Topics in Chemical Engineering
A series edited by R. Hughes, University of Salford, UK
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MODELLING, SIMULATION AND OPTIMIZATION OF INDUSTRIAL FIXED BED CATALYTIC REACTORS by S . S .E.H. Elnashaie and S.S. Elshishini
Volume 7
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THREE-PHASE 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 GAS-SOLID CATALYTIC REACTORS by S.S.E.H. Elnashaie and S.S. Elshishini
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Dynamic Modelling, Bifurcation and Chaotic Behaviour of Gas-Solid 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
•
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Japan • Luxembourg Singapore
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•
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Germany • India
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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 Gas-Solid Catalytic Reactors. - (Topics in Chemical Engineering, ISSN 0277-5883 ; Vol. 9) Title II. Elshishini, S. S . III. Series
I.
660.2995 ISBN
2-88449-078-7
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 (1919-1993) 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 Non-Equilibrium 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 NON-CHAOTIC 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 Poincare-Andrononv-Hopf 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 Three-Dimensional 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 Non-Autonomous S ys tem s and th e Construction of Ex c itati on Diagrams for Non-Autonomous 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 Non-Chaotic Attractors 171 2. 1 9 . 1 Presentation techniq ues 2. 1 9.2 The discrete-time 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 GAS-SOLID CATALYTIC REACTORS C s
CHAPTER 3
3.1
3.2
219
223 Single ataly t Particle 3 .1. 1 Non-porous 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 adsorption-desorption. The lumped parameter adsorption-desorption equilibrium model (LP-ADEM) 257 3.1.2.3 Effect of non-equilibrium adsorption desorption. The lumped parameter non equilibrium adsorption-desorption model (LP-NEADM) 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 Non-isothennal dynamic models 3 .2.2
CHAPTER 4
4. 1
STATIC AND DYNAMIC BIFURCATION BEHAVIOUR AND CHAOS IN SOME GAS-SOLID CATALYTIC REACTORS
Fluidized Bed Catalytic Reactor with Exothermic Consecutive Reactions 4. 1.1 The two-dimensional case with proportional control 4.1 .1.1 The unforced (autonomous) case 4. 1.1.2 The periodically forced (non-autonomous) 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 (non-autonomous) case. Detailed analy s i s of resonance horns, period doubling loci and chaos 438 4 1 2 The three-dimensional 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 Ziegler-Natta 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 NON-POROUS 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 NON-EQUILIBRIUM 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 El-Rifaie (Cairo) who introduced us to the dynamics of chemical engineering systems more than twenty-five 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 gas-solid 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), Al-Khowaiter, Al-Humazi, Alahwan y , Babikr (now with USM, Malaysia), Al-Faris, Al-Zahrani 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 re-emphasized 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 Pre-exponential factor
in surface reaction rate
constant Pre-exponential 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
Pre-exponential fa ctor for reaction Surface reaction rate constant, 1/s
Pseudo-homogeneous 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
Pre-exponential 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 hold-up 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 /3-r 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 gas-solid 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, o-xylene 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 co-operation 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 co-polymers (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 co-workers (Elnashaie and El-Hennawi, 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 co-workers (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 gas-lift 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 co-workers (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 ; Saleh-Alhamed, 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 co-workers (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 co-workers (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 co-operation 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 ill-favoured 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 gas-solid 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 Al-Khayyam, 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' Al-Khayyam, 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 Belousov-Zhabotinsky reaction is a well-known example of this. Oscillating reactions arouse a great deal of interest for chemists and biologists, as they mimic self-organizing dissipative structures and biological rhythms. However, it was shown in the case of the chloride-iodide 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 quasi-steady-state 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 set-up 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 spatio-temporal 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 gas-solid 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 gas-solid 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 gas-solid 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 non-equilibrium adsorption-desorption b y Elnashaie and Cresswell more than twenty years ago, for porous and non-porous 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 gas-solid 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 co-operation 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 co-operation 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 gas-solid 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 physico-chemical 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 non-porous) 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 non-autonomous (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 gas-solid 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 gas-solid catalytic bubbling fluidized bed reactors using Ziegler-Natta 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 gas-solid catalytic reactors. We also hope that applied mathe maticians will find in the book some inspiration for adding some of these rather fascinating gas-solid 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 time-independent 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 non-existing 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 time-independent state for certain given time-independent 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 time-independent 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 NON-EQUILIBRIUM 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 non-equilibrium 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 non-equilibrium 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 non-equilibrium 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 hold-up 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 non-equilibrium 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 time-independent 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 8-1 .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 non-equilibrium 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 non-linear 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 non-monotonic as wi ll be shown later in this c hapter . For higher degrees of non-linear 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 non-homogeneous 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
A-2
( 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 non-linear 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 non-isothermal 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.{, A-2 which are the roots of the characteristic equation: 2 A. ' - ( trA_' ) A. ' + det A_'
=0
( 1 .59)
and the non-homogeneous 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.{, A-2 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.{, A-2 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 time-independent 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 ce-ntrat ion , XA
FIGURE 1.6 Supply (S (XA)) and consumption (C (XA)) functions versus concen tration (XA) for the CSTR with non-monotonic 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 non-equilibrium 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 non-linear 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
<0
( 1 .78)
Equation 1 . 78 can be written in terms of the physical parameters and the steady state concentration as follows, ( 1 .79) This inequality is always satisfied if, ( 1 .80)
ELEMENTARY CHEMICAL REACTORS DYNAMICS
39
That is, ( 1 .8 1) It is easy to show that the maximum of the rate of reactions occurs at XAm where, ( 1 . 82) Therefore all XAss < 11 ae correspond to steady state points to the left of XA m· This is to say that all steady states with XAss < XA m are stable whether they belong to the multiple steady state region (e.g. line 1 on Figure 1 .6) or the unique steady state region (e.g. line 2 on Figure 1 .6). However for steady states to the right of XA m· we have, ( 1 .83) At these points the following inequality applies, ( 1 .84) Consequently for these steady states the second term on the left hand side of inequality ( 1 .79) will always be negative, ( 1 .85) In this case the stability of the steady states will depend upon the absolute values of the two terms on the left hand side of inequality 1 . 79. Therefore the system is stable if, ( 1 .86) and vice versa. It is clear that, ( 1 .87)
40
S . S .E.H. ELNASHAIE and S.S. ELSHISHINI
and, k.
1 - ae . XAss
(1 + ae . XA ss ) 3 -
( 1 .88 )
Therefore the stability condition for XAss > 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 non-isothennal non-adiabatic 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 e-EIJV;T CA ·
( 1 .9 1 )
ELEMENTARY CHEMICAL REACTORS DYNAMICS
41
q c ., T,
c. Cs T
FIGURE 1. 7
Schematic presentation of the non-isothermal, non-adiabatic 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 non-linear algebraic equations ( 1 . 96 and 1 . 97 ) to obtain XA. Y for a certain set of parameters, to the problem of s olving one non-linear 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 non-adiabatic 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_ <
( 1 . 1 1 0)
0
(1. 1 1 1)
It can be shown that the P1 stability condition ( 1 . 1 1 0) is itself the slope condition discussed earlier. This condition can be written as, g1 1g22 - gl 2g2 1
L
Since L is
always
L
>
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 non-adiabatic 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 <0 RA - - +
( 1 . 1 32)
the following form,
RB - 1 + L + -L- RA < 0 ·
)
( 1 . 1 3 3)
50
S . S .E.H. ELNASHAIE and S . S . ELSHISHINI
The static stability condition in this case, is given by,
It is clear that for L > 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 two-dimensional 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 non-oscillatory depending on the nature of the eigenvalues. The stability of these steady states (whether oscillatory or non-oscillatory) 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 non-oscillatory (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 !
c
0
·v. c: �
.§ b Cl
D im e n s i o n less
c o n c e n t ration
,
XA
FIGURE 1.9 Phase planes for steady states 1 and 3 which are stable nodes (hdow temperature steady state, 3shigh temperature steady state and 2 smiddle unstable saddle type steady state. Lines a-2, b-2 (---) are insets (stable manifolds =separatrix) of steady state 2, lines 2-3, 2-1 (_) are outsets (unstable manifolds) of steady state 2, li (x) are initial conditions).
Dim£>ns i o nle ss
FIGURE 1 . 1 0
c o nc £>n t rat i o n
•
XA
Phase plane for steady states 1 and 3 which are stable foci.
52
S . S .E.H. ELNASHAIE and S.S. ELSHISHINI
":'
�..
.. ... :1
lz
a.
E .!!
Ill VI ..
c
0
'iii c ..
.5 0
Dimensionless concentration
•
XA
FIGURE 1.1 1 Phase plane for steady state 1 which is a stable node and 3 which is an unstable node.
When the real parts of the complex eigenvalues are positive ( or the eigenvalues are real and positive) then the steady state is unstable whether oscillatory or non-oscillatory. The non-oscillatory unstable steady state is us u al ly an u n stabl e node (Figure 1 . 1 1 ) while the unstable oscillatory state is called an unstable focus ( Figu re 1 . 1 2) . >.
�
� .... :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 non-oscillatory 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 two-dimensional 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 a-2 and line b-2). 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 a-2-b 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, h-3 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 , 2-3 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 n-dimensional systems, we will restrict our discussion, for simplicity to phenomena well known in two-dimensional autonomous unforced systems. Discussion of forced (non-autonomous) two-dimen sional systems as well as higher dimensional systems will be presented later in the book. In the general n-dimensional case, the set of non-linear 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 non-linear 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 non-monotonic dependence upon reactants concentration(s). For the non-isothermal 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 non-equilibrium 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 Lyapunov-Schmidt 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 non-isothermal 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 Frank-Kamenetskii ( 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 saddle-node 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 saddle-node 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 (two-dimensional system) For the two-dimensional 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 n-dime n sion al system. We start with the original equatio n, f(,!_, J..L ) = 0
(2 . 8 )
where :! is an n-vector 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 on-s ingular (full rank = n), where as for a bifurcation point no such k exists (rank
of k). This exchange of col umns o f the matrix 2. 1 1 has an i nterestin g geo metri c al interpretation. The transition from p arameter J.1 to parameter r means that the branching diagram (static bifurcation diagram) .:! vs. y is obtained from the di agram of .:! vs. J.1 by a rotation of 90°. As illustrated in Figure s 2.3 and 2.4, the turning point disappears when the branch is parametrized by y = xk . This removal of singularity by change of parameter is reflected algebraically in the non-singularity of the "new" Jacobian .
STATIC AND DYNAMIC BIFURCATION
Ill
c
65
u
+
c:: ><
T
FIGURE 2.3 Turning point (static limit point, SLP) when the bifurcation parameter is J.L
FIGURE 2.4 Parameterization of the bifurcation diagram making y( Y Xk ), tbe bifurcation parameter instead of J.t (singularity is removed). =
Trying the same exchange with the matrix associated with a bifurcation point, we obtain Figures 2.5 , 2.6. Here the singularity is not removed by changing the parameter, the bifurcation po int remains . Preserving the rank of the matrix or increasing it by attaching the column E11 to the Jacobian E-x can be described in terms of the range of a matrix. Assuming that the rank of the Jacobian is n - 1 . We first consider the situation of a bifurcation point where there are constants C; such that, i F = zn+i = � � C. z -/1 -
I ). "'
.. =><
Static bifurcation point (SBP) when the bifurcation parameter
FIGURE 2.5
is J.t.
n
i= l
1-
(2. 1 4)
• c X
FIGURE 2.6 Static bifurcation point (SBP) when the bifurcation parameter is r (singularity is not removed).
S . S .E.H. ELNASHAIE and S . S . ELSHISHINI
66
that is, EJl
E
ran ge
of Ex
(2. 1 5 )
because the vectors l:.i span the range of Ex . In contrast, for a turning point, there are no constants C1 such that Ef.l can be expressed as a linear combination of the n column vectors l:.' of the singular matrix Ex , that IS,
( 2. 1 6 )
Loosely speakin g , EJi carries full rank information that is lost by ?:.1 • This is a characteristic feature of a turning point (static limit point). Now the following formal definitions of bifurcation points and turning points (static limit points, SLPs) can be given.
Definition (] )
( :!_0 , J10 ) is
following four conditions
a simple bifurcation point if the are satisfied:
2. E! (:!_0, J1 0 ) has a simple eigenvalue 0 or equivalently 3.
rank (f_! (!_0 , J10 )) = n - l
EJl ( !.o • Jl o ) E range E/!.o,Jl o )
(2. 1 7)
(2. 1 8) (2. 1 9)
4. exactly two branches intersect with two distinct tangents. For more formal analysis of this hypothesis see Seydel ( 1 988).
Definition (2 ) (,!.0,J1 0 ) is a turning point following four conditions hold:
(static limit poin t) if the
2. EJl ( !_0 , J1 0 ) has a simple eigenvalue 0 or equivalently, rank
EJ1 (!_0 , J10 ) = n - 1
3 . f.JI (!_0 , Jl o ) e; E;r (!.o , Jlo )
that
(2.20)
(2.21 )
is
(2.22)
STATIC AND DYNAMIC BIFURCATION
67
4. There is a parameterization _!( CT), Jl ( CT) with _!( CT0 ) = _!0 , Jl ( CT0 ) = J.l 0 and,
d�
d2J1 ( CTo )
:;t:
0
(2.23)
Hypothesis 4 impl i es the "turning property". For more details see Seydel ( 1 988). A simpler and maybe more elegant analysis of the static bifurcation (steady states or sometimes also called bifurcation of fixed points) is nicely presented in a simple manner by W i ggins ( 1990) based on eigenvalues and center manifold theorem (Guckenheimer and Holmes, 1 983). The analysis appli es to high-dimensional systems as well as low dimensional systems. We will briefly present this approach to the analysis of the bifurcation of fixed points . Consider the set of differential equations,
dx dt
-= =
F(x,Jl)
(2.24)
---
where :! is an n x 1 vector of state variables, E is an n x 1 vector of functions and J1 is a p x 1 vector of parameters. E is r times differentiable, where r is deTermined by the need to Taylor expand (2.24). Usually r = 5 will be sufficient. Suppose th at (2.24) has a fixed point (steady state) at ,
(2.25)
in other words, (2.26)
Two qu estio n s ari se
,
1 . Is the fixed poi n t stable or unstable? 2. How does the stability characteristics change as J1 changes?
We will concentrate here on the second conditions and only for static b i furcati on (appli c ation of the procedure to dynamic bi fu rcati on will be di s cussed later in this c h apter). To i n ve st i g ate the problem we linearize equation (2.24) in the neighqourhood of the fixed point (_!0 , !::a ) to obtain, (2.27)
68
S . S .E.H. ELNASHAIE and S.S. ELSHISHINI
where ! = :!: - :!:0 , and the matrix
Ex is given by,
a F = E -X ax
(2.28)
Starting with the simple definition of hyperbol i c fixed points: A hyperbolic fixed point (&, Jl) is a fixed point at which Ex( :!:0 , Jl0 ) has no eigenvalues which lie on the imaginary axis (no eigenvalues with zero real parts). Hyperbolic fixed points are structurally stable, Jl does which means that varying Jl s l ightly in the neighbourhood of -o not change the nature of tile stability of the fixed point. The interesting behaviour (or as Wiggins nicely calls it "the real fun") starts when Ex(:!:0 , Jl ) has some eigenvalues on the imaginary axis . In fact the more eigenvalues of the system on the imaginary axis, the more exotic the behaviour of the system. In the following we will confine ourselves to the situation with one real zero eigenvalues. In this case the structure near the fixed point ( :!:0 , Jl ) can be determined by the associated center manifold, which can bewritten in this case as the following scalar differential equation, ,.. dX " - = x = f(x, J.L ) dt "
(2.29)
where x = x - x0 and [1 = J.L - J10 • In the remaining part of this section we will drop the on x and J1 for simplicity which will represent from now on deviation from X0 and Jlo· Equation (2.29) thus satisfies the two simple relations: A
1 . The fixed point condition
2.
/(0, 0) = 0
(2.30)
The zero eigenvalue condition
a.r (0, 0) - 0 -
ax
(2.3 1 )
I f the re are more parameters i n the problem than the single J.L({l ) in (2.29) we will consider all, except one, as fixed. We will consider in a later section, the i nfluen ce of the simultaneous change of more than one parameter simultaneously . We will follow now some of the well kno wn static bifurcation when one eigenvalue is zero u sing the associated center manifold equation (2.29).
STATIC AND DYNAMIC B IFURCATION
69
a. Saddle-node bifurcation or static limit point (SLP)
Consider equation (2.29) when f( x , J.L ) = J.L - x 2 , that is, x = f(x,J.L ) = j.l - x 2
(2.32)
It is easy to verify that the point (0,0) is non-hyperbolic, of course, f(O, O) = 0
iJf (0, 0 ) = 0
ax
We can also extract more from equation (2.32). The set of all fixed points of (2.32) is given by,
(2.33) Equation (2.33) represents a parabola in the J.L -X plane (the bifurcation diagram with J1 as the bifurcation parameter) as shown in Figure 2.7. It is clear that (2.33) has no fixed points for J1 < 0, and for J1 > 0 it has two fixed points one stable (solid branch of the parabola) and the other unstable (dashed branch of the parabola). This particular type of bifurcation (where on one side of a parameter value there are no fixed points and on the other side there are two fixed points) is usually refered X
� =X
\ \
' , (0,0 )
BP : SN
'
FIGURE ---
=
2.7
2
..... .....
....
... --
� - - -
Bifurcation diagram of fixed points for equation 2.32 (- = stable,
WIStable).
S.S.E.H. ELNASHAIE and S.S. ELSHISHINI
70
to as a saddle-node bifurcation, or as we called it earlier, static limit point (SLP). b. Transcritical bifurcation
Consider the s c alar differential equation, .X = f(x,Jl.) = Jl.. x - x2
(2.34 )
again it is easy to verify that, j(O, O) = 0
and,
h
(0, 0) = 0
(2.35)
(2.36)
The set of all fixed points of (2.34) is given by, X = 0;
X = J1.
(2.37)
which is shown in Figure 2.8. For J1. < 0 there are two fixed po i nts , x = 0 is stable and x = J1. is unstable. The two fixed points c oale sc e at
"
/
/
/
/
/
/
/
/
,"
/
/
(0,0 )
FIGURE 2.8 Bifurcation diagram of fixed points for equation 2.34 (- = stable, - - - = unstable).
STATIC AND DYNAMIC BIFURCATION
71
J1 > 0, x = 0 i s unstable and x = J1 i s stable. An exchange of stability occured at J1 = 0.
J1 = 0 . For
c. Pitchfork bifurcation
Consider the scalar differential equ ati on , it is clear that,
i = f ( x , Jl ) = Jl . x - x
3
(2.38)
/(0, 0) = 0
(2.39)
at (0,0) - 0 -
(2.40)
and,
dx
The set of fixed points of 2.38 are given by, x
=
which is shown in Figure 2.9.
0,
x 2 = J1
(2.4 1 )
X
FIGURE 2.9 Bifurcation diagram of fixed poi nts for equation 2.38 (- = stable, - - - = unstable).
72
S.S.E.H. ELNASHAIE and S .S . ELSHISHINI
For J1 < 0 there is one fixed point, x = 0 which is stable. For J1 > 0, x = 0 is still a fixed point, but two new fixed points have been created at Jl= 0 and are given by x2 = Jl. In the process x = 0 has become unstable for J1 > 0 with the other two points stable.
d. Zero eigenvalue without bifurcation (an inflection point) Consider the scalar differential equation,
it is easy to verify that,
and,
x
= f(x, Jl ) = )1 - x3
( 2 42 ) .
/(0, 0) = 0
(2.43)
z
(2.44)
(0, 0) = 0
Moreover, the set of fixed points of equation 2.42 is given by, (2.45)
which is shown in Figure 2 . 1 0.
X
FIGURE 2.10
Set of fixed points for equation 2.42.
STATIC AND DYNAMIC BIFURCATION
73
In this case, despite equations (2.43) and (2.44), the number of fixed points and their stability characteristics at both sides of the point (0,0) (i.e for Jl > O and Jl < O ) are the same, namely unique stable fixed point. Therefore the point (0,0) is not a bifurcation point. Comparing cases a, b, c on one hand with case d on the otherhand, it is clear that the condition which states that a fixed point (0,0) is non hyperbolic (i.e. j(O,O) = 0 and aj/ax (0,0) = 0), is a necessary condition for bifurcation to occur (at the point (0,0)), but not sufficient. In addition, cases a, b and c show that the non-hyperbolic condition does not determine the type of bifurcation point when it exists. Addi tional conditions should be derived to determine the type of bifurcation. This can easily be done using the implicit function theorem and simple implicit differentiation of the function f The conditions for the three types of static bifurcation discussed above can be summarized as follows: a. The saddle-node bifurcation. The conditions for the occurence of a saddle-node bifurcation at the point (0,0) are:
{�
j(O, O) = 0
(O, O) = 0
}
non-hyperbolic fixed point
(2.46)
and,
jJ__ (O, O) ;t: O
(2.47)
2 a J 2 (O, O) ;t: O aJ.l
(2.48)
aJ.l
-
Equation (2.47) implies that a unique curve of fixed points passes through (Jl, x) = (0, 0), while equation (2.48) implies that the curve lies locally on one side of J1 = 0.0. The sign of the relation, 2
d J1 (0) dx 2
-
2
a t - 7)7 ( 0,0) (# af.l co. o)
- 'l'sN -
( 2.49 )
determines on which side of f.l = 0 the curve lies. The situation in
S . S .E.H. ELNASHAIE and S.S. ELSHISHIN I
74
X
--
--
--
_,
, -'
/
/ ,.
(0,0 )
FIGURE 2.1 1 Saddle-node bifurcati on where lf/s:v (equation 2.49) (- ::: stable, --- ::: unstable).
Figure (2.7) corresponds to the case where 'l'sN > 0, for curve is on the opposite side as shown in Figure 2 . 1 1 .
is negative
'l'sN
< 0 the
b. Transcritical bifurcation
{
}
The conditions for the occurence of transcritical bifurcation at point (0,0) are: j(O, O) = O
dj
ax and,
_
(0, 0) - 0
non-hyperbolic fixed point
z �%
(2.50)
(0, 0) = 0
(2.5 1 )
(0, 0) � 0
(2.52)
d2 f (0, 0 ) � 0 dx 2
(2.53)
The bifurcation diagram shown in Figure 2.8 corresponds to lf/r
where,
>
0,
STATIC AND DYNAMIC B IFURCATION
75
X
( 0,0)
--------��� - -----------+� ... .... .... .... .... ....
{b)
....
.... .... ....
.... .... ....
Transcritical bifurcation where VIr (equation 2.54) is negative (- = stable, - - - = unstable).
FIGURE 2. 12
d _1!_ dx
( 0) =
-
(
i
f (0, 0) 2
�X
-
_j_j__ ( , 0 dxdJl
0 )
= 'I'T
(2.54)
For 'I'T < the curve has the opposite direction as shown in Figure 2. 1 2.
0
c. The pitchfork bifurcation.
{
}
(0,The 0) are:conditions for the occurence of a pitchfork bifurcation at point j (O, O) 0 dj non-hyperbolic fixed point )0 0 0 ( , ax and,
=
_
( 2.55 )
dj (0,0) = dJl
az f
df1 2
0
(2. 56)
(0 , 0 ) = 0
( 2.57)
S .S .E.H. ELNASHAIE and S.S. ELSHISHINI
76
X
FIGURE 2.13 Pitchfork bifurcation where ytp (equation 2.60) is negative (- = stable, --- = unstable).
a2f
3 Jf j1 3 (O, O) :t: O J
The bifurcation diagram shown in Figure 2. 9 corresponds to where,
(2.58)
(2.59) 'I'p
> 0,
(2.60)
For 2.3
'I'p < 0
the curve has the opposite direction as shown in Figure 2 . 1 3 .
SIMPLE DETAILED ANALYSIS OF STEADY STATES ON BIFURCATION DIAGRAMS IN CHEMICAL REACTORS
We will present and discuss in this section four types of static bifurcation diagrams frequently encountered in chemical reactors
STATIC AND DYNAMIC BIFURCATION
77
X
( 2> A FIGURE 2.14
J.l. =
(1 ) B
JJ.
(3)
Simple hysteresis type of static bifurcation, bifurcation parameter.
x =
state variable,
described at steady state by one no n linear algebraic equati o n like the one in equation 2. 1 . -
1.
Hysteresis type bifurcation
Figure 2. 1 4 shows a simple hysteresi s type bifurcati on diagram for equation 2. 1 . For J1 values in the region A-B three steady states x exist. As explained in chapter 1 , the middle one is unstable and is called saddle type, its instability is asserted fro m the steady state analysis without the need to carry out dynami c stability analysis. The other tw o steady states satisfy the steady state condition for stability therefore they will be considered stable till the dynamic analysis proves otherwise. For J1 values outside the re gion A-B , unique steady states (which are stable from a steady state analysis point of view) prevail. The critical points A and B are static limit points (SLP). ,
2.
Pitchfork type bifurcation
Thi s type of bi furc ati on is shown in Fi gure 2. 1 5a. It has one double po i n t bifurcation. There are also other variations of this p itchfork typ e u s u a l l y called imperfect pitchfork as shown in Fi gure 2. 1 5b. For the later case there i s one static limit po i nt (SLP) and o ne static bifurcation point (BP). It has been s h o wn by Elnashaie and co-workers '
78
X
S . S .E.H. ELNASHAIE and S . S . ELSHISHINI
FIGURE 2.15a Pitchfork with one double point (BP + SLP = BSLP) (- = stable,
- - - = unstable).
X
---�--
--�b , -'
J.l. FIGURE 2.15b Imperfect pitchfork with one static bifurcation point (BP, point a) and one static limit point (SLP, poin t b) (- = stable, - - - = unstable),
that this imperfect pitchfork ex i sts for controlled fluidized bed c ata lytic reactors (Elnashaie and Abashar, 1 994; Elnas hai e et al. , 1 995) and for industrial fluid c atalyti c c racking (FCC) units (Elnashaie and Elshishini , 1 993). The same investigators have sho w n that these i m perfect pitchfork s are usually structurally uns tab l e (i.e. they break up i nto two disconnected branches with a slight change in one of the parameters) as will be shown later in ch ap te r 4.
STATIC AND DYNAMIC B IFURCATION
79
X
FIGURE 2.16 Mushroom with four static limit points (SLPs), po ints Ah A2, B1 and B2 (- = stable, - - - = unstable).
3. Mushroom type bifurcation
This type is shown in Figure 2. 1 6 and has been presented in the literature by a number of investigators (Teymour, 1 989; Uppal et al. , 1 974, 1 976). It has four static l i m it points (SLPs).
4.
Isola type bifurcation
This type is shown in Figure 2. 1 7 and has been presented in the literature for polymerization reactions by Teymour ( 1 989), for enzyme X
� SLP2 .. ," ! (_ I I I I
_ _ _
_
__ ..- � �
I
I
FIGURE 2.17 Isola with two static limit points (SLPs), A, B. Characterized by the existence of an isola of solutions disconnected from the rest of the bifurcation diagram (- = stable, - - - = unstable).
S .S .E.H. ELNASHAIE and S . S . ELSHISHINI
80
X
FIGURE 2.18 Hysteresis + isola with four static limit points (SLPs), A, B, C and D (- = stable, - - - = unstable).
systems by Elnash aie et al. ( 1 983) and for a CSTR by Uppal et al. ( 1 974, 1 976). This type of bifurcation has two static limit points (SLPs) like the simple hy steresis, but is characterized b y the existence of an isola of solutions which is disconnected from the re st of the bifurcation diagram. There is a very large number of possible combinations of these basic bifurcation structures giving rise in some regions, to a number of steady state s larger than three such as the case in Figure 2. 1 8, which shows a combination of hysteresis type bifurcation and an isola. In the region B--C, five steady states exist, three stable and two unstable. Stability here means stable from a static analysis point of view, that is the steady state satisfies the static slope condition but has not been checked yet with regard to the dynamic stability conditions. Regions A-B and C-D has three steady states where as for J1 values greater than Jlv and smaller than JlA unique "stable" steady states prevail. Other varieties of the bifurcation di agrams will be presented sequentially in the book in connection with real reaction engineering systems. Many software packages are available for the construction of these static bifurc ation diagrams (Kubicek and Marek, 1 983; Hansen, 1 97 1 , 1 973; Doedel and Kemevez, 1 986). "
2.4
"
DYNAMIC IMPLICATIONS OF THE COEXISTENCE OF MULTIPLE STABLE POINT ATTRACTORS
For Figures 2. 1 4-2 . 1 8 , it was considered that s teady states satisfying the static stability condition discussed in chapter 1 , are actually stable.
STATIC AND DYNAMIC BIFURCATION
81
Thus the steady states represented by the hard lines shown above are stable and can be considered point attractors. The effect of the simultaneous existence of a number of these point attractors on the dynamic characteristics of the system and on its response to external disturbances, will be considered in this s ection A two-dimensional system will be used for simplicity of presentation to clarify the main issues associated with the existence of a number of point attractors. The middle unstable saddle type steady states are usually called repellers. For the two dimensional system,
.
where
,
dx dt
-= =
E= �=
(2.6 1 )
F( x , Jl )
--
(1) (;: )
(2.62) (2.63)
Figure 2. 1 9 shows the two-dimensional phase plane for a case with three ste ad y states, two are stable (point attractors 1 ,3) and one is a b
FIGURE 2. 19 -
Phase plane for a case with three steady states: one stable node (1), one stable focus (3) and one unstable saddle (2). - - - (a-b) = stable manifolds of the saddle (2) = separatrix; 2-1 and 2-3 = unstable manifolds; /h /z = initial conditions.
S . S .E.H. ELNASHAIE a n d S . S . ELSHISHINI
82
saddle type (repeller, 2). This co rresponds to a value of f.1 in the region A-B or C-D in Figu re 2. 14. For any initial condition point above the separatrix ab, such as /1 the system will move with time toward steady state 1 (point attractor no 1 ). The trajectory will go to steady state 1 asymptotically with no oscillations (no spiral motion around the steady state) if it is a node, that is the eigenvalues of the linearized equations are both real (and of course negative) If the eigenvalues are complex (with negative real parts) then the trajectory will approach the steady state (point attractor) sinusoidally (it will go on spirals of diminishing radius around the stable steady state till it settles at the steady state, point attractor"). This is the case shown for steady state 3 with the trajectory starting at h for which the eigenvalu e s of the linearized equations are obviously complex with negative real parts. The line ab called the separatrix, the onset or the stable manifold, represents a region of extreme sensitiv ity to initial conditions. For initial conditions infinitesimally above ab, the trajectory w i ll go to steady state 1 , while for initial conditions infinitesimally below ab the trajectory will go to steady state 3 . If J1 i s chosen to be outside the region A-B (in Figure 2. 14), then the steady state (or point attractor) is unique and trajectories from all initial conditions will go to the unique steady state w ithout oscillations (spiral motion) if the eigenvalues are real and negative (Figure 2.20) or with decaying oscillations (diminishing spiral motion) if the eigenvalues are complex with negative real parts (Figure 2.2 1 ), as discussed in chapter 1 . For the case with the static bifurcation diagram shown i n Figure 2. 1 8 (isola + hysteresis), if J1 is outside the region AD there is a unique steady state (point attractor), while when J1 is in regions AB or CD there are .
.
"
FIGURE 2.20
eigenvalues).
Phase plane for a case with unique stable node (real negative
STATIC AND DYNAMIC BIFURCATION
83
x,
FIGURE 2.21 Phase plane for a case with unique stable focus (complex eigenvalues, with negative real parts).
three steady states with one saddle point and two stable steady states as discussed in the previous paragraph. However a more complex case develops when J1 is in the region BC (Figure 2. 1 8) . In this case five steady states exist with two saddle type steady states, three stable steady states and two separatrices as shown in Figure 2.22 where all three stable steady states 1 ,3,5 are stable nodes. b
d
a
c
- -
- -
- -
x2
FIGURE 2.22 Phase plane for a case with five steady states, three stable nodes and two saddles - - - (a-b, c--d) = separatrices = unstable manifolds of 2 and 4 respectively, 2-1 and 2-3 stable manifolds of 2; 4-3 and 4-5 stable manifolds of 4, x = initial conditions.
S . S .E.H. ELNASHAIE and S . S . ELSHISHINI
84
The two separatri ce s (insets or stable manifolds of the two s addl e type unstable steady states) are ab and cd. The outsets (or unstable manifolds) are lines 1-2, 2-3, 3-4, 4-5 . For any initial condition above the line ab, the dynamics will take the system with time to steady state 1 . For any initial con dition below line cd, the dy namics will take the system with time to steady state 5 . For any initial condition i n the region between the lines ab and cd, the dynamics will take the system w ith time to steady state 3 . Here, the ph ase p lan e is divided into three regions (or domains) of attraction rather than 2 as in the case with three steady states. Both separatrices represent regions of extreme sen s i tivity to initial con ditions as discussed earlier in connecti on with the case with three steady states. For the case with mushroom type bifurcation in Figure 2. 1 6, there are two regions of multi pli city namel y A 1 - B 1 and A2 - B2, while the rest of the re gi on repre s ents unique steady states. The dyn amic behaviour for both the multiplicity re gions and the uniqueness re gion s in thi s case (as well as the cases of the pitchfork in Figure 2. 1 5 and the isola combined with a unique branch in Figure 2. 1 7) follows the same basis discussed for the hysteresis type bifurcation without and with isola. The above are j ust illustrative examples slightly expanded over what was presented in chapter 1 , for the variety and degree of complexity of the bifurcation diagrams are very large. For examp le the pitchfork bifurc ation in Figure 2. 1 5 can break down by varying a parameter other than J..l, into one of the two types shown in Figures 2.23 and 2.24. It is also possible for some systems to have more than five steady states such as the case sh own in Figure 2.25 where in the region AB we X
, __ _ _ _ _ __ __ _
FIGURE 2.23
Broken pitchfork, x (- = stable, - - - = unstable).
=
SLP
J.l
stable variable, Jl = bifurcation parameter,
STATIC AND DYNAMIC B IFURCATION
85
X
SLP
J.1
Broken pitchfork, x : stable variable, JJ : bifurcation parameter, SLP : static limit point (- : stable, - - - : unstable).
FIGURE 2.24
can have seven steady states with three unstable saddle type steady
states and three separatrices separating the phase plane into four domains of attraction. In fact the different possibilities can be endless but most of the cases can be understood and analyzed on the basis discussed in this section. X
,
FIGURE 2.25 A case with seven steady states (double hysteresis + iso la) SLP : static limit point (- : stable, - - - = unstable).
S.S.E.H. ELNASHAIE and S.S. ELSHISHINI
86
The dom ain of attraction of a stable stead y state is not only impor foregoing discussion may suggest, but it is al so practically i mpo rtant with regard to the behaviour of the sy stem in the face of external disturbances . When the system is oper ati n g at a stead y state with a finite dom ain of attraction (i.e. it is not glob al ly stable, the overall state sp ace is divided between different ste ady states) and is e xp ose d to an e xternal disturbance, the behaviour of the system de pends on the m agnitud e and duration of the disturbance. If the mag ni tude and duration of the disturbance do not c ause the system (i.e. the state variables of the system) to move outside the domain of attraction of the operating steady state, then as the disturbance ends, the system will return to the same operatin g steady state. However, if the magni tude and duration of the disturbance cause the system to get outside the domain of attracti on of the operatin g stead y state, then as the external disturbance ends, the system does not go back to its original steady state but it will settle down to the stable steady state having the domain of attraction to which the system has moved under the effect of the tran sient disturbance. Speci al restarti ng policy needs to be implemented to get the s ystem b ack to its ori g inal operating s teady state It w i l l be shown later in the book that i n many cases the middle unstable saddle type ste ady state is the desirable ste ady state, specially for consecutive re act ion s where the intermediate product is the desired product (such as the partial c ata ly ti c o xi dation re actions of s o me h y drocarbon s) In these cases speci al design an d/o r con trol polici e s need to be implemented in ord er to operate at these desirable states of the system In this section we have discussed static bifurcation, as well as the d ynamic behaviour of the different steady states when all the non-saddle s teady states are stable (nodes and/or foci), that is all non-saddle steady states are po i n t attractors. Of course, it is possible for these non -s addle ste ady states to be unstable nodes or foci if they viol ate the d yn amic stability conditions giving rise in most ca ses to p e ri odic attractors . It is poss ible in the multi pl ic ity reg i on that an u nst abl e non-saddle state does not give rise to limit cycles provide d that at least one of the other steady states is s tab le Such a s itu ation is shown in Figure 2.26. In the n ext section we discuss the next import ant type of attractors with a higher de gree of c omple x ity which is the peri odic attractor. But before we proceed to periodic att ractors we present a simple and concise survey for the stab i lity of the s te ady states, which bring to gether most of the preceeding di s c u s s i on in a more formal framework. tant in start-up as the
.
.
.
.
2.5
LOCAL STABILITY OF STEADY STATES
The steady state (fi xed point) is said to be asymptotically stable (i.e. point attractor) if the response to a smal l perturbation (local) approaches
STATIC AND DYNAMIC BIFURCATION
87
x,
FIGURE 2.26 Multiple steady states with a single point attractor and no periodic attractors, • = stable steady state, o = unstable steady state, x = initial conditions.
zero as the time approaches infinity. The local stab il ity of the steady state can be determined by examining the properties of equation 2.61 when it is linearized about the steady state. The characteristic equation for th e linearized system of equations (2.6 1 ), is given by: J
=
det (Ex - A. Jl) = 0 j = 1 , 2, . . . , n
(2.64)
dfi
(2.65)
Where the matrix Ex i s given by,
F = -X
ax .
i = 1 , 2, . . . , n
j = 1, 2, . . . , n
J ss
The characteristic e quati on 2.64 of the matrix Ex , determines the stability of the steady states. The roots of equation 2.64 are the eigenvalues A/j = 1 , 2, . , n) of Ex · The real parts of the eigenvalues of Fx , evaluated at the steady state solutions (.:!ss ), determine in the following way the stability of the two-dimensional system with two eigenvalues A. 1 , A.2 , whereas the richer variety for three-dimensional systems will be discussed later. . .
1 . Purr: real eigenvalues of the same sign
The steady state is called a node and its local stabili ty depends on th e values and signs of the eigenvalues as follows:
S . S .E.H. ELNASHAIE and S . S . ELSHISHINI
88
x,
(a)
( c)
(b)
Xz
Types of steady state nodes. (a), (b) Proper and Improper stable nodes when the eigenvalues are equal; (c) Improper stable node when the eigenvalues are unequal.
FIGURE 2.27
a) Equal eigenvalues: The steady state can be a proper node or an improper node. When the eigenvalues are equal the node is a proper node. For the proper node each trajectory has a different slope (Figure 2.27a) while for the improp er node all trajectories approach the node tangentially with the same slope (Figure 2.27b). If all eigenvalues are negative the node is stabl e. If all eigenvalues are positive the node is unstable. b) Unequal eigenvalues. The steady state is called improper node. If all eigenvalues are negative the improper node is stable (Figure 2.27(c)). If all eigen v alues are posi tive the improper node is un stable. 2.
Pure real eigenvalues of opposite signs
If At < 0 and A2 > 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, A-1,2 = 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 Poincare-Andronov-Hopf 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: A-1 :t A-2 , A.1 . A.2 :t 0 or Re( A ) :t 0, almost always apply (Seydel, 1 988). The cases for which A-1 . A- 2 = 0, A-1 ' 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 two-dimensional 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 non-local 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 n-dimensional 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 ><
• • •
,,
, , "'
.,.-
"'
(§;
x2
ll PlP "' Il "' 11HB
><
x,
( a)
95
"' 1.1 HB
@ Xz
x,
( b) IJ Pl P
IJ HB
IJ
"'z
FIGURE 2.32 Subcritical Hopf bifurcation. Hopf bifurcation <•>. Stable static branch (--), unstable static branch (- - -), stable periodic branch (• • •), unstable periodic branch (o o o). Small figures on the side: (a) stable point attractor (stable focus) for Jl. < Jl.pu, (b) stable periodic attractor + stable point attractor + unstable periodic solution (unstable limit cycle), for Jl.t>LP < Jl. < JI.H8, (c) stable periodic attractor + unstable fixed point, for Jl. > JI.HB· On these figures (a-c), • = 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 two-dimensional 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 (a-c). 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 (a-c) 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
'
>< ><
..... ... _
97
•
- .. - -
•
• • • • --•
••• •
... _
HB2
... .... .....
JJ.
/J l
FIGURE 2.33(a) Bifurcation diagram for equations 2.72-2.74. A case with two Hopf bifurcation points. (-- = stable branch of the bifurcation diagram; - - saddle points; = unstable foci; • = stable limit cycles; HB 1 = Hopf bifur cation points, i = 1,2). · - · -
=
P L P.
( b) • • • •
•
•••
... K X X
A case with FIGURE 2.33(b) Bifurcation diagram for equations 2.72-2.74. two Hopf bifurca tio n points and one periodic limit point. (-- = stable branch = un s tabl e foci; • = stable of the bifurcation diagram; - - - saddle points; limit cycles; o unstable limit cycles; HBi = Hopf bifurcati on points, i = 1,2; PLP peri odic limit point). =
=
=
- . - · -
S . S .E.H. ELNASHAIE and S . S . ELSHIS H I N I
98
{c)
HC ( ! Pl ....•... •
-.. >C .. 0
-·
( "". ...
- - ---
....... ..
.... ....
FIGURE 2.33(c) Bifurcation diagram for equations 2.72-2.74. 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 two-dimensional 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 non-local 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 <11Hc, the limit cycle disappears . This is the second most important type of dynamic bifurcation after the Hopf bifurcation.
STATIC AND DYNAMIC BIFURCATION
101
2
FIGURE 2.36
saddle, HC
=
Xz
Homoclinical orbit. (o = unstable steady state, focus; ® = unstable homoclinical orbit (infinite period, IP, attractor)).
It is important at this early stage of simple analysis to notice that while the Hopf bifurcation is a local phenomenon that can be detennined from the linearized equations near the ste ady state, the homoclinical bifurcation is not. It is characteristic of the non-linear system and cannot be detected from local linearized analysis. At the homoclinical poi nt the periodic attractor has an infmite period. Now after this simple introduction to periodic attractors and three of the most important bifurcation mechanisms in the study of periodic attractors, namely : 1 . Hopf bifurc atio n (HB). 2. Periodic limit points (PLP). 3 . Homoclinic al termination (or infinite period bifurcation).
We notice that two important concepts have come into play and these concepts bear, in a sense, some resemblance to the behaviour of point attractors but on a higher level of complexity. These are:
1 . The existence of unstable l i mi t c yc l es (unstable periodic orbits or pe ri odi c repellers), which neces si t ate s the s tu dy of the stability of limit cycles (periodic orbits) as we did for point attractors 2. The possible existence of multip le attractors, one of them at least is unstable and that the periodic bran ch shows hysteres i s - like .
S . S .E.H. ELNASHAIE and S . S . ELSHISHINI
1 02
behaviour with periodic limit points which in a sense resemble static limit points in the static bifurcation diagrams for poi n t attractors. In addition we introduced (at least pictorially) the first non-local phenomenon that is the homoclinical bifurcation. 2.9
SOME MORE DETAILS ON THE CLASSIFICATION OF SOME TYPES OF DYNAMIC BIFURCATION DIAGRAMS IN THEIR RELATION TO THE STATIC BIFURCATION DIAGRAMS
Now, we try to present a more systematic and comprehensive classi fication of dynamic bifurcation types in relation to the different static bifurcation diagrams. Then we will present a technique for the deter mination of the stab i lity of limit c yc l e s (Floquet multipliers) fol l owed by a p resentation of the continuation technique for the construction of static and dy namic bifurcation diagrams. It is important to notice that when the dynamic bifurcation diagrams are constructed, this means that the periodic behaviour i s superimposed on the static bifurcation behaviour. Therefore the di agram contains both static and dynamic bifurcation characteristics. The only way to obtain a dynamic bifurcation diagram without the static bifurcation diagrams i mpo s ed on it (at least re l ative l y) is w hen the bifurcation parameter is a dynamic parameter not affecting the steady states (point attractors). I n this case, the poi nt attractor(s) will be horizontal line(s) al ong the dynamic bi fu rcatio n diagram. It is almost impossible to present a comprehensive listing of the different types of bifurcation diagrams. This is to be expected since we know that from a static point of view some chemical reaction engineering systems may have a very large number of different po int attractors for one set of parameters. Therefore we will present only some of the most commonly encountered types and the d i ag r am s formed from combinations of simpler types. The best method of classification seems to be the one based on the type of static bifurcation diagrams as follows: ,
,
1.
Dynamic bifurcation diagrams for cases with unique steady states over the entire range of bifurcation parameter
For this class
it i s not possible to have homoclinical (infinite period) bifurcation, because there is no saddle type steady states . Therefore the dynamic bifurcation diagram will take the shapes shown in Fig ures 2.37-2.39. For the case in Figure 2.37(a) for J1 = Jl� > there is a unique
STATIC AND DYNAMIC B IFURCATION
I
1 03
I
N
)( ... 0
-.<
•
T1
•
•
.. . . . i .
.
•
•
•
· .- · -· HBl
..j- · - ·
I
• • ,. .
.
•
I I I
( a)
I
FIGURE 2.37(a) Unique static branch with two Hopf bifurcation (HB) points and neither static nor dynamic limit points <• = HB points; • = stable period attractors; -- = stable steady states, stable foci; - - - = unstable foci). ·
·
point attractor (focus). The corresponding phase plane is shown in Figure 2.37(b ). For f.l = f.12, there is a unique periodic attractor surrounding an unstable focus. The corresponding phase plane is shown in Figure 2.37(c). For f.1 = f.13, there is a unique point attractor (focus). The corresponding phase plane is shown in Figure 2.37(d) .
x,
y X1
).1 : ).1 2
J.l = �.� ,
x,
( b)
1.1 = 1.1 3
!{%) y x,
(C)
x1
x2
(d)
FIGURE 2.37(b-d) Phase planes for Jl.t. Jl.2, p.3 in Figure 2.37(a).
HB 1 and HB2 are the two Hopf bifurcation points havin g pure i maginary complex eigenvalues (eigenvalues with zero real parts). . . A more c om p l ex dynamic behaviour for this case of umque static branch is shown in Figure 2.38(a) where only the maximum value of
1 04
S .S.E.H. ELNASHAIE an d S . S . ELSH ISHINI
.. >< .... 0
•
.•
fo I
o
i. . . . . . . . . . . I ...,... .· - · - ·- · I ,_.--
(a) IJ. FIGURE 2.38(a) Unique static branch with two Hopf bifurcation points and a periodic limit point, PLP at J.l = J.lPLP· (• = stable periodic attractors; o = unstable limit cycles; -- = stable steady states, stable foci ; - - = unstable foci). - ·
·
the oscillations are shown. A periodic limit point (PLP) exists at J1 = J.lPLP· For J1 = )11 . there are two stable attractors, one periodic and the second a fixed point attractor with an unstable limit cycle in between. The corresponding phase plane is shown in Figure 2.3 8(b).
(b)
Phase plane for Jl = p1 • Bistability : stable period attractor and stable point attractor coexisting. (-- = stable periodic attractor and trajectories; - - - - = unstable limit cycle, separatrix; x initial conditions).
FIGURE 2.38(b)
=
STATIC AND DYNAMIC BIFURCATION
.... >< � 0
1 05
HB 2
><
><
FIGURE 2.39(a) A case of a unique static branch witb two Hopf bifurcation points and two periodic limit points at J.ipLPl and J.lPLP2· (• • • = stable periodic attractors; o o o = unstable limit cycles; = unstable foci). - · - · -
It is clear that the unstable limit cycle acts as a separatrix separating the domain of attraction of the point attract or and the periodic attractor. Of course we can have a similar situation developing around HB2 with or without a PLP around HB 1 • These varieties can be easily deducted and are too obvious to be presented. Another possible dynamic bifurcation diagram is shown in Figure 2.39 (a) . This diagram shows two periodic limit po ints (PLPs) at JlPLPi
x,
FIGURE 2.39(b) Phase p lane representing bistability with two stable periodic attractors for J.l = p2 In Figure 2.39(a). (-- = stable periodic attractors and trajectories; - - - unstable limit cycle, separatrix; x initial conditions). =
=
1 06
S .S.E.H. ELNASHAIE and S . S . ELSHISHINI
..
• ..
.
•
:-.---. , . ··,
... 0
•
><
\
(a)
.
•
..
·-
.•. . .. . � · -.... •
..
·,
\
..:._ · .
(b)
(c )
FIGURE 2.40 Bifurcation diagram for a case which has a region of multiplicity (hysteresis) with two SLPs and (a) two HB points; (b) two HB points and one PLP; (c) two HB points and two PLPs. (• • • = stable periodic states; o o o = unstable periodic states (limit cycles); -- = sta ble steady states (stable static branch); = unstable steady states, saddles (unstable static branch)). - · - · -
and /1PLP2· The si tuatio n at J11 i s in principle similar to one of the previous case. However for J12 , there are two stable periodic attractors, one un s table fixed point and one un s tabl e limit c yc le . The p h ase plane for this case is shown in Figure 2.39(b). The unstable limit cycle acts as a separatrix separating the do mai n of attraction of the inner p eri odic attractor from the domain of attract ion of the outer p eri odic attractor. Different combinati o ns of the above three cases will gi ve most of the po s sible dynamic b ifurc ati on cases when the static bifurcation d i agram is a u n i que branch . 2.
Dynamic bifurcation diagrams for cases with multiple steady
states (multiple fixed points)
A case wi th mu ltiple steady states was introduced earl ier to illustrate the Hopf and homoclinical bifurcations. In th is section a more detailed analy s i s of possible dynamic bifurcations for thi s case is g ive n. When there are multi ple steady states then there must be unstable saddle ste ady state (s ) and therefore homoc linic al (and in even more complex cases, hetero c linic) bifurcation(s) are po s s i bl e . Let us consider the case where the steady state bifurcation is a hystere s is type with two static li mit points . A nu mb er of simp le varieties of the c o mplete bifurcation diagram (including periodic branches) are shown in Fi gu re 2.40(a-c) . For these relatively si mp l e three cases, the Hopf bifurcation points are occuring on a unique branch and are actu ally similar to the unique branch discussed in the previous section. For th e case in Figure 2.4 1, there are two Hopf bifurcation (HB) p oi nt s , HB1 and HB2, each on a different static branch. The pe riodic attractors emanating from HB 1 tenninate homoclinically (infinite period
n·
STATIC AND DYNAMIC BIFURCATION
I ''
N >C
...
I I
. . . . . ..... .. ..... . . .. ..... ....... � .
Y' '?I · . l • )' • l :� I .Y\ I I
S odd l � 5 Bran c h
0
-
X
><
1
'•
•
• I ./ • _..... -1 •
:
...
•
/
•
I
I
I
.
',
/f' I
HB t
SLPt
I
.
• •
•
• •
.
I
:
· · · 1 · · · · · r ·� I
1 07
1
:
I
I I I 1 I
J1.
Bifurcation diagram for a case which has a region of multiplicity (hysteresis) with two static limit points (SLPh SLP2), two HB points (HB �o HB2, each on a different branch) and two homoclinical (infinite period) bifurcation points (HC�o HC2). (• • • = stable periodic attractors (stable periodic branch); - = stable steady states (stable static branch); = unstable foci; o = static limit points, SLPs; • = Hopf Bifurcation (HB) points; o = Homoclinical (HC), infinite period (IP) bifurcation points). FIGURE 2.41
- · - · -
bifurcation) at J.1 = HC2 and those emanating from HB2 terminate homo clinically at J.l. = HC1 • For HB1 < J.1 < SLP 1 there is a periodic attractor surrounding an unstable focus as in Fig ure 2.37(c) presented earlier. For SLP 1 < J.1 < HC1 the re is a periodic attractor and a stable point attractor with a separatrix passing through the middle unstable saddle type s teady state separating the domain of attraction of e ach attractor as shown in Fig ure 2.42. In this case of course, there is an unstable focus ins ide the periodic attracto r. For HC 1 < J.1 < HC2 th ere are two periodic attractors, no point aurae tors, one saddle point and two unstable foci each inside one of the periodic attractors . The phas e plane is shown in Figure 2.43. For J.1 close to HC1 . the top periodi c attractor will have a very long peri od because it is close to its i n fi nite period bifurcation po i nt, whi le for J1 close to HC2 it is the other periodic attractor which will have this characteristic. Thus one of the periodic attractors terminates homoc linically at J.I = HC, and the other terminates homoclinically at J.1 = HC2 .
1 08
S . S .E . H . ELNASHAIE and S . S . ELSHISHINI
x, Separotrix
FIGURE 2.42 Phase plane for the case of SLP1 < Jl < HC1 in Figure 2.41 (one stable node + one saddle + one unstable focus + one stable periodic attractor surrounding the unstable focus).
x,
Phase plane for the case of HC1 < Jl < HC2 in Figure 2.41 (two unstable foci + one saddle + two stable periodic attractors each surrounding one of the two unstable foci). FIGURE 2.43
For HCz < J1 < SLP2, there is a periodic attractor and a point attractor. The reader should be able to construct the phase plane for thi s case. For SLPz < J1 < HB2, there is a unique periodic attractor. The reader should be able to construct the phase plane for thi s case. For J1 > HB2. there is a unique point attractor (stable steady state, i.e. stable fixed point).
STATIC AND DYNAMIC B IFURCATION
I I
... . . ...
.
. . . .
(!·-·- ·�: .
\ I ' I \
I I I
_.>, Saddles� i.:._• • • '? I .y, ... 0
><
i
Stanch
I
,I '
'
• · 1I
�
I,
I I
I I I I I 0 •• 0 I A· · . I • . ·).....- ..... I I I • : 1 1 'k 1 I I • • d•• , , 1 II 1 I o• l . . . . •
•.
I
I I · �.
'
•
gOO
...... . ..:.:, · . .
•
•
.
••
1 09
•
. .
.
••• I
,
I
I
I
I
I
• • • •
\
I
I
I
I
I
1
1
1 I 1 I I I 1 I I I I I I I I I I I I
FIGURE 2.44 Bifurcation diagram with two static limit points (SLP., SLP2); two Hopf bifurcation points (HBt. HB1); two homoclinical bifurcation points HC1 , HC2) and four periodic limit points (PLPt. PLP2, PLP3, PLP4), (o o o = stable periodic attractors (stable periodic branches); o o o = unstable periodic solu tions (unstable limit cycles); = stable steady states (stable static branches); - · - · - = unstable foci; • = Hopf Bifurcation (HB) points; o = Homoclinical (HC), infinite period (IP) bifurcation points). --
The situation can be even more complicated if the case in Figure 2.41 involves PLPs as shown in Figure 2.44. In this case there are two static limit poin t s (SLPs), two Hopf bifurcation points (HBs), two infinite period bifurcation po int s (homoclinical, HCs) and four periodic limit poi nts (PLPs). There are eleven different regions with different static and dynamic characteristics. The reader should be able to construct schemati c phase planes for the 1 1 cas es . These regions and their basic characteristic s can be summarized in the following: 1 . For J1 < PLPJ . there i s a unique po int attractor . 2. For PLP1 < fl < HBJ. there is bistability due to a periodic attractor and a po int attractor with their domain of attraction separated by an unstab le saddle type limit c ycle ( w hich is the separatrix in this case).
3. ,
For HB,
1 10
S . S .E.H. ELNASHAIE and S . S . ELS HIS HINI
4. For SLP1 < Jl < PLP2 , there is what we may call tristability due to two periodic attractors as in case 3 in addition to a fixed point attractor (at high values of x) separated from the periodic attractor by a separatrix passing through the static saddle point. The two periodic attractors are separated by an unstable limit cycle. 5. For PLP2 < Jl < HC� , there is bistability due to a periodic attractor and a point attractor with their domains of attraction separated by a separatrix p as s ing through the static saddle type steady state. 6. For HC1 < J1 < HC2 , there is bistability due to two periodic attractors with their domains of attraction separated by a separatrix passing through the static unstable saddle type steady state. Close to HC1 the upper periodic attractor has a very long period and terminates homoclinically at HC1 while close to HC2 the lower periodic attractor has a very long period and terminates homoclinically at HC2. 7. For HC2 < J1 < SLP2, there is bistability due to a periodic attractor and a point attractor with their domains of attraction separated by a separatrix passing through the static unstable saddle type steady state. 8 . For SLP2 <J1
HB2, there is a unique point attractor.
Of course the situation can get more complicated with more static and dynamic limit points and infinite period bifurcations. The reader should try to construct some cases and train himself on the investigation of the different static and dynamic bifurcation behaviours involved and construct schematic representations of the phase planes for different cases. Uppal et al. ( 1 974, 1 976) gave a classification of nine basic phase plots for the two-dimensional CSTR problem (see Figure 45a). However, the most comprehensive classification of possible phase planes published in th e chemical engineering literature seems to be that published by Vaganov et al. ( 1978), where, he gives 35 differe nt types of ph as e planes for a two dimensional system describing a non i sothermal CSTR. Fig ure 2.45b shows the 35 phas e plane cases given by Vaganov et al. ( 1 978) . The reader should excercise himself by following these phase planes and try t o construct the dy nam ic bifurcation diagrams for these -
STATIC AND DYNAMIC B IFURCATION
{NODE} { STABLE } FO�US UNSTABLE {LI M IT } t } CASE
SADDLE POINT
CYCL ES
STABL E
UNSTABLE
TOTAL INVAR I ANTS
111
T Y P I C AL P H A S E P L O T S A
B
I
0
0 0
t 0
0 I
0
c 0 E F G K J I I
I
0
0
I
0
3
t
0
0
0
1
2
3
t
0
2
I
I
0
2
0 I
I
I
t
I
l
0
I
2 I
0
I
0
0 I
1
3
4
4
4
5
I
AB SCISSA - CONCENT RATION
A
�) �
G
H
FIGURE 2.45(a) Classes of different phase plots. Abscissa = conversion, ordinate = temperature.
cases. Of c ourse the 35 cas e s do not exh au s t al l po s s ib i l ities we can have more complex be h av io ur for cases where more than three static attractors ex i st Some of the i nteresting termin ol o gy used in biomedical engi neering stud ie s of heart behaviour is the term "black hole When there is a bi st abil i ty due to a peri odi c attractor and a fix ed p o i n t attractor ( u s ual ly th e sep aratrix is an unstable limit cycle) an d the sy s te m (the heart) is operating on the pe ri o di c attractor (healthy) it is of c ourse p oss ible to knock this stable periodic operation (by external disturbance) to get out of the domain of attractio n of the periodic orbit and into the domain of attrac ti on of the fixed point attract or resulting in the annihilation of the periQdic attractor (death). The re gi on of attraction of the fixed point attrac tor is u s u all y called in this case the "black h ole" (Glass and M acke y 1 988). ,
.
".
,
S . S .E.H. ELNASHAIE and S . S . ELSHISHINI
1 12
-a
( ·' J
�)· _ ...,
:lO
FIGURE 2.45(b) Classes of different phase plots. Abscissa ::: conversion, ordinate ::: temperature.
There are other types of periodic branches; one of the most well known is the periodic isola, where a closed periodic branch exists independent of the Hopf bifurcation points (Teymour, 1 989) . 2.10
MORE O N THE POINCARE-ANDRONONV-HOPF BIFURCATION (HOPF BIFURCATION). A PURE IMAGINARY PAIR OF EIGENVALUES
Earlier we have presented the condition for the existence of a Hopf bifurcation point. In this section we present briefly the techniques used
STATIC AND DYNAMIC BIFURCATION
1 13
for obtaining these conditions and also extend our earlier condition to the detennination of the type of Hopf bifurcation point (i.e. supercritical or subcritical) and the nature of the periodic solution emanating from it (i.e. stable or unstable) . Usually the theory is given the name of "Hopf bifurcation theorem". This is in fact, historically incorrect (Arnold, 1 9 8 3 ; Wiggins, 1 990) . Examples of this type o f bifurcation i n two dimensional systems, are found in the work of Poincare ( 1 892). Also , specific study and formulation of the theorem in two-dimensional systems, is due to Andronov ( 1 929). Hopf ( 1 942) extended the theorem to n-dimensional systems. Consider the set of differential equations,
w here ! i s an
.! = E(!,Jl )
(2 . 8 1 )
n x 1 vector of state variables, E i s an n x 1 vector of functions and J1 is apx 1 vector of parameters. E is r times differentiable, where r is determined by the need to Taylor expand equation 2.8 1 . Suppose that (2.8 1 ) has a fixed point (steady state) at,
(2.82) in other words, (2. 83 )
In order to investigate the behaviour in the neighbourhood of the fixed point (;!_0 ,J1 ), the first thing to do, as we did many times earlier, is to linearize equation (2 . 8 1 ) in the neighbourhood of the fixed point (;! 0 , J1 ) to obtain, -0 (2.84) where,
we can also write,
x -
(2.85)
x - x0 =-
(2. 86) In the- follo wing we will dro p the from g and :!_, !:!_ , and the fixed point will be point (0,0) . A
[l
and write them as
S . S .E.H. ELNASHAIE and S . S . ELSIDSHINI
1 14
Suppose that Ex (O, O) ha s two pure i mag i n ary eigenvalues with the remaining n-2 eigenvalues having non-zero real parts. By the center manifold theorem, we know that the behaviour in the neighbourhood of the fixed point (!_0 , J!:o ) is determined by the set of non-linear differenti al e quati on s (2.8 1 ) restricted to the center manifold. We will restrict the number of parameters to one parameter, J.l. The restriction to the two-dimensional center manifold reduces the set of equations (2.8 1 ) to the following two-dimensional set of differential equations.
( ) ( Re· A (J.l ) x1
x2
=
- Im · A (J.l )
Im· A (J.l )
Re· A (J.l )
) ( x1 ) + ( fJ (x1 , x2 , J.l ) .
)
x2
j2 (x1 , x2 , J.l )
(2. 87)
where A-1 ,2 are the complex eigenvalues of the linearized differential equations (2. 84), which of course form a conjugate p ai r and these eigenvalues are obviously depe ndent upon the bifurcation parameter, J.l, A-1
=
a(J.l ) + i. ro (J.l ),
A- 2 = a(J.l ) - i. ro (J.l )
(2.88)
where i = H, fJ , j2 are n on -line ar functions in x 1 , xz. That these eigenvalues are pure imaginary, means, a(O ) = 0
( 2 89 )
ro ( O) -:�:- 0
(2.90)
.
and
The next step is to transform equation (2. 87) into normal form. The normal form of (2.87) has the form, x1 = a (J.l ) · x1 - ro (J.l ) x2 + (c(J.l ) x1 - d(J.l ) · x2 )(x� + xi )
(2.9 1 )
ro(J.l ) · x 1 + a(J.l ) x2 + (d(J.l ) x 1 + c(J.l ) · x2 )(x� + xi ) + O ( i xl . l xi )
( 2.92)
·
·
+ O ( i xl . l xl )
x2 =
·
·
where 0(i x1 1 5 , 1 xl ) represents higher order term s than order 5 . It i s usu al ly more convenient t o put e q uati ons (2.9 1 ) and (2.92)
in
pol ar coordinates to give, r=
a(j.l ) . r + c(j.l )r3 + 0( r 5 )
iJ = W(f.l ) + d(f.l )r2
+ O(r4 )
(2.93) (2 .94)
STATIC AND DYN AMIC BIFURCATION
1 15
Lin eanzm g equations (2.93) and (2.94) in the neighbourhood of = 0 gives,
J.l
r + c (O ) r3 + 0(J.12r,J.lr3 , r5 ) 2 iJ = co(O) + co'( O ) J.l + d (O)r2 + 0(J.1 , J.l r2 , r4 ) r = a' (O) · 11 ·
(2.95) (2.96)
·
where ( ' ) means differentiation with respect to the bifurcation parameter 11 to distinguish it from 0 which means differentiation with respect to time. Also notice that we used the fact that a (0) = 0.0 in this case of the neighbourhood of the Hopf bifurcation point (0,0) where the complex eigenvalues have zero real parts. In order to understand the dynamics of equations (2.95) and (2.96) for small r and 11 we should,
l . neglect the higher order terms in equation (2. 95) and (2. 96) and study the resulting "truncated" normal form.
2. show that the dynamics exhibited by the truncated normal form are qualitatively unchanged when the neglected higher order terms are included.
We will concentrate on point 1 above and for the proof of point 2 that the neglected terms do not change the qualitative results, can be checked by the reader in any of many standard text books in non-linear dynamics (e.g. Wiggins, 1 990; Guckenheimer and Holmes, 1 983). When the higher terms in equations (2.95) and (2.96) are neglected, we get the following two equations, .
-
r = dJ.l
·
r + c · r3
(2.97) (2.98)
Where for ease of notation we defined,
a ' ( O ) = d,
c(O) = c,
co(O ) = co,
co'(O) = c,
d( O) d =
(2.99)
From equation (2.97) and (2.98) and for r > 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 <
,u.cd
--
<
(2. 1 02)
0
With the above condition 8 (t ) will be, (2. 1 03) From equations (2. 1 0 1 ) and (2. 1 03), it can be shown (prove not given here) that, The periodic solution is stable for c < 0. ii. The periodic solution is unstable for c > 0.
