modelling, simulation and optimization of industrial fix catalytic

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THREE-PHASE CATALYTIC REACTORS by P. A. Ramachandran and R. V. Chaudhari

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MODELLING, SIMULATION AND OPTIMIZATION OF INDUSTRIAL FIXED BED CATALYTIC REACTORS by S. S. E. H. Elnashaie and S. S. Elshishini

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Modelling, Simulation and Optimization of Industrial Fixed Bed Catalytic Reactors

S. S. E. H. Elnashaie and S. S. Elshishini

King Saud University, Riyadh, Saudi Arabia, and Cairo University, Egypt

GORDON AND BREACH SCIENCE PUBUSHERS Switzerland Australia Belgium France Germany Gt B1111 India Japan Malaysia Netherlands Russia Singapon \

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1993 by OPA (Amsterdam) B.V. All rights reserved. Published under license by Gordon and Breach Science Publishers S.A.

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Modelling, simulation, and optimization of industrial fixed bed catalytic reactors I S.S.E. H. Elnashaie and S.S. Elsbisbini.

p. em. -- (Topics in chemical engineering ; v. 7) Includes bibliographical references and index. ISBN 2-88 124-883-7 I. Chemical reactors. 2. Catalysis. I. Elnashaie, S. S. E. H.,

1947- . II. Elshishini, S. S., 1948- . III. Series. TP157.M53 1993 660' .2995--dc20 92-29469

CIP

No J)nrt or th1s book may be reproduced or utilized in any form or by any mcons, clcct&onic 0 1 rncehanicnl, including photocopying and recordi ng, I) I hy uny 111to rmat ion ~torngc or rctricvnl system, without pcrmi,'iinn 111 WI IIIII!.: fmm the publisher. Printed in Sin!(aporc.

8

Contents

Introduction to the Series Preface Notation Acknowledgements for Figures and Tables

INTRODUCTION

CHAPTER 1 SYSTEMS THEORY AND PRINCIPLES FOR DEVELOPING MATHEMATICAL MODELS OF lNDUSTRIAL FIXED BED

I~

XV xvii xix

XXV

CATALYTIC REACfORS 7 1.1 Systems and Mathemat ical Models 8

1.1. 1 A Brief Background 8 1.1.2 Mathematical Model Building: General Concepts 11 1.1.3 Outline of the Procedure for Model Building 12

1.2 Basic Principles of Mathematical Modelling for Industrial Fixed Bed Catalytic Reactors 13

CHAPTER 2 CHEMISORPTION AND CATALYSIS 24 2.1 Adsorption and Catalysis 24 2.2 Physical and Chemical Adsorption 25 2.3 Heats of Adsorption and Desorptjon 26 2.4 The Kinetics of Adsorption and Desorption 26 2.5 Activated and Non-activated Adsorption 27

2.5.1 Activated Adsorption 27 2.5.2 Non-activated Adsorption 27

2.6 Equilibrium and Non-equilibrium Adsorption-Desorption 28 2.6.1 Equilibrium Adsorption-Desorption 28 2.6.2 Steady State Non-equilibrium

Adsorption-Desorption 28 2.7 Adsorption Isotherms 29 2.8 The Effect of Surface Coverage 30

2.8. 1 The Role of the Surface in Heterogeneous Catalysis 30

2.8.2 Heat of Adsorption and Surface Coverage 31

CHAPTER 3 INTRINSIC KINETICS OF GAS-SOLID CATALYTIC REACTIONS 35

3. 1 Kinetic Models for Gas-Solid Catalytit: Reactions 38

VI I l NA\ 111\11 ANI> l . l ~'iiii SIIIN I CUN11 N I :'I '" 3.1.1 Power Law (PL) Kinetic Models 38 3.2.5.1 Reaction network for the partial 3.1.2 Chemisorption-Surface Reaction-Desorption (CSD) oxidation of o-Xylene to phthalic

Kinetic Models for Unimolecular Reactions 39 anhydride 101 3.1.2. 1 The equilibrium adsorption-desorption 3.2.5.2 Kinetic models of the o-Xylene partial

case with negligible product inhibition 40 oxidation reactions 105 3.1.2.2 Effect of product inhibition 42 3.2.5.3 Kinetic models and parameters for the 3. 1.2.3 The steady state assumption (SSA) case partial oxidation of o-Xylene to phthalic

with negligible product inhibition 42 anhydride 108 3.1.3 Chemisorption-Surface Reaction-Desorption (CSD)

Kinetic Models for Bimolecular Reactions 43 CHAPTER 4 PRACllCAL RELEVANCE OF 3.1.4 The Number of Kinetic Parameters to be Estimated for the Bimolecular Case 48

BIFURCATION, INSTABILITY AND

3.2 Chemisorption-Surface Reaction-Desorption (CSD) CHAOS IN CATALYTIC REACTORS 112

Kinetic Models for Some Industrially Important 4.1 Sources of Multiplicity 114

Gas-Solid Catalytic Reactions 50 4.1.1 Isothermal Multiplicity (or Concentration

3.2.1 Steam Reforming of Methane 51 Multiplicity) 114

3.2.1.1 Kinetics of steam reforming reactions 51 4.1.2 Thermal Multiplicity 115

3.2.1.2 A more general chemisorption-surface 4.1.3 Multiplicity Due to the Reactor Configuration 1 16

reaction-desorption (CSD) kinetic model 4.2 Simple Quantitative Discussion of the Multiplicity

for methane steam reforming 60 Phenomenon 116

4.3 Bifurcation and Stability II 7 3.2.1.3 Rate dependence on steam partial 4.3.1 Steady State Analysis 118 pressure 62

3.2.1.4 Reaction rate dependence observed by 4.3.2 Dynamic Analysis 125

earlier investigators 63 3.2. 1.5 Comparison with Bodrov kinetics 65 CHAPTERS EFFECT OF DIFFUSIONAL 3.2.1.6 Comparison with De Deken kinetics 67 RESISTANCES. THE SINGLE PELLET 3.2.1. 7 Implications of non-monotonic kinetics PROBLEM 139

on fixed bed reactors 69 5.1 Non-porous Catalyst Pellets 141 3.2.1.8 The implications of non-monotonic 5.1.1 Isothermal Catalyst Pellets with Linear Kinetics 141

kinetics on the selection of feed 5.1.2 Isothermal Catalyst Pellets with Non-linear conditions for steam reformers 73 Kinetics 144

3.2.1.9 Concluding remarks 77 5.1.3 The Isothermal Effectiveness Factor for Single 3.2.2 High and Low Temperature Water-Gas Shift Reactions (Irreversible and Unimolecular) 146

Reactions 79 5. 1.4 The Non-isothermal Catalyst Pellets with Linear 3.2.2.1 Kinetic models for the high temperature Kinetics (Non-porous Catalyst PeUet with

shift catalysts 79 Unimolecular Reaction) 148 3.2.2.2 Comments on reaction mechanism 85 5.1.5 Non-isothermal Catalyst Pellets with 3.2.2.3 Actual industrial kinetic models for high Non-monotonic Kinetics 151

and low temperature shift catalysts 86 5.1.6 Preliminary Remarks on Complex Reaction 3.2.3 Gas Phase Catalytic Hydrogenation Reactions 87 Networks 151

3.2.3. 1 Hydrogenation of aromatics 87 5.1.7 Preliminary Comments on the Effectiveness 3.2.3.2 Hydrogenation of olefins 91 Factor Concept for Gas-Solid Catalytic Reactions.

3.2.4 Catalytic Ammonia Synthesis 94 Reaction versus Component Effectiveness Factor 152 3.2.4.1 Mechanism of the ammonia reaction 94 5.1.8 Results and Discussion of the Steady State for 3.2.4.2 Rate equations 95 Non-isothermal, Non-porous Catalyst Pellet for a 3.2.4.3 Widely used kinetic models for industrial Single Unimolecular Irreversible Reaction 153

ammonia synthesis 97 5.1.8.1 Practical applications and range of 3.2.5 Partial Oxidation Reactions 100 parameters for industrial catalysts 154

~(.V uv: t ..,._!../1 - -- ...... l J t.. ~ SJ~lUJOJ:)J I nv SlliBlSliOJ wnpQ!J!11b3: uopdJospv UI\!OlS JI!U ISI1 fHII IOJ OJ ~1P.}( ..... .. .. ., J.I. Wa 11o f I f 5.1.8.2 General model with finite pellet thermal 6.2 Optimization of the Performance of Industrial Fixed Bed

conductivity: The symmetrical case 155 Catalytic Reactors 272 5.1.8.3 Numerical solution of the equations 157 6.2.1 The Objective Functions (What Is to Maximize or 5.1.8.4 Simplified models 158 Minimize?) 273 5.1.8.5 Steady state model equations 159 6.2.2 Optimal and Suboptimal Temperature Control

