chemical reactor analysis and design

The Schrödinger Equation 48 Chapter 2: Kinetics of Heterogeneous Catalytic Reactions 2.4.1 The Kinetic Modeling of Commercial Catalytic Processes 87 2.4.2 Generation of the Network of E

Université Catholique de Louvain, Belgium

John Wiley & Sons, Inc

MARKETING MANAGER Christopher Ruel

EDITORIAL ASSISTANT Alexandra Spicehandler

EXECUTIVE MEDIA EDITOR Thomas Kulesa

PRODUCTION MANAGER Micheline Frederick

PRODUCTION EDITOR Amy Weintraub

This book was printed and bound by Hamilton Printing Company The cover was printed by Phoenix Color

This book is printed on acid free paper ∞

Copyright © 2011 John Wiley & Sons, Inc All rights reserved No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic,

mechanical, photocopying, recording, scanning or otherwise, except as permitted under Sections 107 or

108 of the 1976 United States Copyright Act, without either the prior written permission of the

Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc 222 Rosewood Drive, Danvers, MA 01923, website www.copyright.com Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc.,

111 River Street, Hoboken, NJ 07030-5774, (201)748-6011, fax (201)748-6008, website

http://www.wiley.com/go/permissions

“Evaluation copies are provided to qualified academics and professionals for review purposes only, for use in their courses during the next academic year These copies are licensed and may not be sold or transferred to a third party Upon completion of the review period, please return the evaluation copy to Wiley Return instructions and a free of charge return shipping label are available at

www.wiley.com/go/returnlabel Outside of the United States, please contact your local representative.”

Library of Congress Cataloging-in-Publication Data

Gilbert F Froment, Texas A&M University; K.B Bischoff†, University of Delaware; Juray De Wilde, Université Catholique de Louvain

This is the Third Edition of Chemical Reactor Analysis and Design The first

was published by Wiley in 1979 and the second, after a substantial revision, in

1990 When we undertook the third edition in 2008, eighteen years had elapsed since the second edition This is a significant period of time during which chemical reaction engineering has considerably evolved The tremendous growth

of computer power and the easy access to it has significantly contributed to a more comprehensive description of phenomena, operations and equipment, thus enabling the development and application of more fundamental and presumably more accurate models Modern chemical reaction engineering courses should reflect this evolution towards a more scientific approach We have been permanently aware of these trends during the elaboration of the present edition and have largely rewritten the complete text The more fundamental approach has not distracted us, however, from the emphasis on the real world of chemical reaction engineering, one of the main objectives and strengths of the first edition already, widely recognized all over the world

We have maintained the structure of the previous editions, dividing the content into two parts The first part deals with the kinetics of phenomena that are important in reaction engineering: reaction kinetics, both “homogeneous” —

in a single phase — and “heterogeneous,” involving a gas- and a liquid- or solid phase The mechanism of the reactions has been accounted for in greater detail than previously, in an effort to be more realistic, but also more reliable in their kinetic modeling e.g., in thermal cracking, polymerization, hydrocarbon processing and bio-processes The field of reaction kinetics has substantially progressed by the growing availability through commercial software of quantum chemical methods Students of chemical reaction engineering can no longer ignore their potential and they should be taught how to apply them meaningfully

to real processes Chapters 1, 2 and 3 attempt to do that In the heterogeneous reaction case, heat and mass transfer phenomena at the interface and inside the reaction phase have to be considered In modeling these the internal structure of the catalyst has been given more emphasis, starting from insight provided from well developed characterization tools and using advanced techniques like Monte Carlo simulation, Percolation theory and Effective Medium Approximation This approach is further applied in Chapter 4 on gas-solid reactions and Chapter 5 on catalyst deactivation The insertion of more realistic kinetics into structure models of the catalyst has also allowed accounting for the role of catalyst deactivation by coke formation in important commercial hydrocarbon conversion

retained its previous structure

Part II addresses the chemical reactor itself, inserting the kinetic aspects

of Part I into the modeling and simulation of the reactor operation Chapter 7 introduces the fundamental mass-, energy- and momentum balances The Chapters 8, 9, 10 and 11, dealing with the basic types, like the batch, semi-batch, continuous flow reactor with complete mixing and the tubular reactor, filled or not with solid catalyst, have been maintained, of course, and also their strong ties

