A and Bmolecules

m−3 s−1 1.11

Ch Constant in heat transfer correlation Dimensionless 5.33CI Specific heat of impeller J kg−1K−1 Example 14.9Cout(t) Outlet concentration of inert tracer mol m−3 15.1CP Heat capacity in mass units J kg−1K−1 5.15CP Heat capacity in molar units

throughout Section 7.2J mol−1K−1 7.43

CSTR Continuous flow stirred tank reactor Dimensionless Section 1.4

c Concentration of component C mol m−3 1.20


xviii Symbols

cj Gas concentration above j th tray intray column


cJ Gas concentration above last tray intray column


cl Concentration of polymer chainshaving length l

chains m−3 13.7

cpolymer Summed concentration of all polymerchains

chains m−3 13.7

D Component D —D Axial dispersion coefficient m2 s−1 11.9DA Effective diffusivity in membrane m2 s−1 9.18DA Diffusion coefficient for component A m

2 s−1 8.3De Axial dispersion coefficient in

emulsion phasem2 s−1 11.45

Deff Effective diffusivity m2 s−1 10.27

Dg Axial dispersion coefficient for gasphase

m2 s−1 11.34

DI Impeller diameter m 1.60Din Axial dispersion coefficient in

entrance region of open reactorm2 s−1 Figure 9.2

DK Knudsen diffusivity m2 s−1 10.26

Dl Axial dispersion coefficient for liquidphase

m2 s−1 11.33

Dout Axial dispersion coefficient in exitregion of open reactor

m2 s−1 Figure 9.2

DP Diffusion coefficient of product P m2 s−1 10.7

Dr Radial dispersion coefficient in PDEmodels

m2 s−1 9.1, 16.11

Dz Axial dispersion coefficient in PDEmodel

m2 s−1 16.11

d Concentration of component D mol m−3 2.1d j Liquid concentration on j th tray of

tray columnmol m−3 Example 11.7

dp Diameter of particle m 3.21dpore Diameter of pore m Section 10.4.1dt Tube diameter m Sections 3.2, 9.6dtank Tank diameter m Example 1.7dw Incremental mass of polymer being

formedkg 13.48

E Component E — 2.1E Axial dispersion coefficient for heat m2 s−1 9.28E Enhancement factor Dimensionless 11.41


Symbols xix

E0 Concentration of active sites sites m−2 12.1Er Radial dispersion coefficient for heat

in packed bedm2 s−1 9.13

e Concentration of component E mol m−3 Example 2.2e Epoxy concentration mol m−3 Example 14.9

F Arbitrary function Varies Appendix 4.1F Constant value for Fj m3 s−1 Example 11.8F(t) Cumulative distribution of residence

timesDimensionless 15.4

F(r ) Cumulative distribution functionexpressed in terms of tube radiusfor monotonic velocity profile

Dimensionless 15.29

Fa Fanning friction factor Dimensionless 3.16Fj Volumetric flow of gas from j th tray m3 s−1 Example 11.7

f Arbitrary function Varies Appendix 6.1f Initiator efficiency factor Dimensionless 13.39f (l) Number fraction of polymer chains

having length lDimensionless 13.8

f (t) Differential distribution function forresidence times

s−1 15.6

f (t) Differential distribution function forexposure times

s−1 Section 11.1.5

f− Value of function at backward point Varies Appendix 8.3f+ Value of function at forward point Varies Appendix 8.3f0 Value of function at central point Varies Appendix 8.3f ◦A Fugacity of pure component A Pa 7.29f̂ A Fugacity of component A in mixture Pa 7.29fc(tc) Differential distribution of contact

timess−1 15.52

fdead(l) Number fraction of terminatedpolymer chains having length l

Dimensionless Section 13.4.2

fT (tT ) Differential distribution function forthermal times

s−1 15.54

fin, fout Material balance adjustment factors Dimensionless 7.15fR Collision efficiency factor Dimensionless 1.10

