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Topics in Chemical Engineering
A series edited by R. Hughes, University of Salford, UK
Volume 1
HEAT AND MASS TRANSFER IN PACKED BEDS by N. Wakao and S. Kaguei
Volume 2
THREEPHASE CATALYTIC REACTORS by P.A. Ramachandran and R.V. Chaudh ari
Volume 3
DRYING: Principles, Applications and Desi gn by Cz. Strumillo and T. Kudra
Volume 4
THE ANALYSIS OF CHEMICALLY REACTING SYSTEMS : A Stochastic Approach by L.K. Doraiswamy and B .K. Kulkarni CONTROL OF LIQUIDLIQUID EXTRACTION COLUMNS by K. Naj im
Volume 5
CHEMICAL ENGINEERING DESIGN PROJECT: A Case S tu dy Approach by M.S. Ray and D W. Johnston
Volume 6
.
MODELLING, SIMULATION AND OPTIMIZATION OF INDUSTRIAL FIXED BED CATALYTIC REACTORS by S . S .E.H. Elnashaie and S.S. Elshishini
Volume 7
Volume 8
THREEPHASE SPARGED REACTORS edited by K.D.P. Ni ga m and A. S c hu mp e
Volume 9
DYNAMIC MODELLING, BIFURCATION AND CHAOTIC BEHAVIOUR OF GASSOLID CATALYTIC REACTORS by S.S.E.H. Elnashaie and S.S. Elshishini
This book is part of a series. The publisher will accept continuation orders
which may
be
cancelled at any time and which provide for automatic billing
and shipping of each title in the series upon publication. Please write for details.
Dynamic Modelling, Bifurcation and Chaotic Behaviour of GasSolid Catalytic Reactors
S.S.E.H. Elnashaie King Saud University, Riyadh, Saudi Arabia
and S.S. Elshishini Cairo University, Egypt
GORDON AND BREACH PUBLISHERS Australia
•
Canada • China
Japan • Luxembourg Singapore
•
•
•
France
Malaysia
•
•
Germany • India
The Netherlands • Russia
Switzerland • Thailand
•
THE PETROLEUM INSTITUTE UBRARY
United Kingdom
Copyright© 1996 by OPA (Overseas Publishers Association) Amster dam B.V. Published in The Netherlands under license by Gordon and Breach Science Publishers SA. All rights reserved . No part of this book may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying and recording, or by any information storage or retrieval system, without permission in writing from the publisher. Printed in Singapore. Emmaplein 5 1075 AW Amsterdam The Netherlands
British Library Cataloguing in Publication Data
Elnashaie, S . S. E. H. Dynamic Modelling, Bifurcation and Chaotic Behaviour of GasSolid Catalytic Reactors.  (Topics in Chemical Engineering, ISSN 02775883 ; Vol. 9) Title II. Elshishini, S. S . III. Series
I.
660.2995 ISBN
2884490787
The scientist does not study nature because it is useful; he studies it because he delights in it, and he delights in it because it is beautiful. If nature were not beautiful, it would not be worth knowing, and if nature were not worth knowing, life would not be worth living. Poincare He gets full marks who mixes the useful with the beautiful. Horace
To our dear late father Salah Eldin Elnashaie (19191993) A businessman who loved scientific knowledge
Contents
Introduction to the Series Preface Notation
xiii xv xvu 1
INTRODUCTION CHAPTER 1 1.1
1.2 1.3
Stationary Equilibrium States Stationary NonEquilibrium States Main Conclusions of Chapter 1
CHAPTER 2
2.1 2
.
2
2.3
2.4 2.5 2.6 2.7
2.8 2.9
ELEMENTARY CHEMICAL REACTORS DYNAMICS
STATIC AND DYNAMIC BIFURCATION AND THE DIFFERENT TYPES OF NONCHAOTIC ATTRACTORS
Point Attractors (static bifurcation) Summary of some of the M ai n Components of Static Bifurcation Simple Detailed Anal ysi s of Steady States on Bifurcation Diagrams in Chemical Reactors Dynamic Implications of the Coexistence of Multiple Stable Point Attractors Local Stabi l ity of Steady States Basic Pri ncip les of D e generacy and P arametric Dependence . Periodic Attractors of Autonomous Systems 2. 7. 1 Supercritical Hopf bifurcation 2. 7.2 Subcritical Hopf bifurcation Different Types of Periodic Attractors Some more Details on the Classification of some Types of Dynamic B ifurcation Diagrams in thei r Relation to the Static Bifurcation Diagrams 2.9. 1 Dynamic bifurcation diagrams for cases with unique steady states over the entire range of the bifurcation parameter vii
17
17
20
57
59 59 61 76
80 86
89
91 94 94 95 1 02 1 02
Dynamic bifurcation diagrams for cases with multiple steady states (multiple fixed points) 1 06 More on the PoincareAndrononvHopf Bifurcation (Hopf Bifurcation). A pure imaginary pair of ei genv alue s 1 1 2 Co mp utati on of the Period of Periodic Attractors 1 19 121 Stability of Periodic Orbits 1 23 The Two Parameter Continuation Diagram (TPCD) 123 2. 1 3 . 1 Static bifurcation loci on the TPCD 2. 1 3.2 Dynamic bifurcation loci on the TPCD 1 25 Numeri c al Construction of Static and Dynamic Bifurcation Diagrams 131 Some Important Elementary Dynamical Features ( non c haoti c dynamics) of ThreeDimensional S ystems 133 Dege n erate Hopf Bifurcations 1 43 2. 1 6. 1 Type 1 degeneracy :H1 144 2. 1 6.2 Type 2 degeneracy :H2 1 49 2. 1 6.3 Type 3 degeneracy :H3 150 Quasi Periodic Attractors for Non Autonomou s Systems, Periodic Forc i n g of Autonomous Systems with Periodic Attractors 152 158 2. 1 7 . 1 Neimark or secondary Hopf bi furcation 1 60 2. 1 7 . 2 The cyclic fold 161 2.1 7 .3 Flip bifurcation The Stability of Periodic Attractors in Autonomous and NonAutonomous S ys tem s and th e Construction of Ex c itati on Diagrams for NonAutonomous Sy stem s (peri od ic forc ing of aut on om ou s systems with periodic attrac tors) 161 1 68 Strange Chaotic and NonChaotic Attractors 171 2. 1 9 . 1 Presentation techniq ues 2. 1 9.2 The discretetime models and their relevance to the analysis of continuous systems 173 M o del s Based on First Order Difference Equations 1 73 2.20. 1 Conse rv ative and di ssip ative dynamical s y s tems 1 74 2.20.2 Hi g her order continuous dy n ami c al systems ( many to one maps) 1 83 2.20.3 Quantitative uni v ers al i ty and qualitative u ni versality 1 84 2.20.4 More on the characteristics of the l og i stic map 1 86 2.20.5 The control phase space 1 92 2.20.6 The superstable 2n c ycle 1 94 2.20.7 Feigenbaum univ ersality 1 97 2.20.8 Tangent bifurc ati on s intermittencies 203 2.20.9 More on the connecti on b etwee n c ontinu ou s and discrete time s y s tems 206 2.9.2
2.10 2.11 2. 1 2 2. 1 3 2. 14 2. 1 5

2.16
2.17
2. 1 8
2. 1 9
2.20




,
MODELLING AND ELEMENTA RY DYNAMICS OF GASSOLID CATALYTIC REACTORS C s
CHAPTER 3
3.1
3.2
219
223 Single ataly t Particle 3 .1. 1 Nonporous catalyst particle 223 3 . 1 . 1 .1 The symmetrical case 225 3 .1. 1 .2 The asymmetrical case 245 3. 1 .2 Porous catalyst particle. Lumped parameter models 25 1 3 . 1 .2. 1 The importance of surface processes on the dynamic behaviour of catalyst particles 25 1 3 . 1 .2.2 Dynamic modelling of porous catalyst particles with negligible intraparticle, mass and heat transfer resistances and equilibrium adsorptiondesorption. The lumped parameter adsorptiondesorption equilibrium model (LPADEM) 257 3.1.2.3 Effect of nonequilibrium adsorption desorption. The lumped parameter non equilibrium adsorptiondesorption model (LPNEADM) 270 3 . 1 .3 Porous catalyst pellets. Distributed parameter 28 1 models (symmetrical) 28 1 3 . 1 .3 . 1 The dynamic model 285 3 . 1 .3.2 Steady state 3.1.3.3 Brief survey of the main investigations 287 on the subject 3 . 1 .3.4 Application of two numerical techniques for the solution of the dynamic model equations for the distributed parameter, 290 porous catalyst pellet 3 .1.3 .5 Compact presentation of steady state results. The effectiveness factor 296 Thiele modulus diagram 3. 1 .3.6 The effect of adsorption heat release on the dynamic behaviour of the catalyst in different regions of the 297 1J ¢ di agram 304 3.1 .3.7 Simplified stability analysis 308 Fixed Bed Reactors 3.2.1 Classification of mathematical models for fixed 309 bed catalytic reactors
X
3.3
S.S .E.H. ELNASHAIE and S.S. ELSHISHINI
Analysis of fixed bed catalytic reactors using the simple cell model 3.2.2. 1 Mass and heat balance 3.2.2.2 Steady state analysis 3 .2.2.3 Stability analysis 3.2.2.4 Numerical simulation, results and discussion 3.2.3 Analysis of fixed bed catalytic reactors using the radiation cell model 3 .2.3 . 1 Numerical simulation, results and discussion 3 .2.3.2 Stability of the reaction zone to feed disturbances 3 .2.3.3 Stability analysis and wrong directional creep 3.2.3.4 The e ffe ct of intraparticle mass and heat transfer resistances on the velocity of creep of the reaction zone 3 .2.3.5 Static and dynamic bifurcation behaviour of the reactor 3.2.4 Analysis of fixed bed catalytic reactors using continuum models 3 .2.5 Summary and overview of the modelling of fixed bed reactors Fluidized Bed Reactors 3 . 3 . 1 Introduction 3.3.2 Modelling of fluidized bed catalytic reactors 3 . 3 .2. 1 Isothermal steady state models 3.3 .2.2 Nonisothennal dynamic models 3 .2.2
CHAPTER 4
4. 1
STATIC AND DYNAMIC BIFURCATION BEHAVIOUR AND CHAOS IN SOME GASSOLID CATALYTIC REACTORS
Fluidized Bed Catalytic Reactor with Exothermic Consecutive Reactions 4. 1.1 The twodimensional case with proportional control 4.1 .1.1 The unforced (autonomous) case 4. 1.1.2 The periodically forced (nonautonomous) case. Pre li minary presentation of periodic and chaotic characteristics
314 314 318 320 327 33 1 333 342 345 354 358 378 386 3 89 3 89 390 390 392
406 406 411 415
423
CONTENTS
xi
The periodi cally fo rced (nonautonomous) case. Detailed analy s i s of resonance horns, period doubling loci and chaos 438 4 1 2 The threedimensional case 464 The unforced (autonomous) case with proportional control 464 Industrial Fluid Catalytic Cracking (FCC) units 485 4.2.1 Evaluation of some important m athematical models for industrial FCC units 486 4.2.2 Preliminary p resentation of static bifurcation in 508 industrial FCC units 4.2.3 Industrial verification of the steady state model and more on the static bifurcation of 519 in du stri al u n it s 4 2 4 Effect of feedstock composition on static 528 bifurcation and steady state gasoline yield 4.2.5 Effect of fluidization hydrodynamics on static 535 bifurcation and steady state gasoline yield 4.2.6 Preliminary dynamic modellin g and dyn ami c 541 characteristics of i ndu stri al FCC units 4.2.7 Static and dynam ic bifurcation behaviour of 562 industrial FCC units Oscillations and Chaos During the Catalytic Oxidation 575 of Carbon Monoxide 4.3 . 1 Introductory review of experimental and modelling studies on the c atal y tic CO oxidation 576 4.3 .2 Experimental results for th e periodic and chaotic 580 behaviour during catalytic CO oxidation 4.3.3 Mathematical modelling for th e catalytic CO 582 oxidation Static and Dynamic Bifurcation Behaviour of the Industrial UNIPOL Process for the Pro duction o f Polyethylene in Fluidized Bed Reactors with 587 ZieglerNatta Catalyst 4 4 .1 Introduction and description of the process 587 4.4.2 Developme nt of the dyn ami c model 590 4.4.3 Some results and discussion of the static and dynamic bifurc ati on behaviour of industrial UNIPOL units 598 4.4.4 Prelimi nary simple optimization of the industrial
4.1 . 1 .3
4.2
4.3
4.4
.
.
.
.
.
UNIPOL
process
601
xii
S . S .E.H. ELNASHAIE and S.S. ELSHISHINI
APPENDIX A
DERIVATION OF EQUATION 3.2 FOR THE NONPOROUS CATALYST PELLET
APPENDIX B
APPENDIX C
APPENDIX D
APPENDIX E
APPENDIX F
DERIVATION FOR THE INTEGRAL
606
COLLOCATION FORMULATION
608
LOCAL STABILITY ANALYSIS FOR THE NONEQUILIBRIUM SINGLE CATALYST PELLET
610
STABILITY CONDITIONS FOR THE SIMPLE CELL MODEL OF THE FIXED BED CATALYTIC REACTOR
6 13
VELOCITY OF THE CREEPING REACTION ZONE IN FIXED BED CATALYTIC REACTORS
618
COMPUTATION OF LYAPUNOV EXPONENTS
622
REFERENCES
625
INDEX
64 1
Introduction to the Series The subject matter of chemical engineering covers a very wide spectrum of learning and the number of subject areas encompassed in both undergraduate and graduate courses is inevitably increasing each year. This wide variety of subjects makes it difficult to cover the whole subject matter of chemical engineering in a single book. The present series is therefore planned as a number of books covering areas of chemical engineering which, although important, are not treated at any length in graduate and postgraduate standard texts. Additionally, the series will incorporate recent research material which has reached the stage where an overall survey is appropriate, and where sufficient infor mation is available to merit publication in book form for the benefit of the profession as a whole. Inevitably, with a series such as this, constant revision is necessary if the value of the texts for both teaching and research purposes is to be maintained. I would be grateful to individuals for criticisms and for suggestions for future editions.
R. HUGHES
Preface A special debt of gratitude for the initiation and completion of this book goes to our brother Professor Elnaschie (Cornell, Cambridge), executive director of the Pergamon Press Journal Chaos, Solitons and Fractals. He inspired our interest in the chaotic behaviour of ph y sic al systems almost ten years ago and continues to encourage and in sp ire us for every step along the road. The depth, elegance and beauty of the work of Professor Rossler (Tubingen) has always been a source of encouragement insp iring us to try to combine the beautiful with the useful . We will alway s remember and appreciate the continuous encourage ment and valuable friendship of Professors Hughes (Salford) and Aris (Minnesota). We deep ly thank our teacher and dear friend Professor ElRifaie (Cairo) who introduced us to the dynamics of chemical engineering systems more than twentyfive years ago and c ontinues to give us his full v aluable support. Professors Ray (Wisconsin) and Marek (Prague) and Dr Cre sswe ll (ICI, England) introduced us to the field of modellin g and bifurcation an aly sis of gassolid catalytic reactors more than twenty years ago and we thank the m si n ce rel y for that and for their continuous help and encouragement. M any colleague s and students from the chemical engin ee ri ng and c hem istry departments of King Saud University (Saudi Arabia) and C airo U n i ve rsi ty (Egypt) including: Drs Abashar (n ow with USM, M al ay s ia ), Al habdan , Ab asaee d , Wagi al la , Ibrahim, Teymour (now with liT, Chicago), AlKhowaiter, AlHumazi, Alahwan y , Babikr (now with USM, Malaysia), AlFaris, AlZahrani and the late Dr Abdel Hakim and engineers: Noureldeen, Almutlaq, Ali, Moustafa and Elmu hanna, contributed dir ectly and i ndirectly to the completion of this book. They all deserve spe cial g ratitude . The extensive efforts of Dr Abashar with reg ard to the fluidized bed catalytic re ac tor results in C hap ter 4, Dr Abas aeed with regard to the FCC results C hapter 4, Professor Wagialla wi th regard to the fixed bed catalytic reactor results in Chapter 3 and Engineer Nayef Ghasem with regard to the UNIPOL Process in Chapter 4 should be reemphasized with our
in
XV
xvi
S.S.E.H. ELNASHAIE and S.S. ELSHISHINI
great appreciation. Last but certainly not least, the efforts, patience and continuous encouragement of the staff at Gordon and Breach is highly ap preci ated .
Said Elnashaie Shadia Elshishini
Notation •
All symbols h ave the definitions and units gi ven in the fo l lowing list of
Notation except otherwise stated inside the text
.
Fluid mass capacity coefficient, s Fluid (+wall) heat capacity coefficient, s Particle mass capacity coefficient, s Particle heat capacity coefficient, s
Constants for catalytic carbon, additiv e carbon and stri pp able A
hydroc arbons Preexponential factor
in surface reaction rate
constant Preexponential factor in ads orption rate constant Frequency factor in Arrhenius equation for coke b urnin g reacti on Overall frequency factor for crac king Frequency fac tor for deactivation Frequency factor in Arrhenius equ ation for cracking reactions, i = 1 for cracking to gasoline and i 2,3 for cracking to coke Area of bed occupied by bubbles in reactor and regenerator respectively, m2 Area of bed outside bubbles in reactor and regenerator respectively, m2 Forcing amplitude C oncentration Bubble phase concentration of component i in fluidized bed reactors Concentration of reactants in pore space, mol/m3 Concentration of A in the bulk ph ase of cell no. j, mol/m3 Concentration of A in the intraparticle void in partic le =
no. j, mollcm3
Bulk phase concentration, mol/m3 Specific heat of fluid, kcal/g. K
Concentration of gas xvii
oil
and gasoline, kg/m3
xviii
CAlf, CA2f Ccah CSC• Crc Co Crw CPf• Crl• Cps
S.S .E.H. ELNASHAlE and S . S . ELSHISHINI
Feed concentration of gas oil and gasoline, kg/m3 Weight % of catalytic carbon, carbon on spent
catalyst and carbon on regenerated catalyst
Weight ratio of coke necessary for complete de activation of catalyst
3 Concentration of oxygen, mol!m Heat capacity of air, liquid feed, vaporized feed and solids, kcal!kg. K Heat capacity of gases in reactor and regenerator, kcal!mol.K Weight ratio of coke to catalyst in reactor and regenerator, kg/kg Total concentration of active sites on surface, moll kg. catalyst Specific heat of solid, kcal!kg. K
Surface concentration (adsorbed A), mol/kg catalyst
Concentration of adsorbed A in cell no
catalyst
j, mol/kg
Concentration of available active sites, mol/kg catalyst Wall heat capacity, kcallkg. K
Effective diffusion coefficient, m2/s
:Qubble diameter in fluidized bed,
m
Bubble diameter in reactor and regenerator of FCC unit, respectively Activation energy for surface reaction, kcal!mol Activation energy for adsorption, kcal!mol Activation energy for desorption, kcal!mol Activation energy for coke combustion and cracking reactions,
i=
1 for cracking to gasoline, i = 2,3 for
cracking to coke, kcal!mol
Overall activation energy for cracking, J/mol Gas flowrate in the regenerator, kg/s Catalyst circulation rate, kg/s =
FGF (Iscol model  Figure 4.50a, Fd
lb!hr)
FcF, FcM, FAF Ftf F GcF. GAF
GGn, GG/, Gc
=
FCF in
Fresh feed, mixed feed and air feed flowrates, k g/s
Coke formation factor of total feed, kg Coke/m3 oil Volumetric flow rate, m3/s
Volumetric feed flowrate for gas oil and air, m3/s Volumetric regenerator gas flowrate in bubble phase,
dens e phase and overall, m3/s Volumetric
reac
tor gas fl owrate in bubble phase,
dense phase and overall, m3/s
xix
NOTATION
h
H HR,HG (Ml)w (Ml)A (Ml)n (Ml)T M{,,Mf;
Effective film heat transfer coefficient, kcaUm2•
s.K
Dimensionless height of heat transfer unit
Height of fluidized bed in reactor and regenerator, m Heat of adsorption,
kcal/mol
Heat of surface re action , kcal/mol
Overall hea t of reaction, kc aUmol
Heat of coke combustion and cracking reacti ons , i 1 for c rac king to ga soli ne , i = 2,3 for crack i n g to coke, kcal/kg Normalized rates of heat losses from reactor and regenerator, re sp ecti v ely , 1/s Normalized rates of heats of cracking and vapori zation, re spectiv ely, lis Degeneracy Tota l inve nto ry of catalyst, kg Imaginary part of A; Reaction rate const ant Rate constant for adso rp tion, m3/mol . s Reac ti on rate constant for backward reaction =
Desorption rate constant, lis
Reaction rate constant for forward reaction
Effe cti v e film mass transfer coeffi cient, m/s
Preexponential fa ctor for reaction Surface reaction rate constant, 1/s
Pseudohomogeneous rate constant, m3/kg. cata lyst. s K
Normalized proportional controller gain (or nor malized heat transfer c oeffi cien t) for the flu idized
bed reactor
Dimensionless heat transfer coefficient in th e poly
ethylene fluidized bed reactor
Adsorption equilibrium constant, m3/mol
Normalized proportional, integral and differential K* A
controller gai n for FCC unit, respectively
Preexponential factor in ad sorption equilibrium
constant
Rea ction v e locity constant for coke combustion and cracki ng reactions , i 1 for cracking to gasoline, i = 2,3 for cracking to coke =
Overall velocity constant for cracking reactions Lewis number based on adsorptive capacity of the internal surface Lewis number based on void volume of particle
S.S.E.H. ELNASHAIE and S.S. ELSHISHINI
XX
M
Mao• Mas• Me, Me, MA
Nu
·
Nu1 O�g
Phc PR,
Po
Pa
Pr
q
QQ+ Q
ra
R
rs
Rc
Ref
Rcr Rc Rp Re (Aj) Sc
Sh Sv t
Dimensionless height of mass transfer unit Molecular weights of gas oil, gasoline, gases, coke and air Catalyst hold up in reactor and regenerator, kg Molar flow rates of components A and B respec tively Molar holdup of components A and B respectively Gas hold up in reactor and regenerator, mol Number of cells travelled per minute (velocity of creep at time t) Nusselt number= R·hiAe Nusselt number= h Rlk.. Volume percent of oxygen in flue gases Partial pressure of hydrocarbons in stripper, Pa Pressure in reactor and regenerator, Pa Period of limit cycle Prandtl number Volumetric flow rate Heat rejection function Heat generation function Heat removed from regenerator by cooling coil, kcal/s Heat supplied by combustion and heat removed from system, kcal/s Mass and heat transfer interphase coefficient for fluidized bed reactor, 1/s Mass transfer interphase coefficient for reactor and regenerator of FCC unit, l/s Heat loss for unit, kcal/s Rate of reaction Rate of adsorption Rate of surface reaction Dimensionless radiation parameter Rate of coke combustion, kg/s Rate of coke formation, kgls Overall rate of cracking, kg/s Gas constant Particle radius, m Real part of Aj Schmidt number Sherwood number= R kq!D Liquid space velocity, voU vol.s ·
Time, s
Catalyst residence time, s
NOTATION
T
f
TcF. TAF TR, Tc Trt Tn h
1j
T,,
T,,j
v
v Vc VcF. VR, Vc, VAF w
Xss
Xen
XiB, XiD, Xi, Xif
xxi
Temperature, K Bubble phase temperature, K Feed te mperature of gas oil and air, K Reactor and regenerator temperature, K Reference temperature, K Bulk phase temperature, K Feed temperature, K Bulk phase temperature of cell no. j, K Solid tempe ratur e , K Solid temperature of particle no. j, K Bifurcation parameter Active volume, m3 Cell volume, m3 Molar gas flow rate of feed, reactor gases, regene rator gases and air, moVs Refractoriness parameter accounting for differences
in feed stocks for FCC units Weight fraction of coke to coke + gases Volume %of CO to CO+ C02 Weight %of H in coke Vari able Derivative of x with respect to time Deviation variable= x xo Variable evaluated at steady state Fraction of coke burnt Dimensionless concentration of component i, in the bubble phase, dense phase, output and fee d, i= 1 for gas oil, i = 2 for gasol ine Dimensionless concentration = C!Cref
X,,
Yi, YiF
Yo,Yo F y
[Y] Yn
YF
YFu
YRB, YRD· YR
Exit concentration  feed concen tration Dimensionless concentration j ust above the surface Dimensionless bulk concentration = Cn!Crt Coke on catalyst in reactor and regenerator, moV kg. catalyst Dimensionless surface concentration Mole fraction of com ponent i in unit and in feed Mole fraction of oxygen in unit and in feed Dimensionless temperature = T!Tref Exit fluid temperature  feed temperature Dimensionless bulk temperature, Tb!Trf Dimens ionless feed temperature
Base
value of the
dimensionless feed temperature
Dimensionless temperature in bubble phase, dense phase and output of reactor= TITrf
S . S.E.H. ELNASHAIE and S.S . ELSlll SHINI
xxii
Dimensionless temperature in bubble phase, dense
YGh YAF Y, Y.,
z
Greek
f3 /3A, f3a /3r aco
symbols
r
y,, E:
e () A.
Ae J1 J1 PF PL. PF, Pa Ps r r'
phase and output of regenerator=
TITrf
Dimensionless feed temperature of gas oil and air Dimensionless surface temperature Dimensionless
feed vaporization temperature
Axial bed dimension
Coke to oxygen stoichiometric ratio Exothermicity factor=
(!J.H)C,eJI(pCpTref)
Dimensionless heat of adsorption
Dimensionless overall heat of reaction
Dimensionless activation energy for surface reaction
=E,/RcTB Dimensionless activation energy for adsorption= EIRcTB
Exponent in the temperature dependent equilibrium
adsorption
coefficient= (!J.H) JRcTb
Particle voidage
lnterstatial voidage Surface coverage Eigenvalue
Effective solid thermal conductivity, kcallm. s. K Bifurcation parameter
Deviation in bifurcation parameter= J.L J.Lo
Fluid density, kg/m3
Density of liquid feed, vaporized feed and air in FCC unit, kg/m3 Solid density, kg/m3
Dimensionless time
Normalized ti me = t!P u (chapter 2) Thiele modulus Activity coefficein t accounting for catalyst decay
Catalyst activity in reactor and regenerator respec
tively for FCC unit
w Wu
V2
Abbreviations
Forcing frequency
Natu ra l freq uency
Laplacian operator for spherical geometry
BFM
Brute force method
CSTR
Continuous stirred tank reactor
DHB
Degenerate Hopf bifurcation
FCC
Fluid catalytic cracking
NOTATION HB
Hopf bifurcation
HC
Homoclinical
IP
Infinite period
PDB
Period doubling bifurcation
PFM
Principal Floquet multiplier
SBS
Static bifurcation point
SCP
Static cusp point
SLP
Static limit point
SBLP
Static bifurcation limit point
TPCD
Two parameter continuation diagram
TRB
Torus bifurcation
Subscripts
A,B
f
0
ref
Components Feed Initial conditions Reference conditions
xxiii
Introduction Practice has a way of catching up with theory and turning the theorems of yesterday into the criteria of tomorrow
Jorgensen and Aris (1983)
This may be the first book in the che mical engine ering literature devoted to d y n amic model li ng bifurcation and chaotic behaviour of chemical reactors (s pecific ally gas solid c atal yti c reactors). After many years of intensive theoretical inve stig ati on of the static and dynamic bifurcation behaviour of chemical reactors, s peci all y by the Minnesota group of Professors Amundson and Aris and their students, it became evidently clear in the eighties th at a strong gap has developed between the theoretical advances in this field and industrial/experimental practice, not only between academi a and industry but also within ac ade mia itself The researchers of the Minnesota school have certainly achieved a scientificall y honourable success in elevating chemical engineering thinking to higher levels of intellect. However their almost continuous use of the idealized CSTR, which is a must when develop ing the fundamentals of an important theory, had a negative effect on indus trialists and experimentalists with regard to their appreciation of these important ph enomen a This is of course with the e xc epti on of a few fine sc ie n ti st s (mostly graduates from the pioneering Mi nnesota school) who investigated and demonstrated these phenomena experimentally, e.g. Prof. Schmitz and his group at Notre Dame, U.S.A. with s pecial e mphas i s on the g as s oli d catalytic oxidation of CO, Profes s ors Silveston and Hudgi n s and their students at Waterl oo Canada, who investigated a number of gas s ol i d cataly ti c reactions and Professor Ray and his group at Wisconsin, U.S .A. w i th special emphasis on polymerization ,

.
.

,

reactions.
In the late eighties and early nineties it became high time to look at the industrial and practical relevance of these phenomena for important indu s tri al reactors . A limited number of papers appeared in the literature de al ing with the static and dynamic bifurcation behavi ou r of some industrial gassolid catalytic reactors, such as industrial fluid catalytic
2
S . S . E.H. ELNASHAIE and S .S. ELS HISHINI
(FCC) units, polyethylene production in fl ui dized bed cata lytic reactors, oxylene parti al oxidation in fixed bed catalytic reactors . . . etc., in addition of course to the continuous efforts to eluc idate the complex behaviour of the catalytic ox i dati on of carbon monoxide. However, the efforts for bri dgin g the gap between the theoretical and practical inve s tigations of static and dynamic b i furc ation remained limited due to many factors, which include: c racking
continued wrong belief academia and in dustry (and
among many chemical engineers in of course among research granting agencies), that this work is of theoretical value but of little practi cal value. b) the relative spreading of this wrong attitude among some editors and reviewers of chemical en g ineering journals in addition to the spreadi ng of an equally wrong belief among theoretic ally oriented editors and reviewers that investigations of static and dynamic behaviour of industrial units do not add much to the theory and fundamental knowledge developed using ge neric models. Therefore, the limited number of scientists working on bridging the gap between theory and practice in thi s important field, were confro nted from one side with practically oriented colleagues con s idering their work to be too theoretical and also unne cessarily too sophisticated and from the other side with colleagues c onsidering their work to be too practical and thus not theoretical enou gh . c) the attitude of applied mathemati c ians working in this field, usually under the title of " dynami c al system theory" and more recently under the title of "bifurcation theory" , who are almo st always con centrating on systems associated with electric circuits, fluid fl ow , mechanical and structural problems, . . . etc., and rarely on chemical engineering systems. Although this situation has been changi ng s l owly in the last decade, thanks to the efforts of chemical engi neeri n g researchers which led to the unc overi n g of the static and dynamic bifurcation richness of chemical and biochemical sys tems, it is still far from satisfactory . d) the i nadequacy of the undergraduate and postgraduate c urricula in most chemical enginee ring department s worldwide in covering the mathematical methods and tool s needed for the study of static and dynamic bifurcation of chemical engineering processes. This is an important factor contributing to the lack of sufficient appre ciation of these important phenomena among many chemi c al en gineers. a) the
These are some of the factors which we believe are responsible for the relatively wide gap between theoretical and practical work in this important field. Concerted efforts are needed between practically and
INTRODUCTION
3
theoretically oriented chemical engineers to overcome this gap in order for chemical engineering theory and practice to rise continuously to higher levels of scientific knowledge and practical achievements. In the late eighties a new commer entered the arena making the situation even more difficult, that is deterministic chaos. The discovery that deterministic dynamical systems can exhibit different types of chaotic behaviour in addition to previously known static and dynamic bifurcation phenomena, widened the gap further between theoretically and practically oriented chemical engineers. This state of affairs is in contradiction with the fact that a rich variety of such phenomena has been found to exist in chemical and biochemical reaction engineering systems, as clearly shown by a number of investigations (although limited) in the chemical engineering (and related disciplines) literature, in addition to the limited number of journals specialized in bifurcation and chaotic behaviour. The effect of s ome of the factors discussed above became deeper due to the need for new mathematical tools and techniques for the investigation and presentation of chaotic behaviour and to the apparent excessive complexity of the behaviour for some practically oriented chemical engineers. The lack of sufficient investigations demonstrating the practical relevance of this new phenomenon contributed negatively to the situation. The scatter of information, tools and techniques necessary for the understanding of these phenomena, over a wide range of journals and books outside the chemical engineering literature has made the situation even more difficult. We hope that our present book will be a modest contribution towards helping to solve some of the above problems and that it will be followed in the near future by better and more advanced chemical engineering books in this important field. We think it will be important for the reader at this early stage that we list some of the important practical examples of static and dynamic bifurcation and chaotic behaviour of chemical reactors. However, we should warn the reader that due to the limited effort expended in this direction so far, thi s does certainly represent a small part of the tip of the iceberg. The whole of the iceberg can only be discovered through concerted efforts of theoretically and practically oriented chemical engineering researchers in cooperation with the corresponding relevant industries:
1)
The
research of Professor R ay and his students at Wisconsin on liquid phase polymerization reactors has revealed muc h of the richness of static and dynamic bifurcati on and chaotic behaviour of these reactor s for a number of important polymers and copolymers (e.g. Jaisinghai et a/., 1 977; Hamer et al., 1981; Ray, 1981; Schmidt and
S . S . E.H. ELNASHAIE and S.S. ELSHISHINI
4
Ray, 198 1 ; Schmidt et al. , 1 984; B arand iaran et al. , 1 988; Teymour and Ray, 1992). 2) The research of Professor Ray and his students on the gas phase catalytic polymerization of ethylene and propylene in bubbling fluidized bed reactors (the celebrated UNIPOL industrial process of Union Carbide) has demonstrated clearly not only the static and dynamic bifurcation behaviour of this industrially important unit, but also the unstable nature of some of the desired steady states and the effect of the possible oscillatory behaviour of the system on design, operation, control and optimization of these units (Choi and Ray, 1985; Elnashaie and Ghasem, 1 995) . 3) The multiplicity o f the steady states for industrial fluid catalytic cracking (FCC) units has been demonstrated by the pioneering work of Iscol ( 1 970) and the research of Elnashaie, Elshishini and their coworkers (Elnashaie and ElHennawi, 1 979; Elshishini and El nash aie 1990a,b; Elshishini et al., 1 99 2; Elnashaie and Elshishini, 1 993a,b) on industrial type IV FCC units, in addition to the work of Edwards and Kim (1988) and De Lasa and coworkers (e.g. Arandes and De Lasa, 1992) for industrial FCC units with riser reactors. This research work coming from both academia and industry for one of the most important units in th e petroleum refining industry for the production of high octane number gasoline, has shown clearly that multiplicity of the steady states (static bifurcation) covers a very wide range of parameters and that in many cases the operating conditions for maximum gasoline yield correspond to the middle saddle type unstable steady state. Preliminary results for the dynamic bifurcation behaviour (including Hopf bifurcation and homoclinical termination of period attractors) for the industrial type IV FCC units have been presented by Elnashaie et al. ( 1995) and Abasaeed et al. ,
(1995 ).
4) The oscillatory and chaotic behaviour of the catalytic oxidation of CO, which is an important reaction for air pollution control of poisonous CO emission from cars and different industrial plants, has been investigated by many researchers (Jakubith, 1 970; Hugo and Jakubith, 1 972; Dauchot and Van Cakenberghe, 1 973; Plichta, 1976; Plichta and Schmitz, 1 979; Sheintush, 1 98 1 ; Turner et al., 1 9 8 1 ; Beusch et al., 1972a,b; McCarthy, 1 974; McCarthy et al. , 1 9 7 5; Varghese et al., 1978; Cutlip and Kenney, 1 978; Wicke et al., 1 980; Lisa and Wolf, 1 982; Rathousky and Hlavacek, 1982; Elhaderi and Tsotsis, 1982; Lynch and Wanke, 1984a,b; Vaporciyan et al., 1 988; Shanks and Baile y , 1989; Fi c hthom et al., 1989; Boulahouac h e et al., 1992; Onken and Wolf, 1992; Chen et al., 1993; Ehsasi et al., 1993; Vishnevskil et al., 1993; Imbinl, 1993; Gorodetskii et al., 1993; Uddin
et
al., 1993; Boudeville and Wolf, 1993; De Boer
INTRODUCTION
5)
5
et al. , 1993 ; Lauterbach et al. , 1 993; Block et al. , 1 993 ; Boehman et al. , 199 3 ; Krischer et al. , 1 99 3 ; Kapteij n et al. , 1 993). An i nteresti n g , very recent paper by Qin and Wolf ( 1 995) deals with the vibrational control (Meerkov, 1 980; Cinar et al. , 1 987) of the chaotic behaviour of c atalyti c CO o xi dation over Rh/Si02 catalyst . Vleeschhouwer et al. ( 1 992) demonstrated using a simple CSTR
model that the cobalt catalyzed oxo reaction exhibits periodic behaviour. The oxo reaction in vo lves the sequential co nve rsio n of a mixture of olefinic isomers to aldehydes and alcohols: olefins 7 aldehydes 7 alcohols. Despite the fact that the CSTR model is an oversimplification of the industrial oxo reactor, which is a gaslift loop reactor with integral heat exchanger and external recirculation loop, the investigators have demonstrated that the periodic behaviour predicted by this simple model agrees with the behaviour of two industrial oxo reactors. 6) Professor Sheintuch and his coworkers (e.g. Sheintuch, 1 987, 1 989a,b, 1 990; Gutfraind and Sheintuch, 1 99 1; Sheintuch and Wolffberg, 1 99 1 ; Shrartsmond and S hei ntuch 1994) carri ed out an impressive and extensive theoretical and experimental research work on static and dynamic bifurcation in addition to chaotic behaviour and spatiotemporal patterns for catalytic wires and fixed bed catalytic reactors. One of the important practical examples investigated by this research group is the catalytic partial oxidation of ethylene to ethylene oxide ( A daj e and S heintu ch , 1 990; S he i ntuch and Adaje, 1 990) . Ethylene oxide is an i mportan t intermediate for the production of ethylene glycol. 7) The advantage s of unsteady state operation of catalytic reactors by either forced feed oscillation or reverse flow of feed, have been studied by a number of investigators for a number of important catalytic reactions. The po s it i ve effects of this mode of ope rat i on on conversion and yield of the desired product together with the static and dyn ami c bifurcation characteristics and complexities associated with it, have been clearly demonstrated for a number of c ataly tic reactions (e.g. S ilve ston 199 1; Lang et al. , 1 99 1 ; SalehAlhamed, et al. , 1992; Neophytides and Froment, 1 992; Han et al. , 1 992; Matros et al. , 1993 ; Noskov et al. , 1993 ; Vanden Bussche et al. , 1993 ; Metzinger et al. , 1994; E i g enberger et al. , 1 994; Kers h enbaum et al. , 1 994; Purwono et al. , 1 994) . ,
,
Although i n thi s p art o f the i n troduction , we are conce ntrating on p_ractical importance of the ph enomena covered in this book, we cannot conclude this w i thout referring to the important analytical and numerical contributions carrie d out by a number of distinguished research the
groups:
S . S .E.H. ELNASHAIE and S . S . ELSHISHIN I
6
 the extensive analytical work of Luss and Balakotaiah (e g Luss, 1 980, 1 98 1 ; Balakotaiah and Luss, 1 98 1 , 1 982a,b, 1 983, 1 986) usi n g the el e gan t techniques developed by Golubitsky and Schaeffer ( 1 985) who are among the very few app lied mathematicians to use the generic CS TR model as one of their examples.  the extensive numerical work of Hlavacek and coworkers (e.g. Hlavacek and Votruba, 1 978; Hlavacek and Van Rompay, 1 98 1 ; Kubicek and Hlavacek, 1 983; Seydel and Hlavacek, 1 987), and also Marek and his coworkers (Kubicek and Marek, 1 983; Marek and Schreiber, 1 99 1 ) have c ertai nly contributed to the advancements in this fiel d .
.
.
We should also mention a sl i ghtly earlier landmark of theoretical work on the classification of static and dynamic bifurcation in chemical reactors by Uppal et al. ( 1 976). This work does not only repres ent the first i mp ortant breakthrou gh after the pio nee ri n g work of the Minnesota group, but also manifests the great advantages resulting from cooperation between chemical e n g i neers and mathematicians. The "Chemical En g ineering Science" journal published a number of invited review articles dealing with the theoretical and practical aspects of mathematical mod el l in g of chemi cal reactors and their bifurcation and chaotic behaviour. We strongly encourage the reader to read these review articles. In the fo llow ing are some important quota tions from these articles which are most relevant to the purpose of this introduction. In his 1 990 Danckwerts Memorial lecture entitled Manners Makyth Modellers Aris ( 1 990) emphasizes the i mportance of u s i ng simple generic models of chemical reaction systems to discover some of the essential features of these systems and pays tribute to Liljenroth, the first discoverer of multipl icity in chemical reactors (Liljenroth, 1 9 1 8): "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 physi c al matrix. It may be a poor thing, an illfavoured thing when it is by itself. But it must be capable of having this independence. Thus Liljenroth ( 1 9 1 8) in his seminal paper on multiplicity of steady states, can hardly be said to have a mathem atical model, unless a graph ical repre s en tatio n of the case is a model. He w ork s out the slope of the heat removal line from the ratio of numerical values of a heat of reaction and a h eat c apac i ty Certainly he is deali ng with a typical case, and his conclusions are meant to have application beyond this p arti cu l arity but the m ec h an ism of doing this is no t there. To say this , is not to detract from Liljenroth' s paper, which is a l an dm ark of the chemical e n gi neeri ng literature, it is just to notice a matter of s tyl e and the po i n t at which a mathematical model is "born". ,
.
,
INTRODUCTION
7
Professor Aris in his review article no 37 entitled: Ends and beginnings in the mathematical modelling of chemical engineering systems (Aris, 1 993), beautifully and elegantly describes mathematical modelling as a creative activity, much like poetry, and extends the understanding of mathematical modelling from imitating the what does exist towards what may happen. He says: "If we adopt the basically Aristotalian position that poetry is a form of imitation or mimesis, it is easy to accept mathematical modelling as a poetric activity for, in doing it, we are engaged in a form of imitating nature in mathematical terms . There is the obvious first step of representing physical quantities as mathematical variables or parameters, but, beyond this, we need to incorporate physical laws and the constitution of the materials in question. This is done in the faith that the processes of mathematics "imitate", in some sense, the processes of nature and do so in a way that frees them from the accidents of particularity that cling to any experimental investigation. 'From what we have said' , writes Aristotle 1 , 'it will be seen that the poet' s function is to describe, not the thing that has happened, but a kind of thing that might happen, i.e. what is possible as being probable or necessary' . The distinction that Aristotle makes between the poet and the historian, namely that the later describes the thing that has been 2 , whereas the former describes the kind of thing that might be, might serve as the distinction between simulation and modelling. In the former there is a definite attempt to reproduce the detail of reality, as seen through the eyes of the observations that have been made and may yet continue to be made. The model is 'thus something more philosophical and of graver import' than the simulation ' since its statements are of the nature rather of universals ' than 'singulars' 3 • Notice that this has already introduced a final, or teleological, element into the approach to modelling, for it is clear that the purpose of the model has to be considered in its formulation". As Professor Aris draws this beautiful analogy between mathematical modelling and poetry and in the context of this book on the develop ment and use of mathematical models to investigate the behaviour of gassolid catalytic reactors and before we continue with Professor Aris in his beautiful and useful article, we should mention as a short rele vant interruption, that poetry talked about complex dynamics and instability much earlier than science through the rich imagination of Omar AlKhayyam, one of the greatest existentialist poets, born in 1 047 who also contributed a geometrical approach to algebra in his Treatise (see R. Rashed and A. Djebbar, L' oeuvre Algebrique d' AlKhayyam, 1
2 3
Aristotle, Poetrics, 145 1 a, 36. ibid. , 145 l b, 5 .
ibid. , 1 45 1 b,
7.
S . S .E.H. ELNASHAIE and S . S . ELSHIS HINI
8
for a modem translation). In Ruba' iyat he says: "You asked ' what is this transition pattern?' If we tell the truth of it, it will be a long story. It is a pattern that came up out of an ocean and in a moment returned to that ocean ' s depth". He also says: "
Though the five cards o f fortun e support your prop o f stability
And your body life is a fine garment
In the tent of the body which is your shelter Don ' t be secure, its four pegs are unstable"
Back to Professor Aris who goes on describing the essence of the process of model building in the most simple and elegant terms : "A model rests on certain physical laws, usually conservation principles. Thus, most equations are balances of some entity which is created or destroyed in the process being modelled. These laws are quite general. For example, let F be the net flux of some entity (such as mass or enthalpy) into a uniform region, G the total rate of generation of the same entity in the same region and H the total amount of it contained therein. Then, dH dt
= F+G
(1)
This i s a general balance relationship and i s used to acknowledge some law of nature. The relation of F, G and H to one another, or, equivalently, to some common variable, defines the constitution of the particular system within which we are working and are known as constitutive relations. If the entity is a particular chemical species present in the region in uniform concentration C, and V is the volume of the system, then H = V C. If the system is a stirred tank reactor with constant flow rate q, both in and out, and it is perfectly mixed, so that the concentra tion of the effluent is C, then F = q( c1  C). If the reaction rate can be expressed as a function of reactant concentration C, the rate of generation per unit volume is r (C), then G =  V r (C). Substituting, we have ·
·
dC V = q( C1  C)  V · r( C) dt
(2)
Here F, G and H are related to C by the ir constitutive relations, which define the nature of the flux and the kinetics of the reaction". Now we move to another extremely interestin g review article in the same issue of Chemical Engineering Science j ournal by Villermaux
INTRODUCTION
9
( 1 993 ), entitled: "Future challenges for basic research in chemical engineering". Among the many important issues discussed in this excellent article, Professor Villermaux correctly puts the research in nonlinear dynamics of chemical engineering processes in its correct place theoretically and practically, he goes to say: "Complexity and nonlinearity generate a wealth of interesting behaviours". Since the pioneering thesis of Henri Poincare in 1 879 (Sur les courbes decrites par une equation differentielle), it has been recognized that nonlinear systems give rise to problems of stability and multiplicity. No doubt, if Poincare had known chemical reactors, he would have chosen these objects as a support for his work. This application of nonlinear mechanics was revised by Aris and Amundson in 1 950's. In the case of two consecutive exothermic reactions in a CSTR, the reactor may behave as a strange attractor (Jorgensen and Aris, 1983, 1 984). Owing to the coupling between complex kinetics and transfer processes, chemical engineering thereby provides theoreticians with new objects exhibiting a wide range of dynamic behaviours. These can be studied by bifurcation analysis and numerical simulation when a mathematical model is available. Professor Villermaux goes on to ask the important question, which is actually the central theme of this introduction: "Is this only an academic exercise?" and he answers by giving a number of practical examples: "Here are a few examples showing the interest of research in this area. has been reported that industrial reactors were operated in re gions where spontaneous oscillations might appear, and this corres ponded to increased productivity (Fortuin, private communications; Vleeshouwer et al. , 1 992). Research on these phenomena may obviously have a significant impact on safety and control of reactor operation.  Complex chemical systems may exhibit an oscillating behaviour even at constant temperature and without any coupling with heat and mass transfer, owing to feedback loops in the kinetic mechanism. The famous BelousovZhabotinsky reaction is a wellknown example of this. Oscillating reactions arouse a great deal of interest for chemists and biologists, as they mimic selforganizing dissipative structures and biological rhythms. However, it was shown in the case of the chlorideiodide reaction in a CSTR that the existence and nature of the os dllating pattern might be entirely controlled by micromixing of reactants, thereby introducing che mi c al engineering concepts into this area (Fox and Villermaux, 1 990) .  The classical theory of coupling between heat and mass transfer and  It
chemical reaction in
a catalyst
pe llet is based on a deterministic
10
S . S .E.H. ELNASHAIE and S . S . ELSHIS HINI
approach yielding quasisteadystate concentration and temperature profiles. However, oscillating patterns have been observed on cata lytic surfaces by infrared thermography (D' Netto et a !. , 1 984 ; Brown et a l. , 1 985). Looking at these fluctuating hot spots moving at random on the surface, one might wonder whether the treatment proposed in standard textbooks is always relevant ! "
Professor Villerrnaux goes on to express in a brief and concise way, the scientifically exciting nature of the discovery of chaos: "Chaos is one of the most exciting concepts which has emerged during the last I 0 years. Chemical engineering systems are obviously candidates for exhibiting chaotic behaviour as conditions for the setup of this regime  multidimensionality, intermittency and coupling are frequently met. Many 'irreproducible' results observed in the past might perhaps be ascribed to 'chaotic ' behaviour". Next we move to the latest Danckwerts memorial lecture, entitled : "Seamless chemical engineering science: the emerging paradigm" by Mashelkar ( 1 995). An excellent article that should be read very seriously and carefully by all chemical engineers. We quote from it here, only the important parts which are most relevant to this introduction, but the article is certainly much more far reaching than that. Professor Mashelkar stresses the importance of interaction between different disciplines for the advancement of chemical engineering science (CES) and stresses the importance of the discovery of chaos and its effect on opening many exciting possibilities in natural sciences and engineering sciences. He says: "Complexity and nonlinearity reside in much that a chemical engineer encounters. His ability to acquire new tools and knowledge by exploiting the contemporary advances in physics has undergone a sea change in recent years. A wide variety of features ranging from steady state to multiple steady states to oscillations to chaotic dynamics and spatiotemporal patterns (Field and Burger, 1 985; Ott, 1 993) have become more commonplace observations. The advent of increasingly sophisticated experimental tools to detect and analyze microscopic events, new methods of mathematical analy s is and compu tational advances have further accelerated the development in this area. The discovery of dynamical chaos has opened up many exciting possibilities with wide ranging implications in natural sciences and engineering sciences, CES being no exception. The usefulness of chaos comes from the fact that it is a collection of many sets of ord erl y behaviour, none of which dominates under normal circu mstances At first sight an engineer shies aw ay from chaos, because he finds it un 
.
reliable, uncontrollable and therefore unwanted. However, the challenge
for an engineer is to control chaos and make it manageable, exploitable and, in fact, invaluable".
INTRODUCTION
II
We hope that we have succeeded in the last few pages of this intro duction to convince the reader of the theoretical, practical and industrial importance of studying bifurcation and chaos. Going deep enough into this important field of investigation requires many additional mathe mati cal and experimental techniques in addition to those already mastered by the majority of chemical engineers. This book concentrates on the mathematical analysis of bifurcation and chaos for gassolid catalytic reactors, using reliable, physically based dynamic models. Our main obj ective is to attract more chemical engineers to this field by making the entrance of new comers as smooth and easy as possible and also to offer the more experienced chemical engineers in this field a relatively comprehensive text for this important and fascinating subject. These efforts to convince the reader of the practical and industrial importance of this subject should not make us forget that it is tech nologically, scientifically and intellectually dangerous for scientific knowledge to be motivated only by practical and industrial short term usefulness. Henri Poincare, the real father of dynamical systems theory, puts this eternal fact in a very elegant and beautiful way: "The scientist does not study nature because it is useful; he studies it because he delights in it, and he delights in it because it is beautiful. If nature were not beautiful, it would not be worth knowing, and if nature were not worth knowing, life would not be worth living" . In this book we try to bring together in one volume the basic physico chemical principles and mathematical tools necessary for the dynamic modelling of gassolid catalytic reactors and the investigation of their fascinating static and dynamic bifurcation and chaotic characteristics. We also try to highlight the practical and industrial importance of the subject. The dynamic modelling of gassolid catalytic reactors is not as simple as homogeneous or other two phase reactors, for in addition to the nonlinear interaction between mass/heat transfer resistances and rates of reactions (together with their associated heat release or absorption), the surface phenomena, specially the chemisorption processes play a very important role in the dy n amic characteristics of the se systems. This was elucidated in some details for both cases of equilibrium and nonequilibrium adsorptiondesorption b y Elnashaie and Cresswell more than twenty years ago, for porous and nonporous catalyst pellets as well as fixed and fluidized bed c ataly tic reactors (Elnashaie and Cresswell, 1 973b,c,d, 1 974a,b, 1 975 ; Elnashaie, 1 977). It was also elegantly generalized by Aris ( 1 975) in his book on diffu s i on and reaction in porous catalyst p el let s . In the last few years, dynam ic mode ll i n g studies have started to account for these important catalyst surface processes (e.g. Arnold and Sundaresan, 1 989, Il' in and Luss, 1992).
S . S .E.H. ELNASHAIE and S.S. ELSHISH INI
12
We are devoting this book to gassolid catalytic reactors for a number of reasons: a) They represent the dominant and most widespread type of reactors in the petrochemical and petroleum refining industries. b) They are the most complex and the richest in static and dynamic bifurcation as well as chaotic phenomena. c) The basic principles for the modelling and analy sis of these reactors have much in common with other reactors in the microelectronic (Jensen, 1989; Elnashaie, 1 993), electrochemical (Hudson and Tsotsis, 1994) and biochemical (Nielsen and Villadsen, 1 992) industries. d) The modelling of these reactors i s the most complex among the different types of chemical reactors, e.g. in many cases, a model for a homo gene ous reactor can e asily be deduced from that of the catalytic one by simply neglecting the mass and heat transfer resistances.
e) Last but not least, the y represent our main field of specialization and research interest for over twenty five years.
While it is impossible to avoid mathematics for such a subject, the book is written in as simple a style as possible and complicated mathematics are made as simple as possible. The book is written primarily with the aim of attracting as many chemical engineers as possible to this fascinating subject, but it also highlights the importance of cooperation between chemical engineers (specially chemical reaction engineers), chemists (specially surface chemists and catalysis specialists) and applied mathematicians in order to achieve greater advances in the understandin g of these reactors for better design, optimization and control . As much as we are keen to attract more chemical engineers to this field, we also hope to encourage more cooperation between chemical engineers, chemists and applied mathematicians. We are equally hoping to introduce catalytic reactors into the menu of applied mathematicians together with the other models they historically use from mechanical, civil and electrical engineering. The book is divided into four chapters. The first chapter is very small, about forty pages and is devoted to the very b as i c principles of the steady state and dynamic behaviour of closed and open systems. It is aimed at the uninitiated reader in this fi el d Any reader with a reasonable background in process dynamics can safely skip this chapter. Chapter two is a rather large chapter of about one hundred and fifty pages. In this chapter the basic mathematical tools and presentation techniques necessary for the detailed analysis of static and dynamic bifurcation and chaotic behaviour of dynamic systems are pre sente d in a simple manner. The emphasis in this chapter is more on "how to use .
INTRODUCTION
13
the tools and techniques" to analyze these systems rather than on lengthy mathematical proofs which are available in mathematics and applied mathematics books and literature. It is hoped that this approach will make these tools and techniques amenable to most interested chemical engineers and chemists. The chapter covers, using a simple pragmatic approach, point attractors, periodic attractors, quasiperiodic attractors as well as chaotic attractors for both autonomous and non autonomous systems. The different bifurcation degeneracies are also presented, together with the various types of dynamic complexities in the neighbourhood of these degenerate bifurcation points. The ingenious technique of Poincare for the reduction of continuous systems to discrete representation, for autonomous systems (Poincare maps) and for non autonomous systems (stroboscopic maps), is presented in a very simple manner which we hope will be easy to understand and to apply by most chemical engineers and chemists. The main characteristics of finite difference discrete time models and their use in the study of chaotic behaviour as well as their connection to continuous time systems are exp l ained The different numerical techniques for the construction of important bifurcation diagrams and the characterization of the type of attractor, such as Floquet multipliers and Lyapunov exponents, are also explained. When the reader finishes reading and absorbing this chapter carefully, he should have the necessary tools and presentation techniques to start investigating the bifurcation and chaotic characteristics of the chemical reactor system of his choice. Chapter three is again a large chapter of about 180 pages devoted ent irely to the mathematical modelling of gassolid catalytic reactors to gether with the basic elementary static and dynamic characteristics of the developed models. The main assumptions, usually too restrictive, used in some publications in the chemical engineering literature, are critically discussed and their limits of validity explained. In order to bri dge the gap between theory and practice in this field, as discussed at the beginning of this introduction, we must understand clearly what is taking place on the c atal ytic surfaces and express it correctly in the mathematical formulation of the model. The model should be based on a cl e ar physicochemical picture with reliable parameter estimation and mathematical expressions for the rates of the different processes taking place within the system. This chapter is divided into three main sections. The first section deals with the various aspects of accurate reliable modelling of the si ngl e catalyst particle (porous and nonporous) whi c h is the heart of any c atal yti c reactor. The second section deals with fixed bed �atal y ti c reactors. Despite the obvious limitations of the fixed bed configuration in comparison with oth e r configurations for gas solid catalytic reactors, it re mai n s the most dominant in the petrochemical and petroleum refining i n du strie s The third p art is dealing briefly with .
.
14
S . S .E.H. ELNASHAIE and S . S . ELSHIS H I N I
bubbling fluidized bed catalytic reactors which represent a very promising configuration although not dominant in petrochemical and petroleum refining industries inspite of its clear advantages. Chapter four is again a large chapter of about 200 pages which is more than a chapter of case studies. It is divided into four main sec tions. The first section is a detailed bifurcation and chaotic investi gation for a bubbling fluidized bed catalytic reactor with consecutive exothermic reactions, where the intermediate product is the desired product, and for which the desired steady state giving rise to maxi mum yield of the desired product, is the middle saddle type unstable steady state. This desirable unstable steady state is stabilized using a simple SISO proportional negative feedback control loop. It is this closed loop controlled system which is presented in full details in this section for both the autonomous and nonautonomous (externally forced) cases. This case i s used to demonstrate to the re ade r much o f the bifurcation and chaotic patterns of behaviour presented in chapter 2. Also, more details regarding the structure of the chaotic region and the effect of homoclinicity on chaotic behaviour are presented and discussed in as simple a manner as possible. The use of a relatively simple bubbling fluidized bed catalytic reactor as a generic model, is better than the use of the CSTR model for, although it is not mathematically more complicated than the CSTR model, it is physically richer and more relevant to catalytic processes because of the following reasons: 1 ) As briefly stated above, from a mathematical complexity point of view, it is no more complex than the CSTR. The introduction of some physically reasonable assumptions suggested more than twenty years ago by Elnashaie and Yates ( 1 973), Elnashaie and Cresswell ( 1 973a) and Elnashaie ( 1977) and also used by Bukur and Amundson ( 1 975a) and Choi and Ray ( 1 985), allows the solution of the bubble phase equations analytically. This solution is used to evaluate analytically the integral in the dense phase equations reducing the model to a form which is mathematically similar to the CSTR, but with richer physical meaning. 2) Despite the simplifying assumptions, the fluidized bed model used still distinguishes the important physical fact that the effective residence time for each component and for heat, depends upon the rate of mass transfer of each component and the rate of he at transfer between the bubble phase and the dense phase. Therefore, in general, the modified residence time for the different components and for heat, can be different from each other. This feature is obvious ly not possible in the generic CSTR model.
INTRODUCTION
15
3 ) The modified residence times for the bubbling fluidized bed d o not depend upon the feed flow rate linearly as in the case with the CSTR, because of their partial exponential dependence upon the mass and heat transfer coefficients between the bubble and dense phases. Amundson and Aris ( 1 993), in their interesting paper on the occasion of the retirement of Professor Davidson, put this fact quite nicely as fol lows : "We have seen that this model of the bubble bed is essentially the same as the stirred tank when the two sources of feed are recognized. These are the fraction ( 1  /3) 4 that comes with the gas feed at the bottom of the bed and the fraction f3 in the bubbles that feeds the reactor5 at all levels and from a diminishing concentration difference. The latter when referred to the inlet difference (C0  Cp) delivers a fracti on ( 1 exp( Tr)). Thus the total feed minus outlet is: [( 1  /3) + /3( 1  exp ( Tr ))] ( C0 

Cp ) = [1  f3 exp (  Tr) ] ·
and this is what is equated to the reaction rate6 , (k H0 /U) Cp". 4) This model with its simplifying assumptions, has been used success fully to model some important industrial units such as the fluid catalytic cracking (FCC) units (e.g. Elshishini and Elnashaie, 1 990a) and the fluidized bed catalytic reactors for the production of poly ethylene and polypropylene (UNIPOL process of Union Carbide, e.g. Choi, 1 984; Choi and Ray, 1 985). The model has also been used successfully, with some minor modifications , to describe the beha viour of novel fluidized bed steam reformers with and without selective membranes (Elnashaie and Adris, 1 988; Adris et al. , 1 99 1 , 1 994). This practical relevance of the model is a clear advantage over the CSTR model, specially that it is not, mathematically, more complicated. 5) The model being a representation of a gassolid catalytic system, allows the investigation of the effect of different values of Lewis numbers associated with the catalyst chemisorption capacities of the different components involved in the reaction. ·
The second section presents and discusses details of the bifurcation behaviour of industrial fluid catalytic cracking (FCC) units. This process (with the exception of CO catalytic oxidation) is the most studied industrial process with regard to its static and dynamic behaviour. Notice that they are using different notation from the one we are using in this book. 5 The mean the dense phase of the reactor. 6 They refer to the reaction rate in the dense phase, because in this model the reaction in the bubble phase is neglected. 4
y
16
S . S .E.H. ELNASHAIE and S . S . ELSHISHINI
The third section is quite brief, presenting and discussing the basic bifurcation and chaotic behaviour of the most extensively studied system, that is the CO catalytic oxidation. The section is brief for two reasons : 1 . Collected information for this system are available in a large number of review articles. 2. For this reaction, going into more details would require an entire book of the size (and maybe larger) than the present book.
The fourth part is also brief, but for different reasons. It deals with the presentation and discussion of the bifurcation behaviour of the celebrated UNIPOL process of Union Carbide, for the production of polyethylene and polypropylene in gassolid catalytic bubbling fluidized bed reactors using ZieglerNatta catalyst. This section is brief because the static and dynamic bifurcation behaviour of this important process has not yet been adequately investigated in the literature . It is hoped that the approach adopted for this book will make it useful to a wide spectrum of chemical engineers and chemists in academia and industry interested in the dynamic behaviour and control of different types of gassolid catalytic reactors. We also hope that applied mathe maticians will find in the book some inspiration for adding some of these rather fascinating gassolid catalytic processes to their classical menu of models. The book can be used for advanced academic courses to chemical engineers and chemists, on dynamic modelling, bifurcation and chaotic behaviour of chemical reactors as well as training courses for industrial engineers and chemists on the dynamics of chemical reactors. Last but not least, we hope that all our chemical engineering, chemistry and applied mathematics colleagues, in academia and industry, will find the book beneficial and that they will honour us with their criticisms, comments and suggestions.
CHAPTER 1
Elementary Chemical Reactors Dynamics
The study of the dynamic behaviour of any system in its simplest form involves the investigation of the system evolution with time towards a certain timeindependent state. Obviously, if th� system is exposed to continuous external disturbances it may not reach a time independent state. Also, if the system has some inherent instability, then it may not reach a time independent state. However, in order to introduce the subject in the simplest way, let us assume first that the system does not have any inherent instabilities and that external disturbances are either nonexisting or not persistent (i.e. a step change or a square function as shown in Figure 1 . 1 , where Uf is some input parameter such as: feed concentration, feed flowrate, etc.). In addition we assume at the beginning that the system has a unique timeindependent state for certain given timeindependent input parameters. With all these very restrictive assumptions introduced for the sake of simplicity in this introductory part of the book, the system may have two different types of timeindependent states depending on the nature of the system itself.
( b)
(a} Time, t
FIGURE 1 . 1 or variable).
1.1
Time, t
Step (a) and square (b) input functions (u1= input "feed" parameter
STATIONARY EQUILIBRIUM STATES
This is the time independent state when the system is isolated or closed (Elnashaie and Elshishini, 1 993 ; Elnashaie et al. , 1 993) and reaches its 17
18
S . S .E.H. ELNASHAIE and S . S . ELSHISHINI
thermodynamic equilibrium. A simple example for isolated or closed system tending with time towards equilibrium, is the batch reactor. This will be the only example in this book dealing with dosed or isolated systems. Consider the following reversible reaction,
(1.1) taking place in a constant volume batch reactor. The thermodynamic equilibrium corresponds to the situation where both the forward and the backward reaction rates are equal. The reaction will reach equilibrium (i.e. forward rate equals backward rate, thus the net rate is zero) at a certain concentration of A and B. When the reversible rate of the reaction step is extremely slow, the situation approaches the case of the irreversible reaction, ( 1 .2)
and the thermodynamic equilibrium corresponds to the complete con sumption of the reactant component A ( 1 00% conversion of A ) . Consider the batch reactor shown i n Figure 1 .2, where the above simple irreversible reaction (equation 1 .2) is being conducted. Assume that the reactor is isothermal, of constant volume, perfectly mixed and that all the physical properties are not changing with the progress of the reaction. Under the above assumptions, the equations describing the system are given by,
FIGURE 1.2
Schematic presentation of a batch reactor.
ELEMENTARY C HEMICAL REACTORS DYNAMICS
d CA
=
dt
k · CA
19
( 1 .3)
and, ( 1 .4) and the initial conditions are: t=0
at
( 1 .5)
Notice that the reactor volume V, is cancelled from the equations and does not affect the process neither at steady state nor at unsteady state conditions. Equations 1 .3 and 1 .4, when added together give, ( 1 .6)
Calling,
then
Thus the following simple equation in dy = 0 dt
with initial conditions, at
t
=
0
Y( O ) = Yo
y
is obtained, ( 1 .7)
( 1 . 8)
By integration of equation ( 1 . 7) and substitution of the initial condition
( 1 .8), we obtain,
Y(t) = y(O)
S.S.E.H. ELNASHAIE and S . S . ELSHISHINI
20
T i me , t
FIGURE 1.3 The change of CA (t) with time for the batch reactor.
In other w ord s ,
( 1 .9)
It is clear th at the usual rel ati o n between CA . CB is valid during the dynamic change of this u ni t , unlike the CSTR case given thereafter. The dynamic beh av i ou r of the system can thus be completely describ ed by one equation (equation 1 .3) which has the follow i ng s ol uti on ,
CA = CAv exp ( kt) ·
( 1 . 1 0)
A sketch of CA (t) versus time is shown in Figure 1 . 3 . A s t � CA � 0, which sim y means complete conversion. Of course as k the reaction rate constant increases, the system reaches its final time ind epen den t de s i n at o n faste .
pl
t
1 .2
i
oo ,
r
STATIONARY NONEQUILIBRIUM STATES
This is the ti me  i n de p en den t state for open systems (Elnashaie and Elnashaie, 1 993 ; Elna shai e et a/. , 1 993) and is usually called in chemical en gi nee ring "the steady state". These stationary nonequilibrium states
are associated with continuous processes, which is the most common processing m ode nowadays i n the petrochemical and petroleum refining industrie s . Almost all the work presented in this book is related to this case of open systems (dissipative systems ) . A si mple example for a system with stationary nonequilibrium state is the famous idealized continuous s t i rre d tank reactor (CSTR) , where
ELEMENTARY CHEMICAL REACTORS DYNAMICS
21
q FIGURE 1 .4
A schematic diagram for the idealized isothermal CSTR.
the dynamics of the system is described by simple ordinary differential equations and the steady state (stationary nonequilibrium state) is described by algebraic equations. In order to illustrate this s impl e case let us consider the very simple unimolecular irreversible reaction with linear kinetics taking place in a single isothermal CSTR with constant input conditions and no c hange in the flow rate or physical properties due to reaction. Figure 1 .4 shows a schematic diagram for the said reactor. The reaction is, (1.11) where the rate of reaction per unit volume, i s given by, ( 1 . 1 2) Thus, the rate of consumption of component
A is, ( 1 . 1 3)
and the rate of production of component B is ( 1 . 1 4) The si mpl e unsteady state equati on is obtained by performing a material balance on component A , ( 1 . 1 5)
S . S .E.H. ELNASHAIE and S . S . ELSHIS H I N I
22
and material balance on component B, ( 1 . 1 6)
where nA and n8 are the outlet molar flow rates from the reactor; nA and n8 are the molar holdup inside the reactor ; nAJ and n81 are the molar feed flow rates and V is the reactor active volume which is assumed constant. If in addition we use the assumption that the inlet and outlet volumetric flow rates are constant and equal to q m 3/h in addition to the assumption of perfect mixing which implies that concen trations of A, B at all poi nts within the active volume of the reactor are equal and equal to the o u tput concentrations. Then we can write: ( 1 . 1 7)
where
i =A, B
and where Ci' s are exit concentrations and Cif' s are input concentrations all in kmolJm3 .
Substituting equation ( 1 . 17) into equation s ( 1 . 1 5) and ( 1 . 1 6) the following simple equati ons are obtained, ( 1 . 1 8)
( 1 . 1 9) where V, q, CAt• C81, k are the parameters of the system which must be specified before any i nvestigatio n of the steady state or dynamic
behaviour is possible. If some extra restrictive condit i ons
are
imposed,
equations ( 1 . 1 8, 1 . 1 9) can easily be combined into one equation. It will be shown later in this chapter that this is not always po s s i ble However, before we proceed further a couple of lines regarding equati ons 1 . 1 8 .
,
and 1. 1 9 are due. These equation s describe the change of the two state variables CA and C8 with time. Obviously any change with ti me must have a beginning and p o s s ib l y an end (depend ing on how we define "end") . For this first very simple case, it can be positively asserted that the dyn amic s of the system has a simple "end" which is
the stationary nonequilibrium state (i.e. the steady state in common chemical engineering terminology). In other cases to be discussed later, this "end" will sometimes be much more complicated than j ust
ELEMENTARY CHEM ICAL REACTORS DYNAMICS
23
a timeindependent steady state. The beginning is what we usually call the initial conditions, that is the state of the system (the value of CA , Cs in our case ) at some starting time which will be designated as time zero. Therefore equations ( 1 . 1 8, 1 . 1 9) will not completely define the system except when we add to them the initial conditions, at
t=O
( 1 .20)
After the "beginning" has been specified , we now turn to the question of the "end". In the present case, it is very easy to specify the "end". Thi s "end" can either be obtained from the solution of ( 1 . 1 81 .20) as t 4 oo or from the steady state equations which are obtained by setting the unsteady state terms in the left hand side of equations ( 1 . 1 8, 1 . 1 9), to zero. For the present case, this results in algebraic equations which can easily be solved to give the final steady state of the system. This second choice is valid only in such simple cases for which the "end" is at a time independent stationary state. Before solving and analyzing, the two equations ( 1 . 1 8, 1 . 1 9) are combined by add ition to give the single equation,
(V) 
q
d ( CA + Cs )  ( CAf + CBj )  (CA + CB ) dt
( 1 .2 1 )
with the initial conditions, at
t=O
Now, let us call
( 1 .22)
(�) = a
and CA + Cs
= Yo.
=
Y
then CAf + C81 = Yt and CAo + C80 Thus equations 1 .2 1 and 1 .22 can be written as, ( 1 .23) with the initial conditions, at
t=0
( 1 .24)
S . S .E.H. ELNASHAIE and S . S . ELSHISIDNI
24
Equation ( 1 .23) with the initial conditions ( 1 .24) can easily be solved to give, ( 1 .25)
which gives upon back substitution of the above definition of y1 , Yo and y, CA
(t ) + Cs ( t)
At
=
( CAf + CBJ )  (( CAJ + CBJ )  (CAo + Cso n exp ( t I a)
( 1 .26)
steady state (solving the steady state algebraic equation or setting
t � oo in equation 1 .26) the following relation holds,
( 1 .27)
Equation ( 1 .26) means that the relation usually used for steady state analysis ( 1 .27) is not valid under unsteady state conditions. Equation ( 1 .27) can be written as: ( 1 .28)
which implies that the two dimensional system can be reduced to a one dimensional system and that analysis is necessary only in terms of CA (or C8 ) . Equation ( 1 .26) shows the validity of equation ( 1 .27) under steady state condi tions when t � DO and therefore exp ( t I a) � 0 . It also shows that rel ation ( 1 .27) can be valid under unsteady state conditions only under a very restricti ve condition, that is when ( 1 .29) 1bis means in practice that the tank is first filled with the feed before any reaction starts, then the reaction starts from this initial conditions of CAo = CAf· CBo = CBJ therefore CAo + CBo = CAt+ CBJ · An important feature which makes the dynamic system described by equations ( 1 . 1 8 1 .20) quite simple to analyse is that the coupling between equations ( 1 . 1 8, 1 . 1 9) is weak and is in one direction. Equation ( 1 . 1 9) depends upon CA, however equation ( 1 . 1 8) d oe s not depend upon C8, because the reaction is irreversible. Therefore, in both the steady state and dynamic cases, equation ( 1 . 1 8) can be solved independently of equation ( 1 . 1 9) and when C8 is required, the solution of equation ( 1 . 1 8 ) can be substituted into equa tion ( 1 . 1 9) which can then be s o l ved . The s i mple analysis presented above can be c arried out more elegantly when the equations are put in a dimensionless form. Let us exercise
ELEMENTARY CHEMICAL REACTORS DYNAMICS
25
this straightforward procedure for this extremely simple example. Equation ( 1 . 1 8) can be written in the following dimensionless form, ( 1 .30)
at
r = 0,
( 1 .3 1 )
where, =
'r
 CA XA C ef
t·
q =
a=
v '
V·k
q
r
CAo XA  o cref _
,
C,.ef is an arbitrary reference concentration, taken for this simple ana lysis as Cref = CAf·
Analytical solution of equation ( 1 .30) with the in iti al condition ( 1 . 3 1 ) gives, 
XA Cl') =
 ( 1
1+a
+ XAo
_
1
__
1+a
)

exp (  (l + a)r)
( 1 . 32)
oo
Equation ( 1 .32) describes the chan ge of XA with time from the initial condition XA = XAo at r = 0 up to the end when r = .
X
:a:
T
less time 'T.
FIGURE 1 .5
.
1 Aa;,
=
1+ «
D i mens ionless
t ime
The change of dimensionless concentration XA (1') with dimension
S . S .E.H. ELNASHAIE and S . S . ELSHISHINI
26
The steady state solution is given by ( 1 .33) This same end value obtained when r � 00 ' can be obtained (in this simple case, but not always) from the solution of the steady state equation which is obtained by setting dXA I dr = 0 in equation ( 1 .30) to obtain the steady state equation, ( 1 .34) which can easily be solved to give, XAss