5.1.9 Results and Discussion for Complex Reaction Policies 276 Network on Non-porous Catalyst Pellets. The 6.2.3 Optimization for Reversible Reactions 285 Steady State Analysis for the Catalytic Partial 6.2.4 Pontryagin Maximum Principle and the Simple Oxidation of o-Xylene to Phthalic Anhydride 162 Optimality Criterion for Exothermic Reversible 5.1.9.1 The mathematical models for the partial Reactions (The Single Reaction

oxidation of o-Xylene to phthalic Pseudo-homogeneous Model) 287 anhydride on non-porous catalyst 6.3 Modelling, Simulation and Optimization of Industrial pellets for two different kinetic Fixed Bed Catalytic Reactors. Case Studies 290 models 162 6.3.1 Mathematical Modelling of the High Temperature

5.1.9.2 Mathematical expression for the Water-Gas Shift Converter 29 1 effectiveness factors 165 6.3.2 Mathematical Modelling of the Low Temperature

5.1.9.3 Simulation results for the partial Water-Gas Shift Converter 314 oxidation of o-Xylene to phthalic 6.3.3 Ammonia Converters 316 anhydride on non-porous pellets 166 6.3.3.1 Brief literature review for the

5.2 Porous Catalyst Pellets 192 mathematical modelling of industrial 5.2.1 Lumped Models 192 ammonia converters 317

5.2.1.1 The importance of surface processes in 6.3.3.2 Modelling of ammonia converters 320 the dynamic behaviour of porous catalyst 6.3.3.3 The catalyst pellets 320 pellets 193 6.3.3.4 Effectiveness factor for the catalyst

5.2.1.2 Chemisorption and catalysis 193 pellets of ammonia converters 326 5.2. 1.3 Rates of chemisorption 194 6.3.3.5 The overall reactor model for the 5.2.1.4 Heats of adsorption 196 catalyst bed module 327 5.2.1.5 Practical example for the rates of 6.3.3.6 Algorithm for the numerical solution of

activated adsorption 196 the model equations 328 5.2.1.6 Rates of desorption 196 6.3.3.7 Modelling for the different 5.2. 1.7 Equilibrium adsorption desorption 197 configurations of the ammonia 5.2.1.8 The lumped model for porous catalyst converters 329

pellets 198 6.3.3.8 Optimization of the performance of 5.2.2 Distributed Models 204 ammonia converters 331

5.2.2.1 Fickian-type models 208 6.3.3.9 Concluding remarks 336 5.2.2.2 Dusty gas models 229 6.3.3.10 Computer simulation package for 5.2.2.3 The use of the dusty gas model for industrial ammonia converters 338

multiple reversible reaction networks. 6.3.4 Precise Modelling of Industrial Steam Reformers Steam reforming of methane 242 and Methanators 342

6.3.4. 1 Rate expressions 344 CHAPTER 6 THE OVERALL REACTOR MODELS 261 6.3.4.2 Model development for steam reformers 345 6.1 General Classification of Reactor Models 262 6.3.4.3 Modelling of side fired furnaces 347

6.1.1 The Continuum Models 264 6.3.4.4 Modelling of top fired furnaces 349 6.1.1.1 Pseudo-homogeneous models 264 6.3.4.5 Modelling of methanators 350 6.1.1.2 Heterogeneous models 266 6.3.4.6 Modelling of the catalyst pellets for 6.1.1.3 One-dimensional models 267 steam reformers and methanators using 6.1.1.4 Two-dimensional models 269 the dusty gas model 350

6.1.2 The Cell Models 270 6.3.4.7 Computational algorithm 352

uz SJ01::>~~ ::>n.

Preface

A considerable part of this book was written while the authors were living in Riyadh during the Gulf War. With Scud missiles and the threat of chemical warfare, we can only hope that we will see more use of chemical reactions for the production of useful petrochemical products rather than missiles and chemical weapons.

Many colleagues from the Departments of Chemical Engineering and Chemistry at King Saud University and Cairo University, including Drs Alhabdan, Alahwany, Mahfouz, Al-Ubaid, Al-Khowaiter, Soliman, Teymour, AI-Shawarbi, Ettouney and the late Dr Abdei-Hakim, con-tributed directly and indirectly to this book. They all deserve special gratitude. Our students, Engineers Abashar, Elmuhanna and Abdalla, have contributed to many of the practical problems in the book.

Dr Al-Fariss, the chairman of the Department of Chemical Engi-neering at King Saud University and the rest of the staff of the chemical engineering and the chemistry departments, created an excellent academic atmosphere that made this work possible. Their continuous encouragement and help has contributed considerably to the completion of the book under the difficult conditions created by the Gulf War.

A special debt of gratitude goes to Dr Hughes (Salford) and Professor Aris (Minnesota) who encouraged us to undertake the writing of this book. The list would not be complete without mentioning Professor Ray (Wisconsin), Dr Cresswell (now with ICI, England) and Professor Marek (Prague) who introduced us to this field of research more than twenty years ago.

We have also benefited from extensive and enjoyable discussions with Professor Froment (Ghent) and Professor Villadsen (Lyngby) during exchange visits between our departments.

We would also like to thank all the petrochemical and petroleum refining industries in Saudi Arabia, Egypt, UK, Belgium and Canada who provided the industrial data and the necessary interaction with industry which made the industrial verification of the models possible.

Finally, we would like to thank Professor Elnaschie (Cornell, Cambridge) for his continuous encouragement and for suggesting the chapter on the practical relevance of bifurcation, instability and chaos, and Mrs Hala Teymour for her generous help in the typing of the manuscript.

S. S. E. H. Elnashaie and S. S. Elshishini

I

Notation

ac Cell surface area (m2) a, Activity of component i aP External surface area of pellet (m2) an a, Refractory and tube side surface area to furnace free

volume half ratio as Specific surface area of catalyst pellets (m2/kg & m2/m3) A cp Cold plane area (m2) AH Area of heat transfer (m2) AP Cross-sectional area of slab catalyst pellet (m2) AR Refractory surface area (m2) A, Cross-sectional area of bed (m2) C1 Concentration of component i (kmol/m3) c,B Concentration of component i in the bulk phase

(kmollm3) cif Concentration of component i in feed (kmollm3) C;s Concentration of component i at catalyst surface

(kmol/m 3) Cm Total concentration of active sites (kmol/m3) Cp Specific heat (kJ/kmol.K) CP Average specific heat of process gas (kJ/kmol.K) Cp.r Specific heat of catalyst (kJfkg.K) Cref Reference concent ration (kmoVm3) Cs Total concentration at the surface (kmoVm3) Cu Concentration of vacant active sites (kmollm3) dP Catalyst particle diameter (m) d, Tube diameter (m) d1e Tube external d iameter (m) dti Tube internal diameter (m) D Diffusivity (m2/h) De Effective intraparticle diffusivity (m2/h) Dea Axial effective diffusivity in tubular reactor (m2/h) De, Radial effective diffusivity in tubular reactor (m2/h) D1 Bulk d itrusivity of component i (m2fh) D~ Bulk d iffusivity of component i at oc and 1 atm (m2/h) D;e Effective diffusivity of component i (m2/h)

xix

10110d 1SAJtnUO JO Al!f nonpuo:> JHWJ()lJl OA!lO~JJa :'l:m.J sn.t1111lJ:>clw:>l op1s oqn 1.

ltoHIIilll I I 111\1

Variable axial length (m) Characteristic length of pellet (m) Reactor length (m) Total mass flow rate in kg/h

II"

Molar flux of component i in porous catalyst pellets (kmoUh.m2) Diffusive flux of component i (kmoUm2.h) Viscous flux of component i (kmol/m2.h) Total flux (kmoUm2.h) Viscous flux (kmol/m2.h) Partial pressure in bulk gas phase (kPa) Partial pressure of component i (kPa) Total pressure (kPa) Permeability of cell membrane Total pressure at the surface (kPa) Volumetric flow rate (m3/h) Sensible heat of the exit flue gas (kJ/h) Net heat release (kJ/h) Radiation heat transfer (kJ/h) Mean pore radius (m) Rate of reaction at bulk conditions (kmol/kg.cat.h) Rate of heat transfer (kJ/m3h) Rate of reaction j (kmol/kg .. cat.h) Rate of mass transfer (kmol/m2.h) Rate of surface reaction (kmol/kg.cat.b) Universal gas constant Rate of disappearance of component i (kmol/ kg.cat.b) Resistance to mass transfer Overall reaction-diffusion resistance Radius of catalyst pellet (m) Resistance to reaction Sticking probability Active sites Specific surface area of catalyst (m2/kg.cat) External surface area of catalyst (m2) Space velocity (h - 1) Temperature (K) Adiabatic flame temperature (K) Bulk phase temperature (K) Furnace gas temperature (K) Temperature at inlet of reactor (K) Refractory and fuel gas inlet temperature (K) Reference temperature (K) Catalyst pellet surface temperature (K)

I

Tsk T,,o u,. u v jJ

vc Vp wp x, x, x,s

y Yo Yc Y, Y,o Y,,. y if z

z'

z

p y )i 0 (- t:J.H), (- lllf)j e

(m) 1anod JO q12uat ~qspal~U.rntJ:) (W) q12UaJ JU!Xe a{qB!JBA

II NA!illAll AND I I .HI IIIN I

Tube skin temperature (K) Outer tube temperature (K) Superficial velocity (m/h) Overall heat transfer coefficient (kJ/m2K) Reactor volume (m3) Free volume of the reformer (m3) Volume of cell, m3