G Arbitrary function Varies Appendix 4.1G1 Integrals used in variable-viscosity

calculationsm3 kg s−1 8.45

G2 Integrals used in variable-viscositycalculations

m5 kg s−1 8.45

G1, G2 Growth limitation factors forsubstrates 1 and 2

Dimensionless 12.10


xx Symbols

G P Growth limitation factor for product Dimensionless 12.13GS Growth limitation factor for substrate Dimensionless 12.13Gz Graetz number Dimensionless 5.36

g Grass supply kg m−2 Section 2.5.4g Acceleration due to gravity m s−2 Section 4.4g(l) Weight fraction of polymer chains

having length lDimensionless 13.11

g(t) Impulse response function for opensystem

s−1 15.41

g(t)rescaled Impulse response function for opensystem after rescaling so that meanis t̄

s−1 15.41

H Enthalpy in mass units J kg−1 5.1H Enthalpy in molar units throughout

Section 7.2J mol−1 7.42

H Half-height of rectangular duct m 16.1HA, HB , HI Component enthaplies J mol−1 7.20

h Heat transfer coefficient on jacketside

J m−2 s−1 K−1 5.34

h Hydrogen ion concentration mol m−3 Example 14.9hi Interfacial heat transfer coefficient J m−2 s−1 K−1 11.19hr Coefficient for heat transfer to wall

of packed bedJ m−2 s−1 K−1 9.4

I Inert component I — 3.13I System inventory kg 1.2I Number of radial increments Dimensionless 8.12I–IV Reactions I–IV — Section 2.2I0 Initiator concentration at t = 0 mol m−3 13.31[IXn] Concentration of growing polymer

chains of length n that end with Xgroup

chains m−3 Section 13.4.4

[IYn] Concentration of growing polymerchains of length n that end with Ygroup

chains m−3 Section 13.4.4

i Concentration of inerts mol m−3 3.12i Index variable in radial direction Dimensionless 8.12i Concentration of adsorbable inerts

in gas phasemol m−3 10.14

J Number of experimental data Dimensionless 5.2J Number of axial increments Dimensionless Example 8.3J Number of trays Dimensionless Example 11.8


Symbols xxi

j Index variable for axial direction Dimensionless Section 8.3.2j Index variable for data Dimensionless 5.2

K0, K1,K2, K3

Factors for thermodynamicequilibrium constant

Dimensionless 7.35

K1 Equilibrium constant mol Example 14.9K2 Various constants Varies 12.3Ka Kinetic equilibrium constant for

adsorptionmol–1 m3 Example 10.4

Kd Kinetic equilibrium constant fordesorption


Kg Mass transfer coefficient based onoverall gas phase driving force

m s−1 11.2

K H Henry’s law constant Varies 11.1K iH Liguid–gas equilibrium constant at

interfacem s−1 11.4

Kkinetic Kinetic equilibrium constant Varies 7.28Kl Mass transfer coefficient based on

overall liquid phase driving forcem s−1 11.3

Km Mass transfer coefficient betweenemulsion and bubble phases ingas-fluidized bed

m s−1 11.45

KM Michaelis constant mol m−3 12.2K R Kinetic equilibrium constant for

surface reactionDimensionless Example 10.3

Kthermo Thermodynamic equilibriumconstant

Dimensionless 7.29

k Reaction rate constant Varies 1.8k ′ Pseudo-first-order rate constant s−1 Section 1.3k ′′ Linear burn rate m s−1 11.51k* Rate constant Dimensionless 1.64k0 Preexponential rate constant Varies 5.1ka Adsorption rate constant site−1 s−1 10.4k+a Forward rate constant for reversible

adsorption stepsite−1 s−1 Example 10.2

k−a Reverse rate constant for reversibleadsorption step

mol m−1site−1 s−1 Example 10.2

kA, kB ,. . . .

Denominator rate constant forcomponent A

mol−1m3 7.5, 10.20

kA, kB,kC

Rate constants for consecutivereactions

s−1 2.20

kAB Second-order denominator constant mol−2 m6 12.5kc Rate constant for termination by

combinationmol−1m3 s−1 13.39

kd Rate constant for cell death s−1 12.17


xxii Symbols

kd Rate constant for termination bydisproportionation

mol−1m3 s−1 13.39

kd Desorption rate constant mol m−1site−1 s−1 10.6k+d Forward rate constant for

reversible desorption stepmol m−1site−1 s−1 Example 10.2

k−d Reverse rate constant forreversible desorption step

m2 site−1 s−1 Example 10.2

kD Rate constant for catalystdeactivation

s−1 10.35

k f Rate constant for forward reaction Varies 1.15kg Mass transfer coefficient based on