1.
r
2
=
_a �
c
y
0
0
0
/.1. <0
IJ. = O
/1 >0
YL.� 0 0 FIGURE 2.46
Case 1 with if > 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
J.l :O
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 half-plane to the right half-plane, or vice versa) as Jl increases. For d < 0, the eigenvalues cross from the right half-plane to the left half-plane 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 half-plane 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 n-dimensional systems but is ill ustrated here for a two-dimensional 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 two-point 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 2.8-2. 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 saddle-node 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
(quasi-periodic 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 non-autonomous 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 non-linear 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 (a--c). 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
d-t
/(;!_, 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 pseudo-arclength 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 (NON-CHAOTIC DYNAMICS) OF THREE-DIMENSIONAL 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 three-dimensional autonomous system, dx = d-t 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 two-dimensional 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 . Saddle-node of the first kind, when A J , Az, A3 are all real, with Ar < 0, Az < 0 whi le A3 > 0. 6. Saddle-node 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. Spiral-in-saddle, 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 two-dimensional forced (non-autonomous, 2-D) system is a bona fide three-dimensional (3-D) system. In fact it is a s pec ial case of the autonomous three-dimensional (3-D) system, when one unidirectional variable is not affected by the other two variables. Either of the two types of 3-D systems has dynamic s which are much richer than the 2-D system. They both can have more complex dynamic attrac tors than the peri odic attractors pre s ented and analy zed earlier for the 2-D system. 3-D 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 3-D systems, while detailed results and an aly si s for a non-autonomous 2-D system (effectively 3-D) as well as a 3-D 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 · A-3 + A-2 A-3 = M'
(2. 1 30)
•
A- 1 A- 2 • 13 = D
The nature of A.'s for 3-D 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
1 37
0 >0
EB EB ffi G>O@
ffi
FIGURE 2.59 Non-degenerate cases for the three eigenvalues for the 3-D autonomous system.
The eight non-degenerate cases of the eigenvalues AJ , A-2, A-3 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 spiral-out saddle (when G > O). While the four cases with D > 0 include: an unstable node and a saddle-node of the first kind (when G < O); an unstable focus and a spiral-in 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
M >O D : 0
Some degenerate cases.
M>O T > O
D > O
T O o
·"
S.S.E.H. ELNASHAIE and S . S . ELSHISH INI
138
FIGURE 2.61 stable focus.
Three dimensional representation of (a) a stable node and (b) a
A nice way of presentation is used by Abraham and Shaw ( 1 982, 1 983, 1984). They used three dimensional arrows to display motion along a one-dimensional manifold (trajectory) and used flat arrows to denote the motion of family of solutions on the indicated surface (Figure 2.62). The disk represented the stable (for cases a, b, c) and the unstable
StablE' nodE' ( Index 3 )
Sta blt> focus ( Inde-x 3 )
EB
S�ddlt> r10dt> ( I n d ex 2 )
Spiral-out saddle ( I n de-x 1 )
Three-dimensional representation of non-degenerate cases; (a) stable node; (b) saddle-node of the first kind; (c) stable focus; (d) spiral out saddle.
FIGURE 2.62
STATIC AND DYNAMIC BIFURCATION
1 39
Local
FIGURE 2.63 Stable and unstable manifolds (Ws, Wu) and an arbitrary manifold (A) for the saddle-node of the first kind (index = 2).
(for case b) manifolds which are tangent to the eigenvectors with negative (for stable) or positive (for unstable) real parts. The index is used to indicate the dimension of the inset (that is the stable manifold o f the fixed point). Thus for the cases in Figure 2.62 the index is 3 for case (a), the index is 2 for case (b), the index is 3 for case (c) and the index is 1 for case (d).
Stable (or unstable) manifolds ( Ws or Wu) of a .fixed point
These are the collection of states in the phase space which tend to the tixed point at an exponential rate as t ___, oo (for stable) and as t ___, -00 (for unstable). Such manifolds are tangent to the eigenvectors or eigenplane defined by conjugate pairs of eigenvectors with negative (for stable) or positive (for unstable) real parts. If the stable manifold is three dimensional it is called the basin of attraction of the fixed point and this point is called the attractor.
Stable (or unstable) kind (Index 2)
first
manifolds (Ws or
Wu) for a
saddle-node of the
The arbitrary manifold of local trajectories (Figure 2.63) is not a part
of Ws or Wu. The trajectories in thi s manifold do not go to the fixed point because they are not in the stable manifold Ws.
1 40
S . S .E.H. ELNASHAIE and S . S . ELSHIS HINI
FIGURE 2.64 Stable and unstable manifolds (Ws, Wu) and an arbitrary manifold (Wa) for the spiral-in saddle (saddle - focus) with index = 2.
When W s has a stable focus character near the fixed point, the structure as shown in Figure 2.64.
arbitrary manifold takes a more dramatic spiral
Important case (Hopf) In the degenerate case, when the fixed point has a purely imaginary pair of complex eigenvalues, then there may be a local manifold that contains pe riodic solutions of the non-linear equation:
dxf( � J.l ) dt = - ,
;
(2. 1 32)
A manifold which contains only periodic solutions will be called a �enter manifold (We), the flow will thus look like the one shown in Figure 2.65 . Center m anifo ld s can behave quite differently than W s and Wu on global scales . This i s because different radial portion s are not connected by flows, so that the uniqueness of the manifold is not insured by the uniqueness of the solutions . In p artic ul ar We can bifurcate into two branch surfaces as shown in Figure 2 . 66.
STATIC AND DYNAMIC BIFURCATION
FIGURE 2.65
A
center manifold,
141
We.
FIGURE 2.66 A center manifold with two branch surfaces.
Non-linear flows, flows in the neighbourhood of periodic orbits a) Focus limit cycle.
t
An orb i is a stable limit cycle. The flow spirals around the orbit in the process of approaching it as shown in Figure 2 . 67 . b ) Saddle c yc l e .
If th(! flow near a periodic orbit r, has a two dimensional s e t of s olution s which is asymptotic to r as t --7 oo (an R2 stable manifold, Ws) and another R2 set asymptotic to r as t --7 -oo (the unstable manifo l d , Wu), then this is called a saddle cycle. It is an unstable limit cyc l e .
1 42
S . S .E.H. ELNAS HAIE and S . S . ELSHISHINI
FIGURE 2.67
Focus limit cycle (stable periodic attractor).
FIGURE 2.68
Saddle (unstable) limit cycle.
FIGURE 2.69
Nested Tori.
STATIC AND DYNAMIC BIFURCATION
1 43
c) Nested tori. This is another possible flow in the neighbourhood of a periodic orbit. In this case, the flow is on a set of nested tori, and the orbit is a center cycle. If a solution on such a torus is not periodic, then all solutions on that torus cover the surface ergodically (i.e. they come arbitrarily close to any point on the torus).
Of course there is much more to be said about the three-dimensional system (whether an autonomous 3-D or a forced (non-autonomous) 2-D system), specially with regard to homoclinical and heteroclinical bifurcation and also with regard to the chaotic behaviour of the system. However, at this stage, it will be better for the smooth reading of the book, to leave it at this level. Detailed presentation and analysis of the chaotic behaviour of these systems will be given in chapter 4 in con nection with the chaotic behaviour of fluidized bed catalytic reactors. 2.16
DEGENERATE HOPF BIFURCATIONS
One of the most important aspects associated with the dynamic bifur cation and periodic phenomena is degenerate Hopf bifurcation (DHB). By DHB we mean a point in the parameter space where one or more of the hypotheses of the classical Hopf theory fails. The number of parameters that must be varied for this bifurcation to occur is � 2. Therefore it is difficult, if at all possible, to locate these degenerate bifurcations in parameter space by traditional numerical search proce dure (this is where AUT086 comes very handy as will be shown later). In this section the focus will be on degenerate bifurcations ( degene racies or singularities) of codimension 2. The codimension is defined as the number of parameters that must be adjusted for a given bifur cation to occur generically. Most of the results and conclusions pre sented in this section are for situations with codimension = 2. Since codimension 2 is used, therefore the location and analysis of these phenomena will depend mainly on the two parameter continuation diagrams (TPCDs) discussed in the previous section. The intersections or tangential touchings of the different types of bifuraction loci discussed earlier are usually the points of degenerate bifurcation. The intersections or tangential touching of the HB loci with other types of bifurcation loci are the points of degenerate HB. Since AUT086 is able to efficiently construct TPCDs, it will obviously be very useful. There are three types of degenerate H opf bifurcations :
S . S .E.H. ELNASHAIE and S . S . ELSHISHINI
1 44
v
FIGURE 2.70
Double zero-eigenvalues degeneracy, F1•
Type 1 Degeneracy :J{ 1
2. 16.1
This type occurs when more than one eigenvalue (or more than the one complex conjugate pair of eigenvalues for systems of higher order than 3) simultaneously have zero real part. This implies violation of the simple eigenv alu e s condition given earlier (of course it also implies the violation of condition H4 given earlier). Three cases are possible under this type of degeneracy:
Double zero eigenvalues, that is when the Jacobian has two eigenvalues which are equal to zero. This case will be called case F1 • It usually occurs when the locus of the HB points disappear into the locus of the static limit points (SLPs) as shown in Figure 2.70, where the HB locus approach the SLP locus and sink into it. That is to say, HB points exist on one side only of F1 • In th e vicinity of the F1 point (on the HB side), the dynamics of the system are very slow because the value of f3 in the eigenvalues of the HB point where Au = ± i·/3 is approaching zero. It is equal to zero at F1 . This conclusion is clearly consistent with the findings of a number of investigators (Planeaux and Jensen, 1 986) that in the vicinity of F1 ,. infinite period bifurcation occurs and periodic solutions terminate in homoclinical orbit as shown in Figure 2.7 1 . 2 . Pure imaginary pair of eigenvalues and a simple zero eigenvalue, th at is when the Jacobian has a p ure imaginary pair of eigenvalues (± i · b) and a s imp le zero eigenvalue . This case will be called the F2 c a s e It oc c u rs usually when the HB locus crosses the SLP l o cu s and 1.
.
HB point exist on both sides of
F2
(Figure
2.72a) or
w hen the HB
STATIC AND DYNAMIC BIFURCATION
-
fit
>C �
0
""
>C �
0
>C .... >C
/ .
\
'
,.
• •
•
.,- ·
....... . • HB •
IP ' ....
..... ..,
-
•
• •
•
•
1 45
- ._
/).
FIGURE 2.71 foci;
--
Homoclinical termination of a periodic branch (- · - · - = unstable = unstable saddles).
locus touches the SLP locus tangentially and then rebounces so th at points can exist on one side of F2 (above the SLP locus) as shown in Figure 2.72b.
HB
In this case the behaviour in the vicinity of F2 is quite rich. This behaviour depends largely on the type of Hopf bifurcation and the emanating limit cycles before F2 is approached. In the vicinity of F2 secondary bifurcation to Torus may occur if the nearby Hopf bifurcations are supercritical. These secondary bifurcations to Torus do not occur when these nearby Hopf bifurcations are subcritical. To clarify this last point further let us analyze this in a little more detail.
v
v
(b) FIG U RE 2.72 Degenerate Hopf with pure imaginary pair of eigenvalues and a simple zero eigenvalue (F2). (a) HB locus crossing SLP locus; (b) HB locus touching SLP locus tangentially.
1 46
S . S .E.H. ELNASHAIE and S . S . ELSHI S H I N I
H B, HB'. -·-·r-�• •
-
..,
><
...
0
�
FIGURE 2.73
SLP1 ,
S u pe rc ritic al
.... ,
. ·
.... .....
.....
.....
...... _
Hopf case with two HB points and SLP for v = vF2.
a) Secondary bifurcation near F2, supercritical Hopf-saddle-node interaction. For this case and for a fixed value of v = v2, where v2 is smaller than v at the degenerate F2 point vn , the bifurcation diagram will have the structure shown in Figure 2. 73 (Planeaux and Jensen, 1 986), where only the top part of the S-shaped static bifurcation is shown and only the minima of the oscillations . The value of v can be changed so that HB 1 moves towards SLP1 and when HB1 coincides with SLP 1 (at v = vn) then this is an F2 situation. In this case the system undergoes secondary bifurcations to Torus as shown in Figure 2.74 for v > vn (Planeaux and Jensen, 1 986) .
-
.., ><
....
0 N >< .. 0
>( ><
SLPI I ',
,. -
FIGURE 2.74 Supercritical Hopf showing a stab le Torus (+++).
.,.. --- --
·.
•.
..., Jl..o
a' •
�
x
- -----
••
......
.·
_
.
- -
J1
case
with two HB points and SLP for
v = v F2
STATIC AND DYNAMIC BIFURCATION
... >< .... 0 ... )(
.·
0 0
.... 0 ><
FIGURE 2.75
•
•
•
• • • • •
Q r/ '
.......
.......
· � ----
--
--
.....
1 47
.......
......
....... .......
--
Subcritical Hopf case with two HB points, SLP and PLP for
V = VF2•
b)
Subcritical Hopf-saddle-node interaction. In this case secondary bifurcation to Torus does not take place as clear from Figures 2.75 and 2. 76. These figures show the top part of the S-shaped static bifurcation curves and the maxima of the periodic oscillations.
In more general terms both Torus and homoclinical bifurcations are possible in the vicinity of F2 as has been shown by Planeaux and Jensen ( 1 986) and d epicted in Figure 2.77 .
...... ... >< ... 0 ... >< ... 0 >< ><
FIGURE 2.76
.
.
.
:
0 0 0
.
(
.. .
/
\
0
/
'
/
. . . . . ....--- //
.,....., .-
...... ._ O o o oo ...... ....
--
--
Subcritical Hopf case with two HB points
-
and SLP for v = VF2•
S . S .E.H. ELNASHAIE and S . S . ELSHISHINI
148
IP
Tor
IP
Tor
Ia)
X
v
=
v,
X
X /
-"' . .
:
•
--- • •
.. \ <>oo •• ' �\. - - - -
II
1.1
1.1
FIGURE 2.77 Schematic unfolding of F2 bifurcation in J.L and v. (a) Stable Tori (+++); (b) unstable Tori (***). (o o o = stable periodic attractor, o o o = unstable periodic attractor).
3 . Triple zero-eigenvalues. When the system has lt1 = A2 = A3 = 0, this degenerate point in the parameter space will be called G1 • Analysis of G 1 degenerate bifurcation is an imposing task still in its earliest stages (Planeaux and Jensen, 1 986) . Pismen ( 1 985b) gives a normal form for G1 and describes a mechanism whereby chaos arises in its vicinity . The numerical computations of Planeaux and Jensen ( 1 986) for their simple three dimensional model indicated that periodic solutions for the model readily display period doubling sequences and chaos for parameter values chosen near G 1 • They have shown that the local nature of the chaos in the vicinity of G 1 allows the study of s m al l - ampli tu de oscillations for which numerical com p utat ion is greatly facilitated, in contradi sti ncti on to the very stiff chaos investigated by Jorgensen and Aris ( 1 983).
STATIC AND DYNAMIC B I FURCATION
1 49
Im
J.I. = J.I. o
Re
J.l. = /l o
FIGURE 2.78 Jl 2 degeneracy violation of the Hopf bifurcation condition d Re(p.) I dp. I JJ =Jl "" 0.
,
2. 1 6.2
Type 2 Degeneracy :J{2
This type 2 degeneracy takes place when the crossing condition of Hopf bifurcation is violated, that is when the complex eigenvalues encounter the imaginary axis tangentially as shown in Figure 2.78, where J1 is a parameter on the graph, like time on the pha se plane. The simplest case for this de generacy is when two H opf bifurcation points coincide as shown in the two parameter diagram in Figure 2 .79 (the point corresponding to (v)Jl & (Jl )Jl ). The corresponding one paramiter bifurdation diagrams for this case and its neighbourhood (v = v1 and v = v2, v = vJl 2 in Figure 2.79) are shown in Fi g ures 2.80 (a-c). It is clear th at in the v icin i ty of J{2 the occurence of mul tiple Hopf bifurcation poi n t s is to be expected. Also isolas of period i c s oluti o n s c an de ve lop in the vicinity of this degeneracy (Fig ure 2.8 1 ) . For more comp lex behaviours associated with this J£2 degeneracy, the interested reader should consult the excellent paper of Planeaux and Jens en ( 1 986).
S . S .E.H. ELNASHAIE and S . S . ELSHISHINI
1 50
v
\
I I I I
\ \ \ ,\
-
�
- -
- - - - -1-
-
'
-
... , 1
-
-
-/
I
-
,.- -"" - - -
JJ.
FIGURE 2.79. Two parameter diagram for :H2 degeneracy. 2. 16.3
Type 3 Degeneracy Jf 3
This degeneracy occurs when one or more of the coefficients of higher order terms in the Hopf bifurcation normal form vanish simultaneously. This degeneracy is quite important from a practical point of view, for while Jf 2 degeneracy gives rise to multiple Hopf bifurcations, this J{ 3 degeneracy will give rise in its vicinity to individual Hopf points to produce multiple periodic orbits . This was shown earlier in Figures 2. 39a, 2.40c, 2.44 and gives rise to multiplicity of periodic orbits and to periodic limit points. An example is shown in Figure 2 .82.
... .., . ..... ?.-..... ... ..... . - ·--; ...... ..... __,..
\I = V ,
�
0
N ><
-
0
><
><
(c)
'
(�) l.l
FIGURE 2.80 v = vJi , ·
.... �- � ..... .... ;:;
><
�
><
0 ><
"
• •
....... _
\I =Vz
v : \1 112
5N
><
(b)
diagram
for (a)
__...,
"\
J.l
0 ><
� ... ..... --
;:;
� .....
One parameter bifurcation
,., ><
--
,-,
�) J.J
v = v1
, (b)
v =
v2 ,
STATIC AND DYNAMIC BIFURCATION
...... ... X ... 0 .... X .. 0 X
FIGURE
2.81
•
•
•
•
•
151
•
?---•
0 0 0
Isola of periodic solutions.
2 "-'\
•
.
...
><
�
>(
0 -
• • 0 0 0
0O
o
>(
><
• • •
• •
Po 0
0
.·
.
.
0 0 •
•
�1 (a)
IJ. *
2.82(a) Bistability of periodic attractors. A case with two PLPs. At IJ* two stable period attractors co-exist and are obtained from different initial conditions. FIGURE
(c )
(b)
FIGURE 2.82(b,c) Time trace (x1 vs. t) for the two stable limit cycles 1,2 (periodic attractors), at Jl p.• in Figure 2.82a. (b) Periodic attractor 1, with small amplitude. (c) Periodic attractor 2, with large amplitude. =
1 52
2.17
S . S .E.H. ELNASHAIE and S.S. ELSHISHINI
QUASI PERIODIC ATTRACTORS FOR NON AUTONOMOUS SYSTEMS, PERIODIC FORCING OF AUTONOMOUS SYSTEMS WITH PERIODIC ATTRACTORS
Quasi periodic attractors (Torus attractors) are not possible in autonomous (unforced) two dimensional systems. However, their existence is widespread in non-autonomous forced two-dimensional systems. They also exist in autonomous high dimensional (higher than two-dimensional) systems but their existence is not as widespread as in non-autonomous (forced) systems. We will first explain and discuss torus attractors in two-dimensional forced systems. If a two-dimensional autonomous system has a periodic attractor of period T0 and frequency C00 = 2TriT0, T0 is called the natural period of the system and C00 its natural frequency. If the system is sinusoidally forced by changing one of its input variables, say the feed temperature, y1; such as Yt= Yto + A sin rot, then the system, apart from the possibility of chaotic attractors which will be discussed later, will develop certain patterns of behaviour which depend very much on the nature of the ratio ro/a>0• The most convenient presentation of the behaviour is that of the resonance horns (which are sometimes called Arnold tongues) shown on an exitation diagram. To be more specific, let us consider for example the non-isothermal CSTR described by the following two coupled differential equations, (2. 1 32)
(2. 1 33 )
and for sinusoidally forced feed temperature, Yt will be given by, Yt = Yto
+ A sin rot
(2. 1 34)
The system parameters are such that the bifurcation diagram of Xz (the dimensionless temperature) versus Yf for A = 0, (i e Yf = Yfo) , has two Hopf bifurcation points with a stable periodic branch connecting the two Ho pf b ifu rc at i on po i n ts as shown in F i g u re 2 . 83 . For the shown .
.
center o f forcing YfiJ• the autonomous system has a unique stable limit cycle with natural period T0, thu s the natural frequenc y W0 is given by W0 = 2tr/T0•
STATIC AND DYNAMIC BIFURCATION
.. X ... 0
•
•
1 53
••• t • .J.. _.
. .- · - · • • • • • •
�
.
FIGURE 2.83 Bifurcation diagram for the autonomous two-dimensional CSTR and the center of forcing is at Y!o·
Cases with s m all A are considered first. Small is meant in the sense that it does not cause the development of higher complexity than periodic and qu as i - periodic behaviour. It wi ll be shown later that extremely complex behaviour can develop for larger amplitudes. The question of how small is small and how large is large depends upon the bifurcation characteristics of the system and the location of the center of forcing relative to some cri tical points as will be di s c ussed in chapter 4. In simpler terms more complex behaviour such as periodic doubling, chaos , homo clin ical tangle s . . . etc, will not be considered at this stage, but w ill be analyzed in some detail in c hapter 4. The exitation diagram is a two dimensi onal plot of the amp l itude of forcing vers us the forci ng frequency sho wing the regions of periodic i ty and the regions of quasiperiodicity. Quas iperiodi city is a term used to describe some complex dyn amic behaviour that is not periodic and that is also not chaotic (as will be explained in chapter 4). The attrac tor in such a c a se is usually called a toru s and the trajec tory goes around the outersurface of a doughnut as shown in Figure 2.84, which shows a simple rep rese n tati on of a point attractor, a periodic attractor and a quas i period attractor (torus) as well as a strange chaotic attractor which will be discu s sed in chapter 4. On a two- d imensional phase p l ane prese n tat i on , the quasiperiodic
case looks complicated which may confuse it with chaotic behaviour although it is quasiperiodic and not chaotic. The simplest way to distinguish between periodic and quasiperiodic attractors is by u sing what is usually called the stroboscopic map, which is simply a phase
1 54
S.S.E.H. ELNASHAIE and S . S . ELSHISHINI F i x t>d
p o i nt
L i m i t cyclE'
� /
/" 0 •
To r u s
St r a n g E' att ract o r
0 0
FIGURE 2.84
The four main types of attractors.
plane on which the whole trajectory is not pl otted but only the values of x1 , x2 at every one forcing period T. On the stroboscopic map the periodic attractor appears as one point, if the periodic attractor is fully entrained, that is its period is the same as the forcing period. It will appear as a number of points, if it is not fully entrained. Actually T0/T (WIW0) mu st be an integer number otherwise the traj ectory is not periodic
but quasiperiodic. The quasiperiodic (Torus) attractor appears as a circle on this stroboscopic map as shown on Figure 2.85 .
. .. . .
x,
. . .
, ..
...
.. . .
FIGURE 2.85
Stroboscopic map for a quasiperiodic attractor.
STATIC AND DYNAMIC BIFURCATION
FIGURE 2.86
155
A 2-Torus for 2-frequency.
The quasiperiodic (Torus) attractor i s i n fact just
a two-frequency
sol ution of the forced two ordinary differenti al e quati ons of the system where the tw o frequencies are incommensurate as shown in Figure 2.86. After we have introduced in a very simple manner the basic charac teristics of quasiperiodic (torus) attractor and the difference between it and the periodic attrac tor and before discussing the resonnance horns (Arnold tongues) on the excitation diagram, we should explain frequency locking (Iooss and Joseph, 1 990) sometimes called mode-locking (Milonni et al. , 1987). Frequency locking may be said to occur in a dynamical system when oscillations with two independent frequencies influence one another in su ch a way as to produce synchronization of the two oscillations into a periodic oscillation with a common l onge r period (a subharmonic os ci l l ation). The phenomenon of ph ase l oc king in a torus occurs when all the trajectorie s on the torus are captured by one period as Am (the amplitude of the forcing function) increases. Therefore we can say that for quasiperiodic solution a s A m increases, there is a value of Am at which the quasiperiodic soluti on undergoes a frequency locking to a periodic solution. This will prove very useful in understanding the Arnold tongues regions of the excitation diagram. In addition to that, for the frequency ratio wlwo it is easy to see that for rational values of the frequency ratio , 1'a
---=T OJ0 m
ro
n
(n and m are i ntegers) the trajectories on the 2-torus close after P iterations on the stroboscopic map (with the forcing frequency T taken as the strobing time) which defines what we call the frequency locking ,
S.S.E.H. ELNASHAIE an d S . S . ELSH ISHINI
156
Am
a
FIGURE 2.87 Excitation diagram of Am vs. w!Ql, showing two resonance horns (one at wlQl, = 2 and the other at w/Ql, = 5).
state in which the motion is periodic giving P point s on the stroboscopic map. For irrational frequency ratios, however, the motion on the 2-torus is quasiperiodic. Now, we can present, explain and discuss the simplest form of the excitation diagram as shown in Figure 2.87. It i s a plot of the region of quasiperiodic and periodic solutions on a diagram of Am (amplitude of forcing function) versus mlmo (frequency ratio). We show on Figure 2.87 two of the very many resonance horns (or Arnold tongues). Inside the horn, the solution i s periodic (at least for small values of Am). We wi ll show later that period doubling leading to chaos, exists inside some of these horns, whil e outside the horns, the solutions are quasiperiodic. The boundaries of the horns are the bifurcation points between periodic and quasiperiodic solutions. For the case with mlmv = 2, inside the horn, the period of the solution is period two, which means that since for this horn mlmo = Tv IT, we have a periodic solution with period Tv and since Tv IT = 2/1, therefore the peri od of the system i s twice the fo rci n g peri od T. On the strob os copic map with strobing every forcing period (T), any periodic attractor inside the horn will give two points on the map (of course if we strobe every natural perio d (T0), the traj ectory will g i ve only one point on the map). For points inside the horn with a tip at ml mo = TofT= 5/1 a similar s i tu ati o n is encountered where we will have period 5 peri od ic attractors giving 5 points on the stroboscopic map when the strobing is taken for every one forcing period T, an d if the trobing is taken for every one natural period T0, obviously one point is obtained on the stroboscopic map . The practice is to strobe every forcing period, T.
s
usual
STATIC AND DYNAMIC BIFURCATION
1 57
Of course it is not necessary that the denominator is unity. The same applies for any rational number for the frequency ratio. For example for
m (00
1'a T
n 5 = m 3
- - - - -
-
with strobing every forcing period T, we will get 5 points on the stroboscopic map. For when 5 T are passed giving 5 points on the stroboscopic map we will have 3T0 passed and therefore the 5 points are repeated again and so on, i.e. nT = mT0• As mentioned before, the boundaries of the resonance horns represent the loci of the bifurcation points between the quasiperiodic and periodic attractors of the system. Suppose we start at a point "b" where a = n/m is an irrational number (thus at this point the behaviour is quasiperiodic), then with increasing ml (00 till point c is reached on the boundary of the horn, a bifurcation takes place and two stable saddles and two stable nodes are born on the torus as shown in Figure 2.88. Simple analysis shows that in addition to the period 2 (two fixed points SN 1 and SN2, on the stroboscopic map), there is an unstable period one focus (F) and a non-stable period 2 saddle type limit cycles l:" �- Further analysis shows that, as the forcing frequency increases further, eventually the stable period 2 and the unstable saddle period 2 collide and disappear at ml m0, corresponding to point d ( ml m0 I d) on the excitation program. For mlmo > mlmn ld, a quasiperiodic attractor (torus) appears again. This means that the subharmonic period 2 inside
Xz
I
A schematic stroboscopic map for the birth of the saddle node. Only the two points N will appear on the stroboscopic map.
FIGURE 2.88
S . S .E.H. ELNASHAIE and S . S . ELS HISHINI
158
X I '
e,
1 1 .SN,
2 • I2 .S N 2 l • I , .SN 2 i .: I 2 . S N 1
2.0
2.89 Saddle-node bifurcation at the two boundaries of the resonance horn (with tip at (J)/'4, = 2/1), where l:; is the itb saddle and SN; is the jtb stable node. 1 = l:t . SN� > 2 • 'i:Q. . SN2 , 1 ' = l:t . SNz , 2' • 'i:q, . SN1• FIGURE
2.89.
,.
2T
the resonance hom is really an isola of periodic solutions as shown in Figure It is important to notice that each saddle goes to die not with the node with which it was born but with the opposite one. This means that the non-stable limit cycle has a phase difference of 1r with the stable limit cycle when they meet again. For the case of rolroo > 5/1 , the situation is similar but 5 node and 5 saddle limit cycles are born on the boundary of the resonance hom as shown in Figure 2.90. The saddle-node bifurcation is most probably, the simplest mecha nism for the transition (or bifurcation) between quasiperiodic attractors and periodic attractors (as well as vice versa) for forced non-linear oscillators. More complex mechanisms will be discussed in connection with the chaotic behaviour of these systems. A large amount of research has been published regarding the behaviour resulting from forcing naturally oscillating chemical reactors. There are some other interesting types of bifurcations such as the Neimark or secondary Hopf bifurcation (which occurs more frequently in autonomous systems), the cyclic fold and the flip bifurcation. 2. 17. 1
Neimark or Secondary Hopf Bifurcation
have seen that the Hopf bi fu rc atio n of a point attractor is charac terized by th e stable focus type po in t attractor w ith complex eigenvalues h a ving ne gati ve real parts chang i ng into unstable fo cu s ha vi n g c o mp lex eigenvalues with po si tive real parts afte r passing the Hopf bifurcation point where the real parts are zero. We
STATIC AND DYNAMIC BIFURCATION
1 59
X
=
r
: SN
.i 1
FIGURE 2.90 Saddle-node bifurcation at the two boundaries of the resonance horn (with tip at ru/� = 5/l), where I:; is the jib unstable saddle limit cycle and SN; is the ilb stable nodal limit cycle. l = I:1 . SN. , 2 • � . SN2 , 3 = � . SN3 , 4 :: L. . SN4 , S :: � . SN5 , l ' e i:1 . SN2 , 2' = I:s . SN� o 3 ' '!!!! � . SN3 , 4' = � . SN4 , S' • L. . SNs .
To detect a similar bifurcation for a periodic orbit (i.e. further, or secondary, Hopf bifurcation of the periodic orbit itself) , it is necessary to consider at least a three dimensional autonomous system or a two dimensional forced system. Let us consider the autonomous system with order n � 3, i = f_(�.Jl )
(2. 1 35)
which exhibits a periodic attractor at f.l = f.l 1 • S upposing that for this parameter value, the attractor is stable (dissipative), let f.l increase in such a way that at f.l = J12 the behaviour is locally conservative, while at f.l > f.l2, it is area expanding in the neighbourhood of the limit cycle. The analogy with the Hopf bifurcation is evident, so that a bifurcation is e xpected at J1 = J12 · With stabilizing non-linearities we might expect that, as the limit cycle loses its stab ility , an attrac ting torus is born. This phenome non is often refered to as second ary Hopf bifurcation an d can be i mmediately g eneralized to the bi furc ati on of n-torus to (n + 1 ) torus. In some special cases the limit cycle can bifurcate into another periodic orbit of peri od kT where k > 2 . The resulti ng motion lies in a two dimensional torus but
S . S .E.H. ELNASHAIE and S . S . ELSHISHINI
1 60
does not fill the whole surface. This phenomenon is called resonance or phase locking as discussed earlier for the periodically forced system. 2. 1 7.2
The Cyclic Fold
It is one of the interesting features of forced non-linear oscillators (Thompson and Stewart, 1 986). The cyclic fold phenomenon is in a sense, very similar to to the hysteresis (or fold) behaviour presented earlier for static bifurcation and is sometimes called "dynamic bifurcation of autonomous (unforced) systems". For some forced systems we can get a range of frequency ratios wlrov. where two limit cycles co-exist and they approach each other as w!OJ0 is varied. At the bifurcation point they collide. The cyclic fold is often associated with the so-called jump phenomenon or hysteresis, where for a limited parameter range, two stable periodic attractors co exist, separated by an unstable (saddle type) periodic attractor (repeller). This phenomenon has not been adequately investigated for chemical reactors, however, it has been extensively studied for the Duffing equation, x + d x + x + a · x3 ·
=
f cos ro t
(2. 1 36)
·
The results for the Duffing equation with a= 0.05, f= 2.5, and d = 0.2, are shown in Figure 2.9 1 .
w
Wo
FIGURE 2.91
Cyclic fold for a forced system.
STATIC AND DYNAMIC BIFURCATION
161
As w e slowly increase ro form low values t o higher values an d the change is carried out in slow small steps and after each step enough time is left to elapse for the initial transients to die out and the steady response amplitude is recorded, we find that the experimentally observed amplitude of the response is a smooth function of ro/ro0 at all but two values of ro/ ro0• At these values a jump is observed, following the well known hysteresis diagram met earlier in the analysis of static bifurcation behaviour. A single cyclic fold is easily depicted at point F. This is in a sense similar to periodic limit points separating a region of multiple periodic attractors from a region with a unique attractor. 2.17.3
Flip Bifurcation
The other type of bifurcation usually encountered in such systems is period doubling (not leading to chaos) which is sometimes called flip bifurcation (or incomplete period doubling, be c au s e it does not con tinue its sequence of period doubling to chaos). This type of bifur cation will be discussed later in the book in connection with chaotic attractors. 2. 1 8
THE STABILITY O F PERIODIC ATTRACTORS IN AUTONOMOUS AND NON-AUTONOMOUS SYSTEMS AND THE CONSTRUCTION OF EXCITATION DIAGRAMS FOR NON AUTONOMOUS SYSTEMS (Periodic Forcing of Autonomous Systems with Periodic Attractors)
The stability of periodic attractors whether resulting from an autonomous or a non-autonomous systems are determined by Floquet multipliers. This stability analysis of periodic solutions is an integral part of the construction of excitation diagrams for forced oscillators. Excitation diagrams as explained earlier, are in fact nothing but bifurcation diagrams of normalized forcing amplitudes and normalized forcing frequencies (the normalized forcing frequency is the forcing frequency ro, divided by the natural frequency of the oscillator roo). Exitation diagrams are two parameter (Am, ro) bifurcation diagrams (or in more specific terms, two parameter continuation diagrams, TPCD) for the periodically forced system when the autonomous system is periodi c with period ro0• In summary the excitation diagram in its simplest form (more c omplex sides of the excitation diagram will be presented and discussed in chap ter 4 in connection with the chaotic behaviour of forced oscillating fluidized bed catalytic reactors) contains the following two regions:
1 62
1.
S . S .E.H. ELNASHAIE and S.S. ELSHISHINI
Entrainment regions
These as discussed earlier, are regions where frequency l ockin g occurs and periodic trajectories having periods which are i nteger mu l tiple s of the forc i n g pe ri od exist on the surface of a torus . The s e regi on s typic ally look like cusp (M i no rsky , 1 962) and are called resonance horn, Arnold tongues or subharmonic regi on s . Their boundaries are loci of saddle node bifurcation . The t ip of the h orn is a l i mi tin g case (natural system, Am= 0) where the forcing amplitude is zero and the ratio of the frequencies is a rational number (nlm) where n and m are prime integers. The periodic trajectories within the horns have a period nP where P is the forcing period and we refer to them as P-periodic trajec tori e s and the y have m peaks of oscillations in time. The large r the values of n and m the sharper the resonance horn (Kevrekidis et al. , 1 986b ) . 2.
Quasi-periodicity regions
These are reg ion s which are
characterized by at least two frequenci e s is an irrational number) and have a torus stru cture in their phase plane. Their torus is the re s ult of forcing the autonomous peri o dic attractor .
that are incommensurate (their ratio
Construction of the excitation diagrams
The two-dimensional forced system can be repre sen ted by the following two nonlinear first order ordinary d i ffe re nti al equati on s : (2. 1 37)
(2. 1 3 8)
where Am is the forci n g amplitude, ro is the fo rc in g frequency. The simplest re sp on s e of this forced system is the subharmonic response i . e . the system is periodic with a period which is an inte ger multiple o f th e forcing period. The bifurcation boundaries of such response form the boundaries of the excitation di agram s (resonance horns, Arnold tongues) . If a period n so lutio n of the forced system exists at certain values of Am and ro, then :
<1>'1 ( xf , x2 , J.L , Am , ro) = FJ. ( xf , x2 , J.L , Am, ro ) - x f = 0
(2. 1 3 9)
STATIC AND DYNAMIC BIFURCATION
an d ,
where
<1>'2 ( xf , x2 ,,u , Am , w ) = f2 ( xf , x2 , ,u,Am, w) - x2 = 0 F; is obtained
xf , x2 at t = 0 to
by i nte gration
of
1 63
(2. 140)
equations (2. 1 37, 2. 1 38)
from
t = nP where P repre s ents the forcing period. equ ations
(2. 1 39, 2 . 1 40) are two nonlinear algebraic equations with two unknowns (xf , x2 ). These equations can be solved by Newton' s method of iteration whose Jacobian matrix is gi ven by:
!.. =
=
Ji Jxf
Ji Jx2
J2 Jxf
J2 Jx 2
Jl) Jxf
Jf2 Jxf
JFj Jx2 Jf2 Jx2
=
J l) - 1 Jxf
J Fj Jx2
Jf2 Jxf
J f2 -1 axg
-(� �)=
DF(x, .A , Am, w) - I
(2. 1 4 1 )
where I is the ide ntity matrix an d Df_(x, ,u , Am, w) represents the Jacobian of the stroboscopic map . S ince f_((fi !2 )) , is not algebrai c and can on l y be evaluated by integration Therefore the partial deri v ativ e s are obtained by integrating the following four variational equations (2. 1 43-2. 146) simultaneously with equations (2. 1 37, 2. 1 3 8) , let: .
�- = 'J
dX; dXJ0
and the variational equations are obtained by (2 . 1 37, 2. 1 38) with respect to xj giving, dA1 1 dt
_ -
(2. 1 42) diffe renti ating equations
J fi · Jfi · A Jxf AI I + Jx2 2 1
(2. 1 43) (2. 144) (2. 1 45) (2. 1 46)
S .S.E.H. ELNASHAIE an d S . S . ELSHISHINI
1 64
The initial conditions for the above equations are given by:
�j (t = 0) = 8ij
(2. 147)
iJF �-(t = np) = _ , 1
(2. 1 48)
where 8ij is Kronecker delta. At t= nP the variational equations give the elements of the Jacobian DE(x , Jl , A m , ro) as follows: axo1
the sections to follow we will drop the (o) superscript on the initial conditions xj for simplicity and to indicate that they can be any point on the trajectory of the periodic attractor. Upon convergence the eigenvalues of DE( x, J1 , Am , ro) are the Floquet multipliers (FM;) . The Floquet multipliers are obtained by solving the following characteristic equation: In
(FM)2 - Tr [ DF(x,Jl , Am, ro)](FM) + Det [DF(x,Jl,Am, ro)] = O (2. 1 49) where Tr and Det are the trace and determinant of [ D F(x,Jl, A m , ro) ] respectively. For this system there are two Floquet multipliers and the third multiplier which is constrained to be unity is automatically discarded by considering the map and not the whole trajectories. These Floquet multipliers determine the stability of the periodic solutions. The Floquet multiplier with the largest absolute value is called the principal Floquet multiplier (PFM). Detailed discussion of the stability of the limit cycle is presented by Abashar ( 1 994). The bifurcation criterion (<1> '3 ) for this system can be obtained from equation (2. 1 49) as follows: (i)
Saddle-node bifurcation (PFM = 1 ) .
<1> '3 ( x,J1 , A m , ro) = 1 - Tr [ Df(x , Jl, Am, ro ) ]
+ Det [DF(x, Jl , Am, ro )] = 0
(ii) Period doubling bifurcation (PFM = -1 ) .
<1»�1 ( x,J1 , A m, ro) = 1 + Tr [ Df( x , Jl , A m , ro ) ]
(iii)
Hopf
+ De t [ D F( x , Jl , A m , ro)] = 0
bifurcation (norm = 1 ) .
(2. 1 50)
(2. 1 5 1 )
STATIC AND DYNAMIC BIFURCATION
'3 (x,,U, Am , W) = 1 - Det [DE(x, ,U , Am , w)] = 0
1 65
(2. 1 52)
Equations 2. 1 39, 2 . 1 40 and one of the bifurcation criteria equations (equations 1 50- 1 52) form three nonlinear algebraic equations for the two state variables ( xf , x� ) and one of the excitation diagrams parameters (Am, ro). This set of the three nonlinear algebraic equations allows us to trace out the boundaries of the excitation diagram. A point on the boundaries of the excitation diagram at a specified forcing frequency ( ro) is obtained by solving the above three nonlinear algebraic equations using Newton' s method of iteration whose Jacobian matrix is given by: d 'l d x1
]=
d '2 dx1
d '3 d x1
d 'I
__
dx2
dAm
d '2
d 'z
dx2 d '3 dx3
d fl
d 'I
__
dAm
d x1
_1
d'3 dAm
-1
dx2 d '3
dFJ
__
dx2 dF'z
dF'z dx1
=
d fl
dx2
d x1
d fl
dAm d F'z dAm d'3 dAm
(2. 153)
__
The partial derivatives (d F/d Am) and (d F2/d Am) are obtained by integrating the following two variational equations (2. 1 55 , 2. 1 56) simultaneously with equations 2. 1 37, 2. 1 38. For simplicity define B; as, H 1
=
d x; dAm
(2. 1 54)
Differentiating equations 2. 1 37, 2. 1 3 8 with respect to Am gives the two following variational equations: (2. 1 55) and, (2. 1 56)
The i nitial conditions for the above two variational equations are :
B; (t = 0) = 0
i = 1, 2
(2. 1 57)
S.S .E.H. ELNASHAIE and S.S. ELSHISHINI
1 66
At t = nP the integration of the v ariational equations gi ve the following eleme n ts of the J acobian (2. 1 53). d fi B. 1 dAm
i = 1, 2
(2. 1 58)
A large number of differential equations need to be integrated si multaneously for each Newton iteration . Eight differential equations
(2 . 1 37, 2. 1 38, 2. 1 43-2. 146, 2. 1 55 , 2. 1 5 6) are simultaneously integrated to give the top two rows of the Jacobian (2. 1 53). The partial derivatives in the third row are obtained numerically by the simultaneous integration of six differential equations (2. 1 37, 2. 1 3 8, 2 . 1 43-2. 146) four times.
Actually the six differential equations are integrated for three times because these equations were integrated once before for the compu tation of the first and second rows of the Jacobian . Therefore a total of t = 0 to t = nP, are required for each Newton iteration. To avoid convergence difficulties, arc-length continuation
28 integrations from
technique is implemented (Kubicek and Marek, 1 983). The arc-length (L) of the solution curve is taken as a parameter of
l
the method i e the so ution is continued along the solution curve in fou r dimensional space (x1 , x2, Am, co). The total derivatives with respect t o L ar e given by : .
.
d<1>'1 d'1 dx1 = dL dx1 dL •
d<1>'2 dL dcl> '3 dL
= ()<1>'2 dx1
•
dx1
dL
dx = ()«1>'3 . l ()x1
dL
+
d <1>'1 dx2 •
dx2
dL
dx2
+
()<1> '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 59-2. 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 59-2. 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 43-2 . 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 64-2. 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 64-2. 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 68-2. t 7 1 ) are integrated by the
explicit Adams-Bashforth 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 arc-length continuation technique. This starting point is located by Newton' s method as discussed earlier.
2.19
STRANGE CHAOTIC AND NON-CHAOTIC 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 non-autonomous 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 non-autonomous ( 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 non-trivial 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 two-dimensional 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 n-dimensional 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 3-dimensional 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 non-chaotic 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 two-dimensional 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 Discrete-Time 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 discrete-time system. Discrete-time 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 well-known route to chaos, that is period doubling, through the famous discrete model having the well-known logistic map. For one-dimensional discrete processes, it is the simplest non-linear 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 one-dimensional 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 period-doubling 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 one-to-one 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 many-to-one maps. Figure 2.97 represents a two-to-one map. F ( x , J..I )
�-----�--� x • Xn
FIGURE 2.97
xn
The function F (xn. Jl) for a dissipative system
(many-to-one 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 two-dimensional 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 (one-to-one), 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 1-2. 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 1-1 ' ) 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 one-to-one, and therefore no more complexity than periodic attractors should be expected. The above discussion for dissipative systems has been confined to one-to-one 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 many-to-one 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 Two-to-one discrete (iterate) map.
2.20.2
Higher Order Continuous Dynamical Systems (many-to one maps)
A series of bifurcati on s leading to chaos depends on the non-uniqueness 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 many-to-one 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
Two-to-one 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. Two-to-one 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 one-to-one.
The S-map
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, S-maps. 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 S-map. However, the following two-to-one maps are not S-maps. 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*
<
(2. 1 82)
1
Applying thi s very simple stability procedure to the logistic map,
F
(x) = J.LX(l - x)
(2. 1 83a)
The derivative of F (x) is given by,
dF
- = J.L ( l - 2x) dx
(2. 1 83b)
In order to ev aluate,
(2. 1 84) * the value of the fixed point x must be found . This can easily be achieved by solving the simple equation, except for x*
x = Jl.X * (1 - x ) *
*
=
0, the trivial solution , this gives , X
*
1
= l -Jl
(2. 1 85 )
(2 . 1 86)
S . S .E.H. ELNASHAIE and S . S . ELSHISHINI
1 88
Therefore, dF dx
I
x*
(2. 1 87)
= 2 - fl
and the stability condition therefore is that, (2. 1 88) Therefore for x* to be stable we must have, and
f1 > 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 period-two 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<8J (x) map. In general for Figure 2. 1 1 Oa,
f"-.2l(x; ) = F(F(xZ ) ) = xZ . ( k = 1, 2 )
(2. 1 94)
I n general for Figure 2. 1 1 Ob,
p8l (x; ) = F(F(F( F(F(F(F( F(x; )))))))) x; , ( k = 1, 2, . . . , 8 ) (2. 1 95) =
It is clear in both cases that the original fixed point of the map, both figures, is unstable.
Xs
in
Initial transients
In Figure 2. 1 1 Oa, if we start at x' the system follows the shown transients till after the initial number of i terations , it settles on the period 2 attractor alternating between x; and x;(x; = x� ) . The same applies to the period 8 attractor in Figure 2. 1 1 0b. The stability of the fixed point Xs , can thus be easily determined from the first iterate map Xn+l vs. X11, (F(x11) vs. X11) . However, the stability of the period 2 attractor x;, x;, can only be determined from the second iterate map, Xn+ 2 vs. x11, (F< 2l(x11) vs. X11), while the stability of Ps attractor in Figure 2 . 1 1 0b can only be determined from the 8th iterate map Xn+ 8 vs. Xn (P 8 l(x11) vs. X11) and so on. Let us illustrate by analyzing the stability of period 2 attractor x; and x . The stability of the fixed points x; , x2 are linked together. They are stable when,
I :t ' dF (x
i
< 1,
k = 1, 2
(2. 1 9 6)
When either of the two points (x; or x;) loses stability, the other one also loses stability. We can further prove that (2. 1 97)
1 90
S .S .E.H. ELNASHAIE and S . S . ELS HISIDNI
...... )(
.......
)( .....
lL..
lL..
(a)
xi
{ b)
xrr xi
FIGURE 2.1 1 1 Iterate map for Pz attractor. (a) First iterate map Xn+I Second iterate map Xn+2 vs. Xn• (Jackson, 1990).
vs.
X Xn; (b)
which means that when one of these two fixed points on P2l(x) becomes unstable, the other one becomes unstable too. In order to prove that, let us consider the point x0, in Figure 2. l l 1 a (we drop the * from the x for simplicity) . We have, (2. 1 98)
2 dP l = dx2 = dF(x1 ) = dF(x1 ) . dx1 = dF(x1 ) . dF(x0 ) dxo d.xo d.xo dxo d.x1 d.xo dx1
(2. 1 99)
For this period 2 attractor in Figure 2. 1 1 1 a, after two cycles sequence (instead of one cycle sequence) we reach point x; or x0 = x; depending upon the initial conditions. In order to analyze this we draw the two cycles diagram, Xn +Z vs. Xn , that is F(2l(x) vs. x (Figure 2. 1 1 1 b) . It can be shown, by trying different initial conditions, that the basins of attraction of x1 and x2 (x0) are rather complicated as shown in Figure 2. 1 1 1 b. Initial conditions on the horizontal x-axis to the left of the fixed point x 1 are divided into three regions : region � in which each initial condition leads to X J . re �ion IJl in which each initial condition leads to x0 = x2 and region B2 in which each initial condition leads to x1 • Ini ti al conditions lying between x 1 and Xs (region Bi_ ) all lead to x, . In 2 the region lying between x, and x0 = x2 (region BJ ) , all the initial co nditi on s lead to x0 = x2 • The region to the right of x0 = x2 has two regions : one B{' w i th all initial condi tions leading to x0 = x2 and the other Bi with all initial conditions leading to x 1 (the allocation of these
STATIC AND DYNAMIC BIFURCATION
191
Bj regions o n the x-axis o f Figure 2. 1 1 1 b is left a s a simple excercise for the reader). This is a very simple illustration for the complexity of the regions of attraction for these systems when they start to un dergo period doubling bifurcation. Equation 2. 1 99 can be generalized to : (2. 200)
Now we follow the analysis further for specific periodic points, period-n (x0 , x1 . . . , xn-I & xn = x0 which are periodoc points or fixed points for p. nl(x). Let us now check the stability of the fixed points x0 (= x2 ) and x1 in Figure 2 . 1 1 1 a, b. From the general e qu ation 2. 200 it is clear that, ,
dP)J(x0 )
_
dxo Obviously we can also write,
dP 2 l (x1 ) dxl
_
d
dF(x0 ) . dF(x1 ) x0 dxl
dF(x1 ) dF(x0 ) dx0 dxl
(2.20 1 )
(2.202)
Since the right hand side of both equations (2.20 1 , 2.202) are equal, then the left hand sides are equal, (2.203)
In general for any p ai r of fixed points Xk . x , taken from a pool of any number of fixed points (Xo, x,, . . . , Xn ), we have, (2.204) This shows that if any of the fixed points for any n map is s tab le then all the others (except the saddle) are stable, and vice versa. Thu s for the case .of the two fixed points in Figure 2. 1 1 1 , when x , becomes unstable x2 becomes un stable Xs is unstable because it is a saddle in a previou s cycle of bifurcation and it conti nue s to be unstable. .
S . S .E.H. ELNASHAIE and S.S. ELSHISHINI
1 92
s
0·5 0·5
(a)
1 ·0
0.5
X
(b)
1 ·0 X
FIGURE 2.112 Iterate map for P4 attractor. (a) Second iterate map (Xn+2 vs. Xn ); (b) Fourth iterate map (xn+4 vs. Xn).
Now let us follow up the map f'( 2 l(x) as J1 increases to J12 and the points x0 = x2 and x 1 lose stabi lity . The new fixed points are renumbered from x1 � x7 as shown in Figures 2. 1 1 2a,b where now we have four stable fixed po i nts as shown in Figure 2. 1 1 2b (41h iterate map, Xn+4 vs. Xn, with three unstable saddles) . 2.20.5
The Control Phase Space
A plot of the values of x as J1 changes is shown in Figures 2. 1 1 3 , 2. 1 1 4 . The figure shows a pitch - fork (doubl e point) bifurcation structure and X
FIGURE 2.113 A bifurcation diagram of
parameter).
x
(fixed points)
vs.
Jl (bifurcation
STATIC AND DYNAMIC BIFURCATION
1 93
X
I . An allrac tong 1 Cantor s e- t
.�o.l X : 1
,
( .1
{ �{n)}
( n) :={ :t 1 , ± 1, } . . .
FIGURE 2. 1 14 Period doubling sequence till the accumulation point Jl. = J.b when the values of x form a cantor set.
the exchange of stability features of p J1 � Jln , there is a stable period zn cycle and in the limit:
Jl nlim --too n
(a)
St a ble- ( � 2 > 1J.
FIGURE 2.1 1 5 Jlt and
�I
>
P. 1 )
= Jl oo (finite) ( b)
X
S u pe-r s t a bt t' ( � :
iJ.� )
Stable and superstable 2-cycle fixed points. (a) Stab le Jl.2 > Jl
��
<1
(b) Superstable
Jl
=
p,• and
dF'�(xlj
��
=
0.
<
1 94
S . S .E.H. ELNASHAIE and S . S . ELSHISHINI
An accumulation point is reached at a specific value of Jln called f1oo where f1oo is a finite number.
2.20.6
The Superstable 2°-Cycle
In the case of a superstable 2-cycle the stable fixed points of F <2l occur at the maxima and the minima of F<2l. It actually occurs for one special J1 (Jl*), so if J12 > J1 > Jlh that is to say that the two fixed points correspond to the maximum and the minimum of at the same value * of J1 = J1 . This can be easily proven, since for the two fixed points x 1 , x2,
F<2l
(2.205) which was shown earlier (equation 2.204). Thus, if one of these fixed points occurs at the maxima or minima, it will have (dP!dx) x = 0, then,
(d:2)J ( : J XI
=
I
d 2)
=0
(2.206)
Xz
and therefore the other fixed point must occur at a maxima or a minima. This sequence of bifurcations at value Jl� continues from Jl � (bifur cation from 1 -cycle to 2-cycles ), Jl ; (bifurcation from 2-cycles to 4cycles) , . . . etc till the accumulation point Jl� , giving values of x which form an infinite set :! = [x(n) ], stable period-2° (n ---t oo) points obtained as jl ---t floo . This limiting case is very important as will be shown later, it has the following characteristics:
--- .
1. It is an attractor of most of the points in the interval (0, 1 ) . We say most of the points because of course, it does not attract the unstable points ( ) 2. The set :! is nowhere dense because between each two stable points there must be an unstable point. 3 . It follows from point 2 above, that :! is nowhere a continuum, and in fact it is a cantor set.
The values of J1 at the bifurcation points, J1n (we drop the superscript * for simplicity ) . Obviously J1 can b e changed as we w i s h, however what we are looking at now is Jln o the values of J1 at which the bifurcations occur.
STATIC AND DYNAMIC BIFURCATION
1 95
Table 2. 1 Values of Jl,. (JJ. values at the bifurcation points) for the logistic map. f.ln
n
n
3
f.ln
6
3 .569692
2
3.449499
7
3 .56989 1
8
3 .569934
3
3 .544090
4
3 . 5 6440 7
5
3 . 568759
3.569946
Of cour se J.ln up to J1<., depends upon the functional form of F (x, J.l) . For the logistic m ap they have the values given in Table 2. 1 . Notice that J1<., is finite and that the points J.ln b e come denser as n increases. These bifu cation values of J.l are not easy to obtain by any means except by numerical methods. There is however another set of J.l values which we wi ll call Jin , that is more accessible and contains the same topological i n form ati o n about the m ap xn+i = F(xn , J.l ) as the set of bifurcation values J.ln · To show that, let us consider the logistic map : ,
r
F(x) = J1X (1 - x) and try to find the extremum of thi s functi on
(2.207) .
-- = J.l [ x(-1) + (1 - x)]
dF( x)
dx
= J.L [-x + 1 - x] = J.l [l - 2x ]
(2.208)
Since the extremum occurs for:
dF(x) dx Table 2.2
=0
(2.209)
The values of the bifurcation parameters Jin .
n
0
2.00 3.236068
2
3. 49 8 5 6 2
3
3 . 55464 1
4
3.566667
n
5
3.569244
6
3 . 569793
7
3 . 5 699 1 3
R
3 . 5 69939 3. 5 6994 6
S.S .E.H. ELNASHAIE and S.S. ELSHISHINI
1 96
Therefore,
Xm
=
(2.2 1 0)
0.5
Now we try to find the value of J1 point a fixed point:
=
Ji , which makes this extremum
(2.2 1 1 ) Thus,
(2.2 1 2) Therefore, (2.2 1 3) It is important to notice that Xm is also an extremum for F(2n) (x), this can easily be shown by w riti n g equation 2.200 in the form,
(2.2 1 4) Therefore for any point with any dF (xk)ldxk for any k we get,
dFn ) ( x )
---
dx
=0
Thus Xm is also an extremum (maximum or minimum) for any map, therefore for P (x),
2 xm = F (xm , Ji) = Ji[Ji. xm ( l - xm ) ] [1 - Ji. xm (l - xm )]
0. 5 = Ji[Ji X 0.5 X 0 . 5 ] [1 - Ji X 0. 5 X 0.5]
Ji = 3. 236068 Thus Ji1 = 3. 236068 The next v al u e s of Jln (Ji2 , Ji3 , . ) can c o mp uted in a similar manner. The values of lin for the logistic map are given in Table 2.2. They are the J1 values at which bifurcation occurs , but with the fixed points always occuring at stationary points of F, f' 2l, f'4l. The v al ue s of Jln and lin are shifted from each other as shown in Tables 2. 1 , 2.2. .
.
be
STATIC AND DYNAMIC BIFURCATION F
1 97
��Po
(a )
Xm
FIGURE 2.1 16 (a) First iterate map for Ji, = iio and = 0; (b) Second I = 0; (c) Fourth iterate map for Ji, iterate map for Ji, Ji1 and dFc2> Ji2 and = 0.
dFC4l(x)ldx k
2.20.7
=
dF(x)ldxl:r
(x) dxi:r_
=
Feigenbaum Universality
Fei genbau m ( 1 978) considered the logistic map for Jln (values of Jln for which the fixed points are at Xm), given by:
-
.,...( 2 n) Xn+l - l' ' ( Xn ,fl )
(2.2 1 5)
y=x-112
(2.2 1 6)
G ( y,JI) = F(y + 1 I 2,JI) - 1 I 2
(2.2 17)
_
As clear from Figure 2. 1 1 6a,b,c, if one concentrates on the region around xm (here xm = 0.5), the sequence of function F(x,JI0 ), F2 (x,JI1 ), p4 (x, JI2 ) . . . etc . , all look similar except for an inversion process and a scale reduction. The scale redu ct i on occurs both in the range of x and also in the magnitude of variation of F2n (x,JI) over that boxed range of x. To represent this scaling, it is best to ce nter the origin of these figures at (x = 112 , F = 1 /2) by u s ing the simple transformations : and
Thus the maps will have the form shown in Figure 2. 1 17. I t is easy t o show that,
( -)
- ) = -2 5 G<2> - -,flJ y G( y,flo 2.5 ·
(2.2 18)
and that,
(2.2 1 9)
1 98
S .S.E.H. ELNASHAIE and S . S . ELSHIS HINI
+ 0. 5
( a)
(b )
{C )
- 0. 5
FIGURE 2.117 The first (a) and second (b) iterate maps for G(y,ji). This s e quence suggests that there
that:
(
is some scaling parameter
a
such
Jl.- )
Y_ (-at - c< 2 n ) _ (-at ' n
goes to some universal function of y (call it g0 (y)) as n � oo, correctly. Feigenbaum found that for, a
this indeed h appe n s ,
=
a is selected
2.5029078750958928485
(2.220) Feigenbaum found that both a and g0 (y) are independent of the precise form of G (y), provided only that it has a single quadratic maxi mum at y = O. For example,
or,
F(x,Jl. ) = J1. s i n 7rX
for (0 :::; x :::; 1)
(2.22 1 )
G( y,Jl. ) = J.LS in ( n(y + 1 1 2)) - 1 1 2
(2.222)
has the same asymptotic scaling . In otherwords ,
a is
a universal scaling
STATIC AND DYNAMIC BIFURCATION
1 99
and g0 (y) is a universal function. Feigenbaum found from his computer studies that,
where,
lim Jl n+l - Jl n = 8 n�oo Jin+2 - Ji +l n
(2. 223)
8 = 4.66920 1 609 1 029
(2.224)
is again a universal constant. Equation 2.223 can also be written as,
(n -7 oo )
(2 .225 )
The constants Ji"" and A are not universal constants , they depend upon G (y) (or F(x, Jl)) . The result in equation 2.223 or 2.225 is not limited to superstable values of Jl (Ji) in Table 2 .2, but also hold for the period doubling bifurcations at the values of Jln in Table 2. 1 . (2.226) It is clear from the above brief discussion using extremely simple first order finite-difference models, that in addition to the qualitative
..l.. 2
j.J.
� •••
)
FIGURE 2. 1 1 8 a =
lim
n �-
·
The period doubling sequences JJ, and
Ji,.(o
...
= lim n ----+
�. • -+ J
200
S . S .E.H. ELNASHAIE and S . S . ELSHISHINI
FIGURE 2.1 1 9 Period doubling to chaos.
similarity between dynamical systems which have some form of structural stability, there are also quantitative similarity in the bifurcation sequences which also have some structural stability as indicated by the universal numbers a, o. For J1 > Jl=, the logistic map can generate in addition to the infinite number of unstable 2n period solutions (shown in Figure 2. 1 1 9), both: 1 . new stable periodic solutions 3 x 2n, 5 x 2n , etc. 2. aperiodic (non-repeating solutions) depending upon the value of Jl. . . .
Aperiodic solutions
An aperiodic solution involves a point which is mapped sequentially through 2n bands (h � x � Jlh k = 1 ,2, , 22) instead of onto 211 points as in the case of periodic solutions. In other words, the points map periodically through 211 disjoint bands of x values, but are not periodic (or eventually periodic) points (Figure 2. 1 20). . . .
FIGURE 2. 1 20
Schematic representation of aperiodic solution.
STATIC AND DYNAMIC BIFURCATION
20 1
X
FIGURE 2.121 Enlargement of bands for J.l > J.lo.•
( 1 980)
introduced the terminology "semi For this sequence, Lorenz periodic sequence" which is found to occur in Poincare map of the third order differential model of Lorenz. Thus when )l is above f.1oo it is frequently found that points indeed map periodically through such a collection of bands or disjoint x regions . The number of these bands goes to infinity as )l decreases towards f.loo . In otherwords, the number of bands decreases as J1 increases over f.loo . This is called reverse bifurcation. As it is clear from the magnification in Figure the shaded bands widen and then overlap (in pairs) as J1 is increased. Chang and Wright found that this series of serniperiodic bifurcations also satisfy Feigenbaum' s asymptotic relationship, with,
the 2n
2.1 2 1,
(1981)
3.569945671... +A.J-m (m � oo) (2.227) where A = 0 .494454 and 8 4.669 20 1. .. the band is atJ1=3. 67 8 57 . . . , beyond which odd periodic solutions occur. The dashes in Figure 2.121 are meant to convey the fa t dynamics of while periodic in the bands, around in an Am =
. . .
N= O
Xn,
=
spins
c that the erratic
fashion each time it returns to a band. It' s motion, w h i le deterministic, is quite different from the motion of another point which is initially nearby. Such motion is called "chaotic". (Collet and Eckmann, 1 980).
202
S . S .E.H. ELNASHAIE and S.S. ELSHISHINI
On the larger scale of Figure 2. 1 22, many of the details are lost and we can see at most only four of the semiperiodic bands. The density of the points is not uniform, as the sh ading ( d ashed regions) illustrates , and this leads to the rather exciting "overlaping veils" appearance in this increasingly complex region. Contained in this region (the chaotic region in large - not the narrow region we analyzed its bands only) are an infinite number of "windows", so called because the chaotic "veil" disappears in this region and we can "see through a windo w " . The veil is replaced by a simple periodic structure at some value of Jl, and then the map goes through a series of bifurcations much the same as those for J1 below J.L,o, until another "chaotic" region is reached. Only three of these windows are illustrated in Figure 2. 1 22 (namely those that begin with period 6, period 5 and period 3). The remaining win dows will be illustrated and discussed later in connection with the chaotic behaviour of the fluidized bed catalytic reactors . The nature of these periodic solutions (windows) is given by the following important and remarkable theorem (Sharkovsky, 1 964).
Sharkovsky Theorem Let T be the ordered set, 3 < 5 < 7 < . . . < 2 X 3 < 2 X 5 < 2 X 7 <. . . < 22 X 3 < 22
X
5 < 22
X
7 <. . .
<8<4<2<1 Let f: / � I = [0, 1 ] be a smooth map such that f(O) = f( l ) = 0, which has a single critical point. If m < n relative to the order in the set T and f has a periodic point with prime (i.e. the shortest) period m, then f has a periodic point of order n. Thus for the logistic map in Figure 2. 1 22 where Jl = 3.627 . . . there is a period 6 solution, the beginning of a "window". According to S harkov s ky theorem there are periodic (unstable) solutions with periods 2 x 5, 2 x 7 and so forth, fol l ow ing to the right through the set T. Note that the odd periods have not yet occured (they begin at J1 = 3 .67 86 . . . , with a new group of infinite periods) . As J1 is increased, the periodic solutions have periods moving to the left through the T set. When Jl = I + 8 1 12 , the period 3 w i ndow is reached and we are at the first member of the T set (all periods exist) . For further discussion of Sharkovsky theorem the reader is referre d to Guckenheimer ( 1 977) and Osikawa and Oono ( 1 98 1 ). We will show some of these windows for the fluidized be d catalytic re actor in chapter 4 .
STATIC AND DYNAMIC BIFURCATION
203
r X
� 1 ; �rmi - Pf'rlodic moti on :- : ( dr� "r � �i"� n -) 1 : 2" Att r i c h v r b � n d s penOdiC W t n dOW S
1 I
I
FIGURE 2.122 A large scale presentation for J.b > 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 x-value 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 stable-unstable "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 period-three 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 near-tangency 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 near-tangency 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<
FIGURE 2. 124 Tangent bifurcation and intermittency. (a) Third iterate map at J.l. =3.828; (b) Enlargement of box b showing the laminar phase of intermittent
dynamics.
Moreover, this map returns to x = 0.5065 after 1 6 5 iterations, and hence returns to the semiperiodic behaviour for another forty or so iterations. In other words, the dynamics spends one quarter of its time in this semiperiodic state. This percentage increases dramatically as J1 is increased. I f J1 = 3 . 8284 it takes 200 iterates of F(x) before this semi periodic behaviour ends, and after 8 5 more iterates it returns to the semiperiodic dynamics, which now rep resents 70% of its behaviou r Intermittencies of various types are rather common features of "turbulent" states. Thus in steady state turbulence in fluids, it is not uncommon to see orderly ("semiperiodic") motion over a period of time in various regions of space, which then breaks up into much more chaotic behaviour, only to become orderly at another space-time region. The temptation, of course, is to draw some parallels between these different forms of intermi ttency but a solid connection (if it exists) awaits further research. In catalytic chemical reactors dynamics intermittency is also quite common as will be shown for the fluidized bed catalyt ic reactor in chapter 4. As noted above, these tangent bifurcations give birth to the "windows" in th e chaotic dynamics The bifurcation structure within a window is essentially a microcosm of the scheme which occurs for 1 � J1 � 4. but multiplied by the periodicity of the window. This is schematically illustrated in Figure 2. 1 2 5 for the peri od three window, where there are three microcosms. Thus the former period-2n b i furcations now become pe riod 3 x 2n cycle s This is follo wed by three chaotic attracting reg i ons which in tum have th eir pe ri o dic windows. A peri o d nin e (3 x 3 ) wi ndow is illustrated. The end of a w i ndo w is characterized by a bifurc ation in whicl! the chaotic attractor discontinuously increases in size to form a .