Volume of catalyst pellet (m3) Weight of catalyst (kg) Conversion of component i

:>I

I

Dimensionless concentration of component i , C/C,ef Dimensionless concentration of component i in bulk gas phase Dimensionless concentration of component i at catalyst surface Dimensionless temperature, Tl T,t!.f Dimensionless bulk temperature T8 1T, .. 1 Dimensionless cooling jacket temperarure Mole fraction of component i Mole fraction of component i in bulk Mole fraction of component i at catalyst surface Feed mole fraction of component i Depth coordinate of catalyst pellet (distance from centre line of pellet) (m) Dimensionless length coordinate along the axial direction for tubular reactors (z' -IlL). Radial coordinate of tubular reactor (m) Grey and clear gas component distribution factor Bedside heat transfer coefficient Thermicity factor dimensionless activation energy Activity coefficient Thickness of mass transfer film around catalyst pellet (m) Net heat of adsorption of component i (kJ/kmol) Heat of reaction j Void fraction of catalyst pellet Void fraction of bed Stepsize in optimization algorithm Double the adsorption coefficient of the grey gas component Effectiveness factor Fractional surface coverage Fraction of oxidized sites in the redox model for o-Xylene oxidation

'""'c:. , J ' ;-;v; r ;.,--~

(4/zW) aJnlXJW UJ ! lUaUOdWO~ JO hl!AJSD.!J!Q (q/zW) { U! ! lUaUOdWOO JO h}JA!SDlJ!P A.Jeu!q aA!lOalJa

8, ) ...

Pb Ps a ...

:.~ v P"'" ;) """"a :uuuf IJJ ') :r, .. r L r

Acknowledgements for Figures and Tables

The following figures and tables have been reproduced from the papers and books given below with permission from the correspond-ing publishers given in the following list:

1. With Permission from Pergamon Press PLC- Oxford, U.K.

1.1 Figs. 6.4-6.18 and Tables 6.1-6.6 from: Elnashaie, S.S.E.H. and Alhabdan, F.M., Mathematical Modelling

and Computer Simulation of Industrial Water-Gas Shift Converters. Math. and Comput. Modi., Vol. 12, pp. 1017-1034, 1989.

1.2 Figs. 6.21, 6.23-6.32 from: Elnashaie, S.S.E.H. and Albabdan, F.M. , A Computer Software

Package of an Industrial Ammonia Converter Based on a Rigorous Heterogeneous Model, Math. and Comput. Modl., Vol. 12, pp. 1589-1600, 1989.

1.3 Figs. 7.3-7.2 and Table 7.3 from: Elnashaie, S.S.E.H. and Abdel-Hakim, M.N., Optimal Feed Tem-

perature Control for Fixed Bed Non-isothermal Catalytic Reactors Experiencing Catalyst Deactivation. A Heterogeneous Model. Com-puters and Chern. Eng. , Vol. 12, pp. 787-790, 1988.

1.4 Fig. 5.7 and Table 3.14 from: Chandrasekharan, K. and Calderbank, P.H., Prediction of Packed

Bed Performance for a Complex Reaction (Oxidation of o-Xylene to Phthalic Anhydride)., Chern. Eng. Sci., Vol. 34, pp. 1323-1331, 1979.

1.5 Fig. 5.9, Tables 3.15, 3.16, 5.4, 6.41, 6.43 from: Skrzypek, J., Grzesik, M., Galantowicz, M. and Solinski, J., Kinetics

of the Catalytic Air Oxidation of o-Xylene over a Commercial V 20 5 - Ti02 Catalyst, Chern. Eng. Sci., Vol. 40, pp. 611-620, 1985.

1.6 Tables 5.3 a,b from: Elnashaie, S.S.E.H., Soliman, M.A, Abashar, M.E. and Elmuhanna,

S., On the Mathematical Modelling of Diffusion and Reaction Networks. Negative Effectiveness Factors. Math. and Comput. Mod!. , Vol 16, pp. 41 -53, 1992.

I

II I I \',ll-\11 \Jill I I ' ,llf".lll l' /1

1.7 Fig. 5.46 from: Rester, S. and Aris. , R. , Communication on the Theory of Diffusion

and Reaction. II. The Effect of Shape on the Effectiveness Factor. Chern. Eng. Sci., Vol. 24, pp. 793-795, 1969.

1.8 Figs. 5.4 and 5.49 from: Weisz, P.B. and Hicks, J.S., The Behaviour of Porous Catalyst

Particles in View of Internal Mass and Heat Diffusion Effects. Chern. Eng. Sci., Vol. 17, pp. 265-27 5, 1962.

1.9 Figs. 5.51-5.57 from : Elnashaie, S.S.E.H. and Mahfouz, A.T., The Influence of Reactants

Adsorption on the Effectiveness Factor of Porous Catalyst Particles Chern. Eng. Sci. , Vol. 33, pp. 386-390, 1978.

1.10 Figs. 3.3-3.11, Tables 3.2-3.4, 3.6, 3.9 from: Elnashaie, S.S.E.H., Adris, A.M., Al-Ubaid, A.S. and Soliman, M.A.,

On the Non-monotonic Behaviour of Methane Steam Reforming Kinetics. Chern. Eng. Sci., Vol. 45, pp. 491-501 , 1990.

1.11 Table 3.12 from: Zrncevis, S. and Rusic, D., Verification of the Kinetic Model for

Benzene Hydrogenation By Poisoning Experiments. Chern. Eng. Sci., Vol. 43, pp. 763-767, 1988.

1.12 Figs. 5.6, 5.43-5.45 from: Elnashaie, S.S.E.H. and Cresswell, D.L., The Influence of Reactant

Adsorption on the Multiplicity and Stability of the Steady State of Catalyst Particles. Chern. Eng. Sci., Vol. 29, pp. 753-760, 1974.

1.13 Figs. 5.4, 5.5 from: Elnashaie, S.S.E.H. and Cresswell, D.L., Dynamic Behaviour and

Stability of Non-Porous Catalyst Particles, Chern. Eng. Sci., Vol. 28, pp. 1387-1399, 1973.

2. With Permission from Elsevier Science Publishers -Netherlands

2.1 Figs. 5.60-5.64 and Tables 5.11 from: Elnashaie, S.S.E.H. and Abashar, M.E., Steam Reforming and

Methanation Effectiveness Factors Using the Dusty Gas Model Under Industrial Conditions. Chern. Eng. and Processing (in Press, 1993).

2.2 Figs. 5.11, 5.15-5.17, 5.21-5.25, 5.27-5.33 from: Elnashaie, S.S.E.H. and Elmuhanna, S. , Effect of Diffusional Resis-

tance on the Rates of a-Xylene Partial Oxidation over V 20 5 Non-Porous Catalyst Pellets. Studies in Surface Science and Catalysis, Vol. 73, Progress in Catalysis, Proceedings of the 12th Canadian Sympo-

sium on Catalysis, Banff, Alberta, Canada, May 25-28 , 1992, pp. 335-342, Editors, K.J. Smith and E.C. Sanford, Elsevier.

2.3 Fig. 6.22 and Table 6.10 from: Elnashaie, S.S.E.H., Mahfouz, A.T. and Elshishini, S.S. , Digital

Simulation of Industrial Ammonia Reactors. Chern. Eng. and Process-ing, Vol. 23, p. 165-177,1988.

2.4 Fig. 4.14 from : Rossler, O.E., Physics Letters, Vol. 57A, p. 397, 1976.

3. With Permission from the American Chemical Society -Washington, U.S.A.

3.1 Fig. 6.1 from: Froment, G., Analysis and Design of Fixed Bed Catalytic Reactors ,

Advances in Chemistry Series, Vol. 109, pp. 1-34, 1972.

3.2 Tables 6.9, 6.11, 6.12 from: Elnashaie, S.S.E.H., Abashar, M.E. and Al-Ubaid, A.S. , Simulation

and Optimization of an Industrial Ammonia Reactor. 1ndust. Eng. Chern. Research, Vol. 27, pp. 2015-2022, 1988.

3.3 Fig. 5-10 from: Calderbank, P.H., Kinetics and Yields in the Catalytic Oxidation of

a-Xylene to Phthalic Anhydride with V 20 5 Catalyst. Advances in Chemistry Series, Vol. 133, pp. 646-653, 1974.

4. With Permission from Springer-Verlag, New York, U.S.A.

4.1 Fig. 4.15 from: Parker, T.S. and Chua, L-0, Practicale Numerical Algorithms for

Chaotic Systems, Springer-Verlag, New York, 1989.

4.2 Fig. 4.13: Sparrow, C. , The Lorenz Equations: Bifurcation, Chaos and Strange

Attractors, Springer-Verlag, New York, 1982.

5. With Permission from the American Institute of Chemical Engineers- New York, U.S.A.

Hutchings, J. and Carberry, J.J., The Influence of Surface Coverage on Catalytic Effectiveness and Selectivities. The Isothermal and Non-Isothermal Cases. AIChEJ, Vol. 12, pp. 20-23 , 1966.