1
( 1 .35)
I+a
In this simple case the steady state depends upon a, so does the speed of the dynamic behaviour of the system. Thus a affects both the steady state and dynamic behaviour; as a increases XAss decreases and the speed of change of XA with dimensionless time increases. Physically speaking, at a constant flow rate and constant reaction rate, it is V that affects the steady state and the speed of the response. The reactor volume represents from a dynamical point of view, the capacity of the system and from a kinetics point of view, it represents the volume of the active reaction mixture. It is clear from equation ( 1 .32) that as a increases (i.e. V increases) the dynamics of the system becomes faster while intuitively we should expect the opposite. This physically erroneous conclusion is due to the fact that the dimensionless time r, contains v in its definition. Actually, physical evaluation of the speed of response is possible only if the evaluation is done in terms of real time. Equation ( 1 .32) can be written in terms of real time t, as follows,
(
)
·
1 1 XA (t) =  + XA 0   exp ( a t ) l+a l+a
where
a=
( 1 . 36 )
(q/V) + k.
Obviously, for constant q and k, as V increases the parameter a decreases and the dynamics of the system becomes slower in terms of the real ti m e t. The behaviour of this system is quit e simple i.e. unique stationary nonequilibrium stable state (stable ste ad y state) with the dynamics of
the system exponentially approaching this stable steady state . No
ELEMENTARY CHEMICAL REACTORS DYNAMICS
27
i n stabil i ty can appear in such a system, not even decaying oscillations. This is mai nl y due to the fact that the dynamic characteristics are de scri bed by a single linear differential equatio n . Even if the system was described by a nonlinear differenti al equation , the system will still not s h ow any o s cillations as long as it is described by a single diffe ren ti al equ ation . However, the steady state beha vio ur may show some com ple xi ties if the non  l i ne arity is also nonmonotonic as wi ll be shown later in this c hapter . For higher degrees of nonlinear coupling for reactors described by more than one differential equation , dynamic complexities may develop. The capacities associated with the different differential equ ations of such systems will play an important role in the ir dyn amic complexities. A CSTR with slightly stronger coupling
The CSTR inspected earlier has a "one w ay " coupling between the two di fferenti al e quati ons de sc ribin g the system. The present case has a two way coupling where both equations affect each other. Consider the reversible re ac tion ,
The rate of di s appearance (consumption) of A is g i ven by, rA = kj ' CA  kb · C8
( 1 .37)
The rate of appearance (productio n ) of B is given by, ( 1 .38)
If this re action is carried out in a CSTR with the same simpl ify i ng assumptions used earlier, then the mass balance equations will be, ( 1 .39)
( 1 .40) The usual erroneous practice of re l ati ng CA to C8 through the relation in terms of conversion is correct only for steady state but not for dynamics, except for the restrictive initial conditions discussed in the previous section. CA + C8 = CAJ+ C81 or expressing CA , C8
S . S .E.H. ELNASHAIE and S.S. ELSHISHINI
28
Thus, in general the dynamics of the system can only be correctly obtained by solving equations ( 1 .39, 1 .40) simultaneously. The equations can be put in the following form, ( 1 .4 1 ) ( 1 .42) With the initial conditions
at
t = 0,
where XA , X8, XAf• X8t are the dimensionless
and,
al l =  ( a + kf )
a 21 = + kf
a1 2 = + kb
�2 = (a + kb )
concentrations,
b1 = a · XAf b2 = a · X81
a=qiV=lla
Equations 1 .4 1 and 1 .42 represent a very simple case of the more general linear nonhomogeneous matrix differential equation. This subject is covered in any standard textbooks dealing with linear differential equations (e . g . Wylie, 1 966; Boyce and Diprima, 1 969; De o and Raghavendra, 1 980; Kreyszig, 1 988). Equations 1 .41 and 1 .42 can be written in the following matrix form, ( 1 .43)
where :!.
is the state vector given by,
and the initial conditions vector of
the state variables is given by:
ELEMENTARY CHEMICAL REACTORS DYNAMICS
29
the other two matrices and the vector m(t) are given by,
The solution of equation 1 .43 is given by,
where ¢ (t) is the transition matrix given by, f/J (t) = exp (d
·
t)
f/J(t  r) = exp (�(t  'l'))
and
The Sylvester' s fonnulae for evaluating matrix polynomials can be used to e v alu ate r). The fonnulae fur any matrix polynomial "¢(�) is given by:
¢j(t), ¢j(t 
IIi:j__ (�  A i · 1) _ i=l.''_ n
n
t( � ) = I J ( A ) . j =l
II ( Aj  AJ
i=l ,i�j
( 1 .45)
where A1 , A2 , are the eigenvalues of matrix � , thus for the two dimensional system (n = 2) with only two eigenvalues, we get •
•
•
'I' ;; ( 
Thus,
t ) = e = � £.... e At
n
}=!
A,].(
2
II (�  A d)
i=l ,i�j ; 2 ';
II ( Aj  A; )
i=l,i�j
S . S.E.H. ELNASHAIE and S . S . ELSHISHINI
30
and the integral in equation 1 .44 is thus given by,
Thus the solution is given by, :!(t ) =
At
+
�
Az
[i�� (�  Az l}  e A2t (�  A� l)] · :!(O )
!1!!1
( A2  AI )
[(1  (A I) e A1t )
 A2
_
At
(1
_
e�t )
(A I)]  A1 Az
( l .50)
It is clear that the behaviour of the system depends upon the nature of the eigenvalues A1 , A2 which can be obtained from the solution of the algebraic equation: det (�  A l ) = O
( 1 .5 1 )
which on expansion gives, a1 2
az z  A
I
o
( 1 .52)
This determinant gives the following polynomial in A, A2  (tr�)A + det � = 0
( 1 .53)
tr� = (2a + a ' )
( 1 . 5 4)
where,
For the present case,
where,
det A. = a2 + aa
( 1 .55)
( 1 .56)
31
ELEMENTARY CHEMICAL REACTORS DYNAMICS
The eigenvalues are therefore, ( 1 .57)
A 1 =  a
A2
( 1 .58)
= (a + a')
It is clear that both roots are real and negative, therefore the dynamics leads the system exponentially to its steady state solution. When all the parameters q, kt, kb are constant, the speed of the dynamics is inversely proportional to the capacity V as in the earlier example. No oscillations are possible in such a system. Coupling in both cases presented so far, does not cause any complexity in the dynamic behaviour. The steady state of the system which is obviously unique can be determined by putting t � oo in equation ( 1 .50) or by setting dX#dt = dXsfdt = 0 in equations 1 .4 1 , 1 .42 and solving the resulting linear algebraic equations. Note: For nonlinear systems, the stability can be checked by linearizing the system in the neighbourhood of the specific steady state and finding the eigenvalues of the linearized equations. When there is multiplicity of the steady states (as will be discussed later in the book), as t � oo, the system tends to different steady states depending on the initial conditions. When the eigenvalues are complex with negative real parts, the solution is still stable but it approaches the steady state as t � oo in decaying oscillatory fashion. When the eigenvalues are real with at least one positive eigenvalue or complex with positive real parts, then the steady state having these characteristics is not stable and the system will not tend to this steady state as t � oo. Such steady states can only be determined from solving the steady state equations with dx;ldt = O. In some cases and with controlled procedures, these unstable steady states can be obtained by putting t ' = t and letting t � oo .
A two dimensional case
with different mass capacities
The last simple case to be presented is the same case described by equations 1 .39, 1 .40 but with different capacities VA and V8 for com ponents A and B respectively. For homogeneous isothermal cases this is physically unrealistic. However, later on we will show that different capacities for the differential equations is physically realistic for heterogeneous systems as well as nonisothermal systems. In this case matrix A is replaced by A' having the elements, al l = (aA + kr 8A )
lz2. = kr 8s
a{2 = kb  8A al2 =  ( as + kb · 8s )
b{ = aA XAf b2 = as XAf
S . S .E.H. ELNASHAIE and S.S. ELS H ISHINI
32
The solution in this case will basically have the same form as equation 1 .50 of the previous case but AJ , A.2 will be A.{, A2 which are the roots of the characteristic equation: 2 A. '  ( trA_' ) A. ' + det A_'
=0
( 1 .59)
and the nonhomogeneous vector m' is given by m' 
The trA_'
and
(b{) b2
det A_' in equation 1 .59 are given by: ( 1 .60)
and
( 1 .6 1 ) The eigenvalues A.{, A2 are given by: ( 1 .62)
�
± [(iiA +aB)+ (kj " 8A +kb  8B )] 2 4(aA · aB +kb  8B aA +kj " 8A ·aB )
}
In order to ensure that the system goes exponentially (without oscillations) to its steady state values we must have A.{, A2 both real and negative, let us write equation 1 .62 in the simpler form, A.J, 2
where,
=
�(a ± �a 2  4b )
( 1 .63)
a = (aA + aB ) + (kj " 8A + kb  8B ) b = aAaB + aAkb · 8B + aBkf 8A ·
Since b is alway s positive then � a2  4b is al ways less than the value of a thus A.{ 2 will always have negative real parts whi ch means that the dynamic s leads the system to it s steady state not away from it. To ensure that this tendency towards the steady state takes place exponentially w ithou t oscillations, we must prove that these ei genvalues '
ELEMENTARY CHEMICAL
REACTORS DYNAMICS
33
with negative real parts have zero imaginary parts, that is to prove that ( 1 .64) Writing equation 1 .64 in terms of the dimensionless physical para meters, we get after some simple straightforward rearrangements,
It is clear from equation ( 1 .65) that regardless of the values of aA , side of equation ( 1 .65) i s always positive .r Therefore it is not possible to have any complex roots and no oscillations are possible neither sustained nor decaying. It is clear that for this simple linear case, the difference in the capacities of the two differential equations of the two different compo nents does not produce any qualitative changes in the dynamic charac teristics of the system. The dynami c behaviour still leads the system in time from its initial condition toward its final destination at the unique timeindependent stationary state In other more complex systems with strong non linear coupling, this difference in the capacities will change the dynamic characteri stics as will be shown later a8 , k , kb, 8A , 88, the left hand
.
.

.
A simple one dimensional case with complex behaviour
Let us now consider a case where the kinetics of the reaction are non monotonic, a situation which will be shown to give rise to multiple steady states for certain values of the parameters . Consider a CSTR in which a simple reaction is taking place,
and
the rate of reaction is given by, ( 1 .66)
Unsteady state material balance on component A gives, ( 1 .67a) Similarly, unsteady state material balance on component B
gives,
34
S . S .E.H. ELNASHAIE and S.S. ELSHISHINI
( 1 .67b) which can be rearranged in the form, ( 1 .68a)
and, ( 1 .68b) where,
 cB cA a = q I V, XA =  , XB cref cref
•
CBf cAf   • ae = K . cre1·• XAf   XBif 
cref
•
cref
with the initial conditions, at t = 0, XA = XAv. XB = XBo • where, CA ( O ) XAo • Cref
The steady state for this system is given by, ( 1 .69) Simple ph ysi c al analy sis of this equation shows that the left hand side represents the rate of s upp l y of reactants S (XA) to the re ac tor due to the co nt i nuous inflow of reactants, while the right hand side C (XA) repre s ents the rate of reactant consumption in th e reactor due to the chemical reaction. Therefore, they must obviously be equal at steady state for if the rate of supply S (XA) is higher than the rate of con s ump tion C (XA ) then the reactants must accumulate with time and XA must incre as e with time, while if the rate of s uppl y is smaller than the rate of con sump ti on then the reactants must deplete i n side the reac tor and XA mu st decrease with time. Clearly the s ol uti on of the above equation can be obtai ne d nu meri c al l y using any of the well known te c hni que s or it can be solved graphically by p l otti ng S(XA) and C (XA) versus XA on the same graph and the p o int s of intersection are obvi ously the s oluti ons .
ELEMENTARY CHEMICAL REACTORS DYNAMICS
35
Dimensionless Con centrat ion , XA
FIGURE 1.6 Supply (S (XA)) and consumption (C (XA)) functions versus concen tration (XA) for the CSTR with nonmonotonic kinetics, (I) a case with multiple steady states (XAt, XAz, XA3), (2) a case with single (unique) high conversion steady state (XAs)•
It is clear from Fi gure 1 .6 that for a certain range of parameters, multiple steady states exist Now we can easily u s e this s i mple graph together w ith the physical meani ng of S (XA) and C (XA) i n order to discuss some of the ba s ic dy n ami c characteristics of this system without the need for the un s teady state equati ons ( 1 .68a,b). However, before we do that, we mu s t stress the fact that a new situation is e nco untered here, even with this extremely simpl e example, that is the existence of different steady states ( stationary nonequilibrium s tate s ) and therefore it is logical to expect that different i n i ti al conditions will lead to different steady states. We should provide here a very simple definition for the terms stable and unstable steady states which will serve the ne eds of this stage but which wi l l be expanded considerably in the other chapters of the book. When a system at s te ady state is ex po s e d to "infinitesimal" dis turbance s and the disturbances are re mo ve d after a very short time,
and the system returns to this same steady state after some transient ti me then this state i s a stable steady state. On the contrary, if the system d oes not return to the same steady state then this steady state is not stable (unstable).
36
S . S .E.H. ELNASHAIE and S.S. ELSlll S HINI
On the light of this extremely simple and elementary understanding of stability, we will consider the three steady states in Figure 1 .6. If the first steady state XA = XA 1 is disturbed with an infinitesimal disturbance so that the dimensionless concentration increases to XA = XA 1 + 8, it is easy to notice that in contradistinction to the situation at XA 1 itself where S (XA) = C(XA) at XA 1 + 8 it is clear that S (XA ) < C(XA ) . This means that the rate of reactant consumption is larger than the rate of reactant supply and therefore XA will decrease with time, the disturbance decays and XA goes back to the steady state value XA 1 • On the other hand if the disturbance 8, is in the opposite direction, i.e.: XA = XA r  8, it is clear that at this value of XA, S (XA) < C (XA) which means that the rate of supply of reactants is larger than the rate of consumption and therefore XA increases with time, the disturbance decays and XA goes back to the steady state value XA I · The same analysis applied to XA 1 also applies to the third steady state XA3 . Thus we can conclude that steady states XA 1 and XA3 are stable (at least for small disturbances). For the middle steady state, XA2 the situation is actually the opposite. For XA = XA2 + 8, it is clear that S(XA) > C(XA) therefore XA continues to increase with time getting away from XA2 towards XA3 and for XA = XA 2  8, it is clear that S (XA) < C (XA) and therefore XA continues to decrease with time moving away from XAz towards XA I · Thus it can be concluded that XAz is unstable. The above very simple analysis introduces an important concept which will be discussed in details later. This is what is usually called the Region of Asymptotic Stability (RAS) of the steady state or in the terminology of the more modem dynamical systems theory, the domain of attraction of the steady states. Steady states XA J . XA3 c an also be called the attractors and more specifically, point attractors to distinguish between them and other more complex attractors discussed later in the book. The domain of attraction of an attractor represents the region in the state variables space characterized by the simple fact that for any disturbance in the state variable that does not take the state of the system outside this region, the system will return with time to the same attractor after removal of the disturbance. The cases discussed in the previous sections had only one attractor (unique steady state). Therefore for any disturbance of the system (provided it is reasonable and does not destroy the physical integrity of the system), when this disturbance is removed, regardless of the state of the system, it will return to its single, unique attractor. This is sometimes called global stability. However when there is more than one stable steady state (more than one point attractor), it is obvious that none of th e steady state wi ll be g l obally stable. Each steady state will have a region of asymptoti c stabili ty, RAS (i.e. every attractor will have its domain of attraction). In other words, the ste ady states must share the state space. This simple but important ide a can be illustrated very simply u s i n g Fi gure 1 .6 once ag ain. Consider
ELEMENTARY CHEMICAL REACTORS DYNAMICS
37
the disturbance in state variable discussed earlier, by increasing XA 1 to XA = XA I + 8. If 8 is such that XA < XA2. then we notice that S (XA) < C (XA) and therefore XA decreases with time till it goes back to XA I · However, if 8 i s such that XA > XA 2 . then we notice that S (XA) > C (XA) and therefore XA will continue to increase towards the third steady state
XA 3 instead of returning to XA 1 • The same reasoning can be applied to
disturbances applied to the system when it is at the third steady state XA J It is thus clear from Figure 1 .6 and the above simple reasoning, that the region of asymptotic stability, RAS, for steady state XA 1 (in other words the domain of attraction of the point attractor at XA 1 ) extends from XA = 0 to XA = XA2 while that of XA3 extends from XA = XA2 to XA = oo. The size of the domain of attraction of the unstable steady state (attractor) XA 2 is zero, it is the point XA 2 only. Actually, on the phase plane which is a plot of XA vs. X8 with t as a parameter, the middle unstable steady state has a domain of attraction which is a curve, as will be shown later in the book. The point XA 2 is the simplest form of the separatrix, which separates two domains of attraction. After this very simple but enlightening steady state analysis, we go back to the dynamical differential equations ( 1 .68a,b). It is clear that in this simple case the differential equation 1 .68a can be solved inde pendently of equation 1 .68b and therefore the main dynamic charac teristics of the system can be determined using equation 1 .68a alone. However, this equation is nonlinear and cannot be solved analytically but we can determine its behaviour near any of the steady states by linearization using Taylor series expansion with only the first linear term in the series. Equation 1 .68a can be written in the form, dXA
where,
dt
= f( XA )
( 1 .70)
( 1 .7 1 ) Now by linearization around any steady state, the following linear differential equation is obtained, ( 1 .72) A deviation variable can be defined as follows, ( 1.73)
S . S .E. H. ELNASHAIE and S.S. ELSHISHINI
38
At steady state, equation 1 . 70 can be written in the following trivial form, dXA ss
dt
=
j(XA )
ss
( 1 .74)
Subtracting ( 1 .74) from ( 1 .72) and using the definition of y in 1 .73, we obtain, ( 1 .75)
with the initial conditions, at
t=0
y=O
where, ( 1 .76) ss
Solution of equation 1 .75 gives, ( 1 .77) It is clear that for this simple example )., is always real and therefore no oscillatory behaviour can exist. Furthermore, the steady state will be stable if A, is negative and unstable if )., is positive. Thus the condition for stability is, ss
11 ae is satisfied for steady states satisfying the inequality, ss
ss
ss
( 1 . 89)
It is clear from Figure 1 .6 that the steady state XA3 satisfies the stabi lity condition ( 1 . 89), while XA2 violates this stability condition. This simple stability condition is usually called the slope condition. It states that the steady state is stable if the absolute value of the slope of the supply function is greater than the absolute value of the slope of the consumption function. The results of this simple mathematical stabi lity analysis is clearly the same in this case as that obtained from the earlier physical discussion b ased on disturbing the steady states by a small amount o. Since the system is described by a single differential equation , this simple slope condition is sufficient for testing the stability of the steady states. Later in the book it wi ll be shown that in higher dimensional cases this slope condition can only be a necessary condition for stability, but not sufficient. The nonisothennal nonadiabatic CSTR with simple unimolecular irreversible reaction
This is the classical example usually used in the literature to demonstrate the different static and dynamic characteristics of chemical reactors. Although the system is quite simple, it is quite rich in static and dynamic phenomena. Many of the phenomena discovered in this system were found in other more complex systems . Consider the CSTR shown schematically in Figure 1 .7 and assume perfect mixing, constant flow rates, constant volume, constant physical properties and constant cooling coil temperature. The reaction taking place is the simple exothermic reaction, ( 1 .90)
A t B
with th e rate of reaction depending upon temperature and concentration as follows,
r = ko eEIJV;T CA ·
( 1 .9 1 )
ELEMENTARY CHEMICAL REACTORS DYNAMICS
41
q c ., T,
c. Cs T
FIGURE 1. 7
Schematic presentation of the nonisothermal, nonadiabatic CSTR.
Unsteady state material balance gives, ( 1 .92) The unsteady state heat balance equation is given by:
V
volume of the reaction mixture. = volume of the metal parts of the reactor, stirrer and cooling coil. m = density of the reacti on mixture and metal parts respectively. p, Pm Cl', Cpm = specific heat of the reaction mixture and metal parts respectively. = volumetric flow rate of feed. q CAf = feed concentration of reactant A. = feed temperature. Tr Ac = area of heat transfer between the reactor and the cooling coil. U = overall heat transfer coefficient between the reaction mixture
V
Tc
CA
T
=
and cooling coil. = cooling co il temperature = concentration of reactant A inside and at the exit of the reactor. = temperature of reaction mixture inside and at the exit of the reactor .
.
S.S .E.H. ELNASHAIE
42
and
S.S. ELSHISHINI
There are different ways of putting these equations into a dimensi onle ss form; one of the simple and convenient forms is, ( 1 .94)
an d ,
whe re ,
t' = tfa Lm
f3 Yf
= Vm · Pm · Cpm f( V · p · CP )
= ( tili) Cref j(p · CP
= Tj T,.ef
·
T,.ef )
a
r
a0 = V · k0/ q
= Vfq = Ef RcT,.�r
XA = CA/ Cref
Y = T/T,.ef Kc
= Ac · Uj (q · p · CP )
XAf = CA.f /Cr�f
Steady state analysis
The steady state equat ions are given by, ( 1 .96) ( 1 .97)
It is quite easy to reduce the problem of solving the two si multan e ou s nonlinear algebraic equations ( 1 . 96 and 1 . 97 ) to obtain XA. Y for a certain set of parameters, to the problem of s olving one nonlinear equation in Y and an e x pl i c i t equation that g i ve s XA once Y is determined. From equation I . 96 it is strai ghtforward to obtain the s imple rel ati on , ( 1 .98) Substitution of ( 1 .98) into ( 1 .97) to eliminate XA
in
terms of Y gives, ( 1 .99 )
ELEMENTARY CHEMICAL REACTORS DYNAM ICS
43
It is ph y s i cal l y clear that the ri g ht h an d s ide of equ ation ( 1 .99)
( Y), re pre s e nt s the heat gen erati o R ( Y) the heat removal function t s s e n r e re p Y ( (K coil Yc)). cooling the by Y and ) Y (
n function Q
while the left hand side (heat removal by flow
J Equation ( 1 .99) c an be solved graphi call y by plotting G ( Y) and R( Y) vs. Y on the same graph . Obv ious ly the point s of intersection of th e t wo functions are the s olu tions (the steady state temperatures). For e x othermi c reac tion s (Ml > 0) with high enough ex othe rm icity , the heat g e nerati on function has an inflection p oi n t and h a s the general shape shown in Figure 1 .8 . The heat removal function (a straight l i ne in this simple case) can be arranged into the fol lowi n g form, _
R ( Y) = (1 + KJ · Y  ( Y + Kc � ) f ·
The slope o f the line i s 1 + K· an d the inters ecti on with the horizontal axis (R( Y ) = G ( Y ) = 0) is at Y = a , where, ( 1 . 1 00) In Fi gu re 1 . 8,
a2 
Yf + Kc · � 2 1 + Kc
It is clear from Figure 1 . 8 th at for a = a 1 , there are three s te ady state temperatures, the high , middle and low temperature steady states (Y3, Yz and Y1 ). The high te mperature Y3 corre sp ond s to high conversion and the low temperature Y1 corre s pond s to low conversion. For a = a 2 , a single ( u ni que ) high temperature s teady state ( Ys), exists. The simple pseudo  s te ady state te s t for s tabi lity can be perfo rmed here. However s ince the system is described by two di fferenti al equ ati ons, this simpl e te s t g ive s only necessary condition for stabil i ty (however the v i o l ati on of th i s test is a suffi c i ent condition of i n s tabi lity ) . I f for e x ampl e we consider a small i n cre as e o in the temperature of the third s teady state Y3 , we find from Fi g ure 1 . 8 that at this new temperature Y3 + o the he at removal R( Y) is hi gher than the heat gene ration G (Y), thus the temperature decreas e s and goes back to Y3 . If we con sider a s mal l decrease o in the tempe rature we find from Figure 1 . 8 that at this new temperature Y3  o, the heat g ene ration i s h ig her than th e h eat removal and th e re fore the temperature increases and goes back to Y3 . The same applies t o the s te ady state Y 1 • Therefore Y1 and Y3 satisfy the n e c e ss ary condition for s t abi l i ty , that is the slope of the heat removal
44
S.S.E.H. ELNASHAIE and S . S . ELSHISHINI
FIGURE 1.8 Heat generation ( G ( Y)) and heat removal (R ( Y)) functions for the nonadiabatic CSTR, (1) a case with multiple steady states (Yt. Y2, Y3), (2) a case with single (unique) high temperature, high conversion steady state (Y:,).
function is higher than the slope of the heat generation function at these steady states. However, these steady states can still be stable or unstable depending upon the dynamic stability condition which will be derived in the next section from the eigenvalues of the linearized dynamic differential equations in the neighbourhood of the steady states. However, for the middle steady state where the slope of the heat generation function is larger than the slope of the heat removal func tion, the situation is of course different. If Y2 is increased by a small amount 8, we find from Figure 1 .8 that at this new temperature Y2 + 8 the heat generation is higher than the heat removal and the tempera ture continues to increase away from Y2 towards Y3 • On the other hand if the temperature is decreased by a small amount 8, we find from Figure 1 . 8. that at this new temperature Y2  8 the heat removal is higher than the heat generation and the temperature continues to decrease away from Y2 towards Y1 . Therefore, the middle steady state Y2 is unstable regardless of the analysis of the eigenvalues of the linearized dynamic differential equations. Stability and the eigenvalues of the linearized dynamic differential equations The differential equations of the system can be written as,
ELEMENTARY CHEMICAL REACTORS DYNAMICS
45
where, = XA xlf = XAf x1
" Jl ( Xt , Xz
L = l + Lm
x2 = Y Xzt = Yf
) = ( Xtf  xl )  ao · e
· XI
 ylx2
fz (Xt , Xz ) = (x2 f  Xz ) + ao . f3 · e
 yl xz .
Xt  Kc ( xz  Xzc )
Linearizing equati o n s 1 . 1 0 1 , 1 . 102 and writing them in disturbance variables (i.e. deviation of the state variables t' from the state variables at steady state),
terms of the at any time ( 1 . 103) ( 1 . 104)
gives the following coupled two homogeneous linear differential equations: ( 1 . 1 05) and,
( 1 . 1 06) where , JJ;
gij = aX · Equations 1 . 1 05 and 1 . 1 06
r 1
can be
d dt'
( 1 . 1 07) ss
i tten in
wr
!_ A X =

matrix form
as, ( 1 . 1 08)
S . S .E.H. ELNASHAIE and S . S . ELSHISH J NI
46
where,
and,
The characteristic equation is given by: ( 1 . 109) and the eigenvalues are given by, A.
_
1,2 
�
2 trA ± (trA)  4det A 2
and the solution vector is given by,
The stability conditions (that is the eigenvalues of the steady state are either real negative or complex with negative real parts) are thus given by, 1.
det A > 0
2.
trA_
0
( 1 . 1 1 2)
positive, thus the condition becomes , (1.1 13)
ELEMENTARY CHEMICAL REACTORS DYNAMICS
47
From equation ( 1 . 1 07) we find that,
( 1 . 1 1 4) where,
( 1 . 1 1 5) Similarly,
( 1 . 1 1 6) where,
( 1 . 1 1 7) Also,
( 1 . 1 1 8) and,
( 1 . 1 1 9) Thus this stability condition can be written in the form,
 (1 + RA )[  (1 + Kc ) + f3 R8 ] + R8 /3 · RA ·
·
>
0
( 1 . 1 20)
which can be rearranged in the following form,
(1.121) The left hand side is clearly the slope of the heat removal function R ( Y) in equation 1 .99. The heat generation function G ( Y) can be written in terms of x1 (= XA ), x2 (= Y) as follows, _
( 1 . 1 22)
The slope of the heat generation function G (x2) is the total differential of R (XJ, x2) with respect to x2 and is given by:
S . S .E.H. ELNASHAIE and S . S . ELSHISHINI
48
dG(x2 )
_....'.;.....: 
dx2
=
dRR(x1 , x2 ) ()RR dx1 ()RR dx2 = + ax! dx2 ax2 dx2 dx2 

The different terms in equation 1 . 1 23 are gi ven by
dx2 = 1 dx2
()RR = f3 · RA ax!
dx1
dx2
=
_
( 1 . 1 23 )
,
( l + Kc )
f3
()RR = /3 · Ra ax2
( 1 . 1 24)
The slope of the heat generation function is thus given by , ( 1 . 1 25)
Therefore the right hand side of equation 1 . 1 2 1 is the slope o f the heat generation function at the steady state. Thus the first stability con dition obtained from the eigenvalue analysis of the linearized dynamic equation, is the same slope condition obtained from pseudo steady state an aly s i s This stability condition is usually called the static stability condition in contradistinction to the dynamic stability condition to be analyzed in the following section. The dynamic stability condition: .
tr£1 < 0
(1.1 1 1 )
which is given by, ( 1 . 1 26) and from the definition of g 1 1 (equation 1 . 1 14) and g22 (equation 1 . I 1 9), th e above condition can be written as, ( 1 . 1 27) This dy n ami c co n d itio n (e quation 1 . 1 27) i s not n ece ssari l y satisfied when the static condition (equation 1 . 1 2 1 ) is satisfied. This dynamic condition depend s upon the dy namic parameter L. Before we pre s ent a detai led classification of some of the possible
types of instability and the types of s te ady states, we g ive here a number of special cases,
ELEMENTARY CHEMICAL REACTORS DYNAMICS
1.
The case of adiabatic operation (Kc
49
0) and negligible Lm (L = 1 )
=
In this case the dynamic stability condition reduces to, ( 1 . 1 28)
and since the slope condition
in
this case is given by, ( 1 . 1 29)
Thus, from 1 . 1 28 and 1 . 1 29 it is clear that the satisfaction of the slope condition ( 1 . 1 29) in this case implies the satisfaction of the dynamic condition ( 1 . 1 28) and therefore steady states y1 and y3 are always stable (without specifying at this stage yet what type of stable steady states they are). 2.
The case of nonadiabatic operation Kc :t 0 and negligible Lm (L = 1 )
In this case the dynamic stability condition becomes, 1  RA  1  Kc
+ {3 R8 < 0
( 1 . 1 30)
(2 + Kc ) + (/3 R8  RA ) < 0
(1.131)
·
which can be arranged in the form,
·
The slope condition in this case will have the same form as equation 1 . 121, ( 1 . 1 21 )
It is clear that in this case, the satisfaction of the slope condition (equation 1 . 1 2 1 ) does not imply the satisfaction of condition 1 . 1 3 1 and steady states Y J , Y3 may be unstable.
3.
The case of adiabatic operation (Kc = 0) and L :f. 1
In this case the dynamic stability condition becomes, 1

in
L
·
L
( 1 ) (/3
which can be rearranged

1 f3 R8 1 the satisfaction of the static condition implies the satisfaction of condition ( 1 . 1 33) and Y J . y3 are stable. However for L< 1 , Y t . Y3 may become unstable. The condition L < 1 is possible for catalytic reactors as will be shown later. 4. The case
of adiobatic operation Kc :1: 0 and Lm
not negligible,
L> 1
This is the general case where the static stability condition is given by equation ( 1 . 1 2 1 ) and the dynamic stability condition is given by equation ( 1 . 1 27). Clearly the satisfaction of the static condition does not imply the satisfaction of the dynamic stability condition and therefore Y � > Y3 may be unstable. So far we have discussed stability and instability in a very simple manner and have not introduced the different types of stable and un stable steady states. This will be classified and discussed in the coming chapters. However, before we close this elementary part of the book we will give a very brief idea about some of the different types of stable and unstable steady states. For this purpose we will use for pre sentation schematic phase planes. The phase plane is a twodimensional plot of x1 vs. x2, where the changes of X J . x2 are plotted as they change with time, time t ' being a parameter on the graph. Therefore, the initial condition at t ' = 0 (x 1 o, x2o) will appear as a point on the phase plane. The change of x" x2 with time will appear as a trajectory on the phase plane. If the steady state is stable the trajectory starting from a certain initial condition (if the initial condition is within the basin of attraction of the steady state) will move till it settles at the steady state. Steady states y1 and y3 may be oscillatory or nonoscillatory depending on the nature of the eigenvalues. The stability of these steady states (whether oscillatory or nonoscillatory) depends upon the sign of the real part of the eigenvalues (or the sign of the real eigennumber when eigenvalues are real). The oscillatory nature of these steady states (whether they are stable or unstable) will depend upon whether the eigenvalues are real or complex. When Y J , Y3 are stable (the eigenvalues are real negative or complex with negative real parts) , these steady states may be nonoscillatory (usually called stable nodes) as shown in Fi gure 1 .9, if the eigen v al ue s are real . These ste ady states (both or one of them) may be oscillatory (usually called stable foci) as shown in Figure 1. 10 where both Y t . Y3 are stable foci, if the eigenvalues are complex, with nega tive real part.
ELEMENTARY C H EMICA L REACTORS DYNAMICS
>
a
II
3
51
" ... :J
��
Q.
E !
.
�
� .... :1
"' Q.
12
E .!!
Ill Ill "'
c:
0
'iii
c: "'
E a
D i m e n si o n iPss
conc Pntrati on , XA
FIGURE 1 . 12 Phase plane for steady state is an unstable focus.
1 which is a stable node and 3 which
ELEMENTARY CHEMICAL REACTORS DYNAMICS
53
l et:
...

l c:>
>

FIGURE 1 . 13 Unique high temperature generation  heat removal) diagram.
steady
state on the
y Van
Heerden
(heat
For the unstable middle steady state it is always of the saddle type with two real eigenvalues, one positive and the other negative A simpler situation can develop when K·, Yt and Yc are changed so that a unique steady state occurs, as shown in Figure 1 . 1 3 , then it is clear that the slope condition for stability is satisfied for this case. If the dynamic condition is also satisfied and the eigenvalues are real and negative then all initial conditions will lead to this stable steady state in a nonoscillatory manner as shown in Figure 1 . 14 which represents a stable node.
FIGURE 1.14
Phase plane for a unique stable node.
54
S.S.E.H. ELNASHAIE and S . S. ELSH ISHINI
I
FIGURE 1 . 1 5
Phase plane for a unique stable focus.
This unique steady state can also be a stable focus (complex eigen values with negative real parts) as shown in Figure 1 . 1 5 . If it is an unstable focus (complex eigenvalues with positive real parts) then a limit cycle must be fonned around this unstable steady state as shown in Figure 1 . 1 6. The cases presented here for both multiple and unique steady states are given for the uninitiated reader and represent a simple (and certainly
I
FIGURE
1 . 1 6 Phase plane for a unique unstable focus giving rise to limit cycle (periodic attractor).
ELEMENTARY CHEMICAL REACTORS DYNAMICS
55
incomplete) idea about some of the possible phase planes for this two dimensional system. The same will be presented and expanded upon with more details in Chapters 2 and 3 . Saddle type steady states, separatrices and domains of attraction
As we explained earlier, when there are more than one stable steady state there must be different regions of stability (or domains of attrac tion) in the phase plane and if we start with initial conditions within one of these regions it leads to the specific steady state associated with that region and not any of the other steady states. The region in the phase space where all initial conditions lead to a certain steady state, is called the domain of attraction of this steady state (it is also called the basin of attraction of the steady state) . Obviously, when a unique steady state exists, the entire phase plane is the domain of attraction of that unique steady state. For the multiple steady state case, the unstable saddle type middle steady state plays the major role in determining the domain of attraction of the different steady states. This problem is relatively simple for two dimensional systems but can get quite complicated for higher dimensional systems. We will illustrate here a simple twodimensional case in volving two stable nodes as shown in Figure 1 .9. In this case, there are two trajectories mov i n g toward the saddle point 2 (line a2 and line b2). These trajectories are called the inset (or the stable manifold) of the saddle point 2. They are tangent to the eigenvectors corresponding to the negati ve real eigenvalues of the saddle. The line a2b is the separatrix which separates the domain of attraction of the two stable n odes 3 and 1 . Any initial condition lying in the phase plane above this separatrix leads to the stable node 3 (e.g. trajectories /1 3, h3 shown in Figure 1 .9) and any initial condition lying in the phase plane below this separatrix leads to the stable node 1 (e.g. trajectories h 1 , /4 1 shown in Figure 1 .9) . The trajectori es 2 1 , 23 emanating from the saddle and going to the stable nodes 1 and 3 respectively are called the outset (the unstable manifold) of the saddle point 2. They are tangent to the eigenvectors associate d with the positive eigenvalu e s of the saddle steady state. Simple introduction to Lyapunov first stability theorem The extremely simple analysis pre sented in this chapter uses the leas t amount of math em ati c s well known to any undergraduate eng in eering student and provides an introduction to multiplicity and stability characteristics of chemical reactors. This chapter is devoted to the
56
S . S .E.H. ELNASHAIE and S.S. ELS HISHINI
uninitiated in the subject and should certainly be skipped by any reader with previous experience on the subject. A simple stability analysis was presented in an informal manner, therefore it is important to point out that this analysis is based upon Lyapunov first stability theorem which is given here in a brief formal manner. To generalize these simple findings, we present Lyapunov first theorem (Lyapunov, 1 982) . Although our presentation in this brief section is for ndimensional systems, we will restrict our discussion, for simplicity to phenomena well known in twodimensional autonomous unforced systems. Discussion of forced (nonautonomous) twodimen sional systems as well as higher dimensional systems will be presented later in the book. In the general ndimensional case, the set of nonlinear differential equations can be written in the following form, with J1 as a parameter (usually called the bifurcation parameter as will be discussed in the next chapter). i = E.( � . jl )
( 1 . 1 34)
Linearization of the set of nonlinear differential equations around a steady state and introducing a deviation variable as explained earlier results in the following matrix differential equation in the neighbour hood of the steady state (for simplicity we will use � in the following instead of g in order to express the deviation variables). F x = X

F = X
( 1 . 1 35)
·x
diJ
ax!
diJ
ax2 ( 1 . 1 36)
iJfn
ax]
at,
ax2
The Jacobian matrix, ( 1 . 1 37) consists of n2 first order partial derivatives, evaluated at the s teady state for which the stabi l i ty analysis is seeked, dfildx1 where i, j = 1 , 2 , 3 , . , n . The nature of the stati c points can be ch arac te ri ze d on the basis of the eigenvalues A J ,Az, . . . , An of the J acob i an Fx (�. jl ). The number of cases generated by the various combinations of the eigenvalues i ncre ases dramatically with the increase in n. The followin g general stability result is attributed to Lyapunov ( 1 982). .
.
ELEMENTARY CHEMICAL REACTORS DYNAMICS
57
Theorem Suppose E (�) is tw o time s conti nuou sl y differentiable and it ' s Jacobian matrix, L (�) = O. The real parts of the eigenvalues A1 (j = 1 ,2 , . . . , n ) of the Jacobian evaluated at the stationary solution, determines the stabi lity in the fol l o wing way: (a) Re ().1) < 0 for al l j implies asymptotic stability. (b) Re ( A k) > 0 for one (or more) k implies instability.
This theorem establishes the princ ipl e of linearized s tabi l ity . In order to stress the local character of this stabil i ty criterion, this type o f stabili ty is also called co nditional stabi l i ty or linear stability or local stability. 1.3
MAIN CONCLUSIONS OF CHAPTER 1
The main conclusions of this elemen tary introductory ch apter are: 1.
2.
3.
4.
5.
Chemical reactors with a single stable ste ady state will have simple dynamic behaviour where this steady state will be reached from any initi al condition after a certain duration of time which depends upon the cap ac ity of the system. Multiplicity of the steady states may arise for both isothermal and no n isothermal reactors. For isothermal reactors this phenomenon usuall y arises when the rate of reaction shows nonmonotonic dependence upon reactants concentration(s). For the nonisothermal reactor s, the phe nome no n arises even for the s i mple linear kinetics of a unimolecular first order irreversible reaction, when the reacti o n is exothermic. When th ree steady state exists, the middle steady state is always unstable, while the other two steady states may be stable or un stable. For adiab atic operation these two steady states are always stable and in this case each one will have a region of stabil i ty or domain of attraction. The two domains of attraction of the two stabl e ste ady states are on the two sides of the separatrix line which passes through the middle unstable steady state. For non  adi ab ati c operation, it is possible for certain combination of p arameters that one or both of the high and low temperature ste ady states to be u nstable . The possible cases are: (i ) Y3 is unstable and is surrounded by a limit cycle while Y1 is stable. (ii) y3 is unstable and no limit cycles exist and all trajectories go to Y l ·
58
S . S .E.H . ELNASHAIE an d S . S . ELSHISHINI
(iii) The opposite of the above two cases by interchanging the properties of Y l and Y3 · (iv) Both y1 and y3 are unstable and limit cycle (periodic) beha viour dominates the system. The above possibilities are based on local analysis and are therefore limited. A more detailed account of possibilities will be given in the next chapter.
CHAPTER 2
Static and D ynamic Bifurcation and the Different Types of Non Chaotic Attractors
By bifurc ati on we mean a change in the number of solutions of an equation as a parameter (or more) is varied. The equations may be algebrai c , ordinary differenti al equations, partial differential equ ati on s or in fact any type of equ ations show ing a change in the number of solutions as a parameter (or more), in these equations, is varied. The term s ol ution here means static solution, periodic solution or qu asi periodic solution. In a l ater chapter we will extend the concept of "solution" to include chaoti c solution. The term attractor represents a very convenient concept. It is the s o lution at which the system settles after a long transient time, whether st arting from a certain initial condition or after being exposed to some extern al disturbances. In general the attractors can be po int, periodic quasi periodic or strange attractors. The strange attractors are divi ded into two kinds : chaotic and non chaotic as will be discussed in chapter 3. An unstable solution cannot be termed attractor (for in fact it is a repeller as will be shown later). Therefore for a periodic solution we may call it stabl e peri odic s oluti on periodic attractor or stable limi t cycle while for the unstable periodic solution we may call it unstable limit cycl e or unstable perio dic solution ,

,
,
.
2.1
POINT ATTRACTORS (Static Bifurcation)
The attractor most common to chemical engi neers is the point attractor where the system ch anges with time transiently approaching a stationary nonequilibrium state that is a state at which the state variables of the system are stationary, i.e. not varying with time, and therefore the system is represented by a point on the phase space regardless of the dimensions of the phase space of the system. For distributed s y ste ms this is repre sented by a stat i o n ary profile in space, which is constant with time. This profile c an still be c on side re d theoretically, as a point of infinite dimensions . However, practically, it has finite dimensions ,
,
59
60
S . S .E.H. ELNASHAIE and S . S . ELSHISHINI
because most of the profiles are obtained numerically through discretiza tion using of course, finite number of points in the space directions. These types of attractors (point attractors) can be obtained and analyzed using the steady state equations. For a wide variety of equations, including many partial differential equations, problems concerning multiple solutions can be reduced to the study of the variation of the solution (x) of a single scalar equation, g (x, f1 ) = 0
(2. 1 )
with the bifurcation parameter (/1). This simplification depends o n a technique known as LyapunovSchmidt reduction (Golubitsky and Schaeffer, 1 985). The discussion in this part will be restricted to cases where the system can be easily reduced to a single algebraic equation in one variable and one bifurcation parameter. We will mostly use for demonstration purposes the steady state of the nonisothermal CSTR problem discussed in the previous chapter. Since the early work of Van Heerden ( 1 953) and Bilous and Amundson ( 1 955), extensive work has been carried out to find conditions for multiplicity and uniqueness, without actually solving the equations of the system. The first observation for the existence of multiple steady states for such systems has been reported by Liljenroth ( 1 9 1 8). Pioneering analyses of this phenomenon were carrie d out by FrankKamenetskii ( 1 939), Zeldovitch and Zysin ( 1 94 1 ) and Wagner ( 1 945). The subject attracted the attention of the chemical engineers after the publication of the work of Van Heerden ( 1 953) and Bilous and Amundson ( 1 955). Com prehensive reviews have been presented by Schimtz ( 1 975), Endoh et al. ( 1 977), Hlavacek and Vortuba ( 1 978), Eigenberger ( 1 98 1 ) and Luss ( 1 980, 1 98 1 ) Extensive investigations into the subject have been presented by Balakotaiah and Luss ( 1 98 1 , 1 982, 1 983) and others, who made very good use of the work of Golubitski and Schaffer ( 1 979, 1 985) and Golubitsky et al. ( 1 98 1 ) and Golubitsky and Keyfitz ( 1 980) on singularity theory by applying it to the CSTR and catalyst pellet problems. The work of Uppal et al. ( 1 976) has uncovered a fascinating variety of bifurcation diagrams for these systems which includes among many others, the well known hysteresis type bifurcation in addition to the mushroom type and isola types of bifurcation, as well as the pitchfork type bifurcation diagram. The static bifurcation diagrams are simply plots of the steady states of the system as a chosen parameter is varied. This corre s p ond s to solving the equation g (x, J.l) == 0, many times at different v alu e s of J.1 an d plotting the solutions versus the corresponding .
values of J.l.
STATIC AND DYNAMIC B IFURCATION
61
:::
"  )(
H� � A ', / / v; , ; _ "
"
(a)
B i fu r cat i on
( b)
p a ra meot�r , V.
( c)
FIGURE 2.1 (a) Static limit point (SLP); (b) Static bifurcation point (SBP); (c) Static cusp point (SCP). 2.2
SUMMARY OF SOME OF THE MAIN COMPONENTS OF STATIC BIFURCATION
In the following, a summary is given for the basic concepts of static bifurcation behaviour. Multiplicity of the steady states is associated with the existence of more than one solution of equation 2. 1 at certain values of the bifurcation parameters (jl). Recently the singularity theory have been used to analyze the steady state solutions of equation 2. 1 for many lumped parameter systems (Balakotaiah and Luss, 1 982b) to determine: (i) The maximal number of possible steady state solutions and the parameters for which they exist. (ii) The different types of bifurcation diagrams. The commonly encountered steady state bifurcation points (BPs) (loos and Joseph, 1 98 1 ; Kubicek and Marek, 1 983), are: (a) Static Limit Point (SLP) (or turning point or saddlenode point). It is the point at which two branches of the steady state solutions having l imiti ng tangent dp/dx Iss = 0 are joined. At the static limit point two branches of the steady states are born or two branches of the ste ady states annihilate each other. Other name s for the static l_imit point are static tu rni ng point or saddlenode p oi n t . The static limit points are frequently born in pairs, resulting in hysteresis effects i.e. ignition and extinction pheno mena in chemical reaction e ngineering (Figure 2. l a) .
S.S .E.H. ELNASHAIE and S . S . ELSHISHINI
62
� ........ �!� i£ 1   �� :; iii
...
Supercritical
.. iii
I
I
...,
....
Subcritical ......
iii
� "'
'
......
....
   
..... .... ....
( b)
(a ) Bifu rcation
(c)
parameter , J.l
FIGURE 2.2 Perfect and imperfect pitchforks. (a) Supercritical perfect pitchfork; (b) Subcritical perfect pitchfork; (c) Transcritical imperfect pitchfork.
(b) Static Bifurcation point (SBP) . It is the point at which two (and only two) curves possessing distinct tangents cross each other (Figure 2. l b). (c) Static Cusp Point (SCP) . It is the contact point between two curves of steady states having the same tangent as shown in Figure 2. 1 c. (d) Static Bifurcation Limit Point (SBLP). This is a double point at which a static limit point (SLP) and a static bifurcation point (SBP) coincide with each other. This bifurcation point is also called a perfect pitchfork. The terminology of supercritical and subcritical for the classification of pitchfork bifurcations is also widely used. Supercritical perfect pitchfork has stable branches on both sides of the static bifurcation limit point (SBLP) (Figure 2.2a). Subcritical pitchfork has unstable branches on both sides of the static limit point (SBLP) (Figure 2 .2b). When the static limit point (SLP) and the static bifurcation point (SBP) do not coincide with each other, the pitchfork is called transcritical imperfect pitchfork (Figure 2.2c) . Characteristics of turning points and bifurcation (pitchfork) points (twodimensional system) For the twodimensional system,
E(!. , f.l ) = o
(2.2)
STATIC AND DYNAMIC BIFURCATION
63
where, (2.3)
and, (2.4)
We can form the Jacobian, l.'! = E'! (;!_, /1 ) =
()J; axl
()J; axz
dJi axl
dJi axz
=J
(2.6)
For both cases of turni ng points and bifurcation points, thi s Jacobian of the system is singular. Of course for the scalar case where the system is described by equation 2. 1 we will have at these points df/dx = 0. That is, (2.7) where !lo is the bifurcation parameter value at the SLP or bifurcation point and ;!_0 is the vector of state variables at llo· Now we need to distinguish between static limit points (SLPs) and bifurcation points (BPs). The following simple analysis given by Seydel ( 1 988) is elegant, simple and sufficient.
Fornwl definition of SLP and bifurcation points
If we attach to the singular Jacobian matrix Ex (;!_0 ,/10 ) , the vector E_11 ,
e g for a two dimensional system it will have the form, .
.
one obtains an augmented matrix wi th one additional column. This augmented matrix at the bifurcation point is sti ll singular (rank < 2) while at the SLP it has a full rank (rank = 2).
64
S.S .E.H. ELNASHAIE and S . S . ELSHISHINI
This difference between a SLP and a bifurcation point is obviously worth de eper an al ysis in order to reach some calcu latable formal definitions for both types of points. We will present the procedure for an ndime n sion al system. We start with the original equatio n, f(,!_, J..L ) = 0
(2 . 8 )
where :! is an nvector of the state variable. Let us con sider J.1 to be the (n + 1 ) component of this vector, that is, (2.9)
Now, e quati on (2.8) with the new compo nent xn + l • represents a set of n equations in n + 1 unknowns,
(i = 1, 2, . . . , n )
(2. 1 0)
The rectangular matrix of the parti al derivatives consists of n + 1 column s z/ , (2. 1 1 )
We are free to interpret any of the n + 1 components (say the Jeh component) as a parameter. Call this parameter y. (2. 1 2) The dependence of the remai ning
n
comp onents ,
(2. 1 3)
on r is characterized by the "new" Jacobian that results from equati on 2. 1 1 by removing the k1h c olumn . For the SLP, it is poss i ble to find an index k such that the new Jacobian is n ons ingular (full rank = n), where as for a bifurcation point no such k exists (rank
0, the steady state is unstable and is called saddle (Figure 2.28a). The two stable manifolds (insets) that enter the saddle point are called separatrices. The separatrices divide the phase space into attracting basins. ·
3. Complex conjugate eigenvalues
When the ei g envalues are complex conjugates, A1,2 = ai ± i. bi where j = H, the steady state is called focal (spiral) as shown in Figure 2.28b. The local stability of the focal steady state is determined by the
STATIC AND DYNAMIC BIFURCATION
89
x,
(a )
(b)
( c: )
FIGURE 2.28 (a) SaddJe steady state; (b) Stable focal steady state; (c) Limit cycle (when stable is usually called periodic attractor).
sign of the real part aj . If aj is negative the focal steady state is stable. positive then the focal steady state is unstable. For unstable focal the direction of the trajectories is away from the steady state.
I f aj is
4.
Pure imaginary eigenvalues
When the eigenvalues are pure imaginary 1 1 , 2 ±i. bj , the flow is locally a pure rotation about the steady state and forms a close curve called a limit cycle (periodic orbit) as shown in Figure 2.28c. These periodic attractors are the subject of the next section. The first steady state having this characteristic (in addition to some other discussed later) occurs at a point called a Hopf bifurcation (HB). It is the first most important dynamic bifurcation point (in contradistinction to the static bifurcation points discussed earlier). It is the point where periodic attractors start. A historically more accurate name for this point is PoincareAndronovHopf bifurcation point (PAHB), however we will use the commonly used expression Hopf bifurcation (HB). The characteristics of this local dynamic bifurcation point s will be dis cussed in some details later in this chapter.
2.6
=
BASIC PRINCIPLES OF DEGENERACY AND PARAMETRIC DEPENDENCE
As we explained earlier, the steady state (fixed point) is calle d hyperbolic or non degenerate when the matrix Ex has no eigenvalues with zero
90
S . S .E.H. ELNASHAIE and S . S . ELSHISHINI
real parts (Seydel, 1 98 8). The eigenvalues of the Jacobian matrix Ex at a fixed point, in addition to determining the static bifurcation points, they also determine the dynamic behaviour in the neighbourhood of the steady state. This holds for three types of steady states namely: nodal, saddle and focal points. These points are generic in that the assumptions: A1 :t A2 , A.1 . A.2 :t 0 or Re( A ) :t 0, almost always apply (Seydel, 1 988). The cases for which A1 . A 2 = 0, A1 ' 2 = ± i. f3 are non hyperbolic or degenerate (Seydel, 1 988). Usually a differential equation describing a real life problem involves one or more parameters. Denoting one such parameter as usual, by J1 and considering a twodimensional system, the differential equations read: (2.66) (2.67) Because the system ,!. = £( !, J1 ) depends on Jl, we speak of a family of differential equations. Solutions depend on both the independent variable t and on the parameter Jl, ! ( f , Jl )
(2.68)
Consequently, the steady states, the Jacobian matrices and the eigenvalues A, depend on Jl .
A(Jl ) = a( Jl ) + i.f3(J1 )
(2.69)
Upo n varying the parameter Jl, the position and the qualitative features of a steady state can vary. For example, consider a stable focus ( a(Jl ) < 0) for some value of Jl. When J1 exceeds some critical value Jlc, the real part a(Jl ) may change sign and the steady state may convert into an unstable focus. During this transition, a degenerate focus (center) is encountered (e.g. see Figure 2.29). In other words, the degenerate cases can occur "momentarily". Often, qualitative changes such as loss of stability are encountered when a degenerate case is passed. This is obviously a bifurcation point, but a dynamic bifurcation in contrast to the static b i fu rc atio n discussed earlier. One of the most important and w i d e l y enco un tere d l oc al dyn amic bifurcation point is the Hopf bifurcation (HB) point. Before we i ntrodu ce some of the mathematical details of th is dynamic bifurcation type, we will introduce it with the min i mum of mathematical detail s and concentrate in this chapter on a more prag matic bas i s u s in g pictorial methods and correct dynamical common sense for some of the details about periodic attractors. This
STATIC AND DYNAMIC BIFURCATION
(a)
(b)
91
(c)
FIGURE 2.29 Transition from stable focus to unstable focus. (a) stable focus, a (Jl) < 0; (b) center, a (JJ.: ) = 0; (c) unstable focus, a (Jl) > 0.
appeals to chemical engineering sense more than the slightly drier mathematics which will be presented in a simple, easy to follow manner later in the book. 2.7
PERIODIC ATTRACTORS OF AUTONOMOUS SYSTEMS
Point attractors are not the only possible attractors for a dynamical system. There are other attractors, the next in complexity being the periodic attractor where the dynamic behaviour of the system instead of settling down, after the initial transients, to a fixed value of the variables which is invariant with time, it settles down to a trajectory which is changing with time but in a periodic manner. This means that the trajectory repeats itself with time following the relation, �(t + r) = �(t)
(2.70)
where r is called the period of the periodic attractor. A simple two dimensional periodic attractor is shown in Figure 2.30. The case shown in Figure 2 . 30 is a case where there is a unique unstable s te ad y state o (unstable focus). If the system s tarts at any i ni ti al condition outs ide the periodic attractor or inside the periodic attractor, the system state v ariable s x� o x2 change with time till they settle d own
92
S.S.E.H. ELNASHAIE and S . S . ELSHISHINI
x,
FIGURE 2.30
Unique periodic attractor (x a initiaJ conditions).
at the shown periodic attractor which is rotating around the unstable steady state. For a bounded system like the chemical reactor, the condition for the existence of a periodic attractor surrounding a unique steady state as shown in Figure 2.30, is that this unique steady state is unstable. This is the simplest case for the existence of periodic attractors. Later in this chapter we will show more complex situations where there is bistabilities of different types, such as the existence of multiple periodic attractors at certain values of the parameter or the coexistence of a periodic attractor (or more) together with a point attractor (or more). The birth of a periodic attractor at a certain value of the bifurcation parameter Jl, is the most important dynamic phenomenon that connects the static and the dynamic bifurcation of the system. The simplest mechanism by which a periodic attractor is born, as a bifurcation parameter is varied, is the Hopfbifurcation (HB). Seydel ( 1 988) correctly states that: "Hopf bifurcation is the door that opens from the small room of equilibria to the large hall of periodic solutions". Hopf bifurcation represents a mechanism of local birth of periodic attractors which should be distinguished from nonlocal mechanisms such as homoclinical bifurcation which will be discussed later. The basic results regarding the general characterization of Hopf bifurcation were known to Poincare (Minorsky, 1 974); the planar case was handled by Andropov in 1 929. In spite of the s e early results, bifurcation fro m s tati c poin t attractors to period ic attractors is common ly referred to as Hopf bifurcation ( S eydel , 1 988), bec ause it was Hopf who proved the fol l owin g theorem for the ndimensional case in 1 942 (Hopf, 1 942) . The following is a mathematically simple definition of Hopf bifurcation point s which will be expanded upon later in this section.
STATIC AND DYNAMIC BIFURCATION
93
Theorem For the system,
dx d; = E(;!_, Jl )
(2.7 1 )
where J1 i s a varying bifurcation parameter. If the following conditions are satisfied, then a birth of a peri odic solution occurs:
which is the condition for the existence of stationary solution with state vector ;!_0 at the value of the bifurcation parameter J10, and: is the Jacobian matrix formed of the partial derivatives of E at ;!_0 , J10 and has a simple pair of purely imaginary eigenvalues A (J10 ) = ±i./3 and no other eigenvalues with zero real parts. Golubitsky and Schaeffer 3 (book , vol. 1 , p. 72) giv e for this simple eigenvalues c onditi on a more restrictive condition (H4) which is a strengthening of the simple eigen values condition give n in the above lines. This condition (H4) is: If A1 , A2, A.n are the eigenvalues o f the Jacobian at J10, then for Hopf bifurcation we should have, ,
• • •
AI =
+ i.f3,
A2 =
i./3,
Re ( A. ) >
for j = 3, 4, .
0
.
.
,n
Notice that this condition (H4) implies the above condition 2 but the above condition does not imply this condition. 3.
d(Re A (J.l ) )
dJ.l
I
Jl =Jlo
0 ;t
i.e. the slope of Re (A.) versus J1 at Jlo is not zero. These are the condi tions for the birth of limit c yc l es (periodic attractors at C!_0 , J10 )). The i ni tial peri od (of the zero amplitude oscillation) is T, = 2rr I /3. Conditions 1, 2 and 3 above can be viewed as a definition o f Hopf bifurcation. Condition 3 i s the transversality c onditi on , which is usually satisfied except at degenerate points which w il l be discussed later in
this chapter. Now, we should distinguish between bifurcations:
two
basic types of Hopf
S . S .E.H. ELNASHAIE and S.S. ELSHISHINI
94
J.1 "" � H I x,
.... ... 0 
)(
)(
)(
•
•
•
I
__
x2
•
.  ..,_ .... ... . ___... . . . . . . •
•
_ .,.,
I
I I
I
�HB
@ J.1 :=.. J.l. Ha
x,
(!f) Xz
(a)
( b)
FIGURE 2.3 1 Supercritical Hopf bifurcation. Stable static branch (), unstable static branch ( ), dynamic branch with peaks of the p eriodic solution at different values of ll • = HB point. The two small figures (a,b) on the right, are phase planes for ll smaller and greater than ll for the Hopf bifurcation /J.HB , respectively. On figures a, b: • = stable focus, o = unstable focus, x = initial conditions.
1. Supercritical Hopf bifurcation
This gives rise to soft oscillations (in biological sciences it is called soft excitation). This type of bifurcation is shown in Figure 2.3 1 for a case with unique static branch. For these soft oscillations, a periodic attractor arises i nitially with low amplitude as J1 is increased. Figure 2.3 1 (x vs. J1) represents the static stable () and unstable (  ) branches, while the dynamic branch (• • • •) represents the peaks of the periodic oscillations at di fferent values of Jl. The two small figures (a,b) are the phase planes j u st before and just after the Hopf bifurcation point (J1Hs) respectively . 2.
Subcritical Hopf bifurcation
This gi ve s rise to hard oscillations (or hard e xc i tati on ). This type is shown in Figure 2.32. 1t is clear that it gi ve s rise to a re g i on of bistability and a periodic limit poi nt (PLP). The periodic limit points (PLPs) are points where a stable and an unstable peri od i c solution collide. This
STATIC AND DYNAMIC BIFURCATION
1.1
ll "< ll p L P
x,
.... >< ....0 >
LP < Jl. < JI.H8, (c) stable periodic attractor + unstable fixed point, for Jl. > JI.HB· On these figures (ac), • = stable focus, o = unstable focus,    = unstable limit cycle (periodic separatrix), x = initial conditions.
collision occurs at f.1 = f.1PLP as shown in Figure 2.32. It also shows the existence of unstable periodic orbit acting as a separatrix between the domain of attraction of the two attractors in the bistability region. 2.8
DIFFERENT TYPES OF PERIODIC ATTRACTORS
After this simple introduction to the possibility of the existence of a unique unstable steady state surrounded by a limit cycle (or a periodic attractor) and the simple introduction of Hopf bifurcation theorem and the two main types of Hopf bifurcations, let us introduce in a systematic and simple manner the different types of periodic attractors (excluding in this part periodic attractors with higher periodicities than one resulting from period doubling which will be discussed later in this chapter). Because of the strong relationship between p eri od ic and po int attractors, we used the simple technique of presentation based upon impt> sing the periodic attractors on the static bifurcation diagram by plotting for the periodic attrac tor, the value of one of the state vari ables at the maximum and/or the minimum (sometimes for simplicity, we
96
S . S .E.H. ELNASHAIE and S . S . ELSHISHINI
may plot the maximum only or the minimum only of the oscillations as shown in Figures 2.3 1 and 2.32, where only the maxima of the pe riodic oscillations are plotted) of its oscillatory behaviour using black circles for stable limit cycles (periodic attractors) and empty circles for unstable limit cycles (unstable periodic solutions). Dis tinguishing between stable and unstable limit cycles is achieved through the computation of Floquet multipliers The definition of the Floquet multipliers and the technique for computing them, will be explained at the end of this section. For the purpose of this more descriptive part we can simply say that a stable limi t cycle is an attractor while an unstable limit cycle is a repeller (us ually saddle type acting as a separatrix between different attractors ). For simplicity of pre sentation we again consider a twodimensional system with one bifur cation parameter , represented by the two differential equations : .
(2.72) and, (2.73) The steady state of this system is given by : (2.74) We will first consider the case where equation 2.74 has a simple hysteresis type static bifurcation as shown in Figure 2.33 (ac). Notice that in these figures (unlike Figures 2.3 1 , 2.32), both the maxima and the minima of the oscill ations are plotted. From earlier discussions, it is clear that the intermediate static branch is always unstable (saddle points), whereas the upper and lower branches can be stable or unstable d epending on the eigenv alues of the linearized forms of e quations 2.72 and 2.73. The static bifurcation diagram s in Figure s 2.33 (ac) have two static limit points (SLPs), which are sometimes called s addle node bi furc ation points As discussed in chapter 1 , the stability characteristics of the steady state poi nts can be determined from the eigenvalue analysis of the linearized ve rsi ons of equations 2 .72, 2.73 which will have the form: 
.
(2 .75)
STATIC AND DYNAMIC BIFURCATION
HB1
(a)
r
...... ... K .. 0
'
>< >
C .. 0
·
( "". ...
  