gas phase driving forcem s−1 11.6

ki Rate constant for chemicalinitiation

s−1 13.39

kI Denominator rate constantfor inerts I

mol−1m3 10.14

kI, kII Rate constants for reactions I and II mol−1 m3 s−1 2.1kl Mass transfer coefficient based on

liquid phase driving forcem s−1 11.5

kp Propagation rate constant mol−1m3 s−1 13.31kr Rate constant for reverse reaction 1.15kR Rate constant for surface reaction site−1m2 s−1 10.5k+R Forward rate constant for

reversible surface reactions−1 Example 10.2

k−R Reverse rate constant forreversible surface reaction

s−1 Example 10.2

ks Mass transfer coefficient m s−1 10.2kS Rate constant for catalyst

deactivationsite−1m2 s−1 Section 10.4.4

kSI Denominator constant fornoncompetitive inhibition

mol−2 m6 12.6

kX X Rate constant for monomer Xreacting with polymer chainending with X unit

mol−1m3 s−1 Section 13.4.4

kXY Rate constant for monomer Yreacting with polymer chainending with X unit


kY X Rate constant for monomer Xreacting with polymer chainending with Y unit


kY Y Rate constant for monomer Yreacting with polymer chainending with Y unit



Symbols xxiii

L Length of tubular reactor m 1.37L Length of pore m Section 10.4.1L− Location just before reactor outlet m Example 9.3L+ Location just after reactor outlet m Example 9.3l Lynx population lynx m−2 Section 2.5.4l Position within pore m 10.3l Chain length of polymer Dimensionless 13.1l, m, p, q Chain lengths for termination by

combinationDimensionless Section 13.4.2

l̄N Number-average chain length Dimensionless 13.10l̄W Weight-average chain length Dimensionless 13.12

M Monomer Dimensionless 13.1M Any molecule that serves as energy

sourceDimensionless Problem 7.7

M Middle group in condensationpolymerization

Dimensionless Section 13.1

M Number of simultaneous reactions Dimensionless 2.9M Monomer concentration mol m−3 13.32

M0 Monomer charged to system priorto initiation

13.31

MA Molecular weight of diffusingspecies

g mol−1 10.26

M0 Maintenance coefficient for oxygen Table 12.1MS Maintenance coefficient, mass of

substrate per dry cell massper time

s−1 12.15

Mw molecular weight g mol−1 Example 2.9

min Function to select minimum 12.11m Reaction order exponent Dimensionless 1.21m Exponent in Arrhenius equation Dimensionless 5.1m Exponent on product limitation

factorDimensionless 12.13

m Chain length of polymer Dimensionless 13.2m, n, r, s Parameters to be determined in

regression analysisVaries 7.48

m A Mass of A molecule Da 1.11m I Mass of impeller kg Example 14.9

N Vector of component moles(N × 1)

mol 2.47


xxiv Symbols

N Middle group in condensationpolymerization

Dimensionless Section 13.1

N Number of chemical components Dimensionless Section 2.3N Number of tanks in series Dimensionless 4.16bN0 Moles initially present mol Example 7.15NA Moles of component A mol 2.45N.A Molar flow rate of component A mol s−1 3.3

NI Rotational velocity of impeller rev s−1 1.59Ntubes Number of tubes in scaleup Dimensionless Section 3.3Nu Nusselt number Dimensionless 5.33Nzones Number of zones used for

temperature optimizationDimensionless Example 6.5

n Reaction order exponent Dimensionless 1.21n Index variable for number of tanks Dimensionless 4.16bn Zone number dimensionless Example 6.5n Index for moments of distribution Dimensionless 13.9, 15.11

O Operator indicating order ofmagnitude

Dimensionless Example 2.4

ODE Ordinary differential equation Dimensionless

P Product or polymer DimensionlessP Pressure Pa 3.12P• Concentration of growing chains

summed over all lengthschains 13.39

P0 Standard pressure Pa 7.30PDE Partial differential equation Dimensionless Section 8.2Pe Peclet number for PDE model,