,
.
-
-
,
.
-
continous attractor (rather than three regions). This occurs at the value of Jl. where the unstable branches of the tangent bifurcati on interse c ts
206
S . S .E.H. ELNASHAIE and S . S . ELS HISHINI
X
T�ng f'n l
bdurc.o�� t ion
FIGURE 2.125 Schematic representation showing the bifurcation diagram of x J.L and showing intermittency, period doubling of the period-three window and crisis points.
vs.
the chaotic attractors. The intersection bifuraction has been tenned an interior crisis (Grebogi et al., 1 983), and represents a distinctly different type of "bifurcation" (notice that the tenn "bifurcation" has become generalized to include modification of chaotic sets). 2.20.9
More on the Connection Between Continuous and Discrete Time Systems
We have shown earlier the presentation techniques that can be used to present the continuous time system in the fonn of a nu mber of discrete points (section 2. 1 9), namely the Poincare bifurcation diagram return points iterate maps of different orders and return points his tograms The basic principle is simply to strobe the points of the continuous change with time for the state variable(s) of the continuous time system. The resul ting strobed points are used in different combinations to fonn the above list of presentation techniques We here explain this technique in a more formal manner than given in sec tion 2. 1 9, and also discuss some of the practical prob l e ms associated with it. The techniqu e is quite s i mple and helps to reduce the comple x dy n amic s of the continuous time system to a more tractable form, where the peri od 1 attractor is reduced to a single point on the Poincare map an d the p eri od n attractor reduces to n points and the torus reduces to an invari ant circle . In additio n the other presentation techniques u sin g ,
.
.
STATIC AND DYNAMIC BIFURCATION
207
the strobed points help us to analyze the complex behaviour of the continuous time system based on the knowledge of the behaviour of the discrete maps, e.g. the return points iterate maps of different order helps to investigate the bifurcation from period n to intermittent chaos as will be shown later. If the complete discrete iterate map of the continuous system is obtained, then it can be used to investigate the possible behaviours associated with the continuous system at a negligible amount of effort compared with the analysis of the continuous system. The one parameter Poincare bifurcation diagram is a very useful and condensed manner to represent the behaviour of the system (specifically a chosen strobed state variable of the system) over a wide range of the bifurcation parameter Jl. It thus shows regions of periodicity (a region of period n will be usually represented by continuous n lines), regions of period doubling leading to chaos, points of crises where two attrac tors meet and region of tangent bifurcation from intermittent chaos to periodic window, . . . etc. In order to present this extremely u s eful technique in a more formal manner than already done in section 2, 1 9, let us also introduce some more formal dynamical systems theory definitions for some of the simple facts presented informally earlier:
a) Autonomous continuous-time dynamical system An nth-order autonomous continuous-time dynamical system is defined by the state equation, (2.228)
X = E_( X )
where X is the vector of time derivatives (dX1/dt, dXz/dt, . . . , dXnldt) of the system state variables (X� . Xz, . , X11) which are all of course functions of time, t. E is a vector of dimension n and is independent of time in the autonomous case. Equation (2.228) is written in matrix notation, it is common m dynamical system theory to write such equation as, .
x = j(x)
.
(2.229)
where it is stated that x(t) E ��� and f is called the vector field, f:�11 � ��� , which means that x, f are both n-dimensional. We will use in this section the notation of equation 2.229 for simplicity. The solution to equation 2.229 can be written as (/i1 (X0 ) (notice (/)1 is n-dimensional and X0 , the initial conditions vector is also n-dimens ional). The Xv is
S . S .E.H. ELNAS HAIE and S . S . ELSHISHINI
208
included in the solution to indicate clearly that the solution "fr depends upon the initial conditions X0• The one-parameter family of mappings (2.230)
satisfies the two relationships (2.23 1 )
and (2.232)
and is called the flow. The set of points ( "fr (x0 ): - oo < t < oo) is called trajectory throu gh X0• If there is a value T > 0 such that the solution �� = 'ifir+r then the system is called periodic and the smallest such T is the period of this periodic solution. b)
Non-autonomous continuous-time dynamical systems
An n th-order non-autonomous continuous-time dynamical system is defined by the state equation, x
=
f( x, t )
(2.233)
For non-autonomous systems, the vector fieldf depends on time and unlike the autonomous case, the initial time cannot in general, be set to zero. The solution to 2.233 passing through X0 at time t0 is denoted by 'ifi/x0 , t0 ). If there exists a T> O such that f(x, t) = f(x, t + T ) for all x and t then the vector field f is said to be time periodic with period T. The smallest such T is called the minimum period of the function f If there is a value T > 0 such that "fr = � + T then the system is called periodic and the smallest such T is called the period of this periodic solution.
Relationship between autonomous and non-autonomous systems An n1h -order time-periodic non-autonomous system with period T can
always be converted into an (n + 1 )1h-order autonomous system by appending an extra state variable, (2. 234)
STATIC AND DYNAMIC BIFURCATION
209
The equivalent autonomous system is thus given by,
(2.235) and ·
2n
8=
T
8 (to ) = - · to 2n
T
(2.236)
Since f is periodic with period T, the equivalent autonomous system described by equations (2.235 , 2.236) is periodic in e with period 2tr.
c) Discrete-time systems
Any map J: Cft.n � Cft.n de fine s a discrete-time dynamical system by the state equation, n = 0, 1, 2, . . .
(2.237)
where xn E Cft. n is the state andf maps the state Xn to the next state Xn+ 1 · Starting with an initial condition x0, repeated application of f generates a sequence of points {xn };= O called an orbit. Although in this book we are dealing only with continuous-time systems, discrete time systems are very useful, first because the Poincare (or stroboscopic) map which replaces the analysis of the dynamics of continuous systems with the analysis of a discrete-time system is a very powerful tool for studying continuous-time dynamical systems and secondly these maps will be used to illustrate certain important dynamic phenomena in continuous-time dynamical systems which are difficult to analyze using the continuous system.
d) Poincare maps: strobing the continuous trajectory This very powerful technique discussed earlier for analyzing dyna mical systems is due to the great french scientist Henri Poincare ( 1 892, 1 893 , 1 894). In this te chnique we replace the dynamics (flow) of an n1h -order continuous-time system with an (n - 1 )1h -order discrete-time system called the Poincare map. The use fu l ne ss of the Poincare map, as discussed earlier, lies in the redu cti o n of order and the fact that it bridges the gap between continuous- and discrete-time systems .
S . S .E.H. ELNASHAIE an d S . S . ELSHISHINI
210
e) Poincare maps for non-autonomous systems (stroboscopic maps) All of the non-autonomous systems presented in this book are forced using a periodic forcing function of period T. The simplest technique for strobing the continuous-time trajectory of this system is by strobing the values of the state variable every forcing period T, giving Xt at T, x2 at 2T, , Xn at nT forming a discrete set {xn}�=l · If the system is periodic with period one, then we will have, .
.
.
and one point will appear on the Poincare map (Stroboscopic map). it is periodic with period m then we will have,
If
and m points appear on the Poincare map, and so on. In some cases to simplify Poincare bifurcation diagrams (explained earlier), we can strobe at multiples of the forcing period, say kT. These strobed points can be used to construct a number of useful diagrams as discussed earlier. In all cases, if the final attractor is the one to be presented, initial transients should be discarded and the amount of initial transients to be discarded vary widely from one system to the other and also vary with the variation of the position of initial conditions relative to the attractor. The value of the forcing period T must be taken with great accuracy in order not to introduce numerical noise into the strobed points.
FIGURE 2.1 26
Poincare section for period 1 attractor.
STATIC AND DYNAMIC B IFURCATION
21 1
FIGURE 2.127 Poincare section for period 2 attractor.
The Poincare map for autonomous systems with periodic attractors For an n1h-dimensional autonomous continuous-time system, the strobing is taken as the trajectory intersects (in one direction) an (n 1 )-dimensional hyperplane transversal to the trajectory. For this case the period one attractor will give only one point x on this plane as shown in Figure 2. 1 26 (notice that point x' is not taken because we are taking the intersection in one direction only, i.e. increasing x or decreasing x). For a period two attractor, the intersections will be two points x1 ,x2 as shown in Figure 2. 1 27. For a period n-attractor the number of points will be n(x1 , x2 , , xn ). For a strange attractor the number of points will be infinite. These strobed points can be used for the construction of a number of discrete pre s entation techniques as explained earlier. These presentation tech niques are extremely useful in the analysis as will be shown later. The points of intersection of the trajectory with the Poincare hyperplane r must be taken with great accuracy in order not to introduce numerical noise into the strobed points. In order not to be obliged to take very small step sizes in the integration of the continuous system, an efficient ac curate scheme should be used to locate the po i nt on the plane accurately from a known point j ust before the pl an e and a known po i nt j u st after the plane. There are four main techniques of different deg re e s of accuracy for locating the hyperplane crossings, namely : linear interpolation, time - s tep halving, Newton-Raphson and the Henon' s -
. . •
method (Parker and Chua, 1 989) .
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S . S .E.H. ELNASHAIE and S . S . ELSHISHINI
The Poincare map for a continuous system with a quasi-period attractor (torus) A quasi periodic function is one that can be expressed as a countable sum of periodic functions,
x(t) = L � (t)
where hi has a minimum period Ti and frequency fi = 1/Ti , (or CO; = 2Trl Ti) . Furthermore, there must exist a finite set of base frequencies {fi , , JP } with the following two properties: A
. . .
A
1 . it is linearly independent, that is, there does not exist a non-zero set of integers {k1 , . . . , kp } such that: k,fi + . . + kpfp A
A
.
=0
(2.23 8)
2. It forms a finite integral base for the h , that is, for each
l
ii = k� A + +kp ip
for some integers {k1 , . . . , kp }.
. . .
l
i, (2.239)
In words, a quasi-periodic waveform is the sum of periodic wave forms each of whose frequency is one of the variuos sums and differences of a finite set of these frequencies. Consider the two-periodic function,
(2.240)
where h1 and h2 are arbitrary periodic functions with periods T1 and T2, respectively. If we assume that T1 and T2 are incommensurate, that is T, !Tz is irrational, then this function x (t) is quasi-periodic and not periodic and the trajectory lies on the surface of a two-torus T2 : s 1 x s 1 where each s1 represents one of the base frequencies (see Figure 2. 1 28).
FIGURE 2.128 Two-periodic behaviour lies o n a two-torus represents one periodic component.
s
'
x
' s ,
each s '
STATIC AND DYNAMIC BIFURCATION
(a)
(b )
213
FIGURE 2.129 Trajectories on a two-torus. (a) A periodic trajectory with T1 = 2, T2 = 9; (b) A two-periodic (quasi-periodic) trajectory with T1 = 2, T2 -{2. If the simulation was run longer (a) would not change but (b) would be more densely filled in. =
Since a trajectory is a curve and the two-torus a surface, not every point of the torus lies on the trajectory, however, it can be shown that the trajectory repeatedly passes arbitrarily close to every poi nt on the torus. Therefore, the two-torus is the attractor of the two-periodic behaviour (be careful not to mix-up the quasiperiod two-periodic behaviour with the period two periodic behaviour). It should be clear now why two periodic behaviour requires incommensurate frequencies. Consider a trajectory travelling on a torus (Figure 2. 1 29) looping in the '1'1 direction with period T1 and in the '1'2 direction with period T2. If T1 and T2 are commensurate, then there exists positive integers n and m s uch that nT1 = mT2 • It follows that in nT1 seconds (or any other time units), the trajectory will close on itself since it will have made exactly n loops in the first direction and exactly m loops in the second direction (Figure 2. 1 29a) and since nT1 = mT2 thus the trajectory closes up giving a periodic attractor with period nT1 • However if T1 and T2 are incommensurate, no such n and m exist and the trajectory never closes on itself (Figure 2. 1 29b). The Poincare map for the Quasi-periodic trajectory is therefore obviously a closed circle as shown in Figure 2. 1 30.
( b)
(a)
FIGURE 2.130 The quasi periodic attractor on the Poincare map; (a) the limit set is two embedded circles, (b) the limit set is a single embedded circle.
214
S.S .E.H. ELNASHAIE and S . S . ELSHISHINI
Chaotic attractors on the Poincare map for autonomous systems There is no widely accepted definition for chaos. From a practical point of view, it can be defined as an attractor which is not periodic nor quasi periodic. Recently there has been a distinction between strange attractors which are chaotic and tho se which are not chaotic. They both look strange and rather complicated on the phase plane and Poincare map. However, chaotic attractors are characterized by sensitivity to initial conditions and are thus characterized by at least one positive Lyapun o v exponent, while strange non-chaotic attractors do not show sensiti v ity to initial conditions and have no positive Lyapunov exponents (e.g. Brindley, et a!. , 1 99 1 ; Brindley and Kapitaniak, 1 99 1 ; Kapitaniak and El naschi e , 1 99 1 ; Kapitaniak, 1 99 1 ). Hyperchaotic attractors with more than one positive Lyapunov exp onent can also exist (e.g. Klein et al. , 1 99 1 ) .
0. 5 'I
-0. 5 L....-= ... :;;..... ...._ .... ... �c... ... .... .. .. ... .. -15 -1.0 -0.5 QO 0.5 1.0 1.5 x (a) Dutting ' 5 Equation _ _ 0.6 ;· �....�::,... ...-, � .. ....... ..,...�......,
'I
as 0. 4
0. 51
Dutti ng ' 5 Equation
r:.r . . -...,..... ... .... .=... .;... .,.... ,,... ..., ___ .,.., .: r •. -.......,
· · ...·.-,. .
y
�
...<�
0.50 0.49
X
(C)
FIGURE 2.131 Poincare map for the chaotic attractor of the Doffing's equation. (a) Poincare map for the Doffing equation; (b), (c) Successi ve magnifications of (a) displaying the rme structure of the Poincare map of the attractor.
STATIC AND DYNAMIC BIFURCATION
215
A chaotic attractor is not a simple geometri c al object like a circle or a torus, they have quite complic ated geometry (in phase space and on the Poincare map), they are related to cantor sets and possess fractional dimensions. Figures 2. 1 3 1 a-c shows the Poincare map for the chaotic attractor of Duffing' s equati o n Notice that in most of the catalytic processes discussed in this book the Poincare map of the attractors are quite simple due to the fact that these systems are highly dissipative .
.
The construction of a complete iterate map for autonomous systems. The choice of the Poincare hyperplane
Usually it is required to construct one state variable versus one bifurcation parameter (Jl), in other words Poincare bifurcation diagram, like the one shown for the discrete-time system in Figure 2. 1 22. This is usually needed for continuous-time systems in order to get a complete picture of the possible behaviour of the system over a range of one of the important system parameters or input variables. To do that the stro bing of the points using a Poincare plane should be performed at a large number of values of J1 with the attractor (whether periodic, quasi periodic or chaotic) having its specific structure at each of these values. Practical problems sometimes arise when a simple Poincare hyperplane suitable at a certain value of J1 (J11 ) is not suitable at another value of J1 (J12). It is important in this respect to notice that the choice of the bifurcation parameter J1 may be that it does not only affect the dynami c behaviour but also the steady state behaviour of the system. To illustrate, consider the three dimensional system:
(2.241 )
Suppose that the attractor on the x 1 vs. x2 ph as e plane for a value of Jl, is as shown in Fi g ure 2. 1 32, an d the Poincare hype rp l ane r is taken as shown at x2 = x� . thus the Poincare map will give one point P1 as sh o w n If for J1 = Jlz the situation is as shown in Figure 2. 1 3 3 , the same r at x2 = xi will not intersect the new attractor. In this cas e the problem i s easy to solve by taking xi c l o s e to xzss and moving with it as J1 changes. However in some other cases it may be a different problem .
216
S . S .E.H. ELNASHAIE and S . S . ELSHIS HINI
x,
FIGURE 2.132 Period one periodic attractor at JJ
JJ1•
=
requ iring changing the hyperplane r with the change in the bifurcation parameter f.l. Such a situation develops sometimes when the steady state point is not moving much but the attractor is changing its shape or developing higher periodicities in narrow regions of its domain. To illustrate consider the case in Figure 2. 1 32 and the case in Fi gure 2. 1 34 where the period doubling (or adding) develops in a region away from the r p l ane . In this case we will still have one point P on the Poincare plane while
x,
x, ss - - - -
r- - --� I
I
FIGURE 2.133 Period one periodic attractor at JJ
=
JJ2•
STATIC AND DYNAMIC BIFURCATION
217
x,
p
FIGURE 2.134 Period two periodic attractor at
J.L
=
J.L2.
pe ri od doubling (or adding ) has already taken place. Such situations requires some experimentation with the phas e planes over the ran ge of J1 to be studied before deciding on the shape and location of the Po i nc are hy perplane and may also nee d ch anging r with the ch ange in Jl.
The construction of the complete discrete iterate map for the continuous time system
One of th e main adv antages of the use of Poincare maps, as discussed earlier, is to fi n d a co nnection between the continuous-time system and the discrete-time system, so that findings obtained using dis c re te -t i me systems (wh ich are much easier to study and analy ze than continuous time systems) can be u ti l ized in the analysis of continuous time - syste ms . The main presentation techniques which make this connection clear are iterate maps of different orders, that is, for the first iterate map, plotti ng the di screte s trob ed po ints as Xn+ l vs. Xn or p l otting Xn+m vs. Xn for the m1h i te rate map. To cons truct these equivalent iterate maps they should be done without dis c arding the initial transients (and somtimes even using different initial conditions to c omp lete these discrete iterate maps) u n l ike other presentation tech n i qu e s where we d is c ard the initial transients in o rde r to capture the structure of the attractor and its chang e s (e .g. with the change of one or more bifurcation parameters) withou t the interference of strobe d po ints re s ul ting from initial transients .
218
S . S .E.H. ELNASHAIE and S.S. ELSHISHINI
A note on the Poincare bifurcation diagrams for non-autonomous systems The construction of the Poincare map (Stroboscopic map) and other related diagrams for non-autonomous systems is easier than for the autonomous case, because in the non-autonomous case no hyperplane is needed but rather the strobing of the point is taken for every forcing period T. However, when it is required to construct a bifurcation diagram with the forcing frequency ro( ro= 2n/D as the bifurcation parameter and where T is the forcing period, then at each value of ro the strobing must be carried out at this value of ro and its corresponding period T.
C HAPTER 3
Modelling and Elementary Dy namics of Gas-Solid Catalytic Reactors
In the last four decades there has been an incre asing amount of research carried out in order to develop more rational means for the design, operation, control and optimization of gas-solid c atalytic reactors. These efforts have an obvious economic justification since these reactors are in the heart of most chemical plants. This fact, together with the complexity of th e various physico-chemical processes involved makes the study of these reactors a very challenging area for research. Hetero geneous catalytic reactors of the fixed bed or fluidized bed type are used in a wide variety of industrial applications in the petrochemical and petroleum refining industries, ranging through cracking, hydrogenation, oxidation, synthesis reactions and many others. A proper und erstand ing of the complicated steady state and transient behaviour of these reactors is essential for effective design, operation and control. The parametric sensitivity of some of these reactors to s mall variations in operating conditions (which are inevitable in practice) makes the control within safe limits quite difficult. For certain values of the parameters there may be multiple steady states. In such cases steady state informations alone are not sufficient for design, since the final steady state attained by the reactor depends on the dynamic behaviour and initial state of the system. Mathematical models of different degrees of sophistication have been constructed to predict the behaviour of these reactors. With the vast increase in size and speed of modem computers, it is now possible to solve very complicated models and compare the results with those of simp l er ones. There are no general guidelines for depicting the choice of model for a p articular purpose. Ideally, the model should be as simple as po s sible yet still capable of adequate prediction of the essential features of the behav io u r In the case of design this will generally involve the use of steady state models. For control studies, on the other hand, reliable tran s i ent models are nee ded In practice the v al i dity of the model must be established by com ,
.
.
parison with experimental results obtained from pilot plant studies, 219
220
S . S .E.H. ELNASHAIE and S.S. ELSHISHINI
or even better, through the comparison with reliable industrial data. This comparison will suggest improvements to be made in the model . Investigation of the sensitivity of the reactor using an appropriate model helps to identify the important parameters and indicate the degree of precision with which they need to be measured in the laboratory. Such information also provides important feedback in the planning of laboratory experiments. The role of academic research
The academic researcher, not faced with a prompt design or operation problem, can afford to look at these systems more fundamentally to uncover the basic laws that govern their behaviour. Simplifying assumptions introduced into the models should be treated with care since they give misleading results when used outside their region of validity, which is usually unknown. The approach that has been adopted, in general, is to decompose the reactor into its elements and to study each element separately. Such a decomposition of the fixed bed reactor is shown in Figure 3 . 1 . From a steady state point of view, the most
Study of Radial Heal and Mass Transfer in the Bed
Study of Axial Heal and Mass Dispersion i n the Bed
lntraparticle Heat and
Mass Transfer
Resi stances
Combined Internal
and External I leal
and MaS> 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 adsorption-desorption processes taking place on the catalyst pellet surface. lnevitably, in dealing with non-linear 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 non-porous catalyst particle. A comprehensive analysis of the behaviour of a "non-porous" 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 active-site 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 adsorption-desorption 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
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S.S.E.H. ELNASHAIE and S . S . ELSHISHINI
to include the effect of a finite rate of reactant adsorption (i.e. non equilibrium adsorption-desorption). The most noticeable phenomenon associated with relaxing the equilibrium adsorption-desorption 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 active-site 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 start-up with special emphasis on the "travelling reaction zone", is presented together with the effect of feed disturbances on the reactor. The phenomenon of wrong-way 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 Non-Porous Catalyst Particle
Most modelling studies on the non-porous 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 non-porous 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 non-uniform 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
V20s-Al203
23.6
1 098. 1 2
V20s-Al203
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) o-xylene 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
Chrome-Alumina
(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 pre-mature 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 finite-difference solution for the transient equations, with symmetrical boundary conditions is presented using the Crank-Nicholson method together with the Von Rosenberg' s ( 1 969) modification for the non-linear 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 non-porous 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 adsorption-desorption is assumed independent of temperature (or taken as a temperature-independent 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 pre-exponential 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 adsorption-desorption 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 . 3-3.6 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 Crank-Nicholson s cheme was used to discretise the linear equation 3 . 8 ( w ith its non-linear boundary condition at ro = 1 , equati on 3 . 1 0). The set of finite-difference 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 on-lin 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 o--5 ) is nece ss ary . For cases of higher thermal conductivities a larger step size ( I Q-2 ) 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£
:§. ... ..
1.0
No. OF PQUiTS
Fin itr d i l f r r � oc t solut i on
+---'---'----'�-.I...--....I..--L-..I 20 1.2 1.4 1 0 1.6 0.6 08 0.2 0.4 0 t ( l im� ) m i nutrs
FIGURE 3.2a Surface temperature/time response curves. Comparison between finite difference and collocation methods. A case of moderate thermal conductivity. Initial conditions are Y0 (w) = 1, X,0 = 0. 1·16 t : 13 · 0
1 · 14 ..
5 1 - 12
�...
Flnit eo - diftert-nce s o l u t i on
Q.
e 1 · 10 �
CoUoca;tion s.olutlon ( N
.i : 2 x 1 04 r ' 10 7i ' 0 2
Ill Ill ..
c: H le 0
' iii " ..
e
:0 >-
=
5)
a.k : J O '
1 - 06
Cl.m' 50 N u,' 1 0
ii- '
1 ·04
0-02
I : 2 ·0
1 · 02 1 · 00
0
0 ·2 04 W I d i m • ns 1 o n l • . s
0·6
r a d i al
0 ·8
posit1
on )
10
FIGURE 3.2b Internal temperature profiles. Comparison between finite dif· ference and collocation methods. Initial conditions are Y0 (W) = 1 , X,. = O.
MODELLING AND ELEMENTARY DYNAMICS
� ..
-.
�
� .. u
1 16
" Ill
tl
.. c 0
a ' 2 ��o l04 r , 10
1i
' 0·2 so
1 -14
Nul ch
1-12
am� 50
� 1 10
4 x l 0- 3 a k = 1 06
C oUoc-. hQn Mt' lhod ( L � g � n d r P polynomt ai'lo) C u rve No. at point
1 .08
Q) cD cD
;;; 106 c: ..
-�
-o "'
,..
229
1-04 1-02
F�t�•t• d1tft'rl'nc r
i O h.o t • c n
1001-f--r-----,--,--.---.---..--,-�---' 0 01 0-2 0 -3 04 05 0 · 6 0 -7 08 0·9 \ 0 t ( t i m• ) m i n u t • s
FIGURE 3.2c Surface temperature/time response curves. Comparison between finite difference and collocation methods. A case of moderate thermal conductivity. Initial conditions are Yo (CD) = 1, Xso = 0.
surface temperature responses are shown in Figures 3 .2a-c . Figure 3 .2b shows the change of internal temperature profile with time . An extensive study of the dynamics of the system using this finite difference method is impracticable. Therefore a more efficient orthogonal collocation technique is presented to deal with the conduction equation inside the particle together with its non-linear boundary condition.
(ii) The Orthogonal collocation technique This method is based on the choice of a suitable trial series to represent the solution. The coefficients of the trial series are determined by making the equation residual vanish at a set of points, called collocation points, in the solution domain. The orthogonal collocation technique (Villadsen and Stewart, 1 967; Finlayson, 1 972 ; Viladsen and Michelsen, 1 978) provides a systematic basis for choosing the collocation poi nt s . These points are the roots of an orthogonal polynomial of order N. By choosing a suitable weigh i n g function, it is possible to we i gh the s olution accuracy in certain desired regions. Due to the symmetry of the problem in hand, it is appropriate to use the following trial function which satisfies the boundary condition at the centre identically.
Y( ro,t) = Y(l , t) + ( 1 - ro 2 )
L, a; (t)P;(ro2 ) = L, dirozu-o N
N+l
i=l
i=l
(3 . 1 1 )
S . S .E.H. ELNASHAIE and S . S . ELSHISHINI
230
where N is the number of internal collocation points and the polynomials are defined by the relationships:
Il W( co2 ) - � ( co2 ) · lj ( co ) · co2 · dco = c(i) · uu 2
0
�
j = O, .
. .
,i - 1
The usual technique is that the trial function (3. 1 1 ) is substituted into the differential equation which is then satisfied at discrete radial collocation points ro;. This gives a set of ordinary differential equations governing the d; (t). Villadsen and Stewart ( 1 967) and Villadsen and Michelsen ( 1 978) pointed out that the equations can be solved in terms of the solution at the collocation points instead of the coefficients d; (t). This is more convenient and is used here. The Laplacian and the derivative are defined in terms of the solution at the collocation points as follows:
[
and
iP Y
-
a co
2
2 JY
+-
-
]
N+l
co a co W=W;
B; - Y( CO · ) =� � � J
i = 1, . . . , N
(3 1 2) .
; =I
i = l,
. . .
,N
(3 . 1 3)
where the coefficients Au, B;1 are computed using the method described by Vi l ladsen and Stewart ( 1 967). Substituting equation (3. 1 2) into equation (3.8), the conduction equation is replaced by the set of ordinary differential equations,
df
d fi
=
N+l
� B-.I] Yj. -=a �
(3. 14)
j=l
and from 3. 1 0 and 3 . 1 3 the boundary condition at the surface is given by,
II AN+l-JYJ = Nul [ I - YN+I + a . jj . exp ( -r I YN+ I ) . Xs ]
N+l
}=
Owing to the relatively inaccurate formula 3 . 1 3 and
(3 . 1 5)
the steep profile
in the early stages of the response, this procedure requires a large
MODELLING AND ELEMENTARY DYNAMICS
23 1
number of collocation points to accurately approximate the temperature gradient at the surface. Therefore, an alternative procedure based on the more accurate formulae for integrals should be used and thus accurate results can be obtained using 2-5 collocation points. Detail s of the method are given in Appendix B . The improved collocation method is compared with the finite difference method in Figures 3.2a-c. The method gives very accurate results for the surface temperature with relatively few collocation points in only a fraction ( < 1 0%) of the computing time necessary for the finite-difference technique. The collocation points have been chosen to emphasize the accuracy of the solution in the region near to the surface. As shown in Figure 3 .2b, the collocation solution deviates from the exact solution away from the surface. Although this does not present a problem for the non-porous case, it may cause significant difficulties in the case of a porous particle. In this case a greater number of collocation points may be needed or alternatively, a different type of orthogonal function is called for. c) Simplified models
(i) Pseudo-steady state concentration model (PSSCM) When the concentration response of the system is much faster than the temperature response (i.e. the mass capacity is much smaller than the heat capacity, which occurs when KA Cm is small), it is reasonable to assume that the concentration passes through a sequence of pseudo steady states determined by the instantaneous values of temperature. This assumption effectively decouples the mass and heat balance equations, making their solution easier. For the PSSCM model, the concentration is defined by the pseudo steady state relation Xs =
exp ( y I � ) a + exp ( y I � )
(3. 1 6)
obtained from equation 3.7 by neglecting the accumulation term dX5 -· am dt The heat balance equation is thus described by equation 3 . 8 as before,
with the modified boundary condition
[
]
dY = alJ - Nu1 1 - Ys + ---'---d ro a + exp ( Y I 'fs )
at ro = 1
(3 . 1 7)
232
S . S .E.H. ELNASHAIE and S . S . ELSHISHINI
(ii) Infinite Thermal Conductivity Model (/TCM) In this model the particle is assumed to be isothermal and the system is described by equation 3 .7 together with the overall heat balance
(3 . 1 8) d) Steady state equations
The steady state equations are the same for all three models, and the particle is described by two algebraic equations
(1 - Xs ) = a exp - y I � ) Xv
(
( � - 1)
= afJ e xp
·
(- y I � ) · Xs
(3 . 1 9) (3.20)
Equations 3 . 1 9 and 3 .20 are similar to thos e of the adiabatic CSTR, but with different physical meaning of the parameters. These two equations can be decoupled into the single equati on:
(� - 1) = a(l +,B - Yv ) · exp ( - y l � )
(3 . 2 1 )
Equation 3 .21 has been analyzed extensively for uniqueness and multi plicity by many investigators (e.g. Aris, 1969). The following necessary conditions for the existence of multiple steady states were established: (i) ,B ( y - 4) > 4 * (ii) a > a > a * * where
* --a =a
**
exp y ,B y - 1 exp (2 + y l (1 + ,8 )) = 1 + 4(1 + ,B ) I y
--'=----��---'--
Regions of multiplicity can e as ily b�generated numerically and the re sults are shown in Figure 3.3 with y, f3 as variables and a as a parameter. e) Stability analysis In this section local stability of the steady state is studied relative arbitrarily small symmetric perturbations.
to
MODELLING AND ELEMENTARY DYNAMICS
233
1 ·8 1·6 1·4
�f , , 1.2
.
0·8
·�
0· 6
0·4
10 12
FIGURE 3.3
14
16 18
20
22 24 26 28 30
"( (•� )
Regions of multiplicity of the steady states.
(i) Pse udo steady state concentration model (PSSCM) The solution of the linearized equations of this model has the following form,
(3 .22) where x is a small perturbation about the steady state (x = Y - �s ) and An is the n1h root of the equation: (3.23)
wh ere
,
gss =
d�
s
( exp ( - y / � ) / ( l + a e xp ( - y / � ))55
S . S .E.H. ELNASHAIE and S . S . ELS HISHINI
234
A sufficient condition for this system to be stable is that all the roots of equation 3 .23 must be real. Simple analysis of equation 3 .23 shows that for An to be real the following condition must be satisfied:
which reduces to the well known slope condition (Aris, 1 969) for stability: (3.24) Since the slope condition is completely determined from the steady state, the thermal conductivity of the catalyst will have no effect on the local stability of this model, subject to symmetric perturbations.
(ii) Infinite thennal conductivity model (/TCM) This model has been studied by Cardoso and Luss ( 1 969), and they have shown that the slope condition is not sufficient for stability. Therefore some steady states which satisfy the slope condition could be unstable. We present these conditions and link them with the stabi lity of both the PSSCM and the complete model. We also present a condition under which the stability criterion of the PSSCM is valid. The characteristic equation for the eigenvalues of the linearized system is:
(3.25) where,
A = -(au + a23 )
•u
�
( ( ;;J, ) ( %� )
- a,
� 2 = 3 aNu1 aJJ
+ am
s
ss
Suffici ent conditions for local stability are that the roots of equation 3 .25 have negative real parts. This will always be the case if
MODELLING AND ELEMENTARY DYNAMICS
( ) ( )
235
These two conditions can be expanded into the form, a -
and
a1s axs
+ JJ at� a� ss
< ss
1
(3 .26)
(3.27)
By a simple algebraic manipulation it can be shown that condition 3 .26 reduces to the slope condition. It is clear that condition 3 .26 does not imply condition 3 .27 and therefore steady states which satisfy the slope condition could be unstable. It is important to investigate the range of parameters for which the stability criterion of the PSSCM is valid i.e. sufficiency of the slope condition . Sufficient condition for the stability of a steady state that satisfies the slope condition
A sufficient condition for instability is that one of the eigenvalues of e quation 3 .25 has a positive real part. For a steady state that satisfies the slope condition B > 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 j-factor correlation, i.e. (3.32) for gases and vapours. For a specific gas-solid 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
1.8
a.
E �
��
" v oil
1. 6
:a ... c
1. 4
.!i �
1. 2
c
-�
N : "•
0.5
a= 0 . 4
"' >
X s ( d imen sionless s urlcce concen t ra t i o n )
FIGURE 3.4 Effect of thermal conductivity on start-up. A case of high thermal conductivity.
Therefore, a steady state which is unstable under the infinite thermal conductivity assumption is always unstable for any finite value of thermal conductivity. On the other hand if ITCM predicts stability there could be two cases to consider:
1 . am > 3aNu1 The slope co ndit i on is sufficient for stability . Therefore, thermal conducti vity has no effect on local stability. 2. am < 3aNu1 Decreasing the thermal conducti v i ty can lead to insta bility at a certain critical value. The stability should be checked from detailed stability analysis of the complete model.
Although the stability an al ys i s presented here is by no means rigorous it is ve ry useful for quick prediction of pos s ib l e stabil i ty and instability regio n s , as shown in the results presented in the next section. f) Numerical simulation, results and discussion for the non-porous
catalyst pellet. The symmetrical case
Owing to the infinite number of p oss i ble initial states and the number of parameters involved, it is i mpossi ble to pres ent an exhaustive set of results . In all s u b sequ ent computations we s h all present only those cases in which the particle is initially at a uniform temperature (i.e. Yo ( w) = Y,. (0), 0 � w� 1 ) where Ys (0) s i g nifi e s the initial surface temperature.
S .S.E.H. ELNASHAIE and S . S . ELSHISHINI
238
1. 8
P= l . O
1. 6
., u 0 "t: ::> "' "' "' .!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 start-up. A case of low thermal conductivity.
(i) Effect of thermal conductivity Start-up in the region with multiple steady states Figures 3.4-3.6 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 start-up. A case of v ery low thermal conductivity.
MODELLING AND ELEMENTARY DYNAMICS
(j) @
Nu 1 •SO '
N u 1•0S, N u , . so
N u 1•0.! ·
239
• 0 004 - X,(o)TO O: 'r' d o ) o: l 7 ( Jow conduc l i " it y ) • 0 4 : X1(0) 10.0: Y,(o) • l . 1 (hiJh conoJucli,·itrJ � 0.004: X1(o):O.O. Y s(o) al .� (tow conducth·it)) •0 4 - X,to)•O.O: Y,(ob I S (hith col\duwvtt)' )
Pilr.amtttetr s. : Ci : Z a iO � )' : 1 8 Jl = 1 ·0 a,• 1 0 6
� 1-4
�
a.,:
e 1.2
' iii r;:
':§
>-VI
s
1.o-t-..---.--.--,....,,..,___,__,.__,..1 L..::;::::;:=:::;:=== 0 0·2 0·4 0.6 o.s 1 ·0 6·0 s . o 10·0 t ( t ime ) minutu
FIGURE 3.7 Surface temperature/time response curves. Effect of thermal conductivity (X, (0) 0.0). =
actual phase plane of the problem at hand is of infinite dimensions due to the distributed nature of the heat conduction equation. These plots are in fact a compact way of presenti ng results and show the effect of the uniform initial conditions on the final steady state attained by the system . For the start-up of the system at hand, the projection of the initial conditions will usually be on the left hand ordinate of the Ys - Xs diagram. In order to drive the system to the high steady state B the par ticle must be preheated to a sufficiently high temperature before supplying th e reactants, namely Ys (0) = 1 .48 for the particle with high thermal conductivity (Figure 3 .4) and Ys (O) = 1 .65 for the low conductivity case. Some observations of the effect of thennal conductivity on the dynamic response The effe ct of thermal conductivity on the response of the system for a
specific set of parameters varies with the initial conditions. In case (3) of Figure 3 .7, the system with low thermal conductivity is que nched to th e low ste ady state, while the one with high thermal conductivity (case 4) is ignite d to the high s teady state. In cas e 1 , the system with low conductivity shows thermal overshoot. This overshoo t is removed by increasing the conductivity, as in c as e 2. The initial temperature run away is acc en tu ated in Figure 3 . 8 for the initial conditions having Xs (0) = 1 .0. Obviously these represent undesirable start-up c ondition s .
S . S .E.H. ELNASHAIE and S . S . ELSHISHINI
240
(i) (D Q) @ 2 6
�..
N up:0.5· n: . o_4: Y,(n) = 1 � - X,(o) = 1 . 0 (hiJh c:onrluc-tivity ) Nul= 50. ti -: 0 .004. Y,(o) ; 1 4 · X ,fo) = 1-0 ( low conduclivilyl Nu 1aO.� (l' .. n 4 : Y,((l),.1 .8: X,(o) • 1 -0 (hi&h cnnducli \·it)') Nu 1 • .SO· 0 •0 004 · Y �(o)=I .K. X,(o) = 1.0 (lew conductivitt·)
� 2·' ::1
Q.
E 2!
�::1
2-2
..
u
2- 0
..
1·8
P�ram� t E> 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 .4-3 .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 "
3-8
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
, o-1
- - · -- · -
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 : 0-04
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 Pseudo-steady 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 non-porous 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
j �
s
1. 3
I
� 1.2
rl30
N 11 1
li am
,::.. •
=
•
=
2
l
10
0.2
t .!
O.Oj
)04
i 11
4
l
0.06
- 0 .. (0) • 1.12 y.( OJ 'r',(W)((l !:IU$ I) = Yo(o)
101 10·5
2
!"
�
,.. I.O +-�-�-�-.:;.-....::..,...___:c,.--.:...,�"-r--�---l 0 W � � R W � � � � �
FIGURE 3.13 state.
t ( minutu )
Effect of heat of adsorption on the stability of a unique steady
MODELLING AND ELEMENTARY DYNAMICS
245
Notice that we have taken the heat of adsorption into account without making KA function of temperature. This means that KA is considered an average value over the temperature range investigated . Taking the dependence of KA upon temperature into consideration, complicates the model considerably and is still a subject of research. Figure 3 . 1 3 shows the quite dramatic effect of f3A on a unique unstable steady state. The steady state is stabilized when adsorption heat release is included in the model. Moreover, the response is highly damped even for relatively small values of the adsorption exothermicity
/3A·
Figure 3 . 14 shows the case of multiple steady states with the upper state B unattainable for the set of uniform initial conditions with f3A = 0. This steady state is readily attainable for f3A = 0 . 6 . 3. 1 . 1 . 2
The asymmetrical cas e
The asymmetrical behaviour of catalyst particles has attracted con siderable attention in the late sixties and early seventies (Pismen and Kharkats , 1 968; Luss \ et al., 1 972). We present in this section very briefly some of the main interesting features of this problem. For simplicity we deal with the simplest geometry, i.e. slab geometry. Other geometries are considerably more difficult to analyse. An 2.0 .Q ...
Ill ...
a: E � .. u 0
t:
:I Ill
Ill Ill
fi\
\�\ B 'I · i ,J \J
1. 8
1. 6
\ . .
�
1. 4
.� �
Ill >
� -·- 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 non-porous 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 .35-3 .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 S-shaped 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 gas-solid catalytic reactions fail to consider the dynamics of the adsorption-reaction-desorption 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 adsorption-desorption steps is not only related to finite rates of adsorption and desorption, for even when the adsorption-desorption 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 El-Bialey, 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 non-porous 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 22.0-3 3 . 8 1 3 .7-22.4 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) Non-activated 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 0-50 kcal/mol. The activation energy for adsorption, Ead = 1 0-22 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 · 8-3 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 adsorption-desorption
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 active-site theory and assuming equilibrium adsorption-desorption 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 adsorption-desorption is relaxed and the effect of both activated as well as non-activated 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 Adsorption-Desorption. The Lumped Parameter Adsorption-Desorption Equilibrium Model (LP-ADEM)
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 3.56--3.63,
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 adsorption-desorption equilibrium.
T.
Adsorption-desorption 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 pre-exponential 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 pseudo-homogeneous 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 adsorption-desorption relation to eliminate the surface concentartion. The use of the more realistic non-linear 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 pseudo-homogeneous 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 pseudo-homogeneous 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 j-factor correlations,
for gases and vapors. Using this result, we can write,
For the adsorption of ethanol on ion-exchange 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 0-3 for gas-solid systems, Ls is of order 1 , while 4 is of order 1 o-3. 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
1
·'
,
(3 .76)
MODELLING AND ELEMENTARY DYNAMICS
( ) ( )
265
which is simply the slope condition
dQdY
r.
dQ+ dY
>
s
r.
s
as derived by Liu and Amundson ( 1 962) . The second condition can be written as, (3 .77)
For the pseudo-homogeneous model of Liu and Amundson ( 1 962) Ls = 0, and bearing in mind that Lv
1 - > 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 Adsorption-Desorption described by the Lumped Parameter Model (LP-EADM) Figures 3 . 1 9-3 .22 summarize the results of transient computations carried out on the non-linear 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*Id-r 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"Id-r 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 pseudo-homogeneous 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 active-site theory which
MODELLING AND ELEMENTARY DYNAMICS
....'.., ....
--
i 1 ·8
- - - · -
:>
0 :;;
E-
2!
� c: 0
1-6
1-4
.� l 2
· ;;; c
"" ,..
. --
/3A /J A
= 0·0 = 0-2
269
a
P:lr:lmclcrs ·
= 2 x 1 04 p = o. • y • 1 0.0 L. "' 20.0
1·0-k:�-��....--=;:==;::::: ... ::;: =;:==:==;;:::;: =:;: ::: � :: ::::� : ..--..-- � 0 ·12 0 -14 0 -16 0 -18 0-2 0 0 o.o2 o.o4 o.os 0-08 0-1 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 active-site model tends to destabilize the steady state. If the Lewis number Ls of the gas-solid 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 adsorption-desorption 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 non-equilibrium adsorption-desorption. The Lumped Parameter Non-Equilibrium Adsorption-Desorption Model (LP-NEADM)
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 active-site 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 pseudo-homogeneous 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 3-3 .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 adsorption-desorption
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, Y-1
--
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 adsorption-desorption.
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
0.8 0. 7
0.
FIGURE
3.25
10
11
12
15
a•
00
2· 0
1 0
0·5
16
17
18
Multiplicity regions. Non-activated adsorption.
iii) Non-activated adsorption
For non-activated adsorption, the rate of adsorption is insensitive to temperature ()1;1 = 0). Figure 3.25 shows a two parameter continuation diagram (TPCD) for f3 vs. YA for a case in which the rate of adsorption is a weakly decreasing function of temperature ( )1;1 = -2). The effect is entirely different from that of activated adsorption, as might be expected. Adsorption multiplicity is not found for non-activated adsorption. Finite rates of non-activated adsorption serve to decrease existing regio n s of multiplicity predicted from the simpler equilibrium adsorption desorption model.
276
S . S.E.H. ELNASHAIE and S.S. ELSHISHINI
d) Unsteady state equations It is convenient to introduce the normalized concentration variables,
x* = c* 1 C8 ,
and the dimensionless time
3ht r = --
Rpps CPs
in addition to the normalized temperature Y (equation 3.90). Eq u ations 3 . 83-3. 85 may now be written as,
* dx 4 - = 1 - [ 1 + aa exp ( - ra I Y)] X dr -=1
dY dr
•
+ aa [exp ( - yR I Y)]Xs
dX dr
- Y + afJ [exp (- y I Y) ] Xs + f3A L, -s
(3.94)
(3.95)
where aa, ')'a, a, f3, r are previously defined steady state parameters, and YR = Ya - YE· Three additional dynamic parameters are introduced into the analysis: Ls, Lv which are Lewis numbers representing ratios of mass capacitance to heat capacitance of the particle. Ls is based on the adsorption capac ity of the internal surface and Lv based on the void volume.
Ls = PsKAo Cm kg -PsCps h
4
h
= -
-
1
1
--- E.
kg Ps Cps
(internal surface) (void volume)
The remaining parameter f3a is an exothermicity factor for adsorption, f3 = ( -11Ji)a kgCB a hTs
MODELLING AND ELEMENTAR Y DYNAMICS
277
e) Reduction in the order of the model It is pointed out in the preceding part that for practical systems Ls f4 C. I ()3 while L.v C. l in many cases. In general, therefore, the fast responding process of accumulation of reactant in the pore space is in pseudo-steady state. The system can be appro ximated by the two dif ferential equations 3.93 and 3.95 with equation 3.94 replaced by, •
0 = 1 - [l + aa exp ( - ya i Y)]X + a) ex p ( -yR I Y ) ]X5 Combining equations 3.93 and to the pair of equations, '
dY dr
-=1-
.
3.96, the system can finally be reduced
dX 5 d r = {(1 - exp [(-y E I Y)]X . ) I (1 + ( exp (- y I Y)) l a - a exp ( - y I Y) Xs
L
(3 9 6)
'
a
dX Y - a,B [exp (-y I Y)]Xs + ,BALs drs
a
)}
(3 . 97) (3.98)
f) Local stability analysis As explained earlier several times, the reader should now be familiar with the fact that the stability of the steady state to arbitrarily small perturbations can be examined by linearizing equations 3.97 and 3.98 about the steady state and examining the sign of the real part of the resulting eigenvalues. Details of the analysis for this specific case are presented in Appendix C. The main results of this analysis are given here. Two stability conditions must be satisfied for the steady state to be asymptotically stable. The first is the familiar steady state slope condition
dQ- > -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 Non-Equilibrium Adsorption-Desorption Model (NEADM)
i) Non-activated 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 non-activated 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\<'
279 P a r a me te r s :
"
L,
YE
Y
r.
jl p,
=
2 . 1 04 1 4 .5
-5.0 = a
I 0
-z.o 0.6
0
1.2
0·04
0 · 08
0 12
0·16
FIGURE 3.26 Effect of the rate of adsorption on the stability of unique steady state.
the heat of adsorption is equ al to the heat of reaction. This i s much more realistic for chemisorption. For small values of f3,H the steady state can be u n s table above a critical Lew is number. It i s interesting to note that the critical Lewis number passes through a minimum. The steady state is least stable, the refore , at intermediate rates of ad sorption . Figure 3.26 shows the effect of rate of ad s orpti o n on the s tabi li ty character of the steady state in the unique region . For very high and very l o w rates of adsorption, the steady states corre spo nding to both cases are stable. However, for an intermediate value of the rate of adsorption parameter ( aa = 0.2) the resulting unique ste ady state is unstable giving rise to a limit cycle (periodic attractor).
Multiple steady states
Table 3 . 8 s h ow s the effect of decre as i ng the rate of n on- ac ti vate d adsorption on the c ri tical Lewis numbers for the low and hi gh conver sion steady states in the multiple solutions region. The low conversion state is l ocal l y stable for all L,, irre sp ec ti ve of f3a· The l ocal stability of the hi�h conversion state de cre as es conti nu ou s l y as the rate of a�sorp tion decreases to the cri tic al value ( aa = 0.25) produc ing quenchmg to a unique low conversion state . For small values of f3a the critical Lewis
S . S .E.H. ELNASHAIE and S . S . ELS H ISHINI
280
Table 3.8 Critical Lewis numbers for non-activated adsorption. Multiple steady states a= 3.66 x 1()3, {J= 1.3, r = 16, YE = -4, ra = -2.
Critical Lewis Number, L""' High conversion state
Low conversion state
f3a stable for al l Ls
=
0. 6 1 45.0
54.3
stable for all L,
25
70
1
1 3.8
40.8 19
2
0.5
5.6
0.4
3.8
14.5
0.3
1 .9
1 3 .4
number near the quenching point can be quite small and within the practical ran ge . However, this hi gh conversion state is gre atly stabilized for higher values of f3a · Figure 3 .27 shows a phase plane plot for the data in Table 3.8 with an = 0.3 and Ls = 4. The upper stea dy state B is un s table if f3a = 0. All trajectories eventually terminate at the low conversion state A. Extreme transient hot-spot temperatures occur for initi al conditions on the right hand side of the phase plane. For f3a = 0.6, the upper steady state i s stable. No transient hot- spots are observed.
f
>- 1 - 8 41
:;
�1 6
�
�
CD
13a - - - - - f3a --
P a r a m e t er s :
3.66 •
YE
-4
y �
= 0 =
"
0-6
Ya
L,
lla
1-4 � 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
0-6
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 two-point 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( - M-l )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 (pseudo-homogeneous 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 pseudo-homogeneous 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 Sturm-Liouville (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 non-zero 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 finite-difference 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 non-self 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 finite-difference methods such as C ran k-Ni 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 finite-difference 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) Finite-difference techniques We shall first present the solution of the dynamic equations using the finite-difference method (Crank-Nicolson). 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 00-200 finite-difference points were needed in this case to obtain 1 ·0 5 2
1 - 04 8
1-044
1 - 040
1 03 6
a - o - o2 I I f-JA St•ad y sta1• 7/
- · - - f1A =
0
--
I I
;
//
�,-=0-J ___,
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
1-004 1 -000+'-'�"t--.--.--,�--r�---1 0·8 0·6 0 ·4 0-2 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 (non-porous 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* ) =
using the collocation method to approx imate the Laplacian operator we obtain the following set of ordinary differential equ ations at the collo cation points , (3 . 1 29) and (3 . 1 30) The boundary conditions at the s urface (CO= 1 ) can be approximated by, (3 . 1 3 1 ) i=l
and, N+l
L A;,N+Ili = Nu (YB - YN+I ) i=l
( 3 . 1 3 2)
where the BiJ and A iJ are c ompu ted by the method described by Villadsen and Stewart ( 1 967) for Ns 10. For N> 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 0-6 0·4 0·2 0 w -
FIGURE 3.29a Comparison between the finite-difference 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
0-8
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 1-0 w -
x8
y8
• "'
LO
1 .0
FIGURE 3.29b Comparison between the finite-difference 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
•*
=
1 · 02
0
•
y (lll.Q)
1 .0
Bulk. conditions:
I : 0·02
1 · 01
•a
=
Ya
1 .0
1 .0
1 00 -1--�-...:�--.--�-��--,-�....--1 1·0 0 0·8 02 0 6 0 4
FIGURE 3.30a Comp ariso n between the finite-difference and the collocation techniques. Temperature profiles. Low Lewis number.
Figures 3.29a,b show a case with high Lewis number. Nine internal collocation points proved sufficient to produce a converged solution in agreement with the alternative Crank-Nicolson technique using 500 radial increments. Figures 3.30a,b show a similar case but with low Lewis number. Only five interior collocation points were required in 1 ·0
-·-·-
0-8
*
l
X
0-6
I
L,
: 1 ·0
·�
�
�
0 ·2
1 ·0
0·6
.
=
L,
�
f:lr �A y
-
Nu
Sh
·�
0 ·4
00
Pa.rame t e r ! :
N: 3 5 , 9,1 9 ; C / N 500 Sle>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 finite-difference 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 1-3 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 0-20% of the computing time, for the same accuracy of the Crank-Nicolson 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
1-02 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) Start-up start-up 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
1-024
l020 1
016
,. . -
1 · 012
-�:]
l008 l004 1 · 000 -f-o-""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
0-5
=
1 o 1 .0
X
0·3 0· 2
I
0.1
: 0 · 05
o . o+-��....,..._--.---.---r----1 lO 0·8 0·4 02 0 0·6 W ---+
FIGURE 3.33b Effect of beat of adsorption on concentration profiles during start-up. 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 build-up 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 0-6 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 bulk-phase 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 physico-chemical 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
50-500
r
0--0. 1
1 0--40
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 a-xylene 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 wrong-way 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) Pseudo-homogeneous models. b) Heterogeneous models. The pseudo-homogeneous 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). Pseudo-homogeneous 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.
Pseudo-homogeneous 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 non-uniqueness 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 two-dimensional 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 one-dimensional 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 pseudo-homogeneous 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 cross-sectional 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 sub-division exists regarding the parti cle models as in the continuum case i.e. lumped, semi-distributed 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 non-adiabatic. 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 pre-exponential 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
( Xi - Xi ) = M( X1_1 - X1 )
(- r I
(3 . 1 56)
�)
( f; - �i ) = H( f; - 1 - f; )
(3 . 1 57) (3 . 1 5 8) (3 . 1 59)
where j 1 , 2, . . . , N. =
By summing the first J equations of the fluid phase (equations 3 . 1 58, 3 . 1 5 9), an overall energy balance for these J cells, is obtained, (3 . 1 60)
We define Ymax to be the temperature for which X; = 0 to obtain, H ( Ymax - YF ) = f3rMXF
(3 . 1 6 1 )
Then equations 3 . 1 60 and 3 . 1 6 1 can b e combined to give, H( Ymax - Y1 ) = f3 rMX1
(3. 1 62)
Equations 3. 1 56 and 3. 1 57 can be used to eliminate X1 from equation 3 . 1 62 to give, after rearrangement, _
QI -
M ( �; - 0 ) ( Y. y ) H max - J
_
-
0
j
_
- QII
(3 . 1 63 )
MODELLING AND ELEMENTARY DYNAMICS
319
where,
01 = + 1 a exp ( - y I �1 ) a_:xp (- y l �1 )
Equation (3. 1 63) enables us to determine the possible values of YsJ for a given value of Y1 from the intersection between the Q1 and Q11 curves. It is easy to show that all straight lines Q1 pass through a common point S at Y1 = Ymax and Q1 = MIH. Vanderveen et al. ( 1 968) have suggested the following procedure to determine graphically the change of temperature with cell number j.
Graphical marching technique (Vanderveen et al., 1 968) Equation 3 . 1 59 can be rearranged to read,
(3 . 1 64) By writing equation 3. 1 62 for J and J-1 we obtain,
-- ---
Ymax - YJ = X1 = M Ymax - Y1- 1 X1- 1 M + 81
---'=--"----
(3 . 1 65)
Thus by multi plying equation 3 . 1 63 by M ( M + 81 ) -I and use equations 3. 1 64 and 3 . 1 65 one obtains,
QI* -- � 1+H
[
�1 - �1_ 1 Ymax - YJ-1
)
- QII* - M81 M + 81
(3. 1 66)
Thus all the Q; lines must pass through a common point S' for which �.1 = Ymax and Q; = M I (1 + H). A simultaneous use of Qt and Qu curves with Q; and Q;1 curves enables one to determine the temperature profile by the following graphical marching technique (Figure 3.4 1 ).
(i) For a given )j_1 determine by use of Q; and Q;1 curves the value of Ysj • call it b. (ii) Determine from the Qu curves the value of � for Y:,j = b, call i t c. (iii) Draw a straight Q1 li ne through the point s and c to determine the value of lj . ( i v) Repeat for the next cell and so on.
S . S .E.H. ELNASHAIE and S . S . ELSHISHINI
320
5
"H
*
M
"' c
.2 0 c :I
c 0
-
0 a.
· ;;; 11'1
'6
c 0
c ...
111
c Q,o 01
0 Q,o :J:
Yma x
FIGURE 3.41 Marching technique for the solution of steady state equations of the simple cell model.
When multiple intersections occur between the Q1 and Qu curves, this procedure can be used to determine all possible steady states. However, transient computations are necessary to determine which of the se steady states will be obtained from a given initi al c ond iti on Stability considerations can be used to redu ce the number of feasible steady states. .
3. 2. 2 . 3
Stability analysis
In this section we employ the linearization pri nciple to discuss the asymptotic stab ility of the ste ady state of the reactor to arbi trari l y small perturbations. This exercise is useful as a first step in uncovering the important parameters affe cti n g reac tor stability We should bear in mi nd that this analysis leaves unanswered two important proble ms: .
321
MODELLING AND ELEMENTARY DYNAMICS
(i) is the steady state asymptotically stable to large perturbations (i.e. globally asymptotically stable), (ii) if the asymptotic stability is local and not global what is the region of asymptotic stability (R.A.S.), or domain of attraction of the steady state .
These questions are difficult to answer. Invariably we must resort to nume rical methods. Most of the work in reactor s tability dealt with stability of the free system (autonomous system or stability in the sense of Lyapunov) The second class of stability problems input-output stability of forced systems (non-autonomous), is of more practi cal importance We shall focus our attention on this problem later The strategy presented here to give the reader an insight into the asy mptot i c stability character of this model is as follows: ,
.
,
.
.
a. We derive stability conditions for the single particle when only the particle temperature and concentration are disturbed. In other words, we assume the bulk conditions to remain at their steady state values when the particle is infinitesimally disturbed b. We investigate how these conditions vary as the particle condi tions vary along the length of the reactor. Notice that we based the system variables and parameters on constant reference temperature and concentration. This means that on ly the particle temperature and concentration vary along the reactor, the p arameters remain i ng constant c . We then deri ve stability conditions for the whole bed, accounting for transient changes in temperature and concentration of both the solid and fl uid phases. d. Finally we try to disc over wh at addition al stability conditions to those of the single particle must be satisfied to ensure stab i lity of the reactor. .
.
,
I.
Asymptotic stability of the single catalyst particle
The c at aly s t particle equ ations are linearized in the ne i ghbo urhood of the ste ady state and classical s tability analysis is perfo rm ed. The bulk condi ti ons are assumed to remain constant. This analysis gives the following pai r of necessary and sufficient conditions for asymptotic s tability of the catalyst particles in the j'th cell to arbitrarily small perturb ati ons ,
a) Static condition:
(3. 1 67)
S . S .E.H. ELNASHAIE and S.S. ELSHISHINI
322
b) Dynamic condition: (3. 1 68) where,
.fj = exp [( - r I �) ]X; Subscript ss designates steady state value. We s h all refer to condition 3 . 1 67 as the static condition, since it contains only steady state data. Condition 3 . 1 68 on the other hand contains dynamic parameters a3, a4 and f3A and is refered to as the dynamic condition The stabilizing effect of /3A is obvious from 3 . 1 68. Steady states that violate condition 3 . 1 67 are always unstable and w e shall concentrate our attention on the stability of those steady states that satisfy this static conditi on The p se u do homogeneo u s model of Liu and Amundson ( 1 962) assumes a3 = 0 and condition 3 . 1 68 is satisfied automatically. The static condition therefore, is sufficient for stability. For the more physically realistic active site model a3 /a4 can be greater than 1 (c.f. section 3 . 1 .2). In this case equation 3 . 1 67 does not imp ly condition 3 . 1 68. The static condition is necessary but not sufficient for stability It is the critical value of the capacitance ratio a3 /a4 above which the system becomes unstable that concerns us here. In fact, it is easy to show that if, .
.
-
.
(3 . 1 69) then the static condition is sufficient for asymptotic stab ility. However for a3/a4 > 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
1
(3 . 1 72)
Condition 3 . 1 72 is a necessary condition for instability. A sufficient condition for stability, that is somewhat conservative, follows imme diately from 3 . 1 72 by reversing the inequality sign, a91 J
-
1
1
For the simple reaction under consideration, the above stability condition reduces to, a
-3 < 1 +
a4
exp ( y I �; ) '
-
a
(3 . 1 73)
Y,1 changes with }, however we can obtain a conservative sufficient condition by demanding that,
a3
•
(
- < Mm 1 + a4
which g i ve s parameters,
exp ( y I _
a
�))
(3. 1 74)
the following sufficient stability condition in terms of the
(3 . 1 75)
S . S.E.H. ELNAS HAIE and S . S . ELSHISHINI
3 24
where,
Condition 3 . 1 7 5 is an improvement over the previous condition 3 . 1 69. Consider for example the following set of parameters, y = 30,
5 a = 2 x l 0 , {3T = 0.5, M = 4, H = 3,
XF = l . O,
YF = l . O
Condition 3 . 1 70 gives a3/a4 <331 as a sufficient condition for particle stability, which is always fulfilled for any physically reasonable capacity ratio. Note that the effect of transient changes in the temperature and composition of the flowing gas stream on reactor stability have been ignored in this analysis. This more complex problem is now considered. 3.
Stability of the complete bed
This section extends the previous analysis by considering stability with respect to perturbations in both the catalyst particle and the interstitial fluid states. Details of the analysis are given in Appendix D. Only the main results are presented here. We shall concentrate on relating the stability conditions of the whole bed, to the single catalyst particle stability conditions with special reference to those steady states that satisfy both the single catalyst particle stability conditions.
First stability condition From Appendix D the first stability condition is,
If the particle satisfies the two stability conditions derived previously (equations 3 . 1 67 and 3 . 1 68) from the single pellet analysis, then 3 . 176 i s satisfied automatically, since
MODELLING AND ELEMENTARY DYNAMICS
325
Second stability condition
The second stability condition can be
arran
ged to read, (3. 1 78)
If the particle satisfies stability conditions 3 . 1 67 and 3 . 1 68 derived previously, then condition 3 . 1 78 is satified if,
(1 + M) (1 + H) > � + � a3
(3. 179)
a4
Therefore if condition 3 . 1 79 is violated the bed can be unstable although the catalyst particle is stable. It is instructive to examine the physical significance of the parameters in 3 . 1 79. -=
a1 a4
cell mass capacity ( b u lk phase) particle heat capacity
negligible
2 = -----���-��--� -a
a3
cell heat capacity (mostly the wall) particle mass capacity
If the wall heat capacity is negligible then a2/a3 �0 and condition 3. 179 will always be satisfied. On the other hand, if the wall heat capacity is appreciable, condition 3 . 1 79 is not necessari ly satisfied. In fact, wall heat capacity has to be very high indeed to violate 3 . 1 79 and one would expect condition 3 . 1 79 to be always satisfied for all practical cases An important point to notice here is that based on the physically unrealistic pseudo-homogeneous model of Liu and Amundson ( 1 962, 1 963), a3 is ve ry small, therefore even a small wall heat capacity may cause the violation of 3 . 179. .
Third stability condition The third con diti o n can be written as,
(3. 1 80) This condition can be termed the "static stability condition of the bed", since it does not contain any dynamic parameters. The first term i s
S . S .E.H. ELNASHAIE and S . S . ELSHISHINI
326
always positive when the "static" stability condition of the single catalyst particle 3 . 1 67 is satisfied. In this case 3 . 1 80 is satisfied if,
(3 1 8 1 ) .
Condition
3. 1 8 1
can
be
written as,
(3. 1 82) Notice that the left hand side attains its minimum value when Y:f}ss = YF = XF . Therefore the and that the right hand side i s maximum when sufficient condition for 3 . 1 82 to be satisfied (when the particle static condition is satisfied) is that,
x;ss
(3 . 1 83) 4. Summary of stability conditions
Single catalyst particle: Two conditions are necessary and sufficient for local stability o f the steady state a) Static condition
(3. 1 67) b) Dynamic condition:
(3. 1 68) Adiabatic fixed bed reactor Four stability conditions must be satisfied for each cell. Three of these conditions are sati sfied if the single particle conditions a) and b) are satisfied in addition to the following two conditions, c) ( l + M) ( l + H) > az llJ
M
d) gl · > - g2;·f3 r ; H
+� a4
(3 . 1 79) (3 . 1 8 1 )
The fourth condition is exceedingly complex and must be checked
numerically.
MODELLING AND ELEMENTARY DYNAMICS
3. 2.2.4
327
Numerical simulation, results and discussion
Study of the dynamic behaviour and start up of the reactor requires the numerical integration of a large number of differential equations. The number of parameters involved and the large computation time makes it rather difficult to present to the reader a comprehensive coverage of all possible cases. Consequently, special emphasis will be placed on the effects of the particle mass capacity and the heat release due to adsorption. The cases presented are for a bed of 50 cells in length. The physico chemical parameters are such that all the particles are in the multiplicity region. This means the possibility of a great number of steady states for the bed as a whole. Ignition may take place at any cell depending on the initial conditions and the transient behaviour of the system. Figures 3 .42,a,b show the temperature and concentration profiles at different times during start up for a step change in reactant feed composition. Initial conditions are given on the graphs. The bed is only i gn ited in the last two cells. This partially ignited steady s tate will be blown out of the reactor (quenched) for a slight decrease in feed temperature or concentration. There are of course, other possible steady states for this system for which ignition takes place nearer the bed entrance. These steady states can be obtained by preheating the reactor before supplying the reactant feed. The computations in Figures 3 .42,a,b were carrie d out assuming no heat release due to reactant adsorption ({JA = 0.0). 2.4 2.2
F'�rd conditions
: tniLi:al conditions: P' a u m e t e n : 1i'
-- v ,,
---·-
Yj
2.0
»
·;; »
c3
Yr = 1 .0. xF • 1 .0 y1j (o) z Yj (o) = 1 . 0. ,, L o l "' 0, j = 1 .2...... SO 2 .t 10-5 · "!• l l.O. �T • 1 .0 . D ..-. ::: 0.0. al'"' 0. a;z • 0 . 2 . c• '"' 0.25. M • 4. H "' J
•
a
0. 1 5
1 is in minuts
1.8 1. 6
,.
l.J, 1. 2
0. 2 10
20
30
C e- l l n u m be-r
j
40
so
0.0
- xT -· --
0
Xj
10
20
30
C e ll num ber
j
40
50
FIGURE 3.42 Start-up. Simple cell model with /JA = 0. (a) Temperature prof'Lies. (b) Concentration profiles.