to industrial processes Deviations of what was previously called “ideal “ models and behavior are dealt with along entirely new lines, made possible by the progress of CFD — computational fluid dynamics — also made available by commercial software This approach is introduced already in Chapter 11 on fixed bed reactors and consistently applied in Chapter 12, leading to a unified and structured approach of flow, residence time and conversion in the variety of reactors encountered in industrial practice This is another field that has not yet received sufficient attention in chemical engineering curricula Substantial progress and a growing number of applications can be expected in the coming years It is illustrated also in Chapter 13 on fluidized- and transport bed reactors, that enters into greater details than before on the catalytic cracking of heavy oil fractions and reports on simulations based upon computational fluid dynamics

A book like this has to show the path and prepare the future We should not look down, however, upon the correlations derived from experimentation and collected by the profession over the years, be they limited in their range of application There is no way that these could be refined or completely replaced yet by CFD application only Unfortunately, the computational effort involved in the use of CFD in combination with reaction and transport phenomena throughout the entire reactor is overwhelming and its routine-like application to real, practical cases not for the immediate future Chapter 14 on multiphase reactors is evidence for this and illustrates sound and proved engineering practice

Finally, we want to remember Ken Bischoff, who deceased in July 2007 and could not participate in this third edition

Gilbert F Froment Juray De Wilde

Texas A & M University Université Catholique de Louvain December 2009

G.F Froment

Gilbert F Froment received his Ph.D in Chemical Engineering from the University of Gent, Belgium, in 1957 He did post-doctoral work at the University of Darmstadt in Germany and the University of Wisconsin In 1968 he became a full professor of Chemical Engineering in Gent and launched the

“Laboratorium voor Petrochemische Techniek” that became world famous His scientific work centered on fixed bed reactor modeling, kinetic modeling, catalyst deactivation and thermal cracking for olefins production In 1998 he joined the Chemical Engineering Department of Texas A & M University as a Research Professor He has directed the work of 68 Ph.D students and published

350 scientific papers in international journals He presented more than 320 seminars in universities and at international symposia all over the world The

book Chemical Reactor Analysis and Design (with K.B Bischoff) is used

worldwide in graduate courses and industrial research groups and was translated into Chinese He has been on the editorial board of the major chemical engineering journals In his present position, at Texas A & M University, Dr Froment directs the research of a group of Ph.D students and post-docs on Chemical Reaction Engineering aspects of Hydrocarbon Processing in the Petroleum and Petrochemical Industry, more particularly on the kinetic modeling

of complex processes like hydrocracking and hydrotreatment, catalytic cracking, catalytic reforming, methanol-to-olefins, solid acid alkylation, thermal cracking, using single event kinetics, a concept that he launched in the eighties He received the prestigious R.H Wilhelm Award for Chemical Reaction Engineering from the A.I.Ch.E in 1978, the first Villermaux-Medal from the European Federation of Chemical Engineering in 1999 and the 3-yearly Amundson Award of ISCRE in 2007

G.F Froment is a Doctor Honoris Causa of the Technion, Haifa, Israel (1985), of the University of Nancy, France (2001) and an Honorary Professor of the Universidad Nacional de Salta (Argentina) He is a member of the Belgian Academy of Science (1984), the Belgian Academy of Overseas Science (1977), a Foreign Associate of the United States National Academy of Engineering (1999) and a member of the Texas Academy of Medicine, Science and Engineering (2003) He was a member of the Scientific Council of the French Petroleum Institute (1989-1997), of the Technological Council of Rhône-Poulenc (1988-1997) and has intensively consulted for the world’s major petroleum and (petro)chemical companies