' I

' I I I I ' I I ' I ..

' ., . ,. .. ' . I ' I I I .. I ' ' I I '

I I I I' I I il 'I ' ,. ' ,, I .. ' I ,, '

( t . With Per mission rrom the

and successfully verified against a number of industrial units t11en it will be reasonable to assume that the mathematical model describes the unit accurately enough to be used for design and optimization of these reactors.

The main emphasis of this book is on modelling, simulation and optimization. The numerical techniques for the solution of the model equations and for finding optimum operating conditions, alt11ough briefly covered, are not given great emphasis. The techniques needed vary from very simple and almost standard problems of solving non-linear algebrak equations and initial value differential equations which are available as standard subroutines and arc explained in detail in textbooks on numerical analysis, to non-linear two-point boundary value differential equations which can be solved using different techniques, the most efficient of which seem to be the orthogonal collocation techniques.

In addition to the brief discussion of these problems in the book, sufficient references are given for the reader to obtain more informa-tion regarding this important side of the problem.

Modelling of catalytic processes combines knowledge from many disciplines of chemistry and chemical engineering. ll involves the following aspects:

- Surface phenomena responsible for the catalytic activity itself. - Modelling of intrinsic catalytic reaction rates, ie: reaction rates in

the absence of diffusional resistances. - Thermodynamic equilibrium of the reaction mixture for reversible

reactions. - External mass and heat transfer resistances between the bulk gas

phase and the surface of the catalyst pellets which are functions of the fluid flow conditions around the pellets.

- lntraparticle mass diffusion and heat conduction for porous catalyst pellets.

- Pressure drop associated with the flow of the gas mixture through the packed bed.

- Heat evolution (for exothermic reactions) or heat absorption (for endothermic reactions) associated with the reacuon.

- Integration of the above steps into the formulation of the overall reactor model and the inclusion of heat transfer between the catalyst bed and external cooling or heating media.

For such complex systems involving a large number of interacting processes, it seems most appropriate to adopt a system approach for a better organization and economy of knowledge.

The book is written in the spirit of bringing to chemical engineers m both academia and industry a compreltensive text for the develop-ment of reliable mathematical models for industrial fixed bed catalytic reactors from the first step of surface phenomena to the final stage of putting the verified model into a user friendly software package with

colour graphtcs and easy mterlacang bcLwccn Ute U!;t:I and the model. All the models presented are developed and run on standard personal computers which facilitate the wider use of such high fidelity models.

It is important to notice that the use of the words modelling and simulation adopted in this book are different from that used by Aris (1991 ). In his 1990 Dankwerts Memorial Lecture, entitled: " Manners Makyth Modellers" Aris distinguishes between modelling and simu-lation in a special manner which follows the reasoning of Smith ( 1974) in ecological models:

" It is an essential quality in a model that it should be capable of having a life of its own. It may not in practice, need to be sundered from its physical matrix. It may be a poor thing, an ill-favoured thing wlten it is by itself. But it must be capable of having this independence. Thus Liljenroth ( 1918) in his seminal paper on multiplicity of steady states, can hardly be said to have a mathematical model, unless a graphical representation of the case is a model. He works out the slope of the heat removal line from the ratio of numerical values of a heat of reaction and a heat capacity. Certainly he is dealing with a typical case, and his conclusions are meant to have application beyond this particu-larity, but the mechanism for doing this is not there.

To say this is not to detract from Liljenroth's paper, which is a landmark of the chemical engineering literature; it is just to notice a matter of style and the point at wlticlt a mathematical model is born. For in the next papers on the question of multiple steady states, those of Wagner (1945), Denbigh (1944, 1947), Denbigh eta/. (1948) and Van Heerden (1953) we do find more general structures.

How powerful the life that is instinct in a true mathematical model can be seen from the Fourier's theory of heat conduction, where the mathematical equations are fecund of all manner of purely mathematical developments. At the other end of the scale, a model can cease to be a model by becoming too large and too detailed a simulation of a situation whose natural line of development is to the particular rather than the general. It ceases to ha,e a life of its own by becoming dependent for its vitality on its physical realization. (The emphasis is ours). Maynard Smith ( 1974) was, I believe, the first to draw the distinction in ecological models between those that aimed at predjcting the population level with greater and greater accuracy (simulations) and those that seek to disentangle the factors that affect population growth in a more general way (models). The distinc-tion is not a hard and fast one, but it is useful to discern these alternatives".

The basis of the classification given by Aris is interesting and useful. It is typical of the Minnesota school which actually revolutionized the

field of mathematical modelling in chemical engineering with their extensive work in the field since the mid l940s to-date. According to the above definition of Aris, borrowed from Smith (1974), the work of the Minnesota school can be classified under mathematical modelling rather than simulation. They have almost never verified their models (or the new phenomena resulting from it) against experimental or industrial data. In fact this important school in the history of chemical engineering mathematical modelling went through two phases. The first phase was dominated by the study of well established mathemat-ical models such as the homogenous continuous stirred tank reactor (CSTR) and the extraction of interesting new phenomena from the equations of this model. The second phase was characterized by the development of new models for complex processes (fluidized bed reactors, polymerization reactors. etc) and the investigation of their behaviour.

In both phases, experimental verification was almost completely absent. Experimental verifications of the new steady state and dynamic phenomena discovered by the Minnesota school were carried out at other universities, mostly by graduates from Minnesota. The most interesting outcome is the fact that not a single phenomenon discovered theoretically using mathematical models by the Minnesota group, was not experimentally confirmed later. This demonstrates the great power of mathematical modelling as expressed by Aris ( 1991) where the model is stripped of its detail in order to investigate the most fundamental characteristics of the system. The Minnesota school using this approach, has achieved a real revolution in chemical engineering in general and in mathematical modelling in particular.

Like aJl great achievements, those in mathematical modelling of chemical engineering systems with their emphasis on the more fundamental and more general characteristics of the systems, have created some unfavourable reactions from more practically oriented chemical engineers, especially in industry, who considered mathemat-ical modelling to be a theoretical discipline irrelevant to industry!

This situation is changing today due to more cooperation between academia and industry and the extensive introduction of computers into the petrochemical and petroleum refining industries.

In this book our definition (for mathematical modelling and simulation), will be more pragmatic and will serve our specific purpose. This does not mean that we disagree with the definition of Aris (1991 ), but rather that our treatment is directed toward the needs of the petrochemical and petroleum refining industries, and hence we wiU give little, if any, emphasis to the mathematical modelling in the sense of Aris. The definition we will adopt is that mathematical modelling will involve the process of building the model itself, while simulation involves simulating the industrial (or experimental) unit using the developed model. Thus, simulation in this sense is actually closely linked to the verification of the model against experimental and

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industrial units. However, it will also include the use of the verified models to simulate a certain practical situation specific to a unit, a part of a production Une or an entire production line.

The book is divided into seven chapters and four appendices. Chapter 1 introduces the reader to system theory and the classifi-

cation of systems. It also covers the main principles for the develop-ment of mathematical models in general and for fixed bed catalytic reactors in particular.

This chapter is not essential from a strictly technical point of view for the reader to follow the rest of the chapters, however, it is vital for the understanding of the general approach adopted in this book.

Chapter 2 gives a brief discussion of chemisorption and catalysis. Chapter 3 deals with the intrinsic kinetics of gas-solid catalytic

reactions. The basic principles for the development of kinetic models for the intrinsic rates of reactions, which represent the heart of the catalytic process, are presented. Detailed information for a number of industrially important catalytic reactions is given.

ln addition to being a compilation of important information regarding these reactions, this chapter serves as an illustration of the effort needed before adopting a certain kinetic rate model. This effort varies from one reaction to the other, depending on the degree of maturity of the modelling of the kinetics under consideration.

Chapter 4 deals with the practical relevance of bifurcation beha-viour and instability problems in chemically reacting systems. The chapter is brief and simple, however it covers a wide spectrum of information regarding this topic. The importance of this chapter relies on the fact that although theoretical studies on bifurcation behaviour bas advanced tremendously during the last three decades. the indus-trial appreciation remains very limited.

Although the book is not aimed at resolving this gap, it is of great importance that practically oriented chemical engineers dealing with the mathematical modelling of catalytic reactors become aware of this phenomena which is not only of theoretical and academic importance but has important practical implications.

Chapter 5 is dedicated to the single particle problem. the main building block of the overall reactor model. Both porous and non-porous catalyst pellets are considered. The modelling of diffusion and chemical reaction in porous catalyst pellets is treated using two degrees of model sophistication, namely the approximate Fick:ian type description of the difrusion process and the more rigorous formulation based on the Stefan-Maxwell equations for diffusion in multicom-ponent systems.

Chapter 6 is concerned with the integration of the single catalyst pellet model into an overall reactor model and presents a number of detailed examples of the modelling and verification of important industrial catalytic reactors. Optimization of these reactors is also described together with the development of user friendly software packages.