S . S .E.H. ELNASHAIE and S . S . ELSHISHINI
328
:
� "' 2 s 1� . 'f• 1 9.0. h = 1 .0. 1), = o ..-. a1 2 0. Clz = 0. 1 .5 II] • 0.2. a. "' 0 2 5 , M "' •. H = l Feed condllions YF "' 1 .0. lF = 1 .0 lni1ial condilions: y�1 (o) :::o y1 loJ = 1 .0. 1j IOJ = 0 0. j = LL . .50 t IS. in minutu
2.�
Parameters
2.2
2.0 1.8 ,.., .
·;
,..,
· ·-
1.6
><
><
1. �
:
0. 8
.
-- X j - · - -- Xj
0. 6 0. 4
1. 2 l.
0
10
20
30
Crll n u m ber
j
�0
50
o.o +-___.:;:.-..--.,.___.:�-==----1 0 10 20 30 40 so C e U n u m ber
j
3.43 Start-up. Simple cell model with /JA = 0.4. (a) Temperature profiles. (b) Concentration profiles.
FIGURE
Figures 3.43,a,b show the effect of the heat of adsorpti on on start up for the same conditions as in Figures 3 .42 a,b except that f3A = 0 .4. In thi s case the ign ition takes place near the bed entrance (the 9th cell) and the stead y state reaches its maximum adiabatic temperature rise at the exit. The differences in behaviour for the last two cases are quite dramatic. The differences in behaviour are due to the fact that the concentration wave ( which is not accompan ied by any direct heat effects when f3A = 0) in this case is acc omp an ied by heat release due to ads orpti on . Concentration builds up rapidly near the entrance un der the i mpo siti on of a step increase in feed comp o sition . Equation 3 . 1 42 shows that this build up in concentration will have the effect of causing the c atalyst pellet temperature to rise quite rapidly when /3A -:F. 0. The reaction rate and the reaction heat release increase quite dramatically leading eventually to ignition . This kind of behaviour due to the additional co upling between c oncentrat i on and temperature in equation 3 . 1 42 is not observed with simpler models (Vanderveen et al., 1 968), which neglect the heat release due to adsorpti on . It is imp ortant to note that the maj or effect of adsorption heat releas e is to raise the temperature of the bed to a sufficient level that the reacti on ign ites . Once the reaction has been igni ted , the heat of adsorption plays a much s maller role. Figure 3.44a makes this p oint clear. A high temperat ure initial condition (}':,j (0) = 2.0) has bee n chosen. Such a high temperature causes an instantaneous igniti on of the re ac ti on . For this case the adsorption has negligible effect on the temperature profiles . If on the other hand, we start from a hi gh temperature initial condition,
MODELLING AND ELEMENTARY DYNAMICS
2·4
Parameleu :
2-2 Z ·O
>->-
. .,
1 8
i i i
1.6 1-4 1·2 1·0
f
r I
0
t :O-S
I
t : O· O
t : 2 -0
Feed condi\loM l niti•l cond11ions: 1 is 1 n minulcS.
-- Ys, - · - · - y. J
10
20 30 CPU nu mber
li . 2 .II. 10!1 1 1 .0 = y � = 1 .0 P• . 0.4. 0.0 a4 • 0.2S minute •J = 0.2 rniHte 112 = 0. IS minute •• = 0.0 M = 4 H • 3
40
I.F • 1.0, )'p "" 1 .0 y1j (o) "' Yj (D) = 2.0, Aj (O)
329
•
0.0,
j •1 .2,
..•
�
50
FIGURE 3.44a Start-up. Simple cell model with high initial temperature. Negligible effect of heat of adsorption.
x;
but with (0) > 0, these undesirable start- up conditions cause transient temperature runaway. Heat of adsorption damps down these transient temperature runaways, as was the case with the single catalyst pellet. Figure 3 .44b shows the simulation for such a situation. Figure 3 .45a shows the effect of a one-fold increase in the wall heat capacity on the transient behaviour. Figure 3 .45b shows the different steady states obtained by varying the wall heat capacity.
2· 5
Panmeten :
2 ·4
"'
>-
84 II]
2 3
2 z 2. 1
2· 0
il y
ltr
'·
0
' -� -/ ' = o.s
10
'
., ·,
Peed concliliona '- . ,
112 M H
lnilial collditions: 1 is in minutes.
. ...... . ....... .
•
=
=
= =
=
= =
xr
2 1 I fF! 1 8 .0 1 .0
0.25 0.2
II 4 3
=
AJ (o)
•
0
1 .0,
=
1 .0,
YF -= 1 .0 Y•j (o)
=
1 .6S,
j
= l ,l, .....SO
·-------- · - ·
20 30 c"u numbu j
40
50
FIGURE 3.44b Start-up. Simple ceU model. Effect of /JA on temperature runaway. High initial temperature and concentration.
S . S .E . H . ELNASHAIE and S . S . ELSHISHINI
330
2·4
Parameters
:
2 -2
M
2-0
H
••
1 =15-0, stt-ady
8) •2
sta tt-
1-8
••
'F
;;...-
YF
::... 1 - 6
"' >-
Initial con d i tions: t is i n m i nmcs.
1 -4
X ) ()S 1 8.0 1 .0
2 m
z =
:
0.4
4.0
3.0
0.25
0. 2 0.3
0.0
1 .0
1 .0
(o) " Yi (o) x; {o) = 0.0 j = 1 ,2,. . . . . 50
Y•i
=
l .O
--- Ysj - - - - - Yj
1-2 1·0
'l! 'Y liT PA
0
10
20
30
C ell n u m be r
j
40
so
FIGURE 3.45a Start-up. Simple cell model. Effect of wall heat capacity. 2 - 4 -.------,
2-2
5 lf'ady
P a ra m e t e r s :
s ta ti!'S
i'i
'( liT �...
•J
14
2-0
=
=
:
18.0 l .O
0.4
0.2
0.25
0.0 1 .0,
••
YF
Steady states obtai ned for >-lA
I n i tial
1 -5
conditions:
l OS
2 X
1j (o) Yoi
(o)
•F = 1 . 0 = Yi
=
0.0
(o)
=
1 .0
1 -4 1 .2 , . 0 -t--=:::'--r: __,.----.--"""T ------1 0
10
30 20 C e ll n u m be r
40
so
FIGURE 3.45b Simple cell model. Effect of wall heat capacity on the final steady state attained by the reactor.
33 1
MODELLING AND ELEMENTARY DYNAMICS
We now present the more physically meaningful coupled cell model. the presentation will be confined to the case where the cells are coupled by radiation, since at such high temperatures (ignition) radiation i s obviously the predominant mechanism for heat transfer. It will be shown later that conduction will have qualitatively the same effect, but of course quantitatively the effect is weaker than that of radiation. 3.2.3
Analysis of Fixed Bed Catalytic Reactors using the Radiation Cell Model
There are several assumptions which are specifically made in the derivation of the radiation model in addition to the other assumptions used earlier in the simple cell model. These assumptions are as follows: A catalyst pellet in the j th cell is assumed to radiate only to pellets in the adjacent (j - 1 ) and (j + 1 ) cells. (ii) The gas is non absorbing. (iii) The view factor is independent of position. (iv) The emissivity is independent of temperature. (i)
The system is described by the following set of differential equations ,
;
dx * a3 - = x . - x . - a exp df } J _
(-y l f .)x .
_
''J
4
4
+ R [ �J+I + �J-I
-L
dY.
-
1
df
= H ( Y.
J-
1 -
J
- 2�14 ]
Y. ) - ( Y . - f . ) J
J
*
dx;
( - y I Y,; ) X1. + a3f3A --
dX · * a1 1 = M( X · 1 - X . ) - ( X . - X . ) dt jJ J J n�
(3 . 1 49)
J
Sj
d�j a4 - - Y1· - Y.,; + a f3 T exp dt _
*
Sj
•
dt
(3. 1 84) (3 . 1 52)
(3. 1 5 3)
where,
with E' = emissivity, F = view factor, s = Stefan-Boltzman constant.
S . S .E.H. ELNASHAIE and S.S. ELSHISHINI
332
Boundary (entrance and exit) catalyst layers
The first ( entranc e) layer of catalyst partic le s receives radiation only from the second layer. Rad iation losses from this first layer are taken to be,
This gives for the fi rst (entrance) layer, the following di fferential equation, a4 d� � = l'J - � � + a fJ T exp (-y I �� ) x; + a3fJA dx; t dt d +R
[y4
4
s 2 - 2 Ysl +
(
YF + � �
2
)4]
(3 . 1 85a)
The last (exit) layer of catalyst pellets receives radiation from the (N-1 )1h layer. If we ignore radiation losses from this l ast layer , the differential equ ation for the exit layer becomes , a4 d
�N
�
= YN - �N + a fJT exp (-y/ �N)X� +a3fJA
dX� + R [�t-, �t - J �
(3 . 1 85b)
and the initial conditions are given by, y.1 = Yo 1 '
xj = Xjo • �j = �jo• t = O, j = 1,2, . . . , N
(3 . 1 86)
Becau se the j th cell is now coupl ed to the downstream (j + 1 )1h cell through radiation, the solution cannot be obtained by a marching tech nique as for the si mpl e cell model. The co mplete set of equations for the bed must be solved simultaneously at each time step. Numerical integration of the dynamic equations is the only available way at the present time of o btaining an insight into the dynamic b ehaviour of this system. We s h all present to the reader the effect of different physical parameters (radiation, bed mass cap acity , heat capacity and heat of ad s orption) on the dy namic behaviour of the system. After that we present s o me an al ytic al e xpre ssio n s for the c re ep i n g p rofile caused by feed disturbances. In the last part of th e chapter, we first pre s e nt to
the reader some recent results which include some generalization of the bifurcation and instability characteristics of this system and also the
MODELLING AND ELEMENTARY DYNAMICS
333
phenomenon of wrong-directional response which is in principle , different from the phenomenon of wrong-directional creep. Secondly , we present modelling and results based on continuu m models, followed by a critical discussion and comparison between the cell and continuum models from the point of view of their consistancy with the true physic al structure of fixed bed catalytic reactors and the correctness of its physical representation. 3.2. 3. 1
Numerical simulation, results and discussion
We shall basically use a set of parameters which have been used previo u sly by Vanderveen et al. ( 1 968) and later by B erty et al. (1972 ). In all the cases we assume a1 = a2 = 0. Vanderveen et al. ( 1 968) have shown that during start-up the particle temperature profile, after achieving a constant shape , slowly migrates (creeps) upstream. Berty et al.
(1972) have checked this and found no
migration zone. They tried various values of the radiation parameter R and found that the migration zone is only obtained for a radiation parameter forty times greater than the one re ported by Vanderveen et al. (1968). The results shown in Figures 3.46 and 3.47 confirm those of B erty et al. ( 197 2 ) . Figu re 3.46 shows the temperature profiles during start-up for the radi ation parameter reported by Vanderveen et al. (1968). The main feature of the results is the absence of a migrati on zone with the steady state reached in 20 minutes. Fi gure 3.47 shows the same case except that the radiation parameter is increased forty times. The results now show a creeping temperature profile and the stead y state is reached in 2 hours. 1.1 2.0
Paran1�t t r s .
1. � 1. 8
Ysj
1. 7
1. 6
1. 5
loitial cond1110ns: 'raj (0):1.0. j : t,2, ...• 7s, Fe.cl cCII"di tiJns : yF = 1.0, Jl F : 1.0 t i' in minutn .
1. 4
1. 3
1. 2
1. 1
1. 0
04 = 0. 1 6 3 M "' J . 9 1 1J It .. 2 . 3 5 2 a - 4.337 "' 1 0S y , 18.JJ i>r • 0 . 6 I!A. .._ u . o .... R : 8.6, a 10 Cl 1 . a 2 , a, : 0
0
10
FIGURE 3.46
20
30
40
C�ll numb�r j
so
60
70
75
Radiation cell model. Start-up with low radiation parameter.
S . S .E.H. ELNASHAIE and S.S. ELSHISHINI
334 2.1
P:uame1crs :
2.0
1. 9
Ysj
1. 8
1. 7
1. 6
1. 5
Feed condi1ions
1. 4
=
�A
=
Gj
huual coadiuons:
1. 3
0.0
O. l b) ).9 1 9 • 2.352 = 4.::;J7 .\ t OS :::o 1 1 .33 = • 0.6 : J 4S6 1 JO-l .:: � - o1 : o "F "' 1 .0. Yr "" 1.0 yq 101 z 1 . 0 . J = 1 .2� .. - . 73
o� M H i! y 1'1' R
1. 2
L1
10
20
30
40
Ctoll number
j
50
60
70
75
FIGURE 3.47 Radiation ceU model. Start-up with high radiation parameter.
These results of Vanderveen et al. ( 1 968) and Berty et al. (1972), are based on negligible effects of bed mass capacity as well as adsorption heat release. In what follows we shall present to the reader the effects of these parameters on start-up as well as dynamic response to feed disturbances. I. Numerical Simulation o f the Dynamics during Start-up 1. Effect of mass capacity
When the mass capacities of the interstitial gas and the pellet are neglected (a 1 = a3 = 0), the concentration wave propagates in an infinite speed, while the heat wave propagates in a finite speed depending on 2.1
P' u a m e 1 c r s :
2.0
1. 9 1. 8
1. 7
Ysj
1. 6
1.5
Feed
1. 3
� u n u 1 1 10ns.
l n c l i a l conl!itioni:
1. 2
1. 1
0
10
FIGURE 3.48
{JA = 0.
o.. � H a: Y 1'1' R ., a,
1. 4
1. 0
0....
20
30
40
C t>ll num ber j
50
60
70
=
0 0
0 16) -= - J 919 ::. 2 . 3 5 2 4,]]7 1 10� = I UJ = 0 0 = �.456 ,. 1 0- 1 = = 0. 1 2 0.1 = 0 = =
'F
=
=
I 0. .l' F
=
)'1 (ol '" 1 .0 ==- 0.0 j .. 1.2 .. . . 75
y ,1 1o1
� : (o l
1 .0
75
Radiation cell model. Start-up with low pellet mass capacity and
MODELLING AND ELEMENTARY DYNAMICS
2.1 2.0
Parameter! ·
1.9 1. 8
1. 7
R
1.5 1.4
Feed condilJon�
Ql fil l
l n u i � l conditions:
1. 3
1 2
1 �4-0min
1 � z -Omin
1. 1
1. 0
li r l!r
a,
1. 6
Ysj
M H
0
10
20
30
40
C ell nu mber
FIGURE 3.49 Radiation cell
j
model.
50
60
70
335 =
•
3 .9 1 9 1 3S1
: -4 . ) } 7 l 1 0� = JS.ll = 0 6
:: 3 .456 "" 0. 1 6 3 � =
0 "' Dz = O
lr
)'� (OJ
c
•
1
10 2
LO. YF
c:
Yj fO) "' 1 .0 \j iOJ :: 11 0 1 = 1 .'2 ... .. H
l .O
75
Start-up with high
�
capacity
and
/JA
= 0.
the heat capaci ty of the bed. It has been shown in sec tion 3. 1 .2 that the ' particle mass capac ity may be comparable to or even greater than the heat c apacity . In practice , therefore, concentration disturbances travel wi th a finite speed. For the sake of c larity we shall as su me for the momen t neg l igib l e adsorption heat release. The finite mass and heat c ap acities of the bed give rise to some important interactions which are usually overlooked by many investigators. Figure s 3 .47-3 .49 show the e ffe c t of incre as i n g the pellet mass c apacity parameter a3 on the transient temperature profile s for the solid phase . The initial effect is to del ay the time for i gn ition (c . f. compare curves for t = 6 mins ) . However, once ignition does occur, the temperature rise involved may become e xtreme . Figure 3 .49 shows the extremely interes tin g result of very peaked temperature profil es wi th tran si ent " hot- sp ot s " far e x c eeding the ad iab atic temperature rise. This important behaviour is not predicted by models which ne gl e ct th e adsorpti on mass capacity of the catalyst surface. Q uite obviously, this type of behaviour , if occuring in practice, woul d present considerable problems during start-up and dynamic models which do not pred i ct it are ob v iousl y of no real valu e . 2. Effect of adsorption heat release
et
heat release, as discussed earlier for the ear cells e
The e ff c of ads orpti on simple model, is to pr t the ly of the be d at early times and therefore cause ignition to take place nearer to the entrance of the bed. This is shown very c l arly by the c ompari s o n of Figure 3 .48 and
cell
Figure 3 .50.
ehea
Figure 3.48 neglects adsorption heat release. Ignition occurs at t = 6.0 mins towards the tail-end of the reactor. The reaction zone migrates
S . S .E.H. ELNASHAIE and S . S . ELSHISHINI
336 2-1
Parameters
2.0
1.9
:
1. 8
Ysj
1. 7
l6
0. 4
M H 1i y llr �. R OJ a1
L5
1. 4
Feed coadilions
L3
lnilial condl lioR!l'
,_ 2 1. 1
=
•
= =
= �
= ""
"" ::r
0 . 1 63 3 .9 1 9 2.352 4.JJ1 X J OS 1 8.33 0.6 0.3 3.456 X 1 0· 2 0. 1 2 G:z z O =
xf
.;
I 0. YF
•
Yj {o) • 1 .0 fi.O j - 1 .2 • . . . . .75
Y";i (o)
xj (o)
=
1 .0
C PU nu m bPr j
FIGURE 3.50 Radiation cell model. Start-up with low pellet mass capacity and f3A = 0.3.
towards the bed entrance with the steady state being reached after two hours. In Fi gure 3.50 where adsorption heat release is taken into account, the ignition occurs at 2.0 min close to the bed entrance ( 1 Oth cell). The reaction zone mi grates toward the entrance with the steady state reached in 50.0 mins. It is interesting to examine the effect of adsorption heat release on the example with high mass capacity and trans i en t temperature "runaway" (Figure 3.49) . Figure 3.5 1 shows trans i ent temperature profiles for this
case No temperature "runaway" now occurs. Igni tion takes place very close to the entrance. .
2.1
Parameters :
2.0
19
1. 8
1. 7 Ysj
1. 6
1. 5 1.4
Feed conditions
1. J
�A R a, a.1
Initial conditions:
1. 2 L 1
1 0
a.. M H a y llT
0
10
FIGURE 3.51
20
JO
40
C • l l nu m ber
50
60
70
""' •
=
"'
=
=
= •
=
•
0. 1 63 3.9 1 9 2.352 4 . 3 3 7 x J o5 1 8 . 33 0.6
0.3 3 .456 , 1 0 · 2 0.4 a:;. • 0 :t r
•
=
1 .0,
YF
=
YJ {o) = 1 .0 0.0 J � 1 .2, .. ..,75
Yaj (o) xj (o)
•
1 .0
75
Radiation cell model. Start-up with high mass capacity and fJA = 0.3.
MODELLING AND ELEMENTARY DYNAMICS
337
The above examples show very clearly the significance of adsorption mass capacity of the catalyst particle and heat of adsorption associated with the chemisorption process which is an inseparable part from almost all gas-solid catalytic reactions. 3.
Summary
The above series of examples demonstrate clearly the complex character of the dynamic behaviour of the adiabatic fixed bed reactor. Dynamic behaviour is strongly influenced by the radiation coupling between cells. The effect of radiation is to cause the ignited zone to "creep" towards the reactor inlet and the time required to reach steady state is considerably increased. The effect of the catalyst particle adsorption mass capacity, which is neglected in many published work on the dynamics of catalytic systems, is shown to have very important effect within the practical range of parameters . Increasing the pellet mass capacity tends to cause the appearance of severe transient "hot-spots", exceeding the adiabatic temperature rise. The inclusion of adsorption heat release in the model also has a strong influence on dynamic behaviour. This caused the early cells to be preheated and promoted ignition nearer to the bed entrance. Quite complicated behaviour may result when the effects of pellet mass capacity and adsorption heat release are studied together, stressing the need for accurate determination of these parameters. II. Dynamic effects of feed disturbances (creeping reaction zone)
When the reactor is at steady state any change in the feed conditions will cause the reactor to shift to another steady state corresponding to the new feed conditions. If the disturbance is removed and feed conditions are returned to their original state the reactor may or may not return to its original steady state. If the original steady state is globally stable (unique}, then the reactor will return to its original steady state when the feed disturbances are removed. On the other hand, if the steady state is not unique, the disturbances may take the system into the domain of attraction (region of stability) of another steady state and therefore when the disturbance is removed the reactor settles to this later steady state rather than the original one. Several authors (Rhee et al. , 1 97 3 ; Vortmeyer and Janhel, 1 972; Vanderveen et al., 1 968; Padberg and Wicke, 1 967 ; Elnashaie et al. , 1 974) have found, both theoretically and experimentally, that feed disturbances cause the reaction zone to creep forward towards the reactor exit and backward towards the reactor inlet, dependin g on the position of the ignition point and the signs of the disturbances. A decrease in feed temperature and conc e ntrati on causes a "forward creep". Many investigators have reported creep with constant velocity. We
S . S . E . H . ELNASHAIE and S . S . ELSHISHINI
338
show in this section that, in general, the creep is not of constant velocity but that it slows down as the new steady state is approached. Obviously, the velocity of creep at the new steady state must be zero. For practical control purposes, we will present to the reader relations that predict the direction and rate of movement of "creep" in terms of the system parameters, the magnitude and duration of feed disturbances and the observable state of the effluent. We shall first present the effect of different parameters on the "creep velocity" with special emphasis on the effect of catalyst particle adsorption mass capacity and adsorption heat release. Secondly, we shall present an expression of the "creep velocity" on the same lines as that of Rhee et al. ( 1 973) and Elnashaie and Cresswell ( 1 973 or 1 974). We shall then present practical applications and suggest a safe algorithm for the control of such reactors. The effect of intraparticle mass and heat transfer resistances on creep velocity will also be presented. I. Effect of adsorption heat release
Figures 3 . 52, 3 . 5 3 , 3 . 54 show the creeping profiles for a step decrease in feed temperature, a step decrease in feed concentration and simultaneous step decreases in feed temperature and concentration. For these disturbances the reaction zone creeps forward towards the exit of the reactor. The duration of the disturbance is 20 mins and on removing the disturbances the system returns to its original steady state in all three cases. If the disturbance is sustained longer or/and the bed is shorter, the reaction zone may be "blown out" of the reactor. Forward creep is the result of quenching of particles in the reaction zone. Quenching is accompanied by a build up of reactant concentration 2.1
2 .0 1. 9
- · - · - {JA
1. 8
= 0 ·0
F�r-d c ond111ons .. � :: 1 · 0
1.7
Ysj
a,. = 0. 1 63 M = 3 .9 1 9 H "" 2 . 352 ll "' 4.337 � 105 ,. 1 1 . 3 3 "'( 1'1 · • 0 . 6 R .. 3 .4.56 x 1 0-1 CIJ • 0 . 1 2 Ql .. 0:1 = 0.0 Initial conditions: Ys,i (o) = Y•j&s xr(o) =-= Yju j • l .L. ..• JS
P u a m e t er.� :
-- flA = O · J
1.6
l.S
1.4
1. 3
l. 2 ll
l .O
0.9
0
10
20
30
40
C e l l nu m b e r j
so
60
70
75
FIGURE 3.52 Feed temperature disturbance. Effect of adsorption heat release on the velocity of the reaction zone.
MODELLING AND ELEMENTARY DYNAMICS 2.1
2.0
P tr a m c t e r s :
1. 9
--
1. 8
Ysj
1.7
- · - · - {JA
1. 6
: 0· 0
1. 3
2
1. 1
40
0
20
so
60
H
� R
Ql
.-� 0·9 �
1. 4
...
M
(i 1
F � eo d cond i t i o n s : ' 1 ·0 • YF
1. 5
1.
/JA : 0 · 3
339
GJ
lnieiat condition a:
t < min)
=
=
• =
= � =
=
•
0. 1 6 }
3. 9 1 9
2.332
�
0.6
x
4.337 18.)3 3.456
0. 1 2
"7 = 0
10� J (J- 2
xfCO) xj55 =
l'sj (0 >= 's;. j : 1,2,- . • 75
70 7 5
C e l l nu m ber
FIGURE 3.53 Feed concentration disturbance. Effect of adsorption heat release on the velocity of the reaction zone.
in the quenched particles. When ads orption heat release is c on s idered the concentration build up is accompanied by a bui ld up of heat. This will have the effect of sl ow ing down the decre ase in temperature and consequently the rate of forward creep of the pro file By a similar argument, ad sorption heat release also slows dow n the rate of "back ward creep". In general, the effect of adsorption heat release is to stabilize the po siti on of the reaction zone w he n the system is subjected to feed disturbances. This effect is seen most clearly in Figures 3 .52, 3 .53 and 3 .54. ,
.
2.1
21)
� ...- � rU 'J � 0·9 � ·
0
tO I l�ir�)
0
20
Um1nl
1. 8
I
/ i
1.7
Ysj
1. 6
1.5
1. 4
i J-'--1:::-!----'--i
1. 2 11
1. 0
0.9
10
20
30
C•ll
40
numb�r
j
so
/
/ -11. =0·3
i
1.3
,. 0. 1 63 .. 3 . 9 1 9 H • 2 . 3 .52 li "' 4.331 :t ) 1)5 y :::> 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
:
ib--c *b--e :: 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)
I0-2
";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 take-up 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 ignition-extinction 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
and,
where 811 is the temperature decrease at the bed exit at time t = t1 (see curve d in Figure 3.58a,b ). Substituting these relations in 3 . 1 88 gives,
(3 . 1 9 1 )
Obviously N(t1 ) < N (O+) if equation 3 . 1 90 is satisfied. If we define,
and observe that B(t) increases as we approach the new steady state, then we can conclude that N (Q+) is the highest velocity during the transient period. N(Q+) therefore gives an upper bound on the velocity of the reaction zone. Now, if condition 3 . 1 90 is violated we obtain negative value of N(t), which indicates a wrong directional movement of the reaction zone. This is discussed in detail in the following section. 3. 2.3.3
Stability analysis and wrong directional creep
The anaiysis in this section is based on formula 3 . 1 88 and therefore, is subject to the same restrictions implied in its derivation.
346
S . S .E.H. ELNASHAIE and S . S . ELSHISHINI
1. The initial direction of creep The condition described by equation 3 . 1 90 is not always satisfied. This point will be discussed here in detail. First we shall show how the condition in equation 3 . 1 90 was obtained. Write 3 . 1 90 in terms of B (t) at any time t as follows:
(3. 1 92)
where, B(t)
=I I [ X] [ Y]
Let us examine the initial sign of N(t). If the disturbance is a decrease in feed conditions, we shall have, (3. 1 93) Therefore A (O+) in equation 3 . 1 92 is positive. If, (3. 1 94) then C (O+) in equation 3 . 1 92 is positive, and therefore N(Q+) is positive. The creep is in a forward direction towards the new steady state. However, if, (3. 1 95 ) then C (O+) can be either negative or positive. An initial creep may be either backward or forward. When C (0+) is negative the profile is initially creeping i n the opposite direction away from the new steady state. This will be the case if, (3 . 1 96)
MODELLING AND ELEMENTARY DYNAMICS
2.
347
Velocity of "creep "
Let us now tum to the velocity of creep. Differenti ating 3 . 1 92 with respect to B (t) we obtain, dN(t) H A2 - A1 dB(t) = a4 (1 - A2B(t)) 2
(3. 1 97)
where, M A, = f3r H
a3
A2 = (f3r - f3A ) a4
For a step decrease in feed conditions B (t) has its lowest value at the initial instant t= Q+ , and then increases as the system approaches the new steady state. From equation 3 . 1 97 we recognize two situations. (i) The
first
is when the fo llowing condition is satisfied,
(3. 1 98)
This condition makes dN(t)/dB (t) < O. Therefore as B (t) increases N(t) decreases. Thus N(o+) is the highest creep velocity during the transient period and the creep velocity decreases as the new steady state is approached. Condition 3 . 1 98 can be written as,
which is condition 3. 1 90. This condition is also sufficient to guarantee "forward creep" towards the new steady state. (ii) The second situation arises when the following condition is satisfied, (3. 1 99)
In this case we have dN(t)ldB (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 wrong-directional 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 wrong-directional 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 wrong-directional 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
••
1-S
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
1-3 1 ·2
Ot�;;;;;;;;;;�=;:::::..,.. -- r------. ---.---l
=
0.29
xj (o)
y,J (o)
:: xj1 =
)'1j 1 1 1
1 ·1
1·
0-95-f'
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
1-6
= 0
! 0-41 (�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 .<1J
1 ·8 I : 20 · 0 min , s t . s l .
1 -7
;,.-
,..
:a:
=
=
.:::
=-
I : O · O m in
..
(05
3 .• , 6 1 lQ·l J 919 2 lll 0. 1 63 0.29 (Hable)
�J C o ) .: xju 11
X
ll
&
0
Ys j ( o i •Y,j ..
1·6 1 ·5 1 ·4 1·3
1·2 1·1 1 ·0
0
4
8
12
24 16 20 j , C4i'll number
28
32
35
FIGURE 3.60b Effect of catalyst pellet adsorption mass capacity on the stability of the unforced system. Intermediate value of a3.
354
S . S .E.H. ELNASHAIE and S.S. ELSHISHINI
3. 2. 3. 4
The effect of intraparticle mass and heat transfer resistance on the velocity of creep of the reaction zone
In all the preceding sections dealing with the fixed bed catalytic reactor problem, a lumped parameter model for the catalyst particle was used, in which intraparticle resistances were neglected. Clearly this is not always a realistic assumption. The effect of intraparticle resistances on creep velocity should be considered. In order to present the effect of intraparticle resistances a distributed parameter model for the catalyst particle must be used. The distributed parameter model equations which are parabolic partial differential equations in this case, should be solved together with the fluid field equations. The collocation method, due to Villadsen and Stewart ( 1 967) seems to be the most efficient method for solving the partial differential equations of the catalyst particle. The purpose of this section is to present a formula for the computation of initial velocity of creep using a first order approximation (single interior collocation point) and clarify the effect of intraparticle resistances on this velocity. We will use a radiation cell model similar to the one used in the precedin g s ection s . 1. The dynamic model A tran s ient mass and heat balance over the j th cell assuming the
catalyst pellet to be spherical, gives a set of differential equations describing the dynamic behaviour of the system.
axat (a2aoix . aax . ] ax.a ax.a(J) 3a4Nu aar.:= [ aaw2r.� aa�r. )
Catalyst phase mass balance
2 1 = -1 + -1 - 3 aSh exp ( - y / Y ) X 3 as Sh J J (J) ro
(3 . 202)
with boundary conditions,
__
J
at CO = O,
co
=
0
1 = Sh ( X8�- - XSJ ) -
and at ro = 1 .0,
·
( 3 .203)
Catalyst phase heat balance +
2
W
+ 3 afJ TNu · exp (- yj� )Xj + 3 0:JfJA · Nu
aax/
(3 .204)
MODELLING AND ELEMENTARY DYNAMICS
355
with boundary conditions, a y.
at ro = 0,
1 =0
_
o ro
at ro = 1 .0,
Fluid phase mass balance a1 Fluid phase heat
dX8�. = M( X8�.- 1 - X8�. ) - (X8·J - XSj· ) df
(3.206)
--
balance (3 .207)
where,
KR
Sh = _2__!!_ De
Nu =
'
hR
.Ae
P
-
ro = r I RP
R = E'F' c; TJ I 2h,
The subscript B is used to indicate bulk gas phase conditions and all other symbols are the same as in the preceding sections .
2.
Approximate model
Applying the modified collocation method (Villadsen and Stewart, 1 967) to the catalyst pellet eq uations (3.202-3.207) using one interior collocation point and after some lengthly but straightforward algebraic manipulations, we obtain the following differential equations for the dynamics of surface temperature and concentration in the j 'h cell, _
a'l
dXsj- = ( X8 . - X . ) - F
-
. dt
ij
SJ
J
(3 . 208)
356
S . S .E.H. ELNASHAIE and S . S . ELSHISHINI
and,
where, a3 = _
a4
=
Sh + 3. 5
3.5
a3
Nu + 3. 5
3. 5
FJ = a · exp _
a4
-
(-y I �1 )
[Sh
- (XsJ - X8) + XsJ 3. 5
]
(3 .2 1 0)
4 �J = - - [ Nu · Y81 - ( Nu + 3 . 5) �J + R · Nu ( �J+ I - 2 �1 + �1_1 )] 3. 5
- ( 1)
4
4
Assuming no heat losses from either ends of the reactor, the above equations are applicable to the boundary cells (j = 1 , N) by setting Ys0 = Ysl• dYsofdt = dYs l ldt, YsN+I = YrN. dYrN+ddt = dYst/dt. The dynamic behaviour of the system is described by equations 3 .206--3 .2 1 0 together with the appropriate initial conditions. Although the equations look somewhat complicated, the expression for the velocity of creep is similar to that obtained in section 3.2.3.3 except that a3, a4 are replaced by a3 , a4 • For gas- solid systems, the interstitial gas mass capacities a 1 , a2 are negligibly small and in such systems the number of cells travelled per minute is given by the following expression,
(3 .2 1 1 )
where,
N ( t) = velocity of the reaction zone (cell s/min)
[X] = exit concentration - feed concentration (dimensionless) [Y] = exit fluid temperature - feed temperature (dimensionless)
MODELLING AND ELEMENTARY DYNAMICS
357
Since [X]/[Y] is always negative, it is clear from equation 3 . 2 1 1 that an increase in a3 will cause an increase in the velocity of creep. From the definition of a3 and Sh it can be seen that as the intraparticle diffusion coefficient (De) decreases, a3 increases. Thus the velocity of creep increases as the intraparticle diffusion coefficient decreases. Similarly we can show that as the particle conductivity (Ae) increases, the velocity of creep increases. Of course, De and Ae will affect the steady state solution of the system, but in this analysis we are interested in the velocity of creep regardless of the relative position of the reaction zone along the length of the reactor. The effect of De and Ae on the position of ignition can be obtained easily from a steady state analysis. By following the analysis given in section 3 . 2 . 3 . 3 we obtain the expression given below for the initial velocity of creep, (3 .2 1 2)
where, (3 . 2 1 3 )
N (O+ ) = initial velocity of creep (cells/min) XFi = dimensionless feed concentration before disturbance YF.1 = dimensionless feed temperature before disturbance Xp;2 = dimensionless feed concentration after disturbance Yp;2 = dimensionless feed temperature after disturbance
From the preceding discussion and the results in Table 3 . 1 1 , it is clear that Sh and Nu have opposite effects on the velocity of creep. Table 3.1 1 Effect of Sh and Nu on the initial velocity of creep. Computed from Equation 3.196. Pr = 0.6, M = 3.919, H = 2.352, YF.1 = 1. 0, XF.1 = 1. 0, YF.' = 0. 9, XF. = 0. 9, fJA = 0, a3 = 0.0l, a4 = 0.4. '
Nu
Sh
5 5
500 250 250 0
2
0
N (0 +) cells/min 1 .6 1 5 0.695 1 .565 1 . 090
358
S . S .E.H. ELNASHAIE and S.S. ELSHISHINI
The velocity of creep predic ted from the negligible intraparticle resistance model may be higher or lower than that predi cted from the more realistic distributed parameter model depending upon the relative effect of Sh and Nu. For example, in Table 3 . 1 1 we find that the negligible i ntrapartic le resistance model (Sh = 0, Nu = 0) predi cts a velocity higher than that corresponding to (Sh = 250, Nu = 5) b ut lower than that corresponding to (Sh = 500, Nu = 50) and (Sh = 250, Nu = 2). The neg ligi ble intraparticle resistance model therefore, cannot be used to predict an upper bound for the velocity of creep when i ntrapart ic l e resistances are appreciable. The simple analysis pre sented in this section po ints out to the important effect of intraparticle resistance on the velocity of creep. 3. 2.3.5
Static and dynamic bifurcation behaviour of the reactor
In this se c ti on we present to the reader the use of bifurcation
theory and numerical continuation techniques in the investigation of complex static and dynamic characteristics of fixed bed catalytic reactors modelled by a heterogeneous radiation/conduction cell model. The model presented takes into account the thermal coupling between adjacent reactor cells due to radiation and conduction. A rich variety of interesting static phenomena is presented including mu ltipl ici ty with a large number of steady states. A case with a Hopf bifurcation point from which a periodic branch emanates is presented. The peri odic branch terminates with a homoclinical infinite period (IP) bifurcation point. Wrong-way behaviour (which is different from wrong-directional creep pre sented and discussed earlier) and sensi tiv ity of the final steady state profile to initial conditions are also presented and discussed. Throughout this section, the system parameters are assigned the following value s (except otherwise stated): a1 = 0. 0 ;
f3A = 0.0; M = 3. 9 1 9;
0?_ = 0. 0;
f3r = 0. 6;
a = 433700;
a4 = 0. 1 63 H = 2.352 r = 1 8 . 33
In the parametric sensitivity analysis, the adsorptive mass c apac ity parameter for catalyst pellet (a3) is varied from 0.25 to 0.5, the effective radiation parameter (R) is v ari ed from 0.0 to 1 0.0 ( w i th a typ i cal value of 0.03456), while the effective conduction parameter (aH) is varied very widely from 0.0 to 1 00.0. Owing to the very large number of parameters and the infinite number of possible initial conditions, the static and dy nami c bifurcation behaviour of the reactor is presented for the variation of a limited
MODELLING AND ELEMENTARY DYNAMICS
�
0.:
t
.... u�
1 .�
. .t ....
J
C"o
� .. :9;
,.
1.4,
1 .35
l.:t!
I.U
.....���:::,..:�
0. LOU
..•.
i ::::
, �,
1 ' ..
J
� , ...
�
1 .81
Q; , ..7 11 1 .11
. ...
359
�v, ( b)
0.99J2
FIGURE 3.61 Static bifurcation diagrams of solid phase temperature versus feed temperature for cell number 10. (a) The overall bifurcation diagram. (b) Enlargement of upper section of Figure 3.61a. (c) Enlargement of lower section of Figure 3.61a. (d) Enlargement of lower section of Figure 3.61c.
number of the governing sy stem parameters namely the feed temperature ( YF), the ads orption parameter (a3) i n addition to chosen cases of initial conditions of temperature and concentration. I. Static Bifurcation Behaviour
The arclength continuation scheme described by Kubicek and Marek ( 1 983) is used to construct the static bifurc ation di agrams shown in Figure s 3.61-3 .64. The bifurcation p arameter i n Figure s 3.61-3 .67 is the feed temperature ( YF). Figure 3 .6 1 shows the effect of YF variation on the solid phase temperature ( Y5) for cell number 1 0. The bifurcation diagram exhibits an array of hy stere sis (s-shape) behaviour p attern s with ignited intermediate and high - tempe rat u re s t eady states and extinguished low temperature steady states. Below a feed temperature of about 0.975 an d above a value of about 1 .0 1 only one s teady state exists. In between a richness of ste ady state mul tipl i ci ty patterns exist. In particular, a maximum of 29 ste ad y states is shown for values of YF in the neighbourhood of 0.982. As ex plai ned later, this maximum of 29 steady states can increase yet further if the degree of coupling (or feedback) between cells is reduced as a result of say, a reduction in the
) b ( ; n d B=;e .... -..:: ;
360
!..; �
j
�
�
�
S .S.E.H. ELNASHAIE and S . S . ELSHISHINI
08
SLP,
�
0.1
::
0.4
') 3
0.2
(a) Box enlarged in ( d )
c--""
o.o
0.9 7
0.9�
0 98-M
0.99
YF
!
�
.t
� �
�
1.,
1., !(
j
(�)
�
� O.iBOS0 . 5 7
J
IHH
0 -')7�
IU76 r,-
0.971!1
0 979
0.989
uae
0. 98 7 0
�
M6� Q.M�
( b)
0 ...,
0 9111
0 91!11
o 99S
·J esc o c•�
SLP,
/
1 .000
0 0)�
0.0)0
0 01'1 0 01�
O.QTO
c 00�
c ooo 0.970
0
g':$
0.�1!10
0.5155
r,-
0.9ta
0.99�
: .000
FIGURE 3.62 Static bifurcation diagrams of solid phase concentration versus feed temperature for cell number 10. (a) The overall bifurcation diagram. (b) Enlargement of upper section of Figure 3.62a. (c) Enlargement of upper sec tion of Figure 3.6lb. (d) Enlargement of lower section of Figure (2a). R = 0.03456, 0-,t ::: 0.0.
level of radiation and/or conduction. It is important to determine the critical feed conditions below which autothermal operation cannot be sustained. Figures 3.61 b,c,d are enlargements of specific crowded regions of Figure 3 . 6 1 a. It is of interest to refer to the almost indistinguishable and overlapping steady states in the lower sections of Figure 3 . 6 l a as shown in Figures 3.61 c,d. Figure 3.62a and its enl argements 3 .62b,c,d show the corresponding patterns of multiplicities for the solid phase concentration as a function of YF for the same cell number 10. The source of this peculiar static bifurcation behaviour cannot be attributed to a single physical phenomenon. The nonlinear kinetics , h igh activation energy, thermal coupling of reactor cells and the heterogeneous nature of the reactor are all interacting and contributing factors to this complex behaviour. Adaje et al. ( 1 990) noted that when local steady state multiplicity is possible, i.e. when every catalyst pellet may acquire two or more solutions in a certain range of operating conditions, then the global solution may incorporate transitions from one branch of local solution to an other . An u nc ountable number of steady state profiles is
possible in such a reactor when heat conduction (or any other feedback mechanism) through the solid phase, is ignored (Liu and Amundson,
1 962).
MODELLING AND ELEMENTARY DYNAMICS
361
To be more specific and clear, the single catalyst pellet model considered in this investigation can give up to a maximum of three steady states in the range of parameters considered. However, for a simple cell model with no feedback mechanism (R = aH= O), the number of steady states for the entire bed can be quite large; for if cell n has 3 steady states then it is possible that cell n + 1 will have up to 9 steady states (it may have 7 or 5 or 3 steady states), which means that it is possible that cell n + 2 will have up to 27 steady states and so on. Thus although the single catalyst pellet can have a maximum of three steady states, the entire bed can have a very large number of steady states, also different cells can have different number of steady states. However when there is coupling (R > O and/or aH > O), then there is a feedback mechanism in the bed which tends to decrease the number of steady states below the large number corresponding to R = aH= O, and also the phenomenon of different numbers of steady states for the different cells disappears because all cells are coupled through radiation and/or conduction and all cells will have the same number of steady states. An interesting experimental investigation of the relation between the multiplicity patterns of a single catalyst pellet and a fixed bed catalytic reactor for ethylene partial oxidation has been published a few years ago by Adaj e and Sheintuch ( 1 990). More recently Wijngaarden and Westerterp ( 1 992) investigated the effect of the thermal stability of the single pellet on the design of fixed bed catalytic reactors. The multiplicity phenomena in cells 1 , 5, 1 5 and 20 (reactor exit) for a case with R = 0.03456 are shown in Figures 3.63a-d respectively. As discussed above, the number of steady states is equal for all cells. However, the regions crowded with steady states change from one cell to the other depending on the change of the degree of conversion along the reactor length. The multitude of steady states for the reactor inlet and outlet cells is limited to the lower and upper sections respectively, of the bifurcation diagram as shown in Figures 3.63a,d. For the intermediate sections of the reactor, a large number of steady states are observed in the lower as well the upper regions of the bifurcation diagram (cells 5, 1 5 in Figures 3 .63b,c). Figure 3 .64 shows the parametric sensitivity of the static bifurcation plots of solid phase temperature versus feed temperature to the radiation parameter (R ) for cell number 1 0. In this diagram heat conduction through the solid phase is ignored ( aH = O.O ) . For R = O, the number of steady states is quite large as discussed above. From Figure 3 .64, it is clear that as R is increased, the number of steady states decreases, the uppeflllO St ignition temperature is lowered, the reactor becomes ignitable
at progressively lower feed temperatures and the region of multiplicity gets narro wer. The mutitude of static limit points shown for a typic al
S.S.E.H. ELNASHAIE and S . S . ELSHISHINI
362
,. '.
Cell No. 1
0.97
...
'"
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=
�
<:! -<:: �
/
�
� 1 .2
�
3.0 10.0
Yp
FIGURE 3.64 Dependence of the solid phase temperature on the radiation parameter (R) for cell num ber 10. a, = 0.0.
MODELLING AND ELEMENTARY DYNAMICS
363
c--: -�
c.u No. l
"·Q..------.., -�1.1' IO 1,\) ��·� l . OIC' 1.�
Y,-
Y,
,------
7 �, �
(d)
Ce!I No. l5
r,
r,
FIGURE 3.65 Static bifurcation diagrams of solid phase temperature versus feed temperature for different reactor cells. R = 0.0, fZH = 0.0.
value of R = 0.03456, are reduced when R is increased to 0.5, to two
static limit point s (SLP1 and SLP2). As R is further increased to 3 .0 (see Figure 3.64), the two static limit po int s move with different speeds to higher values of feed temperature s and thu s reactor igniti o n occurs at higher values of feed temperatures. At a value of R = I 0.0, the reactor exhibits a u nique stead y state for all valu es of feed temperature s. Figures 3 .65a-d show the static bifurcation diagrams for cells 1, 5, 1 0 and 15 respectively when R= aH = 0.0 with no coupl i n g between cells ( i e no feedbac k mechanism). Cell number one (Figure 3.65a) has only three steady states bu t the number of steady states tends to increase progre ss ivel y down the tube axis as shown in Figures 3.65b-d for cells 5, 10, and 1 5 respectively. Figure s 3.66a-d show the static bifurcation diagrams for the same case in Fig ure 3.65 but with R = 3 0 for cells 1 , 5, 1 0 and 1 5 respectively. These diagrams (in contrast to the corresponding diagrams in Figure s 3 .65a-d) show c l earl y that the number of steady states (3 in this case)
..
.
does not change from one cell to the other when there is coupling between cells with R > 0. Figure 3.67 s how s the structural changes on the bifurcation di agram as a result of the inclusion of both radiation and conduction. The radiation parameter R is maintained constant at 0.03456 while aH i s
3 64
S . S .E.H. ELNAS HAIE and S . S . ELSHISHINI
( b)
( a)
Cell No. I
�
�
� /
3
Ceii No. ! O
"
Cell No. 5
1
'·",¥b '
�---;-:� :----:-;
/�I
l] . .
Ceii No. I S
. .
.
.
.
.
..
FIGURE 3.66 Static bifurcation diagrams of solid phase temperature versus feed temperature for different reactor ceUs. R 3.0, an 0.0. =
=
2.2
,:: ..-
2.0
� 1 .8
� "
�
1 .6
I 2 1 0 0 . 3 ��rr,..,-.,..,-,,.,..,rr�"rr""-,"""r 0 . 90 1 . 00 0 . 80 1.10 1 .20 Yr
FIGURE 3.67 Dependence of the solid phase temperature on the conduction parameter an for cell number 1 0. R 0.03456. =
MODELLING AND ELEMENTARY DYNAMICS
(a)
J ,.
=
YF
L. l
0.7
�
YF
� 1 .1
J ,.
(b) =
0.72
R
: 0.9
(d)
YF
Ju :...
.. ...
I ..
l . � �
"
R
�·
J, .
�
!u
..
R
I
YF
(e) =
1 . 02
� R
(c)
r, . o.n
1.4
l ,. l
�
.. 1:::::::===::.....-�
>:
'!>.• , ,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 x-axes in Figures 3.68a-f 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 = 1-20. 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 a-d 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 .72a-d 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 1-20. 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 = 1-20, 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.
=
I 4
J=
, . Timt, min
1.6
l . 2, . . . , 20
(c)
�---- a1 •
D.lll6
______ IIJ) .. 0.3117
' . Trnw, mi n
1 . ?0
!, ...,.,..,,,.,� ..., ,-., ,� , ...,.��..... ,.... ,.... .. ,.. .. -.� .� .. rr,.... , min
FIGURE 3.73 Dynamics of cell number 10. (a) Effect of different initial conditions. (b) Effect of different values of a3. (c) Example for the sensitivity to a3 variation. R = 0.03456, CZn = 0.0, YF : 1 .0.
MODELLING AND ELEMENTARY DYNAMICS
369
( a)
""
'"' -\-,,....,... ..., .,,... .., ,.-= ., n -,,,-. , .,.,,..,. ,.:--:t ...., ,.--, ,.. niW, m.i..D
FIGURE 3.74 (a)-(c) Effect of o3 on the oscillatory behaviour of cell number 10. (d) Variation of the average conversion with o3 for the periodic branch. R = 0.03456, lXJi 0.0, YF = 1.0. =
(a change of about 0.03%) results in a different final steady state. The initial conditions used for the computations for Figure 3 .73c are flat initial conditions of x; = 0. 0043 and :Yfj = 1 .9307 for j = 1-20. The effect of the model parameter a3 on the magnitude and shape of the dynamic stability in the reactor represented by cell 1 0 is shown in Figure 3.74a. The initial conditions used in the computations for Figures 3.74 1 --c are the flat conditions x; = 0. 0043 and Y�; = 1 .9307 for j = 1 -20. The results suggest that there is a Hopf bifurcation point between a3 = 0.3 and a3 = 0.4 giving birth to periodic behaviour which is getting stiffer as a3 increases to 0.47 and terminates homoclinically near a3 = 0.5 . The corresponding dynamic patterns for the solid phase concentration are shown in Figure 3 .74b. It is of interest to find out whether reactor operation in the region of periodic behaviour has an advantageous or detrimental effect on the reactor performance. For this purpose a plot of conversion versus ti me i s presented in Figure 3 .74c. The horizontal broken line shows the conversion (based on gas phase concentration) for the ignited steady state before and after the limit cycle region. It is clear from Figure 3 .74c that the amplitude of the oscillation increases as a3 increases. Figure 3.74d shows that the average conversion decreases as a3 increases and the homoclinical termin ation approached .
S . S .E.H. ELNASHAIE and S . S . ELSHISHINI
370 4.0
Homo<:linical Te rminallon o r Limit
,_______
,:; �
[i
:;
�
� �
ti ;_: � �
Hopf Bifurcation
""'
I
Point
iI I
. - .- .-..\I / 1.1PPER STATU.: I!Ol,(ITION BRANCH II
2.0
•
..
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0 0
I "' :
Periodic Solution Bran�h
3 0
Cydu
/
MIDDLE A.'olD LOWER STATIC SOLl.TION BRA!Iot :tt �
-f-rn�..,.�,.,r-rr<�rrrr�rr>��rrrnT..,...�orrr�,.,..,.,
0. 1
0.2
0 3
0.4
0.5
GJ
0.6
0.7
0.8
0.9
FIGURE 3.75 Stationary and periodic solution branches for cell number 10. = 0.03456, lXH = 0.0, YF = 1.0.
R
Figure 3 .75 shows the static and dynamic bifurcation with a3 as the bifurcation parameter for cell number 1 0. One static branch is in the upper ignited region and two (al m ost coinciding) static branches in the lower exti nguished region. A Hopf bifurcation point occurs at about a3 = 0.32 and the periodic branch emerging from thi s Hopf bifurcation point terminate s homoclinically (infinite period bifurcation) when the oscillations to u ch the middle unstable saddle type s teady state at an a3 value of about 0.48.
Transient response to step disturbances in feed temperature In this section the transient responses to step disturbances in YF are examined. In parti cular we show the influence of the direction of the step di sturbance, the initial c onditions , the mass adsorption parameter a3 as well as the radiation and conduction parameters R and aH. Fig ures 3 .76a,b show the transient responses to a step decrease in YF from 1 .2 to 0.97 (Figure 3 .76a) and a step increase in YF from 0.97 to 1 .2 (Figure 3 .76b), when radiation and conduction effects are ignored. The initial conditions for Figures 3.76a and 3 .77a for x; are shown in the small box inside Figure 3 .76a and those for Y�J are as indicated on Figures 3 .76a and 3.77a for t = O.O. The initial conditions for Figures 3 .76b and 3 .77b are x; = 0.96 and Y,1 as in di c ated on Figures 3.76b and 3 .77b at t= O.O. As shown on the bifurcation diagram of Figure 3.64 (for
MODELLING AND ELEMENTARY DYNAMICS
, , , � �� /,� {L:/ � ·-· c• -
\ :::< �-� - - - -- - - - - - - - - ---=-=-=-=
0
2
4
6
8
10
1 �
,,
Cell NumMr
16
15
20
37 1
(b) ;::
�
� .:: 'l!
�
:;
0:: � �
2.6 2.1 1 .6
�-----t�_o-� Qy
1.1
_ _ _ _ _ _ _ _
a.• +.-����� 0
�
- _ _t-!f' - - -c
-
2
4
6
B
10
12
Cell Number
14
16
18
20
22
FIGURE 3.76 Transient responses to step disturbances in YF for cell number 10. (a) Step decrease in Yp from 1.2 to 0.97. (b) Step increase in Yp from 0.97 to 1 .2. R = 0.0, � = 0.0, a3 = 0.45.
R = aH = 0.0), feed temperatures of 1 .2 and 0.97 refer to ignited and quenched states respectively. No wrong way behaviour is observed during the step decrease test because the initial conditions do not correspond to an intermediate level of conversion. It has been shown by Mehta et al. ( 1 98 1 ), Pinjala et al. ( 1 988) and Chen and Luss ( 1 989) that a transient temperature rise occurs only in reactors with appreciable conversion. A large transient increase in temperature caused by the sudden increase in YF is shown in Figure 3 .76b. Such transient surges in temperature, whether during step decrease, as in wrong-way behaviour, or step increase in YF, are undesirable because they complicate reactor control policies and may create hot spots which can damage the catalyst. Furthermore, these transient temperature increases may result in (aI ;::
�-
�
�
(b)
2.6
;:: 2.6
�
�
f 2.2
2.1
� 1.8
.!: " 1.6 a
0:: � :s.
3.0
1.1
0
2
4
5
8
10
12
14
16
C1:1l Number
18
20
22
a
10 12 ,, Cell Number
16
1e
20
FIGURE 3.77 Transient responses to step disturbances in YF for cell number 10. (a) Step decrease in Yf- from 1.2 to 0.97. (b) Step increase in YF from 0 . 97 to 1.2.
R = 0.03456, aH = 0.0,
a3
= 0.45.
S . S .E.H. ELNASHAIE and S.S. ELSHISHINI
372
(aI
2.4
l0
lbI
Cell No. 10
2.4
f· =L
Coli No. lD
'"
I.J ._.,
L ) ... . . . . .
.. .. .. ..
..
0.4 '\.-....���� o
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s
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12
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Cell Number
1s
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ro
22
0
.. ...... 0
Cell No. I
Cell No. S
��������.......,. .,..., ...,
3
5
8
10
13
15 18 20 Time, min
23 25
28
30
33
FIGURE 3.78 Dynamic response in the region of multiplicity. Decrease in Y, from 0.982 to 0.96. (a) Solid phase temperature profiles at 1.0, 3.0, 4.0, 6.0 and 20.0 minutes after onset of step decrease in YF at t 0.0. The dotted curve represents the peaks of solid phase temperatures in each cell in the new state at Y, = 0.96. (b) The nature of the periodic oscillations in cells 1, 5, 10 and 20 in the new state at Y, = 0.96. R = 0.03456, an = 1.0, a3 0.325. =
=
undesirable side reactions and may shift the reactor to an undesirable state or lead to temperature runaway (Chen and Luss, 1 989). This calls for special proced ures during startup and shutdown of the reactors. Figures 3.77a,b show the behaviour for a case similar to that in Figure 3.76 but with radiation effects taken into consideration. In comparing Figures 3 .76 and 3 .77, it is clear that axial heat transfer due to radiation has the effect of delaying and m ode rati ng the surge in transient temperature due to a step decrease in YF, but has an o ppo s ite effect of ac c entu ating and speedi ng up the transient increases in temperature caused by a step increase in YF· Figure 3 .78 shows the transient behaviour of the reactor in the region of multiplicity when both radiation and conduction effects are incorporated into the model formulation. The initial conditions for are shown in the small b ox inside Figure 3.78a and those for Y:�j are as shown on Figure 3 .78a for t = O.O. The feed temperature is decreased from 0.982 to 0.96. The lower broken line in Figure 3.78a represents the temperature profile at the new feed temperature of 0.96 after the elapsed time of 20 minutes from the onset of the feed temperature step decrease. Figure 3 .78b shows the oscillatory nature of the temperature dy nami cs at the new feed temperature for cells number 1 , 5, 1 0 and 20 after the transient response of t = 20 minutes has e lapsed To find out whether a temperature overshoot has o c cure d during this trans ie nt , the peak temperature in e ach cell during the periodic oscillations at the n e w feed temperature of 0.96 is plotted in Figure 3 . 7 8 a . The dotted curve in Figure 3 .78a re pre s ent s these maxima and as shown, no overshoots
x;
.
have occured.
b[i_
MODELLING AND ELEMENTARY DYNAMICS
(a)
3.0
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:t '·o
E .::
�
1.5
L !
>: 2.5
:: 2.6
f
(b)
3.0
f l
- - <�> rP - \ '-
��
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ol: ] � 1.0
�
0.5 ���� 0 2 " 6 6
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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 wrong-way 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
1-4
16
18
20
22
FIGURE 3.80 Effect of the size of step decrease in YF on the wrong-way 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
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Cell Number - -- - - - - 1
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Cdl Number
FIGURE
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,
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. Wrong-way 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 a-f, 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 a-f 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 e-f, 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 well-known static and dynamic stability conditions of the single catalyst pellet to the stability of the entire fixed bed. The two well-known 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 c--e 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 wrong-directional creep (which is different from wrong-way 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 s-shaped 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 wrong-way 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 wrong-way 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 non-linear and are thus solved numerically, which means discretizing the differential equations in the axial direction (in addition to the radial direction for two-dimensional 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 two-phase 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 lower-steady 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 non-porous 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 pseudo-homogeneous 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 wrong-way 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, wrong-way 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 non-isothermal 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) a-r
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 = pre-exponential 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 co-workers (Elnashaie and Cresswell, 1 973a-c) . 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 wrong-way behaviour takes place. While in the multiplicity region, the thermal dispersion may enable the wrong-way 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 pseudo-homo geneous, non-isothermal 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_ A-a
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 two-phase 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 2-butene 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 wrong-way 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 co-workers (Elnashaie and Cresswell, 1 973a-c, 1 974a-b, 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 gas-solid 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 self-sustained 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 co-workers (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 wrong-way 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 gas-solid 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 pseudo-homogeneous 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 pseudo-homogeneous 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 fixed-bed 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 self-sustained 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 El-Bialey, 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 2-butene 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 wrong-way behaviour and the expected effect of reactant adsorption on this phenomenon. Wrong-way behaviour is an important pathological phenomenon in fixed-bed 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) . Wrong-way behaviour is different from the phenomenon of wrong-directional creep presented and analyzed in some detail by Elnashaie and Cresswell ( 1 973a). Wrong-way behaviour is simply a transient temperature overshoot with both the initial and final steady states being stable, while wrong-directional 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 wrong-way 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 wrong-way behaviour, while in the multiplicity region, the thermal dispersion may cause the wrong-way 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 wrong-way 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 wrong-way behaviour generates an upstream-moving 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 wrong-way behaviour. They concluded that the two-phase mode l and the pseudo-homogeneous 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 pseudo-homogeneous 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 non-adiabatic tubular reactor. Mehta e t al. ( 1 9 8 1 ) used a pseudo homogeneous plug flow model to demonstrate the wrong-way behaviour of packed-bed 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 wrong-way 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 by-pass. 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 Davidson-Harisson 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 Kunii-Levenspiel 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 Davidson-Harrison 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 cloud-wake 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 cloud-wake then between the cloud-wake and dense phase. ,
III.
The countercurrent backmixing model
This mode l has been developed by Potter and his co-workers 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 nii-Levenspiel 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 cloud-wake 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 gas-liquid 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 pseudo-homogeneous 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
Non-isothermal 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 non-isothermal 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 gas-phase 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). Non-isothermal 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 co-workers) 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 co-workers 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 bubble-cloud 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 two-phase 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 pseudo-steady 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 3.230-3 .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 two-phase 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 two-phase 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 pseudo-steady 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 .236-3 .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 left-hand 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 non-linear 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.
Feed-Back 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 non-adiabatic operation However, another more flexible alternati v e is feasible, that of using a feed-back 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 feed-back 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; "!.
0.4
0.2
0. 0
40 1
Ill
� �
"0
0.
�
'6
..
E
...
�
:;
..
0 2 °
0.6
08
1.0
1.2
1.4
0.0
"0
-;; ' >. >�
FIGURE 3.85 Effect of exchange parameter QE, on the yield and selectivity for a deep bed. Th = 100, at = lOS, az = 1010, Yt = 18.0, rz = 27.0, /JJ. = 0.4, /1 = 0.6.
Stability and dynamic behaviour In order to simplify our presentation of the analysis of the dynamic behaviour of the system, we assume that the adsorption capacity of component A and the heat capacity of the reactor are of the same order of magnitude and that both are much larger than adsorption capacity of components B and C. This assumption allows us to assume that components B and C are in pseudo-steady state governed by the dynamics of component A and the temperature transients of the reactor. Thus the system dynamics reduces to a two-dimensional case for which phase planes are easily constructed and analytical stability conditions are readily obtained. In chapter 4, this assumption which reduces the system to a two-dimensional system will be relaxed and the full three dimensional system will be investigated.
Local stability analysis
We use the steady state equation obtained by setting 1/>n == 0 in equation 3 .237 'to eliminate X8 from equation 3 .239. The resulting equation together with equation 3 . 23 6 are linearized around the steady state and deviation variables are introduced. From the resulting linearized
S.S.E.H. ELNASHAIE and S.S. ELSHISHINI
402
equations, the following couple of stability conditions are easily obtained (Aris, 1 969). (i) Static condition. This condition does not contain any of the dynamic parameters of the system, ( 3 . 249 )
Condition 3.249 can be shown to be equivalent to the well known slope condition. (ii) Dynamic condition. This condition gives a critical value of LS�, above which the system becomes unstable. The condition can be written as, LS' <
B + gi i g22 - B - BK
(3.250)
where LS' is the ratio between the mass capacity and heat capacity ( f/JA/f/JH ) . Notice that in many cases in the literature LS = 1 I LS' is used instead of LS' . 1. 4 .------. �
� ::J
�
�
1.3
1.2
P a r .:L m � t e r s :
ar
111
YF
=
=
J OB· al
0.4 . � 2 = 0.7.'\
=
= 1 06 . )' 1 = 1 8 . )' 2 = 2 7 . 06 . H
=
1 00
.
QE
=
2.0
1 1. 1 �----J �
�
�
Ill
1.0
g .9
]
>:
.
a
r----�--�
. 7 L--L--��--�==��=�� .1 0 .2 .3 .5 .7 1.0 .6 .a .9 · -' XA • d �s ionl�ss dE- n � ph a�
rt-Oi c t ant
concl:'ntration
FIGURE 3.86 Phase plane trajectories for a case with multiple steady states, I/IA = 150.0, 1/ln = 600. 0.
MODELLING AND ELEMENTARY DYNAMICS
403
1 - 4 .-----, Parameters: a1
=
1 08 , a 2
fit = 0 . 4 o YF
=
ll2
=
=
101 0 , y 1 0.6, H
0. 59, Ill stu.dy
S ta blE" limit!'
� 1-0
"'
=
=
1 00, QE
1 8 , y 2 , 27 e
2.0
state t ycleo
c .Q
-�
"'
'0
>:
0 ·9
0 8 -+-------.--r--l 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 start-up 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 : 5-0 ( 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 3.87-3 .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 -�
>-
2-0 .------.. 1 8
\.6
1 ·4
- K = 2-4 ( 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 Gas-Solid 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) two-dimensional systems. However, more complex dynamic behaviour can result when this two-dimensional system is externally forced. This more complex dynamic behaviour includes in addition to periodic attractors, quasi-periodic 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 two-dimensional autonomous (unforced) system followed by the forced (non-autonomous) two-dimensional system, followed by similar (but more complex) investigation of the three-dimensional system. From a dynamical system analysis point of view, the same sequence holds because of the relative simplicity of the two-dimensional 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 Integral-Derivative) 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.<.e
Dense rha.t,c
Pluc now
Perfect mi:dna
l
l
y,
...... t =l '-:
l
!
··<9 o v� t..:::.J -
rf
Y,
FIGURE 4.1 Simulation model for two-phase fluidized bed reactor with single input single output (SISO) proportional feedback control.
taking place in a freely bubbling fluidized bed reactor. The desired product is the intermediate component B (Elanshaie, 1977). The problem of operating the reactor at the middle unstable steady state that gives the maximum yield of the desired intermediate product B in a catalytic consecutive reaction network, is widespread in petrochemical reactions. Partial oxidation of o-xylene to phthalic anhydride is an important candidate (Elnashaie et al. , 1 987, 1990), in addition to other partial oxidation reactions, as well as the oxidative dehydrogenation reactions, such as the oxidative dehydrogenation of butene to butadiene (Wagialla et al. , 199 1 ) . In many of these cases the optimum operating conditions which give the maximum yield of the desired product B correspond to the middle unstable static steady state (saddle type). In order to operate the reactor at this unstable steady state a feed-back control system or a cooling system should be used. A schematic diagram for the two phase model of this reactor with a simple proportional feed back control system installed on it is shown in Figure 4. 1 . The following assumptions are used in the derivation of the dynamic mass and heat balance equations of the model (Elnashaie, 1 977): 1 . Both reactions are first order. 2. The gas in the bubble phase is assumed in plug flow.