....... ..
.... ....
FIGURE 2.33(c) Bifurcation diagram for equations 2.722.74. A case with one Hopf bifurcation point, one periodic limit point and one homoclinical (HC) orbit (infinite period, IP, bifurcation point). ( = stable branch of the bifurcation diagram;    = saddle points;    = unstable foci; • = stable limit cycles; o = unstable limit cycles; HB = Hopf bifurcation point; PLP ;:;; periodic limit point). ·
·
(2.76)
where ,
X; = X;  Xiss
and
( X;.u = X; at steady state) (ss = evaluated at Xiss )
The teristic
where
ei genvalues of equations 2.75, 2.76 are the roots of the charac equation:
the
matrix �
2 A  (tr�). A + (det � ) = 0
is given by:
( 2 .77)
STATIC AND DYNAMIC BIFURCATION
99
The eigenvalues for this twodimensional system, are given by:
A 1,2 _
trA_ ± �(tr�i  4detA 2
(2.79)
The most important dynamic bifurcation point is the Hopf bifurcation point, when AJ , A2 cross the imaginary axis into positive real parts of AJ , A2. This is the point where both roots are purely imaginary and at which trA_ = 0 giving, (2.80) At this point periodic solutions (stable or unstable limit cycles) come into existence as shown in Figure 2.33. The case in Figure 2.33(a) shows two Hopf bifurcation points, HB1 and HB2 (both are supercritical Hopf bifurcation points ), with a branch of stable limit cycles (peri odic attractors) connecting them. Figure 2.34 shows a schematic diagram of the phase plane for this case with Jl = Jli · In this case a stable limit cycle surrounds an unstable focus and the behaviour of typical trajectories is as shown. The case in Figure 2.33(b) has two Hopf bifurcation points (HB2 is a supercritical Hopf bifurcation point, while HB1 is a subcritical Hopf
x,
Phase plane for ll = Ill in Figure 2.33(a). ( = stable limit cycles and trajectories;  = stable manifold, separatrix; 0 = unstable saddle; • = stable steady state (node or focus); o = unstable steady state (node or focus); x = initial conditions. FIGURE 2.34
S . S .E.H. ELNASHAIE and S . S . ELSHISHINI
1 00
'
Xt
'
'
',
'
'
'
'
2
..... ..... .....
....
...... ...... _

FIGURE 2.35 Phase plane for p = p2 in Figure 2.33(b). ( = stable limit cycles and trajectories;  ·  ·  = unstable limit cycle;    = stable manifold, separatrix; ® = unstable saddle; • = stable steady state (node or focus); x = initial conditions.
bifurcation point), a periodic limit point (PLP) and a branch of unstable limit cycles in addition to the stable limit cycles branch. Figure 2.35 shows the phase plane for this case with J.l = J12• In this case there is an unstable limit cycle surrounding a stable focus and the unstable limit cycle is s urrounded by a stable limit cycle. The behaviour of the typical trajectories is as shown in Figure 2.35. The case in Figure 2.33(c) has one Hopf bifurcation point, one periodic limit point and the stable limit cycle terminates at a homoclinical orbit (infinite period bifurcation). Homoclinical orbits are periodic orbits passing by the saddle point and having infinite period (these homoclinical points which are nonlocal bifurcation points, will be discussed in more details later in this chapter) . For J.l = J13, we get a case of an unstable steady state surrounded by a stable limit cycle similar to the case in Figure 2.34. However in this case, as J.l decreases below J13, the limit cycle grows until we reach a limit cycle that passes through the static saddle point as shown in Figure 2.36. This limit cyc le represents a trajectory that starts at the saddle point and ends after "one period", at the same saddle point . This trajec tory is called the homo c li n ical orbit and will occur at some critical value J.l = !lHC · It has an i nfi n i te period and therefore this bifurcation point is called "infinite period bifurcation". For J1 there is a unique
STATIC AND DYNAMIC B IFURCATION
I
1 03
I
N
)( ... 0
.
< � 0
1 05
HB 2
>
C
...
I I
. . . . . ..... .. ..... . . .. ..... ....... � .
Y' '?I · . l • )' • l :� I .Y\ I I
S odd l � 5 Bran c h
0

X
>
, Saddles� i.:._• • • '? I .y, ... 0
>
0, it is easy to see that for a certai n value of J.l (J.l sufficiently small of course) , a periodic solution exists for,
i=O
(2. 1 00a)
and (2. 1 00b)
S. S.E.H. ELNASHAIE and S.S. ELSHISHINI
1 16
For the relation r > O and from equations (2. 1 00a) and (2.97), we can easily see that, for this periodic solution, r(t) =
� ,u�d
(2. 1 0 1 )
Obviously equation (2. 1 0 1 ) has no meaning except i n the following region,  oo
0,
c >
0.
STATIC AND DYNAMIC BIFURCATION
1 17
We can also see that in general, there are four cases of behaviour in the neighbourhood of the fixed point (in all cases here the fixed poi nt is at the origin (0,0) and this fixed point is stable for c < 0 and unstable for c > 0). The four possible cases are: Case 1: d > O, c > O
S ince c > 0, thus as mentioned above, the fixed point at Jl = 0 is unstable. For small changes in Jl we find that the origin is an unstable fixed point for Jl > O and asymptotically stable fixed point surrounded by an unstable periodic orbit for Jl < 0. This case is shown in Figure 2.46. This case corresponds to the local behaviour around HB in Figure 2.3 8a, 2.33b. Case 2: d > O, c < O
In this case the fixed point at Jl = 0 is asymptotically stable. For small changes in Jl we find that the origin is asymptotically stable for Jl < 0, while for Jl > 0 it is unstable and is surrounded by a stable peri odic attractor. This case is shown in Figure 2.47 (W i gg in s 1 990). This case corresponds to the local behaviour around HB1 i n Figure 2.37a and 2.33a. ,

y
0
0
'1  d f.J. r  c
0
YL. � � �
FIG URE 2.47
J..I. < O
Case 2 with d > 0, c < 0 .
J.l > O
S . S .E.H. ELNASHAIE and S.S. ELSHISHINI
1 18
2
Jld
r = c
0
0
/J. < O
JJ =U
Jl>{)
���
'L. FIGURE 2.48
0
Case 3 with d < 0, c > 0 .
Case 3: d < O, c > O
In this case the fixed point at J..l = 0 is unstable. For small changes in J..l we find that for J..l < 0, the fixed point is unstable while for J..l > 0, the fixed point is stable and surrounded with an unstable periodic orbit. This case is shown in Figure 2.48 and corresponds to the local behaviour around HB1 in Fig ure 2.33c. Case
•
4: d
< O, c < O
In this case the fixed point at J..l = 0 is stable. For small changes in J..l we find that for J..l < 0, the fixed point is asymptotically stable while for J..L > O, the fixed point is unstable and surrounded with a stable periodic attractor. This case is shown in Figure 2.49 and corresponds to the local behaviour around HB2 in many cases presented earlier (e.g. Figure 2.3 3a,b, Figure 2.37a, Figure 2.3 8a) . It i s clear from these results and si mple analysis that the s ign of the number c, determines whether the bifurc ating periodic orbit is stable (c < 0) or un s tab l e (c > 0). It is thus clear that for c < 0, the Hopf bifurcation point is s uperc ri t i c al and for c > 0 , it is subcritical . In addition the si gn of d which is a'(O), is given by,
I
d a'(O) = d = ( Re. A (J..L )) d .J.l � =0 
(2. 1 04)
STATIC AND DYNAMIC BIFURCATION
1 19
'I
'L. FIGURE
2.49
0
0
0
J.l O
���
Case 4 with if < 0,
c
< 0.
decides the direction of the crossing of the imaginary axis by the eigenvalues (from the left halfplane to the right halfplane, or vice versa) as Jl increases. For d < 0, the eigenvalues cross from the right halfplane to the left halfplane as 11 increases. This implies that for d < 0, the fixed point (0,0) is unstable for 11 > 0 and stable 11 < 0. While for d > 0, the eigenvalues cross from the left halfplane to the right half plane as Jl increases. This implies that for d > 0, the fixed point (0,0) is stable for J1 > 0 and unstable 11 < 0.
2. 1 1
COMPUTATION OF THE PERIOD OF PERIODIC ATTRACTORS
The unforced autonomous systems discussed in this section, give under certain conditions an unstable steady state surrounded by a stable limit cycle. High accuracy is required in many applications, for the determination of the period (natural period) of the unperturbed oscillatory state . Shooting algorithm using Newton' s method can be used for this purpose. The method can be used for ndimensional systems but is ill ustrated here for a twodimensional system with one parameter 11. for simplicity. The two dimensional autonomous system in vector notation can be written as: (2. 1 05)
1 20
S . S .E.H. ELNAS HAIE and S . S . ELSHISHINI
where r' = t I � with the boundary conditions :
(2 . 1 06)
where :!: ( x1 , x2 ) is the vector of states, ::!:0 is the vector of initial condi tions, f is the vector of non linear functions, 11 is a parameter and P0 is the period of the limit cycle (periodic attractor) . The problem is a twopoint boundary value problem where P0 is unknown and varies with the system parameters. Integration of equation (2.43) from r' = 0 to r' = I, gives:
xJl) = Fj(xf , xf , l�J
i = l, 2
(2. 1 07)
For any periodic solution, equation (2. 1 06) must be satisfied to give:
cf>j( xf , x2 . � ) = xi (l )  xi ( O) = Fj ( xf , x2 , P0 )  xf ( O ) = 0
i = 1, 2
(2. 1 0 8)
equation 2. 1 08 gives two nonlinear algebraic equations with three unknowns xf , x2 and � . An additional anchor equation is required to specify the problem completely. The steady state solutions ( x55 ) of the autonomous system at specified parameter space are given by solving the following set of nonlinear algebraic equations : 0 = l_ ( :!:ss , Jl )
•
(2. 1 09)
Since the autonomous system output is a single unstable steady state surrounded by a stable limit cycle, one of the steady state variables (xu 1 , or x552 ) is fixed. In this case x552 is chosen to be fixed. This represents an anchor equation which is parallel to one of the phase plane axes and intersects the limit cycle phase plane transversally. The anchor equation ensures that the fixed state variable is lying on the limit cycle and eliminates the infinite solutions of the problem, since each point on the limit cycle will coincide with its image after one period of oscillation. The system is now composed of two nonlinear algebraic equations with only two unknowns which Carl be solved by Newton ' s method, whose Jacobian matrix, for fixed xz (x2 = X55 2 ) , is given by:
(2. 1 1 0)
STATIC AND DYNAMIC BIFURCATION
121
Sinc e F; i s not algebraic and can on ly be evaluated by integrati on , the partial derivative s are obtained by integrating the following variational eq uati ons simultaneously with equation (2. 1 05 ) . Let, . UX; Q = '
(2. 1 1 1 )
aP0
Differentiation of equation (2. 1 05) wi th respect to xf and po gives the following v ariational equ at ions :
ddr''¥;1
=
P 0
n=2
ufi . 'Pk l k = l axk
L
i = 1, 2
(2. 1 1 2)
and, (2. 1 1 3) The initial conditions for these equations at n;
=
o
r' = 0,
i = 1. 2
are: (2. 1 14)
oil is the Kronecker delta. Integration of these variational equations simultaneously with equ ation 2. 1 05 to r' 1 gives the elements of the Jacobian as follows : where
=
(2. 1 1 5 )
(2. 1 1 6)
Eq uation 2. 1 1 6
2. 1 1 3.
2.12
show s that it is not necessary to integrate equation
STABILITY OF PERIODIC ORBITS
The st abil i ty of the limi t cycle (periodic orbit) is de termined by the eigenvalues of certain monodromy matrix called characteristic or Floquet multipliers (FM). One of them is c on s trained to be unity (Kevrekidis et al. , 1 986) and thi s may be used as a numerical check of the c o mputed
1 22
S . S .E . H . ELNASHAIE and S . S . ELSHISHINI Jm
Jm
(c )
( b)
(c)
(d )
lm
Im
FIGURE 2.50 The position in the complex plane of the Floquet multipliers. (a) Stable periodic attractor; (b) Periodic limit point (PLP); (c) Period doubling bifurcation (PDB); (d) Torus bifurcation (TRB).
periodic traj e ctories, th e re m aining FMs determine the stability of the periodic orb i t which is stable if, and only if, it lies w ithi n the unit circle in the complex plane (Figure 2.50a). The multiplier with largest absolute value is usu al l y called the principal Floquet multiplier (PFM) . When the PFM cro s s e s the unit circle, as the bifurcation parameter varies, the periodic orbit loses s tabi lity and a dynamic bifurcation occurs. Three typical such dy n ami c bifurcations (most of them have been discussed earlier in section 2.82. 1 0), are well known : 1 . Periodic limit point (PLP). At the periodic limit point (PLP) a stable limit cycle collides with an unstable limit cycle and either the periodic orbit is evaporated or born. The PFM passes the unit circle through (+ I ) (Figure 2.39 (b)) . Th e periodic l imit point (PLP) i s also called saddlenode bi fu rcati on or periodic turning point . 2. Period do ubl i n g bifurcation (PDB). Th i s is the bifurcation of a periodic branch from another. The period of the bifurcated p eri o d i c branch is d oub l ed The PFM crosses the unit circle at ( 1 ) as shown in Fi g ure 2.50c. The p e ri od doubling bifurcation (PDB) can also be cal led p i tc hfo rk bifurcation. The period doubling bifurcation will be discussed in more details in connection w i th the period d ou bl i ng .
STATIC AND DYNAMIC B IFURCATION
1 23
route to chaos later in this chapter and also in the first part of chapter
4, dealing with the chaotic behaviour of fluidized bed catalytic reactors.
3 . Torus bifurcation (TRB) . The periodic orbit bifurcates to a torus
(quasiperiodic attractor) when the Floquet multipliers form a complex conjugate pair crossing the unit circle at an angle as shown in Figure 2 . 50d. A more detailed description of quasi periodic attractors is given in the next section.
The numerical techniques for the computation of Floquet multipl iers is given later in connection with the construction of excitation diagram for forced systems (section 2 . 1 8) . This is because the technique is the same for autonomous and nonautonomous systems. 2.13
THE TWO PARAMETER CONTINUATION DIAGRAM (TPCD)
The two parameter continuation diagram (TPCD) is a more condensed mean for presenting bifurcation information than the bifurcation diagrams . The bifurcation diagrams presented so far can be called one parameter bifurcation diagrams since for these diagrams a chosen state variable is plotted versus a single bifurcation parameter. The TPCD is a plot of the loci of critical bifurcation points as two bifurcation parameters are varied. To make clear this important tool for bifurcation analysis, we will give some simple and detailed explanation of these important diagrams. 2.13. 1
Static Bifurcation Loci on the TPCD
Consider the following nonlinear algebraic equation with two varying parameters,
If v i s fixed at a value follo wing form:
v
g ( x , Jl , v) = 0
(2. 1 1 7)
g(x, Jl ) = 0
(2. 1 1 8)
= c 1 , equation 2. 1 1 7 can be written
in the
Through the solution of this equation for different values of J1 and a cor�stant value of v = C t . a one parameter bifurcation diagram can be constructed as shown in Figure 2.5 1 for the case of a hysteresis type curv e.
1 24
S . S .E.H. ELNASHAIE and S.S. ELS HISHINI
X
FIGURE 2.51
values of v.
« < 'J�: � 1
I
1
I
I 1 I

 ...... ....
 
I ......
Example of bifurcation diagrams of x versus
J.l
for two different
From Figure 2.5 1 , it is clear that the system has two static limit points at J.l= SLP1 and SLP2• If the value of the second parameter v is changed to v = c2 (c2 > c1), and the bifurcation diagram is constructed again, another hysteresis curve is obtained with two different static limit points SLP{, SLP� as shown in Figure 2 .5 1 . If this process is repeated for different values of v the loci of the two static limit points can be drawn on a j.l  v diagram as shown in Figure 2.52. The area between the two curves SLP1 , SLP2 represents the region of multiple steady states while the boundaries of thi s area are the loci of the static limit points . The figure represents the case for which these critical points are corning closer together on the J.1 scale as v is increased. For v = a the two SLPs
C1
Cz
Q
Loci of static limit points (SLPs) on the p. continuation diagram (TPCD).
FIGURE 2.52
 v
two parameter
STATIC AND DYNAMIC BIFURCATION
1 25
coincide and for v > a no SLPs exit and uniquen ess of the steady s tates pre v ail s. Poi �t a repre sents a cri ti c al point which is usually called cusp p oint (Go lubttsky and Schaeffer, 1 985). There is a large variety of possible sh apes for the geometry of these loc i of stati c limit points as will be shown later. The shape in Figure 2. 5 2 is one of the most commonly oc curi ng shape s . Other types of static bifurcation points c an be traced on the two parameter continuation diagrams in the same manner di s c us s ed above. 2. 13.2
Dynamic Bifurcation Loci on the TPCD
The same principle of the two parameter continuation tec hni qu e for stat ic bifurcation points ex plaine d above, can be applied to bifurcation of periodic branches. Consider the dynamic equation,
dx
=
dt
=

(2. 1 1 9)
/(:!_, Jl , v) 
If the second parameter is fixed at a value v = c1 , then we have:
dx d = /( 
t
:!_ , Jl )
(2. 1 20)
From this e qu ati on both the static and the dynamic bi fu rc ation branches can be constructed. Suppose the diagram is as shown in Figure 2.53(a) V
....
>< ... 0
I •
>< X
. ....... . ,_,_ •
•
•
•
· .
• •
=C 1
•
I I I
HB1
I (a)
HB7
/J.
FIGURE 2.53(a) One parameter bifurcation di agram for v = c 1 . A case with unique static branch and two HB points (the m a xi ma and th e minima of the periodic oscillations are shown).
1 26
S . S . E . H . ELNASHAIE and S . S . ELSHISHINI
N
•
)o(
. . ...... . � . .
... 0
•
)o(
•
•
( b)
HBi
FIGURE 2.53(b) One parameter bifurcation diagram for points approach each other as v changes from c 1 to c 2•
v = c 2•
The two HB
with a unique static branch and two Hopf bifurcation points and that by changing v from c1 to c2 Figure 2.53(b) is obtained. It is clear that the two HB points have moved closer to each other. If this process is repeated for different values of v and the HB points are located (when they do exist) on the J1 axis, the loci of the HB points can be traced on a two parameter continuation diagram of J1 versus v as schematically shown in Figure 2.54.
v a
I I
I I I
 t     I 1 I
I
I
I
J.1
FIGURE 2.54. Two parameter continuation diagram for a case with two points.
HB
STATIC AND DYNAMIC B IFURCATION
1 27
v
.
c,
/"
... _.,... _
./
.r·       
FIGURE 2.55 Two parameter continuation diagram (TPCD) representing the = HB points and = PLPs . loci of PLPs and HB points  ·  · 

represents a case where the two HB points are c oming on the J1 scale as v is increased. At v = a, the tw o HB poi nts coincide and for v > a no HB poi nt s exi st There is a large variety of possible shapes for the geometry of th e se loci of HB po i nt s as will be shown later. Other types of periodic bifurcation points ( su ch as peri od ic limit points and infinite period bifurcation points) can be traced on two parameter continuation diagrams in the same manner di scussed above . An important case is whe n PLPs l o c i are fou n d without the coexistence of HB points in thei r neighbourhood. This case suggests the presence of a periodic isola. The two parameter continuation diagrams for some c l as s i c al cases will now be presented and discussed. The first case is a case where no static l i mit points exist and therefore the TPCD is formed of the loci of PLPs and HB p oints as shown in Fi gure 2.55. The one parameter bifurcation di agram s of x versus f.l, for c onstant value s of v (v = C t . c2 and c3), are shown in Figures 2.56 (ac). Fig ure 2 .56(a) i s the one parameter bifurcation d i agram for v= CJ . It is c l ear from the two p aram eter continuation diagram (Figure 2.55), that for v = c 1 there are t w o H B points and one PLP and therefore the bifurc ation diagram wi l l have the structure shown in Figure 2.56(a). This is similar to the cases presented e arl ier in Figures 2. 33b, 2 .38a and 2.40b . The diagram
closer to geth er
.
1 28
S . S .E.H. ELNASHAIE and S . S . ELSHISHINI
V =C 1
PLP
\;. 0 0
•
0
• •
•
•
•
·  · 
·.
( a)
FIGURE 2.56(a) One parameter continuation for a case with one PLP and two HB points corresponding to v = c1 in Figure 2.55. P L P1
�
>< ... ... 0
�. . . 0 0 0
•
•
•
·  ·
•
•
· .
(b)
FIGURE 2.56(b) One parameter continuation for a case with two PLPs and two HB points corresponding to v = c2 in Figure 2.55.
PlP, • • ft"o ...
0
• 0
•
•
•
•
•
0
(c)
FIGURE 2.56(c) One parameter continuation for a case with two PLPs an d no HB points giving rise to a periodic isola corresponding to v = c3 in Figure 2.55.
STATIC AND DYNAMIC B I FURCATION
1 29
fJ.
FIGURE 2.57
Two parameter continuation diagram showing loci of SLPs and and infinite period (IP) bifurcation points. ( :;: SLPs; HB; IP).
HB points
 ·  · 
:;:
 •  •  :;:
For v = c2, it is clear from Figure 2.55 that there are two PLPs and two HB points and therefore the one parameter bifurcation diagram will have the structure shown in Figure 2.56(b) . For v = c3, it is clear from Fi g ure 2.55 that there are two PLPs and no HB points. This suggests the existence of a periodic isola as shown in Fi gu re 2 .56(c).
The next case considered is that of a more complex two parameter continuation diagram involving static limit points (SLPs), HB points and infinite period (IP) bifurcation points as shown in Figure 2.57. For v = C t . it is clear from Figure 2.57 that there are two HB po i n t s on a unique static branch. A s ituatio n very similar to that presented earlier in Figure 2.53. For v = c2 , there are two SLPs, o ne HB point and one IP bifu rc at ion p oint very close to one of the SLPs. The periodic branch emanating from the s i n g l e HB point terminates homoclinically at an infinite pe riod bifurcation point very close to one of the SLPs. The one parameter bifurcation diagram for this case is s h own in Figure 2.58(a). For v = c3, it is clear from Figure 2.57 that there are two SLPs and one HB point as well as an IP bifurcation point. The one parameter
1 30
S . S .E . H . ELNASHAIE and S . S . ELSHISH INI
...... .... X ...
0

(a )
•
•
SLP1
HB
f.J.
FIGURE 2.58(a) Bifurcation diagram with two SLPs and one HB corresponding to v = c2 in Figure 2.57.
....
... 0
, r.: I
(b )
SLP1
IP

••
 
• •
HB
FIGURE 2.58(b) Bifurcation diagram with two SLPs, one HB and one IP bifurcation corresponding to v = c3 in Figure 2.57.
bifurcation diagram for this case will have the shape shown in Figure 2.58(b) which differs from the previous case in the fact that the IP is not close to the SLP2, and that J.lsLP2 > JlHLJ· More complex two parameter continuation diagrams will be presented later in the book in connection with specific cases of the practical behaviour of catalytic reactors .
STATIC AN D DYNAMIC B IFURCATION
2.14
131
NUMERICAL CONSTRUCTION OF STATIC AND DYNAMIC BIFURCATION DIAGRAMS
In principle the c on s tru ct i on of the static and dynamic bifurcation diagrams is simple and straightforw ard . This is because the bifurcati on diagrams are nothing but the solutions of the system equ ati on s for a large number of values of the bifurcation parameter. The diagrams are plotted with the bifurcation parameter J1 as the horizontal axis and a chosen state variable as the vertical axis. For example, a pri mi tiv e way of constructing the static bifurcation diagram is to solve the steady state eq u ation s of the system, g (!_, Jl ) = 0
(2. 1 2 1 )
for many values of Jl, and plot a chosen element of the vector ! versus J1 to obtain the bifurcation d i agram making sure that for each value of J1 all possible solutions are obtained. This is quite simple and can be performed readily and with great simplicity when the s te ady state eq uati on s of the system are reducable through simple algebraic mani pulation to a sing le algebraic equ ation in one variable. However, for higher order systems and/or distributed systems for which the steady state behaviour is described by differential equations, this primitive method is quite tedious and time c on suming but is still, in pri nc iple applic able. We shall call this primitive strai gh tforw ard method the Brute Force Method (BFM). The applic ati on of the BFM for the dynamic bi furc ation is very tedious . It consi sts of solving the dy nam i c eq uati on s for a c ertai n value of J1 to o btai n the periodic orbit by integration over a long time and d i sregarding the initial transients. The values of the chosen state variable at the maximum and minimum of the oscillations are then superimpo s e d on the static b ifu rc ation diagram as circles (or any other notation) . The proc edu re is then repeated for another value of Jl, and so on. In many cases either the maxima or the m ini m a , not neces s arily both, are su perimpo s e d on the bifurcation d iagram The problem wi th this method is not only that it is ti me and effort consuming but it also does not give the unstable periodic orbits in a s trai ghtforw ard manner. Although the BFM has been used for a long time, it is not needed at th e present time because of the existence of more "civilized" technique s for the c on s truc ti on of both the static and the dynamic bifurcation diagrams. Most of these techniques are available in the form of easy to us'e software packages or computer programs listings (e. g . Marek and Schreiber, 1 99 1 ) . ,
,
.
S . S .E.H. ELNASHAIE and S . S . ELSHISHINI
1 32
The most efficient and most widely used bifurcation analysis software is AUT086 of Doedel and Kemevez ( 1 986). The software is available in two versions one for the mainframe and the other for personal com puters (PCs). The method applied is based upon the continuation tech nique (Seydel, 1 988) and the software provides a rich variety of options. A brief description of the capabilities of AUT086 (Doedel and Kemevez, 1 986) is given in the following section.
Efficient construction of bifurcation diagrams using AUT086 The bifurcation diagrams of the autonomous system described by the set of first order ordinary differential equations dx
dt
/(;!_, J.l )
=
(2. 1 22)
can be obtained using the software package AUT086 (Abashar, 1 994). Using AUT086 we can change a chosen bifurcation parameter (J.l) while the rest of the parameters are fixed to compute the complete bifurcation diagram. This package makes use of the bifurcation theory and the numerical continuation techniques to perform the tracing of the branches of solutions. Some of the capabilities of AUT086 are:
l . For the nonlinear system of steady state algebraic equations obtained by dropping the time derivatives terms on the left hand side of equation 2.60, AUTO can: (a) Trace out the entire steady state branches of solutions. (b) Locate the static limit points (SLPs) and continue these in two parameters. To avoid the difficulty of tracing the branches of solutions past singularities (e.g. static limit points), AUTO uses pseudoarclength continuation technique. AUTO computes stable as well as unstable branches of steady state solutions and in order to determine the stability properties of the solutions, the eigenvalues are computed along the solution branches. A known starting steady state point is required for AUTO to start " the computation of the static branches . This starting point is determined by an IMSL routine called ZSPOW (or any other suitable subro utine ) for s ol v i n g a set of nonlinear algebrai c equati ons This algorithm is based on a variation of Newton' s method which uses .
a finite difference approximation to the Jacobian and takes precautions avoid large step sizes or increasing residual s .
to
STATIC AND DYNAMIC BIFURCATION
1 33
2. For the autonomous dynamical system described by the set of first
order ordinary differential equations (2. 1 22), AUTO can: (a) Locate Hopf bifurcation ( HB) points and continue them in two parameters. (b) Trace out branches of stable and unstable periodic solutions and compute Floquet multipliers. Locate periodic limit points, homoclinical orbits and ordinary (c) bifurcation. (d) Continue periodic limit points and homoclinical orbits in two parameters. The starting point for the computation of periodic solutions are generated automatically at Hopf bifurcation points . AUTO computes stable and unstable branches of periodic solutions and the stability properties of the solutions is detennined by computing the Floquet Multipliers along the solution branches. 3 . AUTO can also locate period doubling bifurcation points as well as bifurcation to quasi periodic trajectories (Torus). A guide for the use of AUT086 is given by Abashar ( 1 994). 2.15
SOME IMPORTANT ELEMENTARY DYNAMICAL FEATURES (NONCHAOTIC DYNAMICS) OF THREEDIMENSIONAL SYSTEMS
For the tw�dimensional systems discussed so far, the linearized local stability analysis revealed that there are, from a local stability point of view, five different types of steady states (or fixed points): 1 . Stable node, when the eigenvalues A1 , A2 are both real and negati v e . 2. Unstable node, when the eigenvalues AI . A2 are both real and positive. 3. Stable focus, when the eigenvalues A1 , A2 are complex conjugates (A1 2 = a ± bj, where j = H ) with negative real parts a < 0. 4. Un s table focus, when the eigenvalues A1 . k are complex conjugates ( A 1 2 = a ± bj, where j = !=r) with positive real parts a > 0. 5 . Un stable saddle, when the eigenvalues AJ , A2 are real, one of them is negative and the other one is pos i ti v e .
For the threedimensional autonomous system, dx = dt f(�, f.l )
(2. 1 23)
S . S .E.H. ELNASHAIE and S . S . ELSHISHINI
1 34
where, t=
[J, ]
(2. 1 24)
The linearization around the steady states and the introduction of disturbance variables result in the following type of equations, d
� Ax dt = 
(2. 1 25)
where,
[""
� = llz l a3 1
and,
al l = �� I , �l =�I , I ss
a2 1 = a3 1
I
,
ss
I ss
" n
au
llz 3
a2 2
a3 3
a3 2
dfJ
al z = dx2
d/z 2
llz2 = Jx a, , =
'
]
(2. 1 26)
ai 3 =
ss '
ss
;;: 1.:
:� I 3
dfz
llz3 = dx3 a33 =
df3 Jx3
ss
(2. 1 27) ss
ss
The local stability analysis of such a system reveals that the number of steady state types is much larger than for the twodimensional system, specially with regards to the different types of unstable saddle type steady states. Simple analysis reveals that there are, from a local stability point of view, eight different types of steady states (fixed points): 1 . Stable node, when the eigenvalues A.. 1 , A..2 , A..3 are all real and negative. 2 . Unstable node, when the eigenvalues A.. 1 , A..2 , A..3 are all real and
positive.
3 . Stable focus, when the eigenvalues A.. 1 , A..2 are comple x conj ugates ( A t ,2 = a ± bj) and the third e i g env al ue is real, A3 = c ' and a < 0, c ' < O.
,
STATIC AND DYNAMIC BIFURCATION
1 35
4. Unstable focus, when the e i g env alue s A1 . A2 are c o mpl ex c onj ug ate s
(A1 •2 = a ± b:}) and the third ei genv al u e is re al , A3 = c ', and a > 0, c ' > O. 5 . Saddlenode of the first kind, when A J , Az, A3 are all real, with Ar < 0, Az < 0 whi le A3 > 0. 6. Saddlenode of the second kind, when A 1 , Az, A3 are all real , with A 1 > 0, A2 > 0 whi l e A3 < 0. 7. Spi ral  out saddle, when the eigenvalues A 1 , A2 are complex conjugates ( A , .2 = a ± b·j) and the third eigenvalue is real, A3 = c ', and a > 0,
c ' < 0.
8. Spiralinsaddle, when the eigenvalues A 1 , A2 are complex conjugates (Au = a ± b:}) and the third eigenvalue is real, A3 = c ', and a < 0, c ' > O.
The twodimensional forced (nonautonomous, 2D) system is a bona fide threedimensional (3D) system. In fact it is a s pec ial case of the autonomous threedimensional (3D) system, when one unidirectional variable is not affected by the other two variables. Either of the two types of 3D systems has dynamic s which are much richer than the 2D system. They both can have more complex dynamic attrac tors than the peri odic attractors pre s ented and analy zed earlier for the 2D system. 3D systems can have torus (qu asi period ic ) as well as chaotic attractors . We will now present to the reader some of the basic characteristics of 3D systems, while detailed results and an aly si s for a nonautonomous 2D system (effectively 3D) as well as a 3D autonomous fluidized bed c ataly tic reactor, will be given later in the book, in this chapter and in chapter 4.
Eigenvalues
and eigenvectors
We have shown the reader that after the l ineariz ation of equation (2. 1 23), w e obtained the linearized equation (2. 1 25). The local stability c h arac teris tic s of any of the steady states of the sy ste m are obviously determined by the nature of the eigenvalues of the matrix !1 in equatio n (2. 1 25 ) . The eigenvalues are the ro ots of the characteristic equ ati on ,
det (!1  AD = 0
(2. 1 28a)
which upon solution give s the eigenvalues .:t1 (j= 1 , 2, 3). The characteristic equ at� on can also be written in th e following form,
;t3  T · .:t2 + M · A  D = O
(2. 1 28b)
S . S .E.H. ELNASHAIE and S.S. ELS HISlllN I
1 36
L
T = a 1 1 + �2 + a33 = a1 1 = Tr · 1 M = sum of the diagonal minors. D = det A . The eigenvector, where,
.! =
associated with the eigenvalue equ ati on ,
(�� ] A3 J
� is the vector that satisfies the matrix (2. 1 29)
Obviously because AJ is an eigenvalue (i.e. equ at i on 2 . 1 28 is satisfied), the matrix (1  A, J D is singular and therefore equati o n (2. 1 29) has solutions other than .! = 0. ·
Some important algebraic relations relating the eigenvalues tq, A2, A3 Let us define the follow i n g relations between the three eigen values ,
A 1
•
A I + A, 2 + A, 3 = T 12 + 1 1 · A3 + A2 A3 = M'
(2. 1 30)
•
A 1 A 2 • 13 = D
The nature of A.'s for 3D systems:
•
D, T, M' are real, therefore one A.;
(at least) is real. All ro ot s are real if,
(2. 1 3 1 )
where ,
By substitution, it is clear that all roots
D,
G ( D, T, M') space.
T, M'
=
are real
if:
0 defines the boundary between two regions in the
STATIC AND DYNAMIC BIFURCATION
0 0
EB EB ffi G>O@
ffi
FIGURE 2.59 Nondegenerate cases for the three eigenvalues for the 3D autonomous system.
The eight nondegenerate cases of the eigenvalues AJ , A2, A3 are given in Figure 2.59 which presents the cases with eigenvalues such that Re (A;) # 0 and no real parts are equal except when they are com plex conjugates. The four cases with D < 0 include: a stable node and an unstable saddle node of the second kind (when G < 0); a stable focus and a spiralout saddle (when G > O). While the four cases with D > 0 include: an unstable node and a saddlenode of the first kind (when G < O); an unstable focus and a spiralin saddle (when G > O). Some degenerate cases are shown in Figure 2.60. For the first case from the left, the eigenvalues are all real with one equal to zero; the second case has a complex conjugate pair with positive real parts and the third is zero; the third case has a complex conjugate pair with negative real part and the third is zero; the fourth case has a real positive eigenvalue and a conjugate pair with zero real parts; the fifth case has complex conjugate pair with negative real parts and the third is negative real and equal to the real part of the conjugate pair.
EB EB ® EB ® T : 0 M < O
T > 0 M >O
D : 0
D ::. 0
FIGURE 2.60
T O D : 0
Some degenerate cases.
M>O T > O
D > O
T O o
'3 dL
= ()'2 dx1
•
dx1
dL
dx = ()«1>'3 . l ()x1
dL
+
d '1 dx2 •
dx2
dL
dx2
+
() '2
+
() «1>'3 . dxz
()x2
()x2
•
+
dL
dL
d'1 dAm •
dAm
+
+
dL
+
d «l>'2 _ dA m ()Am dL
()«1>'3 . dA m
()Am
dL
d'1 _ dco
+
+
dco dcl>'2
() co
dL _
=
dco dL
0
=0
()«1> '3  dco = 0 () co dL
(2. 1 59)
(2_ 1 60 )
(2. 1 6 1 )
For each value of L, equations 2. 1 592. 1 6 1 form a set of three linear alge b rai c equations in four unknowns (dx JfdL, dx'lldL, dA.m/dL and dco/dL). The additional equation whi ch determines L the arc length of the abo ve curve of solution, is given by:
as
(2. 1 62) Equations 2. 1 592. 1 6 1 can be written in matrix n otati on as follows:
STATIC AND DYNAMIC BIFURCATION
d'•
d '•
d '•
ax.
ax2
dAm
d '2
d '2
d '2
ax.
dx2
dAm
d '3
__
d'3
d '3 dA m
ax.
a�
dxl
_
dL
� dL
dAm 
1 67
d '1 dm a co dL
=
_
d '2 dm d m dL
_
dL
(2. 1 63)
d '3 dm d m dL
Equation (2. 1 63) can be s ol ved for the three unknowns dx 1/dL, dxydL, dAmldL in form:
the
dx1 =  C t
dm
dL
dx2
dL
(2. 1 64)
(2. 1 65 )
   "'1 dL dL _
r_
dm
dAm _ dm   C3
(2. 1 66)
dL
dL
The coefficients C; are computed by s ol ving the following three linear algebraic equ at i ons by means of Gauss elimination with partial pivoting :
d '•
d 'l
d 'l
ax.
dx2
dAm
d '2
d '2
d '2
dx1
dx2
dAm
d '3
d '3
d '3
ax.
dx2
dAm
(�)�
d '• am
d '2 am
(2. 1 67)
d'3 am
The c oeffic ient matrix is obtained by 26 integrations over a time range of t = nP as shown before. The partial derivatives a<J>';Iam forming the right hand side vec tor are obtained numerically by simultaneous
integration of six differential equations (2. 1 37, 2. 1 3 8 , 2 . 1 432 . 1 46). This is a total of 34 i n tegrati on s for each arc l en gth step size . Substitution of equations (2. 1 642. 1 66) into equ ati on (2. 1 62) gives,
dm dL
= �.,j;:1+"" ::;; c1 "' =+=C�2=+=;C:J ;;=
(2. 1 68)
1 68
S . S .E.H. ELNASHAIE and S.S. ELSHISHINI
Substitution of
droldL into equations (2. 1 642. 1 66) gives, dxl
dL
=
dx2 dL 
dA m =
dL
cl
vft +.C1 + C2 + C) J
l
+ C1
c2 +
Ci + c3
c3 vf l + C1 + C2 + CJ
(2. 1 69)
(2. 1 70)
(2. 1 7 1 )
The set of differential equations (2. 1 682. t 7 1 ) are integrated by the
explicit AdamsBashforth methods (with variable step size for higher accuracy) to trace out the bifurcation boundaries of the resonance horns.
A
starting point lying on the boundaries of the resonance hom is
necessary for this arclength continuation technique. This starting point is located by Newton' s method as discussed earlier.
2.19
STRANGE CHAOTIC AND NONCHAOTIC ATTRACTORS
We have discussed so far, fixed point, peri odic and quasiperiodic
attractors . The first two attractors are quite simple and can be investigated,
to a great extent, using relati v ely simple mathematical and presentation techniques. On a simple phase plane, the fixed point attractor appears
as a point while the periodic attractor appears as a single closed loop (when it is of pe riod one), or as a number of loops forming again a
closed curve (when the periodicity is higher than one) . These periodic attractors obviously have a certain period T which is the time from
a
point on the closed curve till the same point is encountered again as the
variables change continuously with time. Thus, the periodic attractor has a single frequency (whether it is of period one or higher) given by
ro = 2tr/T. Such periodic solution can arise in one dimensional non
autonomous (forced) systems or two (and higher) dimensional autono mous as well as nonautonomous systems. The quasiperiodic attractor differs from the periodic attractor by the fact that its variation with time has more than one incommensurate frequency and therefore the trajectory lies on a higher order surface
usually called "torus". These quasiperiodic solutions arise usually in two di mensional nonautonomous ( forced ) systems or i n three (and higher) di mensional autonomous and non autonomo u s systems. They 
may also (in a trivial sense) arise in one dimensional systems forced
with a quas i periodic forcing function (with two incommensurate
STATIC AND DYNAMIC BIFURCATION
1 69
frequencies such as sin ro, t.sin a>:2 t when the ratio of Wdf»J. is not a ratio of prime numbers). Phase locking may occur as discussed before fonning
lying on the surface of a torus. The word strange used in the title of this section is certainly relative,
a periodic solution
while the word chaotic ha� a specific meaning as will be shown in the following sections. In fact, strange means nontrivial geometrical
structure of the attractor (Grebogi et al. ,
1 984; Brindley and Kapitaniak,
1 99 1 ) .
Quasi periodic solutions when presented
on
th e usual phase plane
will certainly look strange to the eye used to periodic solutions with
period one or even periodic solutions with higher periodicities. How
ever, we are blessed by the great mathematician Poincare who provided
other presentation techniques which do not make the quasiperiodic solution look strange. The idea is simply to change the continuous
output variables of the system into discrete points in successive well
chosen time intervals
on
t, t + &1 , t + 2M2 ,
•
•
•
,t
+ nLYn , where ill is the
strobing time. These points are taken after the initial trajectories settle down
the ultimate attractor under investigation, thus giving the
values of the variables at these discrete times (e.g. for the variable x1
they wi ll be x1 1 , x12,
• . .
, x1n ) .
!1.t; , depends upo n the type
of
The choice of the strobing time intervals the system whether it is autonomous
or nonautonomous. For nonautonomous (periodically forced) systems,
the strobing time ill; , is constant and is taken as the period of the
forcing function T1= 27ri�. For autonomous systems, a well chosen plane is fixed and the
points of transversal intersection (in one direction) of the trajectory with this plane are taken as the strobed values of the variables as shown in
x,
x,
(Q )
( b)
FIGURE 2.92 Period n' attractor on one of the phase planes of the three dimensional systems (or forced twodimensional systems); (a) n' =2 (period 2); (b) n':::::3 (period 3).
FIGURE 2.93 The geometry of the Poincare maps for a periodic attractor (if the shown trajectory repeats itself after the third crossing of the Poincare plane (l:), then it is a period 3 attractor).
Figures 2.92a, 2.93 for a period 2 attractor and Figure 2.92b for a period 3 attractor. In these cases l!!.t; can be different for different values of i. In simpler form, we can say that the Poincare map replaces the ndimensional continuous dynamical system with an (n  1 ) dimensional discrete time system called the Poincare map, through fixing one of the state variables at a well chosen value. For example, for the 3dimensional system with three variables x 1 , x2, x3, one of the possible phase planes is shown in Figure 2.94, where the periodic attractor is clearly of period 2. Thus if we choose the Poincare plane at a value x2 = a, then the
x,
a
b
FIGURE 2.94 A suitable Poincare plane (a) and an unsuitable Poincare plane (b) for the three dimensional system. A system with a periodic attractor of period 2.
traje ctory will intersect this plane transversally (in one direction) in two x po in ts 1 ,2 . Howe � er, if the Poincare � lane j s taken unwisely at 2 = b, map. Pomcare the on appear will ersection no int , X l n and If a special phase plane is drawn for the points X I t . x 1 2 xz t . x22 , , x2n, then obviously the period one attractor will appear as one point that repeats itself indefinitely. The periodic attractor with peri od n will appear as n points and the quasiperiodic attractor will appear as an invariant closed curve of discrete points. The different means of presetting the strobed points are discussed in the following section. These maps are usually called Stroboscopic or Poincare diagrams (or maps) . The name Poincare diagram applies to all cases, however it is usual to call the diagram for the periodically forced system when the points are strobed every forcing period T, the Stroboscopic diagram (or map), whereas for the autonomous system when the strobed points are the points of intersection between the trajectory and the Poincare plane, the diagram is usually called the Poincare diagram (or map) . The strange attractor is called strange because even on the Poincare diagram they appear strange : not a number of points nor an invariant circle but actually the shape is of fractal dimensions as will be discussed later. Such attractors are called strange attractors till someone discovers a way of representing them that makes them do not look "strange". These strange attractors can be chaotic or nonchaotic depending upon what is called "sensitivity to initial conditions". Thi s sensitivity to initial conditions means that two trajectories starting from two very close initial conditions, both leading to the chaotic attractor as t � oo, will diverge exponentially with time. This sensitivity to initial conditions can be identified most easily by the computation of Lyapunov exponents as will be discussed later in this chapter. We should remember that for any system having more than one stable attractor there is a simple form of sensitivity to initial conditions in the neighbourhood of the separatrix (the stable manifold) . After this brief introduction we can move one step forward into the fascinating land of chaotic attractors, starting with a simple exposition of the different presentation techniques to be used in the field of investigating chaotic attractors. Some relevant numerical techniques will be given in later sections. . . . •
. • •
2.19.1
Presentation Techniques
The classical presentation using phase plane and time trace are very well known and do not need to be explained here. After the explanation for strobing points along a trajectory, the different important diagrams constructed using the strobed points are explained. It is then followed
by a more mathematical formal definition and explanation of Poincare diagrams. Accurate numerical techniques must be used to interpolate between points along the time trajectory (in the autonomous case) in order to obtain accurate results for the strobed points without intro ducing external noise into the results . The strobed points are some times called return points. These diagrams are : a) Two dimensional Poincare map
This is a plot of two of the coordinates of the return points (e.g. x1 , x2 ) for a specific value of the bifurcation parameter. b) Poincare bifurcation diagram
This is a plot of one of the coordinates of the return points (e.g. x 1 ) versus the bifurcation parameter, Jl . c) Return points iterate maps of different orders
This is a twodimensional plot of one of the coordinates of nth return (strobed) point versus the coordinate of the (n + i)1h return (strobed) point. The iterate maps are first, second, third, . . . , when i = 1 , 2, 3, . . . This presentation technique is of great importance for it represents, together with the strobing techniques, the link between continuous and discrete systems . The plot of say x (n) versus x (n + 1 ) is a function that can be (if known) written as, x(n + 1) = F( x(n))
(2. 1 72)
This form of discrete model and similar forms have been studied extensively and the knowledge regarding its behaviour gives very good insight into the behaviour of the corresponding continuous system. The function F in equation 2. 1 72 is very difficult to obtain in an analytical form from the model of the continuous system. However, obtaining it numerically is very easy: just solve the continuous model numerically, strobe the points X J . xz, . . . , Xn, . . . , XN and then plot the points Xn+ l versus Xn to obtain the relation numerically. It is usually better to plot the points on a square plot where the scale of the vertical and horizontal axes are the same. Also, it is useful to draw the 45 ° diagonal connecting the bottom left corner to the top right corner. Higher order iterate maps, i.e. Xn+2 versus Xn, andxn+3 versus Xn , are also useful in the recognition of certain modes of bifurcation to chaos as will be shown later. d) Return points histogram
This is a plot of one of the coordinates of the return points versus the corresponding time .
2 . 1 9 .2
The DiscreteTime Models and Their Relevance to the Analysis of Continuous Systems
As we have shown, the behaviour of dynamical continuous systems can be reduced (at least numerically) to the behaviour of some discrete time system by the use of Poincare maps. Although the functional relation relating Xn+ l and X11 is not easily obtained analytically, it is easily obtained numerically. In fact some investigators were able to obtain some analytical form of the function relating Xn+ l and X11 for a limited number of systems, using extensive numerical work to construct iterate maps and use these maps to extract a suitable functional form. Because of this close relation between the continuous system behaviour and the corresponding discrete system behaviour and despite the difficulty of "legalizing" this relation analytically, it has been a successful practice to elucidate many of the complex phenomena in continuous dynamical systems by the use of the corresponding discretetime system. Discretetime systems are obviously much easier to study analytically and numerically. In addition to that, they contain all the interesting dynamic features of the continuous dynamical systems. In the next section, we will introduce the most wellknown route to chaos, that is period doubling, through the famous discrete model having the wellknown logistic map. For onedimensional discrete processes, it is the simplest nonlinear difference equations that has an extraordinary rich dynamical behaviour, i .e. from stable fixed point attractors to chaotic attractors through cascades of stable cycles . One dimensional iterative maps in general, have been studied extensively by several researchers . May ( 1 976) gave an interesting account of this model for problems in the biological, economic and social sciences. Among the early investigators Feigenbaum ( 1 978 , 1 980) studied the universal behaviour of onedimensional systems and quantitatively determined important universal numbers that will be discussed later in this chapter. These numbers represent the threshold values for the onset of chaos from a perioddoubling sequence. Collet et a/. ( 1 98 1 ) generalized the period doubling theory to higher dimensions. Some of the more recent work pertaining to the area of universality in dynamical systems are available in the specialized literature (e.g. Delbourgo and Kenney, 1 986; Chang and Rendley, 1 986; Mao and Heileman, 1 98 8 ; Kim and Hu, 1 990) . 2.20
MODELS BASED ON FIRST ORDER DIFFERENCE EQUATIONS
General considerations
The equation to be studied is:
1 74
S . S .E.H. ELNASHAIE and S . S . ELSHISHINI
F ( X , J.i )
Xn
FIGURE
2.95
Simple function F (x, p.) for a certain value of p..
X
(2. 1 73)
where J1 = control parameter (bifurcation parameter). The dynamics is a sequence of mappings, (2. 1 74)
where F(xn, J.l.) is a single valued continuous function of x. Figure 2.95 shows a very simple case for this function. 2.20. 1 1.
Conservative and Dissipative Dynamical Systems
Conservative systems
In continuous dynamical systems described by differential equations, the conservative systems are Hamiltonian systems which means that the volume elements in the phase space move in such a way as to "conserve" their volume according to the Liouville theorem (Thompson and Stewart, 1 986).
The analogous system in the case of maps (discrete
system) involve functions F(x, J.l.), which keep track of each point in the phase space, by mapping each point to a unique point, in other words : a onetoone map, together with the continuity requirements (as the case shown in Figure 2.95). Such systems can have two steady state points, one stable
STATIC AND DYNAMIC BIFURCATION
1 75

2 2.96 Two solutions S (stable) and (conservative system).
FIGURE
U
Xn
(unstable) for Xn+l = F (xn. Jl)
S and the other unstable U, as shown in Figure 2.96. Such a map F(x, J.l), is called Homeomorphism. All conservative systems must be homeo morphisms but the converse is not always true. 2.
Dissipative systems
In discrete representation these systems are described by manytoone maps. Figure 2.97 represents a twotoone map. F ( x , J..I )
��� x • Xn
FIGURE 2.97
xn
The function F (xn. Jl) for a dissipative system
(manytoone map).
1 76
S . S .E.H. ELNASHAIE and S . S . ELSHISHINI
It is clear that the map has a "compressional effect" on volume elements in the phase space and is in that sense "dissipative" in character. Dissipation in the present sense does not necessarily imply that all states "die" but rather that there is an attraction towards some final, generally dynamic set of states called an attractor. a) Simple continuous systems and Poincare maps
Consider the twodimensional continuous dynamical system with the state variables denoted x1, x2 having two limit cycles as shown in Figure 2.98a, then take a Poincare section at x1 = 0, x1 < 0 to obtain the shown Poincare surface. The set of initial conditions considered for the continuous dynamical system are those in the annulus between limit cycles 1 ,2. These give the segment of F (xn , Jl) shown in Figure 2.98b. In this region, no initial cond iti on goes to limit c y c le 1 . So, as far as this region of space i s concerned this limit cycle is unstable. Notice from Figure 2.98b that, although the map is a homeomorphism (onetoone), the system is not conservative. x,
Poincar� surface
(a)
(b) //
/
/
' • ' � u/1
/ I I
I I
I
I
I I
2
Xn
FIGURE 2.98 A continuous dynamical system with two limit cycles; (a) the limit cycles; (b) the equivalent discrete system (the limit cycles are reduced to two fixed points 1,2, S= stable, U= unstable).
STATIC AND DYNAMIC BIFURCATION
1 77
FIGURE 2.99(a) A continuous dynamical system with one limit cycle. The transient trajectory crosses the Poincare plane at points x � o x2, x3, x4 in its way to settle down on the limit cycle a (the horizontal axis denoted x2 and x1 = 0 is the Poincare plane).
Another case with a s ing l e limit cycle for the continuous dynamical system, is shown in Figure 2.99a. Taking the Poincare surface at x1 = 0 and the intersections of th e trajectory with this pl an for x1 < 0, we get, in principle, the equivalent discrete sy ste m shown in Figure 2.99b.
FIGURE 2.99(b) An equivalent discrete system. The limit cycle appears as a fixed point at Xn+I a. Any starting point like x00 will go through the staircase described by Xn+ l F (xn, p) towards this fixed point. =
=
178
S . S .E.H. ELNASHAIE an d S . S . ELSHISHINI
x,
a
FIGURE 2.100(a) A continuous dynamical system with a stable fixed point attractor and no periodic attractor. The transient trajectory crosses the Poincare plane at points X t . x2, x3, x4 in its way to settle down on the point attractor a .
Figure 2. 1 OOa shows a case where the continuous dynamical system has one fixed point attractor (a) and no periodic attractor. In order to capture this fixed point attractor on the equivalent discrete map, the Poincare plane must be passing exactly at this point (a) and the point appears on the discrete map as a point of intersection in a manner similar to those corresponding to the periodic attractor, as shown in Figure 2. 1 00b. The sequence of points on both Figures 2. 1 00a,b are X0 � x1 � x2 � X3 � a. If another Poincare plane is taken such as the
FIGURE 2.100(b) An equivalent discrete system for the fixed point attractor when the Poincare plane is taken exactly at this fixed point.
STATIC AND DYNAMIC BIFURCATION
1 79
FIGURE 2.101(a) A continuous dynamical system with an unstable fixed point attractor at a and a static point attractor at b.
plane x' in Figure 2. 1 OOa, the sequence of points will be x � � x 1 � x'2 � x'3 � a' and the steady state will not appear at all. Thi s is a simple illustration of the fact that the Poincare pl ane has to be carefully chosen in order to capture as much as possible of the dynamic charac teris tics of the continuous dy n amic al system. Figure 2. 1 0 1 a shows a case of an un stable steady state at poi nt a (an unstable focus) where the trajectory spirals outward to settle at another focus b. Figure 2. 1 0l b shows the equivalent discrete map, which shows the continuous escape of the successive iterations away from the fixed point a.
'
FIGURE 2.10l (b) Figure 2.101a.
An
Xn
equivalent discrete map for the unstable fixed point a in
1 80
S . S .E.H. ELNASHAIE and S . S . ELSHISHINI x,
Slc blt lirM C)'([� 2
FIGURE 2.102(a) A continuous dynamical system with two stable limit cycles and one unstable limit cycle acting as a separatrix between the two stable limit cycles.
The case in Figure 2. 1 02a is characterized by two stable limit cycles separated by an u nstab l e limit cycle 1 '. The equivalent discrete map is shown in Figure 2.1 02b where the Poincare p lane is taken at x1 = 0 and the strobing of the p oints is taken as the trajectory crosses the Poincare plane with x1 < 0. From Figures 2. 1 02a,b it i s clear that each limit cycle has its domain of attraction. The domains of attraction are determined by the sep aratri x (limit cycle 1 ' on Figure 2. 1 02a) and the fixed poi n t 1 ' on the discrete map in Figure 2. 1 02b. From Figure 2 . 1 02b, any initial conditi on X0 to I ,2
FIGURE 2.102(b)
limit cycles.
X0 '
xo
Xn
Equivalent discrete map for the two stable and one unstable
STATIC AND DYNAMIC BIFURCATION
181
x,  limfl cyct. 1
A continuous dynamical system with two unstable limit cycles 1,2 and a stable limit cycle 1 '.
FIGURE 2.103(a).
the right of po i nt 1 ' will go to point 2 (limit cycle), while any initial condition x� to the left of point 1 ' will go to point 1 (limit cycle), except point 1' itself, because the limit cycle 1 ' is an unstable limit cycle. The case shown in Figures 2. 1 03a,b has two unstable limit cycles and one stable limit cycle, it is clear that the domain of attraction of the stable limit cycle 1 ' is the annulus 12. However, it can be noticed that traj ectories visit different regi ons on both sides of the s table limit cycle. In other words a traj ec tory starting at the initial condit ion X0 (Figure 2 . 1 03b) to the left of point 1 ' (in the region 11 ' ) will visit the region
Xn
FIGURE 2.103(b). Equivalent diserete map for the unstable limit cycles 1,2 and one stable limit cycle 1' (stable fixed point on the map).
S . S .E.H. ELNASHAIE and
1 82
S . S . ELSHISHINI
:::t.