Kenneth B Bischoff was the Unidel Professor of Biomedical and Chemical Engineering and past Chairman, Department of Chemical Engineering at the University of Delaware Previously he was Acting Director for the Center for Catalytic Science and Technology He was the Walter R Read Professor of Engineering and Director of the School of Chemical Engineering at Cornell University and had been on the faculties of the Universities of Maryland and Texas (Austin), as well as a Postdoctoral Fellow at the University of Gent, Belgium He had served as a consultant for Exxon Research and Engineering Company, General Foods Company, the National Institutes of Health, W R Grace company, Koppers Company, E I du Pont de Nemours & Co., Inc., and Westvaco Co., and was a registered professional engineer in the State of Texas His research interests were in the areas of chemical reaction engineering and applications to pharmacology and toxicology, resulting in

more than 100 journal articles and two textbooks: Process Analysis and

Simulation (with D.M Himmelblau) (1968); and Chemical Reactor Analysis and Design, (with G.F Froment) (1979) He was elected to the National Academy

of Engineering in 1988, and he received the 1972 Ebert Prize of the Academy of Pharmaceutical Sciences, the 1976 Professional Progress Award, the 1982 Institute Lecture Award, the 1982 Food, Pharmaceutical and Bioengineering Division Award, and the 1987 R H Wilhelm Award In

1987 he was named a Fellow of the American Institute of Chemical Engineers He was a Fellow of AAAS since 1980 Editorial boards on which

he had served include J Pharmacokinetics and Biopharmaceutics, from 1972 on; and ACS Advances in Chemistry Series, 1974 to 1981 In 1981 he became

an Associate Editor of Advances in Chemical Engineering,

Dr Bischoff passed away in 2007

J De Wilde

Juray De Wilde received his Ph.D in Chemical Engineering from the Ghent University, Belgium, in 2001 He did post-doctoral work at the Ghent University and was post-doc research associate at the Chemical Engineering Department of Princeton University, NJ In 2005 he became professor of Chemical Engineering

at the Université catholique de Louvain, Belgium, where he received his tenure

in 2008 Dr De Wilde published more than 30 papers in international journals and served as a member of scientific committees and as a consultant for

numerous companies, including Total Petrochemicals, Tribute Creations, Dow

etc His research interests and expertise include dynamic methods for catalytic kinetics, the modeling and simulation of gas-solid flows, and process

intensification, in particular for fluidized bed processes With A de Broqueville,

he developed the rotating fluidized bed in a static geometry and the rotating chimney technologies

Third edition

G.F Froment, K.B Bischoff, J De Wilde

Chapter 1: Elements of Reaction Kinetics

1.1.1 Rates of Disappearance of Reactants and of Formation 2

1.2.3 Typical Rate Equations for Simple Reactions 9

1.2.3.1 Reversible First-Order Reactions 9 1.2.3.2 Second-Order Reversible Reactions 10

1.3.3 Mixed Parallel-Consecutive Reactions 21

1.4.2 Rate Determining Step of a Sequence of Reactions 22

1.6.2 Free Radical Polymerization Kinetics 38

1.7.2 Quantum Mechanics The Schrödinger Equation 48

Chapter 2: Kinetics of Heterogeneous Catalytic Reactions

2.4.1 The Kinetic Modeling of Commercial Catalytic Processes 87 2.4.2 Generation of the Network of Elementary Steps 89

2.4.3.2 The Evans-Polanyi Relationship for the 94

Activation Energy

2.6 Model Discrimination and Parameter Estimation 104

2.6.1 The Differential Method of Kinetic Analysis 104 2.6.2 The Integral Method of Kinetic Analysis 110 2.6.3 Parameter Estimation and Statistical Testing of Models 112 and Parameters in Single Reactions

2.6.3.1 Models That Are Linear in the Parameters 112 2.6.3.2 Models That Are Nonlinear in the Parameters 117 2.6.4 Parameter Estimation and Statistical Testing of Models 119 and Parameters in Multiple Reactions

Example 2.6.4.A Benzothiophene Hydrogenolysis 123 2.6.5 Physicochemical Tests on the Parameters 126

2.7.1 Sequential Design for Optimal Discrimination between 127 Rival Models

Example 2.7.1.1.A Model Discrimination in the 130

Example 2.7.1.1.B Ethanol Dehydrogenation: 133

Sequential Discrimination using the Integral Method of Kinetic Analysis

2.7.2 Sequential Design for Optimal Parameter Estimation 138

Example 2.7.2.2.A Sequential Design for Optimal 139

Parameter Estimation in Benzo- thiophene Hydrogenolysis

Chapter 3: Transport Processes with Reactions Catalyzed

by Solids

PART ONE INTERFACIAL GRADIENT EFFECTS

3.1 Reaction of a Component of a Fluid at the Surface of a Solid 154

3.2.3 Multicomponent Diffusion in a Fluid 160 Example 3.2.3.A Use of a Mean Binary Diffusivity 162 3.3 Concentration or Partial Pressure and Temperature Differences 163