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C haptct 7 bnclly dcall. with the catalyst deacLtvation problem and presents elementary information on the use of heterogeneous models for the optimization of reactors experiencing catalyst deactivation.

The appendices present the parameters and empirical correlations necessary for the models discussed in the book. They also gtve basic information on the use of the orthogonal coUocation technique for the solution of non-linear two-point boundary value differential equations which arise in the modelling of porous catalyst pellets and the estimation of effecuveness facrors. The application of orthogonal collocation techntques to equations resulting from the Ficldan type model as well as models based on the more rigorous Stefan-Maxwell equations are presented.

The emphasis in this book is clearly on the development of rigorou~ heterogeneous models for petrochemical reactions where the feed-stock, reaction network and products are, to a great extent, well defined. Petroleum refining catalytic processes (e.g. catalytic cracking, hydrocracking etc.) are much more involved due to the great complexhy of the feedstock, reaction networks and products which necessitate the usc of pseudo-components with the associated loss of rigor.

ln cases where enough physico-chemical and kinetic data based on pseudo-components are available, then the same techniques discussed in this book can be used for these petroleum refining catalytic reactions, but cautiously.

It is hoped that the structure of this book will be useful to people in academia and industry mterested in the modelling, simulation and optimization of industrial fixed bed catalytic reactors. The book can be used for advanced academic courses on industrial fixed bed catalytic reactors as well as for advanced training courses for industrial engineers.

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CHA.PTER 1

Systems Theory and Principles for Developing Mathematical Models of Industrial Fixed Bed Catalytic Reactors

PerhapJ maJhematir.s 1s effective in organizing phrsicaf exi,lt(nce because It Is msptred by physical existence.

The pragmatic reality is that mathematics 1s rlw most effect/l'e and mmworthy method that we know for rmder-standing what we see around liS.

Ian Srewan (1989)

Mathematical modelling of diffusion and reaction in petrochemical and petroleum refining systems is a very strong tool for design and research. It leads to a more rational approach for the design of these systems in addition to elucidating many important phenomena associated with the coupling between diffusion and reaction. Rigor-ous highly sophisticated mathematical models of varying degrees of complexity are being used in industrial design as well as academic and industrial research (e.g. Eigenberger, 1981 a,b~ Muller and Hofmann, 1986; Salmi, 1988). An important point to be noticed with regard to the state of the art in these fields, is that steady state modelling is more advanced than unsteady state modelling due to the additional complexities associated with unsteady state behaviour and the additional physico-chemical information necessary (Elnashaie, 1977; Eigenberger, 1981 a,b; Baiker and Bergougnou, 1985).

Based on these advancements, continuous processing has progressed in these two fields over the years and became the dominant processing mode. Computerized design packages have been developed (Einashaie and Alhabdan, 1989a,b) and computer control of units and whole plants has been introduced widely (e.g. Jutan eta/., 1977; Schnelle and Richards, 1986; Richards and Schnelle, 1988). This has been coupled with a considerable increase in the productivity of industrial units and plants with a reduction in manpower and tight control over the product quality.

On the industrial and academic level, these advancements have led to innovative designs and configurations that open new and exciting avenues for developing compact units with very high productivity

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(Boreskov and Martos, 1984; Itoh et al. , 1985; Itoh, 1987; Elnashaie and Adris, 1989).

Despite these advances there is stiU a considerable gap between academic progress and industrial applications regarding the use of rigorous models for the design, simulation and optimization of catalytic reactors. Amundson ( 1984) stresses this by saying:

"The theoretical side of chemical reaction engineering is well in hand or will be shortly. Unfortunately the experimental and practical implementation of these results lags. Universities are not equipped to operate chemical reactors on a scale necessary to validate models appropriate to the industrial scene. The chemical and petroleum companies are loath to present proprietary results and for good reason. This is regrettable, and one cannot fail to notice that there is not one industrial paper presented in this volume".

Amundson wrote this almost 8 years ago, the situation to date is to a considerable extent different with regard to laboratory experimen-tal results, however, on the industrial front the situation remains almost the same. It is hoped that this book with its emphasis on gearing the accumulated fundamental knowledge towards the mod-elling, simulation and optimization of industrial fixed bed reactors will be one step in the direction of bridging this gap between academia and industry.

A system approach will be adopted which treats the fixed bed reactor as a system consisting of subsystems with their properties and interactions giving the overall system (the reactor) its characteristics. Before describing details, it is important to give a brief discussion of system theory and the principles of mathematical models with emphasis on fixed bed catalytic reactors.

1.1 SYSTEMS AND MATHEMATICAL MODElS

Here the basic concepts of system theory and principles for mathematical modelling are presented for the development of diffusion reaction models for fixed bed catalytic reactors.

1.1.1 A Brief Backgound

A system is a whole consisting of elements or subsystems. The concept of system-subsystem-element is relative and depends upon the level of analysis. The system has a boundary that distinguishes it from the environment. It may exchange matter and/or energy with the environment depending upon the type of system from a thermodynamical point of view. A system (or subsystem) is described by its elements (or subsystems), the interaction between the elements

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and its relation with the environment. The elements of the system can be material elements distributed topologically within the bound-aries of the system and giving the configuration of the system, or they can be processes taking place within the boundaries of the system and defining its function. They can also be both, together with their complex interactions. An important property of the system, whole-ness, is related to the principle of the irreducibility of the complex to the simple, or of the whole to its elements, i.e. the whole system will possess properties and qualities not found in its constituent elements. This does not mean that certain information about the behaviour of the system cannot be deduced from the properties of its elements, but it rather adds to it (Biauberg et a/. , 1977).

Systems can be classified on different basis. The most fundamental of whkb is that based on thermodynamic principles and on this basis they can be classified into (Prigogine el a/., 1973; Nicolis and Prigogine, 1977):

1. Isolated systems:

These are systems that exchange neither energy nor matter with the environment. The simplest chemical reaction engineering example is an adiabatic batch reactor. These systems tend towards their thermodynamic equilibrium which is characterized by maximum entropy (highest degree of disorder).

2. Closed systems:

These are systems that exchange energy with the environment through their boundaries but do not exchange matter. The simplest chemical reaction engineering example is a non-adiabatic batch reactor. These systems tend towards their thermodynamic equilibrium which is characterized by maximum entropy (highest degree of disorder), or more precisely they tend toward a state of minimum free energy.

3. Open systems:

These are systems that exchange both energy and matter with the environment through their boundaries. The simplest chemical reaction engineering example is the continuous stirred tank reactor. These systems do not tend toward their thermodynamic equilibrium, but rather towards a state called "stationary non-equilibrium state" and is characterized by minimum entropy production. Open systems near equilibrium have unique stationary non-equilibrium state, regardless of the initial conditions. However far from equilibrium these systems may exhibit multiplicity of stationary states and may also exhibit periodic states.

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It is clear from the above classification that batch processes are usually of the isolated or closed systems type while the continuous processes are usually of the open systems type. For continuous processes a classification from a mathematical point of view is very useful for both model formulation and algorithms for model solution. According to this basis, systems can be classified as (Douglas, 1972):

1. Lumped systems:

These are systems where the state variables describing the system are lumped in space (invariant in all space dimensions). The simplest chemical reaction engineering example is th~ perfectly mixed con-tinuous stirred tank reactor. These systems are described at steady state by algebraic equations while in the unsteady state they are described by initial value ordinary differential equations where time is the independent variable.

2. Distributed systems:

These are systems where the state variables are varying in one or more directions of the space coordinates. The simplest chemical reaction engineering example is the plug flow reactor. These systems are described at steady state either by an ordinary differential equation (where the variation of the state variables is only in one direction of the space coordinates, i.e. one dimensional models, and the independent variable is this space direction), or partial differen-tial equations (when the variation of the state variables is in more than one direction of the space coordinates, i.e. two dimensional models, and the independent variables are these space directions). The ordinary differential equations of the steady state of the one-dimensional distributed model can be either initial value differential equations (e.g. plug flow models) or two-point boundary value differential equations (e.g. models with superimposed axial dispersion). The equations describing the unsteady state of distrib-uted models are invariably partial differential equations.

Another classification of systems which is very important for deciding the algorithm for model solution, is that of linear and non-linear systems. The equations of linear systems can usually be solved analytically, willie the equations of non-linear systems are almost always solved numerically. ln this respect, it is important to recognize the important fact that physical systems are almost always non-linear and linear systems are either an approximation that should be justified, or the equations are intentionally linearized in the neighbourhood of a certain state of the system and are strictly valid only in this neighbourhood.

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A third classification, which is very relevant and important in chemical engineering, is the classification based on the number of phases involved within the boundaries of the system. According to this classification, systems are divided into (Froment and Bischoff, 1979):

1. Homogeneous systems:

These are systems where only one phase is involved in the processes taking place within the boundaries of the system. In chemical reactors, the behaviour of these systems is basically governed by the kinetics of the reactions taking place without the interference of any diffusion processes.