..
c )(
.._

FIGURE 2.104 A discrete map (iterate map Xn vs. Xn +I) for a case where aU 1,2,1 ' are unstable giving a period 2 attractor.
1 '2 and vice versa. This means that the stable limit cycle (periodic attractor) is a focus type limit cycle, i.e. it is approached in an oscillatory manner around it. Let us call the points to the left of 1 ', L and the point to the right of 1 ' R. Then using symbolic dynamic representation, we can easily see that the shown trajectory (Figure 2. 1 03b) follows the sequence LRLRL . . . It is important to emphasize that the maps for two dimensional dissipative continuous dynamical systems are always onetoone, and therefore no more complexity than periodic attractors should be expected. The above discussion for dissipative systems has been confined to onetoone maps. This discussion is sufficient for continuous two dimensional systems when the highest possible complexity is associated with periodic attractors. However, for higher order systems (and forced two dimensional systems) the behaviour may be more complex showing chaotic behaviour. The corresponding discrete maps are manytoone maps. A simple example for these more complex cases is shown in Figure 2. 1 04. In the di screte map of Figure 2. 1 04 all three limit cycles are unstable and starting from any initial conditions (x0 or x� ) the system settles down to a period two attractor. The system alternates continuously between x* and x* . This is the start of a complexity that wi l l be discussed in some detail in the next section.
STATIC AND DYNAMIC BIFURCATION
Xn
xn
1 83
X
FIGURE 2.105 Twotoone discrete (iterate) map.
2.20.2
Higher Order Continuous Dynamical Systems (manyto one maps)
A series of bifurcati on s leading to chaos depends on the nonuniqueness of the inverse of the map. (2. 1 75)
In other words the occurence of chaotic motion depends on the fact that at least two different initial conditions (points xn , x� ) can map to the same point Xn+ 1 as shown in Figure 2 . 1 05 . Such manytoone maps can only occur from some "projection" process which re duces a higher order system to a lower o rder model with non unique dynamics as shown in Figure 2. 1 06. In other words the unique dyn ami c s in three dimensional space i s projected onto no nunique d y nami c s in the (x, y) p lane It is not necessary to have any casual relation between Xm Xn+ l in the x, y plane i.e. there is generall y no reason for a variable x to be exactly self deterministic . Maps like the one shown in Figure 2. 1 04 are crude .
approximation of such projections.
S . S .E.H. ELNASHAIE and S.S. ELSHISHINI
1 84
X
FIGURE
plane.
2.20.3
2.106
Projection of three dimensional dynamics onto a two dimensional
Quantitative Universality and Qualitative Universality
Quantitative values can have a form of structural stability, since these values remain unchanged when F (x, J.l) is smoothly varied. One group of asymptotic quantitative features, which has been identified for a large class of functions, F (x, J.l), has been called "quantitative universality" by Feigenbaum ( 1978, 1 979a,b) .
c )( IJ...
x,
FIGURE 2.107
Xm
X
Twotoone logistic map (Jackson, 1990).
1
Xn
STATIC AND DYNAMIC BIFURCATION
1 85
�
c
)( 
u..
FIGURE 2.108
Xm
1
Tent map or Leaning map (Jackson, 1990).
This discovery was hi storic ally preceeded by the observation of qualitative "universal sequences" generated by an even larger class of fu nc ti ons F (x , J1) and noted by Metropolis et al. ( 1 973). They observed that a l arge class of maps F (x, J1) generate similar qualitati ve patterns. Twotoone maps The Logistic map (Fi gure
2 . 1 07 )
(2. 176) This function has the properties:
1. F(O) = F(l) = 0
2.
on e maximum F(xm ) � 1
This maps the interval [0,
where
it
1 ] i nto a smaller reg ion except for F(xm )
w i ll become onetoone.
The Smap
F (x) has only one maximum and has everywhere a negative "Schwarzian derivative" . This requires that (dFldx )0·5 has a positive second derivative on both s ides of the maximum. We call these m ap s w ith e very wh ere negative Schwarzian derivative, Smaps. Singer ( 1 978) prov e d that in this case for each value of J1 there
is at most one st abl e p eri odic solution of equation 2. 1 76. Also S inger ( 1 978) prov ed that when this periodic solution exists, the iterates of Xm
S.S.E.H. ELNASHAIE and S . S . ELSHISHINI
1 86
(b)
c
"
II
)( �
c
u.
; Li:"
x,
x,
x3 x ·
Xn
xi
xo=xi
X
FIGURE 2.109 The logistic map. (a) A case with period one attractor; (b) A case with period 2 attractor.
tend toward this periodic solution, i.e. Xm is in the basin of attraction of the periodic solution. The logistic map of equation 2 . 1 76 is an Smap. However, the following twotoone maps are not Smaps. The tent or leaning map Figure 2. 1 08 is given by the following equations,
F( x, Jl ) = J.l { l  Xm ) · X F( x , Jl ) = JlXm ( l  x )
X :5 Xm
Xm :5 X :5 1 and JlXm ( l  xm ) :5 1
(2. 1 77)
It may have more than one periodic solution. 2.20.4
More on the Characteristics of the Logistic Map
The Braouwer' s fixed point theorem (Jackson, 1989) states that for any continuous map, F (x), which takes the internal [0, 1 ] into itself (0 � F (x) � 1 ) , there must be at least one fixed point. Stability offixed points
The fixed point x* on Figure 2. 1 09a,b (which is clearly the intersection between the curve F (xn ) and the straight line Xn), can be stable as in Figure 2. 1 09a or unstable as in Figure 2.1 09b. The fixed point x* is stable if, j [df(x) l dx ] jx. < 1
(2. 1 78)
STATIC AND DYNAMIC B IFURCATION
To prove it, take a point such as series to obtain,
F
*
*
F(x + i1) = (x ) + L1
Xo
1 87
= x* + L1 and expand F in a Taylor
(dF)
dx x• = x + L1 *
(d )
dx x*
F
= x1
(2. 1 79)
* * where x* has replaced F (x ) because x * is a fixed point, i .e. F (x* ) = x . The point is stable if, (2. 1 80) From equation (2. 1 79),
(2. 1 8 1 ) Thus condition 2. 1 80 i s satisfied if,
(d ) F
dx
x*
1
(2. 1 89)
f1 < 3
Since for f1 < 1 there exists only the trivial solution, therefore the stability condition reduces to, (2. 1 90)
Behaviour of unstable fixed points In Figure 2. 1 1 Oa the points x; and of F(x), that is,
x; ( x; x�) =
are periodtwo points (2. 1 9 1 )
This means that after two maps their values are repeated, in other words, *
*
F(F(x2 )) = x2 (a)
11 = 3 . 1
;
.J 2)
I"
X n .. 1
*
*
(x2 ) = x2 (b)
(2. 1 92) 11=3 . 8
�� X n
FIGURE 2. 1 1 0 The logistic map for (a) period two attractor for (b) aperiodic attractor for Jl =3.8.
Jl =3. 1 ,
STATIC AND DYNAMIC BIFURCATION
1 89
and
F(F(x1 )) = x1 ; *
*
(2. 1 93 )
Where F (n ) is the nth. iterate of the map F. That is, on the map f". 2 \x), * x1* , x2 are fitxe d pomts. In Figure 2. 1 1 0b the eight points: x;,x; , . . . , x; (x; = x� ) are fixed points on the F ll > J.b for the logistic map.
For J.1 > Jloo the di ffi cu lty in representing periodic vs. aperiodic regio ns of J.1 is related to the following features: The values of J.1 for whi c h the logi s tic map has stable periodi c solutions fall into continuous bands. Two values of J.1 (J.11 , J.lz) w h i ch have no stable periodic solutions are apparently separated by suc h con tinuous bands of J.1 val u e s associated with stable peri odi c sol ution s . Th u s the set of n on  peri odi c J.1 values is apparently nowhere dense. 2.20.8
Tangent Bifurcations, Intermittencies
The appearance of a period three solution is believed to re presen t a type of "benchmark" in the above bi furc ati on process (alternation between chaotic reg i o n s an d peri odic windows for J.1 > J.loo ) . It is on the one hand the first member of Sharkovsky ' s T set, so that all p eriodic solutions are now present. S ec on dl y , the type of bifurcation which generates these odd period i c solutions are quite different from the "p i tchfork" pe ri o d 2n bi furc at i o ns (period doubling). The nature of these ifurcation s can be i nves ti g ated for the peri od three windows, by considering the graph of F 3 (x, J.1) for the two valu es of J.1 s hown in Figure 2. 1 23a. Note , however that this is the last such odd bifurcations as J.1 is increased, whi c h th en c ompletes S h arko v s ky ' s T set. As J.1 incre as es from 3 .7 to 3 .9 the second, fourth and seventh extrema of F3 becomes tange n t, and
b
S . S . E . H . ELNASHAIE and S . S . ELSH ISHINI
204
(a)
( b) 1.0

X I I I I v
o. s
c !;
   u
    \J /
..,   u
\__ s
.,.,.    IJ
0. 5
1
X
'
1 + va
s
J.1.
FIGURE 2.123 (a) The third iterate maps F3 vs. x (xn+3 vs. Xn) for p. = 3.9, 3.7; (b) Bifurcation diagram showing the period three solution starting at p. 1 + v'8. =
then pass through the 45° line, all at the same value of Jl. Thus, in addition to a continuing unstable fixed point, three new fixed points of F3 are "born out of the blue'', and then bifurcate into six fixed points. Only one xvalue of each pair satisfies I dP/dx I < 1, and hence is stable . This tangent bifurcation contrasts with the pitchfork type (period doubling), in which an existing fixed point splits into two stable and one unstable fixed points . In the logistic case the period three bifurcation occurs at }1 = 1 + 8 1 12 , and is illustrated in Figure 2. 1 23b in the control phase space, showing the stable and unstable branches. Note that the symmetry of the stableunstable "horseshoes" is broken by the conti nuing unstable fixed point . Another feature of tangency bifurcation is that, just before the bifurcation, the dynamics of the map produces an effect called intermit tency which is responsible for the "folded veil" appearance in Figure 2. 1 22. Figure 2. 1 24a shows F( 3 l for J1 = 3 . 828, which is slightly lower than the periodthree bifurcation point (}1 = 1 + 8 112 = 3 .828427 . . . ) . The slight opening between F(3) and the straight line (the 45° line) cannot be seen on this scale. When the mapped points come in the v icinity of this neartangency region, the subsequent maps are "held up" (delayed) i n this region (with period three, of course). In other words the maps of F(x) stay sequentially close to the three neartangency re gi o n s , behavi ng much like a semi periodi c motion for a large number of maps. Figure 2. 1 24b illustrates this behaviour in the present case, for the logistic map. It will be noted that i t takes 42 iterations to "break out " of this semiperiodic behaviour.
STATIC AND DYNAMIC BIFURCATION
205
( b)
F 3( > )
0.51.
.U = lB28
0.53
O.Sl 05 1
05
1
0. 50
X
t
0.50
0.5 1
05
Transfer Resistances
FIGURE 3.1
Decomposition of fixed bed reactor into elements.
MODELLING AND ELEMENTARY DYNAMICS
22 1
interesting and practically important problems concern selectivity of complex reactions and the phenomenon of multiplicity of steady states. I n the case of transient behaviour, stability and response to feed disturbances are of special interest.
The objectives of this chapter This chapter deals with the development of dynamic models for gas solid catalytic reactors. Special emphasis is given to the important two phenomena of multiplicity and instability of these reactors prior to the discovery of the chaotic behaviour which will be dealt with in chapter 4. Special emphasis is placed on the heterogeneous nature of the system which leads to a clearer and more rational appreciation of the important role played by the adsorptiondesorption processes taking place on the catalyst pellet surface. lnevitably, in dealing with nonlinear mathematical models, numerical methods are essential. This chapter is divided into three main sections. Section 1: Single catalyst particle. Section II: Fixed bed reactors. Section III: Fluidized bed reactors. Section I is divided into three subsections. The first subsection deals with nonporous catalyst particle. A comprehensive analysis of the behaviour of a "nonporous" catalyst particle with an exothermic reaction taking place on the outer surface is presented and discussed. The effect of finite solid thermal conductivity on both steady state and transient behaviour of the catalyst particle is presented. The effect of heat release due to adsorption of reactants on catalytic sites is also briefly discussed. In the second subsection a lumped parameter model for the industrially more important porous catalyst pellet is developed based upon the activesite theory (Hougen and Watson, 1 943). This model neglects (at this stage) intraparticle concentration and temperature gradients. A comparison of this model, which takes into account the chemisorption process taking place on the catalyst surface, with the pseudo homogeneous model of Liu and Amundson ( 1 962), which neglects the chemisorption phenomenon, shows some important dynamic differences between the two models. The much larger mass capacity of the active site model, due to the large adsorption capacity of the internal surface, tends to destabilize the steady state. For the modelling of this adsorption mass capacity, a linear equilibrium adsorptiondesorption isotherm is used to relate the solid surface and gas phase co ncentrati on s. On the other hand, the heat release due to adsorption on active sites exerts a strong· s tab i l i zi n g influence . The interplay between these two effects can produce complex behaviour. The lumped model is next generalized
222
S.S.E.H. ELNASHAIE and S . S . ELSHISHINI
to include the effect of a finite rate of reactant adsorption (i.e. non equilibrium adsorptiondesorption). The most noticeable phenomenon associated with relaxing the equilibrium adsorptiondesorption assump tion and considering finite rates of adsorption is the appearance of a re gion of multiplicity of the steady states due to chemisorption when the adsorption rate increases strongly with temperature (activated adsorption). In the third subsection, the activesite model is generalized to the distributed parameter case which takes into consideration intraparticle concentration and temperature gradients. The transient behaviour of the particle is presented and discussed over various regions of the parameters, with special emphasis on the effect of heat release due to adsorption. Different numerical techniques for solving the nonlinear differential equations describing the system are compared. In section II, the cell model is used to describe the fixed bed reactor. This section is divided into two subsections. The first subsection introduces th e application of the simple cell model which considers the only coupling between successive cells to be due to the fluid flow itself. Conditions for the stability of the single particle, as well as the whole bed to arbitrary small disturbances are derived and discussed physically. Some simple a priori conditions, for the stability of the bed to arbitrary small disturbances in terms of system parameters, are presented. In the second subsection, the coupled cell model (radiation) is presented. This model takes into account the coupling between cells due to radiation. The effect of radiation on startup with special emphasis on the "travelling reaction zone", is presented together with the effect of feed disturbances on the reactor. The phenomenon of wrongway behaviour is also presented to the reader and is clearly distinguished from the phenomenon of wrong directional creep of the reaction zone. Different static and dynamic bifurcation phenomena (hysteresis, static isolas, Hopf bifurcation and homoclinical termination) for fixed bed reactors, are presented and explained. This section ends up with a discussion of the comparison between cell models and continuum models followed by a review of some of the continuum models used in the literature and some of the new findings on this respect. Section III deals with the dynamic modelling, multiplicity and stability of the steady states in fluidized bed catalytic reactors with consecutive reaction A � B � C. In this section multiplicity and stability of this catalytic fluidized bed system are presented and di scu s s ed for both the open loop as well as the closed loop controlled case with proportional controller to stabilize the op eration of the reactor at the middle unstable saddle type steady s tate s that gi ve the maximum yield of the de s i re d prod uct B . The chaotic behaviour of this system is p re s ent ed to the
reader in chapter 4.
MODELLING AND ELEMENTARY DYNAMICS
3.1 3.1.1
223
SINGLE CATALYST PARTICLE NonPorous Catalyst Particle
Most modelling studies on the nonporous catalyst particle are con cerned with limiting cases of thermal conductivity, i .e. infinite and zero conductivities (Winegardner and Schmitz, 1 967; Lindberg and Schmitz, 1 969). Petersen and Friedly ( 1 964) studied the effect of external gra dients in the fluid boundary layer on the steady state. They found that, unless the concentration and temperature gradients were extreme, a lumped model employing film mass and heat transfer coefficients, agreed well with a more complicated model accounting for distributed mass and heat transfer in the boundary layer, in predicting the overall rate of reaction. The same authors (Friedly and Petersen, 1 964) studied the stability of this problem assuming the solid to be of infinite thermal conduc tivity. Cardoso and Luss ( 1 969) investigated the stability aspects of a chemical reaction occuring on a catalytic wire. Conditions for the existence of multiple steady states and for asymptotic stability have been derived. The multiplicity of the steady states have been demonstrated experimentally for the oxidation of butane and carbon monoxide on a platinum wire. Luss and Ervin ( 1 972) showed the importance of "end effects" on the dynamics of the catalytic wire. In a succeeding paper Ervin and Luss ( 1 972) have investigated the surface temperature fluc tuations of catalytic wires (flickering) caused by the coupling between chemical reaction and fluctuating turbulent transport coefficients. Many investigators have observed and tried to model complex static and dynamic behaviour, including spati a temporal oscillations, on catalytic wires, ribbons, discs and waffers (e.g. Lobban and Luss, 1 989; Cordonier and Schmidt, 1 989; Sheintuch and Adjage, 1 990; Sheintuch, 1 990; Phillipou et al. , 199 1 , 1 993 ; Collin and Balakotaiah, 1 994). The finite thermal conductivity of the catalyst particle can have important effects on the steady state and dynamics of the system. Pismen and Kharkats ( 1 968) have demonstrated the possibility of asymmetrical steady states (i.e. steady states which do not have the symmetry of the particle) in a nonporous catalytic slab of finite thermal conductivity in a uniform environment. In a later paper, Luss et al. ( 1 972) studied in more detail the phe nomen on of asymmetrical s te ady states and presented a graphical method for the determination of all the solutions for the cases of uniform and nonuniform environment. For highly exothermic and fast reactions the catalyst is often deposited on the outer surface of the support which is usually of very low porosity (e.g. V 20s on SiC for o  x yle ne oxidation (Ellis, 1 972)) .
224
Table 3.1
S . S .E.H. ELNASHAIE and S . S . ELS HISHINI
Heats of reactions and activation energies. Catalyst
Reaction
Activation energy,
E,
kcal/mol
Heat of reaction (dH),
kcal/mol
Platinum catalyst
1 2 .0
68. 1 3
Palladium wires
28.5
67.6
V20sAl203
23.6
1 098. 1 2
V20sAl203
20.0
7 89 .08
Adkins catalyst
1 0. 6
32.7
B enzene Hydrogenation
Ni kieselguhr
1 2.29
(Kehoe and
catalyst
H2 Oxidation (Hoiberg et al. , 1 97 1 ) CO Oxidation (Bond, 1 962) oxylene partial Oxidation to Phthalic Anhydride (Froment, 1 967) B enzene partial Oxidation (Hougen and Watson, 1 943) Ethylene Hydrogenation (Furusawa and Kunii, 1 97 1 )
Butt, 1 972)
Butadiene Hydrogenation
ChromeAlumina
(B ond, 1 962)
catalyst
Ethylene Oxidation
Metall ic silver
20.33 1 3 .00
1 52 30.26 332.6
(B ond, 1 962)
In other applications (e . g . ammonia oxidation converters) the catalyst in the form of a woven wire screen ( or gauze) is often s u pporte d on a non  catalyti c pad to prevent premature i g nition (Gille sp ie , 1 972). In the fo l l ow ing table (Table 3 . 1 ) values of the activation energy and heat of reaction for some typical re acti ons are given. For mas s and heat tran sfe r parameters an exc el l ent review is gi ven by Satterfield ( 1 970). In thi s section we consider a model similar to the one used by Cardoso an d Luss ( 1 969) w i th the exception that the assumption of solid isothermality is relax e d . A finitedifference solution for the transient equations, with symmetrical boundary conditions is presented using the CrankNicholson method together with the Von Rosenberg' s ( 1 969) modification for the nonlinear b ound ary conditions. Another, more efficient method of solution is presented. This me thod is based on the orthogo nal collocation method first used b y Villadsen and Stewart ( 1 967) and Fi nl ay so n ( 1 972). The effect of different parameters on the dynami c s and s tab i li ty of the s y ste m are pre sented and d i sc u ssed . Several assumptions for the mo del reductions are pre sen ted . The assymmetrical behaviour is then briefly discussed.
225
MODELLING AND ELEMENTARY DYNAMICS
3. 1 . 1 . 1
The symmetrical case
a) The dynamic Model A spherical nonporous particle of radius Rp is considered, on the ex te rn al surface of which a first order irreversible chemical reaction is occuring. The solid particle has a finite thermal conductivity and is immersed in an infinitre medium. The following assumptions are made in the development of the dynamic model: (i)
(ii)
The temperature and concentration of the gas around the particle are uniform (T8 and C8 respectively). a. The mass transfer rate towards the catalytic surface is equal to kg (C8  C*) per unit surface area where kg is independent of surface temperature and coverage and c* is the gas phase concentration of reactant just above the surface b. Heat transfer rate between the catalytic surfac e and the bulk of the fluid is equal to h (Ts  T8) per unit area. The reaction rate can be expressed as some function of surface temperature and concentration (T.� and Cs respectively). Equi lib rium adsorption desorption is assumed between the surface and the gas just above it. A linear isotherm is assumed and the equilibrium constant for adsorptiondesorption is assumed independent of temperature (or taken as a temperatureindependent average value). The heat capac ity of the film is negligible compare d with the heat capacity of the solid. Heat of adsorption is negligible Transient solutions have the symmetry of the particle. .
(iii) (iv) (v) (vi) (vii) (viii) On
.
k (c.c* ) kc
the particle surface the accumulation of reactant is given by: d
C.� =
dt
a
v
c s KA
(3. 1 )
s
where Cv i s the concentration of vacant active sites per unit area of
surface and C* is the reactant concentration in the gas phase at the catalyst surface; k is the surface reaction rate constant represented by an Arrh enius expression. with k� being the preexponential fa c tor and E the activation energy of surface reaction.
the
S.S .E.H. ELNASHAIE and S . S . ELSHISHINI
226
The rate of accumulation of reactant above the surface is obtained from a lumped parameter form of the diffusion equation (Appendix A) as follows: 8 de* . 2 dt
=
( _s_)
kg ( CB  c* )  k c c* a v
_
KA
(3.2)
where 8 is the film thickness and k8 a mass transfer coefficient. For equilibrium adsorptiondesorption and a linear isotherm c* = CsfKA Cm , equations 3 . 1 and 3.2 can be combined into a single equation ·
Note : of course we can use the adsorption equilibrium isotherm to write the equation in terms of C as in equation 3.3 or we can do the opposite and write it in terms of C*. The rate of heat conduction in the particle is given by,
(3.4) which is subject to the boundary conditions (3.5)
Equations 3 . 33.6 can be written in the normalized form
_l_ am
dt
dX s
dY dt
( 1  Xs )  a exp (  y / Ys ) · Xs
=a(d2Ydol Y =
with the boundary conditions,
d =0 dOJ
+ � dY m dro
)
at w = O
O < ro < l
(3.7) (3 . 8)
(3 .9)
227
MODELLING AND ELEMENTARY DYNAMICS
and,
dY  = Nu 1 [1  Y

s
aco
+ af3 exp (  y / Y·' ) · X' ]
at co = 1
(3 . 1 0)
subje c t to i n itial conditions ,
at t = O
Y( ro,O) = Ya ( ro) w here
am
= kg j(a + 8/2 )
a = k,jp, CpsR; W = rjRP 
/3 =
Cs
=
k9 (tili) T Cb
h  TB
C J r=Rp
ak =
k0 /(1 + 8/2a)
KA Cm X =C s fC8 · a s hR Nu = P k a=
1
s
a = ak jam = k0 · ajkg
r = EfRc · TB � = T, jT8 Ts = TJ r = R
p
b) Solution of the equations
(i) Finite difference techniques Equation 3.7 can be solved step w ise by an Adam M oulton predic tor corrector method. In the corrector scheme Ys is allowed to l ag 1 time step behind X,. At each time s tep equation 3.8 is solved with boundary condition s 3.9 and 3 . 10. A CrankNicholson s cheme was used to discretise the linear equation 3 . 8 ( w ith its nonlinear boundary condition at ro = 1 , equati on 3 . 1 0). The set of finitedifference equati on s can be solved by the Th omas algorithm (Lapidus, 1 9 62 ) . Details of the procedu re for handlin g the no n  linear b oundary condition (3. 1 0) are given in the text by Von Rosen berg ( 1 969).
When this simple proc edure is applied, it works suc c e s sfully but is very s ensitive to step sizes in both time and radial distance as w el l as the accuracy limit in solving the n onlin ear equati on at the surface. Considerable computing time is consumed in iterating at the surface as w ell as evaluating the whole internal tempe rature profile. For low thermal conductivities and at early stage s of the response, a very small step size ( l o5 ) is nece ss ary . For cases of higher thermal conductivities a larger step size ( I Q2 ) c an be taken. Also, as the solution approaches the steady state it is possible to increase the step size. The time s tep sizes are in most cases restricted by the stiffness of equation 3 . 7 . Typical
S.S.E.H. ELNASHAIE and S . S . ELSHISHINI
228
t%:
1.16 �
�
�
c
]
e !
� :::..
1.14 11 2
2 . 103 1 : 10 /J ' 0. 2 a.,' l rf a,.: '50 Hu1 = 1 0 & ' 0 02
1 10
�e
Collocat ion M� t h o d (lrgtndrt Pol y n o m i a l s )
0
c
C URV£
0 this can only be the case if A < 0. This leads to the instability condition (3 .28) From 3 .26 and 3 .28 we obtain a necessary condition for the instability of a steady state that satisfies the slope condition (3 .29)
Condition 3 . 29 should be checked numerically for the specific steady state under consideration. However, we can see from 3 .29 that the condition could never be satisfied if am is greater than 3 a Nu1 • Therefore a s y stem for which (3 . 30)
236
S .S .E . H . ELNASHAIE and S . S . ELSHISHINI
the steady state that satisfies the slope condition could never be unstable. From this we conclude that the system is stable when condition 3 .30 is satisfied. Condition 3 .30 can be written in terms of the parameters of the system as (3 .3 1 )
In practice, kg and h are related through the jfactor correlation, i.e. (3.32) for gases and vapours. For a specific gassolid system and given opera ting conditions, equations 3.3 1 and 3 .32 can be combined to give the condition,
(3 .33) Some general conclusions about the system stability
We will first summarize the stability results presented for the simplified models. 1 . For PSSCM the slope condition is sufficient for stability. Therefore
thermal conductivity has no effect on local stability, relative to symmetric perturbations. 2. Taking concentration transients into account (but using an infinite thermal conductivity model) the slope condition is necessary for stability. If condition 3 .30 is satisfied, the slope condition is also sufficient for stability. Therefore under these conditions, it can be speculated that thermal conductivity will have no effect on local stability. 3. When condition 3.30 is violated, the steady states that satisfy the slope condition could be stable or unstable depending upon 3 .29 and the thermal conductivity of the system. Linking these conclusions with the stabi lity of the complete model of finite thermal c ondu c tiv ity and finite concentration response, we can see that the ITCM is a very useful limiting case. It can be speculated that decreasing the thermal conductivity tends to cause instability.
MODELLING AND ELEMENTARY DYNAMICS
E" �
�
237
P a r "' "' "' .!1
a : 5 '"
1.4
�Ill
E
N • "•
1. 2
E :u
Ill >
50
a = 4 x r o ·3
10 +...,..,.,.......,..� A ( l ow s teady s tate ) 0 0. 2 0.� 06 0. 8 1.0 X 5 ( d im�n siooless surt ace
' o n c � n t ra \ion )
FIGURE 3.5 Effect of thermal conductivity on startup. A case of low thermal conductivity.
(i) Effect of thermal conductivity Startup in the region with multiple steady states Figures 3.43.6 show plots of surface temperature Ys versus surface concentration Xs for different values of the pellet thermal conductivity in the region of parameters where multiplicity of the steady states exist. These plots are not to be mistaken for phase plane diagrams, since the 2.0
�" ..
:;
Q.
E �
u
..
0 "t: ;;J .. Ill "' ..
c 0 · ;;; t E
�
Paramtlers
1. 8
1=18
ii= 1 . 0
1. 6
1. 4
12
.. >
X s ( d im�nsion l e ss
s u r ! ace
c o ncen t r a tion )
FIGURE 3.6 Effect of thermal conductivity on startup. A case of v ery low thermal conductivity.
MODELLING AND ELEMENTARY DYNAMICS
(j) @ r :
1·6
ji
.. "' ..
c 0 · ;;
c: ..
E
H
.. >
1·2
:§
1 0
a = 2 > 10 5
r
ak
=
=
=
1a
1 0
106
O' m = S
0
0·2
0·4
0·6
0·8 1·0 6 0 t ( t imto} m i nutes
8·0
1 0· 0
FIGURE 3.8 Surface temperature/time response curves. Effect of thermal conductivity on temperature overshooting (Xs (0) = 1.0).
From the local stability point of view, the systems in Figures 3.4, 3 .5 and 3.6 satisfy condition 3.30. Therefore we assert that the slope condition is sufficient for local stability and thermal conductivity should have no effect on local stability. Figures 3 .43 .6 bear out this assertion for changes in thermal conductivity over a very wide range. Investigation of a large number of cases has shown that this criterion is always valid. On the other hand, Figure 3 .9a shows a case where there is a unique steady state and condition 3 .30 is violated. For such a case stability depends on thermal conductivity, as speculated earlier. Figures 3.9a, 3 .9b show that decreasing the thermal conductivity destabilises the system, leading to a limit cycle (periodic attractor). Also shown in Figure 3.9b is the limiting case of infinite thermal conductivity which is stable w i th moderate overshooting. ( ii) Effect of heat capacity of the particle
The effect of heat c ap a city on local stability is app are nt from its effect on c on di ti on 3 .30. It has been shown above that a system violating condition 3.30 can be unstable al thou g h it satisfies the slope condition. Cond i tio n 3 . 3 1 sh o w s that this tends to be the case for small particl es of low heat capacity.
MODELLING AND ELEMENTARY DYNAMICS
Q; E �
J
i
1. 5 U
Ci: 2 x i O"
ydO
X s (O ) : O.O Y5 ( 0 ) : 11 2
)i:0.2
N : 50 "'
1.3
10
.,; E !!
J!;
.. ..
f
.. c: ..
� � .. ,..
1. 5 1. 4
24 1
20
t
( lime) mins
30
40
50
p, r,ttJ\C I C f S :
Xs ! O l : 0 Y1( 0 ) : 1 .12
"tt: 2� 1 04
y:IO
�: J .U
a.._ = I 01
1. ]
n,": ( l 05
(j) N01 • 5 .: t0·1 .Q • h:: 1 0_.;
1. 2
(D lntln1Lc: t�nnal conduct1 ..11�
1.1 10
20
30
40
50
60
10
t ( ti ,. )
ao
mi n s
90
model ( N 01 4 0, u ...... ) 100
no
120
130
FIGURE 3.9(a,b) Surface temperature/time response curves. Effect of thermal conductivity on stability (unique steady state).
Stabilization of certain desirable unstable steady states (satisfying the slope condition) can be effected by increasing the heat capacity or the particle diameter. Figure 3 . 1 0 shows a case of a unique steady state violating condition 3 .30 which is unstable. The same sustained oscillation (periodic attractor) is obtained from different initial conditions . By increas ing the heat capacity such that condition 3.30 is satisfied, the steady state becomes stable, as shown in Figure 3 . 1 0. Figure 3 . 1 1 shows the Ys  Xs diagram for a case having the same parameters as those in Figure 3 . 5 except for the heat capacity which is reduced ten folds . The reduced heat capacity causes the upper steady state B to be unattainable from all possible initial conditions, which shows that this steady state is unstable.
S.S.E. H. ELNASHAIE and S.S. ELSHIS HINI
242
!! "
lo . 6
"§..
4.2
.2 :;
J.4
., .. "2
3�
.. ,..
2.2
a. E !! .. v
�
�
0
· ;;; c .. E "
38
2.6
X, ( O l
Pu�mc•cr1;
&:: 2 & 105
: 0. 0
J: lj
P, i .O
a.. :
a rr.=
(DN
I OJ
0. 00�
• dO. U. . 0. 004 •
so. a
:m
, o1
  ·  · 
18 l.lo 1.0
1 80
100
20
2 60
t ( t ime ) m i n e
3 40
4 80
500
FIGURE 3.10 Surface temperature/time response curves. Effect of heat capacity (unique steady state).
2.0
Panmucrs
li• 2 x l 0�
1. 8
" v
Y:l 8
�= 1 . 0
a.k= t o6
1. 6
0 "t:
::J ..
., ..
�... J4
E �
�
"' >
a m=
1. 4
N
: "•
5
50
li : 004
1. 2 0
0. 2
0. 1.
06
o. a
X 5 ( d immsion l e ss s u rface c o ncen t ra t io n )
FIGURE 3.1 1 Effect of heat capacity on the stability of high temperature steady state. A case of multiple steady states.
MODELLING AND ELEMENTARY DYNAMICS
2. 3
.. 5
'§..
a. E
2. 2
20
·g;;;
1.8 1
� �
�
_§ c
..,
.. >
�
r, 
   ·
2.
�
243
C om p lele
P u�omctc r a .
mod e l
Po;eudostwd y
s tole concen t ration mod el
Y=I S ii = I.O oc,. = 10�
1
NII� O�
/X5(0 ) : 1. 0
1�
(1:0.4
1
·  · · ·  ··  ·  · .9 · ·  ·1.6 f ""' :::::.: :..:: =·=·=·=·::�=====.:...... � 1. 7
· ;.;.;
X5(0) = 0.0 .. 1.5 +........! 0
0. 1
0.2
t
( t i me )
0. 3
minutes
0. 1.
o. s
FIGURE 3.12 Surface temperature/time response curves. Comparison between complete model and PSSCM for a case of temperature runaway.
(iii) Effect of the Pseudosteady state concentration assumption Changes in the thermal conductivi ty and heat capacity of the parti c le have no effect on the local st abil i ty of this model. The failure of this simplified model to properly describe the local s tabil ity characteristics and the reg ion s of stability of the steady s tate s is a serious limitation. This simplified model also fail s to predic t the correct response in cases where temperature runaway occurs . This is due to the dec ou pli ng of concentration and temperature responses. It is this coupl i n g which is th e main c ause of runaways. Figure 3 . 1 2 brings o u t this point more clearly. The most accurate predictions of the s implifi e d model are obtained for i niti al c ondition s corres pond i n g to zero su rfac e coverage, Xs (0) = 0.
d) Effect of the heat of adsorption Most of the ad s orption res p on s ible for hete ro geneou s ga s so l id catalysi s is of the chemisorption type, which is almost alw ay s e x othermi c and accompanied by an apprec iab le heat of adsorption. The effect of heat of adsorption on the dynamic behaviour of catalyst particles is pre se nted in detail in the next secti o n which is dealing with the porous catalyst particl es . However, it is worthwhile to check the effect of h e at of ad so rpti o n on the nonporous particle for some of the c as e s presented
earlier.
..
S.S.E.H. ELNASHAIE and S S ELSHISHINI
244
When taking the heat of adsorption into account the model remains exactly the same, except for the heat balance equation at the surface which becomes: dT ks d r = h( TB  � ) + (MI)r rs + (MI)A rA
(r;; + � ) d s
= h ( TB  � ) + (MI)r fs + (MI)A = h(T8 
[
�)+(M/)T'.v
dCs
+ (MI)A dt
f3
In normalised form it becomes, df
_ s
d (J)
=
NuI 1  f + af3 exp ( y / fs ) X
where,
__
s
s
_X l d
+ ___,i_
s
dt
am
k
a = _fL a
at r = RP
at ro = l.O
(3 .24)
m
This case can be solved using the collocation method, as described earlier. IJ l
jJ
Clt
� · fJ. : 0. 6 {J A : 0 . 0 \ ....... .
0
0. 2
Y:l 8
�= 1.0
ak: l06

1. 2
1.0
a : 2 x i 05
.
i ) l
�
.§i
P a r a meters
O.lt
X s (dimension\ess
a m=
....... .
....... .
.......
0. 6
surface
Nu�
5,
CXm : 6
5UO
a = o.o4 ....... .
.......
......
0. 1 conc . ) C s i C b
A ( l ow
1.0
steady state )
FIGURE 3.14 Effect of heat of adsorption on the stability of the high temperature steady state. A case of multiple steady states.
S . S .E . H . ELNASHAIE and S . S . ELSHISHINI
246
�
0
a.
FIGURE 3. 1 5
0.0
1.0
2 .0
w
Asymmetrical steady state in a nonporous catalyst slab.
as y mmetrical steady state in a cataly st slab is shown in Fig ure 3 . 1 5 . The symmetry condition ( d YliJ W= 0 at ro = 1 ) is relaxed. The transient behaviour of the system (assuming pseudo stead y state for the concentration re s pon s e) is g i ven by 0 < ro < 2 . 0
a=
a=
dY daJ
=  Nu1 [ 1  Y + alf I ( a + exp ( y I Y))]
() y = + Nud1  Y + alJ I ( a + exp ( y i Y ))]
dro
(3 .35)
ro = o. o (3.36) ro = 2 . 0 (3.37)
Steady states such as the one shown in Figure 3 . 1 5 for which a>O have a mirror image for which a< 0. Any conclusion with respect to one steady state applie s ex actly to its mirror i mage . Equation s 3 .353 .37 are written for steady states with a> 0. For the mirror image the s i g ns of the right hand sides in 3 .36, 3.37 have to be reversed. ( i) Steady state analysis
The steady state equations are obtained by setting the time derivative in 3 . 3 5 equal to zero. After some integration the steady state must satisfy
MODELLING AND ELEMENTARY DYNAMICS
the following relation
,
( ) ( ) JY
J ro
w=O
247
( 2 )
_ JY _ _ �1  �0 a J ro w= 2
3.36, 3. 3 7 and 3.3 8 �1 �0  2 Nu1 [ 1 t:a
(3.38)
can be u sed to obtain the following non Equations linear relations between Yst. }',o, =
+ af3 I ( a + exp ( y I � 0 ))]
(3. 3 9)
+ af3 I ( a + exp ( y I
(3.40)
similarly, �0 =
�1  2 Nu1 [1  �1
�1 ) )]
(1968),
If we plot Yso versu s Y�1 as su gges ted by Pismen and Khartas we obtain two Sshaped curves as shown in Figure S teady state s olution s correspond to the intersections of the two curves. Steady s tate s 1 are s ymmetri c (since they lie on the line OA and therefore Yso = and are asymmetrical. Ys1) . The remaining six steady states 4, 5, 6, show the effect of increasin g the solid the rm al gu res and c ondu ctivi ty (i.e. decreasing Nu1). Thermal conducti vi ty obviously does not affect the symmetric steady state s , but increasing this thermal cond ucti vity tend s to pu sh the asymmetrical one s towards disappearance at a certain critical value of the thermal conductivity.
3.16.
,2,3 Fi 3.16 3. 1 7
7, 8 9
3 2 �� � a , 2 x 10
ii
2 !
r
2.4
'
=
Nu , : 5 . 0
2.0 Ysr
1. 6 1. 2 0. 8
O . Or�...,.....,..'1 0o.o
FIGURE 3.16
1. 0
,•
0.4
0.8
St•ady
1. 2
1.6
2 .0
2.4
surfa"" temperature a l w : 0 . Y50
2 8
Asymmetrical steady states. Graphical solution.
S . S .E.H. ELNASHAIE and S . S . ELSHISHINI
248
1. 8 Hu,= O. S
1.6
Hu1 = 0. S
Nu1 : 1 . 5
1. 2
18
1.6
1 . 1,
1.2
1.0
2 .0
Ys o
FIGURE 3.17
Asymmetrical steady states. Graphical solution. Effect of thermal
conductivity.
Figure 3. 1 8 shows the asymetrical steady state profiles (mirror images are not shown) for different values of solid thermal conductivity. For Nu1 = 500.0, there are a total of nine steady states, three symmetrical and six asymmetrical. For Nu1 = 1 .5, There are seven steady states, four of them being asymmetrical. For Nu1 = 1 .0, there are five steady states, two of them being asymmetrical while for Nu1 = 0 9 the asymmetrical steady states disappear leaving three symmetrical steady states. .
2.0
Y S3
1. 9
l
YS3
1. 9
1. 8
1. 8
Yso
2.0
1.6
1.5
Ys:
1. �
l. J 1 1
w N.,= �OO
FIGURE 3.18
2.0
1. 8
1. 8
Ys1
1.�
1. 7 1.6
l.S
1. J
1. 1
1 1
1. J
1. 2
1. 2
Nu,= 1 0
Ys2
u.
,_ ,
1. 3
v ,.,
1. 9
1. 9
1.6
1. 2
1. 2
2.0
1.7
1.7
1. 7
,
1. 1
N ,.,r=2
N .., r 1.s
Asymmetrical steady states for different values of Nu 1 .
MODELLING AND ELEMENTARY DYNAMICS
249
( ii) Local stability analysis
It is interesting to examine the stability of these asymmetrical steady state to small perturbations to see if any of the asymmetrical steady states can be stable. After linearizing equations 3 .36 and 3 .37 about the steady state and introducing the disturbance variable, = Y  �s' the system is described by,
x
a x = a x 0 < OJ < 2. 0 at a OJ ax _ = Nu1 x [ a f3gss  1) OJ = 0. 0 a OJ ax = Nu1 x_[_/3 1 OJ = 2 o a gss  ) a OJ +
 = a

.
where, gs s
and
_

( aaisy )
.
ro =O '

( aaisy )
g ss = 
(3 .4 1 )
(3.42)
(3 .43)
ro= 2 .0
fs = exp ( y I Y) I (l + a exp (  y I Y ))
and subscript (ss) signifies steady state. The solution of the linear partial differential equation 3.4 1 with linearized boundary conditions is given by, x
= L (A cos A..j OJ + B sin A..j OJ) j
·
exp
(a · A.] · t)
(3 .44)
where the A../ s are the eigenvalues of the linearized equations, and are the roots of the equation, (3 . 45) where,
250
S.S.E.H. ELNASHAIE and S.S. ELSHISHINI
The stability of the system is dictated by the character of the eigenvalues. It is clear from 3 .44 that the system is asymptotically stable (i.e . .X ? 0 as t � oo) if all the roots of 3.45 are real. Analysis of 3 .45 shows that sufficient conditions for the roots to be real are, h2 > 0, h1 > 0. These two conditions may be written in terms of the parameters as
ali ( gss + gss ) < 2. 0
(3 .46a)
( aJigss  1) ( aJigss  1 ) > 0
(3.46b)
Conditions 3 .46a,b are satisfied if
These conditions can be interpreted as the steady state slope conditions at both faces of the slab. It should be noted that these conditions are sufficient for stability when neglecting the transient accumulation of reactants on the surface. They are therefore only necessary conditions for stability in the general problem. Some of the asymmetrical steady states are stable, notably two for the cases corresponding to the existence of six asymmetrical steady states. For the cases in which there are two and four asymmetrical steady states, all of the asymmetrical states are unstable. Table 3.2 shows the stability character of the asymmetrical steady states for the cases shown in Figure 3 . 1 8. Table 3.2 Nu1 500.0
1 0.0
2.0
Stability of asymmetrical steady states. Yso
Ysl
a
(ipgss
7if3gss
Stability Unstable
1 .0037
1 .45 1 5
0.2239
0.0577808
2. 1 1 57338
1 .0042
1 .95062
0.4732 1
0.05 82383
0. 2 1 7965 3 8
Stable
1 .45 1 7
1 .95 1 0
0.24965
2. 1 1 5 501 2
0.2 1 752756
Unstable
1 .027
1 .47 1 7
0.2223
0.0826302
2.07734 1 8
Unstable
1 .04933
1 . 895
0.423
0. 1 1 442742
0.293 87097
Stable
1 .4297
1 .9 1 978
0.245
2. 1 223444
0.25698449
Unstable
1 . 1 35 1 3
1 .57459
0.2 1 98
0 .3445466
1 .5678754
Unstable
1 . 1 7644
1 .70887
0.2662 1
0.5 392559
0.82023679
Stable
1 . 345 1 1 4
1 .78228
0 .2 1 8
1 .7928329
0.54790309
Unstable
1 .22 1 954
1 . 65074
0.2 1 44
0. 8277 1 79
1 . 1 1 05 3 1 1
Unstable
1 .30907
1 .70843
0. 1 9968
1 .5228468
0.822 1 85 1 5
Unstable
1 .0
1 . 3607 8
1 . 5525 1
0.09586
1 . 8920554
1 .7025722
Unstable
0.95
1 . 3 8449
1 .5 2403
0.06977
2.0 1 24635
1 . 8636775
Unstable
1 .5
MODELLING AND ELEMENTARY DYNAMICS
3.1.2
25 1
Porous Catalyst Particle. Lumped Parameter Models
It is expedient to begin the presentation of the dynamic behaviour of porous catalyst particles by considering the relatively tractable lumped parameter models. Such models neglect intraparticle concentration and temperature gradients, the entire resistance to heat and mass transfer being lumped into a thin film at the surface of the particle. Mathematically, the system is represented by ordinary differential equations in the state variables which are pellet temperature and concentration of reactants (and products in certain cases), with time as the independent variable. This is in contrast to the more realistic, but less tractable, distributed models which account for intraparticle temperature and concentration gradients which will be presented in the next section. Many authors have tried to reduce the complicated distributed parameter model to a simple equivalent lumped parameter formed by a variety of techniques (lllavacek et al. 1 969, 1 970; McGowin, 1 969; Luss and Lee, 1 97 1 ). ,
3. 1 . 2. 1
The importance of suiface processes on the dynamic behaviour of catalyst particles
Many modelling studies of the dynamics of gassolid catalytic reactions fail to consider the dynamics of the adsorptionreactiondesorption steps, as postulated by the active site theory (Hougen and Watson, 1 943). Early in the 60 ' s and the 70 ' s, the work of Kabel et al. ( 1 962, 1 968, 1 970) shows very clearly the importance of the adsorption desorption step on the dynamic response of a tubular heterogeneous catalytic reactor in which the vapour phase dehydration of alcohol is carried out. The importance of the adsorptiondesorption steps is not only related to finite rates of adsorption and desorption, for even when the adsorptiondesorption is very fast and equilibrium adsorption desorption is established very quickly, these processes still affect the mass capacity of the catalyst particle which, in its tum, has a strong effect on the dynamics and stability of the particle. Elnashaie and co workers (e.g. Elnashaie and Cresswell, 1 973a, 1 974; Elmfshaie, 1 977; Elnashaie and ElBialey, 1 980) have shown the importance of these steps on the dynamic behaviour and stability characteristics of single catalyst pellets (refs), as well as fluidized bed catalytic reactors. Relatively recently many researchers started to take these steps in consideration in the modelling of catalytic processes (Arnold and Sundaresan , 1 987, 1 989;, Ill ' in and Luss, 1 992) . Notice that in the previous section dealing with nonporous catalyst particles, we have shown the important effects of adsorption mass
252
S . S .E.H. ELNASHAIE and S . S . ELSHISHINI
capacity and heat of adsorption on the dynamics and stability of the catalyst particle. a) Adsorption and catalysis
One of the oldest theories relating to catalysis by solid surfaces was proposed by Faraday in 1 825 . This theory states that adsorption of reactants must first occur and that the reaction proceeds in the adsorbed fluid film. There is much evidence against this simple view. For instance, the more effective adsorbents are not always the more effective catalysts, and catalytic action is highly specific. That is, certain reactions are influenced only by certain catalysts. The modem view, therefore regards adsorption as a necessary but not sufficient condition for ensuring reaction under the influence of a solid surface Adsorption is due to attraction between the molecules of the surface, called adsorbent, and those of the fluid, called the adsorbate. In some cases the attraction is mild of the same nature as that between like molecules, and is called physical adsorption. In other cases the force of attraction is more nearly akin to the forces involved in the formation of chemical bonds, so the process is called chemical adsorption or chemisorption. b) Rates of adsorption
The adsorption process is generally very fast on the surface of many clean metal films. However in a great many cases of adsorption by a bulk adsorbent, the rate of adsorption is quite slow with equilibrium being reached in a few hours. An excellent review of rates of adsorption is given by Hayward and Trapnell ( 1964) as well as Low ( 1 960). Tables 3.3 and 3 .4 give some example of fast and slow chemisorption respectively. The kinetics of adsorption can be classified roughly into two categories: Table 3.3 Fast chemisorption (equilibrium established in a few seconds). Gas
Adsorbent
Velocity of adsorption Instantaneous reversible adsorption Instantaneous reversible adsorption Rapid adsorption Rapid adsorption
MODELLING AND ELEMENTARY DYNAMICS
253
Table 3.4 Slow chemisorption (equilibrium established in a few hours). Gas
Adsorbent
Velocity of adsorption
ZnO, Crz03 , Alz03 , Vz03 ZnO Crz03 ZnO, Crz03
Slow even at high temperature Slow, Ead 1 5 kcaVmol Slow even at high temperature Very slow even at room temperature =
( i) Activated adsorption The characteristics of this type of adsorption are: 1 . Exponential increase in the rate with increasing temperature. 2. Continuous fall in rate with increasing coverage.
For activated adsorption the sticking probability S, defined as the fraction of collisions between adsorbate molecules and surface resulting in chemisorption, may be defined as, S = af( 8 ) exp
(Ead I ReT)
(3.47)
In this equation a is the condensation coefficient. It is the probability that a molecule is adsorbed, provided it has the necessary activation energy Ead and collides with a vacant site. f( 8) is a function of surface coverage 8 and represents the probability that a collision will take place at an available site. The rate of adsorption is given by,
(3.48) where m is the mass of the molecule, K is the Bolzmann constant, T the temperature and p the partial pressure of adsorbate in the gas above the surface. Table 3.5 gives some typical values of the activation energies for activated adsorption. Table 3.5 Activation energies for activated adsorption (Low, 1960). Gas
Adsorbent
Activation energy, Ead kcal/mol.
Carbon Graphite Diamond
53 22.03 3 . 8 1 3 .722.4 7.0 1 5 .0
Cu z O ZnO
S.S.E.H. ELNASHAIE and S.S. ELSHISHINI
254
Table 3.6 Heats of chemisorption of gases on charcoal (Low, 1960). Gas Heat of adsorption, (Mf'JA, kcal/mol
72.0
3 1 .9
8.4
1 6. 9
9.4
(ii) Nonactivated adsorption This type of adsorption is characterized by,
1 . Weak or zero dependence of adsorption rate on temperature. 2. Initial rate independent on coverage. The adsorption of many gases on clean transition metal films is found to be very fast (see Table 3 . 1 ) and to remain so even at temperatures as low as 78°K (Hayward and Trapnell, 1 964). The activation energy required for this type of adsorption must, clearly, be extremely small. c) Heats of adsorption
Adsorption is an exothermic process. In physical adsorption of gases the heat effect is of the same order of magnitude as the heat of condensation, that is, a few hundred calories per gram mol. In chemisorption the heat effects are more nearly like those accompanying chemical reaction, say 1 0 1 00 kcal/mol. Table 3.6 gives some typical values for heats of chemisorption. d) Example
In the following, parameters for the rate and heat of adsorption of nitrogen on iron are given (Scholten et al., 1 959). The heat of adsorption, (Mf)A = 1 050 kcal/mol. The activation energy for adsorption, Ead = 1 022 kcal/mol. Expressions for rates of adsorption: 1 . For 8= 0.07 to 0.22 (8 is the fractional coverage). 1 ra = 2 1 . 9p · exp (1 32. 48 / J?o ) · exp ((5250  77508) / J?oT) min 
2. For 8 = 0.25 to 0.7 ra
where
= p
I 2. 5 1 X 1 06 p (1  8) 2 · 83 exp (23 / ReT) min·
·
is the partial pressure of adsorbate in em Hg.
MODELLING AND ELEMENTARY DYNAMICS
255
e) Rates of desorption
If it is assumed that desorption may take place from occupied sites, providing the adsorbed molecules possess the necessary activation energy for desorption, then the rate of desorption becomes,
(3.49) where kdo and Ed are the velocity constant and activation energy of desorption respectively, and f' ( 8) is the fraction of sites available for desorption at coverage 8. The activation energy of desorption is related to the heat of adsorption (Ml)A and activation energy of adsorption Ead by the equation,
(3.50) Since adsorption is always exothermic, Ed is appreciable even when Ead = 0. That is desorption is always activated. f) Equilibrium adsorptiondesorption
Adsorption equilibrium can be expressed by isotherms relating the concentration of adsorbate on the surface to that in the gas above the surface. The condition for equilibrium is that the rates of adsorption and desorption are equal. Isotherms may be obtained by equating these rates. Three theoretical isotherms, those of Langmuir (1918), Freundlich (1926) and Temkin (1935, 1942) are important. Each is characterized by certain assumptions, in particular, as to the manner in which the differential heat of adsorption varies with adsorbed concentration of adsorbate, and each is applicable to certain experimental systems. We will present only the Langmuir isotherm, which is the most widely used in catalytic kinetics investigations. Other isotherms are discussed in standard physical chemistry handbooks. The Langmuir isotherm (Langmuir, 1918)
In the simplest case the velocities of adsorption and desorption are given by equations (3.48, 3.49). At equilibrium ra = rd equating 3.48 and 3.49 and noting that Ed= Ead+ (Ml)A, the adsorption isotherm becomes, p
d = k o (j
J2 mnKT J'( () ) exp j( () )
(( M/)A I f?a T )
(3.5 1 )
S . S .E.H. ELNASHAIE and S.S. ELSHISHINI
256
A Langmuir isotherm is obtained if it is assumed that the expression kdo (j
exp (( Ml) A I ReT ) 
is independent of 8. If we then place,
1
= a
where
a
= _Jjg_
k
a
.J2 mnKT exp ( ( M/)A I Re T)
is dependent on temperature alone, this isotherm becomes,
(3.52) For a molecule adsorbed on a single site, f( 8) = 1  8
/'( 8) = 8
(3.53)
and the isotherm becomes, p = a(l  8)
(3.54)
8 = __!!1!__ l + ap
(3.55)
or,
Three important conditions are implied in the derivation of Langmuir isotherm. These are the following, (i)
adsorption is localized and takes place only through collision of gas molecules with vacant sites (ii) each site can accomodate one and only one adsorbed molecule (iii) the energy of an adsorbed molecule is the same at any site on the surface, and is independent of the presence or absence of nearby adsorbed molecules . In this chapter a lumped dynamic model of a porous catalyst pellet is developed on the basis of this activesite theory and assuming equilibrium adsorptiondesorption according to a linear Langmuir isotherm. This model is compared with other models that do not take
these important surface phenomena into consideration such as the pseudo homogeneous model due to Liu and Amundson ( 1 962). Next, the assumption of equilibrium adsorptiondesorption is relaxed and the effect of both activated as well as nonactivated adsorption is presented. The rate of adsorption is treated in very simple terms under the Langmuir postulates as discussed earlier. 3. 1 . 2 . 2
Dynamic Modelling of Porous Catalyst Particles with Negligible Intraparticle Mass and Heat Transfer Resistances and Equilibrium AdsorptionDesorption. The Lumped Parameter AdsorptionDesorption Equilibrium Model (LPADEM)
a) The Model
Consider a simple reaction system A+X � A ·X
adsorption
A·X � B·X
surface reaction
B·X � B+X
desorption
A �B
overall reaction
where X denotes an active site. The rate of accumulation of adsorbed reactant A · X on the internal surface can be equated to the difference between the rate of adsorption rA and the rate of surface reaction,
.
dCs = ra  r.s dt
( 3 5 6)
__
We shall express the rates of adsorption and surface reaction ra . rs as molls . g catalyst. The adsorbed concentration Cs is therefore expressed in mol/g catalyst. According to the active site theory (Hougen and Watson, 1943),
(  c)
ra = ka C* · Cv
where
C*
_
s KA
_
(3.57) (3.58)
is the concentration of reactant in the intraparticle gas
258
S.S .E.H. ELNASHAIE and S.S. ELSHISHINI
above the surface (mol/cm3 ) and ev is the concentration of vacant sites (mol/g). If for simplicity we assume that the percentage coverage of active sites is small compared with the total active sites, em , for both reactant and product, equation 3.57 can be written as,
(  ) *
ra = ka e  em
es
_ _
KA
(3.59)
where em is the total concentration of sites (corresponding to a complete monomolecular layer on the catalyst). The rate of accumulation of reactant in the intraparticle void space per unit volume of pellet can be equated to the difference between the rate of mass transfer from the bulk phase to the particle rM and the rate of adsorption Psra. For a spherical pellet of radius Rp and voidage E, we obtain,
(3 . 60) where,
3
*
rM =  kg (e8  e ) RP
(3.61)
In equation 3.61 e8 refers to the concentration of reactant in the bulk phase. A dynamic heat balance on the particle is given by,
(3.62) where rH is the rate of heat transfer per unit volume of particle be tween the bulk phase (temperature TB) and the particle (temperature n defined by:
3
rH =  h ( T8  T) RP
(3.63)
The second and third terms on the right hand side of equation 3.62 account for the rates of heat generation due to surface reaction and reactant adsorption, respectively. We justify the inclusion of a heat generation term due to adsorption on the basis that the heat of adsorption
MODELLING AND ELEMENTARY DYNAMICS
259
for chemisorption may be comparable to the heat of reaction. For simplicity we have neglected the effect of product adsorption. It is also necessary to state the initial conditions,
C5(0) = Cso C* (0) = C0 T(O) = T0 3.563.63,
The lumped system formulation described by equations comprises three state variables Cs , C*, We shall limit ourselves in this section to the special case of adsorptiondesorption equilibrium.
T.
Adsorptiondesorption equilibrium Eliminating ra between equations of coupled differential equations,
3.56, 3.60 and 3.62, we obtain the pair
E
dC*

dt
+ ps
dC5

dt
= rM

ps rs
(3.64) (3.65)
where
(M>r represents the overall heat of reaction given by
3.64
( !lli ) r = ( !lli ) r + ( !lli ) A
(3.66)
Equation represents an overall mass balance taken over the internal surface and the void space. If we assume equilibrium between the internal surface and the intraparticle gas according to a linear isotherm relationship, then we have,
(3.67) We shall assume further that the equilibrium constant KA is temperature insensitive, although in practice it is a decreasing function of tempera ture (it is physically sound to consider it a temperature invariant average value over the range of temperature covered in the investigation) . Of primary importance, however, is the temperature dependence of the intrinsic reaction rate constant k which can be represented by an Arrhenius expression,
260
S . S .E.H. ELNASHAIE and S . S . ELSHISHINI
with ko being a preexponential factor and E the intrinsic activation energy. The linear isotherm relationship 3.67 allows us to write the overall mass balance (3.64) in terms of the single concentration variable C: . Thus substituting 3.67 into 3.64, we obtain,
(3.69) where k' is a pseudohomogeneous rate constant given by,
(3.70)
with Ea the apparent activation energy. Notice that in the development of the dynamic mass capacity and reaction rate in terms of gas phase concentration we used a linear equilibrium adsorptiondesorption relation to eliminate the surface concentartion. The use of the more realistic nonlinear isotherm relation will not cause much complexity in the rate of reaction term, but will cause considerable complexity in the dynamic mass capacity term and is still a subject of research. Furthermore, in the development of the dynamic mass capacity term, we used an average temperature indepen dent average value for the adsorption equilibrium constant without taking into consideration its variation with temperature. Taking into consideration the temperature dependence of KA in the dynamic mass capacity term causes considerable complications and is also still a subject of research. It is convenient to cast equations 3.69 and 3.65 into a dimensionless form by introducing the normalized variables
and the dimensionless time, 'l' =
Equations
3.69 and 3.65
3ht

Rppscps
may now be written as,
MODELLING AND ELEMENTARY DYNAMICS
26 1
where Ls and 4 are Lewis numbers based upon the separate contributions of the internal surface and the void volume respectively, to the effective mass capacity of the particle. h 1 Cm Ls = K A p s kg Ps Cps h 1 E 4 =kg Ps Cps 

(internal surface) (void volume)
The remaining groups a , {3 , f3A, y are defined by,
RP Pia 3 kg (W)T kg CB {3 = hT8 (W) A kg CB f3A hT.B r = Ea I Rc TB a=
(overall exothermicity factor)
_
( exothermicity factor for adsorption)

b)
(dimensionless activation energy)
The Steady State Model
The steady state equations for the particle are obtained from equations 3.71 and 3.72 by setting the time derivatives equal to zero to give the following two algebraic equations, *
*
(3.73)
1  X5 = a exp ( y i � ) Xs
�  1 = af3 exp ( y / � )x;
(3.74)
where subscript s signifies steady state. Notice that the dynamic adsorption parameters Ls and BA, have no effect on the steady state of the system (except of course with regard to their quantitative effect on a ' f3 and r ). Equations 3.73 and 3.74 can be combined into a single equation which can be written as, 
1
Q =  ( Y,  l ) = exp ( y / � ) ( 1 + f3  Y, ) = Q a
*
(3.75)
S . S .E.H. ELNASHAIE and S . S . ELSHISHINI
262
where Q and Q+" represent the heat removal and heat generation functions for the single catalyst particle. It is clear that the particle may have either one steady state or three steady states depending upon the parameters a, f3 , y. Equation 3 .75 has been analyzed extensively for uniqueness and multiplicity of solutions by Aris ( 1 969). The results are summarized below, (i) Unique solution if r · /3 < 4 ( 1 + /3), irrespective of a. (ii) Multiple solutions if y · f3 > 4 ( 1 + /3) and
where,
y
y
sl ' s2 
(iii) Unique solution if
y(f3 + 2) ± � yf3[ yf3  4(1 + /3)] 2(/3 + y) r · /3 > 4 ( 1
+ /3) and either lla > ¢* or ll a < ¢• .
c) Comparison of the model with simplified models that neglect the mass capacity due to adsorption
Liu and Amundson ( 1 962) investigated the stability of the steady states of a porous catalyst particle subject to small perturbations by employing a pseudohomogeneous model that does not take into consideration neither the adsorption mass capacity nor the heat of adsorption due to the chemisorption process. Their model can be derived from equation 3 .7 1 and 3 .72 by equating the adsorptive capacity term L, = O , (this auto matically cancels the term Ls · f3A (dx*ldr), even if f3A is not equated to zero).
dx * 4  = 1  X*  a exp dT
( y / Y)X
•
d Y 1 Y f3 exp ( y / Y X* =  +a )

dT
They derived the familiar slope condition
MODELLING AND ELEMENTARY DYNAMICS
263
at the point of intersection of the Q, Q+ curves as being necessary and sufficient for local stability of the steady state. From this result they were able to draw the following conclusions. 1 . Uniqueness of the steady state ensures stability. 2 . In the multiple steady state region the low and high temperature
steady states are locally stable while the intermediate steady state is unstable (a saddle type steady state). Comparison of the pseudohomogeneous model of Liu and Amundson with that based upon the active site theory (equations 3 .7 1 , 3 .72) shows up two important dynamic differences: (i) a larger Lewis number in the active site model due to the adsorption capacity of the internal surface (ii) stronger coupling between the heat and mass balance equations by virtue of a heat generation term due to adsorption Both differences have important effects on the transient response and stability of the steady state. Before considering stability in more detail it is useful to obtain an impression of the magnitude of the Lewis numbers L5, 4 in real systems. d) Magnitude of the Lewis numbers
From jfactor correlations,
for gases and vapors. Using this result, we can write,
For the adsorption of ethanol on ionexchange resin catalyst at 1 00°C, Kabel and Johanson ( 1 962), report,
em = 4 X 1 0 3 mol / g. catal . ; 
and
KA = 2. 75 X 1 05 cm 3 I mol
264
S .S.E.H. ELNASHAIE and S.S. ELSHISHINI
Taking Pt ClPs Cps = I 03 for gassolid systems, Ls is of order 1 , while 4 is of order 1 o3. This result shows that accumulation of reactant within the pores will normally be insignificant compared with that on the internal surface. It is also important to notice that for other catalysts and other reactions Cm and KA could be much larger than the above values giving Ls much greater than 1 .0 (Elnashaie et al. , 1 990). e) Local stability analysis
Introducing perturbation variables,
where x; and Ys are steady state values of X* , Y. The linearized forms of equations 3.7 1 and 3.72 for arbitrarily small perturbations can be written as,
where,
1
aF Fx' ; ax· x' aF F.' = � ar � _
F = exp (  y / � )Xs
I
s
As shown several times earlier, the necessary and sufficient conditions that these two equations have asymptotic solutions tending toward zero are: AD  BC > O A+D
s
r.
s
as derived by Liu and Amundson ( 1 962) . The second condition can be written as, (3 .77)
For the pseudohomogeneous model of Liu and Amundson ( 1 962) Ls = 0, and bearing in mind that Lv 4[3 Fr.1  Fx. 1
s
a
"
(3.78)
The first condition 3 .76 implies the second condition 3.78. The slope condition is therefore sufficient for local stability when the adsorption mass capacity is neglected (Ls = 0). In the active site model Ls � Lv and 3 .77 can be simplified to, (3 .79)
Since Ls may be greater than one in real systems, the first condition 3 .76 does not imply the second condition 3 .79. A steady state that satisfies the slope condition may therefore, be unstable. A similar result was obtained by Cardoso and Luss ( 1 969) for reactions occuring on a catalytic wire. f)
Qualitative effects of the dynamic parameters Ls and f3A on stability
It is interesting at this stage to see the qualitative effects of the dynamic parameters Ls and f3A on stability.
i) Effect of the Lewis number Ls
( )
First let us rewrite condition 3 .79 as F�. I I  I +  + s > Fy' (f3  f3A ) L L a s

s
(3.80)
266
S.S .E.H. ELNASHAIE and S . S . ELSHISHINI
It is clear that by increasing the Lewis number we decrease the left hand side of the stability condition 3 . 80, thereby tending to destabilize the steady state. In the limit as Ls � oo, condition 3 . 80 reduces to 1
 > F. a
1
(/3  /3 )
Now if the heat of adsorption is negligible 1
(3 . 8 1 )
A
Y,
(f3A = 0),
3 . 8 1 reduces to
1
 > f3F.r, a
(3 .82)
We can see that a steady state satisfying the slope condition 3 .76 may violate 3 . 82, in which case the steady state becomes unstable. If, on the other hand, the steady state satisfies 3 . 82, it is stable irrespective of the magnitude of the Lewis number.
ii) Effect of the heat of adsorption parameter (/3A)
By increasing f3A we decrease the right hand side of 3 . 80, thereby tending to stabilize the steady state. In the limiting case f3A = f3, (which means that (lllf) r = 0) condition 3 . 80 is satisfied, irrespective of the Lewis number. The slope condition 3 .76 is then sufficient for local stability.
a) Numerical Simulation, Results and Discussion for the Porous Catalyst Pellet with Equilibrium AdsorptionDesorption described by the Lumped Parameter Model (LPEADM) Figures 3 . 1 93 .22 summarize the results of transient computations carried out on the nonlinear model by integrating equations 3 .7 1 and 3 .72 from several different initial conditions. Figure 3 . 1 9 shows the phase plane of a pellet with three steady states, corresponding to the parameters a= 2 x 1 05, y= 1 8, /3= 1 and a Lewis number Ls = 6. The upper steady state A is unstable if f3A = 0. All trajectories eventually terminate at the stable low steady state B . Increasing /3A stabilizes the upper steady state. For f3A = 0.5, the upper steady state is surrounded by a significant region of asymptotic stability. Figure 3 .20 shows the phase plane of a pellet with the same steady state parameters a, f3, y, but having a reduced Lewis number Ls = 1 .2 . The upper steady state is now stable with f3A = 0. Trajectories in the phase plane are strongly dependent on f3A· This behaviour is particularly important for initial conditions on the right hand side of the phase plane, which leads to large transient temperature hot spots. From such points
MODELLING AND ELEMENTARY DYNAMICS
" .... :I ...
0
.... " 0.
E
" 
�
c "
0
' iii
c "
E
'0
>
267
1. 8
1.6
. \
.
1 .4
1.2
\
\.
\ .
\
.\
\ \
 ·  · 
\\
. \ \. \.. \
·, ·· ,
.....:
{J A {J A
= =
0 ·5
0 ·0
����� 8 ·:·� ·� 1 .0 ���1.0 0.8 0. 4 0.6 0 01 X
If
......
._
( d i �nsion l�ss c o n c � n \ ra \ ion )
FIGURE 3.19 Phase plane of a pellet with three steady states. Case of high Lewis number. Effect of PA · (Parameters: Ls = 6, a= 2 x 105, {j = l, r= 18).
" .... :I 
0
.... " a.
E
" 
�
"
c
0
' iii c "
E
'0 >
2 0
·· 
1. 8
{JA : 0 . 5
{J A = O · C
1.6 1 .4
'
1.2 1 .0
· ·
0 X
If
0 .2
.....
0. 4
.....
.....
. ...
.....
06
0.8
( d im�nsion l�ss co n c � n \ ra l ion )
B
1.0
FIGU RE 3.20 Phase plane of a pellet with three steady states. Case of high Lewis number. Effect of fJA · (Parameters: Ls = 1.2, a= 2 x 105, fJ = l , y= 18) .
268
S.S .E.H. ELNASHAIE and S.S. ELSHISHINI
... Col �
::J
c
" a.
E
"
�
c
" 0
Ill c Col
E
'0
>
2 .0 1. 8
:\A
\ \
' �
1.6 1.1.
...... c .......
1.2
1 .0
0
X
If
02
0. 4
( d i mension l e s s
06
� � \ 0.8 I
co n c e n t ra t ion )
B
\
1.0
FIGURE 3.21 Phase plane of a pellet with three steady states. Case of high Lewis number. Effect of /JA· (Parameters: L, = O, a= 2 x l05, fJ = l , y= 18).
the derivative dX*Idr is initially negative. This negative term is intro duced into the heat balance equation by the additional coupling, thereby slowing down the initial temperature response. Transient hot spot temperatures, which arise when the additional coupling is not present (/3A = 0), are greatly damped as f3A increases. Starting from initial con ditions on the left hand side of the phase plane dX"Idr is initially positive. The temperature response is now initially accelerated through the additional coupling. This has the effect of bending the trajectories upwards towards the upper steady state A, thus enlarging the range of initial temperatures on the left hand axis from which the upper steady state A can be realized. Figure 3 .2 1 shows the phase plane for the limiting case Ls = O . This case corresponds to the pseudohomogeneous model of Liu and Amundson ( 1 962) . It is apparent that this model fails to properly describe the local stability characteristics and greatly distorts the phase plane. Figure 3 .22 shows a case of unique unstable steady state (limit cycle) for f3A = 0. For f3A > 0, the limit cycle disappears and the steady state becomes stable. In conclusion, we presented in this section a lumped dynamic model of a porous catalyst particle, based upon the activesite theory which
MODELLING AND ELEMENTARY DYNAMICS
....'.., ....

i 1 ·8
   · 
:>
0 :;;
E
2!
� c: 0
16
14
.� l 2
· ;;; c
"" ,..
. 
/3A /J A
= 0·0 = 02
269
a
P:lr:lmclcrs ·
= 2 x 1 04 p = o. • y • 1 0.0 L. "' 20.0
1·0k:���....=;:==;::::: ... ::;: =;:==:==;;:::;: =:;: ::: � :: ::::� : .... � 0 ·12 0 14 0 16 0 18 02 0 0 o.o2 o.o4 o.os 008 01 0 

x � dimensionless
concentration c "/c b
FIGURE 3.22 Effect of heat of adsorption on the stability of a unique steady state.
takes into account the consequences of the important chemisorption processes taking place on the internal surface area of the catalyst pellet. This model differs from the simple model of Liu and Amundson ( 1 962) in two respects: 1. the effective mass capacity is much greater 2. there is an additional heat generation term due to the adsorption
desorption step
Both differences have important conflicting effects on the dynamic behaviour and stability of the steady state. The relatively large mass capacity of the activesite model tends to destabilize the steady state. If the Lewis number Ls of the gassolid system exceeds unity Ls > 1 it is shown that a steady state satisfying the familiar slope con dition
may be unstable. This critical bound of the Lewis number is well within the practical range. The presence of an additional heat generation term due to the adsorptiondesorption step exerts a strong stabilizing influence on the steady state. The simple model of Liu and Amundson ( 1 962) is a special case of the physically more realistic dynamic model presented in this section, when L_, is set equal to zero.
270
3. 1 . 2 . 3
S . S .E.H. ELNAS HAIE and S . S . ELSHISHINI
Effect of nonequilibrium adsorptiondesorption. The Lumped Parameter NonEquilibrium AdsorptionDesorption Model (LPNEADM)
In the previous part, a lumped dynamic model of a porous catalyst particle, in which an exothermic reaction A 7 B is occuring, was pre sented and was based on the activesite theory with the assumption of an ideal surface, equilibrium adsorption of reactant and linear adsorption isotherm. This model was shown to predict profoundly different dynamic behaviour compared to that of a pseudohomogeneous model. In the more general model presented here, allowance is made for finite rates of reactant adsorption and desorption. Because of the large number of parameters involved in this general model, it is not possible to present all aspects of steady state and dynamic behaviour. Instead, relatively few cases are considered which reveal some interesting features of adsorption resistance on the multiplicity and stability of the steady states.
a) The Dynamic Model Consider the simple reaction system,
A+X H A ·X A · X 7 B · X B·X H B+X A 7 B
adsorption surface reaction desorption overall reaction
where X denotes an active site. For low surface coverage and transport resistances confined to an external surface film, mass and heat balance equations for the particle have been derived in the preceding part. The rate of accumulation of adsorbed reactant on the internal surface is given by,
dC.s dt
= ka
(c*c )
s kC m  KA  s C.
(3. 83)
where the adsorbed concentration C.,. is expressed in mol/g catalyst. The rate of accumulation of reactant in the intraparticle void space the catalyst pellet of radius Rp, is given by, (3.84)
MODELLING AND ELEMENTARY DYNAMICS
27 1
where C*, Cs refer to reactant concentration (mollcm3 ) in the intraparticle fluid and bulk phase, respectively. An unsteady state heat balance for the spherical catalyst particle of radius Rp, gives,
where T, T8 refer to the temperature of the particle and the bulk phase, respectively. The rate constant for adsorption ka, surface reaction k and equilibrium constant for adsorption KA are temperature dependent according to Arrhenius expressions,
ka = kao exp ( Ea I RaT) k = k exp (E I RaT) KA = K� o exp (( MI)a I f?c T) 0
where
(MI)a > 0
The initial conditions are:
c* (O) = C0
T(O) = T0
(3 . 86)
b) Steady state equations The steady state equations for the particle are obtained from equation 3 . 8 33 .85 by setting the time derivatives equal to zero. After some manipulation we obtain the following steady state algebraic relations, (3 .87)
c* = ( 1 1 KA ) + (k l ka )CB F(T) 
3
h(T  T8) = (  Mf) T p,kCm CB I F( T) RP
(3.88) (3 . 89)
where 1
k
R p kCm
KA
ka
3k9
F(T) =  +  +
P
s
where T, C, C* refer to steady state values without adding additional
S . S .E.H. ELNASHAIE and S . S . ELSHISHINI
272
subscripts for simplicity. Equation 3 .89 can be solved iteratively to obtain the pellet temperature T. Then C_,., C* are obtained directly from 3.87 and 3.88. It is convenient to transform equation 3.89 into dimension less form by introducing the normalised temperature,
(3 .90) Equation
3.89
can be written as,
(3 .9 1 ) where {t, Q+ represent the heat removal and heat generation functions for single catalyst particle. The dimensionless parameters are defined by,
a=
= Ya = g1
RP Psko KAokCm
kg exp (  yA I Y) Ea I RG TB
f3
3
C ( = !:Jl ) T kg B
hT8
r = E l RG TB YA = r + r
The steady state temperature depends upon five parameters a, aa, {3, Y,1 and YA · A complete presentation of the general case is hardly possible. Instead, a few cases showing interesting behaviour are considered.
c) Special cases of the general model i) Equilibrium adsorptiondesorption
If the resistance to adsorption is negligible with respect to other steps in the overall process (adsorption, desorption processes are very fast relative to the other reaction and mass transfer processes), the concen tration of A on the catalyst surface is in equilibrium with the concentra tion of A in the intraparticle fluid. In this case of negligible adsorption resistance, aa � oo equation 3.91 simplifies to, Y1

a
= f3 exp ( yA I Y) 1 + a e xp (  rA I Y)
'=''' 
(3 .92)
It is well known that there may be either a single solution or three
solutions of equation 3.92, corresponding to three steady states of the
MODELLING AND ELEMENTARY DYNAMICS
273
2  0 .......
1 8
FIGURE 3.23 Multiplicity regions. Equilibrium adsorptiondesorption.
particle, for particular values of a, f3 , YA · equation 3 .92 has been studied in detail for uniqueness and multiplicity of the solution by Aris ( 1 969). A summary of the results is presented in the previous part. Figure 3 .23 shows regions of multiplicity in ( YA· /3) space for various values of a. It is important to note that this simplified model predicts significant regions of multiplicity within the practical range of parameters.
ii) Activated adsorption In the case of activated adsorption, the rate of adsorption increases strongly with temperature, i.e. rc, � 0. Figure 3 .24 shows a two parameter continuation diagram (TPCD) for f3 vs. Yrl where the loci of the static limit points are shown for a case of moderate activation rc, = 8. It is
S.S .E.H. ELNASHAIE and S. S . ELSHISHINI
274
2  4 �r a = 2 • '()� 2.2
r�
=
e
2·0 1· 1·
1·
0 ·8
• D
FIGURE 3.24
Multipl� Solutoons.
Unique Solution
Multiplicity regions. Activated adsorption.
important to note the dramatic effect of activated adsorption on the multiplicity region for relatively large adsorption resistance ( aa small). Multiple steady states can now occur for a surface reaction having zero apparent activation energy ( YA = 0), though admittedly over a narrow range of exothermicity, /3. If the resistance to adsorption is sufficiently high, the multiplicity region tends to lie outside the practical range of parameters. Such characteristic horizontal bands as shown in Figure 3.24, might be termed "adsorption multiplicity", since they arise primarily from the interaction of external film heat and mass transfer resistances and the adsorption resistance. For smaller adsorption resistance the horizontal bands disappear, giving way to the more familiar multiplicity regions, arising from the interactions of physical transport and surface resistances.
MODELLING AND ELEMENTARY DYNAMICS
275
1.9
1.8
1 .7 a
1.6
r.
= 3 . 6 6 • 1 03
=
_
2
1.5
1.4
Ql.
1.3
1.2
1.1
1 .0
0.9
(i) (I) Gl dQ+ dY dY