ūsdp/DrDimensionless Section 9.1.1

Pe Peclet number for axial dispersionmodel, ūL/D

Dimensionless 9.19

PFR Piston flow reactor Dimensionless Section 1.4Pg Agitator power while gas is being

spargedkw Example 11.18

Pl Polymer of chain length l Dimensionless 13.1Po Power number Dimensionless 1.60Power Agitator power kw 1.60Pr Prandlt number Dimensionless Section 5.3.7PR Probability that molecule will react Dimensionless Section 15.3.1[PS] Concentration of P in adsorbed state mol m−2 10.6

p Concentration of product P mol m−3

p Parameter in analytical solution Dimensionless 9.23p1, p2 Optimization parameters Varies Appendix 6.1pl Concentration of product P at

location l within poremol m−3 10.7


Symbols xxv

pmax Growth-limiting value for productconcentration

mol m−3 12.13

pold Old or current value for optimizationparameter

Varies 6.7

ps Concentration of product P at externalsurface of catalyst

mol m−3 10.8

ptrial Trial value for optimization parameter Varies 6.7

Q Component Q DimensionlessQ Volumetric flow rate m3 s−1 1.3Q0 Volumetric flow at initial steady state m3 s−1 14.9Qfull Volumetric flow rate at steady state m3 s−1 Example 14.4Qg Gas volumetric flow rate m3 s−1 11.12Qin Input volumetric flow rate m3 s−1 1.3Ql Liquid volumetric flow rate m3 s−1 11.11Ql Gas volumetric flow rate m3 s−1 11.11Qmass Mass flow rate m3 s−1 1.2Qout Discharge volumetric flow rate m3 s−1 1.6

q Transpiration volumetric flow per unitlength

3.48

q Recycle rate m3 s−1 Section 4.5.3q Volumetric flow rate into side tank of

side capacity modelm3 s−1 Example 15.7

qgenerated Rate of heat generation 5.31qremoved Rate of heat removal 5.32

R Component R DimensionlessR Radius of tubular reactor m 8.55, Exp. 3.1R Ratio of monomer to polymer density Dimensionless 4.7R Vector of reaction rates (M × 1) mol m−3 s−1 2.42R̄ Average radius of surviving particles m 11.55R ′ Multicomponent, vector form of R′A mol m−3 s−1 3.9R0 Initial particle radius m 11.52RA Rate of formation of component A mol m

−3 s−1 1.6(RA)g Rate of formation of component A in

gas phasemol m−3 s−1 11.12

(RA)l Rate of formation of component A inliquid phase

mol m−3 s−1 11.11

Rdata Experimental values for reaction ratefrom CSTR data

mol m−3 s−1 Section 7.1.1

Re Reynolds number Dimensionless 3.16(Re)impeller Reynolds number based on impeller

diameterDimensionless 4.11

(Re)p Reynolds number based on particlediameter

Dimensionless 3.21


xxvi Symbols

Rg Gas constant J mol−1 K−1 1.11Rh Radius of central hole in cylindrical

catalyst particlem Problem 10.14

RI Rate of reaction I, I = 1, . . . , M mol m−3 s−1 2.8Rmax Maximum growth rate kg m

−3 s−1 12.2Rmodel Reaction rate as predicted by model mol m

−3 s−1 Section 7.1.1RND Random number Dimensionless 6.7Rp Radius of a catalyst particle mol m−3 s−1 Problem 10.14RP Rate of formation of product P mol m

−3 s−1 10.7Rr Rate of reverse reaction mol m

−3 s−1 Example 7.13RS Reaction rate for solid mol m

−3 s−1 Example 11.16RS Reaction rate for substrate kg m

−3 s−1 12.15RX Rate of formation of dry cell mass kg m

−3 s−1 12.8

r Radial coordinate m 8.1r Rabbit population rabbits m−2 Section 2.5.4r Radius, r/R Dimensionless Table 8.1r1 Dummy variable of integration m 8.41, 13.5rA Radius of A molecule m 1.11rB Radius of B molecule m 1.11rp Radial coordinate for catalyst particle m 10.32rX Copolymer reactivity ratio Dimensionless 13.41rY Copolymer reactivity ratio Dimensionless 13.41

S Component S Dimensionless 1.13S Substrate in biological system Dimensionless Section 12.1.1[S] Concentration of vacant sites sites m−2 10.4S Scaling factor for scaleup with

constant t̄Dimensionless 1.58

S Concentration of tracer in side tankof side capacity model


S0 Total concentration of sites, bothoccupied and vacant

sites m−2 Example 10.1

S1, S2 Roots of quadratic equation 2.24SA Root mean residual error for

component AVaries 7.10

SAB Stoichiometric ratio a0/b0 or of Aend groups to B end groups atonset of reaction