2. Heterogeneous systems:

These are systems where more than one phase is involved in the processes taking place. In chemical reactors, the behaviour of these systems is not only governed by the kinetics of the reactions taking place but also by the complex interaction between the kinetics and the relevant diffusion processes. The modelling and analysis of these systems is obviously much more complicated than for homogeneous systems. It is clear that the systems for fixed bed catalytic reactors fall into this category of heterogeneous systems, and more specifically into the category of gas-solid systems and therefore the behaviour of the system is dependent upon a complex interaction between kinetics and diffusion.

The reader interested in more details about system theory and the general concepts of mathematical modelling will find a large number of good books (Dransfield, 1968; Forrester, 1968; Von Bertalantfy, 1968; Robert et a/., 1978).

1.1.2 Mathematical Model Building: General Concepts

Building a mathematical model for a gas-solid reactive system, depends to a large extent on the knowledge of the physical and chemical laws governing the processes taking place within the boundaries of the system. This includes the diffusion mechanism and rates of diffusion of reacting species to the neighbourhood of active centers (or active sites) of reaction, and the chemisorption of the reacting species on these active sites for non-porous pellets, while for porous pellets it also includes the diffusion of reactants through the pores of the catalyst pellets (intraparticle diffusion), the mechanism and kinetic rates of the reaction of these species, the desorption of products and the diffusion of products away from the reaction

centers. It also includes the thermodynamic limitations that decide the feasibility of the process to start with, and also includes heat production and absorption as well as heat transfer rates. Of course, diffusion and heat transfer rates between the bulk gas and lhe surface of the pellet are both dependent to a great extent on the proper description of the fluid flow phenomena in the system. The ideal case is when all these processes are determined separately and then combined into the system's model in a rigorous manner. However, very often this is quite difficult with regard to experimental measurement, therefore special experiments need to be devised, coupled with the necessary mathematical modelling, in order to decouple the different processes interacting in the measurements.

Mathematical models of different degrees of sophistication and rigor were built to take their part in directing design procedure as well as directing scientific research in this field. It is important in this respect to recognize the fact that most mathemtical models are not completely based on rigorous mathematical formulation of the physical and chemical processes taking place in the system. Every mathematical model contains a certain degree of empiricism. The degree of which, of course, limits the generaJity of the model and as our knowledge of the fundamentals of the processes taking place increases, the degree of empiricism decreases and the generality of the model increases. The existing models at any stage, with this stage's appropriate level of empiricism, helps greatly in the advancement of the knowledge of the fundamentals and therefore helps to decrease the degree of empiricism and increase the level of rigor in the mathematical models. In addition any model will contain simplifying assumptions which are believed, by the model builder, not to alfect the predictive nature of the model in any manner that sabotages the purpose of the model.

With a given degree of fundamental knowledge at a certain stage of scientific development, one can build different models with different degrees of sophistication depending upon the purpose of the model building and the level of rigor and accuracy required. The choice of the appropriate level of modelling and the degree of sophistication required in the model is an art that needs a high level of experience. Models which are too simplified will not be reliable and will not serve their purpose. While models which are too sophisticated will present unnecessary and sometimes expensive overburden. Models which are too sophisticated can be tolerated in academia and may sometimes prove to be useful in discovering new phenomena. However, over sophistication in modelling can hardly be tolerated or justified in industrial practice.

1.1.3 Outline of the Procedure for Model Building

The procedure for the development of a mathematical model can be summarized in the following steps:

(1) Identification of the system configuration, its environment and the modes of interaction between them.

(2) Introduction of the necessary justifiable simplifying assumptions. (3) Identification of the relevant state variables that describe the

system. (4) Identification of the processes taking place within the boundaries

of the system. (5) Determination of the quantitative laws governing the rates of the

processes in terms of the state variables. These quantitative laws can be obtained from the literature and/or through an experimen-tal research program coupled with a mathematical modelling program.

(6) Identification of the input variables acting on the system. (7) Formulation of the model equations based on the principles of

mass, energy and momentum balances appropriate to the type of system.

(8) Development of the necessary algorithms for the solution of the model equations.

(9) Validation of the model against experimental results to ensure its reliability and to re-evaluate the simplifying assumptions which may result in imposing new simplifying assumptions or relaxing others.

It is clear that these steps are interactive in nature and the results of each step should lead to a reconsideration of the results of previous steps. In many instances steps 2 and 3 are interchanged in the sequence depending on the nature of the system and the degree of knowledge regarding the processes taking place within its boundaries.

1.2 BASIC PRINCIPLES OF MATHEMATICAL MODELLING FOR INDUSTRIAL FIXED BED CATALYTIC REACTORS

Fixed bed catalytic reactors have a wide variety of configurations ranging from the single bed adiabatic configuration to the multi-tubular, non adiabatic configurations with cocurrent or counter-current cooling or heating. The configurations can sometimes be complex such as the TVA type ammonia converters where there are internal cooling tubes immersed into the catalyst bed as we'll as an external beat exchanger. In other cases, the reactor may be quite simple with an adiabatic single bed or multiple adiabatic beds in series such as the high and low temperature shift converters. Intermediate degrees of complexity of reactor configuration are exemplified by reactors such as the multibed ammonia converters with interstage cooling.

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the development of mathematical models for all (or even an appre-ciable percentage of) the configurations used industrially. The most suitable is to present the basic modelling principles for what may be considered the heart of all these configurations, that is the catalyst bed itself, followed by some details and examples regarding the special features of each configuration. Different configurations of catalytic reactors differ in the arrangement of catalyst beds, the techniques used for the introduction of the feed and the means by which the beat is removed (for exothermic reactions) or added (for endothermic reac-tions).

In the first chapters ( 2, 3, 5) emphasis is given to the link common for all configurations, the catalyst bed itself. In chapter 6 specific industrial configurations are presented, modelled and analyzed.

The catalyst bed consists of the catalyst particles and the bulk gases passing through the voids of the bed. The reactants diffuse from the bulk gas to the surface of the catalyst pellet, then through its pores where it is chemisorbed and reacts forming the products which desorb and diffuse back into the bulk of the fluid.

From a systems point of view, the overall reactor can be considered a system, with the different parts of the entire reactor, e.g. the catalyst bed, cooling (or heating) tubes, external beat exchangers, condensers etc., being considered as subsystems. Obviously in this book (specif-ically in chapters 2, 3, 5), we consider the "subsystem": the catalyst bed, to be our main concern, and in what follows, we will consider it to be "the system". In chapter 6, industrial reactors are discussed and therefore the catalyst bed becomes a subsystem of the overall reactor system.

The catalyst bed as a system can be looked upon from a topological point of view and be considered as formed of subsystems: the catalyst pellets which are stationary and d istributed along the length of catalyst tubes (which can be considered as the second subsystem), and the flowing gases through the packed bed.

In fixed bed catalytic reactors, the function of the system is to transform a raw material A to a product P. This transformation takes place according to a number of consecutive or parallel processes, these processes can be considered to be the elements (or subsystem) of the function of the catalyst bed system. This is equivalent to considering the function of transforming A to P to be an overall process and then distinguishing the steps which lead to the required transformation.

Steps of catalytic reactions

At this stage it is useful to give a simple and relatively detailed verbal description of what is taking place withln the catalyst bed in order to achieve its final function:

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( 1) The reactant molecules transfer from the entrance of the reactor to the neighbourhood of the catalyst pellets. This transfer takes place by convection and/or diffusion. When axial diffusion is negligible and radial diffusion is instantaneous, we get the simplest descrip-tion for the bulk phase, that is one-dimensional plug;ffow.

(2) The reactant molecules diffuse from the bulk of the fl\.iid phase to the surface of the catalyst pellets. This process is usuaily described by mass transfer rate over a hypothetical external mass transfer resistance. The mass transfer coefficient is usually calculated using J-factor correlations (e.g. Hill, 1977). This step is dependent upon the properties of the gas mixture and the flow conditions around the catalyst pellets as well as the size and shape of the catalyst pellets. This step is usually referred to as external mass transfer of reactant molecules.

(3) For non-porous catalyst pellets the reactants are chemisorbed on their external surface. However, for porous pellets the main surface area is distributed inside the pores of the catalyst pellets and the reactant molecules diffuse through these pores in order to reach the internal surface of these pellets. This process is usually called intraparticle diffusion of reactant molecules. The molecules are then chemisorbed on the internal surface of the catalyst pellets. The diffusion through the pores is usually described by Fickian diffusion models together with effective diffusivities that include porosity and tortuosity. Tortuosity accounts for the complex porous structure of the pellet. A more rigorous formulation for multicomponent systems is through the use of Stefan-Maxwell equations for multicomponent diffusion. Chemisorption is de-scribed through the net rate of adsorption (reaction with active sites) and desorption. Equilibrium adsorption isotherms are usu-ally used to relate the gas phase concentrations to the solid surface concentrations.

(4) The chemisorbed molecules, whether on the external surface for non-porous pellets or the internal surface for porous catalyst pellets, undergo surface reaction producing cbemisorbed product molecules. Thls surface reaction is the truly intrinsic reaction step. However, in chemical reaction engineering it is usual practice to consider that intrinsic kinetics include this surface reaction step coupled with the chemisorption steps. This is due to the difficulty of separating these steps experimentally and the ease by which they are combined mathematically in the formulation of the kinetic model.