Between Bulk Fluid and Surface of a Catalyst Particle

Example 3.3.A Interfacial Gradients in Ethanol 165

Dehydrogenation Experiments

PART TWO INTRAPARTICLE GRADIENT EFFECTS

3.4 Molecular, Knudsen, and Surface Diffusion in Pores 172

3.5.1.2 Experimental Determination of Effective 177

Diffusivities of a Component and of the Tortuosity

Example 3.5.1.2.A Experimental 178

Determination of the

a Component and of the Catalyst Tortuosity by Means of the Packed Column Technique Example 3.5.1.2.B Application of the Pellet 180

Technique

3.5.2.2 The Parallel Cross-Linked Pore Model 182

Example 3.5.A Optimization of Catalyst Pore Structure 189 3.5.4 Diffusion in Zeolites Configurational Diffusion 190 3.5.4.1 Molecular Dynamics Simulation 191 3.5.4.2 Dynamic Monte-Carlo Simulation 193 3.6 Diffusion and Reaction in a Catalyst Particle A Continuum 193

Example 3.7.A Effectiveness Factors for Sucrose Inversion 206

in Ion Exchange Resins 3.8 Influence of Diffusion Limitations on the Selectivities of 207

and External Mass Transfer Limitations

Example 3.13.2.A Temperature Gradients Inside the 228

Catalyst Particles in Benzene Hydrogenation

Chapter 4: Noncatalytic Gas-Solid Reactions

4.1 A Qualitative Discussion of Gas-Solid Reactions 240 4.2 General Model with Interfacial and Intraparticle Gradients 243 4.3 Heterogeneous Model with Shrinking Unreacted Core 252

Example 4.3.A Combustion of Coke within Porous Catalyst 255

Particles 4.4 Models Accounting Explicitly for the Structure of the Solid 259 4.5 On the Use of More Complex Kinetic Equations 264

Chapter 5: Catalyst Deactivation

5.2.4 Effect of Shell-Progressive Poisoning on the 280

Selectivity of Simultaneous Reactions 5.3 Kinetics of Catalyst Deactivation by Coke Formation 285

Blockage in the Presence of Diffusion Limitations

5.3.2.5 Deactivation by Site Coverage, Growth of 298

Coke, and Blockage in Networks of Pores

Example 5.3.3.A Application to Industrial Processes: 303

Coke Formation in the Dehydro- genation of 1-Butene into Butadiene Example 5.3.3.B Application to Industrial Processes: 309

Rigorous Kinetic Equations for Catalyst Deactivation by Coke Deposition in the Dehydrogenation

of 1-Butene into Butadiene Example 5.3.3.C Application to Industrial Processes: 312

Coke Formation and Catalyst Deactivation in Steam Reforming

of Natural Gas Example 5.3.3.D Application to Industrial Processes: 316

Coke Formation in the Catalytic Cracking of Vacuum Gas Oil

6.3.3 Single, Instantaneous, and Irreversible Reactions 332 6.3.4 Some Remarks on Boundary Conditions and on 337 Utilization and Enhancement Factors

6.3.5 Extension to Reactions with Higher Orders 340

6.4.2 Single Irreversible (Pseudo)-First-Order Reactions 351 6.4.3 Surface Renewal Models with Surface Elements of 355 Limited Thickness

6.5 Experimental Determination of the Kinetics of Gas-Liquid 356

Reactions

Chapter 7: The Modeling of Chemical Reactors

7.2 Aspects of Mass, Heat and Momentum Balances 367

Chapter 8: The Batch and Semibatch Reactors

Example 8.1.A Example of Derivation of a Kinetic Equation 388

from Batch Data Example 8.1.B Styrene Polymerization in a Batch Reactor 390 Example 8.1.C Production of Gluconic Acid by Aerobic 394

Fermentation of Glucose

Example 8.2.A Decomposition of Acetylated Castor Oil Ester 399

Example 8.3.A Simulation of Semibatch Reactor Operation 403

(with L.H Hosten†) 8.4 Optimal Operation Policies and Control Strategies 407

Example 8.4.1.A Optimum Conversion and Maximum 410

Profit for a First-Order Reaction

Example 8.4.2.A Optimal Temperature Trajectories 412

for First-Order Reversible Reactions Example 8.4.2.B Optimum Temperature Policies for 418