(3.99)
at the point of intersection of the Q and Q+ curves, given by equation 3 .9 1 . This condition is satisfied by all unique steady states in the mul tiple steady state region. The intermediate steady state in the multiplicity region is always unstable (saddle type). A second condition must also be satisfied and can be written in the form, (3. 1 00)
S . S .E.H. ELNAS HAIE and S . S . ELS HISHINI
278
where A and B are constants which are computed from the steady state and defined Appendix C. It can be shown that B is always positive, but A may be positive or negat i ve, depending on the particular steady state. If A < 0, condition 3 . I 00 is redundant and the slope condition 3 .99 is sufficient for stability. On the other hand, if A > O, the Lewis number Ls must also be less than some critical value Lscn g iven by (3. 10 1 ) The critical Lewis number can be readily computed from the steady state. No general conc l usions can be drawn regarding the effect of system parameters, on the stabi li ty of the steady state because of the large number of parameters . However some useful ind i cations will be presented later regard ing the effect of the rates of adsorption on the stability of the catalyst pel l ets ,
.
g) Numerical simulation, results, discussion and stability results for porous catalyst pellet described by the Lumped Parameter Model. The NonEquilibrium AdsorptionDesorption Model (NEADM)
i) Nonactivated adsorption Unique steady state
Table 3.7 s h ow s the effect of decreasing the rate of non activated adsorption (Ya = 2) on the critical Lewis number for stability of the unique steady state. Three values of the exothermicity factor for adsorption (/311) are considered (/311 = 0, 0. 1 , 0.3). The value of f3a = assumes no heat of adso rptio n the entire heat generation being due to surface reaction. The third entry /3= 0.3 assumes 
,
Table 3.7 Critical Lewis numbers for nonactivated adsorption. Unique steady state a = 1.35 x lOZ, /3= 0.6, r = lO, YE = 5, Ya = 2. Critical Lewis Number, Lscr
0.25 0.2 0. 1 8 0. 1
24.9 1 3 .7 1 3 .5 1 3 .5 1 7 .0
37.4 27.7 30.3 32.5 30 1
f3a = 0.3
stable for
all
L,.
MODELLING AND ELEMENTARY DYNAMICS ·
1 8 y...., Cl(l S l f'Ody stolE'  oo
··••
1 .6
A, slab\t'
0 · 2 C ' un slablf'
0·02
B ' s \ ab\ 1  8 41
:;
�1 6
�
�
CD
13a      f3a 
P a r a m e t er s :
3.66 •
YE
4
y �
= 0 =
"
06
Ya
L,
lla
14 � 0
16
J (}J
1 .3
:
2
4
0.3
::1
"' c: 41
E 'ij 1 2
A 0 2
FIGURE 3.27 state.
04
06
dime n s ionless s u r face X s ( X s = X s e l'E )
0 8
1 ·0
concentration
Effect of heat of adsorption on the stability of the high steady
MODELLING AND ELEMENTARY DYNAMICS
281
Table 3.9 Cridcal Lewis numbers for activated adsorption. Multiple steady states a= 2 x 10S, /3= 1 .6, r = S, YE = O, ra = 8. Critical Le wis Number, L ,cr
Low conversion state
1 00
f3a
=
0
stable for all Ls
stable for all Ls
High conversion state
4525
9436
ii) Activated adsorption It i s of interest to examine the stability o f the steady states in the adsorption mul ti plicity region of Figure 3 .24 for activated adsorption. The low conversion state is locally stable for all Ls, irrespective of f3a, as before. The high conversion state is also essentially locally stable, since the critical Lewis numbers are qu ite high. 3.1.3
Porous Catalyst Pellets. Distributed Parameter Models (symmetrical)
The lu mped parameter model presented in section 3 . 1 .2 provides a useful step towards an understanding of the general behaviour of the porous catalyst pellet. However, it is limited in its v al i dity since the true nature of the problem is distributed and internal concentration and temperature gradients have very important effects on steady state as well as the dynamic behaviour for the catalyst pellet. For example, the lumped parameter model predicts multiple steady states for cases for which the distributed system gi ve s a unique solution (Luss, 1 97 1 ) . Both steady state and dynamic analysis o f the distributed parameter model are more di ffic u l t than for the lumped parameter model. The steady state equations of the lumped parameter model are algebraic, those of the distributed model are ordinary differential equations of the twopoint boundary value type. The dynamic behaviour of the lumped model is described by initial value ordinary differential e quation s which can be integrated by standard subroutines. The corresponding de sc ription of the distributed model is in terms of partial differential equations which are more difficult to solve. For the sake of clarity we present the develop ment of the distributed model equations before we proceed to review the main findings reported in the literature re g arding this problem. 3. 1 . 3. 1
Th e Dynamic Model
Consider a spherical porou s catalyst particle i mmersed in an infinite gas
282
S . S .E.H. ELN ASHAIE and S . S . ELSHISHINI
medium of uniform temperature and concentration. We shall assume that intraparticle mass and heat diffusion can adequately be described by Pick' s and Fourier' s laws respectively. We al so assume that diffusion occurs only in the radial direction and that diffusion is i sotropic, i.e. all points with the same radial di stance from the center are having the same temperature and concentration. We shall present a first order irreversible reaction catalyzed by the solid surface following the steps, A+X H A ·X
adsorption surface reaction
A · X t B · X
B·X H B+X A t B
desorption
overall reaction
where X denotes an active site. A mass balance on the internal surface within a differential spherical element of radiu s r and thickness dr gives,
dt
a c,
 = ra  r.s
(3 . 1 02)
where C· is the adsorbed surface concentration of A in mollg catalyst,
is the rate of adsorpti on of re actan t A in mol/g catalyst s and rs is the rate of surface reaction in mol/g catalyst s. According to the active site theory, ra
r,z r�
=
ka( �:)
= kCs
c* cv 
(3 . 1 03 ) (3 . 1 04)
where C" is the local c oncentrati on of reactant A in the intraparticle gas above the surface (mol/cm3 ) and Cv is the concentration of vacant sites (mol/g.catalyst). If, for simplicity, we further assu m e that the number of acti ve sites occupied by the reac tant A is much smaller than the total number of active sites and that the product B is instantaneously desorbed so that the number of acti ve sites it occupies is negligible, then equation 3 . 1 03 can be written as, (3 . 1 05)
MODELLING AND ELEMENTARY DYNAMICS
283
where em is the total concentration of active sites (corresponding to a complete monomolecular layer on the catalyst). For a spherical pellet of radius Rp and voidage E the rate of accumulation of reactant in the intraparticle void space of the pellet can be written as,
(3 . 1 06) where De is the effective diffusion coefficient of A within the porous structure. From 3 . 1 02 and 3. 1 06 we obtain,
(3 . 1 07) If we assume equilibrium adsorption des orption to be established between
the intraparticle gas and the adsorbed gas according to a linear isotherm, then
(3 . 1 08) As before we shall take an averaged value of KA over the range of temperature . This allows us to differentiate 3 . 1 08 into the simple form,
dt
() Cs
= KA
dt
C () C* m
(3 . 1 09)
From equations 3 . 1 07, 3 . 1 08 and 3 . 1 09 we obtain the mass balance equation in terms of the single concentration variable C* .
(3 . 1 10) A dynamic heat balance on the particle gives,
where A., is the effective conductivity of the catalyst pellet. The second and third terms on the right hand side of eq u ation 3 . 1 1 1 account for the rates of heat generation due to surface reaction and reactant adsorption respectively . Eliminating ra between equations 3 . 1 1 1
. ,
284
S .S.E.H. ELNASHAIE and S.S. ELSHISHINI
and 3 1 02 using the relations 3 . 1 08 and 3 . 1 09, equation 3 . 1 1 1 becomes,
where (Ml)T represents the overall heat of reaction (heat of reaction + heat of adsorption) and is given by,
and
k is the intrinsic reaction rate constant given by
It is convenient to write equations 3 . 1 1 0 and 3 . 1 1 2 in the dimensionless
(
form (4
ax *
+L
J 2 x*
2 ax*
= d oi + m d m ,) dr
)
cp exp ( y( l  1 1 Y )) X 2
•
(3. 1 1 3)
where,
(J) =
L
Y = _I_
r
Rp
Trf
Ae E
DePs Cps 2 exp (  y) 2 Rp koKA CmfJ,  cp = 'De "
=
De ( fir =

M/)r Cif
A. Jif

et
A r = =RP2"p, CP,
Ls = AePs KA Cm

DepsCps
r= fJA
=
E
RcTif
De(  Ml )A cif ?. Jif
where C,t and Tif are arbitrary reference concentration and temperature respectively.
MODELLING AND ELEMENTARY DYNAMICS
285
Equations 3 . 1 1 3 and 3 . 1 1 4 are subject to the following boundary conditions, at
w = O.O
(3. 1 1 5)
which is the symmetrical center boundary conditions, at
W = 1 .0
(3. 1 1 6)
where,
Rh
Nu = P
Ae
Initial conditions are, X * ( w, O) = X1 ( co ) *
(3. 1 1 7)
The dynamic behaviour of the system is described by the set of equations 3 . 1 1 3 3 . 1 1 7 . This model differs from the ones usually reported in the literature (pseudohomogeneous model) by the inclusion of the important adsorption mass capacity parameter Ls and a transient heat of adsorption term in 3 . 1 14. This difference does not affect the steady state, but will have important effects on the transient behaviour. The pseudo homogeneous model can be derived from this more general model by setting Ls = 0.
3. 1 . 3. 2
Steady state
The steady state equations are obtained by setti ng the transient terms in equations 3 . 1 1 3 and 3 . 1 1 4 equal to zero. The resulting equations are similar to those reported in the literature.
V2x * = ¢ 2 exp ( y(l  1 1 Y)) · X * V 2 Y = ¢ 2{3T exp ( y(l  1 / Y) ) · x*
(3 . 1 1 8) (3 . 1 1 9)
with the boundary conditions given by equations 3 . 1 1 5 , 3 . 1 1 6, and where,
286

S . S .E.H. ELNASHAIE and S . S . ELSHISHINI
v =
2
,
d2
a
doi
d ro dro
+·
where a = O, 1 , 2 for sl ab c ylinder and sphere respectively. The effectiveness factor is a fre qu ent l y used factor which is conve niently used to express the rel ati vely large number of processes taking place within the catalyst particle and its interac tion with the surro unding in terms of one number. The effectiveness factor 17 is defined as the ratio between the actual rate of reaction and the rate of reaction if all diffusional resistances are negligible, that is,
I
1 .0
(a + 1)
0
17 =
(/)
a
· exp ( y(l  1 1 Y)) ·
,
x* . dro
(3 . 1 20)
exp ( y( l  1 / Y8 )) · X8
For a spheric al pellet for ex ample the effectiveness factor will be, 3
J
1 .0
2 ro · exp ( y(l  l / Y)) · X * · dro (3 . 1 2 1 )
17 = �0�exp ( y(1  1 I Y8 )) X8 ·
There are a number o f formulae for 17 whic h can easily be derived from the ab o ve expression (3 . 1 20) togethe r with the differential equ atio ns and the boundary conditions of the system. For example, for the spherical catalyst pellet, the mas s balance differential equati on can be written as,
( J ,I ,
* d 2 dx  ro ro 2 d (/) d(/) 1
=
2
¢ · exp ( y( l  1 / Y)) · X
*
which upon re arran g ement and integration can be writte n as
dro
w=I
Therefore 17 c an be
=
qy
1 .0
*
ro  exp ( y(l  1 / Y)) · X · dro
0
written as,
·
,
(3 . 1 22)
(3 . 1 23)
287
MODELLING AND ELEMENTARY DYNAMICS
Also from the boundary conditions where effectiveness factor can be written as,
Sh
and
Nu
are finite, the
(3 . 1 25)
i
d
A comprehe n s ve study of the stea y state behaviour is g i ven by Cresswell
d
( 1 969) . For the spe c ial case of Nu 7 oo, Sh 7oo, the s te a y state equa tion s can be decoupled u s i ng the adiabatic relation,
3. 1 . 3 . 3
Y( w) = Y8 + {3T ( X8  x* ( w) )
(3. 1 26)
Brief survey of the 11Ulin investigations on the s ubject
d
We now proceed to present briefly, some of the main fin in g s reported porous catalyst particle using the pseudohomogeneous model (L.. = O).
in the literature regarding the st e a dy state and dynamic behaviour of this
a) Uniqueness and multiplicity of the steady states Gavalas ( 1 966 ), u s i n g
modem top ologic al techniques, was the first to t
obtain sufficient conditions u nder which a unique steady s ate exists for
a chemical reaction in a porous catalyst particle. Luss and Amundson ( 1 969), who c on i n e their attention t o cases with zero he at and mass transfer e s i s ta c es between the p arti c l e and its environment (Nu 7 oo, Sh 7 oo ), were able to obtain similar, but rather less conservative , conditions for uni quen e s s using the s p ectral theorem of the SturmLiouville (Morse and Feshback, 1 953). Subse quently, Luss ( 1 968) s arpened the conditions which guarantee unique ness for particles of all sizes. Cresswell ( 1 970) derived neces sary and sufficient conditions for mu ltip l i ci ty when only internal te mperature gradie ts are ne g lected . Jackson ( 1 972) derived sufficient conditions for uniqueness for catalysts of infinite slab geometry with nonzero heat and mass trans fe r re s is tance s at t e u ac e . The c n stru c ti o n of maximal and min m al solutions for the s t e ady state solutions and effectiveness factors has received some attention. Using the maximum principle for e ll pti c differen t a equati n s , Cohen ( 1 97 1 ) obtained maximal and mi n im al solutions for the problem of c e m i c a l reaction occuring in an adiabatic tubular reactor, in the sense
f d n
r
h
n
o
h s rf
i
i
h
q i
il
o
that any solution of the e u at on describing the temperature profile is never larger h a the maxi mal solution and never smaller than the minimal solution.
t n
288
S . S .E.H. ELNASHAIE and S.S. ELSHISHINI
Vanna and Amundson ( 1 973) used a similar procedure to obtain maximal and m inimal solutions for chemical reaction occuring in a catalyst particle of arbitrary shape with zero external resistances (Nu f oo, Sh f oo ). They showed that these maximal and minimal solutions actually give rise to maximal and minimal effectiveness factors for the exothermic case. If there exists a unique solution, the maximal and minimal effectiveness factors coincide. Villadsen and Stewart ( 1 967) introduced the orthogonal collocation techn ique as an effective method of solving the system equations (both steady state and transient equations). The same authors developed a graphical method based on their collocation approach, for the calculation of multiple steady states and effectiveness factors (Stewart and Vittadsen, 1 969). Paterson and Cresswell ( 1 97 1 ) used the collocation method together with the concept of "effective reaction zone" to develop a simple, yet reliable, method for the computation of effectiveness factors.
b) Stability analysis of the steady states In c ontrast to the
lumped parameter models there is no general approach to the stability problem of distributed parameter systems. Hlavacek et al. ( 1 969) used a finitedifference approach to analyze the transient behaviour and stability of the catalyst particle with zero external resistances to mass and heat transfer (Nu f oo, Sh f oo). Luss and Lee ( 1 970) studied the same problem using the "Galerkin" method to compute the eigenvalues of the linearized equations. Wei ( 1 965) studied the stability problem by means of a Liapunov functional. This stability study was confined to adiabatic perturbations (perturbations that satisfy the adiabatic relation 3 . 1 1 8), thereby allowing the decoupling of the governing differential equ atio n s. Luss and Lee ( 1 968) used the maximum principle for parabolic partial differential equations to determine finite stability regions of the steady states. Although these results are the best reported so far, the method is severely limited to a single partial differential equation aris ing from the rather restrictive assumptions of unit Lewis number and adiabatic perturbations. The more general case of arbitrary Lewis number was treated by Kuo and Amundson ( 1 967), who posed the stability problem as a nonself adj oi nt s pectral problem, but the sufficient conditions for stability involved the computation of eigenvalues by vari ati o nal techniques. Padmanbhan et al. ( 1 97 1 ) questioned the vali d i ty of most of the earlier work wherei n the analysis was restricted to adiabatic perturbations . They showed that this re stricti on yields only "conditional" stab i lity .
MODELLING AND ELEMENTARY DYNAMICS
289
further showed that this restriction i s to tal ly unnecessary and that c arefu l ly chos en Liapunov functional e s tab lishe s previous results without the as s umptio n o f adiabatic pert u rb at io n s . Vanna and Amundson ( 1 972) us in g the co mparis on function approach, as de v el oped by Kasten berg ( 1 967), obtained sufficient conditions that guarantee g l obal asy mptot i c stab il it y of the s te ady state. They also obtained anal yti cal bounds on the growth rate of a perturbation from the ste ady state . The method was sho wn to handle the case of v an ish i ng initial enth alpy (adiabatic perturbations) as well as that of the non van ishi ng residual enthalpy . Denn ( 1 972), using a variational appro ac h t o Ly apunov st abil ity , obtained an estimate of the largest perturb ation for which a s te ady state can be shown to remain stable. He was al s o ab l e , using "Fourier method", to o bta i n an estimate of the s malle s t pe rturbatio n which will cause in st ab ili ty . For a model proble m, he showed that the region of uncert ai nty between these estimates is small . McGowin and Perlmutter ( 1 97 1 ) combined the Lyapunov method with the co l locati on method to generate re gions of asymp toti c stability for the s te ad y states of di strib ut ed parameter s ystem s . They
the
c)
use of a
Numerical techniques for the solution of the model equations
( 1 97 1 ) used the o rtho gonal collocation method to solve both the steady s tate and dynamic equations of six diffe ren t models of the porous particle of i n cre as i ng c omp l e x i ty . He found that only 8 collo c ati on p oints were necessary to obtain accurate results. This leads to a considerable s avin g in c omputi ng ti me c ompare d with the conventional finitedifference methods such as C ran kNi c o l s on . Ferguso n an d Fi n lays on ( 1 970) used the colloc ati on metho d to study a similar probl em . They proved that the method converges to the e x ac t s o lution an d th e y d e mon s trate d c learl y its s uperio ri ty comp ared with the more conventional finitedifference methods. Hansen
d) Experimental investigation In co ntrast to the vast number of theoretical studies, experi me ntal work is relative ly scarce. Hughes and Koh ( 1 970) studied the steady s tate and d ynamic behaviour of a s i ngl e pellet catal y zing the hyd rogenati o n of eth ylene . Kehoe and Bu tt ( 1 972) stu died the hydrogenati o n of benzene , with more emphasis on the modelling a s pects . No m u l tip l e s te ad y states were encountered in e ithe r of these studies . Furusaw a and Kunii ( 1 97 1 ) studied experimentally multiple steady states for the catalytic hydro
S . S .E.H. ELNASHAIE and S . S . ELSHISHINI
290
genation of ethylene on a single pellet of Adkins catalyst. The authors also discussed the effect of initial conditions on the steady state activity. 3. 1 . 3. 4
Application of two numerical techniques for the solution of the dynamic model equations for the distributed parameter, porous catalyst pellet
a) Finitedifference techniques We shall first present the solution of the dynamic equations using the finitedifference method (CrankNicolson). Figures 3 .28a,b show the temperature and concentration profiles at different times for both cases of f3A = 0 and f3A > 0. For f3A = 0 the temperature rises slowly to approach the steady state profile. About I 00200 finitedifference points were needed in this case to obtain 1 ·0 5 2
1  04 8
1044
1  040
1 03 6
a  o  o2 I I fJA St•ad y sta1• 7/
 ·   f1A =
0

I I
;
//
�,=0J ___,
1
Nu
Sh � liT L5
L.
= = =
= =
=
(W,O)
y (11),0)
12.5
25 0
1 .0 0.1 40.0 I0 0 =
=
0.0 1 .0
B u l k conditions: Yo
1 · 02 8
=
Initial conditions :
x•
x8
1  032
>
Paramete r s :
= =
1 .0 1 .0
1 · 024
1·020
1·016
1 · 0 12
_ L  t :0 ·3
1 008
1004 1 000+''�"t..,�r�1 0·8 0·6 0 ·4 02 00 1 ·0 w 
FIGURE 3.28a Internal temperature profiles inside the porous particle. Effect of /JA·
MODELLING AND ELEMENTARY DYNAMICS
29 1
1 0 Nu
0. 9
Sh
•
llT
0. 8
PA y
L,
0. 7
c
t
X
Lv
I n i tial
0. 6
= = = =
1 2.5
25 0 1 .0
=
0. 1
=
40
=
=
0.02
condition s :
X0(1U,0) y ( W,O)
0. 5
1 0
0
= =
0
B u l k condition s :
0.4
XB YB
0.3
=
=
1 .0
1 .0 1 .0
0. 2 0 1
lO
FIGURE 3.28b different times.
0.8
0. 6
0. 2 0.4 W +
00
Internal concentration profiles inside the porous particle at
accurate results. For f3A > 0 a sudden increase in temperature near the surface takes place at early times. About 500 1 000 finite difference points were needed to get accurate results in this case. These results show an example of the important effect of f3A on the dynamic behaviour of the system. However, to investigate the system in some detail we shall present a more efficient numerical method of solution. b) Orthogonal collocation technique
Motivated by earlier success in the use of the orthogonal collocation method (nonporous particle) as well as the published results of Hansen ( 1 97 1 ) and Ferguson and Finlayson ( 1 970), we turn to the use of the orthogonal collocation method. The principles of the orthogonal collo cation method have been discussed in section 3 . 1 . 1 , and in more detail in Finlayson' s text ( 1 972). Equations 3 . 1 1 3 and 3 . 1 1 4 can be written as,
292
S . S .E.H. ELNASHAIE and S . S . ELSHISHINI
( Lv + Ls )
ax * ar
= V 2 x *  f(Y' x * )
(3 . 1 27)
and (3 . 1 28) where
f (Y, x* ) = 1 0 it becomes increasingly more difficult to determine BiJ and Aij by the matrix inversion proc e du re of Villadsen and Stewart ( 1 967). Consequently an alternative method is used as described by Michelsen and Villadsen ( 1 972). Equations 3 . 1 2 1 and 3 . 1 22 form a set of 2N equations in 2N + 2 depende nt v ari able s .
293
MODELLING AND ELEMENTARY DYNAMICS
X�+l and
Therefore w e eliminate 3 . 1 23 and 3 . 1 24 to get,
C4. + L, )
_L dY. }
dr
=
i=l
..
1 BN+ l,j XB  Sh
N
d; = L Bu x;· +
dX*
i=l
and,
N
.(
(
BN+I.j YB Bij Y; +
(
A
YN+l from 3 . 1 2 1 and 3 . 1 22 using
1
N�
N+I,N+I
Nu
(
•=
+1
N
AN+l, N+I• = l +1
Sh
LI �.N+I Y; N
L �.N+ I xi*
)
)
)
)
;
 f( fj , X )
(3. 1 33)
dX*
+ f3Tf( lj , X; ) + L,f3A J Jr (3. 1 34)
which is a set of 2N equ ations with 2N dependent variables and N The su rface temperature and concentration are given by,
) = 1 , 2,
,
.
(3 . 1 35)
and,
(3. 1 36)
The initial conditions are,
lj (O) = }j0
j = 1, 2,
..
,N
In order to check the convergence of the collocation technique, the number of collocation points N is increased successively until the solution stops changing within some specified accuracy. A further
S.S.E.H. ELNAS HAIE and S.S. ELSHISHINI
294
1 · 08 1 ·0 7
 N = S  ·  · 
C/N
=
N : 9�9 ; C / N 5 0 0 stPpS C r a nk  N;colson
1  06
l
Nu
Sh �T
��
1 05
>
P a r a m e 1ers :
L� L�
t
1.04
::
y
0·1
=
•
= =
=
• •
=
1 0.0
0
1 2.5
250 0. 1
0 05 •o.o 1 .0
Initial conditions: x•
((U,O)
y ((11.0)
;
•
0.0
1 .0
Bulk conditions:
XB
103
Yg
1 ·02
•
=
) .0
1 .0
1 01
\ \ 1 00 +....:>,�...��� 1·0 o.s 06 0·4 0·2 0 w 
FIGURE 3.29a Comparison between the finitedifference and the collocation techniques. Temperature profiles. High Lewis number.
check is made by comparing the results with those obtained from an alternative technique, in this case Crank Nicols o n s technique. Many cases have been tested and in all cases, convergence was obtained and good agreement was found between both techniques. Some sample results are shown in Figures 3 .29a,b and 3.30a,b. 
N: 5
l *x
08
N = 9,19; C/N 500 steps
'
Par ameter s : L,
Lv
Nu Sh
liT
0·6
�A y
0·4
=
• =
= =
l0
0
12.5
BO
0. 1
= 0.05  40.0 =
1 .0
Initial conditions: •'
(W.O) y (W,Q)
0·2
=
�
0.0 1 .0
Bulk conditions:
0 . 0 f:lo1oo,"""T,...:::..r.r,....l 0 ·2 0·4 0 0 ·6 0· 8 10 w 
x8
y8
• "'
LO
1 .0
FIGURE 3.29b Comparison between the finitedifference and the collocation techniques. Concentration profiles. High Lewis number.
MODELLING AND ELEMENTARY DYNAMICS
1 · 06 .. N: 3 N :
1·05
5 ,9 ; C I N 500
Sl�ps
t
>
Para m e ters : L, L., Nu
Sh liT �...
1 04
295
r
'
.
1.0 0
� z
l l.j
. ; . . ;
250 O. J O.Oj
40.0 1 .0
Initial condi[ions:
1  03
•* p s
 N:
.
12.5
250
0. 1
o.os
40.0 1.0
•
�
1 .0
0
Initial conditions :
x• (W,O) y ( W,O)
� 0·6
0·4
w 
=
0
1 .0
Bulk conditions: 'B
I : 0 02 :>,.. .
=
YB
0 ·2
. .
1 .0 1 .0
0
FIGURE 3.30b Comparison between the finitedifference and the collocation techniques. Concentration profiles. Low Lewis number.
S . S .E.H. ELNASHAIE and S . S . ELSHISHINI
296
this case. As the profiles approach the steady state the number of collocation points can be further reduced. In the neighbourhood of the steady state as few as 13 collocation points are sufficient is most cases, except for very steep internal gradients in the region of mass transfer controL The collocation technique in all cases required only 1 020% of the computing time, for the same accuracy of the CrankNicolson technique. 3. 1.3.5
Compact presentation of steady state results. The effectiveness factor  Thiele modulus diagram
It is u s ual to account for the effects of transport resistances on the steady state rate of reaction by examining the effectiveness factor of the catalyst. The effectiveness factor, defined as the ratio of the observed steady state rate of reaction to the rate at bulk conditions of concentration and temperature, is given by T7
= f/J 2
•
( ) dx*
3
exp ( y (l  1 I Y8 )) · X8
dm
(3
.
1 3 7)
ss, w=l
where subscript ss denotes steady state value. Without any loss of generality, if the bulk concentration and temperature are used as reference concentration and temperature, then X8 , Y8 1 and T7 can be written as, T]  
3
 f/J 2
( )
=
dx* dm
(3 . 1 3 8) ss,w= l
effectiveness factor T], is plotted as a function of Thiele modulus ¢, for given y; fJr, Sh, Nu, the resulting curve is in general, made up of fo ur regions as shown in Figure 3 . 3 1 . Region 1 corresponds to kinetic control and T7 = 1 . The reactant concentration is uniform throughout the pellet and the pellet is at the same temperature as the bulk gas. For an increased reaction rate, the effectiveness factor falls below unity as a result of "pore" diffusion resistance (Region 2). The heat generation is removed rapidly enough to keep the pellet at a similar temperature to the bulk gas. For even higher rates of reaction (Region 3) the particle becomes progressively hotter than the surrounding gas, although reasonably uniform in temperature. Pore diffusion becomes more pronounced an d most of the reaction occurs in a thin shell close to the catalyst surface. In this range of conditions multiple steady states can occur. I f the

Finally for extreme rates of reaction, the particle becomes so hot that
MODELLING AND ELEMENTARY DYNAMICS
297
.... "l) c .. u
�
dilt u•ion •ffr c t s
FIGURE 3.31
I
I 1
I I
�ng
" pore·· dif1 us1 on r P s i �t.a. n c f
I
I �niplo I
s t f' ,i dy s\.l1ts
I I
I
� �m �
mm l r.ansf�r rct\1!' c o n t rolling
Effectiveness factor Thiele modulus diagram.
the reactants are consumed as they reach the external surface. In this region (Region 4) the supply of reactants becomes the limiting step and mass transfer through the external film c ontrols the rate of reaction. The generalized effectiveness factor diagram of Figure 3 . 3 1 represents a convenient basis with which to explore dynami c behaviour. Severa] cases are presented for partic1es with steady states in Regions 1 , 2, 4 . N o cases are presented in Region 3 ( mu lt i ple steady state region). Particular emphasis is placed on pres entin g the effects of dynamic parameters Ls and f3A·
3. 1.3.6 The effect ofadsorption heat release on the dynamic behaviour of the catalyst in different regions of the 71  ¢ diagram The results shown in section 3 . 1 .3.4 establish clearly the validity and e ffi c ien cy of the coUocation method. We therefore proceed in th i s section to investigate the effect of adsorption heat release in more detail using the collocation method.
298
S . S .E.H. ELNASHAIE and S.S. ELS HISHINI
1 08 ..,.., 1 07
 PA = 0 ·05  ·  ·  /JA : 0 · 0 � •  • � stt"ady statt"
1 . 06
f
>
lOS
Parameters : s.o Ls 0.0 Lv Nu 5.0 250.0 Sh 0. 1 ItT = 1 0.0 "( = 0.2 � I nitial conditions : X0 y
l0 4
((61,0) (W,O)
=
=
0 1 .0
Bulle conditions: )(B
1 ·03
YB
=
=
1 .0
1 .0
102 l01
0·4 0·6 w +
0 ·2
0
FIGURE 3.32 Effect of heat of adsorption on temperature profiles during start� up. Very low Thiele modulus.
a) Startup startup some numeri c al examples are shown, for different values of Thiele modulus.
For
i) Kinetic rate and pore diffusion control region (Regions 1,2)
Figure 3 .32 is for the case of kinetic rate control ( Re gi on 1 ). The effect of ads orpt i on heat release is to cause a sudden increase in temperature near the surface at e arly times. This is due to rapid concentration increase o n the surface at early ti me s . The temperature peak near the surface travels towards the c enter of the partic le by con ducti on . The concentration wave travels towards the center by diffusion c ausing ads orpt i on heat release at e arl y times and reaction he at release at later times . This se quence of events explains the fas t temp erature i ncre ase i n s i de th e p arti c le when the heat of adsorption is t ake n into consideration , compared with the case of /3A 0. Figure 3 .28, presented earlier c orre spo n d s to a case of pore di ffus i o n =
control ( Regi on 2). The effect of adsorption heat re lease is similar to that in Figure 3.32.
MODELLING AND ELEMENTARY DYNAMICS
0 5 2 .,
1
1  04 8 1 04 4
1 ·040

{34 : 0·02
 ·  ·  {JA
P a.ra m e l cr s : Sh
liT y L,
Lv
a
x•
y
l l.S
250  10 "
=
c
. 
I n itial
0.1 40
1 .0
0
co nditions :
( W,O) (W,O)
•
•
0.0
1 .0
B u l k conditions: x8
YB
1 032
>
Nu �
:0
1 03 6
1
299
•
1 .0
10
028
1024
l020 1
016
,. . 
1 · 012
�:]
l008 l004 1 · 000 fo""T=.:.=jlo:c=;,.....L..,,,.,..4 0 0 2 0 ·4 06 0.8 1 0 w ..
FIGURE 3.33a Effect of heat of adsorption on temperature profiles during start up. Low Thiele modulus.
Figures 3.33a,b show a case similar to that in Figure 3 .28, but for lower Lewis number. The effect of adsorpti o n heat release decreases as Lewis number decreases. Figure 3 . 34 shows a case in Reg ion 2 but with a hi gh temperature initial co nd i ti on . The temperature of the system in this case fal ls down w i th time towards the steady state. The effect of ad s orption heat release is to s low down the fall in temperature as well as to eliminate the slight oscillation around the steady state experienced when
f3A
=
0.
S .S.E. H . ELNASHAIE and S . S . ELSHISHINI
300
0 ·9
··

0· 8
BA /JA
= 0· 0
: 0·02
•
� �
Sh
=
L,
�
0
llT y
L,
= =
=
12 '
250 I , 0
0, 1 0
40
1 .0
l n i 1 i.a l cond i t i o n s :
0· 7
l
P a r 1 me1�rs:
Nu
, . ( W.O) y f W.Ol
0 6
�
•
0 . 1)
LO
Bulk. c o n d i t i o n s :
•e
Ye
05
=
1 o 1 .0
X
0·3 0· 2
I
0.1
: 0 · 05
o . o+��....,..._..r1 lO 0·8 0·4 02 0 0·6 W +
FIGURE 3.33b Effect of beat of adsorption on concentration profiles during startup. Low Thiele modulus.
Figure 3 . 35 rep resents a case of high temperature and high concentration initial conditions (the particle is initially saturated with reactants). For this case, when fJA = 0, the temperature rises rapidly to a high temperature and then starts to fall towards the steady state. The effect of adsorption heat release is to slow down this early temperature rise. This is due to th e fact that the rapid temperature increase in this case is accompanied by a decrease in con centration Therefore aX" Ja r is negative and the heat release due to adsorption is negative. .
ii) External mass transfer control reg ion (Region 4) Figu re 3 .36 sho ws a case of mass transfer control. At early times, the concentration buildup c au s e s a temperature increase due to adsorption
heat release for f3A > 0. The temperature profile for f3A > 0 lies above that for f3A = 0. At r= 0.5 the situation is re verse d This is due to the fact that the high temperature at early times for f3A >0 has the effect of accelerating .
MODELLING AND ELEMENTARY DYNAMICS
1  6 5 ..,..  ,94 : 0  05
1·
60
·=��_!.!< :
_O · O
1  55 1 · 50
301
P:lf� m c: te r s : �
1. ,
.. .
. .
Nu
=
•
y
0. 1 1 0
soo
=
Initial ••
2.0
=
liT
Sh
3
0
1 0
conditions:
( (.&).0)
y ( W,O)
=
=
0.0
1 .6
B u l k condi11ons :
1  40
t
:>
x8
Ya
=
1 .0
1 .0
1 · JS 1 · 30 1 · 25 1  20 1 15 , 10 1  05 1 ·00 +�.,�.�.........� 1·0 0·8 0·6 0·4 0 ·2 0 W +
FIGURE 3.34 Effect of heat of adsorption on temperature profiles during start up. Low Thiele modulus. High temperature initial conditions.
consumption of reactants. Therefore the rate of reaction decreases at later time s . It is import�nt to notice that the dynamic profiles retain very nearly the same shape thro ughout the response , unlike those in Region 1 , 2 . From the preceding results it is clear that the effect of f3A on the tran s ient behaviour in regions 1 and 2, is di fferent from that in region 4. This difference is due to the high rate of reaction in region 4. This high rate of reaction has two major effects.
S . S .E.H. ELNASHAlE and S . S . ELSHIS HINl
302
2·1 2·0
 · ·
f3A :Q.Q5 . · ·  ·  ·  · f1 A = 0·0 /
/
P a r ::. m e a e r s : �
L,
•
� Nu Sh y
y
0
5. 1
0. 1 I0
K
�
soo
I0
l n il 1 a l
x•
s
�
L,
c o n.d l l i iJ n s ;
( W.O)
( W.O)
=
�
I 0
1 6
B u l k �.:ondu tcn s :
•u
Yn
=
1 .0
1 .0
1· 2 1.1 1 . 0 +�.�.��.�......; 1·0 0·8 0·6 0 ·4 0 ·2 0 w .
FIGURE 3.35
Effect of heat of adsorption on temperature profiles during start up. High Thiele modulus. High initial temperature and concentration.
1 . high rate of heat production due to reaction which makes adsorption
heat release the secondary process for heat production, 2. high rate of consumption of reactants which decreases accumulation and therefore decreases the transient adsorption heat release. b) Response to disturbances in the bulk phase
i) Kinetic rate and pore diffusion control ref?ion (Regions 1, 2) Due to the similarity in behaviour, with respect to f3A , between regions 1 and 2 we present only one example in region 2.
MODELLING AND ELEMENTARY DYNAMICS
2 . 0 ����=�. S t eady s t a t e
/3A = 0 · 0 fjA : 0  0 5
    
1·9
1·6
··
. ,·· ·  · ··  · 
 ·  ·  · · · ·
1·7
303
P a r ameter s . 10 y "' 0. 1 JiT N u .. 1 0 Sh 250 6.0 • L, 5.0 L, 0.0 I n i t i a l condit i �.> n s : ( W.O) y ( W,O)
x•
1·6
=
=
0.0 1 .0
B u l k c ondit i O n s : ·�
y8
=
I 0
1 .0
1 4 .
1·3
1 .2
1 0+....r1 ·0 0 ·8 .
.,..,r. ..,...;..; � 0 ·4 06 0 ·2 0
.......
FIGURE 3.36 Effect of heat of adsorption on temperature profiles during start up. High Thiele modulus.
Figure 3 .37 shows the system response to a positive square wav e disturbance in the bulk temperature of 0 . 1 uni t s for a period 'fp = 1 .0. Th e temperature of the particle tends to rise more slowly initially if /3A > 0. This is because the i n traparticle concentration of re actants is falling in this peri od (i.e. dX"Id r 0 de cre as e s d Yld r. During thi s period the particle temp erature profile tends to move to a new steady state corresponding to Xn = 1 .0, Y8 = 1 . 1 . When the disturbance is removed, the temperature profile returns to the original steady state. The particle cools down more slowly if fJA > 0. This can be ex pl ai ned by similar reasoning as before . Slight
S . S .E.H. ELNASHAIE and S . S . ELSHISHINI
304
fJA ;:Q.QS fJA ;: Q . Q

 ·  · 
1 ·28
1 · 24
l
5t f'ody
•• 
Parameters: =
L,
�
L
1 ·20
IYl y Nu Sh
0. t
I0 5
250
lniual x•
5
0 2
�
1 ·1 6 ·
c o nd H 1 o n s .
\ W,O)
y \ W.O)
1·08
s t a t E'
=
=
x . ,\ W )
y,< W l
B u l � c o n d i t io n s : •a
1 · 04
1·00+.""T"""".....1 1 0 0 ·8 0 ·6 0 ·4 0 ·2 0
=
l .O
1�� . �bh 0
1
I
W +
FIGURE 3.37 Effect of heat of adsorption on response to bulkphase temperature disturbance. Low Thiele modulus.
oscillation about the steady state occurs if /3A = 0. These oscillations disappear for /3A = 0.05 .
ii) External mass transfer region (Region 4)
Figure 3 . 3 8 shows the system response for a s qu are wave increase in the bulk temperature The results are in s en s itive to the effect of f3A, si nc e ())(*/dT ) 0 over much of the pell et diame ter .
.
3. 1 . 3. 7
Simplified stability analysis
a) Local stability analysis
The stability an al y sis of distributed parameter models is n ot an easy task. Most stability conditions reported are either very conservative or very complicated. As a result, those rigorous conditions are of very l ittle help in exploring the effec t of different physicochemical parameters on s tability Therefore, we choose in this s ection to present to the reader an approx imate method, namely the colloc ation method, and use a first order approximation i.e. sin g l e interior collocation poi nt .
.
MODELLING AND ELEMENTARY DYNAMICS
2 · 0 5 ,, ·  ·  ·  · 
0·
.
.
..... .
·� · ·
s t .st.(X8 = 1 . 0 , y8 : 1 · 1 }
·
=
., �
10
=
0. 1
=
S
=
Sh
2S0
=
•
6.0
=
L,
5.0
=
L\'
5t�ody S l O t t'  ·  ·  {JA : 0 .0 
Parameters : Nu
++ 
2·0 0
305
a
htitjal conditions:
x• (c./,0)
/JA : 0 · 05
= =
x.,( W)
Yu(W) Bullr: conditions:
Y (c./,0) =
••
1 .0
1 . 95
f
>
h .()
1 ·90
0
1
I
1 ·85
1 · 80 +�.,...,,�,j 1.0
0·6
0.8
0·4
w
0 ·2
0
FIGURE 3.38 Effect of heat of adsorption on response to bulk phase temperature disturbance. High Thiele modulus.
By using this first order approximation of the collocation technique, equations 3 . 1 1 3 and 3 . 1 14 are reduced to,
2 5 1_ 0 ._ (l  l'; ) + ,By ¢ [exp ( y( l  1 / }[ ))]X; (1 + 3. 5 / Nu )
__
J x;
_
n L +pA s d 'f
(3. 1 40)
306
S . S .E. H . ELNAS HAIE and S . S . ELSHISHINI
where and Y1 are dimensionless concentration and temperature at the interior collocation point respectively. The surface temperature and concentration are given by,
xt
x"* = Sh +
3. sx;
3. 5 + Sh
3. 5 + Nu
By introducing the following relations, 1 ·· L'v = A �
r' =
where
1 0. 5 r
1 + 3. 5 / Nu
(3 . 1 4 1 )
Nu + 3. 5 1]
Y' =
{3J. = f3r I A .t.'2 'I'
;
=
1 + 3. 5Nu 'I'.�. 2 exp ( r) 1 0. 5
A = (1 + 3. 5 / Sh) / ( 1 + 3. 5 / Nu)
Equations 3 . 1 30 and 3 . 1 3 1 reduce to (L;, + L;)
ax; = 1  x;  ¢'2 exp ( y I l! )Xt a r' ar; 1  y, + {3 ' .t.'2 exp ( y / Y. )X* + {3' L' ax;' a aT =
1
T'l'
1
I
A s
r
(3 . 1 42)
(3. 1 43)
which are the same equations as those of the lumped parameter model but with different physical meaning of parameters. The conditions which are necessary and sufficient that these two equations hav e asymptotic solutions tending towards zero for arbitrarily small perturbations can be obtained in a manner similar to that in section 3 . 1 .2 and can be written as,
(3. 144)
1) where , F = [exp
(y l lJ )] · Xt ;
Yl s
F,'
( aaFr; ) . ' 
ss
Fx�' .
.,
= (}_£_ ax; )
ss
Condition 3 . 1 35 contains no dynamic parameters and can be termed
307
MODELLING AND ELEMENTARY DYNAMICS
Table 3.10
Range of steady state parameters. Nu
Sh
0. 1  1 0
50500
r
00. 1
1 040
the static condition (the slope condition discussed several times earlier) . 2)
1
t/J' 2
( )  + 1 1 L;
>
(/3' {3' )F.' T
A
l},
' Fx;, L;
(3 . 1 45)
where we have noted that L;� £;, . Equation 3 . 1 3 6 is termed the dynamic condition since it includes the dynamic parameters L;, {3� . The first condition does not generally imply the second condition 3 . 1 36 if L; > 1, as shown previously. It is useful to give an impression of the magnitude of the various parameters in various systems.
b) Order of magnitude of the parameters in real systems The ranges that some of the steady state dimensionless groups would cover in actual cases has been discussed in the literature (McGreavy and Cresswell, 1 969). These are summarized in Table 3 . 1 0. Of interest are the additional groups t/J, f3A , Lv, Ls . A reasonable upper bound on t/J that embraces the whole spectrum of behaviour is t/J 5, 1 0 . It should be apparent from Table 3 . 1 0 that /3A must also lie in the range, 0 5, f3A 5, 0. 1 . Lewis numbers reported in the literature refer specifically to values of L, . Ray ( 1 972) reports Lv values in the range, 0.0003 '5, Lv '5, 0. 1 . No collective data on the range of Ls has been assembled. However it was indicated in section 3. 1 .2 that L5 1Lv � 1 . If we take an upper limit of 100 for this ratio, which is quite reasonable, then this puts L� in the range, 0.3 '5, Ls '5, 1 00. Now L;, as defined in the stability condition 3 . 1 36 is related to Ls by,
.
Ls' =
Nu Sh + 3.5 Sh Nu + 3. 5
Ls
Gas solid catal y ti c systems with high surface area and strong chemi sorption, can give very high values of causing instability (Elnashaie
308
S . S .E.H. ELNASHAIE and S.S. ELSHISHINI
et al. , 1 990) as will be shown later for the case of axylene partial oxidation to phthalic anhydride. In s uc h cases, L; may be greater than unity and, as we have shown previously, limit cycl e behaviour (period attractor) is possible. On this respect, the reader should inspect the important and intere sting paper of Berezowski and Burghardt ( 1 993). 3.2
FIXED BED REACTORS
In the prec eding parts of this chapter, we p resented and discussed the dynamic model l ing and the behaviour of a single p article c atalyzing on its surface a first order exothermic reaction. For reactions with l inear ki netics, endothermi c reac tions do not exhib it mul ti plicity and instabi l i ty of the steady states and therefore their static and dynamic beh av i our is rather simple as shown in chapter I . In th is part, s tabil ity and dynamic behaviour of the adiabatic fixed bed reactor are presented with the following obj ec tive s : to show the i mp ortant parameters affecting local stability of the reactor to arbitrarily small disturbances about the steady state. (ii) to characterize reactor stability in a practically useful way when the feed state is subject to disturbance. (iii) to study the phenomena of creeping profiles and wrong directional behaviou r more c losel y. (iv) to discuss and clarify the difference between continuous and discrete models (cell models). (i)
The majority of the chapter concentrates on the use of cell models , it is the opinion of the authors that for fixed bed catalytic reactors, cell models are more physical l y sound in addition to being easier to solve and anal yze. However, in the last part of the chapter, different types of continuous models are pres ented and discussed, preceded by a brief discussion of the two types of models together with some recent important results regarding stability and wrongway behaviour of fixed bed catalytic reactors using a coupl ed cell model. We start with a clas sificati on of the different types of models followed by the presentation and analysis of their static and dynamic characteristics using simple (uncoupled) cell models and coupled (radi ation) cell models. In the adiabatic reac tor rad ial gradi en ts are assu med absent and analysis becomes that much more te nabl e . Stu dy of the ad i abatic fixed bed reactor is more c omp lic ated than that of the s in g l e catalyst particle as a re s u l t of particle/fluid and particle/particle interactions, which arise
from additional he at and mass transfer resistances. General mathematical models have been proposed in the li terature to describe the adiabatic
MODELLING AND ELEMENTARY DYNAMICS
309
fixed bed reactor. A brief review is presented first to set the following presentation in the correct perspective. These models can be broadly classit1ed as,
1 . Continuum models. 2. Cell models. Both of these classes have a similar internal subdivision in terms of heterogeneity, i.e. a) Pseudohomogeneous models. b) Heterogeneous models. The pseudohomogeneous models are subdivided with respect to the problem dimensions and the different mixing mechanisms considered. The heterogeneous models are subdivided with respect to the problem dimensions, different mixing mechanisms and the type of solid particle model (lumped or distributed parameter models). Pseudohomogeneous models assume no concentration or temperature differences between the flowing fluid and the stationary solid phase. Heterogeneous models are obviously more realistic than pseudo homogeneous models, but, are more difficult to solve and contain more parameters. The above discussion applies equally well to both continuum and cell models We shall briefly discuss first, the continuum models and then review the different cell models . 3.2. 1
Classification of Mathematical Mode ls for Fixed Bed Catalytic Reactors
Continuum models (Figure 3.39) 1.
Pseudohomogeneous models
1 . 1 . The basic one dimensional model (plug flow model) The basic or ideal model which is used in the majority of pub lic atio n s on reactors assumes that concentration and temperature gradients occur only in the axial direction. The only transport mechanism operati ng in this direction is the overall flow itself, which is supposed to be ideal. 1 .2.
One dimensional model with axial mixing
In this model mixing in the axial direction due to turbulence and the presence of packing is accounted for by superimposing an "effective"
310
S . S .E.H. ELNASHAIE and S . S . ELSHISHINI
Continuum
P s e u d o  h o mosc:ocg u 1
I
Fluid
Axial
Plu& flow ( c ou p l e d h y pe r b o l i c P.D.E.'s)
dispersion
model ( c o u p l ed
hyperbolic
\
\
\
(I
spat ial
reactor
I.
2.

Distributed """" '
Axial d i s p e r s. i o n ( p a r a bo l i c P.D.E. 's)
( c ou pled O.D.E.)
reaccor
( m i xed O.EI.E. +
mass and thermal d i ff u s i o n
parabolic
(coupled
P.D.E.)
parab o l i c P.D.E.)
s Patial dimensions . reactor lc:nJtb + particle radial vuiablc
2
5.
2 ipatial le ngth +
6.
:?"')
spatial dimension length variable
3.
4.
'
Plug now ( h y pe rb o l i c P.D.E.'s)
phase
I
dimcniion) variable
length
I
2
Models
5patial
length
1
+
spatial
length
dimensions
particle
·
radial
dimcnsiun 
reactor
variable
reactor
uriablc
2 spatial le n sth +
dimensions  reactor p�rticlc: radial variable dimensions particle
·
reactor
radial variable
REACTOR MODE LS
FIGURE 3.39 Classification of continuum models.
diffusion mechanism upon the overall transport by plug flow. The resulting mass and heat fluxes are described by equations analogous to Fick' s law for mass transfer and Fourier' s law for heat transfer. The proportionality constants are effective diffusivities and conductivities. This approach has been discussed in detail by Levenspiel and Bischoff ( 1 963). This model has received considerable attention, the reason being that the introduction of axial mixing terms into the basic equations lead to an entirely new feature, namely the possibility of nonuniqueness of the steady state profile through the reactor (Raymond and Amundson, 1 964; Elnashaie et al. ; Ray et al.). Hlavacek and Hoffman ( 1 970) investigated extensively the multiplicity regions of this model for the adiabatic case and a first order irreversible reaction. Vortmeyer and Jahnel ( 1 97 1 , 1 972) used such a model to investigate the moving reaction zone phenomenon in fixed bed reactors. 2.
Heterogeneous models
When the conditions in the fluid differ from those on the catalyst surface or interior, the models are called heterogeneous. They are of one dimensional or twodimensional type.
MODELLING AND ELEMENTARY DYNAMICS
311
2. 1 . One dimensional model accounting for inteifacial gradients Mass and heat balance equations have to be written for both phases separately. Available correlations for mass and heat transfer between the bulk of the fluid and the solid surface are readily available in the literature (Froment, 1 970). The distinction between conditions in the fluid and on the solid leads to an essential difference with respect to the basic onedimensional model, namely the problem of stability, which is associated with multiple steady states. This aspect was studied first by Wicke ( 1 960) and later by Liu and Amundson ( 1 962). They showed that for a given temperature and reactant concentration in the fluid phase, the catalyst particle may exhibit three steady states for a simple irreversible exothermic reaction over a given range of particle size and fluid flow rate. In terms of fixed bed reactor operation, this finding means that the concentration and temperature profiles are not determined solely by the feed condition but also by the initial conditions from which the reactor was started up. Liu and Amundson ( 1 962) showed from transient computations that typical temperature profiles through the reactor display a discontinous jump from a relatively low temperature, close to the feed temperature, to a high temperature, corresponding to the adiabatic temperature rise. This behaviour, commonly referred to as ignition, is a result of adjacent layers of catalyst particles existing in low and high temperature steady states. Eigenberger ( 1 972) showed that the reactor became ignited at the inlet as a result of heat conduction through the solid phase becoming important at the ignition point. This result is not obtained however, from the cell model which provides a more realistic approach to the ignition problem, involving, as it does, very steep gradients through the reactor. Said add here your own references. 2.2.
One dimensional model accounting for inteifacial gradients and intraparticle gradients
The following model is one dimensional with respect to the fluid field. When the resistances to mass and heat transfer within the catalyst particle are important, as explained in previous sections dealing with the single catalyst pellet problem, the rate of reaction is not uniform throughout the particle. The dynamic behaviour of the catalyst particle is then described by parabolic partial differential equations which have to be integrated together with the fluid field equations. For steady state conditions use is often made of the concept of effectiveness factor. The catalyst effectiveness is a factor which multiplies the re action rate, evaluated at the bulk conditions, to give the overall rate which is actually obtained within the catalyst.
312
S .S.E.H. ELNASHAIE and S . S . ELSHISHINI
Cell Models S t r u c t u re
I
I
P sc u d o  h o m o g c n e o u s Si mple
Cell
Model
SySiem or O.D.E. Solved sequent i a l l y
.
S • mple
c � el l
Hete rogeneous Model
System or O.D.E. S o l ved Sequ e n t i a l l y R adaauon .I '
or
Conducuon Coupling
Coupled system of O.D.E. must be solved simultaneously
FIGURE 3.40 Classification of cell models (with respect to catalyst pellet models, the same classification applies).
Hansen ( 1 97 1 , 1 973) has used the collocation method coupled with the method of characteristics to study the transient behaviour of a fixed bed reactor using such a model.
Cell models (Figure 3.40)
The same classification listed above applies to the cell model. In the cell model, the mixing is described by a series of perfectly mixed cells, rather than in terms of Pick' s and Fourier' s laws. The relation between the number of cells and the Peclet numbers of the continuum model is well known. The principal idea of the cell model is to regard fluid mixing as occuring in a discrete sequence of stages, each stage being a little CSTR. Particle diameter then becomes the natural measure of length and the "void" volume associated with a particle becomes the volume element for the mass balances. We shall present briefly the various cell models that have been used in the literature. These cell models can be either of one phase (pseudo homogeneous) or two phases (heterogeneous) in character. Our presentation will be with reference to the heterogeneous models. The pseudohomogeneous case follows directly by ne g l ecti n g the heat and mass transfer resistances between the two phases.
MODELLING AND ELEMENTARY DYNAMICS
313
One dimentional cell model 1 . Simple cell models (uncoupled) In simple cell models, it is assumed that each layer of particles is immersed in a cell of fluid and each cell is connected to adj acent cells by the fluid flow only. In each cell, the fl uid is assumed perfectly mixed, the volume of the cell being equal to the crosssectional area of the tube multiplied by the particle diameter. This is a simplified model nevertheless it is a useful first step in the investigation of this complicated system. The steady state of this model can be solved by a simple march i ng technique (Vanderveen et al., 1968). Comparison of this model with plug flow c ontinuum models has shown in a limited number of cases a remarkable similarity in the concentration and temperature profiles In fact, it has been shown that the general features of the solution are very similar to experimental results obtained by Padberg and Wicke ( 1967). ,
.
2. Geometrically coupled cell models
In the geometri cally coupled cell model, it is cons idered that a fluid cell has contact with the front half of one particle and the back half of the particle in front of it. This model has been studied by Vanderveen et al. ( 1 968), and later modified by Rhee et al. (1973) to account for thermal co nductivity in the s olid phase .
3. Radiation cell model In reactors which operate at very high temperature radiation of heat from one particle to another may be important. With exothermic reactions, the reaction tends to take place in a very narro w zone with a very large temperature increase over a very short length. Transport of heat by radiation must then be considered and this is most easily done by a cell model. Because of this coupling a strai ghtforw ard marching technique for compu tation is not possible Calculations have shown that the temperature and conce ntration profiles with this model may be considerably different from those of the simple cell model the result being that the reaction zone is moved towards the inlet and the time required to come to steady state is substantially increased . A radiation cell model has been u sed by V anderveen et al. (1968) as well as Berty et al. (1972). The same subdivision exists regarding the parti cle models as in the continuum case i.e. lumped, semidistributed and distributed models. .
,
314
S . S .E.H. ELNASHAIE an d S . S . ELSHISHINI
Analysis of Fixed Bed Catalytic Reactors using the Simple Cell Model
3.2.2
We shall start by presenting the simple cell model because of its simplicity. The simple cell model assumes the reactor behaves as an array of continuous stirred tank reactors, arranged in series. Geo metrically, each stage is a cylinder, the diameter of which is equal to the tube diameter and length equal to a particle diameter. The stages are coupled only by the fluid flow. This simplification allows the computations to be performed sequentially for each cell along the length of the reactor. More realistic though less tractable models allow for greater cellular coupling through particle/particle conduction and radiation heat transfer.
The following assumptions are usually used in simple cell models: (i)
The bed porosity, heat capacity and density of the fluid and of the catalyst particles, as well as the interphase heat and mass transfer coefficients are all constant, i.e. independent of time and spatial position. (ii) Heat and mass transfer between the packing and the fluid stream occur by convective transport across a stagnant fluid "film" at the external surface of the catalyst pellets. (iii) Radial variations in the reactor are negligible. Intraparticle mass and heat transfer resistances can either be neglected or taken into consideration and obviously the reactor can be adiabatic or nonadiabatic. In this section we will present to the reader the simplest case, that is the case with negligible mass and heat resistances and adiabatic reactor operation. 3.2.2. 1
Mass and heat balances
Let us take mass and heat balances on cell number j, for a single irreversible reaction: A � B . 1.
Mass and heat balances on a single catalyst particle
A mass balance on unit void volume of a spherical catalyst particle of radiu s Rp, negle cti ng intraparticle gradients gives the following di fferen tial equation for the particle in cell n umbe r j, (3 . 1 46)
MODELLING AND ELEMENTARY DYNAMICS
315
where,
i s the net rate o f adsorpti on o f reactant A, expressed as mol/s.g catal y st A mass balance on the internal solid surface gives,
dC.vj

dt
= raj  's}
.
(3 . 1 47)
where,
is the intrinsic rate of surface reaction, expressed as molls.g catalyst. c�1 refers to the concentration of reactant in the intraparticle fluid above the surface (mollcm 3 ) . C51 is the adsorbed concentration (mollg catalyst) and CvJ is the concentration of vacant sites. If for simplicity, we restrict our attention to the case of low sur face coverage and equi libri um adsorption desorption (section 3 . 1 .2), then equations (3. 146) and (3. 147) can be combined into the single equation, (3 . 1 48) where Cm is the total concentration of sites and KA the equilibrium adsorption coefficient. Note th at by taking KA outside the differential in equation 3 . 148 we have implicitly assumed an average value over the temperature range in question . The mass capacity of the void volume is always much smaller than that of the internal surface (c.f. section 3 . 1 .2), i.e.
Reactant accumulation in the void space can, therefore, be neg lected The rate constant k appearing in equation 3 . 148 is represented by an Arrheni u s expression, .
with ka being a preexponential factor and E the intrin sic activation energy. Casting equation 3 . 1 48 into a normalized form gives,
316
S.S.E.H. ELNASHAIE and S . S . ELSHISHINI
a3
;
dx
=x J dt
.