3. The extent of the reaction in the bubble-cloud phase is negligible. This assumption is justified for small particle size (dp < 1 50 J..l) and
high flow rates giving rise to fast rising large bubbles and negligible cloud-phase.
STATIC AND DYNAMIC BIFURCATION BEHAVIOUR
409
4. The dense phase gas is assumed to be perfectly mixed. This
5. 6. 7.
8.
assumption is justified for strongly adsorbed gases, also the small height to diameter ratio and large gas flow rate used in this example strengthens this assumption. An average value of the bubble size and hence an average value of the exchange parameter between the dense and bubble phases are used. Negligible mass and heat transfer resistances between the solid particles and the dense phase gas . The dense phase is of uniform temperature, i. e. there is no radial or axial variation of temperature in the dense phase. Negligible heat of adsorption.
The unsteady state material balance equations for the dense phase are thus given by: A> . A 1 'f't A
dC
H -' = G1 ( CIif - CJ ) + QEJ. AC dt
iH ( -C. - C. ) dh - p ( l - E)A1 HR. 0
J
I
S
J
(4. 2 )
Where R; is the rate of disappearance of component i in kmol/kg catalysts. The unsteady state heat balance for the dense phase is given by: A
o ( T - T ) dh t/JH A1 H dt = G1p1 Cpf ( T-1 - T ) + pf Cpf QH Ac JrH
-
dT
+AI H ( l - E )Ps
2
L=l lj (-!llf) J
(4 . 3 )
Whe� r1 are the rates of the two consecutive reactions, A � B, B � C, and T1 is the feed temperature given in the PID feedback controlled case by: T1 = T1 + K( Tm - T) + K1 ( Tm - T) dt + Kd dt
I
-
, dT
I t
(4.4)
0
The bubble phase mass and heat balance equations can be considered at pseudo steady state because of their negligible mass and heat capacities and are given by: Gc
dC. = -QE;�- (C; - C; ) � d dT
-
Gc dh = - QEH Ac ( T - T )
(4. 5) (4.6)
410
S . S .E.H. ELNA SH AIE an d S . S . ELSHISlllN I
With the boundary conditions : at h= 0: (4.7)
Equations 4.5-4.7 can be solved analytically and the resulting solutions used to evaluate analytically the integrals in equations 4.2 and 4.3. Assuming QEA = QEs = QEc = QH = Q£, and casting the equations in dimensionless forms gives :
where: 'f = t I
�H
�H = �H I p1cpf = (1 - e:)ps cps I p1 cpf
The rest of the dimensionless groups and variables are the same as defined for the simpler two dimensional case in chapter 3 . LeA and Le8 are the ratios between the heat capacities and chemisorption capacities of components A and B respectively and the control law is given by, (4 . 1 1 ) B is the reciprocal of the effective residence time of the bed given by :
Where
The yield of the reactor
B=
GI + Gc ( l - e-a ) A1 H
(4 . 1 2)
is given by :
(4 . 1 3)
STATIC AND DYNAMIC BIFURCATION BEHAVIOUR
41 1
Table 4.1 Simple calculation of the two-phase model parameters (Partridge and Rowe, 1966; Chavarie and Grace, 1975).
Gl = A · U - Gc
Ac·
=
E · Gs
( a! - l. O) Umf
A, = A - Ac
where: (4. 1 4) and,
QEHAc a = =-"':.__ -"'Gc
(4. 1 5)
The fluidized bed two-phase model parameters (G,, Gc, A r, a) can be computed from many procedures available in the literature (Davidson et al. , 1 977). We use here the simplest procedure due to Partridge and Rowe ( 1 966) after introducing into it the modification suggested by Chavarie and Grace ( 1 975) regarding the gas flow in the dense phase. However any two-phase model other than the Partridge and Rowe model can be used instead (Bukur, 1 987). Details of these simple cal culations are given in Table 4. 1 and the data used for this model are given in Table 4.2. The two-phase model used in this presentation, although relatively simple, retains the main characteristics of bubbling fluidized bed and has been used successfully to simulate the Type IV industrial fluid catalytic cracking units (Elnashaie and Elshishini, 1 993), as shown later in section 4.2 of this chapter. 4. 1.1
The Two-Dimensional Case with Proportional Control
This case corresponds to, 1 __ = 0 ,
LeB
K;
=
0
,
in the model described by equations 4.8-4. 1 5 .
S . S .E.H. ELNASHAIE and S . S . ELSHISHINI
412
Table 4.2
Data used for the fluidized bed catalytic reactor (Elnashaie, 1977).
Bed cross-section area, A
0.3 m2
Superficial gas velocity (based on unit cross-section of the whole bed), U
0. 1 rn/s
Minimum fluidization velocity (based on unit cross-section of the whole bed), Umf
0.00875 rn/s
a1
1 9. 5
(j)
(= U8 · � 1 U"'r )
Voidage of the dense phase,
QE
1 .8
0.4
�
0.6 s- 1
Bed height, H
1 .0 m ws s- 1
Normalized pre-exponential factor for the reaction A -7 B, a1
Normalized pre-exponential factor for the reaction B -7 C, a2
Dimensionless overall exothermicity factor for the reaction A
-7
B, {31
Dimensionless overall exothermicity factor for the reaction B -7 C,
Dimensionless activation energy for the reaction A -7 B, y1
fh.
l Q I I 8-1
0.4 0.6
1 8 .0
Dimensionless activation energy for the reaction B -7 C. y2 Lewis number of component A, LeA
27 .0
Lewis number of component B, Le8
0.454545
Dimensionless feed concentration of component A , XAf
1 .0
1 .0
Dimensionless feed concentration of component B, X81
0.0
Dimensionless feed concentration of component C , Xq Dimensionless feed temperature to the reactor (base value) r1
0.0
Set point for the controller, Ym
0.92955 1 30
0.55342072
The behaviour of this autonomous two-dimensional case with propor tional control has been presented in the previous chapter. In this section we will concentrate on the presentation of the behaviour for the non autonomous (periodically forced) case. The autonomous case represent a special case of the autonomous when the amplitude of forcing is zero. First we review some of the relevant literature regarding this problem. It should be quite clear to the reader by now that reactive systems exhibit different kinds of nonlinear phenomena s uch as multiplicity of steady states (Balakotaiah and Lu s s , 1 98 1 , 1 983a, b; Aris and Varma, 1 980; Villadsen and Ivanov, 1 978). Many theoretical and experimental investi gations have been carried out to expl ore the complicated interaction of feed disturbances with the non-linearities of chemical re ac tor systems (Bailey, 1 973 ; AI-Taie and Kershenbaum, 1 978; Hoffman and Schadlich, 1 986; Si1veston et al. , 1 986) . Sincic and B ailey ( 1 977) obtained one of
the early evidences of p at ho l o gi c al responses for a periodically forced
STATIC AND DYNAMIC BIFURCATION BEHAVIOUR
413
CSTR with a single irreversible exothermic chemical reaction. Mankin and Hudson ( 1984) studied the transition from quasiperiodicity to chaotic behaviour through a period doubling sequence as well as the existence of bistability in periodically forced CSTR with a single irreversible chemical reaction using the coolant temperature as the forcing variable. The same system was investigated in more extensive and sophisticated manner by Kevrekidis et al. ( 1 985) using two parameters (the forcing amplitude and frequency) excitation diagrams. Kevrekidis et al. ( 1984, 1986) developed efficient techniques for the study of periodically forced systems. McKamin et a/. ( 1 988) presented results for the effect of the distances between the centre of forcing and the Hopf bifurcation point on the excitation diagrams of periodically forced surface reaction model for three different centres of forcing. Very few results are presented in the literature for cases where the centre of forcing is close to the homoclinical orbit of the system (Caneba and Crossey, 1 987). McKarnin et al. ( 1 988) presented a case where the autonomous system has a homoclinical orbit but the centre of forcing was taken close to the Hopf bifurcation point. The responses of nonlinear reaction models to periodic forcing in parameter regions where the unforced system exhibits steady state multiplicity without oscillations have been examined by Cordonier
et al. ( 1 990). Most of the results published in the chemically engineering literature regarding chemically reactive systems concentrate on the low frequency forcing region (e.g. : Kevrekidis et a/. , 1 986; Tambe and Kulkarni, 1993). The dynamic behaviour of the unforced non-isothermal fluidized bed catalytic reactors has been investigated by Elnashaie and co-workers (Elnashaie and Cresswell, 1 973 ; Elnashaie and Yates, 1 973; Elnashaie, 1 977; Elnashaie et al. , 1 980, Elnashaie and Elshishini, 1980; Elnashaie and El-Bialey, 1 980). In the present section we present to the reader the rich variety of complex dynamic phenomena associated with the forced non-isothermal fluidized bed catalytic reactor in the less studied high forcing frequency region, and where the centre of forcing is close to the homoclinical orbit. This is because this region is the richest and helps to present to the reader most of the possible complex dynamic phenomena.
Mathematical formulation of the problem Consider the consecutive solid catalyzed reaction presented by equation 4. 1 , taki ng pl ac e in the fluidized bed shown schematically in Figure 4. 1 . The same assumptions given in the previous section hold. The unsteady state material and heat balance equations are given by equations 4.2 and 4 . 3 .
414
S . S .E.H. ELNASHAIE and S . S . ELSHISHINI
For the proportional feedback controlled case with sinusoidal feed disturbance equation 4.4 becomes:
1J = ( T1 + Am' sin m't) + K ( Tm - T )
(4. 1 6)
where Am' and ro ' are the amplitude and frequency of the forcing periodic signal, K is gain of the proportional controller and Tm is the set point of the controller as shown in Fi gure 4. 1 and T1 is the base value of the feed temperature. The bubble phase mass and heat balance equations are the same as equations 4.5 and 4.6 with the same boundary conditions given by equation 4.7. Assuming QEA = Q£8 = QEc = QH = QE and casting the equations in dimensionless forms with the assumption of pseudo steady state for components B and C and after some simple algebraic manipulations, the three dimensional autonomous dynamical system investi gated is re placed by the two dimensional non-autonomous dynamical system de scribed by equation 4.8 given earlier and equation 4. 1 7 given below.
(4. 1 7) where: and,
iJ = ( Y1 + Am sin ror) + K ( Ym - Y)
(4. 1 8)
Obviously the unforced system is the limiting case of the forced system as the forcing amplitude Am � 0. The concentration of the intermediate product for this case of Le8 � oo, is given by the pseudo steady state relation:
(4. 1 9) B and the yield of the reactor are gi ven by equations 4. 1 2 and 4. 1 3 respectively. The data used in this model is given in Table 1 , and for this two dimensional case Le8 � oo .
STATIC AND DYNAMIC BIFURCATION BEHAVIOUR
4. 1 . 1 . 1
415
The unforced (autonomous) case
This case corresponds to Am � O in equation 4. 1 8 . Numerical methods and computation techniques l . Computation of limit cycle period (natural period) The unforced autonomous system (A m = 0) with the chosen set of parameters for this case, has unstable steady states surrounded by stable limit cycles. High accuracy is required for the determination of the period (natural period) of any of these unperturbed oscillatory state because the value of this period affects the location of forced sub harmonics. Shooting algorithm using Newton method is used for this purpose. The two dimensional autonomous system in vector notation can be written as:
: = P0j ( X,A. )
(4 20) .
with boundary conditions: (4.2 1 ) where r = 'r I P0 , X[XA , Y] i s the vector of state s , xo i s the vector of initial states, f is the vector of nonlinear functions, A i s vector of parameters and Po i s period of the limit cycle. The problem is a two point boundary value problem where P0 i s unknown and varies with the system parameters. Inte grat i on of equation 4.20 from r = 0 to r = l gives :
(4.22) For any periodic solution equation (4.2 1 ) mu st be satisfied to gi ve :
ci>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 physcio-chemical, 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 non-adiabatic 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 one-parameter 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 (saddle-type). 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 (saddle-type). 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 3-0 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)
6-0
10
20
00
1.0
2. 0
2-00
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
3-0
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(a-c) 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 ·;;; <=
A
1. 2
=
/8 0.5 1 9745 8
• • •
00 0
=
c
I
SLP
=
�·c-
· ·· · ·· · · · ··
· · ·· · · · · · · · · · ··
Yr =
0..5 1
0.52
0. .53
)1
••• • •
.��� · · · · · · · · · 0.50
Steady·Jlale brancll Stable .... Unsllblc r.riodic ••• Stable UD5toble 000
0.55343 1 9
0.529373 9
lu
E
D
• • • • • • •• • • • •
I �.
oo HB
0.9
0.8
••
I
42 1
I
0.54
•• • • •
0_ 5500 t
O.SS
··
t
. . . ..
0.5630374 0.5532
SLP =
Yr =
y ( = 0.55342072 O.S6
O.S7
0.58
Y f (dimensionless feed lemperalure)
FIGURE 4.4. One parameter bifurcation diagram of Y vs. Y1 for
K = 3.62744.
upon the value of Yf and whether it is larger or smaller than Yf = 0.55342072. Similar diagrams have been obtained by Elnashaie et al. ( 1 993) for a three dimensional fluidized bed reactor, as will be shown later in this section. The dynamic characteristics of the system also changes with the change of Yf from the base value Yf= 0.55342072. The cases shown in Figures 4.3(ia), 4.3(ib) and 4.3(iia) show relatively complex stable and unstable periodic branches as well as for the two cases of homoclinical (infinite period (IP)) orbits in Figures 4.3(ia, iia). The cases in Figures 4.3(ic), 4.3(iib) do not exhibit periodic phenomena and are therefore of little interest specially for the autonomous case. We now fix the controller gain (K) at a value of 3 . 62744 which is close to the homoclinic orbit in Figure 4.2c, and at this value a one parameter bifurcation diagram for the dimensionless feed temperature ( Yj) is shown in Figure 4.4a. This bifurcation diagram contains a periodic solution originating from the single Hopf bifurcation (HB) at Yf= 0.5293739 and terminates homoclinically. Figures 4.5(a, b) show the sequence of formation of the homoclinic orbit as Yf decreases. As Yf decreases the saddle and node approaches each other till they coincide forming the homoclinical orbit which connects the saddle to itself. In order to show the reader the main characteristics of the system the Y-Yf bifurcation diagram is divided into ·five sections A, B , C, D, E. Typical phase planes for the autonomous system in each section are given in Figures 4.6 (A-E) .
422
S . S .E.H. ELNASHAIE and S . S. ELSHISHINI
1 . 1 5 'T""------�
1.15
· . ._
1.0 5
K-:-:3:-6::-:7:-: 7-:440 ':-:": 00 � Yf : 0. 5 5 6 0 0 000 Ym: O 9 2 9 55 I 30
1. 1 0
1.05
"·
t OO
>
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.---,r---1 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 (A-E). (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 (non-autonomous) 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, quasi-periodic 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
S.S .E. H . ELNASHAIE and S . S . ELSHISHINI
424
1 . 1 1 (1 �-------�
0.100
:.., _···_,. ,·
"
·�.:�'
U
o.ootoo
Pl 0.00200
o.ooaoo
Am. Fon:illg Amplitude
1.050
(o)
.......
·�. · Am. Forciag Amplitude
. ....
I ·1 (d) U 1'511) �•) 1'6/I) PI\Is) .. � .. �.,.,...;:.,... ..:; .... .. ,.rl .,:.. ..... ���.:....:... 0.01 300
0.011100
0.01 1500
0.01100
0.01700
0.01100
Am. Forcin(C Amplitude
FIGURE 4.7 One-parameter stroboscopic Poincare bifurcation diagram (Y1 vs. Am, strobing every forcing period). (a) Am = 0.0-0.004. (b) Am =0.004-0.009. (c) Am 0.009-0.013. (d) Am = 0.013-0.018. =
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 one-parameter stroboscopic b ifurc a ti on di agram is c ons truc ted for this case as shown in Figu re 4.7(a-d). 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
425
· ·-
l .tto
0 ISO . ...
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FIGURE 4.8 Enlargement of the one-parameter stroboscopic Poincare bifur cation diagram (Y1 vs. Am) in region L1 (Figure 4.7). L1(a) The full five stroboscopic branches a-e. L1 (b) Enlargement for the three branches b-d. L1 (c) One-parameter stroboscopic Poincare bifurcation diagram with the stroboscopic points taken every 5 times the forcing period ( Y5) starting at branch e .
a) Region Ll (Am = 0.00001-{). 00016) It is clear from Figure 4.8 that period doubling leading to chaos starts at extremely low values of the forcing amplitudes. It will be shown later that this is characteristic of the case where the centre of forcing is very
426
S . S .E.H. ELNASHAIE and S . S . ELSHISHINI
close to the homoclinical orbit of the autonomous system and ro/(1)0 are relatively high (511 in this case). The limiting case of this region is the autonomous oscillatory un perturbed system (Am = O.O). It is clear from Figure 4.8 that frequency locking occurs only at very small amplitudes forcing i .e five sub harmonic saddles and five nodes are born on the surface of two dimen sional torus and the system is entrained by the forcing frequency. The resonant (entrained) trajectory has periodicity five times the forcing period (frequency locking). Five branches (a-e) emerge on the strobo scopic one parameter bifurcation diagram as shown in Figure 4.8(L l (a)). The Y -axis is the stroboscopic dimensionless dense phase temperature (Y 1 = strobing every one forcing period) . The branches (b, c, d) are very close to each other and therefore this region is further enlarged in Figure 4.8(Ll (b)). It is easy to follow the behaviour of the system in this region through one branch of these five stroboscopic bifurcation branches. Branch e is selected and magnified in Figure 4 . 8(L l (c)). The selection of this branch is equivalent to the strobing of the points at period equal to five times the forcing period, therefore the Y -axis is written as Y5 . It is clear from Figure 4.8 that the system follows the well known route to chaos that is the Feigenbaum period doubling (PD) scenario. The period doubling (PD) occurs when the principal Floquet multiplier passes out of the unit circle through -1 . At these conditions the trajectory losses its stability and a stable trajectory of twice the period is born. When the period doubling sequence starts the torus structure is broken because it is impossible to have a torus structure having more than one periodic trajectory of different periods locked on it (Kevrekidis et al. , 1 986a, b). As mentioned earlier the period doubling (PD) starts at an extremely small amplitude forcing approximately equal to 0.0000725. This value of the amplitude for the present centre of forcing (Yf = 0.55342072) makes Yfmax = 0.55349322 and Yfmin = 0.55334822. Thus during the forcing cycle at the beginning of the period doubling (PD) conditions the system visits regions C and D (Figure 4.4) and therefore obviously crosses the boundary between region C and D (IP orbit) twice for each complete cycle. Chaotic behaviour starts at a very small value of the forcing amplitude equal to = 1 1 .2 x 1 o-5. This occurrence of period doubling and chaos at extremely low values of the forcing amplitude is a clear manifestation that the so called generic behaviour of frequency locking at low amplitudes is very strong l y linked to the characteristics of the system (e s peci al l y the existence of a homoclinical orbit) and the po s i tio n of the centre of forcing relative to this homoclinical orbit. The existence of the homoclinical orbit depends upon the type of system i nv es ti g ated and the specific set of physico-chemical parameters of the s y stem used in the inve stigati o n .
STATIC AND DYNAMIC BIFURCATION B EHAVIOUR
427
l Ull t . tts t.UII t.tU CI 8 U
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FIGURE 4.9 D1(a). Time trace. Iterate map ( Y5 (n + 1) vs. Y5 (n)).
~ 0 140
0 : If)
XA I
Dllbl
0 uo
0 3..,0
D1(b). Stroboscopic map (Y1
vs.
XA 1). D1 (c).
Dynamic simulation of the chaotic trajectory at D 1 (A m = 0.000 1 1 6) is shown in Figure 4.9(D l (a)). The overall stroboscopic map (Y l vs. XA l ) of this chaotic attrac tor is shown in Figure 4.9(D l (b)) . The chaotic trajectory forms segments of a line of an infinite number of poi n t s which are v is ited alternately. The structure of the line segments in Fi gure 4.9(D l (b)) suggests that the system is strongly dissipative and
S .S .E.H. ELNASHAIE and S . S . ELSHISHINI
428
therefore the construction of a meaningful first order iterate map is possible (Fan and Holden, 1 993 ; Thompson and Stewart, 1 989). The fifth stroboscopic map of dimensionless dense phase temperature ( Y5 (n+ 1 ) vs. Y5 (n)) shown in Figure 4.9(D 1 (c)) is equ ivalent to the first iterate map of a single branch (branch e). This map resembles a numbe r of well known one dimensional maps exhibiting chaotic behaviour such as: the logistic map (Doherty and Ottino, 1 988) and the experimental map of the bromide concentration in a well-stirred tank reactor with Belousov-Zhabotinskii reaction (Hudson and Mankin, 1 98 1 ) as wel l as the Rose-Hindmarsh map for neuronal activity (Holden and Fan, 1 992). The Lyapunov exponents were computed using the technique and algorithm of Wolf et al. ( 1 985) (see also chapter 2 and section 4. 1 . 1 .3 in this chapter) and were found to be Ly1 = 0.0 1 0586 and Ly2 = -0.363926. The largest (maximal) Lyapunov exponent (Ly1 ) is positive giving positive proof of the chaotic behaviour of the attractor. The corresponding Lyapu nov fractal dimension of the attractor is equal to 1 .029 1 .
Periodic Windows and Intermittency in region
L1
Existence of an infinite number of periodic windows (PWs) is one of the universal features of the chaotic behaviour (Metropolis et a/. , 1 973). R egio n W l in Figure 4.8 includes a periodic window and therefore the corresponding region of Y5 ve r sus Am is enlarged in Figure 4. 1 0. It is 0 9�4 0 97 2
---...,
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.
.
.
rrtr i i i P I H 1 1 1 1J p • I I P ' l i P I l l
Enlargement of the region W1 in Figure 4.8.
STATIC AND DYNAMIC BIFURCATION BEHAVIOUR
429
evident that period three window (PW3) bifurcates from the chaotic attractor. The mechanism of bifurcation of chaotic trajectory to the period three window (PW3) is investigated by considering the forcing amplitude at point E1 (Am = 0.000 1 228) in Figure 4. 10. It is well known that period doubling bifurcation is not the only route to chaos. Pomeau and Manneville ( 1 980) proposed that the chaotic attractors can develop through three types of intermittencies (Types 1 , 2, 3) .
Dynamic simulation at E1 of the intermittent chaotic trajectory is shown in Figure 4. 1 1 (£1 (a)). The corresponding stroboscopic map ( Y1 vs. XA 1 ) as well as the overall stroboscopic first iterate map (Y1 (n + 1 ) vs. Y1 (n)) are shown in Figures 4. 1 1 (EJ (b), E1 (c)) respectively. Figures 4. 1 2 E1 (d) and E 1 (e) show the overall stroboscopic points histogram and the stroboscopic points histogram corresponding to
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FIGURE 4.1 1 Intermittent chaos i n region W 1 (Figure 4.10) at point E1. E1 (a) Time trace. E1(b) Stroboscopic map ( Y1 vs. XA I ). E1 (c) First iterate map (Y1 (n + 1) vs. Y1 (n)).
S.S .E.H. ELNASHAIE and S.S. ELSHISHINI
430
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E1(d) Stroboscope points histogram. E1(e) Stroboscopic points histogram. E 1 (0 First iterate map. E1(g) Third iterate map. (i), (ii), (iii ) are enlargements of boxes (i), (ii), (iii) in E1(g) respectively.
branch e respectively. It is obvious that the dynamics alternate between region of period three window for each stroboscopic branch (laminar phases) and chaos (bursts). The appearance of period three window (PW3) represents a type of 'bench-mark' in this process because it is the first member of Sharkovsky' s T set, so that all types of p eriodic
STATIC AND DYNAMIC BIFURCATION B EHAV IOUR
43 1
1 . 1 1 0 ..,----------------------.... I .OSO 1 .0,0 1 .020
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I
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· .· .
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0.00 1 40
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FIGURE 4.13 Enlargement of region L2 in Figure 4.7(a).
windows are present (Jackson, 1 989). The first iterate map for the dimensionless dense phase temperature of Y5 (n + 1 ) versus Y5 (n ) preserves its prev iou s general shape as shown in Figure 4. 1 2 E1 (f), however the third iterate map for Y5 shown in Figure 4. 1 2 E1 (g) makes the situation quite clear. The curve approaches the diagonal and almost becomes a t an gent at three distinct points as enlargements of boxes i, ii and iii show. Three channels (trapping regi on s ) through which the movement gives the laminar phases are clearly shown. As soon as the curve becomes tangent to the diagon al the chaotic intermitte nc y dis appears and a stable period three attractor emerges as a result of saddle-node bifurcation i . e. tangent bifurcation (type 1 intermittency). b) Region L2 (Am
=
0. 00125-0. 00150)
The enlargement of region L2 for Y1 is shown in Figure 4. 1 3 . The left hand side of this regio n shows six branche s , th e y go throu gh period halving with incre asing Am gi v ing rise to period three region (PW3) which bifurcates to a thin chaotic strip which is acting as a bifurcation med i um to produce a period two window (PW2) . The mechanisms of bifurcation active in this strip is followed by considering the forci ng amplitude at point A2 (Am = 0.00 1 4 1 ). At A2 the chaotic trajectory and
S . S .E.H. ELNASHAIE and S.S. ELSHISHINI
432
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7 000
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r JlOO
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FIGURE 4.14 Behaviour of the fluidized bed reactor at an amplitude corres ponding to point A2 (Am = 0.00141 ) in Figure 4.13, a case of "sandwich" intermittency (intermittency with two laminar channels of periodicities two and three). A2(a) Time trace. Az{b) Stroboscopic points histogram.
the Y1 stroboscopic points hi s togram are shown in Figures 4.14 (A2(a), A2(b )) . Shown clearly is the co-existence 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
STATIC AND DYNAMIC BIFURCATION B EHAVIOUR 10)0
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A22
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FIGURE 4.15 A2(c). Second iterate map. A21. Enlargement of box A21 o n A2(c). A22. Enlargement of box A22 on A2(c). A2( d). Third iterate map. A23. Enlarge ment of box A23 on A2(d). A24. Enlargement of box A24 on A2(d).
distinct regions as the magnified squares A2 1 and A22 show in Figure 4. 1 5 , whereas the third stroboscopic iterate map (YJ (n + 3) vs. Y, (n)) also sh ow that the curve forms three laminar channels as shown by the enlargement of squares A23 and A24 in Figure 4 . 1 5 . This situation is slightly more complicated than the u s u al bifurcation between periodicity and i ntermittency (type 1 ) through tangent bifurcation (saddle-node). Here the chaotic intermittenc y having two laminar channels bifurcates
434
S . S .E.H. ELNASHAIE and S.S. ELSHISHIN I
to another intermittency thro ugh tangent bifurcation. This new inter mittency with two laminar channels plays the role of ex change medium between two intermittencies, pre-intermittency having laminar phases of periodicity three and post-intermittency hav ing laminar phases of periodicity two. The p re -inte rmittenc y is born by tangent bifurcation from period three w i ndow (PW3) and the post-intermittency bifurcates by tangent bifurcation to period two window (PW2). This type of intermittency · is most probably due to existence of more tangency points .
c) Last Region (Am = 0. 00150-0. 01800)
This repres ents the entire region of amplitudes larger than those in region � in Fi gu re 4.7a. In this region the system alternates between chaotic behaviour which emerges through period doubling (PO) sequence and periodic windows (PW3 , PW4, PW5, . . . . . . ) till the system is fully entrained at high amplitu de as shown by the bifurcation di agrams in Figures 4 . 7 ( a-d). The behaviour of the system in this region is similar to the behaviour of the autonomous three dimensional Rose-Hindmarch model for neuronal activity (Holde n and Fan, 1 992) which exhibits a new type of period adding bifurcation. After the period two window (PW2) in the right hand part of Fi gure 4.7(a) and the left hand part of Figure 4.7(b) an interesting sequence of periodic windows (PWi (f or s), where f and s denote the first and the second period i window res pectively) and chaotic region (C) e merge s . This sequence is as follows : PWrC2-PW3-C3-PW4(f)-C4-PW4(s)-Cs-PWs(f)-C6-PWs(s)-Cr PW6(f)-CR-PW6(s) . . . . . . This shows that some chaotic regi o ns like C2 , C3, C5, C7 separates windows which are different in periodicity by one period (period adding), while other chaotic regions such as C4, C6, C8 separates windows havi ng the same periodicity PWi (f) and PWi (s). All the chaotic regio ns are created as Am increases thro u gh period d oub l i ng sequence and bifurcates to perio dic windows through tangent bifurcation. The final bifurcation to the harmonic trajectory (fully entrained) i s investigated by choosing forcing amp litud e in the enlargement reg ion L3 (Am = 0.01 670-0.0 1 686) at A3 as shown in Figure 4. 1 6. Dynamic simulation is carried out at A 3 (Am = 0.0 1 68 1 8) and the trajectory and stroboscopic histo gram are shown in Fi gu re s 4. 1 6(A3(a), A3(b)). The Lyapunov exponents are fou n d to be of neg ati ve values Ly1 = -0.003595 and Ly2 = -0.945 1 64, s h o wi ng that th e traj e ctory at A3 i s a quasi-periodic attractor Therefore the system bifurcates from the quasi-periodic attractor to the harmonic period 1 attractor thro u gh Hopf bifurcation (M cKami n et al. , 1 98 8 ) . This is als o in agree ment with the conj ec ture 6 m ade by McKarnin et al. ( 1 988) that all tori that do not become phase-locked in the resonance horn undergo a Hopf bifurcation at a period 1 Hopf curve. -
.
.
STATIC AND DYNAMIC B IFURCATION BEHAVIOU R
435
1 1 1 0 ...-----. ' 0�0 I O�Q I 010
.;: 0 990 0 9�0 0 9JO
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n Q<Jfl
0 ...
....,.
() <) \ () n �
() � 1(1 �\
)I)
1 _1 _ ()1)
Enlargement of region L3 (Figure 4.7(d)) and simulation at the point A3 (Am = 0 .01 6818) . (L3). Stroboscopic bifurcation diagram for region L3. A3(a). Time trace at point A3 (Am = 0.01 6818 ) . A3(b). Stroboscopic points histogram at point A3•
I<'IGU�E 4.16
S.S .E.H. ELNASHAIE and S.S. ELSHISHINI
436
;
I 110
l �HO
.;;
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0 '100 1------
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A.m ( rnrc i n a 1mpli1ude)
FIGURE 4.17 One-parameter stroboscopic bifurcation diagrams (strobing every forcing period) for the second case (¥1 ::; 0.5532). (a) Am = 0.0-0.01000 . (b) Am = 0.01000-0.01800. (c) Am = 0.01800-0.1 1 800.
2.
Preliminary Discussion of the Effect of the Position of the Centre of Forcinr: Relative to the Homoclinic Orbit
In this section we present to the reader a prel iminary discussion of the effect of the p o sition of the center of forcing . A more detailed presentatio n and discussion will be given in section 4. 1 . 1 .3. Figure 4. 1 7 shows the strobos c opic bifurcation diagram o f Y 1 v ersu s Am for a case where the centre of forcing is slightly moved away from the homoclinical orbit to the value Y1 = 0.5532. It is c lear from Figure 4. 1 7 th at frequency locking with peri od icity 5 continue up to a value ofAm= 360 * t o--5 (compare this with the previous case at Y1=0.55342072, where period doubling starts at Am "" 7.25 x w-5 and chaos starts at Am "" 1 1 .2 x 1 o-5). The complex behaviour starts at Am "" 360 x 1 o--5 and ends with bifurc ation to period 4 at Am "" 540 x 1 o--5 • When this com plex region is checked by the c omputation o f Lypun ov exponents the values of Ly 1 and Ly2 at different points inside this regi on were found to be negative (e . g . : for Am = 370 x I 0-5 Ly 1 and Ly2 were found to be -0.00673 1 , -0.5447 1 2 respectively). The trajectories in thi s c o mplex reg ion are not chaotic, actually they are tori. The period four trajectory w hich bifurcates from the torus at Am "" 540 X 1 0-5 undergoes period d oubl ing at Am "" 1 060 x 1 o-5 as shown in Figure 4. 1 7. Fully developed
STATIC AND DYNAMIC B IFURCATION BEHAVIOUR
437
1 . 1 10 1 .080 . 1 .0�� 1 .0 2 0
-;
0 Q91) 0 960 0 9 .1 0 0 900 0.870
t----- - - - 0 01
0 0�
-- - - - --
O OJ
_ _
(
O !J.l
om
FIGURE 4.18 One-parameter stroboscopic bifurcation diagram (strobing every forcing period) for the third case (Y1= 0.5500, Am = 0.0-0.03500).
chaos starts at Am "" 1 1 60 x w -5 . The Lyapunov exponents at Am = 1 1 60 w -5 were found to be Ly1 = 0.014945 and Ly2 = -0.598763 which show that the trajectories in this region are chaotic. The chaotic region with its internal windows and complex structure ends at Am ::: 1 300 x 1 0-5 where it bifurcates to a period 3 region which extends till Am "" 5800 x I 0-5 as shown in Figure 4. 1 7(c). At this value of Am a very thin strip of chaos occurs then a period 2 region appears which extends to Am "" 7860x I0-5 where a period 1 (full entrainment dominates for all higher values of Am). These results show a prelimi nary example of the great sensitivity of the behaviour of the system to very small variations in the position of the centre of forcing relative to the homoclinical orbit. Figure 4. 1 8 shows another case where the centre of forcing has been moved further away from the homoclinical orbit to lj·= 0.5500. It is clear that the behaviour is completely different from the previous two cases emphasizing the effect of the position of the centre of forcing on the behaviour of the system. Detailed analysis of the effect of the positi on of the centre of forcing (relative to the homoclinical orbit) on the resonance horns and their internal structure is presen ted and dis cussed in section 4. 1 . 1 .3 . x
Summary of section results
Numerical results for the effect of the forcing amplitude for a case with
S .S .E.H. ELNASHAIE and S . S . ELSHISHINI
438
the centre of forcing very close to the homoclinical orbit has been presented in this section. The investigation has demonstrated the richness and complexity of the dynamic behaviour of this system and that very small amplitudes are sufficient to cause chaos. The usual mechanisms for the routes to chaos, namely period doubling and intermittency, have been presented to the reader and discussed, in addition to a new type of intermittency with two laminar phases which has been shown to form a transitional region between two ordinary intermittencies with single laminar channels. Different mechanisms for the transition between chaotic regions and periodic windows has been identified and analysed in some details including: period doubling, period halving, period adding and tangent bifurcations. It has also been demonstrated that some chaotic regions separate windows which are different in periodicity by one period (period adding), while other chaotic regions separate windows having the same periodicity. It has also been shown that this system exhibits one dimensional stroboscopic iterate maps which resemble the logistic map, the Rose Hindmarsh model map as well as the experimental one dimensional map of bromide concentration for the Belousov-Zhabotinskii reaction in a continuous stirred tank reactor. Preliminary results for the effect of the position of the centre of forcing relative to the homoclinical orbit has also been presented. It was shown that for the forcing frequency used in this presentation ( ro/ (J)0 = 511 ) , as the centre of forcing moves away from the homoclinical orbit the period doubling and chaos disappear from the region of small forcing amplitudes presented in this section. More generalized analysis for this point will be presented to the reader in the next section. 4. 1 . 1 . 3
I-
The periodically forced (non-autonomous) case. Detailed analysis of resonance horns, period doubling loci and chaos
Numerical Methods and Numerical Techniques Construction of the excitation diagram (Resonance Horns)
/-1.
The two-dimensional forced system described by equations (4. 8 , 4 . 1 7-4. 1 9) can be represented by the following two nonlinear first order ordinary differential equations: (4. 3 1 ) (4.32)
STATIC AND DYNAMIC BIFURCATION BEHAVIOUR
439
Equations ( 4.3 1 , 4.32) and one of the bifurcation criterion fonn three nonlinear equations for the two state variables (XI . X2) and one of the excitation diagram parameter (Am or m). This set of the three nonlinear equations allows the tracing of the boundaries of the excitation diagram. Detailed description of this technique is given by Abashar ( 1 994). 1-2.
Computation of Lyapunov
exponents
The average exponential rate of separation of close trajectories (divergence) can be expressed in tenns of Lyapunov exponents (Lys). A positive Lyapunov exponent (Ly) confinns chaotic behaviour. The computation method requires the integration of the variational equations and the reorthonormalization of n vectors. The Gram-Schmidt reortho normalization (GSR) procedure is used. Detail of the computational method and the algorithm used is given by Wolf et al. ( 1 985) and is explained briefly in Appendix F. 1-3.
Computation of the fractal dimension
A chaotic attractor possesses non-integer dimension because its volume is zero and therefore its dimension must be smaller than the dimension of the state space. The dimension of a non-chaotic attractor is always an integer. The dimensions of steady state, limit cycle and two-dimensional torus are 0, 1 and 2 respectively. Kaplan and Yorke ( 1 983) have conjectured that the Lyapunov fractal dimension (Lyp0) is related to the Lyapunov exponents (Lys) by the equation: LyFD = j +
where j is defined by:
± LY;
_ liLy =l J+I I j+I
L LY; < 0
(4.33)
(4.34)
i=l
The number of zero Lyapunov exponents of a non-chaotic attractor indicates the topological dimension of the attractor (Parker and Chua,
1989).
II- Presentation of some more general results for the forced two
dimensional fluidized bed catalytic reactor It is well established in dynamic system theory that for small forcing amplitudes of an autonomous system with a unique steady state and a
440
S . S .E.H. ELNASHAIE and S . S . ELSHISHINI
limit cycle, the autonomous unstable focus steady state becomes an unstable focus limit cycle of period 1 (the same as the forcing period) and the original (natural) limit cycle becomes a two-dimensional toru s (T2) for all forcing frequencies and all positions of the centre of forcing. The main phenomenon associated with small amplitude forcing of the autonomous limit cycles is entrainment. This phenomenon occurs when the ratio of natural and forcing frequency is a rational number and the forcing entrains the system in other words the response of the system becomes periodic and its period becomes an integer multiple of the forcing period i.e. frequency locking takes place between the natural limit cycle and the forcing frequency. For the forced two-dimensional system, the phase surface of the system in this case is toroidal of dimen sion two and the periodic trajectory exists on the surface of this torus. When the natural and forcing frequencies are incommensurate (their ratio is an irrational number) a quasi-periodic trajectory exists and the trajectory winds around the resulting torus surface (Kevrekidis et al. , 1986). For the forced 2-dimensional fluidized bed catalytic reactor described by equations (4.8, 4. 1 7-4. 1 9), the autonomous system has a Hopf bifurcation (HB) point and two static limit points (as shown in Figure 4.4). The periodic branch which emanates from the HB point terminates homoclinically at the entrance from the unique steady state regions to the multiplicity region. As discussed in the previous section, in such a case the position of the centre of forcing becomes an important factor affecting the behaviour of the forced system. Chaotic behaviour in this case, is possible for certain positions of the centre of forcing and disappears for other positions. In the classical cases with neither static nor dynamic limit point (no static or dynamic multiplicities), the position of the centre of forcing is not an important factor and the behaviour is determined by the amplitude and frequency of forcing. 11-1.
Resonance horns
Resonance horns represent a convenient way of presenting the results in the form of a two-parameter bifurcation diagram which describes the dynamic behaviour of the periodically forced system as a function of the forcing amplitude (Am) and the ratio of the forcing and natural frequencies (WIW0). In these two-parameter bifurcation diagrams the entrainment regions appear as hom-like or V -shaped regions called entrainment regions, resonance horns, s ubharrn on ic regions , frequency locking regions or Arnold tongues . The tips of these resonance horns at the zero forc ing amplitude touch the horizontal axis at wl COo = Pof P= nlm, wh e re n and m are prime integers . The larger the primes n and m, the narrower the resonance hom and the sharper its tip (Kevrekidis et al. , 1 986). The re s onant p eri o di c trajectories within the resonance
STATIC AND DYNAMIC BIFURCATION BEHAVIOUR
44 1
horn have a period nP. The boundaries of these resonance horns are loci of saddle-node bifurcation (principal Floquet multiplier = + I ) . At the onset of the entrainment a pair of limit cycles are born, a stable node and an unstable saddle. Such limit cycles are winding around the surface of the torus and appear as n pairs of fixed points on the invariant circle (stroboscopic phase plane). Many investigators have observed experi mentally the subharmonic resonance behaviour (Dolnik et al. , 1986; Dulos and Dekepper, 1 983; Pugh et a!. , 1986). The diagram of resonance horns for different values of the prime numbers n and m is usually called the excitation diagram. At very large forcing amplitudes, only unique period 1 attractors are found having the same period as the forcing period and the system is called fully entrained (the system has become harmonic because the forcing completely overpowered the natural frequency of the system). The two extremes of small and large amplitudes occur in the same way for any forced oscillator, however the region of the intermediate amplitude of forcing is not universal and depend on the inherent non linearity of each system (McKarnin et al. , 1 988). The main types of bifurcation in forced systems are (Kevrekidis et al. , 1984; Cordonier et al. , 1 990) : 1 . Saddle-node bifurcation, where two limit cycles, one stable node
and the other unstable saddle collide and disappear. The principal Floquet multiplier (PPM) passes out the unit circle through ( + 1 ). 2. Period doubling bifurcation, where the principal Floquet multiplier (PPM) passes out the unit circle through (-1 ). 3. Hopf bifurcation of the stroboscopic map, where complex conjugate Floquet multiplier whose norm crosses (+ 1 ) . This results in periodic solution bifurcating to a quasi-periodic solution. On the stroboscopic map this results in the fixed point bifurcating to an invariant circle. Exceptions occur when FMq = 1 (q = 3 , 4, . . . ). These are higher-order (codimension 2) bifurcations and result in periodic resonances of period qP. 4. Saddle-source bifurcation, where the second free multiplier is greater than unity. The main emphasis of the presentation in this section is on the effect of the position of the centre of forcing relative to the homoclinical orbit of the autonomous system on the behaviour of the system for different amplitudes and frequencies of the periodic forcing function. Tl-2. The effect of the position of the centre offorcing relative to the
homoclinical orbit of the autonomol,ls system on the behaviour of the system at high forcing frequency ( ro/roo = 511 )
S . S .E.H. ELNAS HAIE and S.S. ELSHISHINI
442
Table 4.3 Location and corresponding natural periods of the four cases presented in this section. Case
Yt
1
0.55342072 0.55320000 0. 5 5 000000 0.535000000
2 3
4
Po(s) 1 3 1 .7250889408649 63.8 1 908404205 864 3 8 . 46 3 5 56447 377 30 3 1 . 1 6556495626965
In the present section four centres of forcing are chosen, all of them in region C shown in the bifurcation diagram of the two dimensional unforced autonomous system (Figure 4.4). These centres of forcing differ with respect to their distance from the homoclinic orbit (IPB point) of the autonomous system which at Yr" 0.5534245 . Region C contains a single autonomous stable limit cycle surrounding an unstable steady state. Table 4.3 shows the location and the corresponding natural periods of these four cases in region C. The natural periods are determined with a high accuracy by the shooting algorithm using Newton's method as described earlier. The excitation diagrams of the resonance horns for the three cases 1 , 2 and 3 are shown in Figure 4. 1 9(a). The resonance hom of the fourth case is excluded because the hom is very sharp and can not be clearly presented on the scale of this excitation diagram. The resonance horns are constructed using the arc-length continuation technique described earlier. The lower part of the excitation diagram is enlarged in Figure 4. 1 9(b). Detailed investigation of these resonance horns is discussed in the following:
l/-2. 1. The first case (Yt= 0.55342072) This is the case with its centre of forcing closet to the homoclinic orbit of the autonomous system. The period 5 resonance hom for this case (ro/W0 = P0 IP = 51 1 ) is shown in Figure 4.20. At amplitudes higher than 0.0 1 5 this hom rises almost vertically (Figure 4.20a). The boundaries of this hom are loci of saddle-node bifurcation and no meta codimension bifurcation points are detected. These points are usually associated with qualitative changes such as the degenerate Bogdanov points (Bogdanov, 1 98 1 ) which have a double Floquet multipliers at unity (two complex multipliers leaving the unit circle at angle 0°). At the Bogdanov points the loci of the saddle-node, the Hopf bifurcation (two complex c onjugate mu l ti pliers leavin g the unit circle at an angle) and source-saddle (the second free multiplier is greater than unity) meet each other. In the present case, it is reasonable to expect that the hom rises and form a cusp at the 5th root of unity point, ( 1 1 15 or cos 72° ± sin 72°)
STATIC AND DYNAMIC BIFURCATION BEHAVIOUR
443
� 0.01350
.. .,
� s
� 0 . 00000 -�0 ...
' ' :
...e 0.00450
...
0.000003 . 0 0 0 . 00 4 50
� 0 . 00360 �
(b)
,
,• I ,
'•
,
� 0.00 1 80
'·
· ;;
. I I ' I I : I I
0 . 00090 0.00000 4 .40
1 3.00
: ..
., c
e
1 1 .00
. ... 'I . I, ... . :•·
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...
CJ/r..·.
' • ·
.
�
...
,:
- Horn1 - · · · · Horn, - - - Horn,
c::::::
4 . 60
4 .eo
5.00 ', 1 .
· -5 .20
GJ/ r..·.
5.{0
5.60
5.80
6.00
FIGURE 4.19 The exitation diagram o f the resonance horns 1, 2 and 3 for the three cases 1, 2 and 3 at centers of forcing having values Y1= 0.55342072, Y1= 0.5532 and Yt = O.SS respectively.
in this case this will be a period one Hopf bifurc ation curve, according to the conjecture made by M ckarni n et al. ( 1 988) for forced system near a Hopf bifurcation point for period n resonance horn, n '?. 5 . The lower portion o f the horn is enlarged in Figure 4.20b. The concentri c loci of period doubling 1 0 (PD 1 0) and 20 (PD20) were computed and prese nted as shown in Fi g ure 4.20b. The value of Am at which chaotic behaviour starts is also indicated on Figure 4.20b by the le tter "x" . It is clear that the well known Feigenbaum universal sequence of period doubling is the route to the chaotic behaviour as the forcing amplitude is increased at this value of ro/roo. It is interesting to observe the sequence of events when one-parameter cuts are made through this resonance horn. Horizontal (at constant Am) and vertical (at c on stant ro/W0) one-parameter cuts are made.
S . S .E.H. ELNASHAIE and S . S . ELSffiSHINI
444
0 . 04 00 � "0
(a)
.: 0 . 0 3 00 E
c.
<
;! 0 . 0200 c.; 1-
E O. O J OO <
0.0000
1 . 00
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-::: 0 . 0 0 0 3 6 �
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1 3.00
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'
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.. .. ..
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<
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1 7.00
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'. I
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5 . 00
S \', D i f u r c e t P D , 0 B i fu r c a l P D,0 B i fu r c a l Ch�os
6 . 00
on
on on
7.00
(a) Period 5 resonance horn (oir»o = P,/P = S/1) for the first case (¥1=0.55342072). (b) Enlargement of the lower part of the period 5 resonance horn in (a). FIGURE 4.20
STATIC AND DYNAMIC BIFURCATION BEHAVIOUR 1 . 0 100 -r------, 1.0300
;:
445
. ...
0. 9 100 0.9300
..··
(a) Am : 0.0003 �T-l .... .... ... .... .. ... " ...., +-�"""T"�.... 4.90 uo s . oo 5.10 5 20
0.8100
.. 1 . 2 0 .St 1!. :: t oo
i l
r:r 0
!&:
].� �
'i.
It
WIW0
-r------,
--
0.10
s�Q,O,!;��D_!_!:�HE!!� -
4 . 10
4 .90
5.00
W I W0
5 . l0
5 10
0. 6 0
0 40
0. 2 0
0 00
(b)
A m :0.00 00 3
o. a o
4.90
s.oo
WIW0
s.10
s . 2o
FIGURE 4.21
(a) One-parameter stroboscopic cut at Am = 0.00003 through the entrainment region of Figure 4.20 showing the subharmonlc period 5 isola. (b) The PFM of the node trajectories on the isola.
Four horizontal one-parameter cuts are made at different forcing amplitudes. Figure 4.2 1 a shows a one-parameter cut through the entrainment region (below the first period doubling point PD1 0 in Figure 4 . 20b) at Am = 0.00003. The dimensionless temperature co ordinate ( Y1 , taken every one forcing period) of the stroboscopic points is plotted versus the frequency ratio ro/ro0 • In this region a stable period 5 node limit cycle, whose Floquet multipliers are real inside the unit circle, exists. Therefore five stroboscopic branches lying on the stable node period 5 appear in Figure 4.2 1 a. In this region there also exists unstable period I focus (descending from forcing the autonomous unstable focus steady state) in addition to an unstable period 5 saddle type limit cycle, whose Floquet multipliers are real with one i nsid e and one outside the unit circle. These saddle and node limit cycles are born at the boundaries of the resonance horn and therefore as ro/ro0 changes the se saddle-node limit cycles change until they collide and disappear at the other boundary of the resonance horn. Outside these boundaries a quasi-periodic region
446
S . S .E.H. ELNASHAIE and S . S . ELSHISHINI
(a)
1 . 1 25 .,....-----, 1 .075 1 . 02� 0.97� 0 . 9 2 !1
+---�....��,... ... � ....,.. �....,. ..., .._,�_......J u�o.o � 000.0 �750.0 7�00.0 l(f)
0.075
! 00 0
1 . 1 25
I I
J .OiS
>-
(b)
. . \
•,
! . 02� 0 975
·.
'
o.n, 0 075
0.00
0. 1 0
0.20
· "·
XA I
"'- .
__ ""' _ _) O.JO
0.�0
FIGURE 4.22 Numerical simulation in the quasi-periodic region outside the entrainment region (Figure 4.21) at Am : 0.00003 and oir»o : 4.7. (a) A quasi periodic time trace. (b) The stroboscopic phase plane gives a stroboscopic view of the torus (invariant circle).
prevails. Figure 4.2 l b shows the principal Floquet multiplier of the node trajectories. Noti ce that the principal Floquet multiplier become equal to unity at the two boundaries of the resonance hom (subharmonic saddle-node bifurcation). Thus this one-parameter cut through the entrainment region represents subharmonic period 5 isola (Figure 4.2 1 a) in the "ocean" of the quasi-periodic behaviour. Numerical simulation in the quasi-periodic region outside the entrain ment region at Am = 0.0000 3 and OJ/OJ0 = 4.7 is shown i n Fi gu re 4.22. A quasi-periodic time trace is shown in Figure 4 . 22 a . The stroboscopic
STATIC AND DYNAMIC BIFURCATION BEHAVIOUR 1 . 1 300 ......----,
(a)
··!
1 .0800
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1 . 0300
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(b)
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447
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One parameter stroboscopic cut tbrough period 5 resonance horn (Figure 4.21 ) at (a) Am = 0.00009; (b) Am = 0.000 1 05; (c) Am = 0.0001 16.
FIGURE 4.23
ph as e p1ane is shown in Figure 4.22b which give s a strob o sc opic view of the torus (invariant circle). The Ly apu n ov exponents (Lys) are computed using the efficient a1gorithm of Wolf et al. ( 1 985). The Lyapunov exponents are found to be both negative of values equal to Ly1 = -0.00 1 3 1 and Ly2 = -0. 342609, assuring the presence of a quas i
periodic attractor. Figure s 4.23 (a,b,c) show horizontal one-parameter cuts at three va1ues of the forcing amplitude (Am = 0.00009, A m = 0.000 1 05 and Am = 0.000 1 1 6). Figures 4.23(a, b) show that the s ystem exhibits i ncomp lete
S.S .E.H. ELNASHAIE and S . S . ELSHISHINI
448
period doubling cascades. The period 5 stable node on both sides of the cascade is born through saddle-node bifurcation at the boundaries of the resonance horn. At the ftrst period doubling points (PD1 0) , the torus on which frequency locking occurs is broken because it is impossible to have different periodic trajectories locked on the same torus. The smooth invariant circle is broken i. e. the torus structure collapses and the system will have only one frequency of the sustained period 10 oscillations. Figure 4.23(c) shows one-parameter cut at Am = 0.000 1 1 6. Two chaotic regions are clearly shown. As ro/roo increases from left to right the Feigenbaum universal sequence of period doubling (PD5, PD 10, PD20, PD40, . . ) takes place leading to chaos. The starting point for this chaos is denoted by the letter "x" in Figure 4.20(b) . The corresponding Lyapunov exponents of the attractor in Figure 4.23(c) at wlwo = 5/1 are found to be Ly1 = 0.010586 and Ly2 = -0.363926. The largest (maximal) Lyapunov exponent (Ly!) is positive giving positive proof of the chaotic nature of the attractor. The Lyapunov fractal dimensions (LYFD) are computed using the Kaplan and Yorke conjecture ( 1 983) as shown by equations (4.33, 4.34). The corresponding Lyapunov fractal dimension of the attractor is equal to 1 .029 1 . Changing wlw0 the chaotic attractor bifurcates to periodic trajectory having period 10. This period 10 can be considered a periodic window similar to the well known periodic windows that develop within the chaotic regions of autonomous systems (Elnashaie and Abashar, 1 995). As ro/ OJo increases the period 1 0 window bifurcate through period doubling sequence to another thin chaotic region. These chaotic attractors loose their stability through period halving sequence as wlw0 increases further to period 5 at the end of the right hand side of the boundary of the resonance horn. The Figures on the right of Figures 4.23(a-c) represent enlargement of the crowded strips at the middle of these figures. A vertical one-parameter cut through the resonance horn in Figure 4.20(b) at wlw0 = 5/1 (tip of the horn) is made from entrainment region to the chaotic region as shown by the thick arrow in Figure 4.20(b). Five branches (a-e) emerge on the stroboscopic one parameter bifurcation diagram as shown in Figure 4.24.The branches (b, c, d) are very close to each other and therefore this region is further enlarged in Figure 4.24(b) . It is clear from Figure 4.24 that the system follows the well known Feigenbaum period doubling (PD) route to chaos. Figure 4.25 shows the trace of the Floquet multipliers from the tip of the resonance horn till the first peri od doubling point. As the forcing amplitu d e increases the ftrst Floq u et multiplier (FM) de c reas es from unity to zero, while the second Floquet multiplier (FM) remains almost around the zero value. When the ftrst Floquet multiplier approach e s zero and remain around that value, the second Floqu et multiplier .
decreases to negative values and exit from the unit circle at (- 1 ) as
STATIC AND DYNAMIC BIFURCATION BEHAVIOUR
449
. ... 1 - 1 1 0 T"-----------------...... 1 - 080 1 - 050 a
1 -020 ....
>
.
b- d
0 99 0
· ···�·
0 960
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0 - 00000
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b
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0.996
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.
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0 DOOlE
4.24 Vertical one-parameter stroboscopic cut th rough p eriod 5 resonance horn (Figure 4.21) for oi� = S/1.
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
450
0.0000&
" "0 ;:J -
0. 0000 6
'
-- F M - - ---
'
'
a.
E
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0
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t I
{a )
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- 0.50
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1 .00
0 . 50
Nod �
0.00
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R�
- 0 . 50
- 1 . 0 0 t-�........ .. .... .. :;:;.'""'"""';:;;;.....-r,"""' .;: ....... ,.....,..,..f 1 .0 0 0. 50 - 0. 50 0. 00 - 1 .0 0 ( b)
FIGURE 4.25 (a) The trace of the Floquet multipliers from the tip of the period 5 resonance horn (Figures 4.21, 4.24) till the first period doubling point at Am ;;: 0.0000725. (b) The node period 5 PFM crosses the unit circle at (-1) at the period doubling (PD) conditions.
structure is broken because it is impossible to have a torus structure having more than one periodic trajectory of different peri od s locked on it (Kevrekidis et al. , 1 986). The Lyapunov exponents inside the chaotic region at Am = 0.000 1 1 6 are equal to Ly 1 = 0.0 1 05 86 and LY2 = -0.363926. The largest (maximal) Lyapunov expo nent (Ly 1 ) i s po s i tive gi ving posi ti ve proof of the chaotic nature of the attractor at this value of Am. The corresponding Lyapunov fractal dimension of the attractor is equal to 1 .029 1 .
STATIC AND DYNAMIC BIFURCATION BEHAVIOUR 1 . 1 1 0 -r-----.
(a )
1 .0 8 0
45 1
1 . 0 50 1.020
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0 0001.2
0.00 046
A m, Fo r c i n g A m p l it u d e
FIGURE 4.26
Small segment of the one-parameter bifurcation diagram at oi(J)o 5/1 between Am = 0.00034-0.00046 showing clearly that the system exits the period 5 resonance horn (Figure 4.21) to a quasi-periodic region through period halving mechanism. =
The small segment of the bifurcation diagram shown in Figure 4.26 between Am = 0.00034-0.00046 shows clearly that the system crosses the resonance hom to a quasi-periodic region through period halving as is also clear from Figure 4.20(b). The Lyapunov exponents at Am = 0 . 0004 2 are found to be Ly 1 = -0.02249 1 and Ly2 = -0.333234. The two Lyapunov exponents are negative giving positive proof of the quasi periodic behaviour.
452
S . S.E.H. ELNASHAIE and S.S. ELSHISHINI 0 . 0 1 800 -,-----------------, (a)
. .., � 0 . 0 1 350 .. E <
�0. 00900 E � i O . O O < 50 o . ooooo, .+ e o=--��--:-:,-:---: � ·"'"' • o=----=�"6""" o ---:� " . e'o--::6 . o-:! o 4." · c.: .,
o. o�•oo
� 0. 0 . 03800 ��
!
o . o ,�oo
l -----...,.------.,------, rl-:b-:-
·t �0 0 1 800 ro
02700
<
E
0.00900
' . 0.00000 ___,__..,...______,..___,_ _...j 5.20 5.30 5.40 5 . �0 4 . 90 5 . 00 5 10 �:.,: / <--' o
]
0 . 1 2 6 -1 0
� 0 . 0 \J.I � O
c.
�
� O . O f- 3 2 0
"-
;: ;e
0.03 1 60
I
(
//
+r.-�....1�..:t- �,...,--- ...,.-.-��........-.l
0 . 00000 � . eo
FIGURE 4.27
/
(')
:> oo
r...• /r-J.
5.2o
5.40
:> . 6 o
5.80
Resonance horns for: (a) Case 2; (b) Case 3; (c) Case 4.
ll-2.2. The second, third andfourth cases (Yf= 0. 5532, 0.5500, 0.5350)
The corresponding parts of the resonance horns for these cases which are having centres of forcing further away from the homoclinic orbit are shown in Figure 4.27. For these horns (for the regions of low amplitude forcing shown) the situation is very different from that of the first case in Figure 4.20. No period doubling mechanism exists inside the horns (for this range of Am) shown in Figure 4.27, neither leading to chaos nor incomplete period doubling.
STATIC AND DYNAMIC BIFURCATION B EHAVIOUR
453
Therefore both horizontal and vertical cuts through these horns will only show periodicity (period 5), with no occurence of neither period doubling nor chaos. Thus the effect of the position of the centre of forcing relative to the homoclinic orbit at this high frequency forcing hom of ro/ro0 = 511 and small Am, can be summarized by the aid of Figures 4. 1 9 to 4.27 in the following points : 1 . The possibility of chaotic behaviour occuring within the boundaries
of the resonance horns at relatively low forcing amplitudes is more likely when the centre of forcing is closer to the homoclinical orbit (IPB point) of the autonomous system. 2. The further away the position of centre of the forcing from the homoclinic orbit (IPB point) the narrower the resonance hom and the sharper is its tip. 3 . At low forcing amplitudes the position of the centre of forcing relative to the homoclinic orbit (IPB point) affect the veering of the left boundaries of the horns. The nearer the centre of forcing from the homoclinic orbit (IPB point) the larger the left veering of the left boundary of the hom. They tend towards larger periods i.e. towards low frequencies as an indication of homoclinicity (Figure 4. 19b ). 4. At relatively large forcing amplitudes the whole horns tum in opposite sense to the position of their centre of forcing relative to the homoclinic orbit (IPB point) . The nearer the centre of the forcing to the homoclinic orbit (IPB point) the more veering of the whole hom to the right to lower periods i.e. high frequencies (Figure 4. 1 9a). It is clear that the position of the centre of forcing has a profound effect on the entire behaviour of the system. This shows a qualitative difference between systems with SLPs and those without and give the reader a sense of the richness of the dynamics of forced (non-autonomous) systems when both HB points and SLPs co-exist. II-3.
The effect of the position of the centre offorcing relative to the homoclinical orbit of the autonomous system on the behaviour of the system at low frequency forcing ( ro I ro 0 = t)
Three of the previous cases 1 , 2 and 3 (with different position of the centre of forcing relative to the homoclinic orbit) are forced at relatively low frequencies OJ!OJo = P0 1P = 2/1 to demonstrate to the reader the effect of the po siti on of the centre of forcing on the b ehavio ur of the system at low frequency forcing. A resonance hom is constructed for the first c as e and a complete vertical one-parameter bifurcation is made. For the other cases only the vertical one-parameter diagram s are constructed from the entrainment regi on to the harmonic region.
454
S.S.E.H. ELNASHAIE and S.S. ELSHISHINI 0.01 1 00
� 0 . 00860
-- SN, Biturc • l i o n - -- - - PD. Bifurc•lion
:I .....
c..
� 0 . 00 6 6 0 "L c:
�
c t...
....
0 . 004 4 0 0 . 00220 (a) .,... .... .,... ""f'"t.:,:..;,-:;.,..., ... ���,...,. ...., � ...�.......-� 0 . 0 0 0 0 0 ...j-,o.... 2.00 0 . 00 tl . O O G.OO
0 . 0004 5 t.' "=
�
0 . 00036
c..
� 0 . 00027 t o . O OO I O
c 1:...
...:
0 . 00009 (b ) 2 . 00
w /w o
3 .00
4 . 00
FIGURE 4.28 (a) Period 2 resonance horn oir»o = PJP = 211 for the first case (Y1 = 0.55342072. (b) Enlargement of the lower part of the period 2 resonance horn in (a). /1-3. 1. The first case ( Yt= 0.55342072) The part of the resonance hom constructed for this case is shown in Figure 4.28a. The boundaries o f this hom are the loci o f saddle-node bifurcation and n o meta bifurcation points (codimension two bifurcation) were detected. Two different re g i o n s of period d ou bl in g are detected. One region forms a fi nge r near the left saddle-node boundary of the resonance hom. Two segments of this finger at A and B boxes are
STATIC AND DYNAMIC BIFURCATION BEHAVIOUR 0 . 00 2 8 0
� 0 . 00 2 5 2
455
T"""-----r-: -- SNr Bifurcation · - - - -
PO, Bifurcati o n
:;l
_,
c..
� 0 . 00 2 2 4 ., c
� 0 . 00 1 96
c '-'-
E
<
0 . 00 1 66 A 0 0 0 1 4 0 -+-----.--"'----....-----�.... ... .-1 3 . oo 3 .2o 3.40 3 6o 3.ao 4 . oo 0 . 0 0 7 5 0 ------ S S , B i f u rc a t i o n · - · · - P D , B i f u rc a t i o n
c < 0 . 00 7 1 6
� 0. 00702
.;:::
E
<
0 . 0 0 6 /.l (l
O . OOG70 4 GO
FIGURE
B 4 . (i 2
4 .64
4 .66
-1 . 6 !1
4 . 70
4.29 Enlargements of boxes A and B in Figure 28a respectively.
enlarged in Figures 4.29(A, B). It is clear that the period halving branch is very close to the boundary of the resonance hom. The other region of period doubling forms an oval at low forcing amplitude as shown in the enlarged lower part of the resonance hom in Figu re 4.28b. Both period·doubling se quenc e s are clearly incomplete and do not lead to the occurence of chaos inside the hom. The vertical one-parameter cut through this hom is shown by the thick arrow in Figure 4.28b.
456
S.S.E.H. ELNASHAIE an d S . S . ELSHISHINI 1. 1�0 1 . 1 10 1 . 060
1 . 050
;,-
1 . 020 0 . 990
0.960 0 . 11 30 0.900 0. 670
B
A
C D
+,-.,...,.....,...,.,...,,..,...,-rr-:-mTTT-rrr...,--.,-,...,-,+.-.,.f-r-'-r-r,...,-,..,-i
0.00000
0 . 00 0 0 5
0.000 1 0
0.000 1 5
Am .Forc i n g
0 . 00 0 2 0
A m p l i tu d e
0. 00025
0 . 0 00 3 0
FIGURE 4.30 One-parameter stroboscopic bifurcation diagrams (strobing every one forcing period) at oiOJo = P,)P = 2/1 for the first case ( Y1 = 0.55342072).
It is interesting to follow the sequence of the events in this one parameter cut as shown in Figure 4.30. The system crosses the entrain ment region twice through the encircled period doubling region (Figure 4 .2 8b) forming incomplete cascade of period doubling as shown in Figure 4.30. This incomplete cascade of period doubling is shown clearly in the enlarged portion of a horizontal one-parameter bifur cation diagram in Figure 4.3 1 a. Figure 4.3 1b is a further enlargement of Figure 4.3 1 a between Y1 = 0.9950-0.9968. The Lyapunov exponents at Am = 0.000 1 4 are negative, having values equal to Ly1 -0.000333 and Ly2 = -0.363 142, confirming the quasi-periodic behaviour outside the hom. Then the system passes through a very complicated sequence of events which contains periodic windows until the system is fully entrained and becomes periodic with a period equal to the forcing period as shown by the single line in Figure 4.30. The system bifurcates from the quasi-periodic re gion to the harmonic period 1 attractor as the two negative Lyapunov exponents clearly show (Ly1 = -0.002365, Ly2 = -0.336 1 44) at A m = 0.000235 . This bifurcation is a Hopf bifurcation according to the c o nje cture number 6 made by McKarnin et al. ( 1 988) that all tori that do not become pha se- lo cked in the re s o n anc e hom =
undergo a Hopf bifurcation at a period one Hopf curve.