Consecutive and Parallel Reactions

9.1 The Continuity, Energy, and Momentum Equations 427 9.2 Kinetic Studies Using a Tubular Reactor with Plug Flow 432

9.2.1 Kinetic Analysis of Isothermal Data 432 9.2.2 Kinetic Analysis of Nonisothermal Data 435 9.3 Design and Simulation of Tubular Reactors with Plug Flow 438

9.3.2 Design and Simulation of Non-Isothermal Cracking 441 Tubes for Olefins Production

Chapter 10: The Perfectly Mixed Flow Reactor

10.3 Design for Optimum Selectivity in Simultaneous Reactions 461

Example 10.4.2.A Temperature Oscillations in a Mixed 481

Reactor for the Vapor-Phase Chlorination

PART TWO P SEUDOHOMOGENEOUS M ODELS

11.5.2 Design of a Fixed Bed Reactor According to the One- 510 Dimensional Pseudohomogeneous Model

Example 11.5.3.A Application of the First Runaway 519

Criterion of Van Welsenaere and Froment

11.5.5 Fixed Bed Reactors with Heat Exchange Between the 530 Feed and Effluent or Between the Feed and Reacting

Gas “Autothermal Operation”

11.5.6 Nonsteady-State Behavior of Fixed Bed Catalytic 548 Reactors Due to Catalyst Deactivation

11.7.3 Design or Simulation of a Fixed Bed Reactor for 572 Catalytic Hydrocarbon Oxidation

11.7.4 An Equivalent One-Dimensional Model 578 11.7.5 A Two-Dimensional Model Accounting for Radial 579 Variations in the Bed Structure

PART THREE H ETEROGENEOUS M ODELS

11.8 One-Dimensional Model Accounting for Interfacial Gradients 585

11.8.2 Simulation of the Transient Behavior of a Reactor 589 Example 11.8.2.A A Gas-Solid Reaction in a Fixed Bed 591

Reactor 11.9 One-Dimensional Model Accounting for Interfacial and 597

Intraparticle Gradients

Example 11.9.1.A Simulation of a Primary Steam 604

Reformer Example 11.9.1.B Simulation of an Industrial Reactor 614

for 1-Butene Dehydrogenation into Butadiene

Example 11.9.1.C Influence of Internal Diffusion 621

11.10 Two-Dimensional Heterogeneous Models 623

Chapter 12: Complex Flow Patterns

12.4 Micro-Probability Density Function Methods 649

12.4.2 Micro-PDF Methods for Turbulent Flow and Reactions 653 12.5 Micro-PDF Moment Methods: Computational Fluid Dynamics 658

12.5.1 Turbulent Momentum Transport Modeling of the 662

Reynolds-Stresses Annex 12.5.1.A Reynolds-Stress Transport Equations (web) 12.5.2 Turbulent Transport of Species and Heat Modeling of 666

the Scalar Flux Annex 12.5.2.A Scalar Flux Transport Equations (web) 12.5.3 Macro-Scale Averaged Reaction Rates 667

Annex 12.5.3.A Moment Methods: Transport Equa- (web)

tions for the Species Concentration Correlations

12.5.3.1 Models Based upon the Concept of Eddy 668

Dissipation

Example 12.5.A Three Dimensional CFD Simulation of 670

Furnace and Reactor Tubes for the Thermal Cracking of Hydrocarbons

12.6 Macro-PDF / Residence Time Distribution Methods 677

12.6.1 Reactor Scale Balance and Species Continuity 677

Equations Example 12.6.1.A Population Balance Model for 678

Micro-Mixing in a Perfectly Macro-Mixed Reactor: PDF Moment Method

Example 12.6.2.A RTD of a Perfectly Mixed Vessel 688 Example 12.6.2.B Experimental Determination of 689

the RTD 12.6.3 Flow Patterns Derived from the RTD 691

12.6.4 Application of RTD to Reactors 694

Example 12.6.4.A First Order Reaction(s) in 696

Isothermal Completely Mixed Reactors, Plug Flow Reactors, and Series of Completely Stirred Tanks