*
_
*
(3 . 149)
x  a exp ( y / Y  ) XJ. SJ J .
where ,
cA
J. X =J c r
if E = 
Rc Tif
*
CAJ XJ = *
cif
T ySJ. = _!]_ Tif
Tif and Cif are constant reference temperature and concentration.
Taking a heat balance over the particle, assuming no temperature difference between the solid internal surface and the void volume and neglec ting the fluid phase heat capacity with respect to that of the so li d , we obtain,
Eliminating raJ by using equation 3 . 1 47 and the isotherm relation,
Equation (3 . 1 50) can be written in the following normalized form, (3. 1 5 1 ) and where,
TJ . Y = J T.rf '
2. Mass and heat balances on the fluid phase
�j YSJ = T. rf
For the bulk fluid phase we obtain the following normalized mass and heat balance equations,
MODELLING AND ELEMENTARY DYNAMICS
a1
dX1

dt
dYdt
*
= M(X  1  X  )  (X   X  ) J
a2 1 = H(Y.
J
J
J
1
1  Y.)(f.  fSJ. ) J 1
317
(3 . 1 52) (3 . 1 52)
where, + ml1lopwc w 1 vc � az = ""vh
€p1c1
! H = Fp1 C1 � vh
where p1, Pw refer to the dens iti es of the fluid and the wall, CJ. Cw are the specific heats of the fluid and fue wal l , Vc i s the cell volume, d1 is the in side diameter of the reactor, 8 is the wall thickness and v is the external catalyst surface area per unit volume of bed. 3.
Summary of model equations
The dynamics of each cell are represented by four ordinary differential equations which represent the accumulations of reactant mass and heat in the packing, in the void space between packing, and in the wall . These equations are of the form,
dY. 1 = H( Y.1  1  Y.J )  ( Y.  YS1. ) a2 J df
where j =
1, 2, . . . , N and
(3 . 1 53)
the equations are subject to the inlet conditions, (3 . 1 54)
S . S .E.H. ELNASHAIE and S . S . ELSHISHINI
318
where,
and the initial conditions,
lj = Y;o
�j = �jo
3. 2.2. 2
Steady state analysis
xj = xjo
x':J = X)0*
t = O,j = 1, 2, . . . , N
(3. 1 55 )
The steady state equations are obtained by setting the transient terms in equations 3 . 149, 3. 1 5 1 , 3 . 1 52, 3 . 1 53 equal to zero. This yields the following set of algebraic equations for the steady state of the system, * X.J  xJ* = a [ exp ( y I YSJ· )] XJ
1 , the steady state can still be s tabl e because the criteria 3 . 1 69 is conservative. The exact v alue of the critical ratio a3 la4 can be obtained from 3 . 1 68 wh ich can be rearranged in the following form, (3. 1 70) This condition can be checked for each ce ll from the knowledge of the s yste m parameters and steady s tate solution . However, it w i ll be us efu l
to obtain some sufficient stability condition in terms of the s y s te m p arameters only , which i ne vi tably will be conservative compared w ith the ex ac t condition 3 . 1 68 .
MODELLING AND ELEMENTARY DYNAMICS
2.
323
Stability conditions in terms of system parameters only
We concentrate our attention on steady states that satisfy the static condition 3 . 1 67 . A necessary condition for the instability of a steady stste that satisfies the static condition is that, (3 . 1 7 1 ) Obviously this instability condition can never be satisfied if a 3 la4 < 1 since 9 IJ • 921 are always positive. To simplify this condition we note that since 921 > 0, condition 3 . 1 7 1 can only be satisfied if, a 9u
( :: )
>
( :: )
1 8. 3 3 � ... 0 . 6 R • 1.416 l J Q 2 A:J '"' 0. 1 2 .. . 02 : 0.0 = lniti:ll c:ondilions:
Panmetcr1 :
1. 9
 ·  · !1· 60
' 0 ·0
a� M
Jj(O) e lj�
faj(O): '•Ia j :1,2..... ,75
70 75
FIGURE 3.54 Feed temperature and concentration disturbances. Effect of adsorption heat release on the velocity of the reaction zone.
S . S .E.H. ELNASHAIE and S.S. ELSHIS HINI
340
As shown in Appendi x E, the velocity of the "creep" can be given in analytical form as,
(3 . 1 87)
where N (t) = cells travelled per minute, [X] = exit concentration  feed concentration, [ Y] = exit fluid temperature  feed temperature. It is clear from equation 3 . 1 87 that a positive value of f3A causes a decrease in the creep veloc ity N(t), since [X]/[Y) is al way s negative. We notice also that the effect of f3A on N(t) increases as the mass capacity a3 increases. We shall have more to say about the practical use of equation 3. 1 87 later. A last point to observe from these results is that the reaction zone travels much faster for feed temperature disturbances than for feed concentration disturbances. 2. Effect of ca ta lyst pellet mass and heat capacities The effect of increasing the catalyst pellet heat capacity is i ntuitively to decrease the speed of the travelling zone and this is shown in Figure 3.55. F ! IP d
c o ndition 5o
:
ibc *be :: 0·9
0
:
20 t ( minl
o.g
:
0
20 t ( mtn)
2 . 1 ,, t 0 ..0 rn�n 20
1. 9
  ··;:::::
1.8
Ysj
1.7
1.6
i
t = &. O m > n
1.5
1.4
1. 3
1. 2
; !
1
I
\ 1
1.0
0.9
I I
0
10
FIGURE 3.55 heat capacity.
/
:
I
i 1 6 · 0 min f ; j /
20
/
/
30
==.:.
 a4
=· 
Par o m e t ers :
a.l M H
ii l �··
= 0· 815
PA R a,
 ·  ·  0 4 = 0 .163
[nilial conditions:
:IE
� � • 3
• •
=

0. 1 2 3.919
2.352
4.33'7 X l OS 1 8 . 33 0.6
0.3 3 .456
X
, . ..
,l) !o)
h : j (0)
I02
";J�'
Q2 ;;::. 0.0
. x ,., j 1 .2 . 73
I
.:.o
Cell nu m be r j
so
50
70
75
Feed temperature and concentration disturbances. Effect of pellet
MODELLING AND ELEMENTARY DYNAMICS
34 1
� � y.
2.1
0
20 t( min)
0 ·9
•
0
t(rNn)
20
20
1. 9
1. 8
.  ·  · · ..:,...�==:.7:.;. 7':;...:._.:.. /
1.6
 ·  ·  Q ) :0  1 2
//
i
1.7
Ysj
1.5
 a l : 0 48
P arame!cn :
o_.
M
H a y
l'tr
�,
PA
l n u ial conc:Hdo n.s:
1.4
1.3
1. 2
"" 0. 1 6 3 •
J.919 2.351
= =
•.JJ7 , t o> I �.Jl
•
=
0.6 0 .0
c
�
li4�wo
xj co� o xj..
Ys; (0):
j
�
Ysjss
2
1. 2 �.7S
l1
1.0
0.9
0
10
20
JO
40
C e l l number
j
50
60
70 75
FIGURE 3.56 Feed temperature and concentration disturbances. Effect of pellet capacity and fJA = 0.
mass
The effect of catalyst particle adsorption mass capacity is complicated by the accompanying effect of adsorption heat rele ase . We shall first discuss the effect of mass capacity in isolation from the effect of adsorption heat release. As shown in Figure 3 .55 an increase in the pellet mass capacity causes the temperature profile to flatten and increases the "creep" velocity. For the case in Figure 3.55 the reaction zone is "blown out" of the reactor. This can be explained on physical grounds. The "forward creep", as explained earlier, is accompanied by quenching of some particles. Quenching is also accompanied by an increase in reactant concentration in the catalyst particles. The necessary reactants for this concentration build up must come from the fluid. The higher the mass capacity of the catalyst particle, the greater the reactant takeup from the fl uid stream. Consequently, the lower the concentration of reactants reaching the reaction zone during the transient period. Therefore, the reaction zone moves forward faster, as well as flatte ning out, due to lack of sufficient reactants to sustain the high rate of reaction in the reaction zone. Whe n ads o rpti on heat release is considered, the concentration build up in the qu enc hin g partic le s is acc ompanied by heat liberation. The heat release due to ad s orpti on increases as the ads o rpti on mass capacity i ncre ases . Therefore, in this case, higher adsorption mass capacity has two opp os in g effects, one slowing down the travelling reaction zone
S . S .E.H. ELNASHAIE and S . S . ELSHISHINI
342
Fud c ondi\ion1 :
�� r �t::: r:::i_
0 .
0
lO t( rtt i n)
0··
0
2 0 \(min)
2.1 ...,
2 .0
_.. . · �..:::=�
1. 9
. .
1.8 1.7
Ys j
1. 6
 o J = 0  4 8
 ·  ·
1.5 1. 4
D3
= 0 ·1 2
1.3
Parameters :
Q.
=
�
M
�
l l!r
�
a1
=
H
li
�A
l n it1al condi t i o n�:
1. 2
�
�
z
0 . 1 63
3 .9 1 9 2 .3 5 2
4.337 • J 05
1 8.33
0.6 o2
...
a 1 o:: o
(o) • J'sfss xj (ol :Xfss y1j
0
j :1,2 , · · ··,75
t1
1.0
0.9
10
0
20
30
40
c � u number
j
so
60
70 7 5
FIGURE 3.57 Feed temperature and concentration disturbances. Effect of pellet mass capacity and (:J..., = 0.3.
and the other speeding it up and te nding to cause quenching. In Figure 3 .57 the latter has a slightly stronger effect than the former and therefore it causes a slight i ncreas e in the velocity of the creepi n g profile . 3. 2. 3. 2
Stability of the reaction zone to feed disturbances
Stability in this section is used in the ignitionextinction sense and not in the d ynamical sense used in the rest of the book. When th e feed conditions are d i s turbed the reacti on zone may move backwards or forwards depen di n g on the n ature of the disturbance. If the feed disturbance is a step decrease in feed concentration and/or temperature, the reaction zone will move towards the exit of the reactor. The reaction may be quenched depending on the nature of the new s teady state and the duration of the disturbance. The new steady state correspo nding to the new feed conditions, can be obtained by solving the ste ady state equations. All that remains is to calculate the velocity at which the reaction zone moves to ward s the exit of the reactor. To this aim we shall make use of form u l a 3 . 1 87 derived in Appendix E. Some physically reasonable assumptions are made and then an upper bound on the velocity of the reaction zone is obtain ed which doe s not require numerical s i mu lati on However this upper bound is only valid under certain .
MODELLING AND ELEMENTARY DYNAMICS
343
restrictions. Violation of these restrictions leads to an interesting stability analy sis discussed later.
A n upper bound on the creep velocity The formula for the velocity of the reaction zone is given by equation
3 . 1 87 . In what follows we shall first consider the limiting case of "perfect creep" . Subsequently, the inadequacy of the formula will be brought out when natural transients assume importance . The first simpli fication we introduce into 3 . 1 87 is to introduce a physically j ustified as sumption which is negligible heat and mass capacities of the interstitial fluid in the cell i.e. a1
=
a2 =
0.
Then equation 3 . 1 87 reduces to,
(3 . 1 8 8 )
From this relation we can see that an increase in the catalyst pellet mass
capacity parameter a3 will always increase the velocity of the reaction zone (since
/Jr > /3A
and [X]/ [ Y] is negative) . On the other hand, an
increase in adsorption exothermicity
f3A,
decreases the velocity of the
reaction zone . When the reactor is ignited and the system is at steady state, the temperature rise across the bed is almost equal to the adiabatic limit and the exit reactant concentration is almost zero. Under these conditions we have,
M
XF [Y] ={3T · H
[X] = XF
If the feed temperature and/or concentration are changed, the reaction zone either moves forwards or backwards . Thi s motion continues until the system achieves its new steady state. If the new steady state is also "ignited" then the exit concentration will again be almost zero and the exit temperature is equal to the new feed temperature plus the adiabatic temperature rise corresponding to the new feed conditions . D u ri ng the transient period N(t) can be co mpu ted from 3 . 1 88, but this will require the numerical solution of the model e q u ati on s to obtain [X] and [ Y] . However, i n fact, one can obtain an upper bound on the velocity of the c reep in g profile from simple physical arg ume nts without the need
for numerical simulation. Suppose the system is at steady state denoted
344
S . S .E.H. ELNASHAIE and S . S . ELSHISHINI
Yad,
y
Yo d 2
x, . X
'x,,
l
I I I
= bt> d l e n g t h
'
xF,
I
j ,Cell
number
\
\
I
I
1 t 1' j . C"II n u m b " '
FIGURE 3.58a,b Schematic diagram for the derivation of the upper bound on the velocity of creep.
by a in Figures 3 .58a,b and that the feed conditions are Yn , XFJ . Then the exit conditions (if the steady state i s ignited) will be Yad . 0. Consider a simultaneous decrease in feed temperature and concen tration, and suppose that the steady state corresponding to the new feed conditions is also ignited and denoted by b. At very early times, before the disturbance reaches the exit, we obtain the following approximate expression (see curve c on Figure 3.58a,b).
and,
From this we can obtain the velocity of the reaction zone at very early times (at time t= 0+ , which means just after introducing the disturbance at t = O) as,
(3 . 1 89) It is easy to show that 3 . 1 89 gives the highest velocity during
the
MODELLING AND ELEMENTARY DYNAMICS
345
transient period, provided that the following condition is satisfied, ( 3 . 1 90 )
To show this, consider the situation at some time t= t1 where t1 > 0. The time t 1 is defined such that the disturbance reaches the exit at some time t O. Now N (t) increases as B (t) increases. Thus initial creep velocity N(O+) does not correspond to the maxi mu m "creep velocity " during the transient period. Condition 3 . 1 99 can be written as, !!l o  fJA ' fJT ) > M a4
H
which is the necessary condition for wrongdirectional creep (equation 3 . 1 95).
348
3.
S .S .E.H. ELNASHAIE and S . S . ELSHISHINI
Instability of the reaction zone (blow out)
In this section the instability of the reaction zone is defined in the following sense. The reaction zone is unstable if at any time during the transient period, the velocity of the creep becomes infinite. This condition will lead to an instantaneous "blow out" of the reaction zone. Equation 3 . 1 92 is written as,
(3 .200) Referring to the previous part about the velocity of creep, we see that the first condition necessary for instability is that,
(3 . 1 99) Satisfaction of the inequality in equation 3 . 1 99 ensures that the creep velocity is not bounded below the "initial creep velocity" at t = Q+ . Consider the case i n which the disturbances represent step decreases in feed temperature and concentration. This "direction" of disturbances has the effect of moving the reaction zone towards (or even theoretically beyond) the reactor exit, this is leading to a potential instability problem. The initial direction of creep is in a forward direction towards the reactor exit if,
(3 .20 1 ) Now as the reaction zone creeps towards the exit with increasing time, B (t) decreases, tending to its stationary value B ( t) = l!A � o corres ponding to the new steady state. Clearly then (because of condition 3 . 1 99), there will exist some critical time t = tcr at which 1  A2B (t) = 0 and N (t) = oo . For t < tcr the reaction zone will be continuously accelerated until t = tcr, the velocity suddenly becomes infinite and "blow up" occurs instantaneously. It is apparent that the duration of the disturbance now becomes a key factor in the extinction stability problem. 4.
Summary of conclusions
The major conclusions for the case in which disturbances represent step decreases in feed temperature and concentration are:
a) Forward directional creep. If the condition,
a1 1 f3 f3 M (  A / T) < a4 II
·
(3. 1 90)
MODELLING AND ELEMENTARY DYNAMICS
349
is satisfied, the reaction zone moves forward with decreasing velocity which becomes zero at the new steady state. If the new steady state is ignited the reactor remains ignited.
b) Wrong directional creep. If the condition,
(3 . 1 96)
is satisfied, the reaction zone moves initially in the wrong direction, i.e. in a direction away from the n ew steady state.
c) Instability of the reaction zone. If the condition,
(3. 1 95)
and , (3.20 1 )
are satisfied, the reaction zone moves forward with increasing velocity until at some critical time t = tcr, the creep velocity becomes suddenly infinite and instantaneous "blow out" of the reaction zone occurs. For step increases in feed temperature and concentration similar arguments to those in section 3.2.2.5 can be used and the following conclusions apply (notice that in this case B (t) has its highest value at
t = O+).
conditio n s 3 . 1 90 and 3 .201 are satisfied, the reaction zone moves backwards with a stable approach to the new steady state. (ii) If condition 3 . 1 95 is satisfied, the reaction zone moves forward towards the reactor exit (wrong directional creep). (iii) If condition 3 . 1 90 and 3 . 1 96 are satisfied wrongdirectional creep can occur initially.
(i)
If
5.
The effect of natural transients
We are led to question whe ther or not these startling results are phy sical l y realizable or mere ly the product s of an oversi mpli fi e d mathe matical analysis, the ov ers imp lific ati on s being the result of neglecting the natural transients. However, it is indeed worth mentioning here that wrongdirectional response analogous to this wrong dire ctional creep has been found experimentally by Hoiberg et al. ( 1 97 1 ) . 
S.S.E.H. ELNASHAIE and S . S . ELSHISHINI
350
The condition, (3. 1 95)
seems to be the key factor which leads to the more interesting phenomena reported. Let us first examine condition 3 . 1 85 closely. For gas solid system M
2:: 1 and f3A > 0. Therefore the ratio a3/a4 has to be over unity for H condition 3 . 1 85 to be satisfied. In other words, the mass capacity parameter of the pellet has to be higher than the heat capacity parameter. Extensive numerical investigation has shown that in all cases when a3/a4 > M/H the reaction zone changes shape during the creep. This change of shape makes 3 . 1 78 invalid since under these conditions, the natural transients cannot be neglected (see Appendix E). In fact even a slight change in the shape of the reaction zone during creep makes equation 3 . 1 87 invalid and gives a wrong prediction of the sign of the reaction zone velocity. Figures 3 . 59a and 3.59b show such a case. Computations of N(t) from 3 . 1 87 using exact values of [X] and [Y] from the numerical simulation give negative N (t) at all times. Inspection of Figures 3.59a and 3 .59b show that the reaction zone moves forward, i.e. N (t) is positive. Also the previously presented results for high adsorption mass capacity in Figure 3 .56 correspond to such a situation. 2.1 2.0
k
1. 9
0·9
1. 8
0
20
1. 5
1.5
0
u
a liT
r
R
20 l (m o n )
Cl l
l n 1 tial condit rons:
1. 2 1.0
0.9
0
10
FIGURE 3.59a
20
p, y
H
1.3 t 1
04
ll j M
t( min)
� 0·9 �
1.7
Ysj
Parameters :
30
40
C e l l nu m ber
j
so
60
Effect of natural transients.
70 7 5
=
'Z
�
�
•
0
18 ))
J.9 1 9
=
l.Jll
=
o J 7 , 10< 0.6
=
=
•
0. 1 6 3
0 . 48
>'sj
8 . 64
�
10·4
0 2 .. 0.0 (0 )
: )'s.jg
{r ( O J . x i.. j
: 1 , 2 , . . . .7 5
MODELLING AND ELEMENTARY DYNAMICS 2  1 .,,
�:� ��86� � �
Parameters : li
1
,... �A
�
1 6 1 7
R
M
·� 1 · 6
H
••
1S
35 1
= •
�
�
= =
�

·�
4.337 • 1 8 . 33
0.6
0.0
8.64 x t o • 3.3 1 9 2.3S2
0 . 1 63,
lJ
IJ = &2 = 0 Initial conditions:
1 ·4
13 1 ·2
Ot�;;;;;;;;;;�=;:::::..,..  r. .l
=
0.29
xj (o)
y,J (o)
:: xj1 =
)'1j 1 1 1
1 ·1
1·
095f'
0
10
FIGURE 3.59b
6.
20
j , Cell 30
40 50 n u m ber
60
7 0 75
Effect of natural transients.
Evaluation of "correct directional creep "
From the preceding results and discussion it is clear th at condition 3 . 1 8 1 must be satisfied for the analytical formula 3 . 1 7 8 to be of real use. Let us consider the case in Figure 3 .52. For this case, condition 3.181 i s satisfied and we should expect a stable approach to the new steady state. Using formula 3. 1 80 to compute the reaction zone velocity at t = Q+ (upper bound) we get N (0+) = 1 .65 cells/minute . Therefore, the number of cells trave lled in 20 minutes will be Nt20 = 33 cells. The numerical results in Figure 3.52 show that the number of cells travelled in 20 minutes is 30 cells. For the same case but with f3A = 0.3 we obtain N (Q+) = 1 .3 cells/ minute. Therefore, the number of cells travelled in 20 minutes will be Nt20 = 26 cells. The numerical results gi ve Nt 20 = 24 cells. For simultaneous feed temperature and concentration disturbances, let us consider the case in Figure 3.54 and calculate N(O+) from formula 3 . 1 80 which gi ves N (O+) = 2.52 cells/minute. Therefore, Nt 20 = 50.4 cells. The numerical solution gives Nt 20 = 48 cells. 7.
Practical limits on allowable disturbances
In this section we s hall try to answer for the reader, in a practic al sense, the fol l owing question : what are the limits if any, on the duration of step decre ase in feed temperature and/or concentration w h ich lead to "blow out" of the reaction zone ? Part of this ques tion
a)
'An "ignited" new steady state.
can be answered from purely steady state considerations taken together with the analysis of velocity of creep.
S . S .E.H. ELNASHAIE and S.S. ELS H ISHINI
352
If the new steady state corresponding to the new feed conditions, is also "ignited" and if condition 3 . 1 90 is satisfied, then the duration of the disturbance is unimportant. The reaction zone will not be blown out of the reactor.
b)
A
quenched new steady state.
If the new steady state corresponds to "quenching" of the ignition zone, then the duration of disturbance is critical. We can determine a safe limit on the allowable duration of disturbance by using the maximum creep velocity formula 3 . 1 89. This gives a safe limit on the critical duration as,
L F � N(O+ )
t < tc = ,·
where L = reactor length, F = distance of initial igmt10n zone from reactor entrance, N (O+) = initial velocity of creep (number of cells travelled per minute), � = thickness of cell. 8. Effect of catalyst pellet adsorption mass capacity on the unforced
system stability
In this section and after clarifying many issues regarding the ignited reactor and the effect of feed disturbances on ignition, we return to the autonomous unforced system to show that it can also have instability which can cause quenching of the reactor (or continuous oscillations of the profile along the length of the reactor) without any feed disturbances. Figures 3.60a and 3.60b show the effect of catalyst pellet adsorption mass capacity on the stability of the steady state to small disturbances in the state variables for constant feed conditions. For a3 = 0.48 (Figure 3.60a) the steady state is unstable, while for a3 = 0 . 1 2, 0.29 the steady state is stable. For the case of a3 = 0.29 (Figure 3.60b) the approach to the steady state is oscillatory. Checking the critical value of a3 for each particle of the bed, from our previous single particle analysis and the steady state temperature and concentration, it was found that all the particles after the reaction zone (fully ignited) are stable to a11 values of a3• However the particle in the reaction zone (particle no. 1 ) has a critical a3 value of about 0.35. Therefore for this set of parameters, the bed is unstable with sustained oscillations (the whole profile is oscillating) as shown in Figure 3.60a. The instability of the autonomous (with constant feed conditions) bed is discussed in more details in the next sections where Hopf and Homoclinical bifurcations are presented and discussed.
MODELLING AND ELEMENTARY DYNAMICS
2 ·4
Paumcten
... 3 3 1 • 1 01 i:i 1 • I B.3) � = 0.6 p, • 0 0 R 3.4S6 � J Q l M 1: 3 .9 1 9 H • 2 .3 l l a. "' 0 1 6 )
2·2
2·0
;,
,..
353
1·8
a1
c
a2
 "J
�· "J
16
= 0
! 041 (�o�ns1�tM)
=
0·12 ( st. a bllt)
lJ C o l = X i ss
1 ·4 1 ·2 1·0
8
4
0
FIGURE 3.60a
12
j
•
16
Ct\1
20
nu m b e r
24
3S
32
Effect of catalyst pellet adsorption mass capacity on the stability
of the unforced system. High and low values of a3•
�===J
2 ·0 1 .9
P a r il m c t c c s : jj ,. 4 . 3 3 7 1 Ii.Jl 0.6 Pr 0.0 p. R M
H a .
1 .0 1
'"
, .. ' ·'
(c)
�
Cell No. 1 5
'"
0.9i
I.() I
Cell No. S
I .Q�
' ·'
1.0�"'
(b)
,. '.
(a)
(d )
1.4
� 1.2
.. ' l 1 .0!
1 .00+.....=, .. :;,:;:; ; , :;..,.,�...... ..��,.,,..... .,., ,.,
.O>
Y;
FIGURE 3.63 Static bifurcation diagrams of solid phase temperature versus feed temperature for different reactor ceUs. R :;;; 0.03456, lXH = 0.0. 2. 4
2.0 ::0:
1i "!
�"' �
�
; 6
R�
Sll' o
R=
�
:
'!>.• , ,to
f..
l .oo I
365
tJ R
YF
(f) =
1.5
�
...
ii
FIGURE 3.68 Continuation curves with the radiation parameter (R) as bifur cation parameter for cell number 10.
varied from 0.0 to 1 00.0. As shown in Figure 3.67, increasing aH has an effect which is qualitatively similar to the increase of R in Figure 3.64, namely the four observed phenomena of the reduction of the number of steady states, the lowering of the uppermost ignition temperature, the lowering of the feed temperature at which reactor ignition occurs and the narrowing of the range of multiplicity of the steady states. Figure 3 .68 shows the bifurcation diagrams with the radiation parameter R taken as the bifurcation parameter. At a feed temperature YF of 0.7, only a unique quenched steady state for all values of R. Of course the radiation parameter is always positive or zero (R�O), however the xaxes in Figures 3.68af are drawn down to R =  1 in order to show the behaviour near R = 0 specially for Figures 3 .68d,e. At around a YF value of 0. 72, an isola source center emerges and further increase in YF. results in the enlargement of the isola. At YF of about 1 .02, the isola merges with the quenched steady state branch and the bifurcation diagram is represented by a continuous hysteresis curve. At sufficiently high values of YF of around 1 .5 and beyond, a unique steady state exists. Figure 3 .69 ( with aH as free parameter) shows similar patterns of bi fu rcati o n diagrams to those in Figure 3.68. II . Dynamic Behaviour
The initial condi tio n s for the computations shown in Figures 3 .70 and 3.7 1 are flat initial conditions of x; = 0.0043 and Ys = 1 .9307 for
366
S . S .E.H. ELNASHAIE and S . S . ELSHISHINI (a)
� ..
rF
�
0.8
� ··
f ..
I .. � ..
YF
YF
( c) h  0.9
(b) �
0.85
:.!" ....
f
J"
I .. �
D
�
"
O.H
D I>H
(d ) =
0.95
FIGURE 3.69 Continuation curves with the conduction parameter ( Gflf) as bifurcation parameter for cell number 10.
j = 120. At an aH value of 0.25 it is observed that the solid phase temperature profiles do not exhibit appreciable transient oscillations for the first few inlet cells, but for the remaining cells along the reactor axis, appreciable transient oscillations do occur. This is clear in Figures 3.70a,b where the solid phase temperatures are drawn versus time for cells 1 ,5 (Figure 3 .70a) and cells 1 0,20 (Figure 3 .70b) for the same a3 value of 0.25 . However, the profiles show sustained oscillations for all cells at an a3 value of 0.37. Figures 3 . 7 1 ad show clearly the occurence of sustained oscillations in solid phase temperature and concentration (.)
20
I .O
2.00
(b)
1.9:2 0 0
�....� .. .. o.... . ..� .., ,� . 0 ,� , ......., n,., D\ia
,, , r, ..., ... .., ,,,�,,... o , _goo.� .o ""'o.""' • ...,.n, m.iJI
FIGURE 3.70 Dynamics of solid phase temperature with a3 = 0.25, for different reactor cells. R = 0.03456, a,., = 0.0, YF = 1.0.
MODELLING AND ELEMENTARY DYNAMICS LN
(a )
0 .017 5
..: ,; 1 .t2
367
( b)
�
• ···
jII; .... �
� "1 .... ,_,.
i
..:
Cdl No. 10
..
..
(c) Cell No. 10
....
. ..
j' 1.12 .:
� "'
Cdl No. lO
i>r:f"� ""'T ,.""'t" , ·""' · ..,. ,.....,. 2D�1.l ,...,,.. ,.., .... ,.....,. ,...., , n. .m
..
�
0.005
(d)
" � O.DC)t
5a
Cdl No. lO
�
•. ,.,
....
ifI 0.002 , "' � 0.00 1
••
...
t.)
..
FIGURE 3.71 Dynamic behaviour of the reactor with reactor cells. R = 0.03456, an = 0.0, YF 1.0. =
a3
= 0.25, for different
for cells 10 and 20 for an a 3 value of 0.37. Profile oscillations are not observed for values of a3 close to 0.5 and beyond. A general view of oscillation behaviour in the reactor is presented in Figure 3 .72 in the form of phase portraits of solid phase temperature against solid phase concentration for cell number 1 0 for values of a 3 of 0.4, 0.42, 0.45 and 0.47 i n Figures 3 .72ad respectively. The dynamic behaviour in the region of the very large number of steady states is shown in Figure 3.73 with YF set at a value of 0.982. Figure 3.73a shows a case with a3 = 0.325 when different initial conditions give rise to a very large number of steady states and almost each initial condition results in a different final steady state. The initial conditions used for each response in Figure 3 .73 are flat initial conditions of x = 0 . 004 3 and Ysj was set as shown in Figure 3 .73a for j 120. Figure 3 . 7 3 b shows a case with the initial conditions maintained con stant at x = 0.0 and Y�i = 1 .5 for j = 120, when different values of a3 a l s o give rise to a large number of steady states. Figure 3 .73c shows the sensitivity of the dynamic behaviour and the final stead y state re ac hed when a very s m all change in a3 from 0.3386 to 0.3387
;
=
;
368
S . S .E.H. ELNASHAIE and S.S. ELSHISHINI
I
·�._ .. :!:,., :=: . ,'::l .., ��.'::l .... ::"':,:11 . ;;::.� ... ::;.� ..;;,;� Solid Pltal Cottcltt tTtlliorr, ;r
t ; 2.0
:iolldPh �11wtttHI. r
,:: � .i:.�
(C)
:0.: 2. 1
t
J.. 1 �
!i ( ..• ;)! ·�
(d )
t..
a 3 : 0. ' 5
! 1.'.1
"'1.. !...::.%;"�0.IIO�>::, ::...:";;",.::::: :::;,:;: ..,;;00�010 ... ....
"
�
� ,,
'·'
"
o.o.ss
a, = 0. 4 7
...
,,
... ... Solid Pluw C�at1011 , X"
FIGURE 3.72 Phase portraits of solid phase temperature against solid phase concentration. R = 0.03456, lXH : 0.0, YF = 1 .0. us 1 . 56 I .S4
'·'
l
�
l �
1.6
u_
1.0
Y.: 2.5
:: 2.6
f
(b)
3.0
f l
  rP  \ '
��
E .::
ol: ] � 1.0
�
0.5 ���� 0 2 " 6 6

������� '}'} 10 12 14 16 1 8 20
Cell Number
·· .
2.0
373
.
'
. . . .. .. . " . . '  .,
1.5 1.0
o.s +..,.,�����,....,..���� 0
2
4
6
8
10
12
CeU Number
14
15
18
20
22
FIGURE 3.79 Effect of initial conditions on the transient response to a step increase in YF from 0.97 to 1.2 for cell number 10. The transients are shown at 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7 and 0.8 minutes after the step increase at t = 0.0. R 0.03456, ay l.O, a3 = 0.45. =
=
In Figure 3 .79 both radiation (with R = 0.03456) and conduction (with
aH= 1 .0) are taken into consideration in studying the dynamic responses to a step increase in YF from 0.97 to 1 .2 for different initial conditions.
In Figure 3 .79a the initial conditions are flat initial conditions of x; = 0.96 and YsJ as indicated on Figure 3 .79a for t = 0.0, while the initial conditions for Figure 3 .79b are x; as shown in the small box inside Figure 3 .79b and YsJ as indicated on Figure 3 .79b for t = O.O. Depending on initial conditions, the transients can exhibit no overshoots (Figure 3 .79a), or exhibit overshoots (Figure 3 .79b). For the initial conditions used in the computations of Figure 3 .79b, a quenched condition is established at the front of the reactor, a partial conversion at the middle section of the reactor and a relatively high conversion in the exit region of the reactor. Moreover there are no overshoots in the downstream sections of the reactor. It is in fact generally observed that the front section of the reactor takes the brunt of the temperature surges due to disturbances in the feed temperature. For this reason catalyst protection measures and control strategies for the upstream sections of the reactor should be more exacting than those for the downstream sections of the reactor. In Figure 3 . 80 we show the effect of the step decrease in YF on wrongway behaviour. In Fi gure 3.80a, YF is decre ase d from 1 .0 to 0.96 whi l e in Fi gure 3 . 80b the de c rease in YF is from 1 .0 to 0.95 . The i nitial conditions for x; are sho w n in the small box inside Figure  3 . 8 0b and those for Ysj are as indicated on Figure 3.80a for t = 0 . 0 . In b oth cases, the initial conditions are those of appreciable conversion while the final state is a quenched one. As shown in
374
S .S .E.H. ELNASHAIE and S . S . ELSHISHINI
lO
f:L
(b)
(a) 4.5 4.7
5.2
�=
�5
!
...
0
.
.
. . . .. .. . . .
, . .. 12
10
H
16
Time, min
18
20
10
22
12
Time, min
14
16
18
20
22
FIGURE 3.80 Effect of the size of step decrease in YF on the wrongway behaviour for cell number 10 at 1.0, 4.0, 4.5, 4.7, 5.0, 5.2, 5.5 and 7.0 minutes after the onset of step decrease at t = O.O. (a) Step decrease in YF from 1.0 to 0.96. (b) Step decrease in YF from 1 .0 to 0.95. R = 0.03456, «n = O.O, a3 = 0.45.
� ... ,.. 0.
1
� u
'"" ., ,.,.....,"""" • ;,,..., , '"'•"' ...., ,...,. , ,,.,:r !
�..0�  /1/
tu
� .::
.
..
A
ef, .
;::
Cell Number       1
'  .,
� ..
"
j �
..
'
2
a
'
10
u
,,.
Cdl Nambtr
..
11
10
'2
...
Cdl Number
FIGURE
�t ..o O
ii. O
,//
�; 1.1
l .2
� G.8 � u' >:
� 1.1
_ _ ____ _._ _ _ _ _ _ _ _
_ _ __
I
2
(d) I
10
12
14
c,u Numbu
11
�
.::
6
8
"
C�ll Number
.
4
'
�
(b) , o , � ,.;:o; ,.,, , :•"'" • "" � !, ,,.,:r ...., ,...,. , ,,:o:
� Ui
2
0.1
,. 2 .•
;.,
l
..
o.•
� 1.4
,
1.0
....,
.::
(c )
.,
�.r. , ,
    
.. .. ... 
� 1.4
.::
� ... ..
( a)
..
r:
ju
< •> ..
�u if
� 0.8 "'
..
.
•
10
12
1:1
14
,
o..z
( � ···!, ....,....,.. ..r;:.?�...., ,...>"=" "" " 22·"" I 10 12 II,.,; 1 'il 18 :to 1
(I )
Cell Number
o.•
,
Cell Number
of a3 and the step change in YF on the dynamic behaviour In all the Figures, the initial conditi ons are the same at t = 0.0. R 0.03456, fXH = 0.0. (a) a 3 0.45, YF is decreased to 0.97. (b) a3 = 0.4, YF is decreased to 0.97. (c) a3 = 0.45, YF is increased to 1.5. (d) a3 = 0.4, YF is increased to 1 .5. (e) a3 = 0.45, YF is decreased to 0.982. (f) a3 0.4, YF is decreased to 0.982. of cell =
3.8 1
Effect
number 10.
=
=
MODELLING AND ELEMENTARY DYNAMICS
375
Figure 3 . 80a considerable transient temperature overshooting occurs during the relatively smaller step decrease in the feed temperature from 1 .0 to 0 . 96 while the larger step decrease in the feed temperature from 1 .0 to 0.95 does not result in transient temperature overshooting. Wrongway behaviour such as that shown in Figure 3 .80a has been explained earlier as a result of the difference of speed of propagation of concentration and temperature disturbances in the packed bed reactor (Il' in and Luss, 1 992). Figure 3.8 1 shows the impact of a3 and YF on the dynamic behaviour of the reactor. In Figures 3.8 1 af, we start from the same initial conditions of x; as shown on Figure 3.8 1 g and Y�1 as shown in Figures 3 . 8 l af at t= O.O. With YF = 0.97, it is observed that the transients for a3 = 0.35 (Figure 3 . 8 l a) are slower than those for aF OA (Figure 3 . 8 l b), specially in the downstream sections of the reactor. The final steady state for both figures is an extinguished state. The effect of increasing a3 is to speed up the temperature transients. In Figures 3 . 8 1 c and 3 . 8 1 d, the reactor is operated at YF = 1 .5 (giving an ignited state), and as seen from these figures the dynamic behaviour is almost identical, with a slight slow down of the response, to the case of a 3 = 0.4. In the region of multiple steady states (YF = 0.982) shown in Figures 3.8 l ef, it is clear that the final steady states are different for the two values of a3 of 0.35 and 0.4. III. Relation between the Stability of the Catalyst Pellet and the Fixed Bed Reactor
The numerical results presented show two types of dynamic bifurcation, namely, the Hopf bifurcation (HB) and the homoclinical or infinite period bifurcation (IPB) (Figure 3.75) . The HB is a local bifurcation related to the behaviour of the eigenvalues of the linearized differential equations in the neighbourhood of the steady states, while the IPB is a global bifurcation that cannot be determined from local (linearized) stability analysis. The HB corresponds to the point where the complex eigenvalues have zero real parts (at least one conjugate pair of complex eigenvalues cross the imaginary axis from the left to the right with a nonzero speed as a parameter, in this case a 3 , i s changing). It is interesting and useful to relate the stability condition of the single catalyst pellet to the stability of the entire fixed bed. The two wellknown static and dynamic stability conditions of the single catalyst pellet to the stability of the entire fixed bed. The two wellknown static and dynamic stability conditions of the single catalyst pellet given earlier, can be written as, a) Static condition
S . S . E . H . ELN ASHAIE and S.S. ELSHISHINI
376
b) Dynamic condition: (3. 1 68) where,
glj =
( aa}f ) 1
ss
gz j =
( a/ar ) SJ
SS
Jj = exp (

yI
�J ) X;
The static condition (equation 3 . 1 67) is violated for the middle saddle type steady states and are therefore always unstable. For the other steady states of the catalyst pellet, the dynamic condition is always satisfied for a3/a4 < 1 , however for a3/a4 > 1 the condition can be violated for a certain critical value of (a3/a4 )cr giving rise to instability When the entire bed is described by a simple cell model, it has five stability conditions (for the derivation of these conditions see section 3.2), three of them are satisfied if the catalyst particle stability condi tions are satisfied provided that the following three conditions are also satisfied,
.
c) (l + M ) (l + H) > az + � a3 a4 d) Hg11 > Mg2}3 T
e) M H (1 + ag 11  af3 T g2) + (aHg11  aMgz ;f3 T ) > 0 ·
·
(3 . 1 79) (3 . 1 8 1 ) (3. 1 80)
The fifth stability condition (e) for the bed, is complex, difficult to analyze analytically and must be checked numerically. Therefore, it is possible that even though the catalyst particle satisfies its stability con ditions (a, b), the catalyst bed can be unstable if one (or more) of condi tions ce is violated. For the coupled cell model, it has been shown (section 3 .2, Appendix E) that for an ignited bed the velocity of the creep of reaction zone (ignited zone), when the reaction is expo s ed to an external disturbance, is given by,
MODELLING AND ELEMENTARY DYNAMICS
377
where N (t) = number of cells travelled per minute, [X] = exit concentration  feed concentration, [ Y] = exit fluid temperature  feed temperature . It has been shown that wrongdirectional creep (which is different from wrongway behaviour as discussed earlier), corresponds to i nsta bi l ity of th e bed. After some lengthy manipulations of equations it can be shown that the condition for wrong directional creep to occur is that,
(3 . 1 90)
This condition is an extremely useful guide for the instab ility of the fixed bed catalytic reactor. For the present case, with the parameters given earlier, MIH = 1 .6624 1 5, and noting that the HB for the catalyst bed occurs at a3 = 0 . 33 (Figure 3 .75), therefore a3/a4 = 2 .024 which is larger than MIH and thus givi ng rise to i n stability . It is clear that the simple condition 3 . 1 90 gives a good es timate for the rati o of a3/a4 that must be exceeded in order for instab ility to occur for the fixed bed described by the coupled cell model. IV. Summary of the Main Static and Dynamic Bifurcation Characteristics
It has been shown in this section that for a particular set of parameters, three general regions of stati c bifurcation behaviour exist for the heterogeneous fixed bed reactor. In the fi rst regio n , below a critical feed temperature value, a single extinguished steady state exists. In the second region a multitude of steady states exist ranging in number from 3 to a maximum of 29 states, always existing on an odd numerical basis. In the third region, above a certain critical feed temperature value, three steady states exist, the upper one as an ignited state and the other two as low extinguished states. The dynamic analy sis revealed that limit cycles (periodic attrac tors ) exist on the upper ignited static branch over a specific range of the mass adsorption parameter a3• The periodic oscillations start at a Hopf bifurcation point and terminates homoclinically as a 3 increases beyon d a critical value. It was found that reactor operation in the region of periodic oscillations has a detrimental effect on reactor performance. The nature of the transient behaviour in the reactor is sometimes shown to be different along the le ngth of the reactor. For certain values of a3, the tran s ie nt approach to the final steady state was o sc illatory in nature for the upstream (inlet) ce lls and asymptotic for the rest of the do wns tre am cells. This can have serious impl ic ati ons for
reactor operation, should hot spots develop during transient periods. For the case investigated, reactor performance as measured by conversion, is l ower in the periodic region compared to that in the static ignited regton.
378
S . S .E.H . ELNASHAIE and S.S. ELSHISHINI
The increase in radiation and/or conduction effects results in four structural changes in the static bifuraction diagrams namely, the number of steady states decreases, the uppermost ignition temperature is lowered, the reactor becomes ignitable at progressively lower feed temperatures and the region of multiplicity is narro wed. For specific ranges of the radiation and/or conduction parameters, the static bifurcation diagram exhibits an isola branch of solutions over a certain range of feed temperatures. There is a critical value of the feed temperature below which the isola shrinks in size to an isola source point and then disappears leaving unique extinguished steady state solutions . In the intermediate range of feed temperatures, the isola grows in size with the increase in feed temperature. Above a specific critical feed temperature, the isola disappears by merging with the horizontal lower static branch of the bifurcation diagram forming sshaped hysteresis curve. Further increase in YF results in a single ignited steady state for all values of the radiation and conduction parameters. The study of the structural changes introduced in the transient responses to step changes in the feed temperature, subtantiated earlier, reported theoretical and experimental findings regarding wrong way behaviour during the stepdown of the feed temperature. In addition to the conditions reported in the literature for the occurence of wrongway behaviour, it was observed in the present investigation that when multiple steady states exist, the system initial conditions also affect the occurence and the nature of the wrongway behaviour. In all cases the dynamic behaviour of the system is strongly affected by the value of the adsorption mass capacity parameter a3. It has been shown that coupling between cells (e.g. by radiation and/ or conduction) has two effects on the multiplicity of the steady states. Firstly, it reduces the number of steady states in comparison with the simple model in which there is no coupling between cells. Secondly, it equalizes the number of steady states in each cell throughout the reactor length whereas for the simple cell model with no coupling, it is possible for the number of steady states to be different in each cell, such a number tending to be higher in the downstream direction along the reactor axis. Simple instability conditions relating the instability of the single catalyst pellet and the entire fixed bed have been briefly presented and discussed. 3.2.4
Analysis of Fixed Bed C atalyti c Reactors using Continuum Models
In principle and in the final analysis, continuum models when solved are not very different from cell models. This is because most realistic
MODELLING AND ELEMENTARY DYNAMICS
379
catalytic reactors models are nonlinear and are thus solved numerically, which means discretizing the differential equations in the axial direction (in addition to the radial direction for twodimensional models and the time direction for dynamic models). This effectively means the formation of a cell model through discretization. The main problem with this procedure is that this discretization is carri e d out on numerical basis and not on physical basis. This in fact, specially when using the more realistic heterogeneous models, creates a contradiction when the step size is smaller or larger than the catalyst particle size, while at each step the catalyst particle equations are solved based on the physical catalyst particle size. The cell model on the other hand, is actually a physical discretization based upon the catalyst particle size. In this section, we will present to the reader some of the important continuum models used in the literature to simulate fixed bed catalytic reactors. Eigenberger ( 1 972) presented an interesting heterogeneous continuous model taking into consideration the heat conduction along the length of the bed, which is a modification of the continuous two phase model of Liu and Amundson ( 1 962, 1 963). In the twophase con tinuous model, the complex behaviour of the reactor is concentrated on two homogeneous phases, that is in the flowing fluid and in the fixed catalyst. Liu and Amundson ( 1 962) have shown that continuous models may lead in certain regions of parameters to an infinite number of steady states. The infinite multiplicity of the steady states is in fact a confusing conception not proved by experimental results. It claims that it is possible for two adjacent catalyst pellets to be one at the upper and the other at the lowersteady state, whereas the latter can exist at the lower steady state as well. But if one assumes heat conduction to take place not only in the fluid, but also in the catalyst phase, only a small heat conductivity is sufficient to transfer the adjacent pellet to the upper steady state. This is specially so at the ignition zone where dTjdz = oo, for in this region heat conduction in the catalyst phase will considerably decrease the above theoretical possibility which can lead to an infinite number of steady states. The continuous model of Eigenburger ( 1 972) for nonporous catalyst particles or porous catalyst particles with negligible intraparticle mass and heat transfer resistances can be written as follows : For the gas phase
The mass balance equation is given
ac
ar
by,
 + v = A1 ( Cs  C)
dt
dz
(3 . 2 1 4)
380
S .S.E.H. ELNASHAIE and S.S. ELSHIS HINI
while the heat balance equation is given by,
dT + va; d T A2(� T) + A3 ( Tw at = 

T)
(3.2 15)
For the catalyst phase
The mass balance equation is given by, (3 . 2 1 6) and the heat balance equation is given by,
()� = At, (T  � ) + A7 · r( Cp � ) + As �z;t
( 3 .2 1 7 )
The boundary conditions are given by,
(a�)
c(z = O) = Co , ()z
1(z=0) = 1'u
( )
 a� z=O
dz
z= L
o
(3.2 1 8)
where dimensionless parameters are given by,
where kg = mass transfer coefficient (rnls ) ; ko = frequency fac tor ( 1 /s); £= activation energy (kcal/kmol); (MIR) = heat of reaction (kcal/kmol); 59 = specific inner surface area of catalyst particle (m2/m3) ; E,, = void fraction of the catalyst particle; Rp = radius of particle (m) ; CP' = heat c ap acity of catalyst (kcal/m3 K); av = specific outer surface of c ataly s t particle (m2/m3) ; Cpf= heat capa ci ty of flu id (kcal/m 3 K); L = reactor length (m) ; rr reac tor radius (m); E = void fraction of the catalyst bed; •
·
MODELLING AND ELEMENTARY DYNAMICS
38 1
v= interstitial velocity (rnls); C= concentration in the fluid phase (kmol/ m 3 ) ; Cs = concentration just above the surface of the catalyst (kmolfm 3 ) ; CF = feed concentration (kmol/m3 ); Tw = wall temperature (K); T� = catalyst temperature (K); T= fluid temperature (K); lX[c = heat transfer coefficient between the fluid and the surface of the catalyst pellet (kcal/ m2 • h K); �� = heat transfer coefficient between the fluid and the cooling jacket (kcal/m2 · h · K); Aeff = effective axial heat conductivity in the catalyst phase (kcal/m · h K). Eigenburger ( 1 972) has shown, using this model, that the infinite number of steady states predicted by models not accounting for feedback of heat transfer by conduction are reduced to a few steady states when axial heat conduction is taken into consideration. Mehta et a/. ( 1 98 1 ) used a pseudohomogeneous plug flow non isothermal model (which does not predict multiplicity of the steady states), to show that in certain cases a decrease in the feed temperature causes a transient increase in reactor temperature. this behaviour is usually called wrongway behaviour and is very different from wrong directional creep of the ignition zone discussed earlier. While wrong directional creep is associated with the instability of the catalytic bed as shown earlier, wrongway behaviour is a relatively simple phenomenon associated with the interaction of thermal and concentration waves inside the catalyst bed as will be discussed later in this section. It can be predicted by the radiation cell model as shown earlier. In fact, this phenomenon has been predicted earlier to the work of Mehta et al. ( 1 98 1 ), by many investigators (e.g. Boreskov and Slinko, 1 965 ; Hansen and Jorgensen, 1 974; van Deesberg and De Jong, 1 976a,b; Sharma and Hughes, 1 979). However, Mehta et al. ( 198 1 ) identified the key rate processes and parameters which cause this behaviour and developed a simple technique for a priori prediction of the highest transient temperature reached without solving the transient equations . This interesting dynamic feature is caused by the difference in the speed of propagation of the concentration and temperature disturbances. The cold feed cools the upstream section of the reactor and decreases the reaction rate and conversion in that region. The cold fluid with higher than usual concentration of unconverted reactant eventually contacts hot catalyst particles in the downstream section of the bed. This leads to a very rapid reaction and a vigorous rate of heat release, which causes a transient temperature rise. Several years later, Pinjala et al. ( 1 988), used a pseu do  h omogeneous nonisothermal model with superimposed axial dispers i on of m ass and heat, with unequal mass and heat Peclet numbers, Pem :t: Peh. The values for the Peclet numbers used were realistically large, (Pem = 400, Peh = 1 00, and Pem = 800, Peh = 200, and in some cases Pem = 1 60, Peh = 60). ·
·
S.S.E.H. ELNASHAIE and S.S. ELSHISHINI
382
The model dimensionless equations
The dimensionless mass balance equation is given by, 1 ax 1 a2x ax n  ·  =  ·     Da · ( e xp (1 / Y) ) X Le a r Pem az' 2 Jz'
(3 .2 1 9)
The dimensionless heat balance equation is given by, 2
1 ·a Y ay ay n = ;z  , + ,B · Da · (exp ( 1 1 Y))X + U( Yw  Y) ar
Peh az
az
(3 .220)
With the boundary conditions, ax
 = Pe (1  X)
at z = O
az'
m
(3.22 1 ) ax = a r
at z = 1
Jz'
Jz '
=O
(3.222)
The dimensionless parameters and variables are defined as follows:
X = CI C,f , L · v · p1  C1 k
peh = "" ' e
Da =
Y=
Rc;T I E,
z' = z l L
Pem = L · v l Dax
Cs t: · pr crt
Le = 1 + (1  t:) · Ps ·
L · ko  en! rf v
v·t r= · 1 t: · L Le
where k0 = preexponential factor for the reaction constant and Dux = axial dispersion coefficient. The definition of Lewis number Le, shows clearly that the appre ciably large adsorption mass capacity of the catalyst is neglected, while the negligible gas phase mass capacity is taken into consideration and therefore Le is given very large values (e. g . 200, 500, 2000) . This is obviously not physically sound for catalytic reactors as explained clearly
MODELLING AND ELEMENTARY DYNAMICS
383
earlier and demonstrated very c learly in the 70' s and 80' s by Elnashaie and coworkers (Elnashaie and Cresswell, 1 973ac) . Pinj ala et al. ( 1 986) have shown, using the above model that in the region of un ique steady states, the axial dispersion of heat determines the magnitude of temperature excursion when wrongway behaviour takes place. While in the multiplicity region, the thermal dispersion may enable the wrongway behaviour to ignite a low temperature steady state leading to disastrous runaway of the reactor temperature . The calculations of Pinjola et al. ( 1 988) show that the axial dispersion of heat has a major effect even for very large Peclet numbers. Their results also confirm the conclusion of Puszynski et al. ( 1 98 1 ) that multiplicity of the steady states for the axial dispersion model is possible even for high values of the Peclet numbers. A similar pseudohomo geneous, nonisothermal axial dispersion model has been used by Stroh et al. ( 1 990) to i nvestigate the possible number of steady states in non adiabatic tubular reactors.
The influence of interphase mass and heat transport resistances between the gas and the catalyst phases has been i nvestigated by Chen and Luss ( 1 989) using a simple two phase model that distingui shes between the bulk gas phase and the catalyst solid phase. Axial dispersion of mass and heat is taken into consideration for the gas pha se, while for the solid phase axial conduction is neglected (which is not a very realistic assumption). Furthermore, the heat dynamics of the solid is taken into consideration as well as the dynamics of the mass balance for the solid while the intraparticle resistance is accounted for through the effectiveness factor ( 1]), which is evaluated analytically assuming the particle to be isothermal at its surface temperature ( }T,). The solid catalyst particle is assumed to be at pseudo steady state, that is negligible mass capacity of the catalyst pellet, which has been shown earlier to be a physically incorrect assumtion.
The model equations (Chen and Luss, 1 989) The dimensionless mass balance equation for the bulk gas phase: ax 1 a2 x ax +  = M ( Xs  X) Le ar Pem az' az'
··2
1
(3 .223)
The dimensionless heat balance equation for the bulk gas phase is g i ven by ,
S.S .E.H. ELNASHAIE and S.S. ELSHISHINI
3 84
Mass balance for the solid phase,
M(X  Xs ) = 1J, ( Y, ) · X, · exp
(1 _2_J yif
(3 .225 )
Y,
Heat balance for the solid phase,
(3 .226)
The effectiveness factor is given by, ( 3 .227) With the boundary conditions,
(1  X) = 
at z = 0,
1 ax ·Pem az'

ar
1
( Y  Y,:r ) =  · Peh az' ax = a r =O az ' az'
z= 1,
at
(3 .228 )
(3 .229)
The dimensionless parameters and variables are defined as follows: X = C! C , rf
P
e h

· P"1_cc:... rf . _L_·v_ Aa
z'
=zl
L
2hw · L
u
r, · v · p1 · Crf
R
De
De · L
4>s
2
p
=
4>o exp 2
e· L
Le
( 1 1] 
r;.
. v '1 = exp (I I Y, ) = 1

K
0
·V
yif
1
K(Y )
3k" L ( l e)
M
V·t
T = ·
,
K( Y,1 ) = v l L
L·K
0
MODELLING AND ELEMENTARY DYN AMICS
385
Where Yif is a reference temperature chosen by the authors so that is the intraparticle diffusion coefficient and A.a is the axial conductivity of the catalyst bed. The model is not very sound and the assumptions are chosen selectively without good physical justification. However, the model has the advantages of showing the effect of the heterogeneous nature of the catalyst bed on its dynamic behaviour. The results of Chen and Luss ( 1 989) confirm the results of Pinjola et al. ( 1 988) that a sudden feed temperature reduction leads to negligible temperature excursion when the conversion in the reactor is very low or very high, but it can lead to an appreciable temperature excursion for a reactor at the intermediate level of conversion. The results also show that both the twophase model and the pseudo homogeneous axial dispersion model predict the same magnitude of the transient temperature excursion when we select Peh = H, where H is the dimensionless interfacial heat transfer parameter. A large deviation between the predictions of the various models may occur when steady state multiplicity exists for some feed temperatures. Arnold and Sundaresan ( 1 989) have shown experimentally and theo retically, using a suitable mathematical model presented and described in their earlier work (Arnold and Sundaresan, 1 987), that the lattice of an oxide catalyst can act as a reservoir for oxygen, storing and releasing it for oxidation reactions at the catalyst surface under appropriate transient conditions. They investigated the implications of this lattice stored oxygen for the 2butene oxidation over vanadium oxide catalyst. Thermal reactor runaway was observed experimentally for large step increases in the feed butene concentration, following certain catalyst pretreatment. Step increases in gas flow rate and less severe step increases in butene, resulted in the reactor temperature overshooting its final steady state profile. These response characteristics are a direct consequence of the oxygen storage property of the vanadium oxide catalyst. Lattice oxygen has been shown to cause temperature overshoots following step increases in the feed concentration and feed gas flow rate, and to exacerbate the wrongway response following a step decrease in the reactor inlet temperature. Arnold and Sundaresan ( 1 989) investigation is one of the few studies on the dynamics of packed bed catalytic reactors that takes into consideration the catalyst surface adsorption capac ity which was made very clear by Elnashaie and coworkers (Elnashaie and Cresswell, 1 973ac, 1 974ab, Elnashaie, 1 977) several years ago. In most theoretical inves tig ati on s of reactor dy nam ics, it is invariably assumed that the cata lys t responds instantaneously to changes in the c at aly s t temperature and the gas phase composition immediately above the catalyst surface. This is a s eri ou s error in the dynamic mode lling
K (Yif) = v/L, De
386
S . S .E.H. ELNASHAIE and S.S. ELSHISHINI
of gassolid catalytic systems. The dynamics of the catalyst itself has been considered in only a few limited studies of reactor dynamics such as those addressing catalyst deactivation, poisoning or fouling (Ervin and Luss, 1 970; my references) or selfsustained reaction rate oscillations (Razon and Schmitz, 1 986). The only series of work that has consistently taken into consideration the dynamics of the catalyst itself for single catalyst pellet as well as the whole reactor (in fixed and fluidized bed), is the series by Elnashaie and coworkers (e.g.: Elnashaie and Cresswell, 1 973b,c, 1 974a,b, 1 975; Elnashaie, 1 975 ; Elnashaie et al. , 1 990). Ill' in and Luss ( 1 992) have recently recognized the importance of chemisorption dynamics and investigated the influence of reactant adsorption on support, on the wrongway behaviour of fixed bed catalytic reactors. This is certainly a positive step forward in the dynamic modelling of the reactors. However, the adsorption of reactants on the catalyst itself is obviously more important than its adsorption on the support. 3.2.5
Summary and Overview of the Modelling of Fixed Bed Reactors
The steady state and dynamic behaviour of fixed bed gassolid catalytic reactors still attracts considerable research attention because of their complex and interesting behaviour (Juncu et at. , 1 994; Stroh et al. , 1 990; Pita et al. , 1 989). Strong experimental evidence exists for the occurence of multiplicity of the steady state profiles and therefore instability of these reactors specially for highly exothermic reactions. Bistability (the simultaneous existence of two stable steady state profiles) has been experimentally observed in studies of CO oxidation (Sharma and Hughes, 1 979) and CO methanation (Wedel and Luss, 1984) in an adiabatic bed. A large number of steady states (up to 5) were observed by Hedges et a!. ( 1 977) in a shallow bed catalyzing CO oxidation. A number of mathematical models have been developed over the years to describe the behaviour of these important industrial units. These models can be classified on two different bases : 1 . The first classification is based on the number of phases considered; the models that do not distinguish between the bulk gas and the catalyst solid phase are termed pseudohomogeneous models, while those that take into consideration the difference between these two phases are termed heterogeneous models. Within this category of models, there can be different levels of sophistication, e.g. plu g flow and axial dispersion pseudohomogeneous models, heterogeneous models with i n terpha s e mass and heat transfer resistances o nl y and those with interphase as we ll as intraparticle mass and heat resistances, . . . etc .
MODELLING AND ELEMENTARY DYNAMICS
387
2. The second classification is based on the manner by which the axial
variation of the system variables along the length of the reactor is mathematically expressed. The models that consider this variation to be continuous are called the continuum models, while the models that discretize this variation are usually referred to as cell models . Obviously, within this category of models there can be different levels of sophistication as discussed before. Most realistic fixed bed reactor models are nonlinear and hetero geneous, and are thus always solved numerically through the discre tization of the axial variation into discrete steps whose sizes are depen dent upon the accuracy of the numerical solution regardless of the physical system that the model equations describe. These axial step sizes can be larger or smaller than the catalyst particle diameter, while at each step the catalyst pellet equations (in heterogeneous models) are solved based on the catalyst particle diameter. This situation obviously creates a contradiction between the simulation and the actual physical situation, which is not easy to resolve. Therefore, it seems that the cell model which discretizes the length of the reactor to cells having each a length equal to the catalyst pellet diameter (or characteristic length), is in this case more physically sound than the continuum model. Fortunately, cell models are also easier to solve and analyze than con tinuous models. The dynamic modelling of the catalyst pellets in the fixedbed reactor also deserve a short comment. In most theoretical investigations of reactor dynamics, it is almost always assumed that the concentration of species in the catalyst responds instantaneously to changes in the catalyst temperature and the gas phase composition and temperature immediately above the catalyst surface. The concentration dynamics of the catalyst itself has been considered in only a few limited studies of reactor dynamics such as that addressing catalyst deactivation (Ervin and Luss, 1 970) or selfsustained reaction rate oscillations (Razon and Schmitz, 1 987) as well as the work of Elnashaie and coworkers who have shown for single catalyst pellets (Elnashaie and Cresswell, 1 973a), fixed bed reactors (Elnashaie and Cresswell, 1 974) and fluidized bed reactors (Elnashaie, 1 977; Elnashaie and ElBialey, 1 980a; Elnashaie and Elshishini, 1 980b) that the chemisorption process on the surface of the catalyst pell et has an appreciable effect on the dynamics and stability of the system. In additi o n , Arnold and Sundaresan ( 1 989) investigated the effec t of catal y st oxygen c ap ac ity (lattice stored oxygen) on the dyn ami c behaviour of 2butene oxidation over vanadium oxide c atal y st in a fi.� ed bed reactor and showed that this catalyst surface capacity has strong influence on the dynamic behaviour of the system. II ' in and Luss ( 1 992) investigated the effect o f reactant adsorption on the inert catalyst
388
S.S .E.H. ELNASHAIE and S.S. ELSHISHINI
support and found that the reactant adsorption has an important effect on the dynamics of the system. It is thus easy to expect that adsorption of reactant on the catalyst rather than the inert support will have a more profound effect on the dynamic behaviour of the system. Early dynamic modelling work for fixed bed catalytic reactors has been reviewed by Froment ( 1 974) and also a good review till l 972 was given by Elnashaie (1 973b). More recent reviews include those by Jensen and Ray ( 1 982), Razon and Schmitz ( 1 987) and Harold et al. ( 1 987). Before we proceed further, we should define the important dynamic phenomenon of wrongway behaviour and the expected effect of reactant adsorption on this phenomenon. Wrongway behaviour is an important pathological phenomenon in fixedbed catalytic reactors. It refers to situations where a decrease in the feed temperature may cause a transient increase in the reactor temperature. This phenomenon has been observed by a number of investigators (e.g. van Doesborg and De Jong, 1 976a, 1 976b; Sharma and Hughes, 1 979) . Wrongway behaviour is different from the phenomenon of wrongdirectional creep presented and analyzed in some detail by Elnashaie and Cresswell ( 1 973a). Wrongway behaviour is simply a transient temperature overshoot with both the initial and final steady states being stable, while wrongdirectional creep is associated with the reactor instability as clearly shown by Elnashaie and Cresswell ( 1 973a) . Pinjola et al. ( 1 988) investigated the impact of thermal dispersion on wrongway behaviour using a p seudo homogeneous model and have shown that in the unique steady state region, axial dispersion of heat decreases the magnitude of temperature excursion during the wrongway behaviour, while in the multiplicity region, the thermal dispersion may cause the wrongway behaviour to ignite a low temperature steady state to a new high temperature steady state. Il' in and Luss ( 1992) have shown that when the wrongway behaviour leads to the formation of a downstream moving temperature front, reactant adsorption tends to moderate and decrease the maximal transient temperature of these fronts. However, when the wrongway behaviour generates an upstreammoving temperature front, reactant adsorption may exacerbate the transient temperature rise and may thus cause ignition of the reactor. They also found that this reactant adsorption may also lead to surprising dynamic effects upon changes in feed velocity. Chen and Luss ( 1 989) used a relatively simple model to investigate the effect of interphase and intraparticle transport resistances on the wrongway behaviour. They concluded that the twophase mode l and the pseudohomogeneous model predict similar results in the region of unique steady states, while a large devi ation between the predictions of the two models occur when steady state multiplicity exists for some feed temperatures .
MODELLING AND ELEMENTARY DYNAMICS
389
Other theoretical and experimental investigations of fixed bed catalytic reactors include the early work of Eigenberger ( 1 972a,b) who investigated the effect of heat conduction in the catalytic phase on the behaviour of the reactor. Van Doesburg and De Jorg ( 1 976a,b) investigated experimentally and theoretically the behaviour of the fixed bed methanator using a plug flow pseudohomogeneous model. Varma and Amundson ( 1 973) found five steady state solutions for the non adiabatic tubular reactor in which a single exothermic reaction occurs. Sinkule et al. ( 1 976) reported an infinite number of steady states using a heterogeneous model with thermal axial dispersion. Kapila and Poor ( 1 982) reported the existence of a maximum of seven steady states in a nonadiabatic tubular reactor. Mehta e t al. ( 1 9 8 1 ) used a pseudo homogeneous plug flow model to demonstrate the wrongway behaviour of packedbed reactors. Xiao and Yuan ( 1 994) developed an efficient algorithm for the simulation of the dynamic behaviour of a fixed bed reactor with flow reversal for the sulfur dioxide oxidation over vanadium catalyst and discussed the wrongway behaviour of the ignition zone in the reactor. Adaje and Sheintuch ( 1 990) have carried out an extensive theoretical and experimental investigation of the connection between the behaviour of the single catalyst pellet and the fixed bed catalytic reactor and concluded that in many cases, the behaviour of the reactor cannot be predicted from the behaviour of the single catalyst pellet. 3.3
3.3.1
FLUIDIZED BED REACTORS Introduction
Bubbling fluidized beds offer many advantages for conducting gas solid catalytic reactions. A relatively simplified physical picture of the bubbling fluidized bed (Kunii and Levenspiel, 1 969) shows that at a certain gas feed velocity passing vertically upward through a suitable gas distributor into a tube (or a vessel) filled with fine solid particles, the solid becomes suspended and the solid contents of the vessel behaves almost like a liquid. This gas feed velocity is called the minimum fluidi zation Umf· As the gas velocity increases above this value of Umt• almost all the gas entering the bed in excess to the amount necessary for minimum fluidization, passes through the bed in the form of bubbles which start quite small at the distributor and grow in size as they ascend through the solid bed. Thus, the bed is divided into two regions, a dense phase (sometimes called emulsion phase) which contains almost all of the solid (the flow rate of the gas passing through it is almost equ al to that necessary for minimum fluidization) and a bubble phase consisting of discrete bubbles rising along the h e ight of the bed. The bubb le phase ,
390
S . S .E.H. ELNASHAIE and S . S . ELSHISHINI
contains almost all the feed gas in excess to that necessary for minimum fluidization. The bubbles themselves are usually almost free of solid, however, they are surrounded by a cloud phase with a solid concentration much lower than th at of the dense phase. The rising bubbles are followed by a wake with a solid concentration lower than that of the dense phase. Since the cloud and wake phases are associated with the bubbles (and actually in a sense attached to the bubbles), they ris e through the bed with the bubb le velocity Bubble growth along the height of the bed is mainly a result of two factors, namely, the decrease in pressure on the bubble as they rise through the bed, and the coal e sc e n ce of the bubbles as they collide together in their journey up the bed. As the bubbles (with their cloud and wake) reach the upper surface of the bed, they burst and the solids in the cloud and wake fall down into the bed. This process causes continuous circulation of the solid which is on e of the main advantages of fluidized beds. In fluidized bed catalytic reactors, the rising bubbles repre se nt a bypass of a high percentage of the reactan t s without coming into contact with the catalyst in the dense phase, thus tendi n g to decrease the efficiency of the bed as a reactor. However, mass transfer between the bubble and den se phase gases decreases the effect of this bypass. In addition to that, the mass transfer of the re action pro ducts helps to save the products from further reaction to undesirable products and also help to break the thermodynamic equil i brium barrier of reversible reactions by acting as a natural membran e removing the products from the reaction mixture .
.
3.3.2
Modelling of Fluidized Bed Catalytic Reactors
Despite the l arg e number of steady state models for isothermal fluidized bed catalytic reactors, there are only few dynamic models for the non isothermal fluidized bed catalytic reactors. 3.3.2. 1
Isothermal steady state models
There is a l arge number of steady state isothermal models for fluidized b ed catalytic reactors. These models differ from each oth er with respect to the different as sumptions involved in the model d e velopmen t Some of these mod el s are described very briefly in the follo wing : .
I. The DavidsonHarisson model This model was first proposed by Orcutt et al. ( 1 962) but was developed more fully in the book by Davidson and Harri s on ( 1 963 ) . It is based o n the following assumptions,
MODELLING AND ELEMENTARY DYNAMICS
39 1
All the reaction occurs in the dense phase which is considered to be either perfectly mixed or in plug flow. All gas in excess to that necessary for minimum fluidization goes through the bed in the form of bubbles. All bubbles are of equal si z e and are uniformly distributed throughout the bed The bubble phase is in plug flow. Mass transfer takes place between the bubble phase and the dense phase with one mass transfer coefficient (one mass transfer resistance between the bubble phase and the dense phase).
(i) (ii) (iii)
.
(iv) (v)
II.
The KuniiLevenspiel model
The details of the development of this model is given by Kunii and Levenspiel ( 1 979). It is based on some assumptions which are common with the DavidsonHarrison m odel but differs in other aspect s .
(i) Completely mixed dense phase. (ii) Bubbles of uniform si ze rising in plug flow through the bed. (iii) There is a cloudwake region (which is of lower solid density than the dense phase) s urrou n di ng the rising bubbles and where reaction can take place. (iv) Consistent with the above picture mass transfer between the bubble and dense phase takes place in two consecutive steps, between bubble and cloudwake then between the cloudwake and dense phase. ,
III.
The countercurrent backmixing model
This mode l has been developed by Potter and his coworkers and has been discussed in a number of publications which are well reviewed by Potter ( 1 979). This model is more general than that of Ku niiLevenspiel model in a number of respects which can be summarized in the following points : (i) The model takes into account the rate of reaction in the cloudwake phase which is neglected in Kuni i Leven spiel model. (ii) The variat io n of the bubble size along the bed is tak e n into consideration us in g a s i mp l e linear empiric al relation and the initial bubble size at the distributor is esti mate d using another simple empirical relation.
Fryer and Potter ( 1 972) suggested that the model prediction will still be acc urate if the bubble size is taken constant and equal to the bubble size at 40% height of bed.
392
IV.
S.S.E.H. ELNASHAIE and S . S . ELSHIS HINI
The Kato Wen Model
This model was first described by Kato and Wen ( 1 969). As with the countercurrent backmixing model, variation of bubble size with height is allowed for using the empirical relation of Kobayashi et al. ( 1 965), and the initial bubble diameter at the distributor is calculated from the correlation of Cooka et a/. ( 1 968). The bed is then divided into a number of vertical compartments in which their heights are equal to the bubble diameter at the corresponding bed height (i.e. it is a kind of cell model with the size of the cells varying with the height, in accordance with the variation of the bubble size). Two phases are then considered, the dense phase and the bubble phase and the reactant gas is assumed to be perfectly mixed in each phase. The interphase mass transfer coefficient per unit volume of bubble is related to bubble diameter by the expression of Kobayashi et at. ( 1 967).
V. The Werther Model The original model was described by Werther ( 1 980). The model is based on the analogy with gasliquid reactors in which the mass transfer between the two phases of the fluidized bed is represented as an adsorption of reactants from the bubble phase with subsequent pseudohomogeneous reaction in the dense phase. VI.
Other Models
In addition to the few models briefly described above, there are other steady state isothermal models described and reviewed in the literature. The reader is referred to reviews by Pyle ( 1 970), Grace ( 1 972), Rowe ( 1 972), Yates ( 1 975), Horio and Wen ( 1 977), Van Swaaij ( 1 978), Yates ( 1 983) and Grace ( 1 986). The reader should also consult the special topic issue of "Chemical Engineering Research and Design" dedicated to John F. Davidson ( 1 993). 3. 3. 2. 2
Nonisothermal dynamic models
Despite the abundance of isothermal steady state models for fluidized bed catalytic reactors as described above, there are very few non isothermal dynamic models for these reactors. In addition, it is clear that all the above models recognize in this formulation the important role of the bubbles, while some of the nonisothermal dynamic models s urprisingly ignore the bubbles (e.g. Luss and Amundson, 1 968 ; Hatfield and Amundson, 1 97 1 ) and instead take into account the mass and h e at transfer "resistances" between the gasphase and solid particles . These resistances are obviously negl i gi b l e because of the very s m a l l particle sizes used in fluidized be d s in addi t io n to the solid and gas flow ,
MODELLING AND ELEMENTARY DYNAMICS
393
conditions in these reactors which makes these models B .D. (Before Davidson, as Levenspiel ( 1 993) puts it). Nonisothermal dynamic models, which correctly include the bubbles in their formulation, are restricted to the models of Elnashaie and co workers (e.g. Elnashaie and Yates, 1 973a; Elnashaie and Cresswell, 1 973 ; Elnashaie, 1 977; Elnashaie and Elbialy, 1 980; Elnashaie and Elshishini, 1 987). All the models (except that of Elnashaie and coworkers) including the model of Bukur et al. ( 1 974, 1 977) ignore the important adsorption capacity of the solid catalyst in the dense phase, which has a very important effect on the dynamic behaviour of the system (Elnashaie and Cresswell, 1 973b,c, l 974a,b, 1 975 ; Elnashaie, 1 975; Elnashaie et al. , 1 990) as explained in chapter 4. The model of Elnashaie and coworkers has been successfully used to model the steady state and dynamic behaviour of Type IV industrial fluid catalytic cracking units (Elshishini and Elnashaie, 1 990a,b; Elshishini et al. , 1 992; Elnashaie and Elshishini, 1 992). The model has also been used by Choi and Ray ( 1 985) to model and control industrial polyethylene reactors . Consider the consecutive solid catalyzed reaction,
taking place in a freely bubbling fluidized bed. A schematic represen tation of the model is shown in Figure 3.82.
Assumptions used in Model Development The following assumptions are used in the derivation of mass and heat balance equations . (i) (ii)
(iii)
The gas in the bubble phase is assumed to be in plug flow. The extent of reaction in the bubblecloud phase is negligible. This assumption is justified by the experimental evidence of Torr and Calderbank ( 1 968 ), for small particle size (dP < 1 5 0J1) and high flow rates giving rise to fast rising large bubbles and negligible cloud phase. The dense phase gas is as su med to b e perfectly mixed. Important evidence ( Y ates and Constanes, 1 973a,b) shows perfect mixing is approached when the gas is strongly adsorbed on the solid. Also small height to diameter ratio and large gas fl ow rates stre n gthen this assumption (Kunii and Levenspiel, 1 969) . An average value of the bubble size and hence an average value
'
(iv)
S . S .E.H. ELNASHAIE and S . S. ELSHISHINI
394
       