(5) The chernisorbed product molecules desorb from the surface into the gas phase adjacent to the catalyst surface.

(6) The chernisorbed product molecules undergo intraparticle back diffusion toward the surface of the catalyst pellets (for porous pellets).

(7) The product molecules, whether for a porous or non-porous pellet, diffuse through the external mass transfer resistance into the bulk gas phase.

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(8) The product molecules transport away from the pellet by convec-tion and/or diffusion in order to reach the exit of the reactor.

In addition to the above sequence of events, heat transfer resistances also play a part in the overall behaviour of the system. For exothermic reactions, the heat produced at the reaction centers dissipates through the pellet by conduction creating temperature gradient along the depth of the pellet. Then heat gets dissipated from the surface of the pellet to the bulk of the fluid creating temperature difference between the bulk fluid phase and the surface of the pellet. For endothermic reactions, the same sequence of events occurs creating temperature gradients which are opposite to those created for exothermic reactions.

It is this complex and interactive sequence of events that gives the overall behaviour of the reactor. The modeller must model these events in an accurate and rationally integrated manner in order to simulate the behaviour of the industrial reactor.

Some common physically justified simplifying assumptions in modelling industrial catalytic reactors

Fortunately, some of these processes are of such high rates that they can be neglected in the model formulation. A list of commonly encountered situations for straight forward simplifications is given below. However, a word of warning is essential here. There are of course situations where such simplifying assumptions are not valid, therefore the following cases should be considered as "commonly occurring" not " rules";

(1) Usually, dispersion in the axial direction is negligible in industrial reactors. This is because of the high flow rates and long length of the catalyst tubes, resulting in Peclet numbers which are high enough to justify the assumption of plug flow.

(2) Although radial dispersion has been investigated extensively, it seems in many industrial cases that it is appropriate to neglect radial concentration and temperature gradients and use a one dimensional model.

(3) In many cases, the external mass and beat transfer resistances are negligible because of the high gas flow rate that destroys the external resistances (see for example the ammonia converter, ch. 6, sec. 6.3.3, and the steam reformer, ch. 6, sec. 6.3.4). A counter example is the case of the partial oxidation of o-Xylene to phthalic anhydride, ch. 6, sec. 6.3.6, where external mass and heat transfer resistances must be taken into consideration for precise modelling of these reactors.

(4) In many cases the thermal conductivity of the catalyst pellet is

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kinetic model. This task may be as simple as using an intrinsic kinetic rate model together with the kinetic parameters from the literature as in the case of ammonia synthesis, ch. 6, sec. 6.3.3. It may be slightly more difficult necessitating the use of a full reactor model to extract the intrinsic parameters from industrial data as in the case of the water-gas shift reactions, ch. 6, sec. 6.3.1, and the dehydrogenation of ethylbenzene to styrene, ch. 6, sec. 6.3.5. It can also be more difficult, allowing the use of the functional form of the intrinsic kinetic rate model from the literature, while the kinetic parameters have to be estimated for the specific catalyst using laboratory bench scale differential or integral reactors and using an efficient algorithm, such as the Marquardt algorithm (Marquardt, 1963), for parameter estimation. An example for such a case is the partial oxidation of a-Xylene to phthalic anhydride where most of the published data are for specific catalysts prepared in the laboratories of the investigators (ch. 6, sec. 6.3.6.), while the activity as well as the selectivity of such catalysts vary widely from one catalyst to another. The task can be extremely difficult when the chemistry, mechanism and structure of the reaction network are not well established. In this case, an entire experimental research program aiming at establishing these missing links, should be initiated in conjunction with a project for the development of a reactor model.

All the above degrees of complexity apply reasonably well to petrochemical reactions where the feedstock and the products are usually well defined. However, in petroleum refining reactors such as catalytic cracking, hydrocracking, catalytic reforming, etc, the situa-tion is much more complicated and necessitates the use of lumping into pseudo-components (Hasten and Froment, 1984). This class of reactions is beyond the scope of this book, especially since the basic questions regarding the kinetic modelling of these reactions are not settled.

The second step the modeller is faced with in some reactors, is to decide on the suitable j-factor correlations to be used for estimating the external mass and heat transfer resistances between the bulk gas and the surface of the catalyst pellet (external resistances). In many industrial cases the answer to this task is rather trivial, that is, the external mass and heat transfer resistances are negligible. However, in other cases the appropriate choice of the j-factor correlation together with the correlations for estimating the change of physical properties along the length of the reactor (and also radially for two-dimensional models) due to the change of temperature and composition accompa-nying the reaction, are of great importance for the accurate modelling of the reactor.

This external mass and heat transfer problem is closely linked to the appropriate description of the conditions in the bulk phase especially with regard to the hydrodynamics and the axial and radial diffusion mechanisms.

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The third crucial step is that related to intraparticle diffusion of mass and heat. As mentioned earlier, most industrial catalysts will have large enough thermal conductivities to justify the assumption of flat temperature profiles inside the pellet. In the rare cases where this assumption is not valid, Fourier type heat conduction equations can be used with effective thermal conductivity which is best estimated experimentally if not available in the literature for the specific catalyst under consideration. However, intraparticle mass diffusion is quite important and although different types of discrete models are available (Pisarenko and Kafarov, 1991 ), the mosl appropriate and widely used model seems to be the continuum model (Evans et a!., 196la,b, 1962; Mason and Malinauskas, 1964; Malinovskaya et al. , 1975; Jackson, 1977; Kaza et a/., 1980). The continuum model when adopted for the catalyst pellets, has two levels of rigor and sophistication. The first is the Fickian type model which is the most widely used and the second is the dusty gas model based on the Stefan-Maxwell equations for the simultaneous diffusion in multicomponent systems. In bothcases effective diffusivities (molecular and Knudsen) must be used which necessitates estimating not only binary diffusivities, but also the porosity and tortuosity for the catalyst pellets. The porosity is quite easy to measure, but the tortuosity is more difficult and should be carefully estimated experimentally if not available in the literature for the specific catalyst.

This discussion regarding the level of rigor for the modelling of the intraparticle dJffusion can be related to the industrial cases presented in chapter 6 as follows:

( l) for an ammonia converter, section 6.3.3, where the system is formed of a single reaction, the use of the dusty gas model does not necessarily complicate the model. Also the results obtained are close to those obtained using the Fickian type model, and both are very close to those of the industrial reactor.

(2) for the steam reforming and methanation reactions (section 6.3.4.), it is clear that for this multiple reactions system, the model based on the Stefan-Maxwell equations is much more complicated in formulation and in solution compared with the model using Fick's law. The two models give almost identical results in certain regions of parameters, but differ considerably in other regions of parameters.

(3) for the high and low temperature shift converters (sections 6.3.1, 6.3.2), it is clear that the Fickian type model is quite sufficient to obtain very accurate results. Also, because it is a single reaction, the use of the Stefan-Maxwell equations will not add additional complexity.

(4) for the dehydrogenation of ethylbenzene to styrene (section 6.3.5) only the model based on the Stefan-Maxwell equations can be used because of the uncertainities associated with the kinetics of the reaction. Therefore it will not be wise from an industrial point

of view, to introduce extra uncertainties associated with the degree of rigor used in the diffusion-reaction model.

The next step for the modeller is to integrate the catalyst pellet equations with the bulk gas phase equations forming the necessary model equations for the catalyst bed module, the heart of the overall reactor as discussed earlier. The modelling of the bulk gas phase depends upon many factors related to the pellet to reactor tube diameter ratio, the velocity of the gas flow, the uniformity of the catalyst packing, the axial and radial Peclet numbers, the mode of operation (adiabatic, non-adiabatic) etc. It is adequate for most (but not all) industrial reactors to use a one dimensional plug flow model for the bulk gas phase. This is due to the fact that the key physical events taking place are those within the catalyst pellet, and therefore the crucial sides in the reliable modelling of gas phase catalytic reactors are those associated with the catalyst pellets. This does not mean that the gas phase flow conditions are not important, but rather it sets the main emphasis in modelling, for it is of very little use to invariably emphasize that "everything is important".

In the few case where more complex description of the bulk gas phase is necessary, the one-dimensional model of the bulk gas phase can be extended to a two dimensional model or the assumption of plug flow in the axial direction can be relaxed and an axial dispersion term superimposed. However, this should only be carried out when there are strong justifications, for although these extensions of the bulk gas phase modelling are quite simple mathematically, they increase the computational effort considerably and also require the determination of a larger number of parameters.

In addition to the above, the model for the catalyst bed module should include equations for the pressure drop across the bed. Although these equations are reasonably simple and their solution as a part of the model equation is a straightforward exercise, they are of crucial importance for without appreciation of the pressure drop consideration fast increases in reactor productivity can be theoretically estimated using the model with very fine particles. Obviously, this is not practically possible because of the excessive pressure drop associated with fine particles. In fact, it is the excessive pressure drop associated with small catalyst pellets that necessitates the use of relatively large catalyst particles in fixed beds. The use of these relatively large particles in tum, is the reason behind the existence of diffusional resistances and thus all the complexities associated with reliable modelling of industrial fixed bed catalytic reactors.