Example 12.6.4.B Second Order Bimolecular 698

Reaction in Isothermal Completely Mixed Reactors and in a Succession

of Isothermal Plug Flow and Completely Mixed Reactors:

Completely Macro-Mixed versus Completely Macro- and Micro- Mixed

12.7 Semi-Empirical Models for Reactors with Complex Flow 699

Patterns

12.7.2 Axial Dispersion and Tanks-in-Series Models 703

Chapter 13: Fluidized Bed and Transport Reactors

13.2 Technological Aspects of Fluidized Bed and Riser Reactors 720

13.3 Some Features of the Fluidization and Transport of Solids 723

13.5.3 A Hydrodynamic Interpretation of the Interchange 736

Coefficient k I

13.7 Fluidized Bed Reactor Models Considering Detailed Flow 744

Patterns

13.8.1 Kinetic Models for the Catalytic Cracking of Vacuum 749

13.8.2 Simulation of the Catalytic Cracking of Vacuum Gas 753

Oil 13.8.2.1 Fluidized Bed Reactor Two-Phase Model 753

with Ten Lump Reaction Scheme 13.8.2.2 Fluidized Bed Reactor Reynolds-Averaged 756

Navier-Stokes Model with Ten Lump Reaction Scheme

13.8.2.3 Riser Reactor Plug Flow Model with Slip 758

with Reaction Scheme based upon Elementary Steps Single Event Kinetics 13.8.3 Kinetic Models for the Regeneration of a Coked 762

Cracking Catalyst 13.8.4 Simulation of the Regenerator of a Catalytic Cracking 763

Unit 13.8.5 Coupled Simulation of a Fluidized Bed (or Riser) 765

Catalytic Cracker and Regenerator

Chapter 14: Multiphase Flow Reactors

14.2 Design Models for Multiphase Flow Reactors 784

14.2.1 Gas and Liquid Phases Completely Mixed 784 14.2.2 Gas and Liquid Phase in Plug Flow 785 14.2.3 Gas Phase in Plug Flow Liquid Phase Completely 786

Mixed

14.2.6 Models Considering Detailed Flow Patterns 788

Example 14.3.1.A The Simulation or Design of a 793

Packed Bed Absorption Tower Example 14.3.1.B The Absorption of CO2 into a 797

Monoethanolamine (MEA) Solution

Packed Downflow Bubble Reactors Example 14.3.2.A Trickle Bed Hydrocracking of 810

Vacuum Gas Oil 14.3.3 Two-Phase Fixed Bed Catalytic Reactors with 813

Cocurrent Upflow Upflow Packed Bubble Reactors

Example 14.3.4.A The Simulation or Design of a 818

Plate Column for Absorption and Reaction

Example 14.3.4.B The Absorption of CO2 in an 822

Aqueous Solution of Mono- and Diethanolamine (MEA and DEA)

Example 14.3.6.A Simulation of a Bubble Column 830

Reactor Considering Detailed Flow Patterns and a First-Order

Irreversible Reaction Comparison with Conventional Design Models

Example 14.3.7.A Design of a Liquid-Phase 837

o-Xylene Oxidation Reactor

Notation

Great attention has been given to the detailed definition of the units of the different quantities: for example, when a dimension of length is used, it is always clarified as to whether this length concerns the catalyst or the reactor We have found that this greatly promotes insight into the mathematical modeling of a phenomenon and avoids errors

A heat exchange surface in a batch reactor, m²

on the side of the reaction mixture

m

A logarithmic mean of A k and A r or of A b and A r m²

r

A heat exchange surface for a batch reactor, m²

on the side of the heat transfer medium

A frequency factor, for 1st order, e.g s-1

B fictitious component in Wei-Prater analysis

B vector of fictitious components

'

b order of reaction with respect to B

b vector of parameter estimates

in the bulk fluid

fkmol/m

Aeq

fkmol/m

Ai

C molar concentration of A in front of the interface 3

fkmol/m

C , molar concentration of poison in gas phase inside kmol/m 3f

catalyst and at core boundary

S

pkmol/m

D effective diffusivity for transport of A through a m3/mps

fgrain (Chapter 4)

ep

D effective diffusivity for transport of A in the pores m3f/mps

between the grains (Chapter 4)