To DENSE P H A SE
BUBB L E PHASE P LU G F LOW
Gc
PER FECT
1+Q...:..E=+1 M I X I N G
I I I
I
e SET I I I
:
POI NT: T M
I I I I I I
r _ _ _ _ _ _ _j
I
FIGURE 3.82
e
G
Simulation model for the twophase fluidized bed reactor.
for the exchange parameter is used for the whole bed. This assumption is widely used (Tigrel, 1 969; Tigrel and Pyle , 1 97 1 ; Chavarie and Grace, 1 972; Chavarie, 1 973; Bukur and Amundson, 1 975) . Chavarie and Grace ( 1 975) concluded that allowance for axial variation in bubble properties does not have an important effect on the representation of reactor performance. They have found in fact, that the model which best fits their experimental results is one which assumes uniform bubble properties throughout the reactor. (v) For fine solids (dp < 250J.1), the dense phase accomodates more gas than that necessary for minimum fluidization (Godard and Richardson, 1 969; Chavarie and Grace, 1 975). This is accounted for by the dense phase expansion parameter ¢ (Elnashaie, 1 977). (vi) Negligible mass and heat transfer resistances be tween the solid particles and the dense phase gas . (vii) The dense phase is of uniform temp erature, i.e. there is no rad i al or axial variation of temperature in the d e n s e p h as e
(viii) Both reactions are first order. (ix) Negligible heat of adsorption.
.
MODELLING AND ELEMENTARY DYNAMICS
395
Dense phase milSS and heat balance equations
Unsteady state material balance on component A in the dense phase gives,
¢�
dCA &
= Gl ( CAf  CA ) + QEA · Ac f ( CA  CA ) dz H
p5 (1  E)A1 H · k{ · CA ·
0
(3 .230)
Unsteady state material balance on component B i n the dense phase gives,
(3 .23 1 )
Unsteady state material balance o n component C in the dense phase gives,
(3.232)
Unsteady state heat balance on the dense phase assuming the bed to be adiabatic gives,
dT dt
P s Cps A I H(l  E) = GI p f · Cpf (Tf ·

T) + pf
J H

· Cpf · QH · AC (T  T) dz 0
+ A1 · H ·p5 ( 1  E) [k; · CA ( Ml1 ) + k; · C8 (Mf2 )]
(3.233)
Bubble phase milss and heat balance equations B ecause of the rel ativ ely high he at cap acity and adsorption capacity of the solid in the dense phase, the bubble phase can be assumed in pseudo steady state in relation to the dense phase dynamics. This assumption allows us to write pseudosteady state mass and heat balance equations for the bubble phase. The mass balance equation for the bubble phase
S . S . E . H . ELNASHAIE and S . S . ELSHISHINI
396
can be written as, (3 .234)
The heat balance equation can also be written as, (3 .235)
Reduced Model
If we further assu me that, QEA = QEB = QEc = QE = QH and then integrate equations 3.234 and 3 .235 and u se the result to evaluate the integrals in equations 3.2303 .233, we obtain the following set of normalized equations describing the dynamics of the fluidized bed reactor. Dense phase equations f/J A
dXA

· dt = B ( XAJ  XA )  a1 (exp (  y1 / Y)) · XA
f/J 8 ·
dX8
dt

= B (X81  X8 ) + a1 (exp ( y1 / Y)) · XA  a2 (exp ( y2 / Y)) · X8
dXc f/Jc · dt = B ( Xq
dY
(3 . 236)

Xc ) + a2 (exp ( y2 /
(3 .237)
Y)) · X8
(3.238)
f/JH · dt = B ( Y1  Y) + a1 · /31 (exp ( y1 I Y)) · XA 
+ a2 · /32 · (exp (  y2 / Y )) · X8
(3 .239)
where li is the reciprocal of the effective resistance time of the bed given by,
The bubble phase concentration and temperature profiles are given by,
MODELLING AND ELEMENTARY DYNAMICS
397
and,
�( ro , t) = Y(t) + ( Y1  Y(t)) · exp ( a · ro)
(3.242)
where OJ= zJH and a = QE Ac H!Gc . The selectivity and yield of the reactor are given by, ·
S=
GIXs + GcXBH G, ( Xs + Xc ) + Gc ( XBH + XcH )
(3.243)
and,
Y = G1X8 + GcXsH
(3.244)
( GI + Gc ) · XAf
where XAt= 1 .0. The twophase model parameters can be computed by many procedure s based on different hydrodynamic models and available in the literature (Orcutt and Davidson, 1 962; Partridge and Rowe, 1 966; Kunii and Levenspiel, 1 968a,b; Godard and Richardson, 1 969; Yates et al. , 1 970; Rowe, 1 972; Chavarie and Grace, 1 975). Comparison and discussion of these various procedures is beyond the scope of this section which deals with the phenomenon of thermal instability of the reactor. We will use the simplest procedure due to Partridge and Rowe ( 1 966) after introducing the modifications suggested by Chavarie and Grace ( 1 975) regarding the gas flow in the dense phase. However, any twophase model other than that of Partridge and Rowe, can be used instead. It is clear that after the use of the physically justified assumptions and the mathematical manipulation of equations (mainly the analytical solution of the linear pseudosteady state mass and heat balance equations for the bubble phase and the analytical evaluation of the mass and heat transfer integrals in the dense phase), the model reduces to a form which mathematically resembles a CSTR, however, physically it is clearly very different from the CSTR.
Steady state analysis The steady state equations can be obtained by setting the transient terms in equati ons 3 .2363 .239 equal to zero. After some manipulation of the steady state e quati on s we obtain the following equati on for computing the steady state dense phase temperature (for XAt= 1 .0, Xst= 0.0) , G( �5 . ) = �s  Yf =
(
EX1 /32 · EX2 {3, + B + EX2 + B EX1
) ..
= H( �, ) (3.245)
S . S .E.H. ELNASHAIE and S.S. ELSHISHINI
398
where ,
The lefthand side of equatio n 3.245, G (Ys;) is proporti o nal to the rate of heat remov al and is represented by a straight line having a sl ope equal to one and intersecti ng the temperature axis at Y1. The right hand side i s proporti onal to the rate of heat generation and is repre s ented by a hi ghl y nonlinear function. Equ ati o n 3.245 is known to have up to five s te ady states for certain values of the parameters (Aris, 1 969) and can be solved graphically as shown in Figure 3 . 83a where two re gion s of multiplicity exist. Once Yss has been found graphically, the corre spondin g yield and selectivity can be readily computed from the steady state equations together with equ ati on s 3.243 and 3.246. The result of such computations is shown in Figure 3.83b. The desired operating temperature Yss for which the yield is higher than 0.97 lies between Y:�.� = 1 . 1 and Y55 = 1 .4. The feed temperatures c orre spondi n g to those desirable operating ��
.C i � .. c
1.0 .8
.. 0
l5 t;" . 6
� .2
·
� ... .. >
ao o E .. � :X:
;;
·"
P : H :t m e t e r ' :
0 1 : 1 0" · 0: 1 y, �. H
=
=
=
=
I � . y1 = 0.4 . p , 1 00 . (/E
10,
27
=
•
0.6 .
2.0
.2 0 1. 0
9
1.1
13
1.5
1. 7
1.9
2.1
2.3
y, dimt"'fls ionlt"SS dt"n� pt\a S fi' � t"mperat urr
FIGURE 3.83 Multiplicity, selectivity and yield for a case with two regions of multiplicity. Relatively slow second reaction B � c.
MODELLING AND ELEMENTARY DYNAMICS
399
temperatures lie between Y1= 0.725 and Y1= 1 .0. The desirable steady states in this case lie on a stable branch of the heat gene ration curve, i.e. they satisfy the slope condition for stability and therefore are stable pro v ided they satisfy the dynamic condition discussed earlier. For the case in Figu re 3.83, if YF 0.75 three steady states are possible (A, B, C); the hi gh temperature s teady state, C, is the desired operati ng point . Therefore for this case preheating of the reactor at startup is necessary to attain the desi red steady state . For Yt = 1 .025, three steady states exist; the low tempe rature one is the desired steady state and there fore preheatin g should be avoided in thi s case. For 0.77 < Yt< 1 .075, un ique steady state exists and any startup policy will lead to the desired steady state. When the second reaction B � C, is fast, the i ntermediate branch of the heat generati on curve disappears. This is shown in Figure 3 .84b. In this case, the desired steady state does not satisfy the slope condition and therefore it is unattainable. This problem will be dealt with later in this secti on . : �i; L
� "' .. g
31! �
�� 0' 0
 e .. .. .. I
1. 0
.8
/
.6 .4
/
/
/0 )
(a)
.2 0 1.0
.. :; '0 .. E t a> ' .:
.8
.;
.2
!5 ., :5
_
�K �..
"' vi
.9 ., .7
.6
.5
.4
0
(b)
7
1 .3
'1 . dimll!'f"lsionless d t t15e p � Q se t e m pe r o t urt
1.5
.3 .1
i
j E
' c
i
!
,...
FIGURE 3.84 Multiplicity, selectivity and yield for a case with relatively fast _ second reaction B � C. (I) Adiabatic heat r va line (K 0). (II) Feed back heat r oval line (K 2. 4).
em
=
emo l
=
400
S . S .E.H. ELNASHAIE and S.S. ELSHISHINI
The effect of exchange parameter, QE Increasing the value of the exchange parameter, Q£, causes an increase in the value of the maximum yield of the reactor. It also causes the optimum operating temperature to be shifted to higher values. The effect of QE on yield and selectivity is shown in Figure 3 . 84.
FeedBack Controlled System When the rate of the second reaction B 7 C , is fast, we have seen that the desired steady state does not satisfy the slope condi ti on and therefore it is not attain able (Figure 3.84). To stabilize this steady state, we can us e nonadiabatic operation However, another more flexible alternati v e is feasible, that of using a feedback control system shown by the dashed lines in Fi gure 3 . 82. For this control system, the measured output temperature is the controlled variable and the temperature of the middle steady state, Tm . is the set point for the proportional controller. The manipulated variable is the heat input to the feed heater. If we assume that the dynamic lags of the measuring instrument and the feed heater are negligible and further assu me that the measured output temperature is almost equal to the dense phase temperature, then the heat balance equation becomes, .
dY 
¢ H · dt = B ( Y1

Y) + a 1 · {31 ( exp (  y1
I Y ) ) · XA
+ a2 · /32 (exp ( y2 I Y)) XB + BK ( Ym  Y) ·

·
(3.247)
where K is the gain of the controller multiplied by the steady state ga i n of the feed heater.
For this case, the equation for the computation of the steady state dense phase temperature is, (3 .248)
The righ t hand side of equation 3 .248 is the same as that of e q uation 3 . 245 . H oweve r the heat removal line h as a s lope of (1 + K) and intersects the temperature axis at the point ( + KYm ) I ( 1 + K). Fo r K = 0, i.e. uncontrolled system, the heat removal line is denoted (I) i n Figure 3.84a, multiple steady states exist (A, B , C ) and the de sired ste ady state (B) is unstable. For the feedback controlled system with K = 2.4, the heat removal line is denoted (II) on Figure 3.84a. For this ,
Y1
case only steady state (B) exists and it satisfies the slope condition . Therefore the desired steady state is stable provided it satisfies the dynamic condition discussed in the following section.
MODELLING AND ELEMENTARY DYNAMICS
Ill
. u
:l "0 0 ... 0.
�
..
"0 ..
E t
.�
1. 0
0.8
0. 6
1. 0
H : 20. 0
( I ) Q E : 0. 0 ( 2 ) 9 E = 05 ( 3 ) Q E : 1 0.00
0. 8
0. 6
2 0. 4
·s � ..
>
t; "!. :
0 ·9
0 8 +.rl 0 0·1 0 ·3 0 ·2
FIGURE 3.87 Phase plane for unstable steady state giving rise to a limit cycle, K = 2. 4, 'A = 1000.0, � 400.0. =
For any steady state that satisfies the slope condition 3 .249, we compute the c ritical LS' from e qu ation 3.250. The functions that appear in equations 3.249 and 3 .250 are defined in the nomenclature.
Dynamic behaviour
Consider for example, the steady state C in Figure 3 .83a which satisfies the slope condition. The phase plane trajectories for thi s case with LS' = 0. 25 (th e Ls;, for the steady state is 28.8), are sho w n in Figure 3 .86. It is c lear that preheating at startup is necessary to attain the desi red steady state (C). For the case show n in Figure 3.84, the desired steady state (B) satisfies the slope condition for K � 2. 4. For K = 2. 4, the LS�, computed from equation 3.250 gives LS�, = 1. 76. Therefore if LS' > 1. 76, this unique steady state will be unstable and a limit cycle is obtai ned on the phase plane as shown in Fig ure 3 .87. The c orresponding time variations of XA, Y and the yield are plotted in Fi g ure s 3 . 88 and 3.89. It is i ntere s tin g to notice that the yi e l d of B exceeds 1 .0 for a short ti me during each c ycle . This behaviour is not possible for homogeneous re acto rs where all components have the same capacitance (D oug l as , 1 972) .' However for the system c ons id ered here, the capac i tan ce of component A is taken higher than that of component B and therefore this increase in the yield of component B above 1 .0 is achieved at the
S.S.E. H . ELNASHAIE and S.S. ELSHISHINI
404
K = 2 · 4 ( u n s t ablt ) K = 3 2 ( un s t a btt )      K : 50 ( s t ablt )
�n\i I
!
0
200
400
600 t , t i m e ( m i n )
BOO
1 0 00
1 200
FIGURE 3.88 Concentration and temperature fluctuations with time for the cases shown in Figures 3.87 and 3.90.
expense of component A adsorbed on the catalyst content of the reactor. It is also noticed that the limit cycles shrink with increasing controller gain, till the system becomes stable as shown in Figure 3 .90. For all the limit cycles shown in Figu res 3.873 .90, the time average yield of component B is lower than that of the optimum steady state yield. However, there may be, of course, a certain combination of parameters for which the limit cycle time average yield is higher than that of the steady state yield (Dorawala and Douglas, 1 97 1 ). It is a tedious task to search for such cases without some analytical guide. A
2
�
v
1:
0 :2 �
>
20 ... 1 8
\.6
1 ·4
 K = 24 ( u n s t abtt ) K = 3 · 2 ( un s ta t> • • I    K = S 0 ( s t a bt t I
··
� � It I'
1 ·2
1 0
r.
I'
jl
0.8
>. 0.6
0· 4 0 2
FIGURE 3.89
1
, l i m P { m in )
Yield fluctuations with time for the cases shown in Figure 3.88.
MODELLING AND ELEMENTARY DYNAMICS
405
,_ 4 .., j(
:
2.4
R : 3 .2 j( = 3 6
R : S. O
( slob l l! s l . 5\ )
0.9+r.r'
0
01
0. 1
0.2
FIGURE 3.90 Effect of controller gain K on the size of the limit cycle. The same parameters as in Figures 3.88 and 3.89.
considerable advance along this road has been achieved by the extensive work of Douglas ( 1 972) . However the techniques developed do not accurately predict the behaviour for cases with large amplitudes of oscillation.
Summary of the main points in this section A simplified dynamic model for the fluidized bed reactor has been developed and presented in this section to study some of the problems associated with catalytic exothermic consecutive reactions taking place in a fluidized bed reactor. Attention has been focussed on the implication of the multiplicity phenomenon on the yield of the intermediate product . It has been shown that the maximum yield of the reactor may correspond to an unstable steady state of the system . A simple proportional feedback control system is suggested and it was shown that for this closed loop system a unique unstable steady state is possible giving rise to limit cycle behaviour . During each cycle of th e oscillation, the yield of the reactor exceeds unity for a short period. This behaviour is characteristic of heterogeneous catalytic systems with unequal capacitance for the various reacting components but this phenomenon is not po s sib le for homogeneous systems. The l imit cyc les di sappe ar by increas i n g the gain of the proportional controller. More details regarding this problem will be given i n chapter 4.
CHAPTER 4
Static and Dy namic Bifurcation Behaviour and Chaos in Some GasSolid Catal ytic Reactors
Four cases will be covered in th i s c hapter . The first case is a case of consecutive exothermic reactions takin g place in freely bubbling fluidized bed catalytic reactors. In this first case, detailed results for static and dynamic bifurcation as well as di ffe ren t types of chaos will be presented and discussed for both the three dimensional model and the two dimensional e x tern al l y periodi call y forced model. The externally forced three di men s i onal system will not be covered because it involves external fo rci n g of chaotic attractors which is not investigated enough in the literature and is still a research topic. The second case will deal with static and dynamic bifurcation of type IV industrial fluid catalytic cracking (FCC) units, which are i mp ortant units in the petroleum refining industry. Chaotic behaviour of these units has not been yet investigated in the literature and is still a research topic. The third case deals very briefly with the extensively in ve sti gate d (theoretically and e xp erim entall y ) case of the catalytic oxidati on of carbon monoxide to carbon dioxide, an imp ort ant reaction for air pollu tion control. The fourth and last case in this chapter involves the industrial fluidized bed catalytic reactors for the production of po ly ethy lene
which is a very important unit in the petrochemical industry. The pre sentation will i nvo l ve preliminary static and dy n ami c bifurcation re sults . The full investigation of the bifurcation and chaotic behaviour of this in du s tri ally important unit is still a research subj ect .
4. 1
FLUIDIZED BED CA TA LY T I C REACTOR WITH EXOTHERMIC CONSECUTIVE REACTIONS
The e leme ntary dyn ami c al an al y s i s of the fluidized bed catalytic reactors pre s ented and discussed earlier in chapter 3, has shown that the two dimen s ional case (where the ch e m i sorpti on c apacity for the desired intermediate product B is negligible compared with the chemisorption 406
STATIC AND DYNAMIC B IFURCATION BEHAVIOUR
407
capacity of the reactant component A and the heat capacity) can have oscillatory behaviour for certain regions of parameters. The existence of such periodic attractors is actually the highest degree of complexity for these autonomous (unforced) twodimensional systems. However, more complex dynamic behaviour can result when this twodimensional system is externally forced. This more complex dynamic behaviour includes in addition to periodic attractors, quasiperiodic as well as chaotic attractors. On the other hand, the three dimensional system (resulting from relaxing the assumption of negligible chemisorption capacity of component B ) will show, even for the autonomous (unforced) case, complex dynamic behaviour including in addition to periodic attractors, quasiperiodic and chaotic attractors. Forcing the three dimensional system is clearly quite complicated since in certain regions of parameters we will be forcing chaotic attractors. Forcing chaotic attractors is not very well studied and is still a subject of research. From a physical point of view, a high percentage of the efficient catalysts for consecutive catalytic reactions will have the characteristics of weak chemisorption of the intermediate desired product B, since strong chemisorption of B will imply in most cases high rate of reaction of this component and therefore lower yield of the intermediate desired product B . Thus from a physical point of view, it is clear that the appropriate sequence of investigation should be the investigation of the twodimensional autonomous (unforced) system followed by the forced (nonautonomous) twodimensional system, followed by similar (but more complex) investigation of the threedimensional system. From a dynamical system analysis point of view, the same sequence holds because of the relative simplicity of the twodimensional system compared to the three dimensional system. This physically and mathe matically justified sequence of presentation and discussion will be used in this section for the sake of simplicity of presentation for the reader. We start by the derivation of the model for the more general case of the three dimensional system with PID Control, since all the other cases are special cases of this more general system. The Model of the Three Dimensional case with PID (Proportional IntegralDerivative) Control . Mathematical formulation of the problem
The problem i nv e sti g ated is that of the catalytic exothermic consecutive reaction netw ork:
408
S . S .E.H. ELNASHAIE and S . S . ELSHISHINI y
·  · · · · · · · · ·  · · · · ·  · · · · · · ·
T
T
Bubble pha.j( X1° , XY , P0 ) = Fj ( X? , xf , P0 )  XP (O) = Xi (l)  XP (O) = 0
i = 1,2 (4.23)
Equation 4.23 gi ve s two nonlinear algebraic equations w ith three unknowns X 1 , X2 and Po · Additional anchor equation is required to specify the problem completely . The steady state solution (X5) of the autonomous sy st em (Am = O ) at specified parameter sp ace is given by: (4.24)
416
S . S .E.H . ELNASHAIE and S . S . ELSHISHINI
Since the autonomous system output is a single unstable steady state surrounded by stable limit cycle , one of the steady state variables (X1s , X2s) is fixed. This represent an anchor equation which is parallel to one of the phase plane axes and intersects the limit cycle phase plane transversally. The anchor equation ensures that the fixed state variable is lying on the limit cycle and eliminates the infinite solutions of the problem, since each point on the limit cycle will coincide with its i mages after one period of oscillation. The system is now composed of two nonlinear algebraic equations with only two unknowns which can be solved by Newton' s method, whose Jacobian matrix (for fixed X2 [X2 = XzsD is given by:
[
d Fi 1  axr Jd F'z axf 
(4 .25 )
Since F; is not algebraic and can only be evaluated by integration, the partial derivatives are obtained by integrating the following variational equations simultaneously with equation 4.20. Let:
ax; ' dPo
n. =
Differentiation of equation 4.20 with respect to following variational equations: n =2 a £iqlil  p, "' _1L_ '¥ =  � :�x k t u 'r
and,
o
k=I
a
k
•
dO; "' ax iJ!; .Q k , = = JiF. + p,o � n=2
u'
k=I
k
The initial co ndit ion s of these equations at where c5il is
the Kronecker c5.
(4.26)
Xf and P0 gives the
i = 1, 2
i = 1, 2
r=0
(4.27)
(4.28)
are: (4.29)
STATIC AND DYNAMIC BI FURCATION BEHAVIOUR
417
Integration o f these variational equations simultaneou sly with equation 4.20 to 'f = 1 give the elements of the Jacobian as follows :
dF
ax]o I
and
Ti]
_ UI

() p,, = Q i = _t; (X(l), l )
JF
i = l,2
(4.30)
0
Equ ati on 4.30 shows that it is not necessary to i ntegrate equation 4. 1 6.
2. Computation Techniques The bifurcation diagrams for the auto no mou s system and the results for the two parameters continuation analysis can be obtained by using the software package A UTO of Doedel and Kemevez ( 1986). This package is able to perform both steady state and Hopf bifurcation analyses, inclu ding the determination of entire periodic solution branches. DGEAR s ub rou tine (Gear, 1 97 1 ; IMSL user manual, 1 985), with automatic step size to ensure accuracy, for stiff differential equation can be used for numerical simulation of the different attractors. The differential equations of the present system are quite stiff and in many cases abound on allowable error as small as w 1 2 should be introduced to obtain accurate results. Needless to say, double precision should be used throughout all the computations.
Analysis of the unforced (autonomous) system The two dimensional unforced (autonomous) and uncontrolled system has a large parameter space ( 1 2 parameters), therefore it is almost impossible to investigate the entire static and dynamic bifurcation behaviour of this system over the entire twelve dimensional parameter space. Instead we chose in this part of the book to present to the reader a set of physciochemical, design and operating parameters which were used earlier by Elnashaie ( 1 977) and are given in Table 4.2. The addi tional parameters of the control system (or the coo l in g system for the nonadiabatic case) are two, namely K and Ym. The value of Ym which is the set point for the controller (or the coolant te mperature for the non adiabatic system) is kept constant at the temperature corresponding to the maximum yield of the desired intermediate product B (El n ash aie et al. , 1 993). The richness and complexity of the behaviour associated with the oneparameter investigation presented in this section, strongly j ustifies this severe reduction in the parameter s pac e d i men sion s .
S . S .E . H . ELNASHAIE and S.S. ELSHISHINI
418
a: E
" t
�
11.
�
�
0
j
· ;;;
�
0 ,.:
3.5 J. o 2.5
Ym : 1 . 0 1 6 6 1 13 5 Y I : O U U 72 2 4
2 0 1. 5 ·  ·   
10 o.s
. .. _ _ _ .. .. .  . .  . .. (o )
0. 0  0. 7 5
 0. 2 5
0.25 C o n t r o ll e r
a:
E �
j
a.
�
c " a "' "' .!1
�"' c "
�
a
,..
a:
�
c
2 "' c
0. 7 5
12 5
2.5 2.0
Y m : 0. 5 8 2 2 7 1 4 6 Y l : 0. 9 7 7 2 7 7 6 1
1.5
�
1.0 0.5
00
 1 00
0.00
{b )
1.00
3 00
2 .00
C on t ro l l e r G ain ( K )
2.2 5
ad iabatic
I
e �
0 ..c ll. " "' c " a .. "' "
Goin ( K )
1 . 50
?E�
b r a nc h Sta b l e Un1fa b l t Puio d i c b r a n c h s ta b l e Un s to b t e
S t . st
: · ·�
.. ..,
075
�
6 ,.. 0 00 1 0
I
w .
10
Ym : 0. 9 2 9551 3 0 Y l : 0. 5 5 3 4 2071
.
J.o
C o n t r o l !� • G a i n
(c ) 5.0
7.0
9 .0
(K )
FIGURE 4.2 Bifurcation diagrams for the autonomous (unforced) nuidized bed reactor. (a) a2 108, (b) a2 = 1010, (c) a2 = 101 1 • =
STATIC AND DYNAMIC BIFURCATION BEHAVIOUR
419
Since the system contains two independent chemical reactions the maximum possible distinct steady states are five according to the relation (2N + 1 ) where N is the number of independent chemical reactions (Aris, 1969), of which N, two in this case, are certainly unstable (saddletype). When the rate of the second reaction (B � C ) . is very fast, the maximum possible number of steady states are three, of which only one is certainly unstable (saddletype). The effect of the second reaction speed (a2 = 1 08, a2 = 1 0 10, a2 = 1 0 1 1 ) on the possible number of steady states are displayed on the one parameter bifurcation diagrams developed by using the normalized controller gain (K) as the bifurcation parameter as shown in Figures 4.2a, b, c. The second reaction speed does not only affect the number of steady states, it also affect the dynamic of the reactor as shown clearly by the number of the Hopf bifurcation (HB) points and their locations (Figures 4.2b, c). Figures 4.2a, b, c show the bifurcation diagram for three values of a2. For each case Yf is chosen to give the maximum yield of the desired intermediate product B . For the three cases this maximum yield correspond to saddle type unstable steady state. The value of Ym (controller set point or cooling coil temperature) is chosen equal to the temperature corresponding to the maximum yield of B for each case. Figure 4.2a shows the case for a relatively slow second reaction, a2 = 1 08• In this case we notice the existence of a region of five steady states where the steady state with maximum yield of B is stable for all values of K (even for some negative values of K) and therefore the reactor can be operated adiabatically without control (K = 0), at this desired state of the system. Fiqure 4.2b is for a case of higher rate of the second reaction, a2 = 1010• In this case it is clear that the system cannot be operated adiabatically without control at the middle desirable state, because for K = 0, this steady state is unstable. For this case a region of five steady states exits, however it is shifted to higher values of K compared with the case in Figure 4.2a. Figure 4.2c shows a case of high a2 = 1 0 1 1 , in this case the region of five steady states disappears. The desired middle steady state is unstable upto high values of the normalized controller gain, K z 4.658. Stable limit cycles emerge from the second H B point close to K = 5, with a PLP as shown in the enlargement box. These limit cycles terminate homoclinically on the left very close to the tip of the i mp erfect pi tchfork Unstable limit cycles emanate from the other HB point and terminate homoclinic al ly at the middle unstable saddle type state. The presentation in this s e c ti o n of the book focuses on the bifur cation diagram of Figure 4.2c because it allows investigating the .
forcing of an autonomous periodic branch. For the case in Figure 4.2c any change in Yf causes a break in the imperfect pitchfork, which shows that the imperfect pitchfork in Figure 4.2c is structurally unstable.
S.S.E.H. ELNASHAIE and S . S . ELSHISHINI
420 .. 5 e � e
,., "' .. � .. ..
�
j
...
>"
0 6 50
ic
0 62 5
, . ··,·') '
o wo
0. 5 7 5
0. 550
o. m
I.
�'a"
..  ·
I
R a ion i Rtogion II
iio ilb
0.500 +,'f..,..j 10 4.0 30 so 60 2 .0 Control ! or Gai n ( K ) (ia)
1
so
> 1 .00
0.50
0.0 0  1.0
00
10
20
2 .00
K
30
40
1 . 50 .
Y t : QS U 5 s.o
( ib)
60
10
20
00
1.0
2. 0
200
K
3 0
40
50
60
J.O
4.0
5o
6 0
1.50
> 1 . 00
1 .00
0 50
000 10
00
00
10
20
30
K 2.50
'f' 1 : 0. 58 75
4.0
5.0
050
oro
 1. 0
6.0
( ic )
2.00
K
1 . 50 1 00 'I'
=
0. 6.4 0 0
O.sot'o�,,,,1  1 .0 00 1.0 2 .0 3 0 l.. O 5.0 6 . 0 K
FIGURE 4_3 Two parameter bifurcation diagram for static limit points (SLPs) locus and Hopf bifurcation points (HBs) locus and the corresponding one parameter bifurcation diagram at regions ia, ib, ic, iia and lib.
Figure 4.3 shows a two parameters ( YF ve rs u s K) continuation diagram which exhibits the loci of static limit point s (SLPs) and Hopf bifur cation points (HBs). When Yf= 0.55342072 the pitch fork bifurcation diagram breaks up into two disconnected curves and the variety of bifurcation diagrams corresponding to region i and ii are shown in Figures 4.3i(ac) and 4.3ii(a, b). The nature of these diagrams depends
STATIC AND DYNAMIC BIFURCATION BEHAVIOUR
� " i! K. E �
.. "" = .s::;
11'�
c u "'0 "' "' u
�
� )
I
e
I
1.1
:
• •
1 .0 PLP
c
0 ·;;;
0.95 0.9 0
.
e�� �
• • • •
O BS 0.00
.
 
stable steady stale
1 table manifold
0.20 XA
0 10
...
095
4.5 Sequence o f formation Yt= 0.5560. (b) Yt= 0.5535.
·.
�
·· ·· s t able sloody ••·�· · · · ·  . o o o o u nctableo $teady stah! . . . . . uno l ct> � morif o l d  s lc bl e man i f o ld ••••
O. SS
0.00
0. 40
0 30
FIGURE (a)
. '4
0 90 (a)
K : ] . 627 4 4 0 0 0 Y f : 0. S S J S 0 0 00 Ym : 0 9 2 9 5 5 1 JO
(·
1 .00
>
�=��, ��;.�tP · ·. .
1 .10
010
0.20 XA
···
(b)
0.30
040
o f the homoclinic orbit as Y1 decreases.
1 2 0 ,,
10
1
.0
0
1 . OD 0.
..· · •
• • .
0 0 o •
·· · · ·· · · · · •·
o� o o O ' ' ' ' ''' ' ' ' ''''' ' ''pO �' 9 0 t..::,:;;,F. : . .. . . . .. .. . . •
I
I
t.l()() �b I
I
0 .80+...,...'r,.,'....L.,r1 o. 5 o o 51 o 52 o .5 J o.54 o.ss o s& o 57 o.ss 1.�5
>
· 770 ·0 6�.oo
0 I 155
0.935
0 880
' �:·: � 0 93 5
0·825
0· 00 0 10
O · •o
o 20
0 ·2 0
O JO
0 · 40
0
825 0.00
o. &o
0 990
0 935 o.ee 0
8 �·'; . o:n o '�:;;�::7.:� d
� 0·10
0 20
O JO
0 40
0 82 5 ''....�___, .. 0 ·00 0 · 1 0 0 20 0 · 1 0 0·40
FIGURE 4.6 One parameter bifurcation diagram ( Y vs. Y1) for K = 3. 62744 and the corresponding phase planes in the different regions (AE). (A) Stable focal steady state. (B) Stable steady state and stable limit cycle separated by an unstable limit cycle. (C) Full time oscillator  unique stable limit cycle surrounding an unstable steady state. (D) Three steady states  one stable and two unstable. (E) Improper node (two negative unequal real eigenvalues). The boundary between region C and D is an IP orbit.
STATIC AND DYNAMIC B IFURCATION B EHAVIOUR
4. 1 . 1.2
423
The periodically forced (nonautonomous) case. Preliminary pres entation of periodic and chaotic characteristics
The case presented in this section is for a set of parameters giving a maximum of three steady states and with K= 3 . 62744 at Ym= 0.92955 1 3 . Figure 4.4 shows the static and dynamic bifurcation diagram with the bifurcation parameter being the feed temperature Yf This case is characterized by two static limit points (SLP), one Hopf bifurcation point (HB) and one periodic limit point (PLP). The periodic branch starting from the HB point terminates homoclinically near the SLP with Yf= 0.55343 1 9 . When the value of Yf during the forcing cycle reaches either the static limit point (at Yf= 0.55343 1 9) or the Hopf bifurcation (HB) point (at Yf= 0.5293739) we are not forcing a full time oscillator (region C) any more. During the forcing cycle we can force three steady states case (Region D), homoclinic IP orbit case (the boundary between region C and D), improper node case (two negative unequal real eigen values) (Region E), a bistability case of a stable steady state and a stable limit cycle separated by an unstable limit cycle (Region B) and a stable focal steady state case (Region A). The types of the phase planes of the autonomous system included during the forcing cycle depend on the position of the centre of forcing as well as the magnitude of the forcing amplitude. In this case, we chose a centre of forcing in region C very close to the homoclinical orbit (i. e. close to the boundary between regions C, D). The forcing frequency ( ro= 2II/P) used in this case is relatively high, where the ratio of the forced frequency to the natural frequency of the (0 p 5 = _£_ =  . system is a rational number of a value equal to 5 (0(} p 1 Therefore at low amplitude forcing the phase surface of the system is toroidal of dimension two and entrainment region (phased locking or resonant trajectory) prevails. However it is important at this early stage to make it clear that with regard to phase locking the statement "low amplitude forcing" is very relative and depends on the forcing frequency and the position of the centre of forcing relative to the homoclinical orbit as will be shown later. The possible attractors for periodically forced systems are periodic, quasiperiodic and chaotic attractors. When the forced system is periodic, its period is an integer multiple of the forcing period In the present section we present the results for three centres of forcin g �1 in regio n C but differ with respect to their distances from the homoc l i n ic i n fi n i te pe riod (IP) orbit. These cases correspond at the centre of forcing to a single autonomous stabl e limit cycle surroun di ng
(.
J
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FIGURE 4.7 Oneparameter stroboscopic Poincare bifurcation diagram (Y1 vs. Am, strobing every forcing period). (a) Am = 0.00.004. (b) Am =0.0040.009. (c) Am 0.0090.013. (d) Am = 0.0130.018. =
u n s table s teady state. The first case is for Y1 = 0.55342072 which is very close to the homoclini c (IP) orbit at Yr 0.5534245 . The second case is for Y1= 0.5532 which is slightly moved to the left of the homoclinic (IP) orbit, while the third case is for lj·= 0.5500 which is moved considerably to the left of the homoclinic (IP) orbit The positions of these three cases are clearly shown on Figure 4.4. .
The case of Yt = 0. 55342072
In th i s case the perio d of the unforced sy stem i s equal to 1 3 1 . 7250889408649s. A compl ete oneparameter stroboscopic b ifurc a ti on di agram is c ons truc ted for this case as shown in Figu re 4.7(ad). These figures are actually one bifurcation diagram which are pre sented in large sc ale s in order to be able to i dentify and ana ly s e the fin e structure of th e behaviour. It is c l ear that the behaviour is quite complex
and more enlargement of specific regions is needed.
STATIC AND DYNAMIC BIFURCATION BEHAVIOUR
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the Y1 stroboscopic points hi s togram are shown in Figures 4.14 (A2(a), A2(b )) . Shown clearly is the coexistence of two types of nearly periodic behaviour (laminar phases) of periodicities three and two and that the dynamics alternate between these two regions of periodicity and the chaotic behaviour (bursts). This new type of intermittency is not very common. Both the second and third stroboscopic iterate maps for this case are constructed in order to elucidate this new type of intermittency. For the second stroboscopic iterate map ( Y, (n + 2) vs. Y, (n)) the curve approaches the diagonal at two
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FIGURE
shown in Figure 4.25 (b) at the first period doubling point at a forcing amplitude equal to Am = 0.0000725. The second Floquet multiplier in this case is called the Principal Floquet Multiplier (PFM). At these conditions the trajectory loses its stability and stable traj ectory of twice the period is born. When the period doubling sequence starts, the torus
S . S .E.H. ELNASHAIE and S . S . ELSHISHINI
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FIGURE 4.31 (a) Enlargement ofthe part of the stroboscopic bifurcation diagram in Figure 4.30 (Am = 0.0000250.0000 85) showing incomplete cascade of period doubling. (b) Further enlargement of part of the stroboscopic bifurcation in (a) (Y1 = 0.99500.9968).
The periodicity of the trajectories at forcing amplitudes (A, B, C, D in Figure 4.30) is 2, 3, 4 and 5 respectively. These show a sort of period addingsequence as observed earlier by some investigators for a physically different system, namely the RoseHindmarsh model for neuronal activity (Holden and Fan, 1 992) .
S . S .E.H. ELNASHAIE and S . S . ELSHISHINI
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FIGURE 4.32 (a) Oneparameter stroboscopic bifurcation diagrams (strobing every one forcing period) at oir»o = PofP = 21l for the second case (Yt= 0.5532). (b) Oneparameter stroboscopic bifurcation diagrams (strobing every one forcing period) at oir»o = PofP = 21l for the third case (¥1 = 0.5500).
l/3.2.
The second and third cases
( Yf= 0.5532
and
0 .55 )
The complete oneparameter bifurcation diagram for the second and the third cases are shown in Figures 4.32a and 4.32b respectively. In Figure 4.32a the system goes from the entrainment region to the harmonic region through a compli c ated structure characterized by the large periodic regions. These periodic regions have sequence of periodi citi es 2, 3, 4, 5, 6, 7 , . show i ng another type of period adding sequenc e The bifurcation mechanisms of these period adding sequences are not fully studied in the literature. Figure 4.32b shows a complete oneparameter bifurcation diagram for the third case. The system exhibited a very complicated behaviour . . ,
.
STATIC AND DYNAMIC BIFURCATION BEHAVIOUR
0.0 1 100
,.,  H .,   H :�o2 . . . . . . . . . . PO,
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(K)
FIGURE 4.38 Bifurcation diagram of Y vs. K for ¥1= 0.55342072 and the rest of the parameters as in Table 4.2. HB h HB2 and HB3 are Hopf bifurcation points, SLP is a static limit point and SBP is a static bifurcation point.
the heat removal lines (for different values of K) Hr ( Y) vers u s Y. It is clear that for the uncontrolled adi ab atic case (K = 0) three steady states exist and that the maximum y ield of B corre s pond s to the middle unstable saddle type s teady state (Ym). Changing K whi l e keeping the Ym con stant causes the heat removal line to rotate around the point Ym. The rotation of the heat removal line follows the form shown in Fi gu re 4.37b provided the dimensionless feed temperature Y1 is kept constant at i t s base value corresponding to K = 0. Fi gure 4.38, which is pro duc ed using the software packag e AUT086 of Doedel and Kemevez ( 1986), shows that for this base value of Y1, the bifurcation diagram of Y vs. K appears as an imperfect pitchfork with one stat i c limit point (SLP); three Hopf bifu rcation (HB) points and one static bifurc ati on point (SBP) . At the SBP with incre a sing the v al ue of K the middle saddle type steady state be c omes a l o w temperature steady s tate . When Y1 :;:. 0.55342072 the pi tc hfork bi furcation diagram breaks at the SBP up into two disconnected curves and a variety of bifurcation di a gram s are ob t ai n ed The n ature of each of these di ag ram depends upon the value of Y1 and whether it is larger or smaller than 0.55342072. Such diagrams have also been obtained by El n ash aie and Elshishini ( 1 993) for industrial fluid c atalytic cracking (FCC) u n i ts as will be shown later in the section dealing with bifurcation behaviour of '
.
FCC units (section 4.2) .
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S ta b l e U n sta b l e
C o n t ro U e r G a in ( K )
FIGURE 4.39
Bifurcation diagram for Y1= 0.8.
Fig ure 4.39 shows the bifurcation diagram for Yf= 0.8 >0.55342072. This c as e with two Hopf bifurcat ion po int s and no homoclinical orbit is the one to be analyzed and disc u ssed in this secti on . I2.
Periodic Attractors
From the dynamic point of view, this nonchaotic bifurcation diag ram for Yf= 0.8 (Fi gure 4.39) is characterized for K > 0 by two HB points . The two HB poi nts are connected by a branch of periodic attractors as sh o wn in Figure 4.39. The behaviour around HB1 wi th K = 3 . 660462 is rather simple; for K < 3 . 660462 static stable ste ady states exist down till K= 0, and for K> 3. 660462 a period one (P 1) peri odi c attractors exi st and their ampl itude i ncrea s e s as K i n cre as es , but in the neighbourhood of HB2 the amp li tu de drops sharply. The behaviour around HB2 (K = 7 .775400) is much more complex. En l argements of the rec tangle around HB2 (Figure 4.39) are shown in the next fi gure . Figure 4.40b shows clearly that with increasing K the first horn (H1 ) with period one (P 1 ) attractors is followed by a g ap after which a period two (P2) hom (H2) is born. H2 is fol l owed again by a gap after which a period four (P4) horn (H3) is born. It is not po s si ble u s ing A UTO (Doedel and Kemevez, 1 986) to obtai n higher p eri odi c ity horns. This point wi l l be discussed later in this section in connection with chaotic behaviour and perio di c windows for this system . Fi gure s 4.4 l a, b s h o w that a small P2 s ubh om (SH l ) is attac hed to the P I main hom H1 and a small P4 subhom (SH2) is attached to the P2 hom (H2).
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FIGURE 4.40 Magnification of the rectangle around HB2 in Figure 4.39. (a) Poincare bifurcation diagram of Y; (intersection of Y with the Poincare plane) vs. K. (b) Bifurcation diagram Y vs_ K (obtained by AUTO). 13.
Chaotic attractors and the connection between chaotic and periodic attractors
The P oinc are bifurcation diagram for the region around HB2 (the box in Figure 439) is shown in Figure 4.40a, and on this scale we can only observe that a period doub ling sequence leading to choas emanates from the periodic branch which originates from HB2 and that the gaps between the horns are occupied by chaos interrupted by periodic windows which
are not quite visible on thi s scale (except for the P3 window near the left hand side of the diagram) _ The two special windows of P2 and P4 associated with the periodic horns H2 and H3 are clearly visible and seems to be dividing the diagram into regions of different chaotic
470
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FIGURE 4.41 Further enlargements of the periodic branches around HB2• (a) Enlargement of the region enclosing the gap between H1 and H2· (b) Enlargement of the region enclosing the gap between H2 and H3•
densities . In order to achieve a better visualisation and analysis of this region in a well organized and detailed fashion, a number of important
subregions are enlarged and analyzed in some details . /3. 1 .
Region A
This region shown in Figures 4.42a, b extends from K = 7.7 1 (to the left of HB2) down to K 7 .685 . From Figures 4.42a, b it is clear that the P l per i odic branch startin g at HB2 loses its stability through a PD poi n t at K = 7 .703 877 an d a stable P2 attractor is born. The pe ri o d dou b ling sequence continues P2, P4, P8, . till the accumulation point at K� 7.6898 where chaos starts . The chaotic attractor born at K� =
_ . . _
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alternate chaotically between an infinite number of bands forming a Cantor set. The number of bands, N, decreases as K decreases (the bands widen and overlap in pairs). This is referred to usually as reverse bifurcation (Lorenz, 1 980). Chang and Wright ( 1 98 1 ) found that this se ri e s of s emi  pe ri od ic bifurcation also satisfies the Feigenbaum' s asymptotic relation ( 8 = 4.6692 . . . and a = 0.494454). From Figure 4.42a we can notice a re gio n of four bands chaos for K slightly less than K�. The se four bands continue till the first interior crisis point. After that only two bands exist as shown in Figure 4 .4 2 for K= 7.689. This twoband chaos term inates at the interi or crisis point (K;., = 7.688 1 ) and a fully developed oneband chaos prevails (Grebogi et al. ,
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FIGURE 4.43 Detailed dynamic characteristics for a case with K = 7.68 9 in Figure 4.42a (intermittency). At;{a). Time trace, Y vs. r). A6(b). Return points histogram, return points first iterate map for the case of K = 7.6854 (intermittency, po in t A6 in Figure 4.42a). A6(c). Return points third iterate map (Y(n+3) vs. Y(n)). (i) Enlargement of box i in A6(c). (i) Enlargement of box ii in A6(c).
STATIC AND DYNAMIC B IFURCATION BEHAVIOUR
473
1 983 ). Near the end of this region at K = 7 .6854 this fully developed chaos turns into intermittency and eventually loses its stability to P3 stable attractor creating a P3 window. Pomeau and Manneveille ( 1 980) proposed that chaotic attractors can lose their stability through intermittency (tangent bifurcation). The term intermittency refers to oscillations that are periodic for certain time intervals (laminar phases) interrupted by intermittent erratic bursts of aperiodic oscillations of finite duration. After the bursts the system returns to the laminar phase again until the next episode occurs (Pomeau and Manneville, 1 980 ) . This intermittent behaviour is shown for K= 7.6854 in Figure 4.43. It is clear in Figures 4 .43 A6(a, b) that the dynamics alternate between regions of P3 and chaos. The sequence of iterate maps in Figure 4.43 A6 (c) makes this fact clearer. The third iterate map shown in Figure 4.43 A6(c) makes the situation very clear. The curve approaches the diagonal and almost becoming a tangent at three distinct points as the enlargements of boxes i, ii show. It is these small gaps that create the three laminar channels giving rise to intermittency. As soon as the curve touches the diagonal and then intersects it, the intermittency disappears and a P3 stable attractor is born together with a P3 unstable orbit. /3. 2.
Region B
This region shown in Figures 4.44 (a, b) extends from K = 7.685 down to K = 7 . 6 83 and covers another part of the region between HB2 and the PD point on H3 • In this region with the decrease of K the intermittent chaos loses its stability to P3 attractor (a P3 window for K = 7 .6840 1 ) , which goes through a period doubling sequence to chaos (P3 , P6, P l 2, . . . chaos) as K decreases further. The resulting chaos is banded and at an interior crisis point it becomes a fully developed chaos, which again loses its stability giving a P7 window, which bifurcate again with decreasing K to give another chaotic attractor which extends (interrupted by other narrow periodic windows) down to the left hand side end of this region at K = 7 .683 . 13. 3.
Region C
This region shown in Figures 4.45a, b extends from K = 7.6830 down to K = 7.6770. The region covers the remaining part of the gap between HB2 and the PD point of H3 together with the entire H3 hom as well as a part of the gap between H3 and H2 • If we start our discussion from the PD point of H3 at K = 7.68 1578 as K increases (region C(i) in Figure 4.45a), we will find that a period doub ling sequence P4, P8, P 1 6 . . . takes place over a very small range of K gi ving rise to a narro w chaotic regi on ' which bifurcates giving ri s e to a P7 window . The P7 orbit bifurcates again to a chaotic region followed by a periodic window and
S . S .E.H. ELNASHAIE and S . S . ELS H I S HINI
474
�
g
� �
�
� � .
f
�
c • Q
B(ii)
0 . 990
0 980
0. 970
:�_;;s:; ;·
 · · ·:·: · · .. .. . : : �
0 960
0 950
. .:. . �·::::: : ::m:�f':::�= 
·· ·  . . . . _ _ . .
.
.
.
::
; •· · · .
B3
. . .r· · .
.. =
• ·· ··•
. ... . .. .... ... ... ... ,.. , •1_ ....... .. ··· ·· '" • • •• ••••
82 B '
(a)
7 . 6830
7 6835
7.68�0
7 . 684 5
Reg i o n
8 7 . 6 850
� t sa r;:=====:;, St. sr. )rue. cb> E lJ B I U b l t ru le d l c ll r a u lli E' t 4 o U a & U blt 000 ..
..
!
:: 1 30
.. "" ..
:: 1 . 2 0 " ..
Q " "' "
"
11 0 � tt t t� ! t� 9 9 t t ! t t t B ! � t t t ! ! tt t ttt t 9 t t tt t t t t t
.!! 1 . 00 f: o • • · · · · · · · · · · · · · · · · ·· · · · · · · · ·· · · · · · · · · •• e • �
                                                               
5 Q
�
0. 90+,."""T'"T"'""""""' 7 . 6 83 0 7 . 6 83 5 7 . 6 e� o 7 . 6845 i . EooJ C o n t r ol l r r G a i n ( K )
FIGURE 4.44 Enlargement of another part of the gap between HB2 and H3 (Region B). (a) Poincare bifurcation diagram. (b) Bifurcation diagram.
so on as K increases . The most important window is the P8 window near the end of the right hand side of this region. This P8 window bifurcates as K increases to the chaotic region extending till the right hand end of this region. This chaotic region is the extension of the chaotic region on the left hand end of the region B (ii) in Figure 4.44a. Thus the connection between the period doubling sequences leading to chaos emerging from HB2 with decreasing K and from the PD point of H3 with increasing K is achieved through a complex sequence of chaotic reg i ons and periodic windows where the dominate mechanisms of bifurcation are period doubling (or halving) and tangent bifurcation. A wide variety of periodicities (odd and even) is encountered in this
region.
STATIC AND DYNAMIC BIFURCATION BEHAVIOUR �
�
�
..

"
,
,
STATIC AND DYNAMIC B IFURCATION BEHAVIOUR
497
( 1967) and Denn ( 1 986) who have shown that under certain assu mpti ons and when vapors are fl owing i s oth erm al ly in plug flow, such a depen dence can be obtained. However, as shown in equation 4.64, the term is made also dependent on the c at al y st circulation rate (Fe), which is not a s tate variable and cannot be directl y related to the rate of catalyt i c carbon formation. The inclus i on of such an input variable in such a manner negati vely affects the l ogical structure of the model and therefore affects th e c on sistenc y and the conclusions obtained using this model. The model also repre s ents Ec (the conversion) in a manner which is, to a great extent, i ndependent of the rate of formation of catalytic carbon. In fact the rate of formation of catalytic carbon and the conversion are strongly interlinked. The model is not expected to show any mu l tipl ic ity of the steady states since the authors use the rate equations of Kurihara (1967) for which the constants are fitted to a specific steady state. As explained earlier, these rates are not valid at any other steady state and the solution of the steady state model will always give the assumed steady state. /5.
Elnashaie and ElHennawi model (E&E model)
Elnashaie and ElHennawi (1979) deve loped a model that overcomes some of the limitati on s of the p rev i ou s models in order to check the c ontradictory results reported in the literature, spec i fical ly the opposite conclusions of Iscol (1970) on one hand and Lee and Kugelman (1973) on the other hand. The model of Eln ash aie and ElHennawi (1979) take s into consi deration the twophase nature of the fluidized beds in the reactor and the regenerator albeit in a simple form. The y us ed the three p s eudo components kinetic model of Weekman (1968, 1969):
Gas Oil (A1 )
�
�
Gasoline (A2 )
� Coke + Gases (A3 )
f
K
The rates of reactions in g . converted/g.catalysts., for this conse cutiveparallel network are given by Weekman and Nace ( 1970) as: Rate of disappearance of g as oil:
(4.69) Rate of appe aran ce of gasoline: R2
=
lfl R
(1  E)p , ·
€
( K l CA2 l

Kc ) 2
A2
(4.7 0)
498
..
S . S .E.H. ELNASHAIE and S S ELSHISHINI
Rate of appearance of coke:
(4 . 7 1 )
and the temperature dependence of the rate constants given by,
Ki (i = 1  3)
is
For the coke burning in the regenerator, Elnashaie and ElHennawi ( 1 979) used the continuous reaction model for solids of unchanging size presented by Kunii and Levenspiel ( 1 969) because of its simplicity. In this simple coke burning model, the gaseous reactant (oxygen) is assumed to be present throughout the particle at constant concentration (no diffusional resistance) and reacts with solid reactant B (coke deposit) accordi ng to the rate equation: (4.72)
The model equations, based on the above relations, are given in dimensionless form (details of the derivations are given by Elnashaie and ElHennawi, 1979) as follows: Reactor dense pha se equations:
Gas oil balance:
BR ( x1 1  xw )
Gasoline balance:
 VIR · xw2 (
a1
·
e  yl / YRT! + a3 · e  y3 / YRD ) = Q (4 . 73)
(4.74) Coke balance: (4.75) Heat balance:
Reactor bubble phase equations: Gas oil balance:
STATIC AND DYNAMIC BIFURCATION B EHAVIOUR
499
Gasoline balance: (4.78) Heat balance: (4 . 79) Regenerator dense phase equations:
Coke balance:
(4.80) Heat balance: Regenerator bubble phase equations:
Heat balance:
(4.82) where: a,
=
Fe
cps I ARI HR c,if
pf a 2 = FcM Cpi I ARI H R Crf p f a, = F e cps I AGI HG • cpa ' p .
.
.
.
'
.
and,
.
.
.
.
a
a R = QER · A RB I GRI
a c = QEG · Acn I Gel yi = Ei I Rc Tif ·
where A ; stands for the preexponential factor for reaction i. The space velocities BR and Be in the reactor and the regenerator, are given by: 
BR 
Ba
=
 R HR ) GRI + GRB (1  e a A RI · HR
) GGI + GaB( l  e � HG AGI · Ha
(4.83) (4.84)
S .S.E.H. ELNASHAIE and S . S . ELSHISHINI
500
The rate of coke combustion in dimensionless variables is gi ven as: Rc = ac · erJ YG · (1  'uc ) .,
where :
(4.85)
ac = � · Co · (l  E)
and the exothermicity
Yc =
Ec I Rc . Tif
factor for the coke burning reaction i s gi ven by: (4.86)
The rate of coke formation is given by:
Ref = a3 . e  r3 1 YRn . x[D + a2 . e
(
The
by :
r21YRo . x2D)
(4.87)
endothermicity factors for the three cracking reactions are given /3; = cAlf . CM( ) t (Tif . cpf . PJ )
(4.88)
i = 1,2,3
the overall endothermic heat of cracking is given by:
and
M/cr = 'If
R(
and
,
/31 . xfD . e  yi / YRD + a2 . f3z . XzD . e  y2 / YRD 2 xw · e  y3 / YRD )
al . + a3 . f33 .
.
(4 8 9)
CR = Fe · Cm I AR1 · HR  CA!f
C(; = Fe I AGI Hc ·
This model was used by Elnashaie and El Hennaw i ( 1 979) to investigate the mul tipl icity phenomenon in fluid catalytic cracking units. Since the model involves a reaction kin etic network, it became possible to investigate the g asoline yield of the system. The investigation of E l n ashaie and ElHennawi ( 1 979) u s i n g this relatively more rigorous model confirmed the conclusions of Iscol ( 1 970) and c o n trad i c te d the conclusions of Lee and Kugelman ( 1 973), since they found that the multiplicity regi on covers a very wide range of operating conditions (Figure 4.50b). They also found that the maximum gasoline yie ld corresponds to the midd l e unstable branc h of the bifurcation diagram
STATIC AND DYNAMIC BIFURCATION BEHAVIOUR
501
and that operati on outside this unstable branch at any of the other s tabl e st ead y states g i ve s very low gas ol i n e yield.
Evaluation of Elnashaie and ElHennawi model
The model of Elnashaie and ElHennawi ( 1 979) rec ognize s the two phase nature of the fluidized b ed s in both the reactor and the regenerator albeit in a simp le manner. The model uses a reaction network that relates the different steps of the crackin g reactions to each other and to the reactor model. From a h ydrodynami cs p oint of view a simplified tw oph ase model is used and many ass umpti on s were introduced to simplify the overall model of the system . With regard to the kinetics of the cracking reaction, althou gh the approach adopted, based on lumping several components into three pseudocomponents, is reasonably sound for the purpose of investigating bifurcation behaviour and g as o l i ne yield, the network used has a strong defici ency in the lumping of coke with light hydrocarbons in one component while they have comple tely different roles. A fourcomponent network (Lee et al. , 1989) may improve the re liability of the model. However, a muc h more detailed network is needed for c ompu ti ng produc t distribution and gas oline oc tane number (J acob et al. , 1 976). It is also important to notice that the activation energies of the ki netic reaction network are obtained from the work ofWeekman ( 1 968, 1 969) and Weekman and Nace ( 1 970) which g ive s the rate constants at only two temperatures. Therefore the reliability of the se activation energies is not high and more experimental work is required to accurately deter mine the activation energies and rate c on stants for different gas oil feedstocks.
Another weakness with regard to this model, is the fact th at it has not been systematically checked against the performance of commercial or even pilot p l ant FCC units.
The model used by Edwards and Kim ( 1 988) is a proprietary , however the authors gave a brief summary of the concepts and relations employed by them in the use of the model to i n vestigate the multipli city phen o menon. The model consists of an o verall carbon balance for the wh ol e unit (reac tor + regenerator) in the following form: 16.
Edwards and Kim model (E&K model)
(4 .90) In this model the capacity of the reactor is considered negligible in comparison to the capacity of the regenerator. This by
is justified
S .S.E.H. ELNASHAIE and S . S . ELSHISHINI
502
Edwards and Kim ( 1 98 8 ) by the fact that modemday FCC units reactor normally amounts to less than 1 % of the total catalyst in the system. The energy balance of the regenerator is given as: (4 .9 1 )
where the heat capacity of the reactor has been assumed negligible as compared to the heat capacity of the regenerator; Qc is the heat generated by coke combustion in the regenerator and Qr is the heat removed from the regenerator. (4.92)
The rate of carbon formation, which appears in the right hand side of equation 4.90 is given by:
Ref = Rcat (tc , TR , Co ) · Fe + R.�r ( T,; , fhc ) · Fe + Rad ( TGF • Tc, Ct ) FcF (4 . 93 )
The first term on the right hand side of equation 4.93 represents catalytic carbon formed by cracking, the second term represents the rate of coke entrainment in the stripper and the third term is the rate of laydown of "additive coke". A formula for coke burning similar to that used by Iscol ( 1 970) and by Lee and Kugelman ( 1 973), has been used in this model. Edwards and Kim (1988) use the following pseudosteady state relation for the computation of the catalyst circulation rate when the gas oil conversion is known, The gas oil conversion is given by: (4 .95 )
The authors used this model to investigate the behaviour of a com mercial FCC unit and concluded that the phenomenon of the multiplicity of the steady state does exist (Figure 4.50c) and they explained some of the pathological behaviour of the unit due to the existence of multiple steady states. Evaluation of the model
reme
a view from
It is ext l y useful to have industry . The model and the paper give lots of extra insight into the behaviour of the complex fluid catalytic cracking units.
STATIC AND DYNAMIC BIFURCATION BEHAVIOUR
503
The neglec t of the carbon and the heat capacity of the reactor makes the model valid only for systems where the inventory of cataly st in the reactor is much smaller than in the regenerator. However in such systems, the reaction is taking place in the riser where the cata lyst and gases are moving in plug flow and hence the lumped para meter formulation in terms of CSTR may not be valid. The model does not recognize the tw o ph ase nature of the fluidized beds in both the reactor and the regenerator and is similar in this respect to the models of Kurihara (1967), Iscol ( 1 970) and Lee and Kuge lman (1973). In the expre ssion for the rate of coke formation Ref (equation 4.93), the important first term expressing the rate of c atal ytic coke formation is made depende nt upon Fc. the rate of c ataly st circulation. In fact, Fc is neither a state variable nor a capacitance term. It is an input output parameter and therefore the rate of a proc ess taking place inside the s y stem should not be expressed as a di re ct function of it. The relation 4. 95 for €c , seems to be highly empirical whe re Ec is a function of (TR, Ta, Ca. Fc. Fm} In fact, any consistent mass and heat balance formulation will show that Ec is not a direct fu ncti on of Fe or Ta or Fc;F· Ac tu al ly these parameters, th rou gh the overall mass and heat balance, will affect TR in different ways depending upon the confi g uration , the size and other conditions of the s y stem TR will then affect €c through the Arrhenius dependenc e of the rate constants upon TR. Edwards and Kim ( 1988) reach oppos i te conclusions to those of Lee and Kugelman (1973). They conclude the existence of mu ltiple steady states, however, their conclusi ons regarding the bifurcation behaviour of the system need to be clarified in two important points 

,
.
.
1.
The relation between multiplicity of the steady states and catalyst circulation rate FC
Heat is communi c ated from the regenerator to the reactor by means of the c atal y st circulation rate which is also responsible for tran sporting
the carbon formed in the reactor to the regenerator. Therefore, too low a c at al y st circulation rate will deprive the endothermic reactions in the reactor from the necessary heat. The cracking reac ti ons will s top and consequently carbon formation in the reactor will stop which results in a sh arp drop of the c a rbo n supply to the re g ene rator and the whole s y stem wi ll quench On the other hand, too high a catalyst circulation rate may s upp ly too mu ch heat to the reactor rai sing its temperature and cau s ing overc rac ki ng "
".
with large carbon formation. This in tum, will r aise the amount of carbon recirculated to the regenerator and "ignition" occurs causing complete loss in gasoline yield.
504
S . S .E.H. ELNASHAIE and S . S . ELSHISHINI
For intermediate values of the catalyst circulation rate, which is the range in which most industrial units are operating, multiple steady states exist. Actually three steady states: a quenched low temperature steady state with almost no reactions taking place, an ignited high temperature steady state with almost zero gasoline yield and very high rate of carbon formation, and an intermediate unstable steady state with a moderate temperature and high gasoline yield. This last steady state is the one at which most industrial units are operating. For any fixed value of catalyst circulation rate, in this region, multiple steady states do exist. Therefore, the conclusion of Edwards and Kim (1988) that, for an open loop system, there exists one unique steady state solution when the catalyst circulation rate is fixed, is not correct. A unique steady state can occur if the chosen fixed value for the catalyst circulation rate lies outside the multipl icity region since no system will exhibit multiple steady states over the entire range of parameters. A unique steady state can be obtained in a cl osed loop system say with a feedback proportional controller. The occurence of multiplicity will depend upon the c on tro ller gain (Elnashaie and Elshishini, 1993 ) For certain values of the controller gain, the middle steady state may become un i que and stable and the multiplicity may disappear altogether. .
2.
Multiplicity and the degree of combustion completion in the regenerator
As clearly explained in the first chapter of this book, there are two sources of multiplicity in chemically reacting systems:
a) isothermal (or concentration) multiplicity resulting from the non monotonic dependence of the rate of reaction upon the concentration of one or more of the reactants. b) thermal multiplicity, resulting from the exotherrnicity of the reaction (the reaction should be highly exothermic and the activation energy of the reaction should be high). In FCC units, obviously the cracking reactions cannot be the source of multiplicity, because the kinetics of the reactions are monotonic and the main reactions are endothermic. Therefore, neither thermal nor isothermal multiplicity are possible in the reactor. The only possible source of multiplicity is the regenerator where a highly exothermic reaction is taking place, that is carbon burning. Therefore, parti al combustion is not the source of multiplicity as stated by Edwards and Kim (1988). Ac tually complete combustion of the c arb on in the regenerator g i v e s larger regions of multiplicity . This has been proved to be true by E l shi sh i n i and El nas h ai e ( 1 990b) (Figure 4.5 1 ) and will be discussed in more details in section 4 . 24 (13 ) .
STATIC AND DYNAMIC B IFURCATION BEHAVIOUR
"i a:
>
::l
e
4 .0 0
8. J . o o
! .....
� 0
£
m
·�� Ci
505
,.
2 . 00
I I
1 .00
0. 00
/

/


  