STATIC AND DYNAMIC B IFURCATION BEHAVIOUR 1 . 11.0 1 . 1 10
(a)
-r-----.
1.090
1.05 0
·····
1 . 020
>
457
0.9 90
. . . . . . . . . . '
· ·· ················· · ·· ···· .. . . . . . . . . . . .
�
I ' • •
. . . . . ' . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
0.9GO
0 · 930
0.900
... .. __,.,......,. ..., O. B70 t-........ ______---i _ ..
0 -00002 5
o . oooo 4 s
o . ooo o G s
Am , Fo r c i ng A m p l i t u d e
o . ooooes
0 .9 956 0 . 9953
-4 .. .. ---------...... ... 0 - 995 0-f-....-.--... Q . 0 0 0 025
0- 00001.5
0- 0000 6 5
A m , Forc i n g A m p l i t u d£>
o . o oooes
FIGURE 4.31 (a) Enlargement ofthe part of the stroboscopic bifurcation diagram in Figure 4.30 (Am = 0.000025-0.0000 85) showing incomplete cascade of period doubling. (b) Further enlargement of part of the stroboscopic bifurcation in (a) (Y1 = 0.9950-0.9968).
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 adding-sequence as observed earlier by some investigators for a physically different system, namely the Rose-Hindmarsh model for neuronal activity (Holden and Fan, 1 992) .
S . S .E.H. ELNASHAIE and S . S . ELSHISHINI
45 8
• . • ,0
(a)
J .l lO
1 . 0&0 1 .050
1 . 020
>
0.990
0.860 0.930 O.SMI(J
A
!
0.870
O.OOQ(IO
O.OOtOO
J . O&.O
B I
I I ll
O.O
0 . 0 1 000
(b )
1 .020 0.990
>=
0.980 (1 _931) O.liltJU
A
o.e7o 0.02000
0.02500
0 . 0 3 000
0.03500
0 . 0 4 000
Am , F o r c i n g A m p l i t u d e
0.04:.00
FIGURE 4.32 (a) One-parameter stroboscopic bifurcation diagrams (strobing every one forcing period) at oir»o = PofP = 21l for the second case (Yt= 0.5532). (b) One-parameter 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 one-parameter 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 one-parameter 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,
� 0.00880 "
Bifurcation
'ii. E o. oo55o
'
<
.. " '!j � o . oouo ": E <
0 {10220 o. ooooo
459
'
, -1:.o��z;:;; ..; ��;..;,.� .. "'�"' .o�� e .� o ��7�.o��e�.o�.,..,.,.,. g.o .l . o ;::;:.� (a)
GJ/c.J,
o.ooc•� .,....-----,---,-,
� 0.00036
� 0.00027
..3 'ii.
s <
I
/' .; ./
...
0 . 00009 o.ocooo
I
I
.. 1': ·o; � 0.000 1 8
.·
(b)
. /·
/
,_ , ·
,1''
.· · J ,', ,
"'
, ' - H,. - - - Hu · · - - - - - PD, Bifurcation
. 5 --�3-.o�,...,_,;:!-3-':.5===l , 'l:.o---,� . �--..:;:,.,.-;:;.......� .:-. z .:. ,.o w/ c.J,
FIGURE 4.33 (a) Excitation diagram for the 2:1 resonance horn (H21) and 5:2 resonance horn (H52). (b) Enlargement of the lower part of the exitation diagram in (a).
during its pathway from the subharmonic region to the harmonic region. The bifurcation diagram is characterized by periodic regions such as regions (A-E in Figure 4.32b). These periodic regions do not show period adding sequence like the previous two cases, however they exhibited period doubling sequence. The system is fully entrained at forcing amplitudes larger than the previous two cases. In this case the entrainment region is also larger than the previous two cases indicating that the further away the position of the centre of forcing from the homoclinical orbit th e sharper the resonance hom. II-4. lnteraction of the resonance horns (WIWo = 211 and 512) The complete description of the excitation diag ram and the ful l
analysis of interaction between resonance horns is very difficult because
S . S .E . H . ELNASHAIE and S.S. ELSHISHINI
460
1 . 1 50 0 ,.-----., (a) J . l ODD
;:
! '.
i !
��-/
1 .0500 J .OOOO
.
-
�-----+--+��
0 . 9 :"> 0 0
\
0. 9 0 0 0 0 . B 5 00
l
.·.
A m , = O.DDOO I
1 .96
I
<.
.
�; ;.
+.-....,.,. ... .,_..,. ..,.. .,..,...,. ....,. .,. ... "T"'""'...,. ..,... ,. ..,.., .. ...,.. ,..,. .,.,... ,. .,... .,... ,..,...,....j 0
1 . 1 �:10 .-------., (b)
w/wo 2.00
2. 1 0
2.20
.
_.:
;::
� •�=o :z • •
Am1 : C. OOOO I
--' +---�-�-..--z �0 W / w0
FIGURE 4.34 (a) One-parameter stroboscopic cut at Am1 = 0.0000 1 through the entrainment region of Figure 4.33b showing the sub-harmonic period 2 isola for 2:1 resonance horn (H21). (b) One-parameter stroboscopic cut at Am1 = 0.0000 1 through the entrainment region of Figure 4.33b showing the sub-harmonic period 5 isola for 5:2 resonance horn (H52).
of the infinite number of resonance horns due to the infinity of rational numbers . Therefore we concentrate on the interaction between two resonance horns . The i nv e s ti g ati on of the interaction of the resonance horns is co nducted through the co n s tru ct i on of the ex citat i o n di agram for centre of forcing close to the IPB point ( Yr= 0.55342072) at two values of ro/roo namely 2/ 1 and 5/2, shown in Fi g ure 4.33 . For clarity the very lower portion of the Fi gu re 4.33a is e nlarg ed in Fi gu re 4.33b. The boundaries of these parts of the ho rn s are l o ci of saddle-node bifurcation. At very small forcing amplitudes these two horns do not overlap and the e ntrainme nt regi on s prevail in each horn with periodicities being integer multiples of the forcing period for all frequency ratios. The two horns intersect at point x as sh o wn in Figure 4 .33b.
STATIC AND DYNAMIC BIFURCATION B EHAVIOU R
46 1
l . lliOO .,...-----,
(a)
1 . 1 000
;;::
I .OliOO 1 . 0000
··-.._
0.1�00 0 .8000
Am,=0.0004 .,. .... .,.... .,... ..,.. .., .,...� .., ,...� ..,. ,...�,... ... .,.,.,. ..., ..,� O.BMO +,-......,. 3 . 80 3.8� 3.70 3.7� 3.80 3.85 3.80 3.15
t.J/c.J.
...,.,..,j
4 .00
l . lMO ;-------,
(b)
1 . 1 000 1 .0500
;;:: 1 .0000
0 . 1!'>00
0 . 9000 0 . 8500
Am1•0.0004
. ...�. - - .
�-.,-...,--.,..... �-,...... �_,...,-....-� ...
3.60
3.65
3 . 70
3 . 75
3 . 80
3.85
3.90
3.95
4 . 00
r.: /c.;.
FIGURE 4.35 One-parameter stroboscopic cuts through the 2:1 resonance horn (H21) and the 5:2 resonance horn (H52) in Figure 4.33b at Am3 = 0.0004 for different
initial conditions showing clearly the bistability of periodicity 2 and 5 simultaneously.
Fi gures 4.34a, b show horizontal one-parameter cuts at Am= 0.0000 1 . Fig ure 4.34a s ho w s periodicity 2 w ithi n the hom 2/ 1 while Figure 4.34b shows periodicity 5 within the hom 5/2. However for the larger ampli tudes the two horns and the period doubling oval interfere with each other as ex pl ai ned above giving rise to different regions of bi s tabi l ity . Figu re s 4.35a, b show the on e -p arameter bifurcation diagrams at Am = 0.0004 (Figure 4.33b) obtai ned from different initial conditions. Bista bility exists, i. e. periodicity 2 and periodic ity 5 e xi s t simultaneously and each trajectory can be obtained by different initial conditions . Two and five distinct horizontal lines do exist. It is clear from Figures 4.35a, b that for ro!OJo = 3 . 65 two horizontal lines (periodicity 2) exist on Figure 4.35a �nd five h ori zontal lines (periodicity 5) exist on Fi gu re 4 .35b and rolroo = 3 . 67 five hori zon tal lines (periodicity 5) e x ist on Fig ure 4.35a and two horizontal lines (p eriodic ity 2) e xi st on Figure 4.35b.
S.S.E.H. ELNASHAIE and S.S. ELSHISHINI
462
(Q)
1 . 1 4 ""T"-----, 1 .1 0 1.05 >-
1 .0 1 0.96
0. 92 •
Stroboscopic
Points
........ .. ... .. ""'"T .. ........ ... .... .. .,... .. . ........ .. I""P"T" ... .. ... 0.8 7 -t-........
3140.0
1 .1 4
326!5.0
3515.0
3390.0
T(s)
3640.0
-r----...,
1.10 1.05
>-
1.01 0.9 6
, 1� 1 �
0. 9 2
0.1 7 ������ 31 40.0
•
Stroboscopic Po i nts
3315.0
349Q O
3665.0
,. ( s )
3840.0
FIGURE 4.36 Time traces at Am3 ;;; 0.0004 and oiat� "' 3.65 for different initial conditions. (a) Time trace of periodicity 2. (b) Time trace of periodic ity 5. Time traces at Am = 0.0004 and ro/ro0 = 3. 65 are made for different initial conditions (Figure 4.36). The stroboscopic points on the time trace i dentify the periodicity of the trajectory . The trajectory on Figure 4.36a has a periodicity 2 while the traj ectory on Figure 4. 36b has a
periodicity 5.
III-
Summary of results of this section
Detailed numerical results have been presented and discussed in this section for the effect of the forcing amplitude frequency and the position of the centre of forcing, relative to the homoclinical orbit of the ,
STATIC AND DYNAMIC BIFURCATION BEHAVIOUR
463
autonomous system, on the behaviour of an oscillatory fluidized bed reactor. The results, presentation and discussion are more generalized than the preliminary presentation in the previous section. At low forcing amplitude the response of the system is dominated by quasi-periodicity with interspersed regions of subharmonic resonance having periodicities which are integer multiples of the forcing period for all forcing frequencies. At very high amplit ude forcing a harmonic region prevails having periodicity which is the same as the forcing period because the system is overpowered by the external forcing. These two limits are in a sense universal for all forced oscillators. However, even in these universal limiting regions the behaviour of the system is affected by the position of the centre of forcing. The position of the centre of forcing relative to the homoclinic infinite period orbit of the autonomous system affects the width and the length of the frequency locking regions and the position of the harmonic regions. The effect of the position of the centre of forcing relative to the homoclinical orbit can be summarized for these limiting regions as follows: ,
1 . The further away the position of the centre of the forcing from the homoclinic IP orbit the narrower the resonance hom and sh arper its tip. 2. The further away the position of the centre of forcing from the homoclinic IP orbit the higher the forcing amplitudes at which harmonic regions prevail .
The response of the system for intermediate values of the forcing amplitude is not generic and depends on the inherent nonlinearity of each system. At this region the centre of forcing as well as the magnitude of forcing amplitude and frequency have profound effect on the behaviour of the forced oscillatory fluidized bed reactor. All types of complications can occur in this region from period doubling to chaos. The major types of complex behaviour shown in this section for this region, can be summarized as follows: (a) For the cases with the centre of forcing very close to the homoclinical orbit (IPB point) of the autonomous system, with high frequency forcing mlmo = 5/1 ) and for small forcing amplitudes, there are concentric ovals of period doubling sequence inside the resonance horns leading to chaotic behaviour. Thus very small ampl itude s are sufficie nt to cause chaotic behaviour of the system. These small amplitudes resemble the natural disturbances and external noises that any ph ysi cal system may be s ubjecte d to in practical situations. H owever for the same case when the centre of forcing is moved away from the homoclinical orbit the period doubling and ch ao s
464
S . S .E.H. ELNASHAIE and S . S . ELSHISHINI
inside the resonance hom both disappear for these small amplitudes . (b) Large number o f periodic windows exist within the chaotic regions. (c) For high amplitude values a number of cases presented show the new type of period adding bifurcation. (d) For the cases with the centre of forcing very close to the homoclinical orbit but with relatively low forcing frequency ( ro/ W0 = 211 ) there are period doubling ovals inside the resonance hom giving incomplete period doubling in contradistinction to the case of ro/ro0 = 5/l which give sequence of period doubling leading to chaos. However similar to the case with ro/W0 = 5/ 1 as the centre of forcing is moved away from the IPB point the period doubling ovals disappear. (e) The overlap of two of the resonance horns investigated leads to hi stability, in general overlap for many resonance horns may lead to mutlistability . The qualitative and quantitative methods proved that the strange attractors presented are chaotic with one positive Lyapunov exponent and Lyapunov fractal dimension between 1 and 2. There are two bifurcation mechanisms to harmonic region. The first mechanism i s Hopf bifurcation at which two Floquet multipliers leave the unit circle at an angle . The second mechanism is period halving. The two mecha nisms agree with the results of Kevrekidis et al. ( 1 986) and conjectures made by McKarnin et al. ( 1 98 8) for forced system near Hopf bifurcation points. The excitation diagram can be used to define the operating ranges for the forcing amplitude and frequency to yield response free from chaotic oscillations when they are harmful to the reactor or to make use of them when they are beneficial . 4.1.2
The Three-Dimensional Case
We will present to the reader in this section the main features of the dynamics of the three-dimensional case of the fluidized bed catalytic reactor where the adsorption capacity of the desired product B is not negligible . It is shown that in this case chaotic behaviour of the autonomous system is possible. The presentation and discussion will be restricted to the autonomous case with SISO proportional controller, since the non-autonomous case involves forcing chaotic attractors which is still a research subject and not enough is known about it.
The unforced (autonomous) case with proportional control
For this part, 1 /LeA :;t: O, l!Le8 :t: O, K :;t: O, Kt = KJ = O.
STATIC AND DYNAMIC B IFURCATION BEHAVIOUR
465
The model equations (4.8-4. 1 1 ) represent both the proportional feed back controlled system as well as the non-adiabatic cooled system with only a change in the physical meaning of the parameters in equation (4. 1 1 ) For the feed back controlled system, K represents the normalized proportional controller gain and Ym represents the dimensionless con troller set point. For the non-adiabatic case, K represents the normalized heat transfer coefficient of the cooling jacket (or coil) and Ym represents the dimensionless coolant temperature. The case with K equals to zero represents the uncontrolled adiabatic case and K can be less than zero only for the controlled case with positive feed-back control. The parameters LeA and Le8 are two dynamic parameters which do not affect the steady state solutions of the system, however they have a profound effect on the dynamic characteristic of the system (e.g. Ray and Hastings, 1 980; Berezowski and Burghart, 1 993). LeA and Le8 represent the ratio between the heat capacity of the system and the mass capacities for components A and B respectively. For liquid phase reactors LeA and Le8 w ill almost always be close to unity. For gas-solid catalytic systems the mass capacities of a strongly chemisorbed component may exceed the heat capacity of the system and therefore LeA and Le8 may both vary strongly from unity (Aris, 1 975; Elnashaie and Cresswell, 1 973a-c, 1 974; Elnashaie et al. , 1 990). Dynamic models which assume negligible chemisorption capacities of the different components on the catalyst results in very large values for LeA, Le8 therefore the mass balance equations (4.8, 4.9) predict extremely fast response for XA and X8 compared with the response of Y. In this case XA and X8 can be assumed at pseudo-steady state all the time. Under these physically un justified assumptions the system becomes dynamically one-dimensional and most of the dynamic characteristic of the system are therefore lost. The important effect of chemisorption capacities on the dynamic characteristics of gas-solid catalytic systems has been recognized by a number of investigators (Elnashaie and Cresswell, 1 973a-c, 1 974; Elnashie, 1 977; Arnold and Sundersan, 1 989; Elnashaie et al. , 1 990; Il ' in and Luss, 1 992; Elnashaie and Elshishini, 1 994), but has not been fully investigated yet. In the present model the chemisorption capacity of the solid catalyst in the dense phase is taken into consideration and the richness of the dynamics associated with the relaxation of the physically unjustified assumption of neglecting these important mass capacities, is demonstrated. The parameters used in this presentation are given in Table 4.2. .
I- Results and discussion concerning the bifurcation and chaotic behaviour ofthe three-dimensionalfluidized bed catalytic reactor /-1. Static and dynamic bifurcation diagram With all parameters kept constant except K and Y1, s ti ll a rich variety .
S . S .E.H. ELNASHAIE and S . S . ELSHIS HINI
466
(a)
� 0 . 8 r----,
�
0 )o. 0... o . e .,
ii..
a , ., 1 08 a 2 = 1 01 1 )' o = 1 8 Jz= 27 ,8 , = 0 4 fJ 2 = 0 . 6 l f = 0. 5 53 42 0 72 l m = 0. 92955 1 3 0
t:!
E..
�
0.4
10 ..... 0 . 2 "t:!
-.: ::::
1.2
(b)
1 . 350
1 .3
1 .000
';::- o.eso :::::::
� '-
�
0 . 300
. . . . .. . . . . . . . . . .
::: ::: ::
:.::.:) :
.·
..
K= - 1 . 0
-o o5o
....�� .. ....� ... ..__._....��.... ... .. ...._ ..__._.... ..._ .._,��....... 0.6 0.7 0.9 Ym 1 . 0 1 .2 1 .3 }', Dimens ionless Dense Pil cr.s e Temp e r a ture
- 0 . 4 00
FIGURE 4.37
Yield and Van Heerden diagram for the data in Table 4.2. (a) Yield of intermediate product B vs. dimensionless dense phase temperature Y, where Ym is Y at the maximum yield of B. (b) Van Heerden diagram showing the heat generation function of the system Hg(Y) and heat removal line Hr(Y) for different values of K vs. the dimensionless dense phase temperature.
of bifurcation behaviour is encountered. The parameters of the control system (or the cooling system for the non-adiabatic case) are three, name l y K, lj; Ym . The value of Ym which is the set point for the controller (or the coolant temperature for the non-adiabatic sy s tem) is kept con stant at the temperature corresponding to the maximum yield of the desired intermediate product B (Figure 4.37). Figure 4.37a shows the yield of the desired in termedi ate product B versus the dimensionless dense phase te mperature Y. Figure 4.37b shows the Van Heerden diagram (Elna shai e , 1 977) co mp o s ed of the heat generation function Hg ( Y) and
.. ';; n .. ..
�
� c
.. .. � �
STATIC AND DYNAMIC BIFU RCATION BEHAVIOUR
2.5
2·0
.,------;:=====::::;1 Sf.
0 0
1·5
c ..
1·0
c 0 · ;;;
o.s
..
:::
..c
e
q >
� 1 . b r .1 n c h - S r a b l • • � • • t.l n 5 I :s b I f t n :. n c h rulod ic • • • S r o b l • 00 0 U n s r • � l •
HRI•J./8Jl12
�
Q
467
- -- - -- ----------- --- -------
1
I
_ .., _ _ _ _
..
� · ' · e->"-"--= ·
----
H BJ � J . 6 /509J
ssr-J.8JHn
II B 1 • 1 . J / U / 8
0 · 0 +-----�--�--��3-0 5-0 - 1 ·0 0·0 1·0 2 ·0 6 0 C o n t roller G a in
(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) .
S.S.E.H. ELNASHAIE and S . S . ELSHISHINI
468 �
"§ � :J
E �
.. .. Cl £ !l.
�
c; .. 0
.. ..
� �
�
-� E i:5
>-.
2 . 00 ,-------, (ib )
1 . 50 1 .00 --
0.50
----
•• • •
JJ:
•; ;- - - - - - - - - - • • • • • • • • •
• • 0
0 0
R B I : J . 66046 2 . . . · · · · · · H.B.h
.
7 7 5400
S t . sL b ra n c h - S ta b le - - - - u n .table Pe r i o d i c branch
0.00 +-----L"""T"-" -----..---------:-,_____::=::::= ::::; ===---! - 2. 5 5. 0 2.5 10. 0 7. 5 o. o • •• ooo
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 . I-2.
Periodic Attractors
From the dynamic point of view, this non-chaotic 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).
STATIC AND DYNAMIC BIFURCATION B EHAVIOUR 0.990
�
�...
.::
.------�
I
0 . 965
E
�
....
0.980
�. "' c:..
.
0 . 97 5
�
�
::l
�
0.970
�
0
c
;;;
" �
;..
0 960 7.55
.... 1. 5 0 ::0
"
.. ... 1 4 0
'- � ': ' �\ r
�
I!
• •
e ee •
;. .. 1 . 3 0 ..
::..
::: c:
� � ..
� � "
1. 1 0 1 .00
• "' H;
••
le•e e e
·. :
•
ee
• •
••
,.,iod I
eo .,
� ��
(a) 7.70
I
R•'• / .
1
Roflt
/
•• ••
7.75
J
St. 11. --
Pt r lo d l c
-
.. .. ..
�
� 1 20
7 . 65
7 . 60
e
HB2 = 7 . 7 7 5 4 0 0
.
0 . 9G5
c •
469
�''H'd 1
�
oCo
• • •
frflod
v
� �� O oo o c o o o o oo��
- - - - - - - - - �;,- �-;; � -; �- oo UMu-
7 . !" " l
lb r u c b Stablr
tuuch !Habit
4
(b)
�-:-:-i-! :-% - B ·I C-t '!'--1
� · · · · · -�· · · · · · · �· H 8 1 • 1 . 1 1 J tD O E 90+---....,..---"""T----.--,.---l 0 . i:5 7 .70 7 . 65 7 . 60 7 . 55 7 . 75 7 . ; :; > Co n l r o l l o r G a i n ( K )
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). 1-3.
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
S . S .E.H. ELNASHAIE and S.S. ELSHISHINI
�
� ft
1. so .-------------;ll"'·-··...,l.-----;ll ;•-•�·1,---. -. --. -. -. -. ...---. . 1 40
• ••• • •• • •
• •
• aa
•
/
•••
:: 1.30
• z: II.
�::" Q
1 20
� ·�" e
1 .00
· - -
�� �m
•••
'l l
• ••
.
. .
.
.
(a)
\ : �
...� '
.. ., 1.1 0
Subhra 1
0
0
0
n°o Q,-o �� � �� ��
11 or
��Foo o o o o o o o o o o o o o o o o o o o o o o ------·-
o oooooo ooa
- - - - - - - - - - - - . - - - - - - - - - - - - - - . - - - - - - - - - - - - - - - - - - - - - - - - - - - -
Q > 090 +---.----.--�---r--� 7. 6 2 0 7. 600 7 590 7.6�0 7. 650 7. 5 80 7 630 1 61 0 Con trollor G a i n ( K )
�
;;..
.. ... E .. !-
lSO 140
,..---------u1 -------------, 110::7 , ::0 • •
•
•• • • • • • • • • • • •
•
::
t. 1.3 0 "
.. �
�
Q .. .. � "
·� e "
i5 >
frriod 2
1.20
•
/
.r,
. ····
·.
I . I.
Hora J
s . . .... l
1 ! .l '" ".
.
.
:
�
.\o'>;�o� <S>d'��tf
1 10
(b)
0 0
.
y �
c s o00
� 0 0�==
+---r-----r---.--.--r---4 7. 650 7 60 7. 6 7 0 7 69 0 7. 680
1 00
Con r rolltr G a i n ( K )
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
sub-regions 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� =
_ . . _
=
STATIC AND DYNAM IC B IFURCATION B EHAVIOUR .... 0 . 9 68 ; f 0.966 .. ... e � 0 984 �
t. .." �
c
"
� c: 0
�
�
� ,.
ltr-:r :
0.962 0.960
I
0.978 0.975 0 .974
AS
7.665
�4
A3
/."�= 7. 6 8 9 R
I. Cris i s = 7. 6 8 8 1 7.690
R•gion -- -
--
A
-- - - - ·-- . . - -- - --
A2
7.695
A 1
; _;oo
47 1
7 705
(n) 7 .· n o
.. � 1.03 0 -.-------, ';;
� 1 . 0 20
;.
1.01 0
t.
�
0 9 90
�
0.970
e
::: .. ..
1 00 0
::: 0.980 ..
·;;;
:; E
••••••••• o o o o o o o o o o o o o o o o o o o ������ ������ �· [""'
..r..L>.C � · · · · · · ·
7 . �90
(b)
Q 0 9 60 +------r----.------.-----�------l
;,:
7 . 6 85
7 . 695 7 . 7 00 C o n trolltr G •in (K)
7 . 7 05
7 . 71 0
FIGURE 4.42 Further enlargement of the region very close to HB2• A part of the gap between HB2 and H3 (Region A). (a) Poincare bifurcation diagram. (b) Bifur cation diagram.
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 two-band chaos term inates at the interi or crisis point (K;., = 7.688 1 ) and a fully developed one-band chaos prevails (Grebogi et al. ,
472
S . S .E.H. ELNASHAIE and S.S. ELSHISHIN I 1 .04 ....-----.
0.96
Q94 +-----�----�--�-� 5000 0 58000 6600.0 740QO 8200 0 9000 0 T 0.988 .....----..., 0 986
:>
0.984 0.98 2 0.980
0 9 78 ����----�--��---�-� 5000. 0 6 4 00.0 7 800.0 9 2 000 10600.0 12000 .0 T 0.988 Q986 ..,
r.-;;-;:-��---;:::=: :;::: 71 :; A6 ( c )
0.984
5 0.9 8 2
0.9 8 0
0.9 78 -t<--.----�-.--.---1 0 . 9 78 Q982 0986 y (n J
M
.
c:
>-
09875
0.9803
0.986 7
09 7 9 9
0.9859
'-: 09 794
09851
>-
09843 0 9 8 43
c
0.9859 Y (nl
0.9875
ii
097
0. 9794
y( n)
0.9 803
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 . 1-3. 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 �
�
�
.. < �
�
�
. -"
475
0.990
0 .980
Q.
:: 0. 970
c .. 0
0.960
(a)
�
;; ..
7 . 6 770
1 . 50
� I LO E
f!.
:: 1
" .t:: ...
:::
="
ft ft
� c
-�
e c
30
1. 2 0 11 0 1 00
cs
C4
7 . 6 7 BO
I
C/
CJ
7 . 6790
7 . 6800
H d nt
(b)
�
7.66 1 0
J
"'"'\ .............. . ..... . :
.
• • 0 " 0 00
Ptriod
0o00
PlP•7 6 7 9 1 4 2
,.
co o o o 6'
C2
4
Region
$ 1.
II.
...-
•. • • • •
P c r lu t lr: ••• 000
b n n c a. Vu••••' lll n • c • SU,It U•stalllt
/\
P 0 • 7 61 1 !- "1 1
PLP•7.61 :l" o °o0 oc o O o0
oo o o o o �8888&8�g�ggoooooocoooooooooo
-- - -- - - --- -1t"t tii t"t
7 6830
7 . 68 2 0
I
�� ���������� ������������� %% � % % � � % � � � � � � � � � - - - . - - - .. --- - -·
c 0. 9 0 7 . 5770 >
7 . 67 80
tifi-i t1iliiii!'� ·�-
7 . 6810 7 . 6800 7 . 67 90 C o n l ro l l t r G a i n ( K )
7 . 6820
7 . €E ) �
FIGURE 4.45 Enlargement enclosing the remaining part of the gap between HB2 and H3 and the entire H3 as well as a part of the gap between H3 and H2 (Region C). (a) Poincare bifurcation diagram. (b) Bifurcation diagram.
The region between C2 and C4 in Figure 4.45a ( which corresponds to the H3 width from PD of H3 at K = 7 .68 1 5 7 8 down to the PLP of H3 at K = 7 .67 8076) is dominated completely, as should be e xpec te d , by the P4 periodi c i ty of H3 • The comp l e x behaviour starts again at the PLP of H3 at K = 7.678076 which repres ents the right hand side b ou nd ary of the gap bet w ee n H3 and H2• At this PLP bifurcation to c haotic intermittency type I takes place through tangent bifurcation. At this PLP an un s tab l e P4 attracto r (saddle) coalesce with a stable P4 attractor (node) and evaporate ( saddl e - node bifurcation). At this p o int the Fl oqu et multiplier (FM) cro s ses the unit circle through + 1 . This is a nice numerical il lu stration of the birth of chaos at such a point through
476
S.S .E.H. ELNASHAIE and S.S. ELSHISHINI
Table 4.4 Lyapunov exponents and Lyapunov fractaJ dimensions computed at different values of the controUer gain (K). Normalized Controller gain, K
7.790000
Ly1
Ly2
Ly3
-0.007 1 72
-0.008506
- 1 .077928
LYFD 0
7.707500 (A 1 )
0.000000
-0.037660
- 1 . 1 1 2487
7.695000 (A2)
0.000000
-0.06 1988
- 1 .099795
1
7 .689000
0.01 2076
0.000000
- 1 . 1 8 1 087
2.0 1 02
7.686500 (A5)
0.026200
0.000000
- 1 . 1 99000
2.02 1 9
7.685400 (A6)
0.0 1 9936
0.000000
-1 .942950
2.0 1 03
7.683500 (B3)
0.026276
0.000000
- 1 .299090
2.0202
7.678080 (C4)
0.000000
-0. 1 1 0652
- 1 . 8 83694
1
7.678072 (C5)
0.0 1 9520
0.000000
- 1 . 1 83400
2.0 1 65
7 .675027 (D 1 )
0.0 1 7348
0.000000
- 1 . 1 89094
2.0 1 46
7.674500 (D4)
0.000000
-0.44267 8
-2.686900
7.620 1 00 (Fl )
1
0.000000
-0.234549
-2.533382
0.000000
- 1 .976656
2.0 1 3 3
7.620080 (F2)
0.026276
7 . 6 1 6625 (0 1 )
0.01 6076
0.000000
- 1 . 1 7 1 057
2.01 37
7.61 6000 (04)
0.000000
-0. 1 02891
- 1 .976656
1
tangent bifurcation. At C4 (Figure 4.45a) just to the right of PLP (K = 7 .678080), a P4 attractor exists, while for C5 just to the left of the PLP (K = 7 .678072) we have a chaotic attractor. The nature of the P4 attractor and chaotic attractors at C4 and C5 are well identified by the Lyapunov exponents shown in Table 4.4. Decreasing K further we obtain a P7 window followed by a chaotic regions, then another P7 window followed by a chaotic region, then a P6 window which extends till the left-hand end of this region at K = 7 .677. In the regions C(i) and C(ii) in Figure 4.45a, it is interesting to notice that each of the chaotic regions separating the periodic windows are actually mixtures of nearby periodic states. The branches of the special P4 window associated with H3 are always contributing as a part of the periodicity of the periodic windows and also as part of the mixtures producing the chaotic regions. This behaviour is very similar to the alternating periodic-chaotic sequence discussed by Swinney ( 1 983) and Roux ( 1 983). 1-3.4. Region D
This region shown in Figures 4 . 46 a, b extends from K = 7.677 down to K = 7.674. It covers the remaining part of the gap between Hz and H3 together with a part of Hz. We start our discussion for the subregion
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7 . 6770
FIGURE 4.46 Enlargement enclosing the remaining part of the gap between H3 and H2 as well as a part of H2 (Region D). (a) Poincare bifurcation diagram. (b) Bifurcation dia gram.
D(i) from the PD point on H2 at K = 7 .674976. At this point a period doubl i ng sequence, with the i ncre ase in K, takes place leading to chaos (P4, P8 , P 1 6, . . . chaos). The chaotic region is very narrow followed by a series of alternating periodic windows and chaotic regions. The most important peri odic window is the relatively large window of P5 which loses its stability to another narrow chaotic region which in turn loses its stabi l ity to another window and so on till we reach the peri odi c window P6 at the right-hand end of this re gio n which is an extension of th e P6 wind o w at the left-hand side of regi on C(ii) in Figure 4.45a. The remaining re gion (between the le ft-h and boundary of D(i) and the left-h.and end of reg ion D) is domin ated by P2 attractors of H2. The behaviour in reg io n D(i) is dominated by an alternating periodic-chaotic sequence (Swinney, 1 983 ; Roux, 1 983) similar to that discussed in
S . S .E.H. ELNASHAIE and S.S. ELSHISHINI
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connection with C(i) and C(ii) in Figure 4.45a. In this case both the P2 att ractor to the left and the P6 to the right play a special role in contributing as parts of the periodic windows and the chaotic "mixtures" in this regi on . 1-3.5. Region E Thi s reg io n extends from K = 7 . 674 down to K = 7 . 624 and co vers another part of Hz . N othi n g exciting takes place in this region, P2 stab l e attractors associated with Hz dominate the whole region.
1-3. 6. Region F This reg i o n shown in Figures 4 .47a, b extends from K 7 .624 d o wn to K = 7 .6 1 8 and covers the remaining part of Hz and a part of the gap =
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between H1 and H2• The subregion from the right hand end of this regi on down to the PLP at K= 7.620097 is dominated by the P2 stable attractors associated with H2. At the PLP a tangent bifurcation takes place as K decreases giving rise to intermittency type I chaotic attractor, which covers a very narrow range of K and loses its stab ility very quickly as K decrease s giving a P4 window, whic h in tum loses i ts stab i lity to a chaotic attractor which extends over an e xtremely narrow range of K, then loses its stability as K decre ases further to a P3 attractor. The P3 window persists all the way down to the left hand end of this reg i on .
/-3. 7. Region
G
This last region shown in Figures 4.48a, b extends from K = 7 .6 1 8 down to K 7 . 6 1 55 and covers the remaining p art o f the gap between =
480
S.S .E.H. ELNASHAIE and S . S . ELSHISHINI
H1 and H2 as well as a part of the main hom H1 • We start our discussion at the PD point on H1 at K = 7.6 1 6554 and follow it as K increases. The period doubling sequence (P I , P2, P4, . . . chaos) gives a very narrow
chaotic region which survives over an extremely small region of K and then loses its stability as K increases infinitesimally to a relatively wide P3 window which prevails till the right hand side of this region. This P3 is clearly the extension of the P3 window at the left hand end of region F in Figure 4.47a. For K smaller than that of the PD point the P I stable attractor associated with H1 dominates till the left hand side end of this region (actually it continues till HB � . in Figure 4.39). It is clear that the P3 attractor in the right hand side of this range can not bifurcate directly (with decreasing K) to a Pl attractor through any known mechanism, and therefore the extremely narro w chaotic region is created in order for the periodicity of the system to change from P3 to P l . 1-4.
Lyapunov exponents and dimensions for the chaotic attractors
Relatively recently a new class of strange attracting sets were found which look topologically strange but are non-chaotic (Romeiras and Ott, I 987; Kapitaniak et al. , I 990 ; Kapitaniak and Elnaschie, I 990) . In other words these strange non-chaotic attractors have fractal structure but they do not show sensitivity to initial conditions (i. e. nearby orbits do not diverge exponentially with time) therefore it is important to test the strange attractors obtained in this study to find out whether they are chaotic or non-chaotic . One well known quantitative method o f characterizing the attractors is by calculating the Lyapunov exponents which measure the rate of divergence of nearby trajectories . Chaotic attractors exhibit exponential divergence of nearby trajectories (sensitivity to initial conditions) , which expresses itself through the occurrence of positive values for the maximal Lyapunov exponents. On the contrary strange non-chaotic attractors do not show sensitivity to initial conditions and therefore do not have positive Lyapunov exponents . For dissipative systems the sum of all Lyapunov exponents is always negative and therefore the post transient motion must occur on a zero volume limit set, an attractor. This is obviously incompatible with the occurrence of positive Lyapunov exponents for chaotic attractors unless some sort of folding proces s merges widely separated traj ectories. The Lyapunov exponents (Lyi) and the dimensions (LYFD) for a chosen number of attractors for thi s fluidized bed catalytic reactor at different values of the controller gain K are given in Table . 4.4. Each attractor is designated by its s y mb ol in Fig u re s 4.42a, 4.44a, 4 .45a, 4.46a, 4.47a, 4.48a. The L y apu nov exponents are computed using the well known and efficient algorithm of Wolf et a!. ( 1 985). The Lyapunov
STATIC AND DYNAMIC BIFURCATION B EHAVIOUR
48 1
Table 4.5 Possible sign spectrums for different types of attractors in 3-dimensional dynamical system. Attractor type
Steady State
Sign spectrum
(-,-,-)
Limit cycle
(0 ,-,-)
Two-dimensional torus
(0,0,-)
Chaotic
(+,0,-)
fractal dimensions (LyFD) are computed using the conjecture of Kaplan and Yorke ( 1 983). In a three dimensional continuous dissipative dyna mical system like the one we are studying, the only possible spectra of Lyapunov exponents and the attractors they describe are as shown in Table 4.5. It is clear from the above classification (Table 4.5) that any system without steady state will have at least one zero Lyapunov exponent. This was proven some years ago by Haken ( 1 983). It is clear from Table 4.4 that all the strange attractors (K = 7 .689000, A 6, B3, C5, D 1 , F2, G 1 ) are chaotic having positive Lyapunov exponents ranging from 0.0 1 2076 for attractor A4 to 0.026276 for attractor F2. Obviously all periodic attractors (A 1 , A2, C4, D4, F l , G4 ) have the largest Lyapunov exponents equal to zero (Ly1 = 0) . For the steady state attractor all the Lyapunov exponents are negative. The Lyapunov fractal dimension (LyFD) for the steady state attractor is clearly zero, while for the periodic attractor it is equal to 1 .0. For the chaotic attractors the Lyapunov fractal dimension are slightly larger than 2 (showing clearly that chaos is only possible in dimensions greater than 2). The Lyapunov fractal dimension (LYFo) for the chaotic attractors range from 2.0 1 02 for attractor A4 to 2.02 1 9 for attractor AS . The magnitude of the positive Lyapunov exponent reflects the time scale on which system dynamics become unpredictable (Shaw, 198 1 ) . It actually gives the rate of divergence of nearby trajectories, o r the degree of sensitivity to initial conditions. The small positive values of Ly1 for the fluidized bed catalytic reactor given in Table 4.4 indicates that the sensitivity to initial conditions is not very strong, i . e. divergence of nearby trajectories is not very fast, at least in terms of the normalized time scale 'f, used in this investigation (Normalized Time Unit, NTU). If we consider the long term evolution of an infinitesimal three dimensional sphere of the initial conditions, the sphere will become an elli p soi d due to the local deforming nature of the flow of trajectories ini tiati ng from initial conditions sphere. The Lyapunov e x ponents are defined in te rms of the length of the ellipsoidal axes, and thus the Lyapu no v exponents are related to the expanding (diverging) or
482
S . S .E.H. ELNASHAIE and S.S. ELSHISHINI
contracting (converging) nature of the different directions in phase space. The axis that is in the average expanding corresponds to the positive Lyapunov exponent. Axes that are in the average contracting correspond to negative Lyapunov exponents. Thus for the chaotic attractors gi ve n in Table 4.4, the expanding axis is expanding by the rate e Ly. r (which is not very large because Ly1 ' s are small) while one of the other axes is contracting by the rate e Ly3 r w hich is relatively large. The third axis is always constant because Ly2 is always equal to zero. From an information theory point of view the Lyapunov exponents measure the rate at which the system creates or destroys information. Thus the Lyapunov exponents are expressed in term of bit/NTU. For the chaotic attractors of the fluidized bed catalytic reactor the positiv e Lyapunov exponents given in Table 4.4 vary from 0.01 2076 to 0.0262776 bits/NTU. Hence if an initial point was specified with an accuracy of one part per million (20 bits), the future behaviour can not be predicted after about 1 656 NTU (20 bits/0.0 1 2076 bits/NTU) for the chaotic attractor A4 and after about 76 1 NTU (20 bits/0 02 627 6 bits/NTU) for the chaotic attractor F2. After these times ( 1 65 6 NTU for A4 and 76 1 for F2) the small initial uncertainties will essentially cover the entire attractor reflecting 20 bits of new information that can be gained from an additional measurement of the system. This process is sometime called an information g ain On the other hand the average rate at which information contained in transients is lost can be determined from the neg ative Lyapunov exponents. .
.
II-
Summary of results for this section
A small part of the tip of the iceberg of the complex dynamic behaviour of this system h as been presented to the reader in thi s section. The three dimensional, fourteen parameters system (the entire iceberg) has been presented here in a very restricted portion of the parameters space with the variation of one parameter only, namely K. The values of the other thirteen parameters have been fixed at values which are restricted to cases with a maximum of three steady states. The case presented is characterized by a bifurcation diagram with two unconnected static bifurcation branches and on the right hand side branch (for K>O) two HB points exist and are connected by a P l branch of stable periodic attractors (Figure 4. 39). The dynamic behaviour arou n d the second HB poi nt (HB2) proved to be qu ite rich. Three periodic horns, H1 , H2 and H3, ch aracteri zed by periodicities P l , P2 and P4 respectively, w ere located in this regi on of K. The periodic attractors of the three horns are b ounded from the right for each hom by a PD point. From the left the periodic attractors of the horns are bo u n ded by PLP for H2 and H3 and by HB 1 for H1 • The horns shrink in width very fast from one hom to the next as the periodicity of the horns increases -
STATIC AND DYNAMIC BIFURCATION B EHAVIOUR
483
in the direction of increasing K. Gaps exist between the horns and the width of these gaps shrinks slowly from one gap to the next in the direction of increasing K. A period doubling sequence leading to chaos ( P 1 , P2, P4, . chaos) emanates from HB2 as K decreases . Period doubling sequences leading to chaos emanate with increasing K from the three PD points bounding the stable periodic attractors of the three horns (HI . H2. H3) from the right. Bifurcation to intermittent chaos through tangent bifurcation emanates with decreasing K from the two PLP' s bounding the stable attractors of the horns H2, H3 from left. The gaps between HB2 and the PD point of H3 and between the PLP of H3 and the PD point of H2 and between the PLP of H2 and the PD point of H1 are covered by different types of chaotic regions (banded chaos, fully developed chaos and intermittent chaos) interrupted by periodic windows of different periodicities. The most important of these periodic windows is the P3 window. The dominant mechanisms for bifurcation in these regions are period doubling (or halving) and tangent bifurcation. From a certain global point of view the horns H1 , H2 , H3 obtained using A UTO can be considered nothing but relatively wide periodic windows among the sea of chaotic attractors and narrower periodic windows covering this region. However closer inspection of this complex region as a whole will show certain characteristics which make these "periodic windows" or "periodic horns" different from other periodic windows: .
.
1 . They are much wider (in terms of the region of the bifurcation
parameter K they cover) than other windows.
2. They have a highly organized period doubling sequence from P I to P2 to P4 and if it was not for the fast rate of shrinking of these horns
and the ceiling of numerical limitation we could have had most probably, P8, P 1 6, . . . etc horns leading to chaos. Therefore it seems that they have their own period doubling sequence and most probably their own accumulation point. Their period doubling sequence has a very unique characteristic, that is, it is interrupted by gaps covered with chaotic regions and periodic windows. On the otherhand the other periodic windows although interrupted by chaotic regions are not ordered in such a period doubling sequence as these periodic horns. 3 . Their origin is clearly connected to HB1 and HB2, for ex amp l e H1 i s connected from the left hand side to HB 1 (Figure 4. 39) and from the right-hand side its formation actually comes from the P l attractor which loses i t s stability at the first PD p oint on the periodic branch e man ati n g from HB2 which cont i n u es to the left as K decreases, reaches an unstable PLP turns over and moves to the right with .
484
S . S .E.H. ELNASHAIE and S.S. ELSHISHINI
increasing K (Figure 4.40b) till it reaches another unstable PLP at K = 7 .6I 7957 (Figure 4.48b) where it meets the unstable PI attractor emanating from the PD point on H1 at K = 7 . 6 I 655 4 (Figure 4.48b). The other horn s H2 , H3 can also be traced to unstable periodic orbits emanating from the next PD points on the periodic branch starting from HB2 and other unstable branches born from the unstable branches of H1 (for Hz) and Hz (for H3). The small subhoms (SH I and SH2) attached to H1 and Hz at their PD points are, from all points of view, nothing but the periodic attractors in the first period doubling cycle in the sequence of period doubling leading to chaos. The inability of AUTO to locate the next 'subhom' is due only to the numerical limitation of AUTO and the computer used, in this fast changing narrow region following the second PD point. This can be seen from Figures 4.46a,b and 4.48a,b where the period doubling sequences reach chaos over a very narro w range of K. When the ordinary periodic windows interrupting the chaotic re gions loses their stability to chaotic attractors through period doubling sequences, these sequences are called U-sequences according to Metropolis et al. ( I 973), who found several years before the universal scaling properties of one dimensional quadratic maps were discovered by Feigenbaum ( 1 979), that one-dimensional maps with a single extremum exhibit universal dynamics as a function of the bifurcation parameter. They discovered that beyond the period doubling sequence (that is for K
STATIC AND DYNAMIC BIFURCATION BEHAVIOUR
485
The density of the chaotic regions as compared with the density of periodic windows decreases considerably as K decreases from HB2 towards HB 1 • The chaotic regions disappear completely close to point G3 (Figure 4.48a) at K = 7 . 6 1 6625 below which the chaotic attractor loses its stability through period halving to a P I attractor which prevails till HB 1 at K= 3 .660462. For K values lower than this value periodicity disappears all together giving a stable unique steady state down to K = O.O. Computation of the Lyapunov exponents and the Lyapunov fractal dimensions for a number of strange attractors at different values of the controller gain K shows that these attractors are chaotic having positive Lyapunov exponents and Lyapunov fractal dimensions. Simple analysis of the Lyapunov exponents and fractal dimensions points to some of the main characteristics of these chaotic attractors. It may be too early in this section to comment on the practical implications of chaotic behaviour on the design and control of such reactors. However, it is important to notice that many important reactions such as the gas phase catalytic partial oxidation of hydrocarbons (such as o-xylene to produce phthalic anhydride) and oxidative dehydro genation reactions (such as ethylbenzene to styrene) have high enough activation energies , heat of reactions and chemisorption capacities to give rise to similar complex static and dynamic bifurcation behaviour as the ones reported in this section. Preliminary results from our research on industrial units are indicating that fluid catalytic cracking units and fluidized bed polyethylene units can have such complex static and dynamic bifurcation behaviour (see section 4.2 and 4.4 in this chapter). It has also been known for a long time that the important carbon monoxide reaction (discussed briefly in section 4.3) shows similar complex behaviour (Yip, 1 989). A short review of the practical relevance of bifurcation and chaos is given by Elnashaie and Elshishini ( 1 994), in an earlier book. Such complex behaviour may be beneficial as shown by B ailey ( 1 977) and others (Al-Taie et al. , 1 978; Silveston et al. , 1 986; Hoffman and Schadlich, 1 986) . At this early stage, it can be stated with enough theoretical and practical justifications that the investigation of this behaviour (apart from its fundamental importance) is practically useful to be avoided (or suppressed) when it is harmful or to be exploited to the most when it is beneficial.
4.2
INDUSTRIAL FLUID CATALYTIC CRACKING (FCC) UNITS
Most of the work dealing with the modelling of industrial fluid catalytic cracking units is based upon a highly empirical approach that helps in building units and operating them, but does not help to elucidate the
S . S .E . H . ELNASHAIE and S . S . ELSHISHINI
486
MIX
FIGURE 4.49
Schematic representation of an FCC unit (Type IV).
main features and characteristics of these units for better design, control, and optimization. It is shown in this section, after reviewing most of the important models, that a better insight into the behaviour of these units can be achieved through the use of less empirical mathematical models coupled with industrial verification and cross verification of these models. The present section concentrates on the bifurcation behaviour of these important industrial units in the petroleum refining industry. 4.2. 1
Evaluation of Some Important Mathematical Models for Industrial FCC Units
A Fluid Catalytic Cracking unit is made up of a reactor, in which gas
oil is catalytically cracked to gasoline, coke and light hydrocarbon gases and a regenerator, where the catalyst is regenerated by using air to bum the coke deposited on the catalyst in the reactor. The catalyst circulates between the reactor and the regenerator (Figure 4.49). All the main reactions taki n g place in the reactor are usually endothermic and the kinetics are monotonic. Therefore the reactor cannot be the source of the b i fu rc ati o n behaviour of the s y stem It is the regenerator with its high l y exothermic carbon burning re ac ti o n .
which is the source of the bifurcation behaviour of these industrial units. The mathematical modelling of industrial fluid catalytic cracking units
STATIC AND DYNAMIC BIFURCATION B EHAVIOUR
487
Table 4.6 Summary of some of the basic characteristics of the models used to simulate industrial Fluid Catalytic Cracking (FCC) units. Verified Characteristics Model
L&L (Luyben
and Lamb, 1 963)
K ( Kurihara, 1 967) I (lscol,
1 970)
Two-
Kinetics
Kinetics
Kinetics
Phase
of gasoline
of carbon
of carbon
Predicts
industrial
Models
formation
formation
burning
multiplicity
data
No
Yes
Yes
Yes
No
No
No
No
(Lee and Kugelman, 1 973)
No
1 979)
Yes
E&K (Edward
No
L&K
E&E (Elnashaie and EI-Hennawi, and Kim, 1 988) E&E modified (Eishishini and E1nashaie,
1990)
Yes
No No
No
Yes No
Yes
Yes
Ye s
Yes
Yes Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
against
No
No
Yes No
Yes
Ye s
No
Ye s
Yes
Yes Yes
Yes
is complicated by the fact that each of the two vessels of the unit is formed of at least two phases. Two-phase models are the simplest physically accepte d formulation, for certainly the flow patterns in these units are more complicated, specially in fluidization regimes beyo nd the bubbling regimes. One of the most complicated problems is that associated with riser-reactor type units because of the complex flow patterns in the riser. Recent research in fast fl ui dization regime and circulating fluidized beds helps to elucidate the complex nature of the flow in these risers (Galtier et al. , 1 989; King et al. , 1 989; Gi ane tto et al. , 1 990; Kraemer et al. , 1 990) . Grace ( 1 986, 1 990) argues that it is not difficult to revise two-phase fluid bed reactor models, originally intended for the bubbling bed regime , to be applied to other regimes of fluidization.
I- Evaluation ofsome ofthe most important models presented in the literature Seven different models of different degrees of sophistication for the FCC unit are presented in this section . The models differ from each other in s ome basic characteristics as w e l l as in their structure. The degree of empiricism in expressing the rates of the different processes also dl ffer from one model to the other. Table 4.6 shows a summary of the basic characteristics of the seven models presented.
488
S . S .E.H. ELNASHAIE and S.S. ELSHISHINI
l-1.
Luyben and Lamb model
(L&L
model)
Luyben and Lamb model ( 1 963) was described by Douglas ( 1 972) as being a crude model that predicts the direction of change of the most important dependent variables and at least give an order of magnitude estimate of the response time of these variables. It is assumed in this model that the following reaction occurs in the reactor:
A � B + O. l C where A, B and C represent gas oil, gasoline and coke respectively. The coke combustion taking place in the regenerator is given by:
C + mO � P where 0 and P represent oxygen and combustion products respectively and m is an empirical sto i c h iometric coefficient depending upon the degree of completion of carbon combustion and the percentage of hydrogen in coke. The model is also based on the following assumptions:
1 . The reactor and the regenerator are perfectly mixed. 2. The heat capacities are independent of composition and temperature. 3 . The catalyst holdup remains constant in each vessel. 4. The reaction rates are first order and the rate of coke combustion is independent of the amount of coke. 5 . The reaction network in the reactor is formed of one reaction. The material and energy balance equations for this model can be summarized as follows:
Reactor:
dNR I dt = VGF - VR d(NRYz ) I dt = VGF Y2F - VRY2 + NR ( l - Y2 )A1 e - £1 1 Rc TR Et T MRdXR I dt = Fc XG - FcXR + 0. 1 NR (1 - y2 )A1 e - i Rc R cps MRdTR I dt = cpRVGF TGF + cps FcTG - crR VR TR - cps Fc TR +( - LV/I ) NR ( l - y2 )A1 e- £1 Rc TR
(4.35) (4.36) (4.37) (4.38)
Regenerator:
dNG I dt = VAF - VG d ( NGYo ) I dt = VAFYOF - VGYo - mNGYo� e - EJ Rc;Tc
(4 .39) (4.40)
STATIC AND DYNAMIC BI FURCATION B EHAVIOUR
McdXc I dt = FcXR - FcXc - NcYoJ\e -EJ ReTe
489
(4.4 1 )
cp McdTc I dt = cpGVAFTA F + cp Fc TR - CpcGcTc - cp�FcTc +( - Ml2 ) NcYoJ\e-EJRcTc + Q (4.42) s
s
where all the variables and parameters are defined in the nomenclature of this chapter. Using the ideal gas law and simplifying the equations, the model was reduced to a set of six coupled, non-linear, ordinary differential equations which can be solved numerically.
Evaluation of the model The model was not tested experimentally and using such a simplified model for estimating the dynamic characteristics of the system may give misleading results. However, by introducing some modifications on the kinetic scheme used for the cracking reactions and on the rate of carbon burning it can provide a starting point for developing a good approximate model. /-2.
Kurihara model (K model)
Denn in his book (Denn , 1 986) considers Kurihara model (Kurihara, 1 967; Gould et al. , 1 970) to be the best model available in the open literature. The model considers two different carbon species. The carbon that is deposited on the catalyst during the course of the cracking reaction is denoted catalytic carbon (Ccat) . The remaining (additive) carbon is generally present in gas oil and is deposited without reaction. Carbon that has passed into or through the regenerator on the spent or regenerated catalyst has a different chemical activity from the catalytic carbon. This species is identified as Crc or Csn depending on whether the c atalyst has been regenerated or spent. The mass balance on total carbon on the spent catalyst in the reactor is given in this model by: (4.43) The term FcCc is simply the rate at which catalyti c carbon leaves the reactor, while FcCsr is the rate at which carbon remai n in g on the regenerated catalyst is suppl ied to the reactor through the catalyst circu lation stream. The specific rate of carbon formation is described by:
(4 .44)
S . S .E.H. ELNASHAIE and S . S . ELSHISHINI
490
where
(4.45) The first term in equation ( 4.44) represents the rate of production of catalytic carbon from the cracking reaction and the second term represents the rate of formation of additive carbon which is proportional to the feed
rate GcF·
The remaining reactor mass balance is written for Ccat w hich is the catalytic carbon (in weight %) obtained from the cracking reaction ,
(4.46)
The energy equation for the reactor is written as :
cps . MR · dTR I dt = cps . Fc ( Tc - TR } + cpJPJGGF(TcF - TR ) (4.47) +(-Mlv}PJGGF + (-Mlcr )I\.r
The term containing the enthalpy of vaporization accounts for the fact that the gas oil enters as a liquid and is vaporized inside the system. The overall specific rate of cracking reaction Rcr is given by:
(4.48) (4.49)
and
The regenerator mass and energy balance equations are given by:
(4.50)
The c arb on burning rate
Rc
is given by:
(4.52) where (4. 5 3)
STATIC AND DYNAMIC BIFURCATION BEHAVIOUR
49 1
and C1 is a stoichiometric coefficient that reflects the selectivity of the combustion reaction to C02 and CO products and (2 1-018)/1 00 is the oxygen conversion, since the volume fraction of oxygen in the air feed is 0.2 1 .
Evaluation of Kurihara model The model is distinguished from other models by the fact that it differentiates between catalytic carbon which deposits on the catalyst due to catalytic cracking and non-catalytic carbon which is generally present in gas oil and is deposited without reaction. In the solution of the model Kurihara assumes additive carbon to be zero (Ftt= 0). Other than that, the model is in principle, very similar to the model of Iscol ( 1 970), and that of Lee and Kugelman ( 1 973) discussed later, specially with regard to the model structure, the degree of empiricism involved in the different g enerati o n and c on s umpti on functions i n s erted in the model equations. Kurihara model (Kurihara, 1 967 ; Gould et al. , 1 970) does not recognize the two-phase nature (bubble and dense phases) of the two fluidized beds in the reactor and the regenerator. Although the solid and gas in the dense phase can be considered as having the same conditions (pseudo-homogeneous model for the dense- or emulsion- phase), the differentiation between dense phase as a whole and bubble phase as a whole, in such bubbling or turbulent fluidized beds can be quite important for accurate mass and heat balance formulation of the system. Kurihara ( 1 967) uses a very simplified kinetic scheme given by the following equation: Gas oil
Cracking
Product gas + C oke (additive
+
catalytic)
This network is similar to that used by Luyben and Lamb ( 1 963) and has the disadvantage of lumping gasoline, coke and light hydrocarbons. Consequently, there is no way to obtain the gasoline yield which is the most important design variable. The most serious limitation in this model is that the constants A2 (for the rate of catalytic carbon formation) an d A 1 (for the rate of catalytic cracking) have been fitted for a certain assumed s teady state. Kurihara [24, 43] assumes that the reactor is op erated at a certain probable steady state condition (e.g. MR = 54432 Kg, PR = 275 KPa, TR = 499°C, GcF = 0. 1 84 m3/s, conversion on fresh feed = 0.6), tog ether with a certain probable steady state condition of the regenarator and he evaluates A2 and A 1 by setting the time d eriv ati ve s to zero and solving the steady state mass balance equations. This was justified for optimal control,
492
S . S .E.H. ELNASHAIE and S . S . ELS H I S H I N I
since the investigation was concentrating on the problem of small disturbances around the ste ady state condition as Kurihara puts it in his thesis: "Since it is not only difficult but also useless for our purpose of simulation to estimate accurately these constants by purely chemical analysis of feed composition and catalyst condition, this kind of closed-loop method was used" (Kurihara, 1 967) . However, these rate equations are not valid at any other conditions away from the assumed steady state (simple calculations using this model show that for a reactor temperature of 593°C the percent carbon formed is much higher than 1 00% ! ) . Consequently, they cannot be used to investigate multiplicity of the steady states since the model will always give one steady state (the assumed steady state).
l-3.
lscol model (! model)
Iscol ( 1 970) carried out his modelling excercise to investigate the steady state and dynamic characteristics of Model IV Auid Catalytic Crackers with special emphasis on trying to explain the continuous drift of a commercial unit away from its operating point at high gasoline yield. Iscol model con s ists of the following mass and energy balance equatio n s together with the necessary generation and consumption fu n ct i ons .
Reactor enthalpy balance
Regenerator enthalpy balance
Mc Cp 5 dTc I dt = 3 0 2 X 1 0 7 Rc - Fc Cps (Tc - TR ) .
-75 1 72(Tc - 288.6) - 1. 2 X 1 0 7
(4. 5 5 )
Reactor coke balance (4. 5 6)
Regenerator coke balance (4. 5 7) Coke make rate
(reactor) (4. 5 8)
STATIC AND DYNAMIC BIFURCATION BEHAVIOUR
493
Coke bum rate correlation (regenerator) Rc =
6.45 [1 - exp [ -Ca (747 + 14.94 (Ta - 894) + 0.107 (Ta - 894)2 )]] (4.59)
The coke make and bum rates are given by empirical correlations, which as Iscol states in conj unction to the coke burning rate: "It should be noted that equation (4.59) will vary among different FCC units and depends upon catalyst type, chemical composition of the coke and actual catalyst flow pattern, and thus vessel geometry".
Evaluation of Iscol Model The empirical nature of the coke forming coke burning and heat losses from the vessels is apparent. Also, the coke make rate is expressed in te rms of the input variable FGF (gas oil feed flow rate) which is not very rigorous since all processes in a physical model should be expressed in terms of state variables in the model formulation. Empirical formulation of a rate process does not dictate the physically inconsistent use of input variables in these empirical relations, e.g. the rate of coke burning, altho ugh it is expressed in a semi-empirical form, it is actually e xpressed as a function of the regenerator' s state variables Cc and Tc with no input variables appearing in the equations. The Iscol model does not involve any reaction network entailing all the consequences discussed earlier and it als o does not recognize the two phase nature of the fluidi zed beds in the reactor and the regenerator. lscol ( 1 970) investigated the steady state characteristics of his model and showed that multiplicity of the steady states (bifurcation behaviour) does exist over a relatively wide range of parameters (Figure 4.50a). He also concluded that the operating steady state is the middle unstable ste ady state and that the instability of the system manifests itself as a slow drift from this unstable steady state due to the very h igh heat capacity of the system. ,
-
/-4. Lee and Kugebrum model (L&K model) The model of Lee and Kugelman ( 1 973) was published three years after Iscol model to rein vestigate the proble m of multi pl i city of the steady states in fluid catalytic cracking units. The model consists of the fo l l owi ng equations:
Reactor carbon balance
(4.60)
S . S .E.H. ELNASHAIE and S . S . ELSHISHINI
494
..
§
.Y ..
d
JC
ODIJ
p
�
L6
(a)
�
i J
00 11
i
6'0
650
FD I L B I H R 200
u
.. "
..., " "' .. E
0
660
1
r::----
0 ....-----,
(b)
------
12
_ _ _ .... -
08
0.
4
.; 0.0 0 . l5
670
...... . ,
- - - -
1.2
FAF -:=: 11 2 .5 119 1s F C ::1066ll l.25 1r.gls - FGF : 1400 0 k9ls - - - FGf' : 600 0 k91S - · - FG F = I 2 0 kgls FGr = '0 0 lr. g ls - ·· '8
60
12
VAF Diml'nsionl�ss A i r Ft!'d Te-mP. Rtgtf"le'ratot
(c )
19 0
/
2o
1 80
1 60
lo
-- H�at Generation
Parl
- - - He-at Rl'moval
R eg m rrotor B�d
Total
Combustion
Te-mp�at ure , °K
FIGURE 4.50 Examples of diagrams showing multiplicity of the steady states for industrial Fluid Catalytic Cracking Units (FCC) obtained by different investigators. (a) Iscol (1970) (bifurcation diagram with gas oil feed flow rate as the bifurcation parameter). (b) Elnashaie and El-Hennawi (1979) (bifurcation diagrams at different gas oil flow rates with air feed temperature as the bifurcation parameter). (c) Edwards and Kim (1988) (Van Heerden diagram - heat generation, heat removal diagram - for one set of parameters showing multiplicity of the steady states).
Reactor energy balance d( MRCpJR ) I dt = F;, Cp.J Tc - TR ) + F0M Cp1 ( TcF - TR ) - FcM (M/v + €c · Mlcr )
Regenerator carbon balance d(M0 C0 ) I dt = Regenerator energy balance
f'c(CR - C0 ) - 1\:
(4.61) (4.62)
STATIC AND DYNAMIC B IFURCATION B EHAVIOUR
495
The carbon formation in the reactor is given by Lee and Kugelman
(1973) as :
(4.64) Lee and Kugelman ( 1973) state that the first term on the right hand side of equation 4.64 corresponds to the cracking reaction and is based on the assumption that coke formation follows the so-called "Voorhies relationship" (1945) in which the weight fraction of coke on catalyst is proportional to the n1h power of the catalyst residence time in the reactor. The coke formation rate constant arr is taken by Lee and Kugelman (1973) to be a function of feedstock and catalyst. Lee and Kugelman (1973) determine am E2 and n by fitting to experimental laboratory and plant data and therefore their values strongly depend upon catalyst, feedstock and the characteristics of the particular FCC unit. In addition to catalytic coke, Lee and Kugelman consider additive coke (e.g. Conradson-type coke) which is included in the second term of equation 4.64. They also take into consideration the efficiency of the stripper by adding a term which depends upon the fact that the hydrocarbons interspersed among the catalyst particles are to be completely stripped by countercurrently flowing steam. However, due to the poor efficiency of the stripper some fraction of the strippable hydrocarbons remains and is carried over to the regenerator. This effect is included in the last term of equation 4.64, the amount of strippable hydrocarbons to be burned being proportional to an [ lh power of catalyst circulation rate. In fact, this term should not be included in the carbon formation equation for it does not affect the catalyst activity the same way carbon does. It should be accounted for in the heat balance of the regenerator for in actual fact it gets burned in the regenerator and contributes to the amount of heat released in the regenerator. The gas oil overall cracking conversion €c is given by the model as:
(4.65 ) where:
(4.66) and
A=
fPJ�r e-Ec,IRJN Pt Sv
(4.67)
496
S.S .E.H. ELNASHAIE and S . S . ELSHISHINI
where Ad, Am Ecn Ed are experimentally determined parameters charac terizi ng the combination of the partic ular c atalys t and feed s tocks . The coke burnin g rate is given by,
(4.68)
[
-- l
is the oxygen mole fraction at the exit of the regenerator which is c omputed from the relation:
where Ofg
0
Jg
= 21
exp
-A c e
-(E' I RG 1a )
CG · MA · MG · aCO
0. 2 1 FA G Me
Lee and Kugelman ( 1 973) used this model to i nves tigate the steady state characteristics of the system and the y reached an oppo site conclusion to that of Iscol ( 1970) with regard to mu ltipli c ity Lee and Kugelman ( 1973) attributed the multiplic i ty results obtained by lscol ( 1 970) to the cubic temperature dependence in the expression for Kt u se d by Is co l Denn (1986) c omments on this contradiction by emph asizing the need to study the sensiti vi ty of the model pre di c tion s to structural as well as .
.
parametric changes.
Evaluation of Lee and Kugelman model The model does not consider the two-phase nature of the two fluidized beds which is similar to previous models. However, it d i s ti nguishe s between c atalyti c carbon and additive carbon, which is in principle comparable to Kuri hara model ( Kurih ara, 1967; Gould et al. , 1970). The model takes into consideration the stripping efficiency, which accounts for what is know n as c at-to o i l coke. This is a measure of unstri p pe d but potentially strippable hydroc arbon s that are carried out to the regenerator (Venuto and Habib, 197 9 ) because of the non-ideal s tripp in g in the commercial process leav ing residual hydrocarbons on the catalyst which are burned in the regenerator. Obvio u s ly this is not rea11 y coke. The inclusion of this stripping effic i en cy is an advantage to the model bu i l d ing however the au th or s include this term with the carbon formation rate in the reactor (equation 4.64) ! This causes some errors in the carbon balance in the reactor which affect the co mputation of the catalyst activity and the heat produced in the regenerator. The first term in e q u ation 4.64, wh i ch is supposed to be the term for the formation of catalytic carbon, has a dependence over t� which is the c atal y s t residence time i n the reactor. This depen de nc e can be j ustified by V oorh ie s work [44] and also by the derivations of Kurihara "
-
"
,
,
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 El-Hennawi model (E&E model)
Elnashaie and El-Hennawi (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 El-Hennawi (1979) take s into consi deration the two-phase 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 cutive-parallel 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 El-Hennawi ( 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 El-Hennawi, 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 pre-exponential 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 · e-rJ 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 . C-M( ) 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 El-Hennawi ( 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 El-Hennawi model
The model of Elnashaie and El-Hennawi ( 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 o-ph 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 pseudo-components, 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 four-component 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: 1-6.