e

D effective diffusivity for transport through completely m3/mps

freacted solid (Chapter 4)

jm

D effective molecular diffusivity of j in a

Deff effective diffusivity, a combination of molecular

and turbulent diffusivities in a fluid m3/mf s

also internal energy;

also energy of the particle, consisting of

potential and kinetic contributions;

also total energy, consisting of internal and

   d

E residence time distribution function

  x

Ei exponential integral function

Eo intrinsic activation barrier of a reference step of a

b

Eö Eötvös number, based on bubble diameter, dbρLg/σ

  r

Ê number of pore mouths per network on a sphere

at a distance r from the center of the particle

  

erf error function

  

erfc complementary error function, 1 -erf 

F ratio of variances, used to test model adequacy

or used to select the best out of a number of

competing models (Chapter 2)

also single-particle- or one-point joint micro-

probability density function

I   d internal age distribution function

I internal distribution function

I initiator; also intermediate species; inert;

J matrix of partial derivatives of function with

respect to parameters (Chapter 2); Jacobian matrix

l

J , molar flux of species j in l direction, with respect kmol/m²s

to mass average velocity

D

j j-factor for mass transfer, g m fA Sc2/3

G

p M k

H

j j-factor for heat transfer,  Pr 2/3

p

f G c h

k, rate coefficient for a catalytic reaction;

f

k rate coefficient for a reaction 3n 1 n

f kmolA

between a fluid reactant A (kmol S)-m mp3m

(order n) and a solid or solid component S (kg part)-1 s-1

k mass transfer coefficient from gas to liquid kmol/m2bar s

interface, based upon partial pressure driving force

k I bubble-emulsion phase interchange coefficient m /m3s

r

3 f

L

k mass transfer coefficient from interface to

liquid bulk, based on concentration driving force m /m2 s

k mass transfer coefficient from gas to solid interface

when based on concentration driving force m /m2s

i

3 fwhen based upon partial pressure driving force kmol/m2bar s

l

k mass transfer coefficient between liquid and m3L/m2i s

catalyst surface, referred to unit interfacial area

k elutriation rate coefficient (Chapter 13) kg/m² s

k mass transfer coefficient between stagnant

liquid and catalyst surface in a multiphase reactor m /m3s

r 3 L

  kbi b mass transfer coefficient from bubble to interchange m /m3s

b

3 fzone, referred to unit bubble volume

  kbe b overall mass transfer coefficient from bubble to m /m3s

b

3 femulsion, referred to unit bubble volume

  kie b mass transfer coefficient from interchange zone to m /m3s

b

3 femulsion, referred to unit bubble volume

  kce c mass transfer coefficient from bubble + interchange m3G/m3cs

zone to emulsion, referred to unit bubble

+ interchange zone volume

also distance from center to surface of catalyst pellet mp

Lw modified Lewis number, e /s c ps D e

M ratio of initial concentrations C Bo / C Ao

m Henry’s coefficient based on mole fractions;

t

j

N (t) instantaneous molar absorption rate in element

of age t per unit gas-liquid interfacial area kmol/m2s

i

ref p ref ep

C R

C t D

n number of single events;

also number of replicated experiments

also probability that a site is accessible (Chapter 5);

also reaction product

Pr Prandtl number, c p/

Prt turbulent Prandtl number, Prt t c P t

Pi probability that a molecule is in the i-th

quantum state with energy level E i

P mass averaged degree of polymerization

p probability of adding another monomer unit

to a chain; also number of parameters

Pk production of turbulent kinetic energy kg/(m·s3)

rN/m

t

q order of reaction with respect to Q j

qt, qr, qv, qel translational, rotational, vibrational and

electronic contributions to the total partition

function Q

also radius of a spherical particle (Chapters 4 and 5); mp

also reaction component

also radial position in spherical particle; mp

also stoichiometric coefficient;

also space vector (Chapter 1)

A

r rate of reaction of component A per unit volume kmol/m3f s

for homogeneous reaction

or per unit catalyst mass for heterogeneous reaction kmol/kg cat s

r rate of reaction of S, reactive component of solid, in kmol/kg part s

gas-solid reactions or rate of reaction of solid itself

r~ rate of reaction of component A in terms

of the variation of its mass fraction kg A/(kg total·s)