     
,
           


    7 5 ' /, C O M P L E T E C O M B US T I O N 
QOO
100'/, C O M PL E T E C O M BU S T ION
0. 1.0
0.80
1 . 20
Dimensionl� ss Catalyst C i rculation Ra � Fe /Fcrl'f
FIGURE 4.51 Effect of degree of coke combustion to C02 on bifurcation diagrams for reactor temperature vs. catalyst circulation rate.
17.
Elshishini and Elnashaie model (E&E modified model)
The model developed earlier by Elnashaie and ElHennawi ( 1 979) was further modified (Elsh i shi n i and Elnashaie, 1990a) to account for:
1. The change in volumetric gas flow rates between inlet and outlet of reactor and regenerator. The volumetric gas flow rate at outlet of reactor is:
The volumetric gas flowrate at outlet of the regenerator is: Gc
=
T F (0. 5 W:H + 0. 5 W.co ) �Re Te + A FRc c lOOPc 29Pc
(4.97)
where FGM and FAF are the feed mass flow rate of gas oil (to the reactor) and air (to the regenerator) respectively (in kg/s). 2.
The pa rtia l cracking of ga soline and gas oil to lighter hydrocarbons.
The heats of cracking re porte d earlier (ElHennawi, 1 977) were based on complete cracking of hydrocarbons to carbon and hydrogen and thus,
506
S . S.E.H. ELNASHAIE and S.S. ELSHISHINI
were overestimated. In this model the heats of cracking were modi fied. The overall heat of cracking obtained after introducing these modifications was found to be very close to the industrial value obtained empirically (Nelson, 1 964; Maatschappij , 1 975). 3. The lumping of light hydrocarbon gases with coke. Weekman ( 1 968, 1 969) kinetic scheme lumps light gases with coke. This is not suitable for realistic modelling of commercial units, therefore the amount of light gases was obtained from the weight ratio of coke to (coke + gases) given in plant data.
FCC
The recycle stream. Weekman ( 1 968, 1 969) kinetic scheme does not account for the formation of (heavy cycle oil), (light cycle oil) and (clarified slurry oil), which are the fractions produced in addition to gasoline. In the refineries, the only fraction recycled is and to calculate it, the ratio of + + is obtained from plant data.
4.
CSO
HCO
HCO
LCO
HCO/(HCO LCO CSO)
Evaluation of Elshishini and Elnashaie model
FCC
The mod el was successfully compared with two industrial units and it was found that both the units are operated at the mid dle steady state (Figure 4.52) and that the multiplicity region covers a very wide range of parameters . The model can be used to investigate the effect of feed composition on the performance of units (Elshishini and Elnashaie, 1 990b; Elshishini et al. , 1 992) as will be shown in the next section (section 4.2.2). Figure 4.52 shows the Van Heerden (heat generationheat removal versus reactor temperature, diagrams (Figure 4.52a) and the unreacted gas oil and gasoline yield profiles (Figure 4.52b) for both industrial units simulated. It is clear that the maximum gasoline yield for both industrial units occur at the multiplicity region and specifically at the unstable saddle type state . It is also clear from the figure that both units are not operating at their optimum con ditions with the operating point slightly shifted from the maximum gasoline yield. Simple manipulation of the operating variables can shift the units to their maximum gasoline yield with a considerable im provement in the productivity of the unit (Elshishini and Elnashaie, 1 990a, b). The model needs to be further improved to avoid the empirical evaluation of the ratio of carbon to light hydrocarbons from industrial measurements, by using a 4lump kinetic network (Lee et al. , 1 989). Also, the effect of recyle on the kinetics needs to be incorporated into the model in a more rational way than its present empirical form. A
FCC
YR)
STATIC AND DYNAMIC B IFURCATION BEHAVIOUR
507
4 . 0 ,......,.. I N D U ST R I A L U N I T I (a) _ _ __ I N O U S T R I A L U NIT 2
�
2.0
'0
� .§ "'
0 0 .1+'>r;ri
u
5
u.
1i  2 . 0
:I:
4.0
L......L...II ........ .. 1 .1.
0.00
0
1 . 00
....
(.!) >< o � ..
.; ""0
��
� ..§ .!! a ::l
3  00
4 .00
1 . 0 0 ,...,.,  I N DU ST R I A L U N I T I     I N DU S T R I A L U N I T 2
"' � 0.8 0 c
2 .00
..
� s � � E �
( b)
0. 60 0.40 Q20 0. 00
L......L.lll...l ....______._____:=::::...._ '
0.00
1 . 00
2 0 0
O imensioni!!SS Reactor
3 00
4 . 00
Temperot ur!! ( Y R )
FIGURE 4.52 (a) Heat functions for units 1 and 2. (b) Conversion and gasoline yield for units l and 2.
more rigorous kinetic model for coke and hydrogen burning in the regenerator, needs to be incorporated instead of the simple kinetic model used. The model also needs further development to include modem riser type FCC units. Since the units are usually operated at the middle unstable steady state, extensive effort needs to be carried out for the analysis of the dyna mic behaviour of open loop and closed loop systems (Elnashaie and Elshishini, 1 993) as will be shown later in this chapter (section 4.2.5).
508
II
S.S.E.H. ELNASHAIE and S . S . ELSHIS HINI
Summary of the state of the art in modelling industrial FCC units
Empirical models can be useful in the simulation of a unit operating at the conditions for which the empirical model was derived but these models cannot simulate the perfonnance of the reactor at any other conditions. These models cannot be used reliably for design purposes or for the investigation of bifurcation behaviour as was demonstrated in the evaluation of these models presented in this section. Furthennore, the introduction of the twophase model of fluidization is not an unnecessary sophistication, since the role of bubbles is not only to achieve better mixing, but also to supply reactants to the active dense phase and to remove intennediate products from the active dense phase to the inactive bubble phase. Thus, bubbles have an important and complex effect on conversion, yield and selectivity of the system. They also affect the heat balance of the system, specially in the regenerator which is the source of heat to the system. Much experimental and theoretical work is needed in cooperation between industry and academia, to develop more rigorous models of highfidelity for this industrially important system. Both the steady state and the dynamic behaviour of the unit need to be simulated as will be discussed in the next section. From a fundamental point of view, such a research will have a strong impact on the understanding of bifurcation behaviour in reactor regenerator systems. A richness of steady state and dynamic phenomena, exceeding those found for the classical CSTR problem, is certainly lying there waiting to be discovered. The preliminary unsteady state results presented in section 4.2.5 show very interesting static and dynamic bifurcation behaviour. 4.2.2
Preliminary Presentation of Static Bifurcation in Industrial FCC Units
The twophase nature of the reactor and regenerator should always be recognized. The gas entering the bed splits into two parts, one rising between the individual particles of the catalyst giving the dense phase gas, and the other rising through the bed in the fonn of bubbles forming the bubble phase. There is an exchange of mass and heat between the two phases (Kunii and Levenspiel, 1 969; Chavarie and Grace, 1 97 5 ; Bukur and Amundson, 1 975 ). In ad diti on to the familiar a ss u m ption s for modelling bubbl i ng fluidized bed catalytic reactors (Elnashaie and Yates, 1 973 ; Elnashaie and Cresswell, 1 973 ; Elnashaie, 1 977), the fo l lo w ing assumptions sp ec ifi c to FCC units are also used in the model derivation :
STATIC AND DYNAMIC B IFURCATION BEHAVIOUR
509
change of number of moles with the extent of reaction is negligible. The heats of reaction and physical properties of gases and solids are constant. The above two assumptions will be relaxed in section 4.2.3 dealing with the industrial verification of this steady state model. The refractivity parameter W (Pachovsky et al. , 1 973) defined by equation 4.98, is taken equal to 1 .0.
1 ) The 2)
3)
(4.98)
and ko are the feed reactivities at concentration CA and CAo (fresh feed concentration) respectively. Excess air is used in the regenerator, i.e. the oxygen concentration in the regenerator is constant. where k
4)
I The Steady State Model
The mass and heat balance equations for the steady state model are given in the previous section (section 4.2. 1 , equations 4.694.89). In the following we give a simple procedure for the computation of the model parameters. /1 .
Computation of twophase parameters
two phase parameters are computed using the Kunii and Leven spiel model ( 1 969). The bubble velocity Ub (cmls) is computed from, The
(4.99) where,
(4. 1 00)
and D8 is the bubble diameter in em. In a bed where the bubbles are fast and large, the net upward velocity of the bubble is, (4. 1 0 1 ) Hence, the
volume fraction of bubbles is, (4. 1 02)
S.S.E.H. ELNASHAIE and S . S. ELSHISHINI
510
Therefore, the area occupied by the bubble (and cloud) phase would be,
� = A · c5
(4. 103)
where A is the cross sectional area of the bed. Consequently, the area outside the bubble phase (i.e. the dense phase) would be, AI = A  � = ( 1  D · A
)
(4. 1 04)
The bubble flow rate is given by,
(4. 1 05) and the dense phase gas flow rate which flows at mi nimum fluidization velocity is give n by,
(4 . 106) coke
12. of cracking and burning The kinetic scheme used is formed of three components,
Kinetics
G as Oil (A1 ) � Gasoli ne (A2 )
t��K21
K2
Coke + Gases (A3 )
�f
____________
where A 1 represents gas oil, A2 represents gasoline and A3 repre s e n ts coke and dry gases. The rate of disappearance of gas oil, the rate of appearance of gasoline and the rate of appearance of coke and light gases are gi ve n by equations and respectively. The rate constants can be written in the Arrhenius form as follows,
4.69, 4.70
4.71
K1 = 0.095 x 10 2 x exp ( 21321.664 / Rc · TR) m 3 / kg. s. K2 = 0.077 X 102 X exp (70466.93 / Rc TR ) l i s
K3 =
·
473.8 x 102 x exp (109313.28 / Rc
·
TR )
m
3
I kg. s.
where the activation energies are i n units of kJ/krnol.K. It is important to notice that these acti v ati on e n e rgie s were obtained from the work of Weekman and co  w orkers (Weekman, Weekman and Nace, N ac e et which gives the rate constants at o n l y two temperatures, therefore the reliability of the se activation energies is not very high. Relatively more rel i ab l e values of preexponential factors and activatio n energies based on slightly more
1970;
al. , 1 97 1 )
1968, 1969 ;
extensive experimental data will be used as starting values in
the
STATIC AND DYNAMIC B IFURCATION BEHAVIOUR
511
industrial verification section (section 4.2.3). These starting values are used as initial guesses for fitting the model to industrial data and the preexponential factors are changed to obtain the best fit. This is due to the fact that the kinetic parameters depend upon the specific characteristics of the catalyst and the gas oil feedstock. It is also due to the inherent difficulties in accurate modelling in petroleum refining processes in contradistinction to petrochemical processes. This point will be discussed in more details in section 4.2.3 and is clearly related to the use of pseudocomponents which is the only realistic approach available todate for such complex mixtures. Gas oil and gasoline cracking are assumed to be second order and first order respectively. The justification for these reaction orders has been discussed by Weekman ( 1 969) . Wojciechowski (Pachovsky et al. , 1 973) introduced the parameter W, defined as the refractoriness of the feedstock which accounts for the fact that not all molecules in the charge stock have the same crackability. This problem is not encountered in p ure hydrocarbon cracking because of the homogeneity of the charge. It was also found that Wranges between 00 . 7 approximately. Weekman et al. (Weekman, 1 968, 1 969 ; Weekman and Nace, 1 970; Nace et al. , 1 97 1 ) take W equal unity for all charge stocks. The rate of coke burning is given in section 4.2. 1 (1.5) by equation 4.72 and the value of the preexponential factor and activation energy are as follows (Kunii and Levenspiel, 1 969), � exp (Ec I ReT) m 3 I kmol.s �· = 1. 682 1 X 1 08 , Ec = 28. 444 X 1 0 3 kJ I kmol. K
Kc
13.
=
·
Steady state mass and heat balance equations
The reactor and regenerator mass and heat balance equations for the dense phase and the bubble phase are given by equations 4.734.89 . The catalyst activities in the reactor and regenerator are defined by the following two relations, (4. 1 07)
(4. 1 08)
where,
Cm = Total amount of coke deposits necessary for complete deactivation/ total amount of catalyst.
512
S .S.E.H. ELNASHAIE and S . S . ELSHISHINI
Xce
= Total amount of coke deposits in regenerator/total amount of catalyst in regenerator. XcR = Total amount of coke deposits in reactor/total amount of catalyst in reactor.
II Solution of the steady state equations The steady state equations can be manipulated in such a manner as to put them in the form of heat generation and heat removal functions ( i.e. a modified van Heerden diagram). This manipulation can be carried out in different ways, all leading of course to the same results and it is chosen here to obtain the heat generation and removal functions of the regenerator as a function of reactor temperature. The solution procedure in this case is as follows :
1)
2) 3)
Choose a value of
YRo
within the desired range
Assume a value of VIR as an initial guess.
Compute x w from the following equation which can easily be obtained from equation 4.73, 2
 BR + BR + 4 '1'R · BR · x 11 · ( a 1 exp (y1 / YR0 ) + a3 exp ( y3 / YR0 )) 2 'I' · (a1 exp (  y1 / YR0 ) + a3 exp (  y3 I YR0 )) R
(4. 1 09)
4.74 which can easily be put in
4)
Then calculate x20 from equation the following e xplicit form,
5)
Calculate the rate of coke formation Ref• from equation VIe from equation 4.75 . Calculate Mlcr from equation and Yen from equation Calculate the rate of coke burning from equation Check the correctness of the assumed value 'I'R· by computing its value from equation If the residual is too large, correct the value of VIR using Newton Raphson iteration technique then repeat steps 38 . If the residual is small enough (< l Q6), substitute in the heat balance equation of the regenerator which can be written as,
6)
7)
8)
9)
4.87,
4.89
4.85.
4.76.
4.80.
4.8 1
R(YRv) = Be . YAJ + a3 . YRD =(Be + a3)· Yco f3cRc +l!ilhe = G ( YRo)
(4. 1 1 1 )
STATIC AND DYNAMIC BIFURCATION BEHAVIOUR
Table 4.7
513
Data for the fluid catalytic cracking unit.
Height and diameter of regenerator' s bed respectively
25.05, 1 3.50 m
Height and diameter of reactor' s bed respectively
25 .50, 8.25 m
Pressure in both vessels
1 .5 atm
Umf •
0.4
Voidage in both vessels in regenerator and reactor respectively
Density and average size of catalyst respectively
1 6 x 1 0""4 , 4 x I 04 rnls 450 kg!m 3 , 60 Jlm
Average molecular weight of gas oil and gasoline respectively
1 80, 1 1 0
Boiling point of gas oil and gasoline respectively.
539 K, 4 1 1 K 8 . 1 26 x 1 Q3 kJ/kg
Heat of reaction of gas oil to coke f:Jl3 Heat of reaction of gas oil to coke f:Jl 1 Heat of reaction of gasoline to coke f:Jl2 (by difference) Latent heat of vaporization of gas oil Unless otheiWise stated, Heat of combustion of coke
Reference temperature, (Tif)
6.036 X I ()2 kJ/kg 7520 kJ/kg
QER = QEc = oo
264.7 kJ/kg
30. 1 9 X 1Q3 kJ/kg 500 K
10) Change the value of YRn and repeat steps 29 until the required range of reactor temperature has been covered. Ill
Steady state simulation results for an industrial size FCC unit
For the results presented in this section, all parameters not stated on the figures are fixed according to Table 4.7. Ill/.
Steady state determination for given sets of parameters
In the present section simulation results will be presented to the reader and discussed in a relatively simple framework for simplicity. Therefore, the four modifications in the model ofElshishini and Elnashaie (1990a) given in section 4.2. 1 (part 13), will not be used in the present section but rather the model of Elnashaie and ElHennawi (1979) given in section 4.2. 1 (part 15). On the basis of the assumptions of Elnashaie and ElHennawi model (1979), the results in this section are based on manipulating equation 4.1 1 1, so that the heat removal line becomes independent of feed temperature and its slope independent of reactor temperature and other variables, so that the slope of the heat removal line becomes constant. The gas flow rate in the reactor and regenerator are assumed constant and are calculated by the ideal gas law from their feed mass flow rates (FcM, FAF) at vaporization conditions for the gas oil to the reactor and at feed conditions for the air to the regenerator.
514
S . S .E.H. ELNASHAIE an d S . S . ELSHISHINI
�, t� �'• • " m . s ••'" ( a) JSO
0 .;:(!)
JOO
Fe :: 1ll5  FGN : 1100     FGN :: 1000 ··FG M : 200
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jl
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(a )
 Ao
t
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':"": ,"';"";' .:. �."7':';:: :":":'.�.":": .. . . . . . .
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c .....
!!!
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� ..., �
li iii
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( b)
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1 3 '
I  H IG H T E I4! S TE A DY STAT E
2  II IOOLE ST EADY STATE S T EM�' S TAT E
3  L OW TE ... P .
0. 40
Fc o
0.�
O I M E H S I O H L ESS C AT A L Y S T R ATE
1 20
C I R C U l AT I O N
( Fe f Fc , r. r l
FIGURE 4.62 Effect of charge stock composition on bifurcation diagrams. (a) reactor temperature vs. catalyst circulation rate. (b) gasoline yield vs. catalyst circulation rate.
for the case of 75% C02/(C02 + CO) and the case of 100% C02. It is clear that the multiplicity region increases with the degree of combustion completion. The quenching poi nt occurs at FCIFC ref of 0. 15 for the case of 75% C02 and 0. 1 for the case of 100% C02. The ignition point lies away from the region under consideration. The heat of coke combustion is calculated from equation 4 . 1 1 4 and it increases from 3 1 246 kJ/kg at 75% C02 to 37 1 2 1 kJ/kg at 1 00% C02 •
This increase in the exothermicity o f the reaction explains the increase in the size of the multiplicity region .
STATIC AND DYNAMIC BIFURCATION BEHAVIOUR
535
/1
Summary of section results By applying the model developed earlier to four different feedstocks,
it is shown that the performance of an industrial FCC unit is greatly affected by the charge stock composition. The reactor must, invariably, be operated at the middle unstable steady state. The maximum gasoline yield can be attained by changing the catalyst circulation rate. However, this change must be affected with care since the system can approach the bifurcation point where quenching occurs. Gasoline yield is strongly affected by the charge stock composition and for each charge stock an optimum catalyst circulation rate must be used to obtain optimum gasoline yield. However, the corresponding regenerator temperature should be inspected so that it does not exceed the suitable temperature for the regeneration of the FCC catalyst (as specified by the catalyst manufacturers). The model can be used to find this optimum circulation rate and the necessary input and output heats to be used in the non adiabatic operation to keep the regenerator temperature down to acceptable limits while operating at low catalyst circulation rates and maximum gasoline yield. 4.2.5
Effect of Fluidization Hydrodynamics on Static Bifurcation and Steady State Gasoline Yield
It has been shown so far in the previous sections that the industrial FCC units presented were operating at the middle unstable steady state in the multiplicity region and therefore, changes in different parameters will have an effect on this middle steady state which is opposite to their effect on other steady states (Elnashaie et al. , 1 972; Elnashaie and Elbialey, 1 980). This is a general characteristic of middle unstable saddle type steady state in different fluidized bed catalytic reactors. The gas flow rates industrially used for both the reactor (vaporized gas oil) and the regenerator (air), are quite high compared with the minimum fluidization velocity of the catalyst. The minimum fluidization gas which goes directly into the gas phase is in the order of 24% of the total flow for both the reactor and the regenerator. Therefore most of the gas will be in the bubble phase from which it is supplied by mass transfer to the dense phase and react there. In addition, the products formed during the cracking reactions in the reactor, diffuse to the bubble phase. For gasoline, this step enhances the gasoline yield by removing gasoline from the active dense phase protecting it from further cracking. In the regenerator, the bubble phase supplies the necessary air to the dense phase to achieve the burning of the deposited carbon. If this supply is not sufficient, the catalyst will not be fully generated which will not only cause catalyst activity decline, but will also disturb the thermal balance of �he whole system and may cause quenching. Heat
536
S . S .E.H. ELNASHAIE and S.S. ELSHISHINI
transfer between the phases in both vessels also affects the temperature of the dense phases as well as the overall thermal balance of the system . Because of the crucial role of this mass and heat exchange between the phases on the behaviour of the system, we will focus in this section on this process, by showing the effect of bubble diameters in both vessels. 1
Effect of bubble diameter in reactor and regenerator on the behaviour of FCC units
We present the effect of changing the bubble size in each vessel separately. The model developed and verified against a number of industrial units which was presented and discussed in the previou s sec tions, is used in this presentation. The discussion is based on all important state variables, and on one type of bifurcation diagram, with the dimen sionless catalyst circulation rate FCD as bifurcation parameter. 11 .
Effect of bubble size in the regenerator
In the present model, the oxygen in the dense phase of the regenerator is assumed to be in excess and therefore, it is the heat tran sfer between the two phases that will affect the system. The exchange coefficients between the bubble phase and the dense phase were computed from the correlation of Sit and Grace (1978) for mass transfer and that of Kunii and Levenspiel (1977) for heat transfer The increase in the regenerator bubble diameter causes a decrease in the heat transfer from the hot dense phase ( where the exotherm ic combustion reaction is taki ng place) to the bubble phase Therefore, the dense phase temperature should increase and the bubble phase temperature decrease. However, the conclusion based on this physical argument is completely reversed when dealing with the middle steady state. This pecularity of the middle steady state was discussed in the literature for different types of catalytic reactors (Eln ashai e et al. , 1972; Elnashaie and ElHennawi, 1979; Elnashaie and ElBialey, 1980; Elshishini and Elnashaie, 1990a, b). Figure 4.63a sho w s the bifurcation diagram with the regenerator dimensionless dense phase temperature as the state variable. The three static branches are shown for three different regenerator bubble dia meters. It is clear from the figure that the high temperature steady state branch moves to higher values of Y GD as the bubble diameter increases, while the middle steady state branch moves to lower values of YGD. The low temperature steady s tate branch is not affected by the bubb le diameter. The region of multiplic i ty extends to lower values of FCD as the regenerator bubble diameter increases. The exit temperature from the regenerator i s the mean (Figure 4.63c) between the bubble (Fig ure 4.63b) and dense phase (Figure 4.63a) temperatures and since the bubble phase gas flow rate is much higher, we notice that the behaviour .
.
STATIC AND DYNAMIC B IFURCATION B EHAVIOUR OBA 1 C.l m  O I C, a 0  1 M O tG w 0 . 1 rn      ti B C. a O.l "'
1 _ Low ltrnp. s l eodr s t a t t L W i d d l t l t f'l'lp . l l t a el p 11Gtr l_ Hlth I • "'P t l t a d y 'I IG I I
 ·
(b)
·�
0
�
l .Q
(/ \\
/ ./
l.C 1.0
� � � /c l• )
 1.0
....    
(;)
0.1
'
.......
'
... . ... .
  
·
\ ,_ ' ::"'.:.:=:.·=:..: : · '"':"'  
1 0
(I )
10
� '�...: � =:... =;.. =..:..� ...
00 L....L''...' o.o,__....___,__'''' 01
,r,_·� ,
f ! '.... I \ \\
'
u
�
(< I
c .o
• .o
o.o
0 c
537
I�)
1.0 0� ...___,____,__.:...___.__.
(i)
•.. 0 I 00 00
FCO
FCO
FCD
FIGURE 4.63 Effect of bubble diameter in the regenerator on the performance of an industrial FCC unit (YGo = dimensionless output regenerator temperature = YG, YRO =dimensionless output reactor temperature= YR• X2o =dimensionless output gasoline concentration, D8R = bubble diameter in the reactor, DBG = Bubble diameter in the regenerator).
of exit temperature follows that of the bubble phase. The events in regenerator dense phase affect the reactor, since the circulated catalyst from the regenerator comes from the dense phase. Therefore, the temperature in the reactor follows closely the trend of the regenerator dense phase temperature as shown in Figure 4.63df.
the the
Figure 4.63gi show that the gasoline yield for the middle steady state increases as the regenerator bubble size increases. However, this observation should not be generalized because, since this effect on gasoline yield is due to the increase in temperature of the reactor, it will
S . S .E.H. ELNASHAIE and S . S . ELSHISHINI
538
•
. !:: 0
�
... c 0 r::
i!
.!!
c
• ... c
.s a
FIGURE 4.64 Schematic diagram of gas oil conversion (X1) and gasoline yield (X2) versus reactor temperature (YR) in FCC units.
depen d on the relative location of the operating temperature with regard to the maximum gasoline yield point as shown in Figure 4.64. For, if the operating temperature is to the left of point a, an increase in tem perature will increase gasoline yield. However, if the operating tempera ture is to the right of point a, then increasing the temperature will cause the gasoline yield to decrease. This is in contradistinction to the fact that increas ing the temperature al ways increas es the gas oil conversion. /2.
Effect of the bubble size in the reactor
Figure 4.65ai shows the effect of reactor bubble size. This is obviously quite complex due to the interaction of the units. An increase of bubble size in the reactor leads to a decrease in the rate of mass and heat transfer between the bubble and dense phase s Although the reactions are endothermic, the dense phase is usually hotter than the bubble phase because it is the phas e th at exchan ges the solid c atalyst with the much hotter regenerator. An increase in the bubble diameter in the reactor tends to c aus e an increase in the den se p hase temperature and affe cts the rate of supply of reactant to the dens e p h as e as well as the exchange between the dense and bubble phases. The temperature of the dense .
phase increases or decreases depen ding upon the balance between these effects, in addition to the effect on the regenerator and the feedback
STATIC AND DYNAMIC BIFURCATION BEHAVIOUR OBG
:
0.1
m
 O I A : 0 . 1 m  ·  ·  D B A : 0 . 1 "'      D D A : 0 . ) m
�� · · ·
6,0
(o )
4.0

, ___ _ _ _ 
lO
lO 10

�c �=:..    �=
30
:

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  :..: :..=
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Low l•mp. • t • o d y stal• M l lfcllt • l l ea4y at ol t H i g h t r m p . t. l �ady 1 1 ate
(b)
�.
539
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(c)
\ >;     
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""'..:: _ _ _ _ _ _ _ _
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Ol
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(h)
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o.oL...___;:::::�::J� ooL�5;_; ::._�;;[:;;:;;�;;::����;;;;,� O O L',::,,__.,.== 0.0
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I2
FIGURE 4.65 Effect o f bubble diameter in the reactor o n the performance of an industrial FCC unit (YGO = dimensionless output regenerator temperature = Ya,
YRO = dimensionless output reactor temperature = YR• Xw "' dimensionless output gasoline concentration, D8R = bubble diameter in the reactor, D8a = Bubble diameter in the regenerator).
effect of the regenerator on the reactor. The increase of the reactor dense ase temperature will obviously increase the conversion of gas oil, while the gasoline yield is related to the conditions being to the left or righ t of p oint a in Fi gure 4.64. At higher conversions and decreasing gasoline yield , the carbon formation increases. This tends to increase the endothermic heat absorbed and therefore tends to decrease the reactor dense phase temperature. But more carbon means higher rate of carbon burning in the regenerator which when fedback to the reactor dense phas e tends to give higher temperature. The opposite trend will be seen if the i ncre a se in conversion is accompanied by a relatively higher increase in gasoline yield. ph
S . S .E.H. ELNASHAIE and S . S . ELSHISHINI
540
Figure 4.65ai shows the effect of a change of reactor bubble diameter on the different state variables of the system. Figure 4.65a shows that an increase in the reactor bubble diameter causes a decrease in the regenerator dense phase temperature for all three steady states, with different degrees of sensitivity for each branch of the bifurcation diagram. The multiplicity region increases slightly as DBR changes from 0. 1 to 0.2 m, but decreases again as DBR changes from 0.2 to 0.3 m. The regenerator exit temperature follows the same trend as shown in Figure 4.65c. For the reactor dense phase temperature, Figure 4.65d shows a complex behaviour due to the complex interaction of all the factors dis cussed earlier. Only the low temperature steady state shows undirectional changes: the temperature decreases as the bubble diameter increases. For the middle and high temperature steady states, the curves for the different bubble diameters change position relative to each other as FCD is changed. For the reactor bubble phase exit temperature, the high and the middle temperature steady states show that an increase of bubble diameter is equivalent to decreasing temperature, while for the low temperature steady state it has the opposite effect (Figure 4.65e). Figure 3.65f shows the reactor exit temperature which follows the same trend as the bubble phase exit temperature. Figure 4.65g shows the effect of reactor bubble diameter on the yield of gasoline in the dense phase. For the low temperature steady state, the gasoline yield in the dense phase increases with increasing bubble diameter. However, this branch is not interesting because the temperature is below the vaporization temperature of the gas oil. For the middle steady state, increasing the bubble diameter from 0. 1 to 0.2 m gives a small increase in the gasoline yield. However, increasing the bubble diameter further to 0.3 m causes the gasoline yield in the dense phase to decrease. For the high temperature steady state branch, the increase of bubble size from 0. 1 to 0.2 m causes the gasoline yield to decrease whereas further increase in the bubble diameter to 0.3 m causes the gasoline yield to increase. Figure 4.65h,i shows the bubble phase and reactor exit gasoline yields. For the three steady state branches, the gasoline yield always decreases as the bubble diameter increases. II
Summary of section results
We have shown in this section, some of the complex interactions between the two phases in each vessel of the FCC units as well as the interaction between the two vessels. The steady state response of the sy stem to the change of one parameter is obviously quite complex. The desired op erati ng state is the middle unstable steady state which tends to ha e a dependence upon parameters which is opposite to other steady states. These complexities demonstrate the difficulties as so c i ate d with v
STATIC AND DYNAMIC B IFURCATION BEHAVIOUR
54 1
the use of chemical engineering intuition and/or operators experience to predict the response of the FCC units to changes in the parameters and emphasizes the need for reliable mathematical models. It is also important to emphasize the fact that the introduction of the twophase model for fluidization is not an unnecessary sophistication since the bubbles have important and complex effects on temperature, conversion and yield. ,
4.2.6 1
Preliminary Dynamic Modelling and Characteristics of Industrial FCC Units
The Dynamic Model
The dynamic model presented in this section, is developed on the same basis and assumptions as the steady state model developed earlier, with the inclusion of the necessary unsteady state dynamic terms, giving a set of differential equations that describe the dynamic behaviour of the system. Both heat and coke capacitances are taken into consideration, while vapour phase capacitances in both dense and bubble phase are assumed negligibl e in this section and therefore the corresponding mas s balance equations are assumed at pseudo steady state. This last assumption will be relaxed in secti on 4.2.7 and the chemisorption capacities of gas oil and gasoline on the surface of the catalyst will be accounted for, albeit in a simple manner. In addition, the heat and mass capacities of the bubble phases are assumed to be negligible and thus the bubble phases are assumed at pseudosteady state. Based on these assumptions, the dynamics of the system is controlled by the thermal and coke dynamics in the dense phases of the reactor and regenerator. 11 .
Unsteady state heat balance for the reactor
After some simple manipulations the unsteady state heat balance equations for the reactor can be written as follows. ,
Dense Phase
The unsteady thermal behaviour of the dense phase is described by the following nonlinear ordinary differential equation .
542
S . S .E.H. ELNA�HAIE and S . S . ELSHISHINI
with the initial conditions, at t = O Bubble phase (pseudo steady state)
The assumption of pseudo steady state in the bubble phase in addition to the assumptions introduced earlier in steady state modelling (Elnashaie and ElHennawi, 1979; Elshishini and Elnashaie 1990) allow us to describe the bubble phase temperature profile using the simple algebraic relation 4.79 in section 4.2. 1 . ,
12.
Unsteady state heat balance for the regenerator
Simple manipulation similar to that used for the reactor gives the following heat balance equation for the regenerator. Dense phase
The therm al behaviour of the regenerator is described by the following nonlinear ordinary differential equation : EHG
d�D

dt = BG ( YAF  YGD ) + f3c · R, + a 3 ( YRD  YGD )  MJLG
(4. 1 17)
with the initial condition, at t = O Bubble phase
The assumption of pseudo steady state in the bubble phase, in addition to the assumption described earlier in relation to the s teady state model (Elnashaie and ElHennawi, 1979; Elshishini and Elnashaie, 1 990), allow writing the bubble phase temperature profile in the simple algebraic form given by equation 4.82, section 4.2. 1 . 13.
Unsteady state carbon balance in the reactor
The dynamics of carbon inventory in the reactor, expressed in terms of catalyst activity 'I'R· is described by the following nonlinear ordinary differential equation, (4. 1 1 8 ) where
Ref
is given by equation 4.87.
STATIC AND DYNAMIC BIFURCATION BEHAVIOUR
543
With the initial conditions: at
t= O
14.
Unsteady state carbon balance in the regenerator
The dynamics of carbon inventory in the regenerator, expressed in terms of catalyst activity 1Jfc, is described by the following nonlinear ordinary differential equation, (4. 1 19) where Rc is given by equation 4.85. With the initial conditions: at
t= O 15.
1Jic = 1flc ( O)
Pseudo steady state gas oil and gasoline mass balances in the reactor
Equations 4.73 and 4.77 represent the pseudosteady state gas oil balances for the dense and bubble phases respectively, while equations 4.74 and 4.78 are for gasoline. Equations 4.76, 4.79 are for the steady state dense and bubble phases temperatures for the reactor and equations 4.8 1 , 4.82 for the dense and bubble phases temperatures in the regenerator. The exit concentrations and temperatures from the reactor and the regenerator are the result of mixing of gas from the dense and the bubble phases at the exit of the two subunits. Exit gas oil dimensionless concentration: (4. 120) Exit gasoline dimensionless concentration: (4.121) Exit reactor dimensionless temperature:
(4.122) Exit regenerator dimensionless temperature: .Yc rO
=
GIG · P1c · CpiG · Yeo + Gn; · Pee · CPec · YcB GIG . PIG . Cp/G + Gee . Pee . CpcG
=
Yc
(4. 1 23)
S . S .E.H. ELNASHAIE and S . S . ELSHISHINI
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o . lO O .td 0.)0 T i m e i n dey l
....
FIGURE 4.73 Trajectories for Figure 4.72. (a) Dimensionless reactor dense phase temperature. (b) Dimensionless regenerator dense phase temperature. (c) Dimensionless gasoline dense phase concentration. (d) Catalyst activity in reactor.
Fi gure 4.73 sh ow s the time traj ectorie s ofthe different state variables, and the resemblance to Figure 4.7 1 is evident. Most important is the increase of conversion and g a so l i ne yield along the trajectories tending to the low temperature steady state, followed with qu enc hi ng at times varyin g between 0.20 . 3 dependi ng upo n the specific initial c ondi ti on s vector Ob v i ou sly this sensitivity to 1/fR (O) is due to the fact that the des ired operating state is the middle unstable steady state ( saddle ) which lies on the separatri x There fore, s tarting i nfinite simally to the ri ght or to the "left" of this steady state leads to different directions of drift. This is applicable to all four state variables of the system. However, what is s pe ci al ly important abou t 1/fR (O) is that the whole range of 1/fR between the high and low temperature steady states is very narrow, from 1/fR = 0.998 for the low temperature steady state to 1/fR = 0.9596 for the hi gh tempe rature steady state with 1/fR for the middle steady state lying on the separatrix in this very narrow reg i o n with 1/fR 0.989. This is physically due to the fact that the re generator is very efficient and the catalyst circulation rate is quite high; therefore, small change s in 1/fR correspond to apprec iab l e amounts of carbon. The use of other variables to express the amo unt of carbon does not improve the numerical s en s itivity to any appre ci abl e extent but sacrifices the elegant presentation of the model in term s of catalyst activiti es varying between zero and .
.
"
=
uni ty
.
"
S .S.E.H. ELNASHAIE and S.S. ELSHISHINI
554
_ _ _ _ _
.. j5
�0 >
.. ;; Vi
.
• • •
.
J
_ _ _ _ _
6
 6
..
,., . A , B i f ur c a t i o n
P or o m • l • r
FIGURE 4.74 Schematic diagram for simple switching policy around the middle unstable steady state.
1/4.
Simple control strategies for the stabilization of the middle unstable steady state
In order to av oi d the continous drift of the system away from the unstable steady state with its high gasoline yield, two simple c ontrol s trategies are discussed in this section: to operate the system dynamically around this unstable steady state or to use a suitable feedback control system that stabilizes th i s unstable steady state. Both alternatives will be investigated i n the following parts of this section. /l4. 1 .
Dynamic operation around the middle unstable steady state
S w itc h ing the feed conditions arou nd the values c orrespondi ng to the desired middle steady state has been suggested by many investigators B le y , Bruns and Bailey, Douglas ,
( ai
1973, 1977;
1975, 1977;
1972;
Cinar et at. , 1 987). The idea in princ iple i s qu ite s imple and can be illustrated si mpl y by * Figure 4.74 . If as in Figure 4 . 7 4 , we have a desired state x corresponding to a feed variable (bifurcation parameter) A.,* and the desired state x* is unstable, then it i s possible in princ ip le to o pe rate the system as near as poss i b le to the unstable state x* by switching the feed c on di ti on para * meter bet ween the values AH, AL to keep the s y s t em near x . Specifically * * when x goes dynamically below x to a value of say x  8 then A is
STATIC AND DYNAMIC BIFURCATION BEHAVIOUR
555
switched to AH which corresponds to a unique state XH then the system will be forced towards increasing x till it crosses x * and reaches a value x * + 8 then A. is switched to AL corresponding to the state XL and thus the system is forced to a decreasing x and so on. The switching law i s thus: A = AH
and for A = AL
and for
for
*
x ::;; x  8 (for any dx I dt)
x* + 8 > x > x· for *
 8 (for dx I dt > 0)
x � x + 8 (for any dx I dt) •
x +8>x>x
*

8 (for dx I dt < 0)
Of course the specific dynamic behaviour of the system depends upon the values of AH, 1\.L, 8 which can be varied to get the best desirable response. Notice that in Figure 4.74 we are taking AL < Acrl (the first static limit point) and AH> A.cr2 (the second static limit point). In principle AL, AH can be taken between Acrh Acr2• however, this w i l l complicate the problem, as it has been shown by Cordonier et al. 1 990). The possibility of chaotic behaviour in this industrially important system together w i th a detailed investigation of the optimal switching policy as well as vibrational control (Bellman et al. , 1 983) is not covered in this book. Taking AH, AL outside the region Acrl Acr2 seems to be the less troublesome choice at the present level of knowledge regarding the dynamic behaviour of industrial FCC units . However, in contradistinction from the simple switching policy in Figure 4.74, the FCC problem has the added complication that changing one input parameter is not sufficient to control the system because of the specific shape of the bifurcation diagrams shown in Figure 4.66b,c. It is clear that when the dense phase reactor temperature goes above the desired middle steady state temperature we can force it down by switching FCD to lower values corresponding to a unique low temperature steady state, while when it goes below the middle steady state temperature it cannot be brought up by switching FCD alone for that will require an unrealistic high value of FCD to give unique high temperature steady state . The exact opposite applies when using Yfi, as switching parameter. Therefore, a switching policy employing both FCD and lfa, with YRo as the measured variable, should be implemented. The switching control law chosen here is expressed as follows: ·

Yfa = 0. 872,
Yfa =
YfaH•
with FCDL
Yfacr ·
YRD � YRD ( M . st. st. ) + 8
YRD
S
YRD ( M . st. st. )  8
S . S . E. H . ELNASHAIE and S . S . ELSHISHINI
556
ou
/ JJ
. )...
(c)
o.n
16
..
01
� OJ8
u
OJ6
OJ •
l l
l l .·
(b)
/9
��J��� J R L�0 OJ Ol OJ 01 OJ 06
'( ( Tim� i n days }
(d! � 1S L......L...L...L.l 0 0.1 Ol OJ 0.1 OJ 06
't (Tim�
in
days!
FIGURE 4.75 Dynamic response of the system under the switching policy 0=0.1, YfaH = 2.0, FCD1• = 0.2. (a) Dimensionless reactor dense phase temperature. (b) Dimensionless regenerator dense phase temperature. (c) Dimensionless gas oil dense phase concentration. (d) Dimensionless gasoline dense phase concentration.
Figure 4.75ad shows the behaviour of the system using a switching policy with the following parameters: 8 = 0. 1 , FCDL = 0.2, YtaH = 2.0. Although the reactor dense phase temperature is allowed to vary by a value of ± 8 and 8 = 0. 1 (50°C), the dense phase reactor temperature (Figure 4.75a) oscillates between 1 .745 and 1 .345 that is an oscillation with an amplitude of about 0.4 (200°C). As for the regenerator dense phase temperature (Figure 4.75b), although the amplitude of oscillations is settling to a low value, the centre line of these oscillations is drifting away from the steady state regenerator temperature to a very high value, above 2.4 ( 1 200 K). These results show that this switching policy has a pathological effect on the regenerator temperature which drifts to high values. The unreacted gas oil in the dense phase oscillates as shown in Figure 4.75c. The dense phase gasoline yield, X2 D oscillates strongly between values higher than 0.42 and values lower than 0.32 (Figure 4.75d). The steady state gasoline yield for this case is x2D = 0.387. Fi gure 4.76ad shows the behaviour of the system using a switching policy wi th another set of p arameters 8= 0.03, FCDL = 0.28, YtaH= 1 .85. Althoug h 8 for this case (8= 0.03 , i. e. l 5 ° C) is much smaller than for
STATIC AND DYNAMIC BIFURCATION BEHAVIOUR 0.42
1 7
).,
:
��
OJ&
14
OJ6
OJ7
). ( ..
0.(
0 .]9
/J
H
Q
1J
0.11
(b)
1 .1
>
_6
00
I
,
 2 00 K ( . D• m t n $i on \ t ' �
, Pro por \ i o n a l
,
2
I
00
6 co
( on l r ol l t r (, o i n
FIGURE 4.78 Pitchfork bifurcation for the closedloop system with simple proportional co ntroller (YRD vs. Kc).
absolute values greater than about 5.5, the steady state temperatures other th an the middle steady state, tend to infinity. Our presentation and discussion will be restricted to the more practically relevant range of Kc > O. /l5.
Effect of controller law parameters
The control algorithm contains 3 parameters Kc, lfass• YRDss· Kc is obviously the main parameter of the control algorithm. It is clear from Fig u re 4.78 that Kc must exceed a certain critical value be fore the controller is able to stabilize the middle unstabl e steady state. Elnashaie ( 1 977) has shown earlier, for a much simpler system, that Kc must exceed the value c orresponding to u ni queness of the middle steady state. For the present system with its i mperfect pitchfork shown in Figure 4. 78, it is possible to achieve stability of the desired steady state (middle stead y state) for values of Kc lower than Kc co rrespon din g to uniqueness of the steady state . In the regio n between point a, b o n Figure 4.78 the desired s te ady state becomes a low temperature steady state and is therefore stable (from a static point of view, i . e. it satisfies the steady state condition for stability). The portrait of the bifurcation diagram of YRv vs. Kc c an change with the change of any of the other two parameters in the control algorithm.
560
S . S .E.H. ELNASHAIE and S . S . ELS HISHINI
s . oo
4 . 00
0
�
3 . 00
(a)
Yras a = 0. 11 2
5 . 00
4.00
0 . 00
s . 00
4 . 00
(b)
y fau • O . I B
0 a: >
5 . 00 4 .00
(d )
v,.,. • • 1 . 5
b
O . 0�00 L2..L..00.,..,,.0_0_'5.0 0 I< C
O.OO L..'....l.' 0.00 2 .00 4.00 s . oo
KC
FIGURE 4.79 Effect of increasing yfass on the bifurcation diagram for the closed loop system with proportional control ( YRD vs. Kc).
We chose to alter Yrass · Fi gures 4. 79ad show the change of the bifurcation diagram with the change of lfass· Figure 4.79a i s the base case with Ytass = 0.872 which corre sp ond s to the middle steady state YRvss = 1 .5627 used (and kept constant) in the control algorithm. Figure 4.79b shows the effect of i ncreasing Ytass s l i gh tly to 0.875 (e qui valent to an increase of 1 .5°C), the point "a" spl its into points a 1 , a2 and the bifurcation diagram is composed of two disconnected curves , I is a hysteresis curve and II is a half "isola" with a small regio n of 5 steady states. Multiplicity of the steady states exist for Kc in the region from zero till the value of Kc corre sponding to the point a 1 , then uniqueness prevails (unique high temperature steady state) till Kc corresponding to point a2, then multiplicity starts again at a2 and ends at b. Notice that this dramatic chan ge took place for a change in Ytass equal to 1 .5°C ! Further increase of l':"tass causes points a h a2 to get further away from each other as shown in Fi gu res 4.79c,d. Fi gure 4.79d shows the continuous disappearance of multiplicity for very high values of lfass · Fig ure 4. 80 shows the effect of decreasing lfim on the bifurcation diagram. The change s in the bifurcation diagram , in a sense, are opposite to those observed with increasing Ytass in Fig ure 4.79. In this case, curve I is the half "isola" shown in Figure 4. 80b, while curve II is the
STATIC AND DYNAMIC BIFURCATION BEHAVIOUR
5.00
(a)
4.00
0 ..
>
y , .. . .
3.00
5.00
4 . 00
= 0 . 17 2
0 00
5 . 00
(b)
4.00
0
y tau
l.OO
c
4 .00
0.1
2 .0 0
1 .00
0.00
0,00
Yr.u s
(d)
KC
4 .00
6.00
0.00
0.00
= 0,1
v, ....
1.00 2 .00
: 0.1
11
0.00
5.0 0
>"'
(c )
3.00
2.00
56 1
2 .00
II
KC
4.00
s.oo
FIGURE 4.80 Effect of decreasing yfass on the bifurcation diagram for the closed loop system with proportional control (YRD vs. Kc).
Decreasing Ytass further, shrinks the half "isola" to the left, moves the multiplicity region on curve II to the right (which also shrinks) . For lfass = 0. 1 n o region of 5 steady states is observed. The above relatively simple analysis shows that the sen sitivity problem of this system is certai nly not solved by introducing proportional feed back control and s tab i l izi ng the des ired unstable steady state. The closed loop system is still "rich" in bifurc ation and sensitivity phenomena Other modes of control (PI, PID) need to be investigated. hysteresis curve.
.
Ill Summary of section results
i
s ri l
The relatively rigorous steady state model for ndu t a FCC units presen t in earlier sections, is e t en ed in this section to the unsteady state case in order to present to the reader the main characteristics of · the d namic behaviour the system. In this section we use a 4dimensional model which assumes that the adsorption mass capacities
ed
y
x d
of
562
S . S .E.H. ELNASHAIE and S.S. ELSHISHINI
of gas oil and gasoline are negligible. In the next section these assumptions are relaxed and a more rigorous 6dimensional model is presented. The drift of the system away from the operating middle steady state is presented and the extreme sensitivity of the system to initial values of catalyst activity is demonstrated. It is also shown that the reliability of the model predictions is very sensitive to the dimensionality of the model. Any reduction in the system dimensionality should be treated with extreme care specially in the industrially important region near the middle steady state. The simulation results presented show clearly that reduction of the dimensionality of the dynamical system in this case to 2dimensional gives erroneous predictions not only quantitatively but also qualitatively. The 4dimensional model represents the lowest acceptable dimensionality in order to obtain reasonably accurate results. Most control studies on FCC units (e.g. Kurihara, 1 967 and Nakano, 1 97 1 ) as well as the usual industrial practice (e.g. Huang et al. , 1 984) are based on suppressing the system's response to external disturbances, using a wide variety of control policies, while in fact it is shown in this section that the main control problem associated with FCC units is the stabilization of the unstable saddle type operating state of the system. Two control policies are presented in this section, both aiming at operating the industrial unit close to the middle unstable steady state region characterized by high gasoline yield. The switching policy, which is necessarily using two feed parameters because of the nature of the bifurcation behaviour of the system, shows some interesting results and can be problematic in industrial applications because of the drop in average gasoline yield and the drift of the regenerator temperature to very high values. The simple proportional feedback control policy shows very interesting pitchfork type bifurcation. The policy seems to be more practically sound than the switching control. However, the bifurcation behaviour of the system is extremely sensitive to very small variations in the control parameters. Further details regarding the static and dynamic bifurcation behaviour of the FCC units with SISO proportional feed back control, are given in the next section. 4.2.7
Static and Dynamic Bifurcation Behaviour of Industrial FCC Units
The s te ady state version of the model used in this sec ti on has been pre se n ted in the pre v i o u s sections (Elnashaie and ElHennawi, 1 979 ; El s his hi n i and Elnashaie, 1 990a,b ). A preliminary investigation of the dynamic behaviour of these important units is pre sen te d in section 4.2.6
STATIC AND DYNAMIC BIFU RCATION BEHAVIOUR
563
(Elnashaie and Elshishini, 1993) whereas the dynamic bifurcation investigation will be presented in this section. The model used is more general than the one used for the preliminary investigation since it relaxes the assumption of negligible mass capacity of gas oil and gasoline in the dense catalyst phase. The relaxation of these assumptions is based upon taking into consideration the catalyst chemisorption capa cities (Elnashaie and Cresswell, l973a, 1974; Arnold and Sundaseran, 1989; Elnashaie et al. , 1990; ll'in and Luss, 1992). I The dynamic model
The dynamic model can be written in the following vector form: (4. 1 24) The initial conditions are given by:
at
r = 0,
(4. 125)
X = =o X

The vectors X and X0 contain the state variables and the initial conditions vector respectively while Yta and Kc represent the bifurcation parameters for the open loop (uncontrolled, Kc = O) and the closed loop (controlled, lfi1 = constant) systems respectively; whereas i = t I EHR . The vectors X and X0 are given by, (4. 126) The vector E is given by, BR (Yv  YRD ) + a2 (YGF  YJ  Mv + M,.r + a , ( YGD  YRD )  fJILR
E=
N, [BG ( Yfas + Kc (�P  YRD )  YGD ) + f3c . Rc + a3 ( YR/J  YGD )  fJILG ] /YR Nz BR (X3f  XI/) )  \}1R a , · e yi / YRD + a3 · e y3 D x�D
[
( ( N3[ BR (X4f  XzD )  ( x2D \}1R
a2 .
N4 [ C� (\}J
G

)}
· e  y2 / YRD  a , .
\}1R )  \}1R . R,j ]
Ns [ R,.  c;; (\}1G  'I'R ) ]
J
x�D  e yl/YRD )]
(4. 1 28)
564
S . S .E.H. ELNASHAIE and S . S . ELSHISHINI
where, N, _ EHR
EHG
_ EHR 4EMR
N

f/lm3 and lflm4 are the mass capacitances due to the chemisorption of gas oil and gasoline respectively on the catalyst which has high surface area. These parameters can be very appreciable and have important dynamic implications on the system as has been shown clearly by Elnashaie and Cresswell (1974) and Il'in and Luss (1992). The values of N1, N2 and N5 are calculated from the operating and physical parameters of the industrial unit given in Table 4. 10 and 4. 1 1 . For the parameters lflm3 and f/lm4 they are taken to be equal to fHR i.e. N2 = N3 = 1 .0. The effect of higher and lower (than unity) values of N2 and N3 on the dynamic behaviour of this system needs to be investigated in order to establish the effect of chemisorption capacities of the catalyst on the dynamic behaviour of the system. Such an investigation has not been carried out yet and is not available in the open literature. II
Some static and dynamic bifurcation results and their discussion
The dimensionless air feed temperature used for this unit is Ytas = 0.872 and the dimensionless operating temperature of the reactor dense phase is YRv = 1 .5627. This operating condition corresponds to a steady state on the intermediate branch B of Figure 4.8 la (point a) in the multiplicity region with three steady states. Thus this operating point is an unstable point (saddle type steady state) as discussed ear1ier. It is clear from Figures 4.81b,c that this operating point does not correspond to the maximum gasoline yield, x2D· The alteration of the operating condition to operate at the maximum gasoline yield has been presented and discussed in the previous sections. In all cases the steady states are saddle type unstable steady states. As explained in the previous section, one of the simple ways to stabilize such steady states is by using nega tive feed back proportional control. The static bifurcation characteristics of this closed loop system has been discussed in the previous section and it was shown that the bifurcation diagram of the reactor dense phase dimensionless temperature, YRD versus the controller gain Kc is a pitchfork whic h is structurally unstable when any of the system parameters are altered even very sl ightly . In thi s section both the i ndustrially operating steady state as wel l as the steady state giving maximum gasoline yield will be presented for both the open loop and the closed l oop cases.
STATIC AND DYNAMIC BIFURCATION B EHAVIOUR
565
3 · 5 r... (a)
0 a >
2· 5 1· 5
     
.Y.'�
_ _
,_., } ..
,
'· =
0·5 0·0 0·5
v,
>
4.228964), steady states on branch D become unique and globally stable.
5.
.
.
,
ll3.
Behaviour of the system with feed back proportional control around the same YRD = 1. 5627 and with Yta disturbed from its value of lfa� = 0. 872 (Ysp 1. 562 7, Yjas = 0. 7) The i mperfect pitchfork in Figure 4 8 2 is structurally unstable, i e. =
any c h an ge in Ysp or Yfas c aus e s the bifurcation diagram to bre ak into two disconnected parts. This is shown in Fi g ure 4.84a where YJas has been changed from 0.87 2 to 0.7, while the set point Ysp remains at the same value Ysp = 1 .5627. .
.
STATIC AND DYNAMIC BIFURCATION BEHAVIOUR
57 1
1 · 80 "T". (b)
 U n 11�01t  S 1 �bto • ••• H B •• • • sPa
0 . 0t.""'""1rrro..',_....,.._,..,.t 0
3 Kc
i.
s
6
O · J S   ... 2 ·122
 


2 ·1 2 7 Kc:


2 ·132
FIGURE 4.84 Closed loop bifurcation diagrams for Yfas = 0.1 and Ysp = 1.5627. (a) Dimensionless reactor dense phase temperature vs. controller gain. (b) Enlargement of rectangle 1 in Figure 4.84a. (c) Dimensionless reactor dense phase gasoline concentration vs. controller gain. (d) Enlargement of rectangle 1 in Figure 4.84c. (HB = Hopf bifurcation point, SPB = Static periodic branch).
1.5627
In this case the steady state reactor temperature YRn = does not exist as a solution except as a saddle type unstable steady state at Kc = 0.0. In addition, the e x hange of sta l ity in Fi g ure 4.82, is broken and the bifurcation diagram breaks into two parts. The middle steady state branch D does not continue throughout the range of Kc and as shown in F gure 4.84, never corresponds to the maximum gasol ne yield. A Hopf bifurcation point and a terminating homo clini al ly, exist for values of Kc higher than the case in Figure 4.82. Figure suggests that, as for the previous case, the periodic solution may give average gasoline yield higher than the maximum of th e static branch however, operating under such conditions, introduces the danger of igni ti n to branch E as discussed earlier. The hi g h gasoline yield on the stable of branch decreases slightly as Kc increases. For Kc higher than that corresponding to the HB point, high gas oline yield is achieved on a stable branch. Between points 1 and the operating points are not globally stable, however for Kc greater than that corre s po nding to point 4, the steady states on branch C are unique and globally stable . .
c
i
c
bi
it
at point 5
i
periodic branch
4.84d
C,
o part
C
4,
�
S . S .E.H. ELNASHAIE and S . S . ELSHISHINI
572
5.0
4. 0
�
a
J.O 2 0 1. 0
'
D
. (' )
\'
rY )
• . .
. . .. .. ... � 1' .....,    • •• n., n .... ..t...._. ··· ·
·
D .. / ... .... �.._.L 'v · ; · (I)
0. 0 +..,.�....,.i 0. 5 ., (b) · · · ·•· · • · · · · · · · · ·r ;r .._._   . ••T'W'T•,... .... • .a. a.. • ( 1 l 0. 4 7_ I ' •• D � \ • D I I C 0. 3 \ � .
� 0. 2 a
"'T",
0. 1
B
r;;
'
�I I
I

(1 )1
••••
• • • •
Unstobf� S ta b I • HB SP8
0 . 0 +..:;;..,T.a.....,r...,.4 5 10 15 20 25 JO 35 5 0 Kc
FIGURE 4.85 Closed loop bifurcation diagrams for Y1..., = 1.419213 and Ysp = 1.5627. (a) Dimensionless reactor dense phase temperature vs. controller gain. (b) Dimensionless reactor dense phase gasoline concentration vs. controller gain. (HB = Hopf bifurcation point, SPB = Static periodic branch).
/14.
Operation at the maximum gasoline yield
As was mentioned earlier in this section, the industrial unit is not operating at its maximum gasoline yield. From Figures 4.8 1 ac, it is clear that the maximum gasoline yield occurs at YRD = 1 . 193 14 (about 60°C above the vaporization of the gas oil used in the unit). For the unit to operate at this temperature, the air feed temperature must be increased considerably to a value of Yta = 1 .419213. This is because the saddle type unstable steady state has a dependence over system parameters which is opposite to other steady states (Elnashaie and Cresswell, l973b; Elnashaie and ElBialey, 1980). The gasoline yield for this steady state is X2v= 0.437885 while for the case in Figure 4.82, it was X2v=0.38088, with a percentage increase of about 1 5 % . However, again this high yield is not attainable in a stable fashion without stabilizing this saddle type steady state using a proportional feed back control. Ytas is equal to 1 .419213 and the set point of the controller is the desired steady state temperature Yw 1 . 1 93 14. The bifurcation diagrams for this case are shown in Figure 4.85. Figure 4.85a shows th at basically the YRv  Kc diagram is an imperfec t pitchfork. =
STATIC AND DYNAMIC BIFURCATION BEHAVIOUR
( 
573
0 . 5 0 .,... 0 .4 5
0
N
>(
x, :0.4378 8 5
0.1. 0 0. 3 5
0. 3 0
0. 2 5
..
0
Average cone . fro m � riodic b r a n c h     Co n e . a t mid d l e u n s t a ble s ti!'Cdy s tate 5
10
15
Kc
20
25
30
FIGURE 4.86
Comparison of reactor dense phase gasoline concentration at the middle unstable steady state with the average yield from the periodic oscillations for Y1 = 1.419213 and Ysp = 1. 19314 at various values of controller gain.
Figure 4.85b shows that there is no static point or branch that gives X20 higher than branch D, this is because Ysp was chosen from Figure 4.8 1 as the steady state temperature corresponding to maximum static X20. The degenerate Hopf bifurcation point, 2, which is on the static limit point on the boundary of the five steady states region in Figures 4. 82a and 4.84a, also lies here on the same relative location. However, in the present case, the Hopfbifurcation point I lies on branch D at a very low value of Kc = 27.5 1 577. The amplitude of the periodic solutions emanating from HB point 1, grows as Kc decreases and terminates homoclinically before reaching the static limit point 4. The homoclinical termination occurs at Kc = 2. 89905 , with very large amplitude of the periodic oscillations. It is clear that in order to operate the reactor for this case at stable steady states on branch D, Kc must be very large (Kc > 27 .5 1 577), which may prove to be a practical control hardware problem. It is interesting to notice from Figure 4.86 that for Kc as low as 10, the average gasoline yield for the periodic oscillations is very close to the maximum static Xzo on branch D (X20 = 0.437885). Obviously it is of practical importance to investigate the maximum and minimum temperature of these oscillations to ensure that it does not exceed safe limits when rising and does not go as low as the vaporization temperature of gas oil when falling.
1 .9 8 0 . � . .. . 2.0 B o �' EJ s il l] o s fm 2o . · . .. · . l EJ lliil . 7� m . 000
574
�
>
S . S .E.H. ELNASHAIE and S . S . ELSHISHINI
(a)
: ::
1 .0
0.8
a :·:
O.B
2.0
0
�00
600
(c )
o
250
sao
0 cr > 1 . 5
1 .0
�0
>
 \ )
1 .0
0 . 46
( e)
.
41
0 �6
�00
0
.
o "' 0 . 4 2 >
1 .3 3 3 which makes it unfeasible to work in this region. 4. A region between U01Umf = 1 0.081 2.6636. In this region there is an unstable steady state surrounded by a periodic solution. The steady state is at YD < 1 .333, whereas the oscillatory temperature reaches above 1 .333. It is feasible to work at the steady state in presence of a temperature controller. At UoiUmF 1 0.08, PE produc tion rate is 1 0.87 kg/s and a productivity of 78.264 kgPE/gcat is achieved which is a high value compared to the usual industrial range of 220 kgPE/gcat. The value of 1 0.08 for the relative gas velocity is quite reasonable. Table 4.23 Effect of relative gas velocity on the different variables at the unstable and osciUatory steady states for optimum conditions. YD (min) .
YD (max)
X
(average)
PE, g/s
1 29 1 2.96
1 207
1 . 69 8
1 . 208
1 .67 4
0.245
1 3204 . 82
1 .0 2 8
1 .644
0.237
1 3 362.36 1 3440.96
1 .2 1 1
0.252
1 . 598
0.228
1 2 24
1 .493
0.2 1 2
1 3028.00
1 . 305
1 . 308
0. 1 89
1 2238 .06
.
605
STATIC AND DYNAMIC BIFURCATION BEHAVIOUR 1 5000 . 000
1 4000 . 000
.. 1 3000. 000 "'
w Q.
1 2000. 000
1 1 000. 000
�
1 CXXXJ. OOO
. ..
1 .333 1 .323
0 >
1 .3 1 3 1 . 303
1 .293
1 .243
1 273 1 . 26:3
I Q .OOO
1 1 .000
1 2.000
1 3.000
1 4 .000
Uo I Umf
15 000
16 000
17 000
IS 000
FIGURE 4.97 Effect of gas velocity on: (a) polyethylene production rate, (b) reaction temperature.
A region between U01Umf= 1 2 .6636 to HB2 = 1 2.706. This region consists of an unstable steady state and a periodic solution. The oscillations are below Yv = 1 .3 3 which makes it possible to operate the system in this region as regard to the melting point of poly ethylene. Furthermore, a temperature controller is not required for the periodic solution since there is no need to avoid the tempera ture oscillations. An average PE production rate of 1 2.2492 kg/s which corresponds to a productivity of 8 8 . 2 kgPE/gcat is obtained while operating at the periodic solution, whereas a value of 1 2. 1 4 1 1 7 which corresponds to a productivity of 87.4 kgPE/gcat i s reached while working at the unstable steady state in the presence of a stabi lizing controller. 6. A region for Uc/Umt > 1 2 .706. In this region there is a unique stable steady state with high PE production rate. The operating dimension less temperature is below 1 .333 but the relative gas velocity is rela tively high causing solid entrainment problems.
5.
R egi ons 4, 5 and 6 (till U01Umt = 1 8 .0) are en larged for Yo and PE in Figure 4.97 for the static branch only. It is clear th at the most preferable operating conditi on is at U01Umf = 1 0.08 due to higher convers i on , reasonable gas v e l oc ity and high PE pro duc tio n rate. However, it may still pose a catalyst e ntrai nment prob l em .
Certainly more j oint research work between ac ademi a and industry is needed to discover the exciting pos sib i l ities of this process.
APPENDIX A
Derivation of equation 3.2 for the NonPorous Catalyst Pellet
Consider a film of thickness 8 surrounding a nonporous spherical particle of radius Rp, on the surface of which a single irreversible reaction is occuring . The accumulation of reactant in a thin spherical layer of the film is described by the diffusion equation,
(A. l )
Where D i s the reactant diffusion coefficient. The boundary condition at the outer surface of the film is given by, (A. 2) A mass balance taken over unit area of the particle surface plus the adjacent fluid layer at r = R; gives,
mean reactant concentration C
Now the rate of change of the in the film can be obtained by i ntegrati ng equation A . l between the limits r = Rp and r= Rp + 8. Noting that,
We obtain,
606
APPENDIX A
607
If we use the concentration difference (Cb  C') divided by the film 8 as an approximation of the gradient (() Cf() r) r = Rp + 8 , where C* is the reactant concentration in the gas phase surface, A.4 can simplified to,
thicknes s
then
at at the catalyst
be
(A. 5)
 (
where we have observed that
1
Noting
that
C = ( Cb
1
2
�2 + 82 R R ·8 p p
+ c* ) / 2 ,
* 8 cs  de = k (Cb  C* )  k Cv C*  a g KA 2 dt
where kg is
)
equation A.5 becomes,
a mass transfer coefficient, defined by kg = D/8.
(A.6)
APPENDIX B
Derivation for the Integral Collocation Formulation
in
n
By tegrati g the conduction e obtain,
w
a (il

dt
0
equation 3 .8 between the limits 0 and 1 ,
1 ) [ aYJ 2
=
2 Yw dw = a w 
aw
0
(B. 1 )
Defining, (B.2) From 3 . 1 0, B . l and B.2
we find,
u i the integration formula given by Villadsen and Stewart ( 1 967),
Now s ng
r Yw 2 dw = Y = £..J � w, y, Jo N+l
I
(B.4)
i=l
W; form
[� ) [= LN (N+I l J
where the £..i i=l
"'i
dY
'
dt
+ WN+I
a set of weighing factors . From B . 3 and B .4.
s =aNu1 (1 � + a /3 (exp ( y / � )) · X5 )
dY

_
dt
(B. 5)
From 3 . 1 4 and B . 5 , a
i=l
W;
L BulJ i=l
d� + WN+1  = a Nu1 (1  � + a J3 (exp ( y i � )) · X5 )
dt
=
608
_ 
APPENDIX B
609
which reduces to,
The system of equations
of the complete system.
3.7, 3 . 1 4 and B.6 are solved for the dynamics
APPENDIX C
Local Stability Analysis for the Non Equilibrium Single Pellet
First write equations 3 .97 and 3 .98 in the form,
dXs ·' dr
L
=
1  J; (Y ) X,
l + f2 (Y) I a
and,
af ( Y ) X u
3
(C. l )
s
where, fi ( Y) = exp ( yE I Y)
f3 ( Y)
Introducing perturbation variables,
=
exp (  r I Y)
XI = Xs  Xsp x2 = Y  YP where Xsp and Yp are steady state values of Xs and Y, the linearized form
of equations written as,
C.l
and
C.2
for arbitrarily small perturbations can be
( C. 3 ) ( C. 4 ) Where,
610
APPENDIX C
1
cl = L s
(f4,X,p I

61 1
afs.x,p ) I
a f3f; x,P + f3A u: x,P  af;x,P ) C4 = a f3J;y +f3A u:y  af;y ) 1 C3 =
•
+ 
J4 
p
1  fJ Xs
' p
• •
p

1 + (fi I aa )
fs = f3 Xs and for example,
The conditions necessary and sufficient that equations C. l and C.2 have asymptotic solutions tending towards zero are: ( C. 5 )
(C. 6 )
Noting that
Condition C.5 can be reduced to,
_!__ > f3 (f{h  fJi)  (a/3 I aa )f32fi. a
[fi + af3(1 + !2 / aa )] 2
612
S . S .E.H. ELNASHAIE and S . S . ELSHISHINI
which is simply the slope condition ( C. 7 )
at the point of intersection of the steady state {F, Q+ curves, given by equation 3 .91 . The second condition C.6 can be written in the form,
ALs < B
where, A
= a/3.fj5, '
B = aj.s .
�
r p
x,p
+ f3a ( f4, Yp  af5,• rp )  1 •

{4, x,.
( C. 8 )
APPENDIX D
Stability Conditions for the Simple Cell Model
of the Fixed Bed Catalytic Reactor
n
The system is described by the following set of nonlinear ordi ary differential equations given in chapter 3 :
Catalyst particle equations
;
dx
a3  = XJ·  X ·  a exp (  y I YSJ. ) · X}. J f d
*

*
(3. 1 40) *
�i = y.  y . + _ a 13 exp (  r I y . ) . X . a4 Sj T Sj } J d
+ f3A a3
df
*
dX1 df

(3 . 1 42)
Gas phase equations a1
dX . * 1 = M( X ·  1  XJ. )  ( XJ.  XJ. ) ] dt

and, n�
L.
and j = 1, 2, Defining,
. . .
N,
dY. df

where
1
=
H( Y._1  Y.} )  ( Y·}  fSj. ) }
N is the number of cells. X z j = Ysj X4 } =
fjl
g zJ
613

�jss
lJ  l}ss
= fjss + g ,j xlj + g 2j x2j
= ( :�. ] SJ
SS
(3 . 143)
(3 . 1 44)
S.S.E.H. ELNASHAIE an d S . S . ELSHISHINI
614
where fi1 is the linearized form ofjj obtained by Taylor series expansion for arbitrarily small perturbations about the steady state equ ations 3 . 1 40, 3 . 142, 3 . 1 43, and 3 . 1 44 can be linearized into the form,
a4
d/ = �1x11 + �1 x21 + �1x31 + B41x41
(D. 2)
d
(D. 3)
dx2 .
dx3 . a1 t 1 = C1J· x1 J· + C2J.x2J· + C3J.x3J. + C4J.x4J· + Mx3)·  1
where, At J
A31
�J
=
=
=
�j = c,1
(1 + agt)
A2J = ag2J A41 = 0
1
agt/f3r  f3A )  f3A fJA 1
Dtj =
B41
=
=
ag2/f3r  f3A )  l
1
c2 1 = o C41 = o
c31 = (l + M ) =
�J
�} = 1
0
n. . = 0
D4 1 =  ( l + M )
''3)
and X4o =
0
This set of linearized equations can be put in the matrix form,
d BX C dtX = 

where,
(D. 5)
APPENDIX D
[
Au Az}
A  = �J
_,
el i
�j
Bz j
Cz J
Dzj
A� )
615
A,]
8:3} B4
�
c4J
c3J
�j
D4J
Necessary and sufficient conditions for the asymptotic stability of system 0.5 is that all eigenvalues of det (� 1 �  /ll) = 0, have negative real parts. Owing to the special form of the matrix �. the characteristic equation can be written in the form, det (� 1 �  ll D =
n
IJdet (a1A1  Ill) = O
(D. 6)
j= l
After some lengthy algebraic manipulations the determinant in (D.6) reduces to, det (c 1 !1.  A. D
= 11 A.4 + Q11A.3 + Q12A.2 + Q13A. + Q14 n
j=!
S.S.E.H. ELNASHAIE and S.S. ELSHISHINI
616
where,
Qj2 =
( 1 + ag1 1 )( 1 + M) al a3 
( 1 + aglj )[ ag 2j ( {JT  fJA )  1 ]
( 1 + M)( 1 + H ) Q1 3 = a l a2 _
1 + ag11
_
ag2/f3T  fJA )  1
[
a3
a4
[
] [ [
+
]
1+H a2
]
1 _ 1 + M + 1 + aglj + ag2j agl/fJT  {JA )  {JA · a3 a4a2 al a4 a3
· l+M + 1+ H
Q1 4 =
[
(1 + M)( l + H) + ���
( l + agi J ) ag2/fJT  fJA )  l 1 + M _ . a4 al a3
__
[
( 1 + ag11 ")(1 + H) +  a3a2
a1 1
al a2 a3 a4
a2
]
+
ag2 j fJA
l  ag2 J ( fJT  fJA )  1 + · a3a1 a3a4a1 a4
[( 1 + M) ( l + H) ( 1 + ag1 1  ag21{JT )
]
l+H a2
]
(1 + H ) (1  ag21{J T ) � ( I + M) (1 + ag11 ) + 1 ]
For all the roots of 0.6 to have negative real parts the following conditions must be satisfied
1 ) QiJ > O
which can be written in terms of parameters as, l+H
2 ) Q2j
>
�
0
+
ag2/fJT  fJA )  l a4
+
1 + ag1J a3
+
l+M >O al
(D. ?)
APPENDIX D
617
which can be written in terms of parameters (after some manipulation) as,
3)
Qj4 > 0
when expanded this condition gives, M · H · (1 + ag11  af37g21 ) + (11ag11  MCig21{37) > 0
4)
QJIQJ2QJ3 > Q}1QJ 4 + Q}3
(D. 9) (D. l O)
We will not expand this condition because of its extreme complexity.
APPENDIX E
Velocity of the Creeping Reaction Zone in
Fixed Bed Catalytic Reactors
(1973)
Rhee et al. have derived an analytical expression for the veloci ty of the creeping reaction zone for the geometrically coupled cell model. Their erivati on is not generally valid, and includes an impli c it assumption which is not appreciated in their paper. We derive a similar expression for the radiation model, showing the assumptions implied and discussing their limitations. For the moving reaction zone, we can define a moving coordinate system,
d
i_ = j  v
'r = t
�
v
i s the ve loci ty of the moving coordinates (velocity of creep),