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 modem-day 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 pseudo-steady 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 (1-3 ) .
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.
1-7.
Elshishini and Elnashaie model (E&E modified model)
The model developed earlier by Elnashaie and El-Hennawi ( 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 (El-Hennawi, 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 generation-heat 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 4-lump 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---;r--i
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 two-phase 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 co-operation between industry and academia, to develop more rigorous models of high-fidelity 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 two-phase 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 4.69-4.89). In the following we give a simple procedure for the computation of the model parameters. /-1 .
Computation of two-phase 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
1-2. 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 pre-exponential 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 pre-exponential 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 pseudo-components which is the only realistic approach available to-date 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 0--0 . 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 pre-exponential 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
1-3.
=
·
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 4.73-4.89 . 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 3-8 . If the residual is small enough (< l Q-6), 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 0-4 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 2-9 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 1-3), will not be used in the present section but rather the model of Elnashaie and El-Hennawi (1979) given in section 4.2. 1 (part 1-5). On the basis of the assumptions of Elnashaie and El-Hennawi 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
k g ls kgts trg15 kglt
0·4
0·8
2 50
200 1 50
0·6
10
YR o
1·1
1·4
1·6
1 ·8
1 0
(b)
1·00 0 96
O·a4
D
>(
�
><
0 72
0 60
'
o •a
o.J6
0·24
0 12
I
\
...._.__,__.L... .. .... .J ...._ _j
0 00 I..-L-'--'-"-/-'-"'=:::o.L. :t:l .._._.__._ 00 02 0 4 0 · 6 0 8 H) 1 · 2
YRo
'
1 ·4
1·6
1 8
2 0
FIGURE 4.53
Effect of gas oil flow rate. (a) Heat generation function and heat removal line vs. reactor temperature. (b) Yield and conversion vs. reactor temperature.
Figure 4.53 shows the effect of gas oil flow rate on the modified heat generation function, G ( YRD ) (Figure 4.53a) as well as conversion, 1 - xw, and gasoline yield x2o (Figure 4.53b ) . It is interesting to notice that the relation between gasoline yield and reactor temperature shows a number of maxima. This is a result of the difference between the activation energies of the different reactions taking place in the reactor and the regenerator. For FGM = 200 kg/s, three such maxima occur. The third one gives the maximum gasoline yield at a reactor dimensionless temperature of 1 .2. For FGM l 000 and 1 800 kg/s, two maxima occur and the maximum gasoline yield occurs at a re ac tor dimensionless temperature of 1 .45 and 1 .5 respectively. For the given c o ndi ti o s three =
n
STATIC AND DYNAMIC BIFURCATION BEHAVIOUR 1.1
I I I I I I I I I I ' I I
&.4
s 6
q 0 '-' >
4.0
).1
I
1 .4 1.6
I I
,
01 0.0
0
I
' r
1
Fcs = 1 060 k g / s Fc iFco Y R O 0.2$
1 -25
r
i r
I
2.50
I1
10 - 0
: , 'f' I I I
oa
515
I I
x , o 'I' R nH
X2oo� 'fi!D opt
1 .()
0 . 1 7 5 0- 006 0.06 nit 5 . 15 nil ni1 0 -255 I 0
0 -1 7 2
1 - 35
0 - 117
1 . 45
0 -191
1-5
0 · 198
1-5
0 - 71 0 o .oos 0 -0J 1 5]5 0 - 1 16 0 -91 nil 1.0 0 25
0 ·7 7 5 0 005 0 · 1 1 1 - 505 0 - 1 92 0 ·96 n il 1.0 0-25
0 -711 0 0 - 004 0 · 09E 1 -475 0 . 195 0 · 99
FAF = 1 1 2 . 5 k g ls Fe;� = 1 4 0 0 k g l s
' I i I ..,
t&
Q.JI
- - - - Fe
u
1. 2
=
26 5
- Fe = 1 3 25 - - · - Fe = 2625 .....,. � : 1 0600
40
kg r s
kgl s kgls k IS
FIGURE 4.54 Regenerator temperature vs. reactor temperature. Effect of catalyst circulation rate. Tbe enclosed table shows tbe effect of c on the steady state temperature, yield and catalyst activity.
F
steady states are obtained for Fc;M = 200 kg/s at YRo = 0. 19, 0.62, 1 .26 and X2o = 0.0, 0. 18, 0.63. For FcM = 1 000 kg/s at YRo= 0.28, 0.77, 1 .5 and x2o= O.O, 0.02, 0.26 and for FcM= 1 800 kg/s at YRo= 0.27, 0.78, 1 .66 and Xzo 0.0, 0.0 1 , 0. 1 1 . From these limited results we notice that the increase in gas oil flow rate at cons tant operating conditions causes an increase in the temperature of the system together with a decrease in gasoline yield. It can be noticed that as Fc;M increase s the corresponding Yco (for the same value of YRo) increases since the increase in FGM causes an increase in the amount of coke formed, thus causing a larger amount of coke to enter the regenerator and get burned, and so leads to the evolution of a larger quantity of heat of combustion. The i ncrease in YGD may exceed allowable limits, in such a ca se the regenerator temperature can be decreased by increasing the rate of catalyst circulation, Fc as shown in Figure 4.54. But of course, by changing Fc the system steady states are changed and consequently the gasoline yield. This shows the comple xity of the interaction between the various variables in this system and the need for rational mathematical model s to use in the optimization of the operation of such a system, taking into accoupt the various parameters simultaneously . W e notice however that i n the case o f Figure 4.54, the increase of =
the catalyst circulation rate increases the yield by decreasing the
516
S . S .E.H. ELNASHAIE and S . S . ELSHISHINI 7 2
FA F : 1 6 8 ·7 5 k g / s
fG� =
6 �
1 0 Soc_ , I OE G : 0 · 5 S e c- 1 - - - O E R : 1 0 Soc-1 • QEG = 2 - 0 S • c- \ - - - O E R : 0 · 58 S•c- 1 ' Q EG : 0 · 81 S r < - 1 ..
4 8-
T � fi rst
OER: I O
4· 0
QEG :
3 2-
05
1 0 OEG = 2 · 0 OE R :
2 �
YR
\wo
x2
c ,a s r s
1/JR
c o i nc idf'.
Xz op\ YR ol)\
1-0 0 ·1 8 6 0 74 0 · 006 0 · 0" 1 60 0 · 1 78 0 · 846 nil 0 26 1 - 0 0·186 0 · 72 0 · 0 3 0 · 02� 0 · 26
I
1t
2 2
4 63
1·6
nit
1 · 63
0
0 8
16
2·4
\. 5
1·5
0 · 1 3 8 0 · 744 0 084 0 · 21 nit
nil
1 · 0 0 · 1 98 QEA : 0 5 8 0 2 6 n i t 0 E R = 0 · 81 0 74 0 005 0 · 4 2
0·8 0 0
kgls kgts
1400
- QER :
5· 6
�
: 1 060
"c
0 168
1 ·S
0 83 0
3·2
FIGURE 4.55 Effect of exchange rates between the bubble and dense phases in reactor and regenerator on the system behaviour.
overheating of the system which destroys the gasoline to coke and dry gases. Therefore, it may be postulated that increasing the catalyst circulation rate in order to control the regenerator temperature also has a favourable effect on gasoline yield. Notice that for the results in Figures 4.5 3 , 4.54 , QER. QEG = oo and therefore the dense phase and bubble phase conditions are identical and are equal to the output conditions of the reactor and the regenerator. For the cases with finite exchange rates between the bubble and dense phases in the reactor and regenerator, the output conditions from the reactor and the regenerator are obtained by simple mass and heat balances for the concentration and temperature of both phases and are expressed with the same symbols as before but without the subscript D (for dense phase) used in the previous results. It is interesting to notice from Figure 4 .55 that the change in the rate of exchange of mass and heat between dense phase and bubble phase in both vessels has little effect on the YGzrYRv diagram. However, they have an important effect on the multiplicity and yield. For the case of QER = 1 . 0, and QEG = 0.5, we get three steady states that with gasoline yields of 0.0, 0.006, 0. 1 7 8 , while increasing QEG 4 fold s (keeping QER constant) shows five possible steady states with gasoline yields of 0.0, 0. 03 , 0. 1 38, 0. 084, 0. 0. For QER = 0.58, QEG 0.8 1 , w e obtain again 3 steady states with =
yields: 0.0, 0.006, 0. 1 68. These last two values of the exchange parameters
STATIC AND DYNAMIC B IFURCATION BEHAVIOUR
517
were computed by the following correlation (Kunii and Levenspiel, 1 969), for both reactor and regenerator, (4. 1 1 2)
(
where D is the diffusivity which can be calculated by Gilliland correlation (Perry, 1 9 84), em 2 1 1 (4. 1 1 3) + sec M1 M2 More on the effect of exchange rate between bubble and dense phases will be presented and discussed in section 4 .2.3 (part 11-3.2) for one of the industrial units modelled, verified and cross-verified against industrial data in section 4.2.3 (parts I, I-1). /11-2.
J
Determination of the air feed temperature for maximum gasoline yield
In this section, equation 4 . 1 1 1 is used without modification, and the air feed temperature is used as the manipulated variable to obtain the maximum gasoline yield. Any other variable can be used instead of the air feed temperature, but the equation should be rearranged such that the chosen variable would appear in the linear side to facilitate the simple graphical solution, otherwise numerical optimization techniques should be employed. In this case, the maximum gasoline yield and its corresponding reactor temperature are determined from a plot of x20 vs. YRD· Then by projecting YRo over the heat generation function - heat removal curve, a point on the G (YRo) curve is determined and from this point, a line is drawn with slope a3 to intersect the y-axis. From this intersection, the air feed temperature can be determined as it is clear from equation
4. 1 1 1 .
Table 4.8 gives the optimum air feed temperature obtained by the above graphical method at different values ofFc. for the given parameters, FAF = 1 1 2.5 kg/s, FcM = 1 400 kg/s and Qc:R = QEc = 00 • It is i mportant to notice that the optimum TFA decreases sharply as Fc increases. This is due to the fact that increased quantities of heat are removed from the regenerator as the catalyst circulation rate increases. ///-3.
Hysteresis loops
hysteresis phenomenon (static bifurcation) associated with the of the steady states plays an important role in start-up,
The multiplicity
518
S .S .E.H. ELNASHAIE and S . S . ELSHISIDNI
Table 4.8 Optimum air feed temperature at different values of catalyst circulation rate, F"" = 1060 kg/s.
Fc/Fcs
TFA K
1 .25 1 .875 2.5 5.0
1 958.5 665 .0 308.6 282.6
control and stability of chemical reactors. The desired optimum steady state may be within the multiplicity region, and hence special start-up procedures must be followed so as to attain the desired steady state. As discussed in chapters 2 and 3, when such conditions prevail, a large disturbance may shift the reactor outside the region of stability of an optimum steady state, towards the stability region of other steady states. When such a disturbance is removed, the reactor does not restore its original steady state hence an irreversible drop in productivity arises. Figure 4.56 shows the hysteresis loops YRv vs. YFA · for various values of FG· It is clear that the multiplicity range is in general quite large and increases as the gas oil flow rate FGM increases. These results verify the results obtained by Iscol ( 1 970) using a simpler model, i.e. the multiplicity of the steady states dominates the behaviour of the FCC unit. 1 - 8 �-------, 1 .6 1
4 .
1 2
- --- - -
--- -- -
.
--· --
-
--
1 ·0
0
·
FA F : 1 1 2 ·S
kg/•
:
kgts
Fe = 1 3 2 5 - FG t-4 : 1400
0 6
--- fG�
- · --
4
-·
0·2 oo
O·G
1 2
,
a
z. t.
J.o
3·6
YFA
-
-
FG '-4 : I"G '-4 :
t. 2
600
kgl•
120 40
4·8
s �o
kgts
kgts kg10
6 0
6·6
6·1 2
FIGURE 4.56 Hysteresis loops, reactor temperature vs. air feed temperature. Effect of gas oil flow rate.
STATIC AND DYNAMIC BIFURCATION B EHAVIOUR
5 19
IV- Summary of section results
A relatively simple phenomenological mathematical model for the description of the steady state behaviour of an (FCC) unit has been presented in this section. The model uses the kinetic scheme ofWeekman (1 968, 1969) which lumps the reactants and products into three groups only. This model takes into account the two-phase nature of the reactor and the regenerator using the hydrodynamics presented by Kunii and Levenspiel ( 1969). It also takes into account the complex interaction between the reactor and the regenerator due to catalyst circulation. The effect of some parameters on the steady state behaviour is presented in this section and a simple graphical technique has been presented for the determination of the optimum air feed temperature. It is noticed that this optimum air feed temperature may exceed allowable limits, hence the catalyst circulation rate has to be adjusted to bring down the optimum air feed temperature and regenerator temperature. The results obtained with such a model provide a better understanding of th e complex interaction between the reactor and the regenerator and its effect on the overall behaviour of the system than using empirical models. The model can be used for optimization and steady state control of the FCC unit. One obvious advantage of this model is the ease by which the model equations can be solved graphically using a modified van Heerden diagram. This model of Elnashaie and El-Hennawi ( 1969) involves some as sumptions which need to be relaxed in order to use the model for accu rate simulation of industrial units as was discussed in connection with Elshishini and Elnashaie model (1990a), in section 4.2 . 1 (part I-7). Industrial verification of the steady state model and more on static bifurcation of industrial units. 4.2.3
Industrial Verification of the Steady State Model and More on Static Bifurcation of Industrial Units
The model of Elshishini and Elnashaie (1990a) presented in section will now be used for the simulation of some industrial FCC units. It may be useful here, to summarize once again the main remaining assumptions for this model after relaxing some of the assumptions used in Elnashaie and El-Hennawi model ( 1 979): 1. There is no reaction in the bubble phase. 2. The dense phase is perfectly mixed and the bubble phase is in plug flow with respect to mass and heat. 3. Interchange between bubble phase gas and dense phase gas is by bulk flow and diffusion.
4.2. 1 (part 1-7),
S.S.E.H. ELNASHAIE and S.S. ELSHISHINI
520
Table 4.9 Some basic data for the two industrial units and the characteristics of their feedstock, products and catalyst. 1 80, 1 1 0
Average molecular weight of gas oil and gasoline, respectively B oiling point of gas oil and gasoline, respectively
539, 4 1 1 K
Latent heat of vaporization of gas oil
265 . 1 5 kJ/k.g
Reference temperature, Trt
500 K
Concentration of gasoline in gas oil feed, CA2f
0
Voidage in both vessels
0.4
He i g ht of expanded bed in reactor and regenerator, for the operatin g conditions of unit I .
7, 7.2 m
An average mean bubble diameter is used throughout the bed. Heats of reaction and physical properties are constant except for diffusivities, de n si ty of vaporized gas oil, and de n s ity of air, which are functions of temperature. 6. The Excess air is used in the regenerator. s tarting values of the kinetic rate constants used and the heats of the three reactions are as follows, 4. 5.
= .468 exp (- 08 83. 3 / ReT) m3 / kg.s; Mil =603.9 kJ / kg M/2 = kJ I kg ReT) 1 / s; =1 0773. 5 exp M/3 = 8 1 6. 0 / kg = 1 7. 09 exp (-376 0 / ReT) The heat of combustion of coke calculated from the following correlation (Nels on, 1 964 ), )+3370 (H)]x2.326 / kg ( -Mfc) =[4100 + 10100 ( ) ( K1 K2
4
2
(-75240 /
2
K3
2
7520.0
m3 / kg . s;
2
is
COz C02 + CO
C
kJ
kJ
4. 1 1 4
where C02/( C02 + CO) refers to the relative volume of these gases in the flue gases from the regenerator and H/C is the atomic ratio of hydrogen to carbon in the coke formed. Heat losses in the reactor are assumed to be 0.5% of the heat entering with the catalyst, and the heat losses in the regenerator are assumed to be of the heat supplied by coke burning. The rest of the data is given in Table 4.9. The two industrial fluid c ataly tic crackers considered in this section, are of the type IV model (with U-bends). The two units vary in i n pu t parameters which leads to different output parameters. Table 4. 1 0 gives the plant data for the two commercial FCC u nit s . Additional plant data
0.5%
521
STATIC AND DYNAMIC BIFURCATION BEHAVIOUR
Plant data for both units.
Table 4.10
Fresh feed flow rate, kg/s Re cycle HCO flow rate, kg/s
Combined feed ratio, CFR Air flow rate, kg/s
C ataly st circulation rate , kg/s
Unit
1
Unit 2
1 6.78 2
1 3 .476
2. 1 08
2. 1 1 1
1 . 1 25 6
1 . 1 566
1 0. 670
1 0. 670
8 8 . 605
1 1 7. 1 1 3
Combined feed temperature, K
527
538
Air feed temperature, K
436
433
Hydrogen in coke, wt%
4. 1 7
6.79
HCO/(HCO + CSO + LCO)
0.286
0.4 1 9
0. 2 2 1
0.207
C02/(C02 + CO), m3/m3
0.75
0.6 1 3
Coke/Coke + gases
(-Mf)c '
kJ/kg
3 1 .24 x 1 03
30.78 x 1 Q3
common to both units is given in Table 4. 1 1 . The two industrial units are relatively small in comparison with more modern, large capacity riser-reactor units. I- Simulation Procedure (verification and cross verification)
The model was fitted to the first FCC unit using correction factors for the frequency factors. The modified frequency factors are: K{0 = 1 .5 K1u. Kz 0 = 0.5K20, K3 o = 1.5K30 • Of course, these correcti on factors will vary from one catalyst to the other depending upon the type of Table 4.1 1
Plant data common to both units
Regenerator dimensi ons
5 .334 m ID x 14.859 m TT
Reactor dimensions
3 .048 m ID x 1 2.760 m TT
C atal y st retention in reactor
1 7.5 metric tons
C atalys t retention in regenerator
50 metric to n s
API of raw oil feed
28.7
Reactor pressure
225.4938 kPa
Regenerator pressure
254.8708 kPa
Average particle size
0.00072
Pore v o l ume of catal y st
m
0.0003 1 m3fkg
Appare nt bulk density
800 kg/m3
Catal yst Surface Area
2 1 5 m2/g
S .S .E.H. ELNASHAIE and S .S . ELSHISHINI
5 22
1. . 0
.------.---..., I H O U $ T I U A � UN I T t
..
�
'""
2. 0
I N O U S T II I i· l U " l f l
- 2.0
(a )
\II � Cl
"' ; o. e o u-
--
0 �
5
- - - -
c o 1>' . a: IU
l "' O U S TIII .ol l UN I T I
I " O U S T A I A l liN I T 2
;: 0 6 0
<�� "'
\II ! "' J Z: 0
.O Vl �
iii "
:x l5 g 0. 1 0 z: � "' "
lb l 1. 00
2 .00
3.0 0
4 . 00
OI N E HS I ON L E SS R E A C l O R T H I P E R A l U RE ( Y ) FIGURE 4.57
(a) Heat function for units 1, 2.(b) Conversion and gasoline yield
for units 1, 2.
catalyst, its activity and age as well as the type of feedstock used. To make sure that these correction factors are not just empirical numbers that fit only this specific FCC unit at these specific operating conditions, a cross verification of the model should be carried out. The model, with the correction factors, is used without any modifications to simulate the second FCC unit which has a very similar catalyst with almost the same activity and age and a very similar feedstock. The success of this second simulation (cross-verification) ensures to a large extent that the developed model is a reasonably good general representation of these FCC units. However, the catalyst activity should be regularly checked to ensure the validity of the pre-exponential factors used in the simulation.
STATIC AND DYNAMIC BIFURCATION BEHAVIOUR
Table 4.12
Simulation results of FCC units 1 ,2. Model results
Parameters
Unit 1
Plant data
Unit 2
Unit 1
Unit 2
1 .55
1 . 57
1 .55
1 .57
YG
1 .98
1 . 85
1 .9
1 .88
xi
0.382
0.355
0. 3 9
0.322
y
5 23
x2
0.39
0 . 39
0.4 1 5
0 .4 1
Coke kg/s
4.2
4.083
3.68 1
4 . 07 3
HCO kg/s
2.057
2.354
2. 1 09
2. 1 1 1
% error Unit 1
0
Unit 2 0
+4. 2 1
- 1 .6
-2.05
+ 1 0.25
-6.02 + 1 4. 1
-2.45
--4.9 +0 . 25
- 1 1 .6
II-
Some simulation and bifurcation results and discussion for the two industrial FCC units Il-l. Comparison of model results with plant data
Figure 4. 57a shows the heat function v s. reactor temperature for both units 1 , 2 and Figure 4.57b shows the corresponding unreacted gas oil and gasoline yield. Table 4. 12 gives the values of the output variables obtained from the model and from plant data for both units 1 , 2. It is clear that the agreement is quite good. The simulation can be improved if more accurate kinetic data is provided for the same kinetic scheme and can be improved further if the kinetic scheme is extended to include separate routes for the formation of coke and light gases. It is clear from Table 4. 12 and Figures 4.57a,b that both units are operating at the middle unstable steady state and that both are not operating at their maximum gasoline yield. II-2.
Effect of some operating parameters
The units being operated at the middle unstable steady state, it should be expected that the steady state response of the system to variation in operating conditions will be somewhat unusual as has been shown some time ago for the first time by Elnashaie et al. (1 972) for a much simpler system. Therefore, a detailed parametric investigation of the system is presented using the present model, in order to give the reader a deeper insight into the behaviour of this rather complex system. It is obvious that the optimization problem in the neighbourhood of the unstable middle steady state will be different from that in the neighbourhood of the stable high temperature steady state. Also, there will be an obvious trading between the gasoline yield of the system and its stability, i.e. in some cases, higher yields correspond to an operating point closer to the critical bifurcation point. The parametric presentation is simplified by the fact that the dense phase gas oil, x w and gasoline dimensionless concentrations, x2v vs. Y diagrams do not change with variations in Fc,
S.S.E.H. ELNASHAIE and S . S . ELSHIS HINI
524
4 0 *
2.0
... ...
�
z 0 -
\U l:
...
't ' ' ' . ]Q J ' t , . 'c : 1 1 , , 01 ' ' '' Fe : 1 1 7 . 1 1 0 ,., ,,
0 . 0 t----,��':"'T--f--rl----1
:I ...
�
0 ..----�..,..--.
-
2.0
( (I J
T1 0
� ..
2 .0
... ...
•
HO
,,6 : 5 t J
t 1 0 , nt
:·
•
K
z 0 ;: \.1 z
2 :(
... X
- 2 .0
(bl
OI � E: N S I O Nl E SS R EA C T O R T E � PE R A T tJ � E ( Y )
FIGURE 4.58 (a) Effect of catalyst circulation rate on heat function for unit 1. (b) Effect of gas oil feed tempeature on heat function for unit 1.
YAt or YGf; the only changes being in the operating temperatures of the reactor and regenerator. For any operating temperature the corresponding xw and x20 can be obtained from the same curves (Figure 4.57b). The parametric investigation presented here to the reader is confined to FCC unit 1 with its x w - x20 vs. Y diagram shown in Figure 4.57b. 11-2. 1 .
Effect of catalyst circulation rate
Figure 4.58a shows that, for the middle steady state, the reactor temperature (YR) decreases from 1 .75 to 1 .56 to 1 .5 as the catalyst circulation rate (FC) increases from 44.302 to 88.605 to 1 77.2 1 kg/s. The corresponding conversion ( 1 -xw) decreases while the corresponding gasoline yield (x20) inc rease s from 0.3 14 to 0.385 to 0.4 1 . It is clear from Figures 4.5 8a, b that the response of the middle steady state to the change in any parameter is opposite to the response of the other steady
STATIC AND DYNAMIC BIFURCATION BEHAVIOUR
525
D l to� t. � 5 1 0 N l £ S S A I R HEO T t: M P t AA T U A E (Y'1c:a )
FIGURE 4.59 (a) Effect of catalyst circulation rate on the bifurcation diagram of Y(= YRD) vs. YAf• (b) Effect of catalyst circulation rate on the bifurcation diagram of x2 vs. YAf• (c) Effect of DB on the bifurcation diagram for YRn vs. YAf• (d) Effect of DB on the bifurcation diagram of X2 vs. YFA [ l = low temperature steady state; 2 = Middle steady state; 3 = High temperature steady state].
states (Elnashaie et al. , 1972). This is clear from Figures 4.58a,b. This fact is of great importance with respect to the optimization and control of FCC units and will be discussed in some details in the next parts of this section .
II-2.2.
Effect of gas oil feed temperature
Figure 4.58b shows that for the middle steady state the increase in the gas oil feed temperature (YGf) from 450K to 527K to 539K causes the dimensionless reactor temperature to decrease from 1 .76 to 1 .56 to 1 .07. This increase in YGf causes a decrease in conversion and an increase in gasoline yield from 0.3 14, to 0.3857, to 0.397 1 (Figure 4 . 5 7 b) . The o pp os ite response of th e middle steady state as compared with the response of the other steady states is again evident ll-3. Bifurcation diagrams
.
It is clear from the previous results that the reactor is not operated at the optimum operating conditions . These optimum operating conditions cannot be e sti mated unless the mul tipli ci ty regions are specified for different parameters, bec au se as the system approache s maximum y ield it gets closer to the bifurcation point where ig nition of the reactor can
526
S .S . E. H . ELNASHAIE and S.S. ELSHISHINI
occur. In this section the reactor temperatures at the different steady states are obtained by solving the heat balance equation using the very simple bisectional method. In later sections, more sophisticated tech niques for the construction of the bifurcation diagram (the continuation technique discussed in chapter 3) will be used. l/-3. 1 .
Effect of FC on the bifurcation behaviour
Figure 4.59a shows the bifurcation diagram for 3 different values of catalyst circulation rate (FC): 177 .2 1 , 88.605, 44.303 kg/s. The multi plicity region increases as Fe decreases. As the air temperature increases, the yield corresponding to the middle steady state increases (Figure 4.59b ). However, the system is approaching the bifurcation point and at an air temperature higher than 1 . 7 (for the case of Fe = 44.302 ), ignition occurs i.e. the reactor operates at the high temperature steady state to which corresonds a negligible gasoline yield represented by the lower branch of the bifurcation diagram (Figure 4.59b ) At h i gh er circulation rates, the drop in gasoline yield occurs at Yta = 1 .65 for Fe = 88.605 and Yfi, = 1 .6 for Fe = 1 77.2 1 . The plant is operated at Fe= 88.605 and ffi1 = 0.875 (with a gasoline yield of 0.39) which means that a higher yield can be obtained either by increasing Fe or increasing the air feed temperature. The optimum yield occurs at YAt= 1 .5 for Fc = 88.605 and YAt= 1 .46 for Fe= 1 77.2 1 . For Fc= 44.302 the highest yield is at YAf= 1 .7 but this corresponds to the bifurcation point above which ignition occurs. A gasoline yield of 0.47 (20% improvement) can be achieved by increasing both Fc and air temperature. .
/l-3. 2.
Effect of DB on the bifurcation behaviour
In the model presented in this section, the bubble diameter is taken as a constant value throughout the bed and is calculated from the following equation (Fryer and Potter, 1 972): (4. 1 1 5)
and was found to be 0. 1 m. The effect of bubble diameter on bifurcation is presented to the reader in order to highlight the sensitivity of the system to bubble diameter and the possible need to incorporate more rigorous means of calculating an average value for the bubble diameter or the need to take into consideration bubble size variation along the height of the reactor. Figures 4.59c,d show the bifurcation diagrams for the dense phase for different average bubble sizes. Fi gure 4.59c shows the increase in the range of the multiplicity re g io n as D8 in c reases whereas, Figure 4.59d shows the corresponding gasoline yield. It is
527
STATIC AND DYNAMIC B IFURCATION BEHAVIOUR
clear that the system is qu ite sensitive to bubble di ameter For the mi ddle s te ady state, as D8 increases from 0. 1 to 0.2 the gasoli ne yi eld in the dense phase also increases, but a further increase to D8 = 0.3 causes a decrease in the dense phase gas ol in e yield. For the high temperature steady state as D8 in crease s x20 dec re as es The effect of D8 on the bifurcation behaviour based on the exit reactor conditions (dense phase + bubble phase) is quite complex and will be pres en ted in m ore details in section 4.2.5 dealing entirely with the effect of hydrodyn amic parameters on the performanc e of FCC units. An improve m e nt in the model simulation of the industrial units can be achieved by obtaining a more acc urate experimental estimate of the average bubble size. Taking into consideration bubble variation alo ng the hei ght of the bed will complicate the model considerably and is not expected to affect the predicti on of the model in co mparison with a model u sing a good esti mate of constant average bubble size (Fryer and Potter, 1972). .
.
III-
Summary of section results
The model presented has prov ed to simulate the industrial data quite well. It is shown that the reactor is operated at the middle unstable s teady state for both industrial FCC units considered. The verified model has been used to pre sent to the reader a parametric investigation that has shown that the response of the middle steady s tate to variation in the input parameters is opposite to that of the high temperature steady state. The bifurcation results have shown that the multip lic ity regio n is quite large and the regions of se n siti vity near the bifurcation po int s have been elucidated. It has also been shown that the commercial units are not operating at their optimum c onditions of maximum gasoline yield and suggestions have been presented to improve the gasoline yield without pres enti ng a formal optimizati on investigation yet. Table 4.13 Kinetic parameters for the various charge stocks (Pachovsky and Wojciechowski, 1975d). Reaction Stock
1
A l WO
E
0 mol% Wax
ko
A2W 1 7 . 8 mol% Wax
E
ko
3
2 1 .4 X 10'1 1 . 98 X 1 01 4
26.8 X 1 0'1 2.2 X 1 02 1
1 9.7 x 1 04 1 .53 X 1Q l7
20. 3 X ! 04
26.2 X 1 04 8 .67 X 1 020
20.5 X 1 04 3 . 05 X 1 01 4
20.98 X 1 04 5 . 5 1 X 1 01 7
A 1 W2 1 8 .5 mol% Wax
ko
1 9.9 x 1 04 9.00 x 1 0 1 7
AOW 1 26.8 mol% Wax
E ko
1 9.6 x 1 04 1 .04 X l Q I R
E
2
8 .99 x 1 0 1 3
20.3 X 1 04 3. 1 4 X 1 014
26.2 x 1 04 2.07 X I Q2 1 26.3 x 1 04 3 .09 x 1 02 1
528
S . S .E.H. ELNASHAIE and S . S . ELSHISHINI
Table 4.14 Physical properties of the various charge stocks (John and Wojciechowski, 1976).
Stock
mol% Wax
Molecular
weight
w
API
Boiling Temp., K
A I WO
0
387
3 . 220+
30.3
744
A2W l
7 .8
3 .303+
1 8.5
3 . 590+
3 1 .2+ 32.4 *
745*
A I W2
390.8 ' 396.0*
AOW I
26.8
400
3 . 666+
33.4
747
746*
' interpolated values average values in the range of operating temperatures
t
4.2.4
Effect of Feedstock Composition on Static Bifurcation and Steady State Gasoline Yield
In order to present to the reader a preliminary idea about the effect of feed stock composition on the behaviour of FCC units, four different feed stocks varying in wax content are used. The values of the frequency factors and activation energies for the various feed stocks (Pachovsky and Wojciechowski, 1975d) are given in Table (4.13). The refractoriness parameter (W) introduced by Kemp and Wojcie chowski (1974) is used together with an activity coefficient ( 1/f) which accounts for the catalyst decay. The rate equations used are:
These rate equations were incorporated into Elshishini and Elnashaie model (1990) using the same heats of reaction and coke burning rate. The refractoriness parameter (W) is a function of feed composition and temperature. For each feedstock an average v al ue of W is used over the range of reactor temperatures of practical importance . These values are given in Table 4. 1 4. 1- Some results for the effect of feedstock on FCC units The results obtained using the plant data gi v n in Tables
are
same
e
4 . 1 0-4. 1 1 while the catalyst circulation rate is the manipulated variable used to adj ust the operation of the unit to obtain maximum gasoline
STATIC AND DYNAMIC BIFURCATION B EHAVIOUR
529
2 . 0 .-----.---., (a)
., ' S!
- 2 .0
-- A 1 ... 2 --- - - ,�� , . . . . .11 0
W0
w, w2 w,
- 4 . 0 ����-���-�-� 1.00 .----.--., (b) A , W0 "' 2 w ,
... , w2 ... . w ,
(c )
,, I I .
1
.
'\�� /_<· ·
-4 . 0 2. 0
I
· - Fc _ _ _ Fe - - - Fe · Fe
... . w ,
= " - 7 67 = l2- 1 5 1 : 44. 30 2 : 8 &. 6 0 5
k g /s k g /s kg l s kgI>
'---.L---'-----..1.--...J----.�---r--r-�---., .-
- 4 . 0 L--....lL--.L---J----L---L--� 6.00 � -00 0.00 2.00 O I,. E N S I OHlESS R EACTO R
T E M P ERATU R E ( Y R l
FIGURE 4.60 (a) Effect o r charge stock composition o n heat function. (b) Effect of chatge stock composition on unreacted gas oil and gasoline yield. (c) Effect of catalyst circulation rate on heat function for oil AOWl. (d) Effect of catalyst circulation rate on heat function for oil Al WO.
S .S .E.H. ELNASHAIE and S.S. ELSHISHINI
530
Table 4.15 Reactor temperature and gasoline yield corresponding to the middle steady states for the various charge stocks. Reactor Temp. ( YR)
Gasoline Yield (X2)
A l WO
1 .62
0 . 1 573
A2W I
1 .65
0.224
A 1 W2
1 .69
0.332
AOW l
1 .72
0.37 1 1
Stock
yield in the face of changes in the feedstocks. Non-adiabatic operation with deliberate cooling of the regenerator and heating of the reactor is also considered. 1-1 .
Effect of Charge Stock Composition on Conversion and Gasoline Yield
The heat function can be easily calculated and plotted as a function of output reactor temperature for the four feed stocks (Figure 4.60a). The corresponding dimensionless gas oil concentration (X1) and dimen sionless gasoline yield (X2) are presented in (Figure 4.60b ). It is clear that the reactor must be operated at the middle steady state since almost zero gasoline yield is obtained at the low and high temperature steady states. At the high temperature steady state, the gasoline yield is almost zero and the gas oil concentration is also small. This is due to the fact that at high reactor temperatures, the coking reaction is favored. It is also clear from Figure 4.60b and Table 4. 1 5 that the reactor operating temperature and gasoline yield increase as the wax percent increases. The increase in wax percent in the feed stock, with all the other feed conditions kept constant, causes the operating reactor temperature to vary from 807K for feedstock AIWO (0% wax) to 863K for feedstock AOW l (26.8% wax). Also the corresponding gasoline yield varies from XF 0. 1573 for AlWO to 0.37 1 1 for AOWI, more than double the value. The effect of wax content on regenerator temperature is even more pronounced than its effect on reactor temperature. For A 1 WO, TG = 948K, while for AOWl , TG = 1 064K (Table 4. 15). From Figures 4.60a,b it can be noticed, by careful inspection, that the operating temperatures of the reactor for different feedstocks do not correspond to the maximum gasoline yield. For feedstock A I WO in Figure 4.60a,b, a vertical line (not shown) drawn from the intersection of the he at function with the horizontal axis at point in Figure 4.60a to point b in Figure 4.60b shows the steady state temperature to be shifted considerably to the left of the optimum temperature, giving a gasoline yield which is much lower than the maximum gas o line yield. a
STATIC AND DYNAMIC BIFURCATION BEHAVIOUR
53 1
Table 4.16 Values of the different parameters at the middle steady state for the various charge stocks, at different circulation rates.
FCD
YR
0. 1 0.2 0.3 0.5 1 .0 1.1
1 . 8 1 44 1 .6664 1 .64 1 6 1 .6242 1 .6 1 3 1 1 .6 1 25
0.2 0.3 0.5 1 .0 1.1
A I W2
AOW 1
Stock
A 1 WO
A2W 1
YG
XI
x2
lJIR
1 1 .6842 4.5295 3 . 2608 2.4260 1 . 8970 1 .8536
0.4699 0.6252 0.6635 0.6930 0.7 1 27 0.7 1 39
0. 1 375 0. 1 673 0. 1 635 0. 1 58 1 0. 1 534 0. 1 53 1
0.5037 0. 8935 0.9438 0.972 1 0.9878 0.9890
1 .7 1 37 1 .6824 1 .6584 1 .6445 1 .6445
5 .0273 3 .5458 2.5738 1 .9796 1 .9336
0.5503 0.5 869 0.6 1 90 0.6389 0.6389
0.2227 0.2262 0.2243 0.22 1 2 0.22 1 3
0.8783 0.9366 0.9695 0.9868 0.9880
0.2 0.3 0.5 1 .0 l.1
1 .7853 1 .7424 1 .7 1 1 9 1 .6877 1 .6868
5 . 8480 4.0029 2.8328 2. 1 030 2.0477
0.4408 0.4708 0.4963 0.5 1 94 0.5252
0.3003 0.32 1 0 0.3296 0.332 1 0.33 2 1
0. 8529 0.9252 0.9640 0.985 1 0.9866
0.2 0.3 0.5 1 .0 l.1
1 . 8 1 56 1 .7736 1 .7382 1 .7255 1 .7 1 7 1
6.2077 4.2438 2.9537 2.2 1 42 2. 1 289
0.4 1 0 1 0.4330 0.4565 0.4656 0.4723
0.3 1 64 0.347 1 0.3656 0.3704 0. 3727
0.84 1 5 0.9 1 94 0.9626 0.9830 0.9856
It is possible to manipulate some parameters to obtain the operating temperature corresponding to the maximum gasoline yield. We chose here to use the catalyst circulation rate as this manipulated variable. Table 4. 16 shows the effect of manipulating FC, expressed as FCD = FCIFCref (where FCrer= 88.605 kg/s). Table 4.17 summarizes the optimum results in Table 4. 1 6. It is clear that for feedstocks AlWO and A2W l there are maximum values for x2 occuring at relatively low values of FCD (0.2 and 0.3 respectively). For feedstocks A1W2 and AOWI a monotonic dependence of x2 on FCD is observed and the highest gasoTable 4.17 Maximum gasoline yield and corresponding reactor temperature for the various charge stocks. Stock
A 1 WO A 2W 1
Maximum X2
0. 1 673
0 . 22 62
Op timum YR
Op timum FCD
1 .6664
0.2
1 .6824
0.3
A I W2
0.332 1
1 . 6877
1 .0
AOW l
0.3727
1 .7 1 7 1
1.1
S . S .E.H. ELNASHAIE and S.S. ELSHISHINI
532
IX >-
0.: �
11.1 1-
2 .00
1. 9 0
3 1 .80 1-
Fc o Fc o Fc o
=
Fc o
= =
=
0. 2 0. 3 o. s t.o
Fc o = 1 . 1
-
v
..:
-·-
1&1 a:
� 1.70 1&1 �
z 0 0/1
%
1&1
:1
0
1.60
1 . 50 0.00
·
__.,. . ...... .
-·
- · ---
O P ER AT I ON AT MAX I M U M G AS O L I N E Y I E L D
OJ O
0. 2 0
MO L E C R A C T I O M WA X
QJO
FIGURE 4.61
Effect of charge stock composition on reactor operating temperature for different catalyst circulation rates.
line yi eld is approached asymptotically. The lowest value of FCD at whichx2 reaches its asymptotically highest value is taken as the optimum value since a further increase in FCD above this value will not affect x2 (however it will help to bring the regenerator temperature down as will be discussed later). It is clear that close to the asymptotic values, large changes in FCD, result in small changes in x2 (e.g. in Tabl e 4. 1 7, for A 1 W2, FCD = 0.5 and 10 give practi cally the same gasoline yield). Figures 4.60c,d show the effect of catal yst circulation rate on the heat function for two of the four types of oil A l WO and AOW l respectively. The diagrams for xl ' x2 VS YR do not change with variations in catalyst circulation rate (FC), but only with variations in wax content. Thus, Figure 4.60b is common to all the cases presented for different values of FC. The maximum gasoline yield for each type of charge stock is given in Table 4. 17 with the corresponding reactor temperature. It can be noticed that the difference between the max i m u m gasoline yield for the different feedstocks is quite considerable. This means that a reactor operated with a certain feedstock, at the maximum gas o line yield will not operate at its optimum if the feed stock is changed. Consequently the catalyst circulation rate must be manipulated to attain the maximum yield for this new feedstock. Figure 4.6 1 is a grap hic al summary of the optimization results in
Table 4 . 1 6. It gives the operating reactor temperature as a function of
STATIC AND DYNAMIC B IFURCATION BEHAVIOUR
533
wax content for different values of catalyst circulation rate, with the solid line representing optimum reactor operation. It is clear from Figure 4.61 and Table 4. 16 that for some feedstocks (in our case A 1 WO and A2W1) the optimum reactor temperature giving rise to the maximum gasoline yield corresponds to low values of FC, giving rise in tum to a high regenerator temperature (for A 1 WO, TG 2265 K and for A2W 1 , TG = 1773K). The se temperatures are obviously too high and this problem can be solved in either of two ways: =
I ) Sacrificing some of the gasoline yield by varying FC. For example, for A1 WO an FC of about 5 times the optimum value would result i n TG = 948.5K which is a reasonable value. However, the gasoline yield decreases from 0. 1673 to 0.1534 (which represents a decrease of about 8.3% of the maximum value). 2) Keeping the optimum FC and introducing non-adiabatic operation by deliberately removing an amount of heat Q 1 from the regenerator and adding an amount of heat Q2 to the reactor. In the case of feedstock AlWO, Q1 = 3.2778 x 104 kJ/s and Q2 x 2.349 x 104 kJ/s, giving rise to a reactor temperature of 1 .6468, with gasoline yield 0.1 654 and a regenerator temperature YG = 2.03 (TG = 10 15K), which is a reasonable operating temperature for the regenerator. The loss in gasoline yield is only 1 . 13% from the maximum value. l-2.
Bifurcation Diagrams
The bifurcation diagrams are drawn for the reactor temperature (Figure 4.62a) and gasoline yield (Figure 4.62b) as a function of catalyst circulation rate for the four feedstocks under consideration. The low and high temperature steady states for the four feeds coincide, giving almost zero gasoline yield. For the middle steady state branch, which is the operating branch, the operating reactor temperatures vary for the different feedstocks. The maximas in the x2-FCD diagram for feedstocks A 1 WO and A2Wl for operation at the middle steady state, are apparent in Figure 4.62b. /-3.
Effect of Degree of Coke Combustion in the Regenerator
As discussed earlier in connection with Edwards and Kim (1988) model, the authors indication that the middle steady state corresponds to the partial combustion of the carbon deposits in the regenerator, while the high temperature steady state corresponds to complete c ombusti on , is not correct. Mu ltipl i ci ty of the steady state exists for both partial combustion and complete comb u s ti on In fact, for complete combustion the multiplicity region increases in size. Figure 4.5 1 , in s ec ti on 4.2. 1 represent s the bifurcation diagram for the reactor temperature vs. the dimensionless catalyst circulation rate .
534
S . S .E.H. ELNASHAIE and S.S. ELSHISHINI
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li iii
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I - H IG H T E I4! S TE A DY STAT E
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3 - L OW TE ... P .
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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 2--4% 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. 1-1 .
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 El-Hennawi, 1979; Elnashaie and El-Bialey, 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 t-G w 0 . 1 rn - - - - - ti B C. a O.l "'
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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 4.63d-f.
the the
Figure 4.63g-i 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 4.65a-i 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
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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 fed-back 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 4.65a-i 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 two-phase 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 pseudo-steady 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. 1-1 .
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 non-linear 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 El-Hennawi, 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 . ,
1-2.
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 non-linear 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 El-Hennawi, 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 . 1-3.
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 non-linear 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
1-4.
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 non-linear ordinary differential equation, (4. 1 19) where Rc is given by equation 4.85. With the initial conditions: at
t= O 1-5.
1Jic = 1flc ( O)
Pseudo steady state gas oil and gasoline mass balances in the reactor
Equations 4.73 and 4.77 represent the pseudo-steady 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
544 ., .
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FIGURE 4.66 (a) Gasoline yield vs. Dimensionless reactor dense phase temperature. (b) Bifurcation diagram, Dimensionless reactor dense phase tem perature vs. Dimensionless air feed temperature to the regenerator. (c) Bifurcation diagram, Dimensionless reactor dense phase temperature vs. Dimensionless catalyst circulation rate.
The above dynamic model equations are defined in terms of the same state variables as the steady state model. The parameters are also the same as those of the steady state model except for the additional four dynamic parameter EHR , EHG • EMR • EMG · II-
Some results for the dynamic behaviour of FCC units and their relation to the static bifurcation characteristics of the units
1
The dy namic behaviour of in d u s trial Unit (for which the steady s tate behaviour has been presented in the previous section) is pres e nte d
in this section. Both open-loop and closed-loop feed-back controlled configurations are presented.
STATIC AND DYNAMIC BIFURCATION BEHAVIOUR
545
Figure 4.66a shows the gasoline yield x2v vs. dimensionless dense ase temperature YRD· The heat generation function for this case with ph Yfi1 = 0.872 and FCD= 1 .0 is shown in Figure 4.77 (Kc = O) . The bifurcation diagrams are shown in Figure 4.66b,c. The bifurcation diagram with FCD as the bifurcation parameter has very different characteristics com pared to the diagram with Yta as the bifurcation parameter. For the latter case, at constant FCD, Y1a can be varied within the physically reasonable range to give a unique ignited steady state, but it is not possible to obtain a uniq u e quenched state (Figure 4.66b ). When FCD is the bifurcation parameter at constant Yra. FCD can be varied to give a quenched unique state, but it is not possible to obtain a unique ignited state, within the physically reasonable range of parameters (Figure 4.66c). There is also a narro w region of five steady states (Figure 4.66b ) . We have chosen the steady state with Yfi1 = 0 . 8 7 2 and FCD = 1 .0 giving a dense phase reactor temperature of YRv= 1 .5627 (Figure 4.66b,c) and a dense phase gasoline yield of x20 = 0.387 (Figure 4.66a). This is the ste ady state around whic h most of our dynamic analysis will be concentrated for both open-loop and closed-loop systems. A variety of important issues regarding the dynamic modelling, behaviour and control will be addressed in this paper, namely: the effect of numerical sensitivity on the results, the effect of model dimensionality on the reliability of the model predictions , sensitivity to initial cataly s t activity in the reactor and the direction and speed of the system' s drift from the middle steady s tate The investigation will also address the problem of stabilizing the middle (desirable) unstable steady state using a switching policy as well as a simple proportional feed-back control. .
Il-l . Numerical sensitivity The set of four non-linear ordinary differential equations describing the system are qu i te sensitive numerically. Extreme care should be exercised in order to obtain reliable results. A subroutine equipped with automatic step size control for accuracy is used where the required accuracy (TOL) is specified and the step s i ze (DT) for integration is adjusted automatically at every integration step in order to achieve the targeted accuracy. Despite that, it will be shown that the results are quite sensitive to the specified accuracy limit (TOL) and to the method of integration. The subroutine also provides different integration methods depending on the stiffness of the differential eq u ations In many cases a very small accuracy limit (TOL< t o-8) correspond i ng to a step size of DT < 1 o-8 mu s t be spe c ified in order to obtain accurate results U s i n g a value of TOL as small as 1 Q---6 still, in many cases, does not gi v e accurate resu l t s Actually this is a very tricky point for with the advent of chaos in many physical systems as discussed in chapter 2 and 3, it is quite .
.
.
,
,
S . S .E.H. ELNASHAIE and S . S . ELSHISHINI
546
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0 •t
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FIGURE 4.67
Numerically chaotic results. (a) Dimensionless reactor dense phase temp erature (YRn ). (b) Dim ensi onle ss gasoline dense phase concentration (Xw).
difficult to specify the definition of "accurate results". In some cases with TOL = 1 Q-6 giving rise to DT = 1 o-5 a chaotic like response of the system is obtained as shown in Figure 4.67 . In fact, such chaotic behaviour of the results persists over a range of TOL and its related DT. For example for TOL= I 0-7 and TOL= I 0-8 chaotic looking trajectories were obtained, which may suggest the existence of chaotic behaviour. However, with tighter control over the accuracy using TOL = I0- 10 giving rise to D T as small as I0-9, this chaotic behaviour of the solution trajectory disappears, as shown in Figure 4.68. The conclusion here is
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FIGURE 4.68 Effect of Model Dimensionality. (a) Dimensionless reactor dense phase temperature. (b) Dimensionless regenerator dense phase temperature. (c) Dimensionless gasoline dense phase concentration. (d) C atal y st activity in reactor. ---- 4-dimensional model, - - - 2-dimensional model.
STATIC AND DYNAMIC BIFURCATION BEHAVIOUR
547
that the results for the case in Figure 4.67 are clearly due to numerical inaccuracy and the correct results are those of Figure 4.68. This conclusion has been reached after trying different integrating routines of varying degrees of s ophistication Each i ntegrating routine showed a different degree of tightness of the tolerance allowed in order to obtai n the correct res u lts without numerical instability. However, when each integrating routine supplied with the appropriate tolerance, was used, they all gave the same correct res ults .
.
/1-2.
Effect of Model Dimensionality
The dimensionality of the dynam ic model is an important factor for the reli ability of the model to simulate the dynamic behaviour. The assumption of pseudo-steady state condition for the gas phase mass and heat balances in both the dense and bubble phases reduces the dimen sionality of the system considerably, from (because of the di stributed parameter nature of the bubble phase model) to 4 dimensions. It is shown in Figure 4.68 that using the 4-dimensional model for a certain initial condition close to the middle unstable steady state the system drifts toward the low temperature s teady state (quenching occurs and the temperature goes below the vaporization temperature of gas oil Yv = 1 . 078). The conversion and gasoline yield decrease and so does the catalyst activity in both the reactor and the regenerator. The time of drift to the low temperature quenched state is about 0.25 day (approx. 6 hrs), the gasoline y ield drop s strongl y after about 0. 1 8 day (approx. 4 hrs). It is important to notice that the steady state towards whi ch the system drifts depends on the initial conditions and i s very sensitive to it, as will be shown later in the section dealing with sensitivity to initial values of cataly st activity The results shown in Figure 4.68 are for the initial conditions vector: YRv = 1 .6, Yco = 2.0, li'R 0.9902, ljfc 1 .0. The system dimensionality may be further reduced to 2 dimensions by ne glecting the capacitance terms in the carbon balance equations for the reactor and the regenerator. The 2-dimensional model is certainly much easier to solve and analyze than the 4-dimensional model. However, for the 2-dimensional model two degree s of freedom with regard to the system and its initial conditi ons are lost For in this case, li'R · ljfc will be related to YRD· Ycv directly through pseudo- steady state relations, and at initial conditions they are not specified but are automatically determined by spec ifying YRD• YCD at initial conditions. It will be shown in the next section that the system is very sensitive to initial conditions speci al l y lf/R (O) . Fi gure 4.68 shows a sample of the results where the 2-dimensional model does not o n l y deviate quantitatively from the more physi call y sound 4-dimensional model but it also gives the wrong direction of the drift. Therefore, the 2-dimensional model is not a suitable representation for the dynamics of th i s system. oo
,
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548
II-3.
S.S.E.H. ELNASHAIE and S . S . ELSHISHINI
Sensitivity to the initial values of catalyst activity
In the neighbourhood of the middle steady state, which is the desirable operating state, the dynamic behaviour is very sensitive to values of the initial catalyst activity in the reactor lJIR (0). This type of sensitivity cannot be explored using the two dimensional model because in this model lJIR (0) is automatically specified when both YRD (0) and YGD (0) are specified. Although during most of the runs lJIG was always very close to unity (indicating efficient regeneration of the catalyst in the regenerator), the reduction of the 4-dimensional model to a 3-dimensional model by setting lf/G ( r) = 1 .0 all the time, resulted in strong deviations, even in the direction of the system response. This indicates a strong sensitivity of the system to variations of lf/G with time even when these variations are very close to unity. The 4-dimensional model on the other hand, catches these effects which are important for the precise dynamic modelling of the system. In addition, analytical analysis of the three-dimensional model with lf!c = 1 .0, shows that this model is not analytically consistent and should not be used to simulate the model. A full investigation of the initial conditions means varying the initial conditions in 4-dimensions lf/R (0), lJic (0), YRD (0), YGD (0). In order to simplify the task, we present the effect of initial conditions using the 4-dimensional dynamic model but varying the initial conditions in two dimensions only: lJIR (O), YRD (O), while setting ljlc {O) , Ycv (O) at their values at the middle steady state. This reduces the 4-dimensional initial conditions space to the 2-dimensional initial conditions space shown in Figure 4.69. A sample of the initial conditions sensitivity analysis is shown in Figures 4.69 and 4.70. Figure 4.69 shows only the direction of the response, Lss meaning the trajectory is going to the low temperature quenched steady state having the state vector: YRv=0.97 1 8, Ycv=0.9650, lf/R = 0.998, lflc = 1 .0 and Hss means that the trajectory is going to the high temperature steady state with the state vector: YRv = 2.85 1 5, Ycv = 4.4325 , 1JIR = 0.9596, lJIG = 1 .0. From Figure 4.70 it is clear that, for YRv = 1 .6 which is higher than the middle steady state value of 1 .5627, the lf/R of the middle steady state ( lf/R = 0.989) represents to some extent a dividing line, where a lf/R (0) higher than 0.989 causes the system to quench to the low temperature steady state, while a lf/R (O) lower than 0.989 causes the system to drift to the high temperature steady state. For YRv = 1 .4 (which is lower than the value of YRv at the middle steady state) and lJIR (0) as high as 0.98 the system tends to the high temperature steady state, while for lf/R (O) of 0.986 (which i s still lower than the middle steady state value of 0.989) the system tends towards the low tempe ratu re steady s tate . A clo s er investigation of this prob l e m will be carried out after d isc uss i n g the transient profiles corresponding to the cases in Figure 4.69, which are shown in Figures 4.70 and 4.7 1 .
STATIC AND DYNAMIC BIFURCATION BEHAVIOUR
0 -;; 0 ..
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FIGURE 4.69 Sensitivity to initial catalyst activi ty of reactor. Effect of YRV (O) on direction of drift. ---- approximate location of the separatrix.
1. 'I'R (0) = 0.986, YfG (O) = l.O, YRo (0) = 1.6, fc0 (0) = 2.0074 2. VfR (0) = 0.991, VfG (O) = l.O, YRD (0) = 1.6, YGn (0) = 2.0074 3. V'R (0) = 0.986, VfG (O) = 1.0, YRn (O) = 1.4, Yc0 (0) = 2.0074 4. lJIR (0) = 0.96, VfG (O) = 1.0, YRD (0) = 1.4, fGo (0) = 2.0074 5. VIR (0) = 0.975, YfG (O) = 1.0, YRD (O) = 1.4, YGo (O) = 2.0074 6. VfR (0) = 0.98, VfG (O) = l .O, YRD (0) = 1.4, fGn (0) = 2.0074
O.H
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FIGURE 4. 70
. ..
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. ..
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(d)
6,UOO
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0.,0 0 . 31) O . :JQ 0 .10 r , T i rn • i n d O 't l
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Effect of initial conditions o n the system trajectories for cases in Figure 4.69. (a) Dimensionless reactor dense phase temperature. (b) Dimensionless regenerator dense phase temperature. (c) Dimensionless gasoline dense phase concentration. (d) ClJtalyst activity in reactor.
550
S .S .E.H. ELNASHAIE and S.S. ELSHISHINI
Figure 4.70 shows the transient profiles for the different variables for the six cases in Fi gure 4.69. It is interesting to notice that runs 2, 3 , which are te nding towards the low temperature steady state, tend to give higher gasoline yield during their dynamic trip to the low temperature steady state. This can be understood by careful in spec ti on of Figure 4.66 where the opti m u m reactor tempe rature g i v in g the maximum gasoline yield is lower than the middle steady state temperature for this case. Therefore the system on its trajectory of d ecre asing reactor temperature tends towards the maximum gasoline yield, but then it quenches in about 0.25 day (6 hrs) because the reactor temperature becomes lower than the vaporizati on temperature of the feed ga s oil. The trajec tori e s for cases 2, 3 are therefore not continued after thi s poi nt because below this temperature the system will actuall y correspond to liqui d gas oil flooding the bed. Of course, if the ste ady state des ign was at the maximum gasoline yield, such an increase of gasoline yield with the drift of the system towards the low temperature steady state will not be observed . In section 4.2.4 (part I-1 ) we have presented cases where it was not possi ble to operate the system at its maximum gasoline y iel d wi thout inflicting a regenerator temperature which is too high for this type of c ataly st , and they suggested a special type of non-adiabatic operation to run the unit as close as possible to the maximum gasoline yield. The pre limi nary results discussed above suggest that dynamic operation can also be an alternative provided that when the s y stem approaches quenching, it is switched away to a higher temperature and left to drift beneficially as shown in runs 2, 3. C a s es 1, 4-6 are drifti ng with different speeds towards the high temperature s teady s tate s , whi ch co rrespon d to very high reactor and regenerator temperatures and very low gasoline yi eld . The slowest drift is that of case 1 and the fastest drift is that of case 4 with the lowest lf/R (0). Figure 4.70 can be confusing because the terminal points of the curves at time zero do not seem to correspond c orrectl y to th ei r stated initial conditions . This is due to the scale used (in days) in order to capture the overall dynamics of the s y stem , c o up led with the fast compl ex responses at early times of the transient behaviour of the process. Figure 4.7 1 is an enl arg eme nt of the e arly parts of the co mp l ete transients shown in Figure 4.70. It is clear from Figure 4.7 1 that the different cases exc h ang e their relative positions at the early stages of the transients e speciall y with regard to YRD• and Xzo . For run 4 w ith YRn (O) = 1 .4 and for run 2 with YRn (O) = 1 .6 after a time less than 0.002 days (2. 88 mi nu te s ) run 4 h as a higher temperature than run 2, because the combin ati on of YRv (0), lf/R (0) for run 4 c aus e s the unit to tend to the high temperature ste ady state, while the combination for run 2 c au se s it to tend to the low
temperature steady state. Comparing run
4 with run
1 , it is clear from
STATIC AND DYNAMIC BIFURCATION BEHAVIOUR
55 1
0.4 4
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FIGURE 4.71
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in d a y &
Enlargement of the early times of Figure 4.70.
Figure 4.69 that both are tending to the high temperature steady state and that even though run 1 starts at YRD (0) = 1 .6 and run 4 at YRo (0) = 1 .4, the low lJIR (0) of run 4 causes it to move much faster to the high temperature steady state while for run 1 the temperature decreases first then it rises to the high temperature steady state. Therefore after a time of less than 0.002 day, run 4 has a higher temperature than run 1 . The high sensitivity of the dynamic behaviour to lJIR (0) and the qualitative difference of the effect of lJIR (0) for different values of YRo (0) suggests that we look further into this sensitivity around the operating middle steady state. In this case instead of taking two values of YRo (O) (around the middle steady state) and changing lJIR (O) along it, we do the opposite, we take two values of lJIR ( O) around the middle steady state and change YRo (0) along it. The two dimensional initial conditions space in this case is shown in Figure 4.72, which shows only the direction of the system response, either towards the low temperature steady state (Lss ) or the high temperature steady state (Hs5) . The extreme sensitivity of the system to lJIR (0) is evident, e.g. for YRD (0) = 1 . 5727 we notice that when lJIR (0) 0.986, th e system drifts to the high tem pe rature steady state while for the same YRo (O) but w ith lf/R (0) = 0 . 99 1 the system drifts to the low temperature steady state. Also it is important to notice that the coordinates of the middle steady state are not very helpfu l in decidi ng the direction of the drift. For =
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S . S .E.H. ELNAS HAIE and S . S . ELSHISHINI
552
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FIGURE 4.72 Effect of initial catalyst activity of reactor on the direction of drift. ------ approximate location of the separatrix
1. VIR (0) = 0.986, VldO) = 1.0, yRD (0) = 1.5727, yGD (0) = 2.0074 =
2. VIR (O) 0.986, VIG (O) = 1.0, YRD (O) ;; 1.5527, Y Gn (O) ;; 2.0074 3. VIR (0) = 0.986, VldO) = 1.0, yRD (0) = 1.42, y GD (0) = 2.0074 4. VIR (0) = 0.991, lp'G (O) = 1.0, YRD (0) ;; 1 .5727, YGo (0) : 2 . 0074 5. lp'R (0) = 0.991, lp'G (0) = 1.0, YRD(0) = 1.59, YGo(0) = 2.0074
6. lp'R (0) = 0.991, lp'G (0) = 1.0, YRn (0) = 1.7, YGn (0) = 2.0074
Aux = Auxiliary point used in order to locate the approximate position of tbe separatrix more accurately VIR (0) 0.986, VldO) = 1.0, YRD (O) = 1.5, YGn(O) = 2.0074 =
l/fR (0) 0.986 the shift from a drift to high temperature steady state to a drift to low temperature steady state lies between YRv(O) 1 .5 and 1 .42 (closer to 1 .42), while for l/fR (0) = 0.991 this sh ift occurs for YRv (0) greater than 1 .59. The dashed lines passing by the middle stead y state =
=
represents the approximate location of the separatrix for this system. From Figures 4.69 and 4. 72, it seems very difficult, if not impossible, at the present level of modelling, to pred i c t a priori the direction of drift of the system. For, although the present model (El shi shin i and Elnashaie, 1990) is more rigorous than previous models (Kurihara, 1967; Is col , 1 970; Lee and Kugelman, 1 973, Ed wards and Kim, 1988), it still con tains a degree of u ncertainty and empiricism that makes it unable to accurately model such extremely high sensitivity to initial values of catalyst activ i ty . Much theoretical and experimental work is needed through co-operation between academia and industry to develop more rig orous and more accurate dynamic models in order to achieve the fine dy namic simulation c ap ab le of predicting a pri ori , the direction of drift
at any time.
STATIC AND DY NAMIC B IFURCATION BEHAVIOUR
553
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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 0.2-0 . 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 .
.
"
=
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.
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S .S.E.H. ELNASHAIE and S.S. ELSHISHINI
554
_ _ _ _ _
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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 feed-back control system that stabilizes th i s unstable steady state. Both alternatives will be investigated i n the following parts of this section. /l-4. 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
<
FCD = FCDL
if
FCD = l . O
if
FCDcr and YfaH
>
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
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l l .-------·
(b)
/9
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'( ( 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 4.75a--d 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 4.76a--d 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
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Dynamic response of the system under the switching policy o;:: FCDL = 0.28. (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 4.76
0.03,
Ytan ;:: 1,85,
the case in Figure 4.75 , the dense phase reactor temperature still has a large amplitude. It varies between 1 .65 and 1 .46, this is an amplitude of 0. 1 9 (95°C). For the regenerator dense phase temperature (Figure 4.76b), again like in the case presented in Figure 4.76b, the amplitude settles to a low value, but the center line of oscillations drifts away from the steady state dense phase temperature of the regenerator to a very high value above 2.3 ( 1 1 50 K). The unreacted gas oil in the dense phase oscillates as shown in Figure 4.76c. The dense phase gasoline yield, x2D oscillates strongly between values higher than 0.4 1 and values lower than 0.36 (Figure 4.76d). /l--4. 2. Feed-back control of the system Operation at the middle unstable steady state can also be achieved by stabilizing the unstable steady state using simple feed-back control loop (Elnashaie, 1 977). For the sake of simplicity we present the use of a .simple SISO proportional feed-back control loop, where the dense phase temperature of the reactor is taken as the controlled (measured variable) while the manipulated variable can be any of the input variables
S . S . E. H . ELNASHAIE and S . S . ELSHISHINI
558
� a:
�
kC
0
I
•
0,1 ,2 ,l , 4 , S , 6
20000.00
j1;
t
c: • "
c; .. :z:: ..
j
1
II.
0
• :z::
,..:
II.
- 4 0000. 0 0 L..lo.'-L. ...l .& .... .... .. .... ... .... ... .... ... .... .. .... ... '-'-" ... ........ .. .... ... U-.I....L. ... ....L. .L ..L...L. ..I.. ....... L..L.J� 1 . &0 1.30 0. & 0 2 so 2 .30 y i\ D
FIGURE 4.77 Effect of Kc on the heat function vs. YRD diagram.
of the system Yfa • FCD, . . . We chose here to use Y1a as the manipulated v ariable. The set-point to the proportional controller is thus the dense phase reactor temperature at the desired middle steady state. The simple SISO control law is thus Yra U) = Ytass + Kc ( YRDss - YRD (t)) where Kc is the gain of the proportional controller. The control loop in this case, affects both the static behaviour as well as the dynamic behaviour of the system. The main objective here, is to stabilize the unstable saddle type steady state of the system. In the above control law the following steady state values were used Yfass = 0.872, YRDss = 1 .5627 (Figures 4.66a--c). The new bifurcation diagram corresponding to this closed-loop case is then constructed. The heat function is obviously altered by the introduction of this control loop as shown in (Figure 4. 77), with the high and low temperature steady states changing with the change of the proportional gain of the controller, Kc. The middle steady state, of course, does not change with Kc since it is the set point of the proportional controller. The change of the heat function with Kc is rather complex and a better presentation can be given using the bifurcation diagram (Figure 4.78). It is clear that the bifurcation diagram represents an imperfect pitchfork diagram, where the middle steady state persists over the entire range of Kc, ev en for negative values of Kc (positive feed back control, which would have the effect of destabilizing the system). For ne g ati ve values of Kc with
STATIC AND DYNAM IC BIFURCATION BEHAVIOUR
V 11 0 •
.. .. ;J
i.
v,,., . .