S reaction component also dimensionless group,

 ;

Sc Schmidt number, μ/ρD

S internal surface area per unit mass of catalyst m² cat./kg cat

p

m

m

Sh modified Sherwood number for liquid film, k L / A v D A

Sh' modified Sherwood number, k g L/D e (Chapter 3)

 standard entropy of adsorption of a component kJ/kmol K

s stoichiometric coefficient; also parameter in s-1

Danckwerts’ age distribution function;

also Laplace transform variable

t tabulated  /2 percentage point of the

t-distribution with n-p degrees of freedom

  s, transfer function of flow model (Chapter 12)

also functional expressing the interaction between

t

V reactor volume or volume of considered "point" m3r

v elements of inverse of matrix V ε

x conversion of A, B … for constant density kmol/m³

X matrix of settings of independent variables

T

X transpose matrix of X

Y vector of species mass fractions

y radius of grain in grain model of Sohn and m

Szekely (Chapter 4)

yˆ estimated value of dependent variable

y coordinate perpendicular to gas-liquid interface; m

also radial position inside a grain in grain

model of Sohn and Szekely (Chapter 4)

y vector of mole fractions

y arithmetic mean of n e replicate observations

y vector of observations of dependent variable

y weight fractions of gas oil, gasoline (Chapter 5)

Z compressibility factor; also

c

Z critical compressibility factor

z spatial coordinate vector

Greek Symbols

also profit resulting from the conversion of $/kmol

1 kmole of A into desired product

 vector of flow model parameters

 convective heat transfer coefficient on the side kJ/m²sK

of the reaction mixture

r

 convective heat transfer coefficient on the side kJ/m²sK

of the heat transfer medium

u

 convective heat transfer coefficient for a packed kJ/m²sK

bed on the side of the heat transfer medium

also weighting factor in objective function (Chapter 2);

also stoichiometric coefficient (Chapter 5);

also cost of 1 kg of catalyst (Chapter 11);

also dimensionless adiabatic temperature rise,

(T ad – To)/ To;

also thermal expansion coefficient;

also interphase momentum transfer coefficient; kg/(m3s)

r

s e

s s

e C /λ T D

 Hatta number, for first order reaction kD A / k L , A;

for reaction with order m with respect to A

and n with respect to B: n A L , A

Bi

m

Ai C D / k kC

m

11

also dimensionless activation energy, E/RT

(Chapters 3 and 11);

also dissipation of pseudo-thermal energy (solid

phase) by inelastic particle-particle collisions kg/(m s3)

r/sm

 column vector of n experimental errors

r

3

g/mm

r

3

s/mm

bends in pipes

m

 quantity of fictitious component

 effectiveness factor for solid particle

 global utilization factor; also effectiveness

factor for particle + film

 fractional coverage of catalyst surface;

also dimensionless time, D e t/L² (Chapter 3),

ak’C A t (Chapter 4);

angle between pore and radial at

distance r from center of spherical particle rad

κ conductivity pseudo-thermal energy (solid

 effective thermal conductivity in a solid particle kJ/m s K

λeff effective thermal conductivity, a combination

of molecular and turbulent conductivities kJ/m s K

er

ea 

 , effective thermal conductivity in a packed bed kJ/mr s K

in axial, respectively radial direction

l

 effective thermal conductivity in l direction kJ/m s K

m

 negative of eigenvalue of rate coefficient matrix K;

 , effective thermal conductivity for the fluid phase, kJ/mrsK

respectively solid phase in a packed bed

 L

 probability density function of pore length

 dynamic viscosity; also type of radical in a

 extent of reaction; also reduced length, z/L kmol

or reduced radial position inside a particle, r/R

 reduced radial position of core boundary

i

e

ekg/m

mf

 bulk density of fluidized bed at minimum

rkg/m

s

pcat./mkg

 standard deviation; also active alumina site

also symmetry number

 sorption distribution coefficient (Chapter 5)

 tortuosity factor for catalyst;

also time scale or decay time (Chapter 12); s

τji shear stress tensor, jith component kg/(m s2)

 Thiele modulus for 1st order

eA

s / D k S /

 T s s C s / D e k

S /

φ internal coordinate vector

C

A 

 , deactivation functions for main and coking

reactions (site coverage)