The integration of the catalyst bed module into the overall reactor model depends largely on the configuration of the reactor system and the mode of its operation. In principle, the adiabatic single bed reactor is the simplest since for this configuration and mode of operation the catalyst bed module represents the overall

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model of the reactor system, while usually the most complicated cases are those associated with non-adiabatic operations with counter-current cooling or heating.

The next step the modeller faces is the determination of all physico-chemical parameters and the suitable correlations for com-puting their changes with the variations in composition. temperature and pressure at different points in the reactor (in general axially and radially) and also along the depth of the catalyst pellets. These parameters include physical parameters such as specific heats, densi-ties, viscosities etc.; transport parameters such as diffusivities and thermal conductivities; kinetic parameters as discussed earlier as well as thermodynamic parameters such as equilibrium constants and heats of reactions.

When the modeller built his model to the chosen degree of sophistication, including the appropriate assumptions and approxima-tions and all physico-chemical parameters and their correlations are determined, he is then faced with solving the model. That is for a certain design configuration with its design parameters and feed conditions, what are the output conditions and the changes in the values of the variables along the reactor length? Tbis is the usual simulation problem. The design problem will consist of identifying the output conditions and determining for a certain configuration and design parameters what feed conditions will produce the desired output. The situation may also be that most of the feed parameters are determined from previous units and it is required to find say the volume of the reactor that gives the desired output. The problem may also be a combination of the above situations. Of course, it is also usually necessary to find the optimum operating conditions and/or design parameters that will give the maximum desired product, or to maximize a certain profit function or minimize energy consumption.

ln all cases, it is essential to solve the model equations efficiently and accurately. Some techniques are discussed in this book and in the appendices, for the solution of the highly non-linear algebraic differential and integral equations arising in the modelling of fixed bed catalytic reactors. The most difficult equations to solve are usually the equations for diffusion and reaction in the porous catalyst pellets, especially when diffusional limitations are severe. The orthogonal collocation technique has proved to be very efficient in the solution of this problem in most cases. In cases of extremely steep concentration and temperature profiles inside the pellet, the effective reaction zone method and its more advanced generalization, the spline collocation technique, prove to be very efficient.

With todays computers and the state of the art regarding numerical techniques, it does not seem that the numerical solution of the model equations presents any serious problems. With the fast development of computer hardware and software, this problem wiU become almost trivial in the near future.

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Before the developed model can be reliably put to any useful purpose, it has to be verified against industrial or pilot plant units.

The verification process is quite difficult, the first problem being associated with obtaining data from industry which is usually very conservative regarding this matter. Although Amundson (1984) states this problem and indicates some kind of sympathy with industry regarding this attitude, we do not think it right either in the short or long term. ln fact most of the "secrets" that industry is worried about are not publishable. What the model verification requires is the input-output, reactors dimensions and pellet sizes used. The investi-gator does not need the secret recipe for the catalyst preparation nor the specific secrets of the process. In fact, there are a number of industrially oriented publications where the omission of a few little details keeps the "interests" of the industrial side completely intact. The close co-operation between industry and researchers and the problems of exposing certain process secrets to the researcher can always be solved through "secrecy agreements" similar to that between the company and its employees. Inspection of the literature reveals an extreme shortage of industrial data which indicates that industry is not ready yet for this non-conservative attitude. Industry has a lot to gain in the short and long term in becoming more flexible and pragmatic regarding this issue.

When the first obstacle is overcome and the industrial data is in hand, the modeller has work to do. First he has to make sure that the data is consistent. This may be a simple consistency test on the mass and heat balances of the overall system, or it may go deeper into the process and the events taking place according to the modeller's physical visualization of the system in order to verify the consistency of the data, otherwise he will have to go back and discuss the data with the industrial people operating the specific reactor. Suitable physico-chemical data must then be found to put into the model. Usually this step does not represent a serious problem except with regard to the intrinsic kinetics of the specific catalyst. This was discussed earlier and may range from using intrinsic kinetic model with its kinetic parameters from the literature to developing a whole experimental kinetic modelling program in order to obtain the necessary kinetic model and its parameters.

Using the supposedly efficient numerical algorithm, the model equations with the operating and design parameters of the industrial unit are solved and the output is compared with the output of the industrial unit. It is not unusual even with the utmost care in model formulation, choice of the physico-chemical parameters and the use of an accurate solution algorithm, that the predictions of the model differ from the industrial data. Blind empirical fitting using one or more adjustable parameters will make the model Jose almost all its

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CHAPTER 2

Chemisorption and Catalysis

Although catalysis is a subject of tremendous industrial importance, it has long been regarded by the gent>ral scientific population as being the last strong hold of ale/rem).

David Trimm (1980)

Three main subjects are discussed in this chapter: the effect of adsorption on catalysis, the distinction and the differences between physical adsorption and chemisorption with regard to rates of adsorption, and heats of adsorption.

As the surface of the catalyst plays an important role in the catalytic reaction the effect of surface coverage especially on the heat of adsorption is discussed in the second part.

2.1 ADSORPTION AND CATALYSIS

During the nineteenth century, attempts to explain the action of heterogeneous catalysts were based on one or other of two general theories.

The intermediate compound theory (Ashmor, 1963) proposed that the reaction took place betw~~n the bulk solid and the reactant to give an intermediate compound. This intermediate compound decomposed or reacted with any other necessary reactant to give the product of the main reaction and to regenerate the catalyst. As long as the interme-diates were considered as bulk compounds, the intermediate com-pound theory was of limited applicability.

The other theory had its origin in ideas put forward in 1834 by Faraday (Ashmor, 1963). He assumed the condition of adhesion between solid and fluid. This condition leads, under favourable circumstances, to the combination of bodies simultaneously subjected to the attraction. The chief evidence that this simple idea is inadequate, is that some substances can decompose to give quite different products in the presence of different catalysts.

The nature of this specific interaction became clear after Langmuir's work on adsorption and its application to chemical reactions. Exper-iments showed, however, that the adsorption often reached a constant

24

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maximum value as the pressure increased. Langmuir attributed this kind of adsorption with saturation to short range attractions between surface and adsorbent giving bonds which are essentially chemical in nature and limited in number by the number of sites available for bonding on the surface. This type of adsorption has become univer-sally known as chemisorption to distinguish it from physical adsorp-tion (Langmuir, 1929; Taylor and Langmuir, 1933).

In principle, therefore, the study of catalyzed reactions is intimately bound up with studies of chemisorption. The rate of the reaction may be controlled by the rate of chemisorption of the reactants, or the rate of reaction between chemisorbed molecules or by the rate of desorp-tion of the product.

2.2 PHYSICAL AND CHEMICAL ADSORYflON

Adsorption is due to attraction between the molecules of the surface and those of the fluid. If this attraction is mild (of tbe same nature as that between like molecules) this is called physical adsorption . In other cases, the forces of attraction are more nearly akin to the forces involved in the formation of chemical compounds; this is called chemical adsorption or chemisorption.

The adsorbing molecule loses entropy since its motion on the surface is more restricted than in the gas phase. The free energy of the system also decreases as the surface valencies become saturated so it can be concluded that the adsorption process is always exothermic. There is no single criterion which distinguishes between physical adsorption and chemisorption in aU systems, but there are a few which are generally valid:

(i) The magnitude of the heat of adsorption: when the bonds are physical, the beat is a little more than the latem beat of condensation of the sorbate, usually about 1.5 times the latent heat. For chemisorption however, the heat may be as large as ten times the latent heat.

(ii) The rate of the process: physical adsorption is fast while the chemisorption process bas an appreciable energy of activation which limits the rate at low temperature and leads to a rapid increase in rate with temperature. Physical adsorption is unlikely to occur to any appreciable extent at temperatures above the boiling point of the sorbate.

(iii) Since electrons are shared between solid and sorbate molecules, no more chemisorption can occur when the surface of the solid has been covered with enough sorbate to satisfy the residual valency requirement of the surface atoms. Chemisorption cannot result in more than one layer of sorbate molecules although the situation does not preclude additional layers of physically ad-sorbed molecules forming.

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2.3 HEATS OF ADSORPTION AND DESORPTION

In all physical adsorption and in most chemisorptions heat is evolved. Heat release upon spontaneous adsorption would be expected for the following reasons. There must be a descrease in the free energy of the system adsorbent-adsorbate for the spontaneous process at constant temperature and pressure, and as the adsorbate is more localized it loses some of its translational entropy and some of its rotational entropy. Thus !lG and Mare both negative and since ll.H !lG + TM then 6.H must also be negative and therefore heat is released.

The activation energy of desorption Ed is related to the heat ( - 6H)11 and the activation energy of adsorption E0 by the equation,

(2.1}

Since adsorption is always exothermic, Ed is appreciable even when E11 - 0. That is desorption is always activated.

2.4 THE KINETICS OF ADSORPTION AND DESORPTION

The adsorption and desorption process may be expressed by:

where S defines an active

modelling, simulation and optimization of industrial fix catalytic

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