Dimensionless 1.65, 13.3

Sc Schmidt number, μ/(ρDA) Dimensionless Section 9.1.1Sinventory Scaling factor for inventory Dimensionless 1.56SL Scaleup factor for tube length Dimensionless Section 3.2.1SR Scaleup factor for tube radius Dimensionless Section 3.2.1SS2 Sum-of-squares errors Varies 5.2SS2A, SS

2B, SS

2C Sum of squares for individual

componentsVaries 7.14


Symbols xxvii

SS2residual Sum of squares after data fit Varies 7.10Sthroughput Scaling factor for throughput Dimensionless 1.55Svolume Scaling factor for volume Dimensionless 1.57SX Scaling factor for property X Dimensionless Section 3.2.1

s Root mean residual error Varies 7.10s Substrate concentration kg m−3 s−1 12.1s Sulfate concentration mol m−3 Example 14.9s Laplace transform parameter s−1 Example 15.2s0 Initial substrate concentration Example 12.5

T Dimensionless temperature Dimensionless Table 8.1Tburn Time required to burn particle s Section 11.17Text External temperature K 5.13Tg Temperature in gas phase K Section 11.1.1Tl Temperature in liquid phase K Section 11.1.1Tn Temperature in nth zone K Example 6.5Tref Reference temperature for

enthalpy calculationsK 5.14

Ts Temperature at external surfaceof catalyst particle

K Section 10.4.3

Tset Temperature set point K Example 14.8

t Time s 1.2t Residence time associated with

streamline, L/Vz(r )s Section 8.1.4

t̄ Mean residence time s 1.40t1 Dummy variable of integration s 11.49t0 Time at end of induction phase s 12.9t1/2 Reaction half-life s 1.28tb Residence time for segregated

group of moleculess Section 15.3.2

tc Contact time in heterogeneousreactor

s 15.52

tempty Time when reactor becomes empty s 14.10tfirst First appearance time when W (t)

first goes below 1s Section 15.2.1

tfull Time to fill reactor s Example 14.3thold Holding time following fast fill s 14.6t̄loop Mean residence time for single

pass through loops 5.34

tmix Mixing time s Section 1.5t̄n Residence time in nth zone s Example 6.5ts Time constant in packed bed, L/ūs s 9.9tT Thermal time s 15.53

U Overall heat transfer coefficient J m−2 s−1 K−1 5.13


xxviii Symbols

U ′ Heat transfer group s−1 Example 7.8

ū Average axial velocity m s−1 1.34ub Gas velocity in bubble phase m s−1 11.46ue Gas velocity in emulsion phase in

fluidized bedm s−1 11.45

ub Gas velocity in bubble phase influidized bed

m s−1 11.46

ūg Average gas velocity in gas–liquidPFR

m s−1 11.28

ūl Average liquid velocity in gas–liquidPFR

m s−1 11.27

umin Minimum fluidization velocity m s−1 Section 11.3ūs Superficial velocity in packed bed m s−1 3.21(ūs)g Superficial gas velocity, Qg/Ac m s−1 Example 11.18

V Time-average velocity vector m s−1 9.16V Volume m3 1.3V 0 Velocity at centerline m s−1 Problem 8.2VA Molar volume of component A m3mol−1 7.32Vact Activation volume m3 Problem 5.4Vfull Full volume of reactor m3 Example 14.3Vg Volume of gas phase m3 Section 11.1.1Vl Volume of liquid phase m3 Section 11.1.1Vm Volume of main tank in side capacity

modelm3 Example 15.7

Vr Radial velocity m s−1 8.49Vr Dimensionless radial velocity

component, Vr/ūDimensionless 13.49

VS Volumetric consumption rate for solid 11.50VS Volume of side tank in side capacity

modelm3 Example 15.7

Vz Axial component of velocity m s−1 8.1Vz Dimensionless axial velocity, Vz/ū Dimensionless Table 8.1Vz(r ) Axial component of velocity in tube m s−1 8.1Vz(y) Axial velocity in slit flow m s−1 Example 16.3

v Velocity vector in turbulent flow m s−1 9.16

W Mass flow rate kg s−1 Example 6.1W (t) Washout function Dimensionless 15.2W (θ, t) Washout function for unsteady system Dimensionless 15.51W1, W2 Randomly selected values for

washout functionDimensionless Example 15.6

w1, w2 Weight of polymer aliquots kg 13.14

chemical reactor design,...

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