0
Fc o u
e
.. ....
�
• • =
559
t. U l l
0.1 7 7 t .o
Cl " 0. .. ..
! � :;
b
Cl .. cz: .. .. ..
c
0 ·;; <: ..
E i5
0 " >
_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 closed-loop 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. /l-5.
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 L--2..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. 79a-d 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 6-dimensional 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 2-dimensional gives erroneous predictions not only quantitatively but also qualitatively. The 4-dimensional 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 feed-back 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 El-Hennawi, 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,
><
0·
3
I
.
' ' .
- ---
\
\
'
• •••
Unstobl• S t abt • HI
0· 3
Yt a
I
I
. . . .
,' .. � - � .. .. ... ,
2· 5
2- 0
1 ·5
(C) '
.
. . .
'
. . '
0 2
'
.
j .
o· 1 0· 0
1 ·0
0·5
o· 5
...
_) '
O· O O·O
a
. .
0· 1
><
\
.
0· 2
0· '
(b)
I
---
...
2 .s
2 -0
..
_ _,
0· ' 0
1 ·5
1 ·0
0 ·5
OHB
0·0
0 ·5
1 ·0
' ,""'
1 -5 Yoo
2 ·0
2 ·5
FIGURE 4.81 Open loop bifurcation diagram. (a) Dimensionless reactor dense phase temperature vs. dimensionless air feed temperature to regenerator. (b) Dimensionless reactor dense phase gasoline concentration vs. dimensionless air feed temperature to regenerator. (c) Dimensionless reactor dense phase gasoline concentration vs. dimensionless reactor dense phase temperature.
//-1.
Behaviour of the open-loop (uncontrolled) unit
Figure 4.8 1 shows the bifurcation diagram of YRo , X2o versus Yfi, (Figures 4.8 1 a,b) for the open loop unit as well as the Xw versus YRo diagram (Figure 4. 8 1 c), where all the parameters other than the air feed temper!lture lfa, are assigned their industrial values given in Table 4. 1 0. The value of Yta for the industrial unit is 0.872 and the corresponding dimensionless reactor dense phase temperature YRo is 1 .5627 as shown in Figure 4.8 l a. It is <:lear from Figures 4.8 l b,c that this condition does
S . S .E.H. ELNASHAIE and S.S. ELSHISHINI
566
not correspond to the maximum gasoline yield, it actually corresponds to The maximum gasoline yield is = and
X2f)= 0.38088.
X2/) 0.437885,
occurs as shown on Figure 4. 8 1 b, at lfa = 1 .4 1 92 1 3 and YRo = 1 . 1 93 1 4.
Therefore operating at this maximum gasoline yield results in a 1 5 % increase i n gasoline yield . There are two important observations deduced from Figure 4 . 8 1 : 1 . The operating conditions for the maximum gasoline yield i s very close to the critical points, specifically static limit points (sometimes called turning points) as shown in Figures 4 . 8 1 a,b. It is in the region of three steady states, but very close to the boundary between the region of three steady states and the region of five steady states . It should
2.
be
expected that controlling the unit at this steady state will
not be easy . There
is
a degenerate Hopf bifurcation point (shown by the symbol •
on the three Figures 4 . 8 1 a-c and denoted DHB on Figure 4. 8 1 a) . As discussed in chapters 2 and
3, this degenerate Hopf bifurcation point
involves interaction between a Hopf bifurcation point and a static
limit point. This type of degeneracy may give rise in its vicinity to a homocl inical (infinite period) bifurcation, bifurcation to torus or period doubling leading to chaos . It is important to notice that the optimum Yta is quite close to Yta giving rise to the degenerate Hopf point, i.e. the condition for max imum gasoline yield is in the vicinity of this degenerate Hopf point.
11-2.
Behaviour of the system with feed back proportional control at the unstable industrial operating point YRo = 1 . 562 7 (Ysp = 1 . 5627, Yjas
=
0. 872)
In this part a proportional feed backcontrol system is used where the manipulated variable is the air feed temperature ( Yfa), the controlled variable is the dimensionless reactor dense phase temperature ( YRo), and the dimensionless controller gain, Kc, is the bifurcation parameter.
The controlled steady state in the first case is the unstable industrial operating point with the value of the dimensionless air feed temperature being Yfas = and the set point of the controller is taken as Ysp =
1 .5627
0.872
which is exactly the middle steady state temperature. Figure 4 . 82a shows the static and dynamic bifurcation diagrams for YRo against
Kc for this case. The static bifurcation diagram is an imperfect pitchfork
5
C
with a small region of steady state s . On branch there is a Hopf bifurcation point (HB) from which a periodic branch emanates with an increasing amplitude as Kc increases. An enlargement of rectangle 1 in Figure 4 2 a is shown in Figure 4. 82b where it is clear that for a region of Ke to the left of the HB point, the amplitude of the periodic solution increases slowly as Kc decreases. .
8
STATIC AND DYNAMIC BIFURCATION BEHAVIOUR 3-5 3- 0
1- 70
(a)
E
1 ·60
2-5 0 cr >
2-0 1-5 1-0 0-5
0
0 a: >
0
�::�-�
l5)
0- J
- _ .,.. _ _ _ _ _ _ _ _ _ _ _
1 -40 1 · 30
.
---
�- �. --- .! - :- � ,• . . . .
1- 10
0- 0
0 "' ><
( b)
1 ·50
1 - 20
A
0·5 0-4
567
D
B
\_
./!_?.'-Cal
0-2 0·1
' ' ' I ' '
0 0
0-0
''
\-v- B
' I
1·0
'
0
(C)
',�c '
0·45
r-- -.-:-:-1 -1 ""' 0 .... ><
( 4)
- -- - U n t U . b t lf
2· 0
- S l l blt
••••
....
3-0
Kc
4· 0
HB
5 P8
5 ·0
6-0
td l
... _
0·4 1
--
0 -37 1. 616
..&... - - - - - - -
1-618 Kc
-- - -
1 ·6 2 1
Closed loop bifurcation diagrams for Yfas = 0.872 and Ysp = 1.5627. (a) Dimensionless reactor dense phase temperature vs. controller gain. (b) E nlarge ment of rectangle 1 in Figure 4.82a. (c) Dimensionless reactor dense phase gasoline concentration vs. controller gain. (d) Enlargement of rectangle 1 in Figure 4.82c. (HB = Hopf bifurcation point, SPB = Static periodic branch).
FIGURE 4.82
However, close to Kc = 1 .617, the amplitude increases sharply with the decrease of Kc and after a very short distan ce on the Kc scale, the perio d ic branch terminates homoclinically, leaving only t he high temperature steady states as the unique solution on branch E till point 2 is reached where another stable branch A appears. Branch A is not physically feasible because it corresponds to reactor temperatures below the vaporization temperature of the gas oil feedstock used in this unit. Figure 4.82c show s the bifurcation diagram of gasol in e yield x2 D versus the computer gain Kc. Because the operating point at which the unit is controlled is not the opti mum operating point corresponding to maximum X2v (as di sc u sse d earlier) therefore it is noti ced that the controlled steady state (the horizontal branch D) does not al w ay s give the maximum gas ol ine yie ld . In the region of Kc between points (a) and (b) branches C ' and B give higher values of X2D than those for b ranc h D. The h ighe s t static stable value of X2D on branch C' is 0.438755 compared with 0.38088 for branch D. This represents a percentage increase of abo ut 1 5% .
568
S.S.E.H. ELNASHAIE and S . S . ELSHISHINI
It is also interesting to notice , from the enlarg ement of rectangle 1 in Figure 4.82c shown in Figure 4.82d, that the periodic solution in certain reg ions of the values of Kc (e.g. Kc = 1 .61705) gi v es average gasoline yie ld slightly higher than the maximum X2D on branch C' (ab ou t 0.9%) and o f c ourse higher than the yie ld on branch D ( ab ou t 16%). From a practic al point of view, it is extremely difficult to exploit this po ss ibility because the periodic attractors with gasol ine y ield s higher than the static point attractors, are very clo se to the region where the system could go to the high temperature, low conversion, branch (E). However there may be other region s of paramete rs where this i mprovement is more pron ou nce d and the d an ger of i gniti on is minimal. It is clear from Fi gure 4.82 that there is a number of region s of bistability which can be summ ari zed as follows,
From Kc= 0.0 to Kc= 0.8897925 ( c orresponding to the static limit point, SLP, 3), the system has three steady states, two stable on branch A and E and the third unstable on branch D. From Figure 4.82c , it is clear that in this region , the unstable steady state on branch D gives the h ighe st g aso line yie ld (X2D = 0.38088) while the other two stable steady states both give very low gas oline yield (X2D < 0.05). S te ady states on branch A are obvi ous ly ph y sic al ly unreali stic bec ause they corre spond to reactor temperatures lower than the v apori z ati on temperature of the feed gas oil used in this un it . 2. In the region from Kc= 0.8897925 (SLP, point 3) to Kc = 1 . 173691 ( corre spon di n g to degenerate Hopf bifurcation, point 2), th e system has five steady states. However, only the hi gh temperature steady states on branch E an d the low te mpe rature steady s t ate s on branch A are s table . Both branche s gi ve very l ow gas ol in e yield (X2J)<0. 1) as shown in Figure 4.82c. 3. In the region from Kc = 1 .173691 (c orrespondin g to degenerate Hopf bifurc ation point) to Kc = 1 .61703 (corre sponding to homoclinical termination of the periodic b ranch) , there are on ly three ste ady state branches , the only s table ste ady states are the ones on the high temperature branch E. Th e stable branch E i n this region corresponds to low gasol ine yield . However, the unstable branch B corresponds to gasoline yiel ds highe r than those of branch D around which the system is being controlled. 4. From Kc = 1 .61703 to Kc 1 .620444 the system has a periodic attrac to r c oe xis ti n g with the static high temperature branch E. Figure 4.82c shows that it is po ssible that the periodic branch gives, for certain v alues of Kc, an average gasoline yield higher than th at for branches D and C ' . Figure 4.83a shows the periodic oscillation of the reactor 1.
=
temperature X1 at Kc = 1 .6 1 75 which is obtained when the system
569
STATIC AND DYNAMIC BIFURCATION BEHAVIOUR 1 .60
(a)
1 .50
.j!-
0
=
Kc
1 . 40
0 N ><
1.30 1 20 1. 1 0
Q.t..t.3
1. 6 1 1 5
2000
1000
0
J.()()
0.439
2 .0 0
0 a:: > 1 . 50
0 N
>
K c • l. 6 1 7 5
0.200
0.0
200
0
J . oo
0 a::
(d)
>< 0 . 1 00
1 . 00
0 . 50
4 000
3000
2(XX)
O . .t.
K t o 1. 6 1 7 5
Kt : U I 7 5
O.H1
0.437 1000
3 000
(c )
2.50
(b)
600
400
(e)
zoo /
0
800
Kc : 2 . 2a
800
600
400
200
Q
( f ) K t : 2 . 20
1 . 00 0.00 1 . 37
.: 1 . 3 6
0
400
200
0
600
(g)
Kt : 2 . 20
0.41.
r""'
(h)
0.435
Kc
=
2. 20
0.42 5-
1 . 35 0
600
200
:< 0.4300
0
0
500
1000
T i m�. s�c
1500
0.420
0
5-00
1000
T im�. u c
1500
FIGURE 4.83 Dynamic response of the system under feed back proportional control for YJas = 0.872 and Ysp = 1.5627 at various values of controller gain. (a, b) YRn (0) = 1 .5, Yco (O) = 1.55, Xw (0) = 0.449, Xw (0) = 0.10, VIR (0) = 0.993 and Vlc (O) = 0.9890. (c, d) YRD (O) = l.S, Yco (0) = 1.60, Xw (0) = 0.449, X20 (0) = 0.10, VIR (0) = 0.993 and � (0) = 0.9890. (e, t) YRo (0) ;:: 1.30, Ycn (O) = 1.70, Xw (0) = 0.350, X20 (0) = 0.38, VIR (0) = 0.970 and Vlc (0) = 0.9890. (g, h) YRo (0) = 1.35, Yc0 (0) = 1.665, Xw (0) = 0.400, Xw (0) = 0.44, VIR (0) = 0.970 and V'o (0) = 0.9997.
has the initial conditions shown. Figure 4.83b shows the oscillations of the gasoline yiel d. For different initial conditions (essentially a slightly regenerator temperature), the system tends to the high temperature steady state (Figure 4. 83c) with very low gasoline yiel d (Figure 4 . 8 3d).
higher
570
S .S .E.H. ELNASHAIE and S.S. ELSHISHINI
For values of the controller gain from Kc = 1 . 62 044 to Kc = 3.28 1 3 (corresponding to the exchange of stability, point 5 on Figure 4.82a), the system has three steady state branches, the high temperature branch E, the middle branch D (still unstable saddle) and the low temperature branch C' (which is physically realistic because it corres ponds to temperature wel l above the vaporization temperature of the gas oil used as feedstock for this unit) and this branch C' is stable. From Figure 4.82c, it is clear that this branch gives higher gasoline yield than branch D. The temperature of the reactor o n branch C' increases and the gasoline yield decreases as Kc increases till point (5) on Figure 4.82a, is reached where the exchange of stability takes place between C' and C. Dynamic simulation for Kc = 2.2 (in a region of bistability, branche s C' and E) s how the effect of the initial conditions on the ste ady state towards wh ich the system tends. Figures 4.83e,f show the reactor dense phas e dimensionless tern perature (YRD) and the gasoline yield (X2D) respectively at Kc = 2 2 for the initial conditions shown. For di fferent initial conditions shown on Figures 4.83g, h and the same Kc, the system moves from the high temperature steady state (Figure 4 83 e) with low gasoline yield, X2D < 0. 1 , as seen on Figure 4.83f, to the low tempe rature ste ad y state (Figure 4.83g) with higher gasoline y ield , X2D=:0.43, as seen in Figure 4 . 8 3 h . 6. From the value of Kc = 3.28 1 3 (at the exchange of stability point 5 on Figure 4.82a), till Kc= 4.228964 at the static limit point 4, the branch D becomes stable for the first time. Therefore the value of Kc = 3 .28 1 3 is the critical value for stabilizing the saddle type unstable middle steady state wh ich is the originally declared purpose of installing this proportional feed back control loop. It is clear from Figure 4.82c, that these, now stable, steady states on branch D give the highe st gasoline yield (X2D = 0.38088), the other two branches C and E give lower values of X2D. The reactor operation at steady states on bran c h D i n this region, can be quite sensitive to small disturbances involving an increase in the reactor temperature YRD· This sensiti vi ty decreases when moving from point 5 to point 4. For Kc values higher than those corresponding to point 4 (Kc>4.228964), steady states on branch D become unique and globally stable.
5.
.
.
,
ll-3.
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 . 0-t---.""'""1----r--r-r--o-..-',_....,.._,..,.-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
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4,
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S . S .E.H. ELNASHAIE and S . S . ELSHISHINI
572
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4. 0
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. . .. .. ... � 1-'- .....,-- - - - -• •• n., n ...-. ..-t...._. ··· ·
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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 \ � .
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0. 1
B
r;;
'
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I
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(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).
/1-4.
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 a-c, 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 El-Bialey, 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
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x, :0.4378 8 5
0.1. 0 0. 3 5
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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.
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574
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S . S .E.H. ELNASHAIE and S . S . ELSHISHINI
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' FIGURE 4.87 Dynamic response of the system under feedback proportional - - -· control for Yfas = l.419213, Y,p = l.l 9314 and initiaJ conditions of YRD (0) = - 1.3, � YGD (0) = 1.6, Xw (0) = 0.4, Xw (0) = 0.4, lfR (0) = 0.9 and � (0) = 0.99. (a, b) Kc = 26. (c, d) Kc = 15. (e, f) Kc = 9. (g, h) Kc = 4. (i, j) Kc = 3. (k, I) Kc = 2.9. ·
Figures 4.87a,b show YRD and X20 oscillations with time, for a· · value of Kc close to the HB point 1 (Kc 26), the oscillations are clearly ' smooth, with small amplitudes that do not push the system up to prohibitively high temperature nor down below the vaporization temperature of the gas oil. Figures 4.87c,d show the behaviour of the system for Kc= 15. The amplitude of the oscillations increases but is still practically acceptable. Figures 4.87e,f (Kc= 9.0) show a considerable increase in the ampli tude size and the sharpness of the oscillations. As Kc decreases further to Kc 4.0, the oscillations become very sharp as shown on Figures 4 87g,h and the temperature rises to unacceptable high values and falls to values very close to the vaporization temperature of the fedd gas oil. The gasoline yield, X4, also decreases sharply. The sharpness of oscillations increases further as Kc decreases and approaches the homoclinical bifurcation at Kc = 2.89905, as shown in Figures 4.87i,j for Kc= 3.0 and Figures 4.87k,l for Kc= 2.9. The bursting oscillations (sometimes called hard oscillations) shown in Figures 4.87g,h-k,l resemble those observed in the Rose-Hindmarsh model for neuronal activity (Holden and Fan, 1 992; Fan and Holden, 1 993). The phenomena associated with bursting oscillations were studied extensively by Holden and Fan ( 1 993), such as transition from simple to simple bursting oscillations via chaos, transition from simple to complex bursting =
.
=
STATIC AND DYNAMIC BIFURCATION BEHAVIOUR
575
oscillations via intermittent chaos, transition from periodicity to a crisis induced chaos and back to periodicity by a saddle-node bifurcation, transition from simple oscillations to bursting oscillations without chaotic transition regions. From these phenomena, only the latter was found to occur in this analysis as shown on Figures 4 . 87a-l. III-
Summary of section results
The bifurcation analysis presented in this section for one of the industrial FCC units, using static and dynamic bifurcation diagrams constructed by the efficient software AUT086 (Doedel and Kemevez, 1 986), has shown a number of important and interesting features of these industrially important units. The industrial open-loop (uncontrolled) unit is not operating at its optimum conditions. Increasing the air feed temperature can give considerable increase in gasoline yield. The maximum gasoline yield occurs quite close to the degenerate Hopf bifurcation point and gives rise in its vicinity to more c ompl e x dynamic phenomena. The control of the unit at the industrial operating point shows that there is a stable branch giving higher gasoline yield than the industrial operating point. A Hopf bifurcation point develops at relatively low values of the controller gain, Kc and the periodic branch emanating from this HB point terminates homoclinically. For some values of Kc, periodic attractors give average gasoline yield higher than all possible static attractors. However this mode of operation, in this range of parameters, poses difficult practical problems. The change of Yra shows the structural instability of the pitchfork static bifurcation diagram. Control of the system around the maximum gasoline yield at the unstable operating conditions shows that the stabilization of this maximum gasoline yield requires a very high value of the controller gain. However, almost the same high gasoline yield can be achieved on a periodic attractor for a much lower values of the controller gain. 4.3
OSCILLATIONS AND CHAOS DURING THE CATALYTIC OXIDATION OF CARBON MONOXIDE
One of th e earliest and most extensively i n ve s ti g ated catalytic systems which shows complex dynamic behaviour is the isothermal or nearly isothermal c atalyti c oxidation of carbon monoxide over pl atinu m cata lysts as w el l as some other catalysts as will be di s c us s ed in this secti on This complex dynamic behaviour can occur over a certain range of .
constant operating conditions. The nature of this complex behaviour
576
S . S .E.H. ELNASHAIE and S.S. ELSHISHINI
and the conditions under which it occurs vary wi dely among different studies on apparently similar c atalysts . B oth periodic as well as chaotic behaviours are possible for this system as will be shown very briefly in this section. 4.3. 1
Introductory Review of Experimental and Modelling Studies on the Catalytic CO Oxidation
In this section some of the important studies on the dynamic beh aviour of the c atalytic oxidation of carbon monoxide to carbon dioxide will be presented. The reader can also find in the literature some good reviews (Sheintuch and Schmitz, 1 977; Slin'ko and Slin'ko, 1 978, 1 980; Razon and Schmitz, 1 986) . Jakubith ( 1 970) and Hugo and J acobith ( 1 972) observed self-sustained oscillations aro un d high conversion states over a platinum gauze. They offered an explanation for the o sc ill atory states in terms of two adsorbed species of CO, only one of which (i.e. the bridged form, x- C08, where x denotes an active site) reacts to form carbon dioxide. They then postulated that, as the surface becomes increasing ly saturated with chemi sorbed CO, the reverse of the surface step 2 x - COL � x - C08 + COg, is enhan ced . They speculated that oscillations were due to a slow reversible conversion of the linear form, x- COL, i nto the bridged form, x - C08, which at certain carbon m onoxide p artial pressures, leads to a decrease of the reactive intermediate x - C08, which is c apable of re acting to form carbon dioxide. Dauchot and Van Cakenberghe ( 1 973) s h owed that temperature and concentration oscillations could occur on a thin platinum wire surface. The oscillations were recorded as re s i stance change s caused by reaction heat evolution on the surface of the wire. The Langmuir-Hinshelwood mechanism that they postulated, was apparently autoc atalytic and could thus lead to o sc il lation s provided experimental c onditi ons were suitable. Plichta ( 1 976) and Plichta and Schmitz ( 1 979) studied the behaviour of the oxidati on reaction over a platinum foil, and observed that the experimental behaviour of the reaction was characterized by non osc il lato ry stable steady states in some cases and by osc i llatory states with con stant ampl itudes and periods in other cases. Three distinct regimes were encountered. On the basis of their apparent relative reactivities, these were designated as low , high and intermediate activity. No multiple steady states were observed in the low activity regime but were found in the high activity regime . The interme di ate regi me , i.e. the transition regime , was characteriz ed by rapi d deactivation and a non-reproducible oscillatory regime. A lthou gh a detailed mathematical an al y sis of the oscillations was propo s ed , the resulting theoretical model
n
did not satisfactorily represent the e xperime tal results.
STATIC AND DYNAMIC BIFURCATION B EHAVIOUR
577
Sheintush (198 1 ) using platinum foil had also observed three characteristic regimes. At low CO concentrations as well as hi gh CO concentrations, the system was stable, while at intermediate concen trations oscillatory states were detected. In the oscillatory regime, two sub-regimes also existed: one where the oscillatory states were simple (one peak per cycle) and a second one where multi-peak oscillations were observed. Several observations of the multiplicity of oscillatory states and non stationary aperiodic (chaotic) states were cited. Analysis of the observations and experimental limit cycles indicated that: a) The simplest system of differential equations necessary to describe th e observations of simple oscillations was a second order one accounting for two dynamic variables, namely, a gas-phase concen tration and a surface concentration. b) A third or higher order system was necessary to describe the obser vations of multi-peak oscillations and chaotic behaviour.
Turner et al. ( 1981) observed oscillations on a poly crystalline platinum wire. The oscillations were reproducible in both the rate of carbon dioxide production and the temperature of the platinum wire, over a wide vari ety of gas compositions and gas temperatures. They postulated that oscillations occured between two branches of a Langmuir-Hinshel wood reaction mechanism and were caused by the slow formation and removal of subsurface oxygen. A simple model based on this hypothesis was developed which gave excellent qualitative agreement with the observed oscillations. Beusch et al. (1972a,b) studied the oxidation reaction on a single pellet of a supported Pt catalyst. They observed stepwise instabilities, corresponding to "ignition" and "extinction" as well as oscillatory behaviour. It was concluded that these oscillations were mainly caused by the kinetic nature of the system rather than by mass and heat transfer resistances. They postulated that the reaction rate decreased due to the blocking of reaction sites with carbon monoxide. At temperatures above 523K, no further oscillations were observed. They proposed that several states of chemisorbed CO took part in the oxidation reaction. M cCarthy (1974) and McCarthy et al. (1975) detected isothermal oscillations on Ah03 supported pl atinum catalyst. They suggested that a likely cause of the oscillatory state was the interaction between two mechanistic steps of comparable rate. These steps were proposed to be: a) An Eley-Rideal type of reaction
cherni s orb ed oxygen. b) The chemisorption of oxygen.
between the
gas
phase and the
578
S . S .E.H. ELNASHAIE and S.S. ELSHISHINI
Varghese et al. (1978) observed isothermal oscillations over Ah03 supported platinum catalyst. They found a surprising effect of hydrocarbon impurities on the oscillatory behaviour. Oscillations were observed when the "impure" oxygen was used and disappeared when it was replaced by "ultrapure" oxygen. The only apparent difference between the "impure" and "ultrapure" oxygen was a 30 ppm hydrocarbon impurities. Cutlip and Kenney ( 1978) detected dramatic reproducible oscillations over Ah03 supported platinum catalyst only when 1 % 1-butene was added to the reactant feed mixture. They attributed the oscillatory phenomenon to simultaneous oxidation of carbon monoxide and 1butene. Wicke et al. ( 1980) proposed a mechanism of CO clustering and CO removal to interpret the isothermal oscillatory behaviour. Lisa and Wolf (1982) studied systematically the effect of reaction environment and catalyst surface on self-sustained oscillations on an Ah03 supported platinum catalyst. Rathousky and Hlavacek (1982) studied oscillatory behaviour over a long and a short bed of Ah03 supported platinum catalyst. They observed multiplicity of the steady states and periodic activity. For a long bed, oscillations were periodic while for a short bed, a complex interaction of individual oscillations was caused by individual particles. Chaotic behaviour was also observed for a certain concentration of carbon monoxide in a narrow range of temperatures. They postulated that the chaotic behaviour was caused by an interaction of two oscillatory processes. Elhaderi and Tsotsis (1982) observed reaction rate oscillations over Al203 supported platinum catalysts. They postulated that the oscillations observed in their system were of a true kinetic nature or the result of interaction between competing physicochemical processes and were not caused by the presence of impurities. Lynch and Wanke (1984a) investigated the occurence of self-sustained oscillatory behaviour over Ah03 supported platinum catalyst. They suggested that the oscillatory behaviour and the stable steady state activities were affected by the history of the catalyst, i.e. prior treatment and reaction conditions. Subsequent work done by the same workers (Lynch and Wanke, 1984b ), examined the effect of reactor operating conditions (recycle ratio, gas phase temperature, feed composition, feed flow rate and reactor pressure). They were careful to ensure that the catalytic activity remained constant throughout the investigation. The results indicated that oscillations were caused by changes in the rate controlling step coupled with heating of the metal crystallites to temperatures above those of the support and gas phase.
STATIC AND DYNAMIC B IFURCATION B EHAVIOUR
579
Vaporciyan et al. (1988) investigated forced reactant cycling of carbon monoxide-nitrogen stream over Pt/Sn02 catalyst in a differential reactor and observed an enhancement of the reaction rate of up to 9 times that of the optimal steady state. They also observed quasi-periodic behaviour. Their transient response experiments confirmed a Langmuir Hinshelwood type mechanism and their spectral analysis of the quasi periodic patterns identified three time constants of different orders of magnitude. The nature of the patterns they obtained also indicated a reaction mechanism which included one or more "reservoirs" of reactants and/or reaction intermediates. Shanks and Bailey ( 1989) applied the feedback-induced bifurcation method to CO oxidation on Rh-Ah03 catalyst. They obtained experi mental Hopf bifurcation data from operating the reaction system under feedback control based on measurements of the gas phase concentration of CO. Fichthorn et al. (1989) used Monte Carlo simulation technique to study the dynamic behaviour of a hypothetical bimolecular catalytic reaction which includes the elementary steps of adsorption and desorption of reactants A and B , surface reaction through Langmuir-Hinshelwood mechanism and desorption of product AB . The reaction resembles the catalytic CO oxidation. The investigators found regions of oscillatory as well as chaotic behaviour. Onken and Wolf ( 1992) investigated the total oxidation of a mixture of ethylene and CO by oxygen on silica-supported platinum using a continuous flow reactor. They observed self-sustained reaction rate oscillations under oxygen rich conditions for a broad range of CO/ ethylene feed ratio. Chen et al. ( 1993) investigated the thermal coherent structures of spatia-temporal dynamics during CO oxidation on Rh black/Si02 wafer. They analyzed the dynamics and associated it with the classical Melnikov mechanism for intermittent chaos. Ehsasi et al. ( 1993) followed the formation of spatia-temporal patterns during catalytic CO oxidation on a Pt(21 0) surface using photoemission electron microscopy (PEEM). They found that depending on the choice of reaction parameters (e.g. flow rate, sample temperature), the reaction exhibited both steady state and oscillatory rates. Vishnevskil et al. (1993) constructed a kinetic model formed of six differential equations which accounts for the dynamics of adsorbates on the catalyst surface during the catalytic oxidation of CO on a Pt ( l l O) surface, the model was found to reproduce the qualitative features of the reaction dynamic s under fixed parameters (T, Pco· l'_q.) In addition,
the model was found to be able to predict the possibility of bursting regime s (intermittence) . Imbinl ( 1 993) investigated the catalytic oxidation of CO on various
580
S.S.E.H. ELNASHAIE and S.S. ELSHISHINI
Pt and Pd orientations and found that the catalytic reaction system exhibited a wide variety of interesting phenomena including spatio temporal pattern formation, existence of Turing structures as well as the appearance of deterministic chaos and chemical turbulence. He investi gated the mechanistic steps leading to the observed phenomena and formulated a mathematical model which has been analyzed using bifur cation theory. Other important investigations on this reaction includes for example, the work of Gorodetskii et al. (1993), Uddin et al. (1993), Boudeville and Wolf (1993), de Boer et al. (1993), Lauterbach et a l. (1993), Block et al. (1993), Boehman et al. (1993) Krischer et al. (1993) and Kapteijn et al. (1993), just to mention a few. 4.3.2
Experimental Results for the Periodic and Chaotic Behaviour During Catalytic CO Oxidation
It is clear from the previous short review that extensive results have been reported in the literature regarding this reaction. We are presenting in this section some illustrative results. ( 0 )
( b )
roo� � :'
N�
( c )
80
ULJl..JLJlJ
0�--:,�,0� 1�-2:;';0,.::,2 ; � 5 ,,-;;0 -.! , T l lto4 ( ( m in I ( d )
i !i��r� r r u w u
T I M E (min)
FIGURE 4.88
Examples of relaxation (Type I) oscillations.
STATIC AND DYNAMIC BIFURCATION BEHAVIOUR
581
( c )
:t� 0
2
6 8 { mi n )
4
T I ME
10
12
( b ) 3 6 B.7 1- s e c . ----1
( c I 16 z 0
ii\
� a::
z 0 u
�
�
,1
v
15
TIME
FIGURE 4.89
v v 25
35
( s ec )
Examples of harmonic (Type II) oscillations.
Razon and Schmitz ( 1986) classified periodic oscillations into two major classes and called them either relaxation or harmonic. They further subdivised them into subclasses shown in Figures 4.88 and 4.89. Type I in Figure 4.88 are relaxation oscillations (Turner et al. , 198 1 ; Lynch and Wanke, 1982; Beusch et a/. , 1 972; Liao and Wolf, 1982). These are defmed as having abrupt transitions in their wave form. Type II in Figure 4.89 are harmonic oscillations (Ertl et al. , 1982; Sheintuch, 198 1 ; Keil and Wicke, 1 980). These oscillations do not have abrupt sharp transition points in their waveform The · co catalytic oxi dation also shows aperiodic (or chaotic as discussed in the previous parts of this book) oscillations as shown in .
S . S .E . H . ELNASHAIE and S . S . ELS H I S H I N I
582 0 �
� 0
0. 2
a: <&J
0. 1
0 u .j ...
0 z - 0 tI.J "' 4[ a: lo..
> z 0 u
0
�
�
In,
��
'V
������� ��� 0 10 20 30 40
50
60
T
FIGURE 4.90
w
L..
�
�
.__.__����--._�
__
70
60
I�E ( min )
90
100
1 10
1 20
I�
Example of aperiodic osciUations of CO catalytic oxidation.
Figure 4.90 (Razon and Schmitz, 1981). Chaotic behaviour of CO catalytic oxidation has been observed by many investigators as discussed in the introduction to this section. 4.3.3
Mathematical Modelling for the Catalytic CO Oxidation
Self sustained oscillatory behaviour has been frequently observed during the heterogeneously catalyzed oxidation of carbon monoxide over a platinum catalyst. Throughout the years many models (Sheintuch and Schmitz, 1977; Slin'ko and Slin'ko, 1978, 1980; Dagonnier and Nuyts, 1976; Eigenberger, 1976; Pikios and Luss, 1977; Strozier et al, 1979; Lagos et al., 1979; Dagonnier et al., 1980; Ivanov et al. , 1980; Chang and Aluko, 1 98 1 ; Takoudis et al. , 198 1 ; Rathausky and Hlavacek, 198 1 ; Lynch et al. , 1 982; Collins et al., 1987; Vaporciyan et al., 1988; Shanks and Bailey, 1989; Fichtharn et al., 1989; Chen et al., 1993; Vishnevskil et al., 1993; lmbinl, 1 993; Boudeville and Wolf, 1993; Beehman et al., 1993; Krischer et al., 1993; Kapteijn et al., 1993; Yip, 1989) have been proposed to explain this self sustained oscillatory behaviour (periodic behaviour). A very large number of models have been developed in the literature to describe the behaviour of CO catalytic oxidation. Razon and Schmitz ( 1986) summarized at that time, up to 90 different models formed of different combinations of: 6 different possible steps for CO chemisorption and desorption; 7 different possible steps for 02 chemisorption and desorption; 10 different possible steps for surface reaction; 5 different possible steps for changes in the form of the adsorbates; 2 different possible steps for adsorption of inactive species; 3 different possible assumptions. Th e simp lest model, which Razon and Schmitz ( 1 986) call "the basic model", consists of the fol lowing three steps :
STATIC AND DYNAM IC BIFURCATION BEHAVIOUR
A + X ¢:=:h> Ax
583
k
k_l
kz Bz + 2x <=> 2 & k_z
(4. 129) (4.130)
k3
Ax + Bx ----'--1 C + 2x
(4. 131) C02 , x = active sites of pl ati n um catalyst.
where A = CO, B = 0, C = The mathematical model representing the normalized surface concen trations of x, Ax and Bx (expressed as 80, 81 . fh respectively) is given by the two differential equations,
(4.132) and
(4. 133) where 80 = 1 - 81 - fh, CA and C8 are the gas phase concentrations of A and B re spectivel y Clearly this two-dimensional system cannot predict higher complexity than periodic behaviour. As mentioned before there are hundreds of models available now trying to describe the behaviour of the catalytic oxidation of CO. We present here a simple model developed by Yip (1989) for the c atalytic oxidation of CO over a single spherical supported platinum catalyst pellet (0.5% Ptly- A}z03) . The model simulates the experimentally observed periodic and chaotic behaviour, but does not predict correctly the magnitude of the oscillations and their periods. In this model the small rise in the catalyst surface temperature over the gas phase temperature, is neglected. The following assumptions are used in the derivation of this model, .
1. 2. 3.
A uniform environment surrounding the catalyst pellet in the experimental reactor. The feed gas to the reactor contains CO, 02 and inert i.e. no traces of C02 are present in the feed gas. The bulk gas phase temperature is constant and the feed gas stream enters the reactor at the reactor temperature.
nitrogen,
584
S . S .E.H. ELNASHAIE and S . S . ELSHISIDNI
4. The reactor pressure is constant at one atmosphere. 5. Adsorption of CO, 02 and C02 do not occur as activated adsorption. 6 . The carbon monoxide molecule adsorbs in the linear form on the catalyst surface and not the bridged form (Hayne and Tompkins, 1966) and thus each carbon monoxide molecule occupies a single metal catalyst site. 7. Oxygen dissociates upon adsorption on the surface (Ducros and Merrill, 1976; Gland, 1980; Campbell et al. , 198 1). Each oxygen molecule thus requires two metal catalyst sites for adsorption. 8. Desorption of atomic oxygen is neglected below lOOOK (Winter bottom, 1973a,b; Dagonnier and Nuyts, 1976). 9. The conversion of carbon monoxide to carbon dioxide proceeds primarily through the surface reaction of carbon monoxide and oxygen surface species, i.e. via a Langmuir-Hinshelwood kinetics (Palmer and Smith, 1974; Matsushima, 1978, 1979; Engel and Etrl, 1 979; Dwyer and Bennett, 1982; Gland and Kollin, 1983; Lindstrom and Tsotsis, 1 98 5 ) . 10. Product C02 desorbs from platinum immediately on formation, but is rapidly adsorbed by the alumina support from which it desorbs into the gas phase. 1 1 . Product C02 does not dissociate upon adsorption on the support sites (y.Alz03) and thus requires one support site for adsorption. 1 2 . The catalyst sites are conserved. 13. The activation energies are independent of surface coverage.
From the above assumptions, the kinetics of the reaction sequence can be described by the following mechanistic steps, A( g ) + x � A - x k
k_ ,
k
(chemisorption of CO)
B( g ) + 2x � 2B - x k3
(dissociative chemisorption of 02 )
A - x + B - x � C( g ) + 2 x k4
C( g ) + S � C- S k-4
(surface reaction)
( adsorption of C02 by support )
(4. 1 34 ) (4. 1 35) (4 . 1 36) (4 .
1 37)
where S = active site of alumina support. With these mechanistic steps (4. 134-4.137) and the use of the above assumptions; six ordinary differential equations can be formu lated, namely three gas phase mass balances for Oz, CO and COz (B , A and C respectively), and three surface phase species balances for
A-x, B-x and C-S . These six equations are represented as follows,
STATIC AND DYNAMIC BIFURCATION B EHAVIOUR
585
Mass balance on gas phase B, A, and C gives, V
deA = %eAJ - q · eA - am · k1 eA · ex + am · k_ 1 eAx --;]!
(4. 1 38)
V
d� 2 = qo eBf - q · en - am · kz . eB . eX -;}t
(4 . 1 39)
V
dec - a s · k 4 · ec . es + a s k-4 · ecs ---;u- = -q · ec + a m · k3 · eAX . eBX
( . 14 )
•
•
4 0
Mass balances on chemisorbed species A-x, B-x and C-S , d
e dt = am kl eA . ex - amk_ , eAx - amk3eAx . eBx
am
Ax
(4. 1 4 1 )
de8x - 2am� B . x2 - amk3 dt eAx . eBx e e _
am as
decs dt = a5k4 ec
·
(4. 142) (4. 143)
es - as k-4ecs
where, k_1 = k�1 exp ( -£_1 I
(4. 1 44)
Rr; T)
(4. 1 45 )
k3 = kf exp ( -£3
I Rr; T) k-4 = k� exp ( - E-4 I RrJT)
(4. 1 46)
ex = em - eAx - eBx
(4. 1 47 )
es = C.� - ecs
(4. 1 48) 2
q = q0 - ( RG T I P)[am ki eA ex - am k_1 eAx + a m k2 e8 ex -am k3eAx eBx + a5k4eces - a5k_4 ec5 ]
(4. 1 49 )
where em and es are the total active sites for catalyst and support respectively in kmol/m2 catalyst surface. These equations 4. 1 3 8-4 . 1 49 can be transformed into dimensionless form by introducing the following dimensionless terms. XA = CA I CAJ
9, = C. f C, 01 = Ccs f�
0 = CHf f CAf
X8 = C8 1 C81
F = q f qn
1'= tq0 / V
91 = CAx f Cm
Xc = Cc f CA/
9., = Cx f C., 92 = C8x f C., V.:. = VCAf f a"'C"'
S.S.E.H. ELNASHAIE and S.S. ELSHISHINI
586
rJ = EJ IRcT
K_l = am Cmk-l"e xp(-yl )/qoCA/ K -4 = a, 'C, k-4.,exp (-y4 )/ q0CAf
8, = I- 83
8., = 1 - 81 - 82
2
F= q / q0 = l - (RG T/ P)(K1 CA/ -xA (l - 81 - 8) -K_ 1 CA/ 81 + K2C8/x8(1 - 81 - 82 ) -K3CAf8182
+ K4 CA1 ·xc
( 4. 1 50)
where q 0 and q are the inlet and outlet volumetric gas flow rates in m3/s . The dimensionless terms are inserted into equations to
4. 138-4. 143,
g1ve, dXA
dr d� dr
:�
= 1 - F · XA - K1 · XA (1 - 81 - 82 ) + K_ 1 81 = 1 - F · X8 - K2 · X8 ( 1 - 81 - 82 )
2
= -P. Xc + K3 81 82 - K4 Xc ( l - 83 ) + K--4 83 ·
dr = Vm [ K1 · XA ( l - 81 - 82 ) - KjJ1 - K3 81 82 ]
d 81
d82 dr d 8-,
d�
-
-
2
(4. 1 5 1 ) (4. 1 5 2) (4. 1 5 3 ) (4. 1 54)
= Vm [2 X8 · us: · K2 (1 - 81 - 82 ) ) - K3 81 82 ]
(4. 1 5 5 )
= � [ K4 · Xc ( l - 83 ) - K--4 83 ]
(4. 1 56)
Equations 4. 1 5 1 -4. 1 5 6 were integrated numerically and both periodic and chaotic dynamic characteristics were obtained for different CO concentrations. In the mechanism proposed by Yip ( 1 989), a mechanistic step of adsorption-desorption of C02 concentration on the support sites (i.e. yAh03), was introduced. The adsorption/desorption of C02 was included because supports such as y-Al203 ofte n exhibit a large capacity for C02 adsorption (Lynch et al. , 1 982). However, it is important to note that the introduction of support adsorption/desorption for C02 does not affect the catal yti c production of the C02, but it does affect the dynamics of the gas-phase C02 concentrations. Wi th this step, the model was able to predict most of the oscillatory behaviour found in the experimental
results (Yip, 1 989).
STATIC AND DYNAMIC B IFURCATION BEHAVIOUR
587
Catalyct Fuder
� Reactor
Timer
Filter
G u F e� Compre&sor
FIGURE 4.91 Simplified flow diagram of Union Carbide "UNIPOL" fluidized bed catalytic process for the production of polyethylene. (V = valve).
4.4
4.4.1
STATIC AND DYNAMIC BIFURCATION BEHAVIOUR OF THE INDUSTRIAL UNIPOL PROCESS FOR THE PRODUCTION OF POLYETHYLENE IN FLUIDIZED BED REACTORS WITH ZIEGLER - NATTA CATALYST Introduction and Description of the Process
The production of polyethylene using the bubbling fluidized bed reactor technol ogy (UNIPOL process) of Union Carbide and employing Ziegler - Natta catalyst, offers the most versatile process capable of producing HDPE and LDPE in tQe same reactor through a relatively si mple change in the operating conditions. The reactor (shown in Figure 4.9 1 ) is formed of a reaction zone which consists of a bubbling bed of polymer particl e s growing around very small catalyst particles and both are fluidized by the continuous flow of gaseous comp onents in the form of make-up feed and rec ycl e gas. The gas flow rate through the bed must
588
S . S .E.H. ELNASHAIE and S.S. ELSHISHINI
be well above the minimum flow required for minimum fluidization (usually from 3 to 6 times the minimum fluidization velocity is used). It is essential that the bed always contains particles to prevent the formation of localized hot spots and to entrap and distribute the particu late catalyst through the reaction zone (Wagner et a/. , 1 98 1 ). Ethylene which is the main reactant, is used for fluidizing the solid particles and also for cooling. Reaction temperature is maintained between 1 00°C to 1 30°C and the pressure is maintained at 300 psi while catalyst productivity is in the range of 2-20 kg PE/g-cat. Polymer is formed as small granules of 0.5 to 1 .0 mm diameter. The granular product is dis charged intermittently into a tank were unreacted gases are removed then after blending and stabilization, the product is ready for pro cessing or pelletizing. Molecular weight distribution is controlled by catalyst selection. A density range of 0.9 1 5 to 0.97 g/cm3 is produced and controlled by the amount of comonomer added. This versatile process which avoids the use of hydrocarbon diluents or solvents has proved to be a very succesful technology. In comparison to the con ventional high pressure LOPE process, Union Carbide has reported significant savings in investment and energy requirements for this gas phase technology (Kirk Othmer, 1 980). The importance of a more complete understanding of this gas phase polymerization process has been reasonably recognized by the academic community. Some experimental and reactor modelling work has been carried out in recent years. Galvan and Tirrel ( 1 986) and McAuley et al. ( 1990) investigated the means of predicting the molecular weight distribution for this process. Modelling at the particle level to predict polymer particle growth rate, has been investigated by Hutchinson et al. ( 1 992) . Hutchinson and Ray ( 1 990) developed a thermodynamic model to predict equilibrium monomer concentration at the catalyst sites in terms of external gas phase monomer concentration in the vicinity of polymer particles. A fluidized bed polyethylene reactor model was developed by Choi and Ray ( 1 985) who used the model to investigate the different control policies for this reactor. A simplified well-mixed model which describes the system in the limiting case where there is no difference between bubbles and emulsion gas tem perature and concentration, was developed by McAuley et a/. ( 1 994) . Xie et al. ( 1 994) presented a review of relevant macroscopic and micro scopic processes for gas phase ethylene polymerization, both chemically and physically. Polyethylene polymerization is a highly exothermic reaction and it has been observed by Elnashaie and El-Bialey ( 1 980), Bukur and Amundson ( 1 982) and Bukur et al. ( 1 974) that when highly exothermic reactions occur in fluidized bed reactors, unusual steady state and dynamic behaviour may occur.
STATIC AND DYNAMIC BIFURCATION BEHAVIOUR
589
The Polyt>thylt>ne Re a c t o r M od t>l
qc
Dense
B u bble P h a se
..-....oof ..
( E m u l s io n ) Pn a se -----il----
Gb
FIGURE 4.92
G
CMo
ao
Gl
Schematic diagram for the polyethylene reactor model.
In the present section, the dynamics and bifurcation behaviour of the system, with special emphasis on the effect of some operating variables, are presented to the reader through the use of the two parameter conti nuation diagrams which give a good and condensed presentation of the basic characteristics of the bifurcation behaviour. One parameter bifur cation diagrams are also presented to show the effect of some parameters on temperature, conversion and polyethylene (PE) production rate. Figure 4.92 shows a schematic diagram of the freely bubbling fluidized bed for polyethylene production. The monomer is introduced at the bottom of the reactor through a distributor which plays an important role in the operation of the reactor. Since polymer particles are hot and possibly active, they must be prevented from settling to avoid agglo meration, so it is essential that the bed always contains particles to prevent the formation of localized hot spots. During start-up, the reaction zone is usually charged with a bed of polyethylene particles before gas flow is initiated. The catalyst is stored in a catalyst feeder under a nitrogen blanket and is injected to the bed at a rate equal to its consumption rate. Makeup gas is fed to the bed at a mass rate equal to the mass rate at which polymer particles are withdrawn. Polymer product is conti nuously withdrawn from the reactor at a rate such that the bed height is held constant.
S . S .E.H. ELNASHAIE and S . S . ELSHISHINI
590
4.4.2
1-
Development of the Dynamic Model
Assumptions
phase) is perfectly mixed and is at incipient fluidization conditions with constant voidage. 2. The flow of gas in excess of minimum fluidization passes through the bed in the form of bubbles. 3 . The bubbles are spherical, of uniform size and are in plug flow. 4. The bubble phase is always at quasi-steady state. 5 . Polymerization occurs only in the emulsion phase and no reaction occurs in the bubble phase. 6. Mass and heat transfer between the bubble and the emulsion phases occur at uniform rates over the bed height. 7. An average value of bubble size and hence average values of the mass and heat exchange parameters are used in the comp utati on . 8. Mass and heat transfer resistance between the solid polymer particles and the emulsion phase are negligible. 9. No elutriation of solids occurs. 1 0. Catalyst injectio n is continuous at constant rate. 1 1 . The product withdrawal rate Q0, is always adjusted to maintain the 1 . The emulsion phase (dense
bed height H, constant.
1 2. There is no catalyst deactivation. II-
Hydrodynamic relations
Based on the two-phase model of fluidization, the following relations hold, GI =
Umf · A = UI · AI *
AI = A ( 1 - o )
As = A o* A = AI + As
UI = umf I (1 - 8* ) ue = UI I €mf by
(4. 1 57) (4. 1 5 8)
(4 . 1 5 9 )
(4 . 1 60 ) (4 . 1 6 1 )
(4. 162)
The superficial ve loci ty at minimum flu idi z ing conditions is g i ven the fol lowi ng correlation (Kunii and Levenspiel, 1 969), (4. 1 63)
STATIC AND DYNAMIC BIFURCATION BEHAVIOUR
591
The bubble rising velocity i s given by (Kunii and Levenspiel, 1969), (4. 1 64)
uB = uo - umf + 0. 7 1 1-Jg · dB The
1 969),
relative fraction of bubble phase is given by (Kunii and Levenspiel, (4.165)
The velocity of the emulsion gas is g i ven by (Bukur et al. , 1974) , (4. 166) The bed voidage at minimum (Broadhurst and Becker, 1975),
fluidization
conditio n s
is given
by
(4. 167) u b diameter is obtai ned from Mori and Wen (1975) corre
The b b l e
lation,
DB = DBM - ( DBM - DBo ) · exp ( -0. 3 H / 2 D)
(4.168)
where, DBM = 0 . 652 [ A . ( Uo - umf )]0.4
an
d
DBo = 0. 00376 (Uo - umf )2
The heat transfer with the wall is given by (Wen and Leva, 1956), (4. 169) where, Pr =
Jl · Cpg Kg
equal a
Lmf Lf
=
UB UB - Uo + Umf
and ¢,.. is assumed to 0.6. The mass and he t transfer coefficients between the bubble an d dense phases are obtained from the following relations (Kunii and Levenspiel, 1 969) :
S.S.E.H. ELNASHAIE and S . S . ELSHISHINI
592
for the mass transfer coefficient,
-1 -1 Kbe
where,
=
Kbc
1
+-
6. 78 nr
Emf D us
for the heat transfer coefficient,
1
1
(4. 1 7 1 )
(4. 1 72)
1
= -- + Hbe Hbc Hce
where,
Hk = 4 Hce _
III-
) 0· 5
( -- . 5 ( Pg pg) +5. 85 ( . )0.5 (Emf · Us ) 6.7 8 (P Kce =
(4. 170)
Kce
Um
f
·C
·
g 0 5 KP
D
B
g · Cpg · Kg
·
Pg
d5 B
D�
Model equations
·
Cpg
]0 . 5
.
(4 1 73 )
(4. 1 74)
(. 5
4 17 )
Bubble phase monomer mass balances A
mass balance on the inactive bubble phase for the monomer gives:
(4. 1 76)
at
z*
= 0.0,
which can
--.- -
be rearranged as follows, dCMb dz
Kbe · Aa GII
( CMb - CMe )
(4. 1 77)
STATIC AND DYNAMIC B IFURCATION BEHAVIOUR
593
Defining dimensionless height z = z*IH, K8 = Kbe · HI U8 and U8 = G81A8 and integrating the differential equation gives,
the
as
linear
(4. 1 78)
Using th following d imensionless groups : xl B = CMb iCo, X l = o CMo fC0, X1 = CMe iCo. equation 4. 1 78 can the following dimensionless form,
e
be put in
(4. 1 79)
Dense (emulsion) phase monomer mass balances
dCMe -
AI . H . €mf -----;Jf - GI ( CMo - CM) +
Khe . A8 .
( CMb - CMe ) dz
*
(4 . 1 8 0)
- Kp(I'e) · CMe · A1 · H · ( l - Emf ) · Ps · Xcat - � · CMe · Emf
The int gr l eq a o of the linear differential equation of the
JH
JH 0
e a in the above u ti n can be obtained from the solution bubble phase (4. 1 59),
( CMb - CMe ) dz * =
0
(
--1- K ( T ) € mf
p
e
·
Kb< 8*
( 1 - 0 )Emf
(4. 1 8 1 )
using the hydrodynamic relations
Substituting into equation 4. 1 80 and
given in section II, we obtain, dCMe = Ue ( CMo - CMe ) + dt H
)
( CM - C ) H o M ( l - exp (- KB )) K8
·
(
)
( CM - CM) a · ( l - exp
KB
CMe · (1 - Emf ) p·' · Xcat -
Qo · CMe
A . H . (l - 8
Putting the equation in d imensionless form
•
)
(- K )) 8
(4. 1 82)
gives,
(4. 183) where,
594
S . S .E.H. ELNASHAIE and S.S. ELSHIS HINI
= B
K H Ks -- be · Us
Ue · to R = --
H
a=
Ps 0 - Emf ) Co · Emf
Ya =
S=
s · to U = -H · Emf -
t = -t to
Ea
/?a · T,.ef
lo · Kpo · Ps · ( 1 - Emf )
Emf
8* f = -* 1 8
U
Catalyst mass balance in the dense (emulsion) phase
(4 . 1 84) which can be reduced to the following equation, (4. 1 85) Using the following dimensionless group, (4. 1 86) and using the assumption of constant bed height (H) which implies that the volumetric rate of product removal (Q0), is equal to the rate of increase in bed volume due to polymerization,
{k . (ps ( l - Emf) + E,..j . CMe ) = A . H(l - 8* )(1 - Emf ) . Ps . Kp('Fe ) · CMe . Xcat
(4. 1 87)
Rearranging equation 4. 1 87 give s , {k · !0
* A - H (l - 8 )
= 10(1 - Emf )Ps · Kp('Fe) CMe ·
·
Ps ( l - Emf ) + Emf " CMe
Xcat = fo
(4. 1 88)
which can be put in the following dimensionless form, /,
"=
t, (l - EmJ )Ps · Ke C Fe ) · CMe · Xcat = ' · X1 · X2 · exp ( - Ya I Yo)
Ps O - Emf ) + Emf " CMe
(a + X1 )
(4. 1 89)
STATIC AND DYNAMIC B IFURCATION BEHAVIOUR
The dimensionless form of the catalyst mass balance equation (4. 1 84) becomes,
dX2 -
dl
to .
,
595
qc
' A - H( 1 - 8 ) (1 - Enif ) · p,
Energy balance on the bubble phase (4. 1 9 1 ) z* = 0
and at
T = Jbo
Since the bubbles rise much faster than the gas in the emulsion phase, we take an average value CMb (CMb ) to simplify equation 4 . 1 89,
for
(4. 1 92) Putting the equation in dimensionless form leads to,
(4. 1 93) Integrating equation 4. 1 93 gives,
(4. 1 94) where, r;Bo _ y; Bo _ _ _ F
y;
y; Ts s --
T,-ef
The average value of Y8, is,
_
T,-ef
(4. 1 95 ) The average value of X18 is obtained from the bubble phase mon ome r balance equation 4. 1 79 and is give as follows,
f
� 8 = Xw dz = o
n
X1 +( XIo - ) XI
Ks
(1 - exp ( - K8 ))
(4. 1 96)
S . S .E.H. ELNASHAIE and S.S. ELSHISHINI
596
Energy balance on the Emulsion phase
-G1 · CMe · Cpg ( J;_ - � ) + A8
•
Hb_
J ( I;, - J;_ ) dz• H
+AI · H (-!lli, )Kp (T, ) ·ps(l - Emf ) · Xcar · CMe -Q)l -Emf ) · Ps · Cps (Te - T,�f )
-Q
o
· E
mf
0
· CM e · Cpg ( Te - Tre..f ) - n · D · H · h ( Te - Tw ) w
(4. 1 97)
The energy balance equation can be put in the following form,
+(-Mfr ) Kpo · ex p ( - ya I YD ) · p-� { l - Emf ) · X1 · Co · X2
Qo ( l - Emf )·Ps · CP,.( Yv A · H · ( l - o* )
Jr· D· hw
- 1) T,.ef
(4 . 1 9 8 )
D - Yw ) Tr�f
A-(1 - 8*) (Y where,
YF = TFI Tref
Yw =
Yv = Te i Tref
Yn =
Twi T,�r Th i Tref
The integral in e u at on 4. 1 98 can be evaluated from the inert bubble phase energy balance equation 4. 1 94 to give the following relation,
q i
f I
0
( YB - YD ) dz =
I I
0
[< Yno
-
Y
D
)
·
x
e p ( - KH
· z l xlB )] dz
= ( Y80 - YD ) · �1 8 (1 - ex p ( - KH I X1 8 )) H
(4. 1 99)
STATIC AND DYNAMIC BIFURCATION BEHAVIOUR
597
Replacing the integral into equation 4. 1 98 and using the following dimensionless groups, b'
=
( 1 - Emf )p" Cps ·
E mf · Co · Cpy
( ) (
Tb Y. = Tref
B
K=
-
8
•
KH · U · f = -- · 1-8
to · hw
D · Emj. · Co
( )
*
Hbe · to
Co · Emf cpg
*
· C · (1 - 8 ) py
the final dense phase heat balance equation is obtained, d Y!' + YD - 1 dt b' + XI +K H
·U -r
d�l = B YF - YD R XI dt b' + X1
( YBo - YvXX,B / KH ) (1 - exp (-KH I x,B)) b ' + X,
)
y-S'- X1 - X2 e xp (- ya / YD) '· Xr X2 (YD - l) exp (- ya / YD ) +�����--�-=� V+� a+� For a specific reactor design (H; D), the operating parameters which can easily be altered are: the feed temperature (T1), the wall temperature
( Tw). the catalyst injection rate (qc), the fluidizing gas velocity ( U0).
Table 4.18 Base set of values for the design parameters, operating variables and physical and chemical properties. Cps , cal/g . K
Cpg . cal/g . K glcm3
Ps •
p9 , glcm3
3 Peat . g/cm
K9 , cal!cm . s . K Kpo , cm 3/g-cat . s
Ea . cal/mol
Jl, g/cm . s
(-Mir). dp , em
cal/g
Da, cm2/s
l/Jw
C0,
g/cm 3
Tref• K lm S
0.456
T1, K
0.44
Uof Umf
0.95
Umf• crnls
0.029
U0 , crnls
26
2.37
U8 , cmls
80
7 . 6 x w-5
4. 1 67 X 1 ()6
9000
l . l 6 X I (}-4
916
0.05
6.0 x I 0- 3 0.6
Ue , cmls
Tbo • K
T, (O), K
Tw ,
K
qn gfh
ds , em
Emf
0
300 5 5.2
1 5 .4 330
300 340 150 50 0.38 0. 1 1 5
0.029
D, cm
250
300
H . cm
600
1 .08 X J ()5
S.S.E.H. ELNASHAIE and S.S. ELSHISHINI
598
-1·0
us..-------,
I 00
• . so
1 · 50
( 01 )
7 00 1 -7!t
>
1 -2�
Q.)�
�
o.�o
> ---
. . . . . . . . --....___ 0-'25 o.oo�-....-----.---1
J
·
25 70 \0 1 ') QC 1 � ..----, 1 40 Ic) 1 . )0 1 -20
no
1-00
0.90
0.80
o .7Q
060
0 000
1 ��
1 00 0 7�
0 10
0-010
FIGURE 4.93
0 020
0 -0JO CMa
0 - 0l.O
0 050
0 - 060
2 Uo f Um i
1 . !) 0
· · · .· · �� - � �
O l<>!---...----,.--..--,---1
(b)
1 - 75
1 75 ( d )
0 90
\.00
1 10
1 -2 0
1 )0
I 40
1 ·SO
Two parameter continuation diagram.
In this section of the book we will present to the reader the effect of some of the s e p arameter , together with a simple optimization exercise in order to show the possibilities and phy sic constraints on important i ndustri a l process an d the impl ications of the static and dynamic bifurcation behaviour on the op erat i o n and optimization of these units. Table 4. 1 8 presents the base set of values for the design parameters operat i n g variables and physic al and che m i c al p rop erti es .
s
4.4.3
al
this
Some Results and Discussion of the Static and Dynamic Bifurcation Behaviour of Industrial UNIPOL Units
The two parameter continuation diagrams in Figures 4.93, a-d s how the loci of the static limit points (SLPs) and the H op f bifurcation (HB ) points for the change of some i mpo rtant operating parameters. All three diagrams (Figures 4.93, a-c) show intersection betwee n the loci of the HB point s and the loci of the SLPs , and al so show a cusp point on the SLP loci. On the other hand, Figure 4.93d which is drawn for YJ vers us Yw. diffe rs by the fact that all the loci are a lmost parallel. Thi s is due to the fact that the effect of both parameters is almost the same since an increase in feed te mpe ratu re ( Y1) c au se s an increase in the heat adde d by the inlet gas feed and the increase in wall temperature ( Yw) re s ult s in less cool ing rate for the reactor, which is e quiv a le nt to heat added to the system. Table 4 . 1 9 presents the b as e values of the parameters us ed in Figure 4.93 .
STATIC AND DYNAMIC BIFURCATION BEHAVIOUR
Table 4.19
599
Base values of the parameters used i n Figure 4.93. 4. 93c
Parameter
4. 93a
4. 93b
Yr ( Trll'ref)
variable
variable
1 . 1 75
variable
variable
250
250
250
2.8
variable
UJUmf CMo
Qc
Yw ( Twll'ref)
4. 93d
2.8
0.025
0.03
variable
1 .05
1 .05
variable
2.8 0.03 variable
A one parameter bifurcation cut at Yw = 1 . 5, is taken on Figure 4.93c and the resulting diagrams are shown on Figure 4.94, with the feed monomer concentration ( CMo) as the bifurcation parameter. As expected from the two parameter continuation diagram, this gives a bifurcation diagram with two HB points and without any static limit points. The diagram shows the effect of (CMo) on X1 . X2 and Yv. ••
• •:-:.7. 1 . 0 0 -r-----. -: , .c:-:-:.-:-: . ,c;c · '"" · ----., . ·"'" • •
0.3 0
0
------�
'
\I
I I \
0. 60
0 40 -
0. 20
.
. .... . .
'
g g0 g o o o
0 0 • o
.
0.00040 N
><
QOOOJO QOOOlD
000010
2.25 0 >
2 00
•,•'
1. 7 5
150
1 25
---
.·
·.
•'
- - -- � -- -- '
• '
. . ..
oj
/g 10
'
,'
,'
.. . . . . . . . . . . . .
0
HS 2
g
0
- �g
1 00 +---.--....,.---.,..---.--�--,,.-.--t 0 040 0000 0.030 0.010 0 020
FIGURE
C Mo (c;� t c':n )
4.94 Bifurcation diagram with
CMn as bifurcation parameter.
S . S .E.H. ELNASHAIE and S.S. ELSHISHINI
600
Table 4.20 CMo
Low temperature stable steady state for Figure 4.94. Yo
0.0 1 465
1 . 293
0. 0 1 25 5
1 . 273
O.Q l l 425
1 . 263
0.00687
1 .225
0.000 1 52
1 . 1 56
XI
0 . 83 2
Conv.
PE
0. 1 68
1 754.0
0 . 8 34
0. 1 66
1483 .67
0. 1 66
0. 824
0. 1 76
0.420
0 5 80
86 1 . 1
0.834
.
1 3 49 . 0
72.88
The entire region for this case is shown in Figure 4.94 and can be divided into four s ubre g ions : 1 . A stable steady state region for CMo < HB 1 = 1 .465 x I o-2 • This region falls below Yv= 1 .333 (the melting point of the polymer) but represents low polyethylene production rate (PE), with a maxi mum value of 1 754 g/s as shown in Table 4.20. 2. region between HB 1 and (CMo)cr = 1 . 8 x 1 Q-2 (at Yv = 1 .333 which is the maximum allowable temperature). This region consists of one unstable steady state and a periodic solu tion . Op erati n g the unit at this unstable steady state, gives higher conversion than in region 1 , but a controller i s necessary to avoid the oscillations with temperatures above the poly ethy le n e melting point. The min i mu m and maxi mum Yv are ( Yv ) m in = 1 . 1 84 and ( Yv)max = 1 .93. PE production rate at (CMo)cr is 2444 . 5 g/s, with an increase of 39.4% over that at HB l . The Yv and PE are shown on Figure 4.95 for the static bran ch on l y . 3. reg i on between CMo = (CMo)cr and CMo = HB2 = 2.832 1 X l Q-2. In this region there is an unstable steady state and a periodic attractor. The unstable steady state temperature is above YD = 1 .333, the melting point of PE which m akes operati on in this region unfeasible . 4. A regi on between CMo = HB2 = 2.832 1 x l 0-2 and (CMo)H = 2 . 875 x l Q-2 which corresponds to a periodic limit point (PLP). In this region there is a stable steady state and a stable and an u n stabl e periodic solution. Working in this region is unfeasible due to the h igh reaction temperature s attai n ed .
A
A
Details of re gions below critical v alues are shown in Table 4.2 1 whi ch contains the PE produ ct o n rate, minimum and maximum v alu e s of the temperature in the o s ci l l at ory region along with the average PE production rate in the o sc i llatory region . It is cle ar th at the average PE production rate in the oscillatory re g ion is h i gher than that at the s te ady states but it is characterized by high te mperature s relative to the melting point of polyethylene. Op e rating at the unstable steady st ate s with a
i
STATIC AND DYNAMIC BIFURCATION B EHAVIOUR
{a)
1 . 333
1 .3 1 3
1 .293 1 .273 1 .253
0 >
1 .233 1 .213 1 . 1 93 1 . 1 73 1 .1 53 2500
� "' ..... Q.
601
(b)
2000 1 500
HB1
1 000 500 0
0
0.002
0 . 004
C ill o
•
o . 006
o . ooe
o.o1
0.01 2
o.O I 4
fe e d monomer concentration ,
o.o 1 s
g/cm'
o .o 1 e
o 02
FIGURE 4.95 Effect of feed monomer concentration at low reaction temperature on: (a) reaction temperature, (b) polyethylene production rate.
suitable controller would give higher polyethylene production rate than that corresponding to the stable steady states. 4.4.4
Preliminary Simple Optimization of the Industrial UNIPOL Process
In this section we will present to the reader a simple exercise of trying to find the optimum combination of operating parameters (Tt, U0 , qc, Tw and CM0), with a constraint only on the reactor temperature which Table 4.21 Low temperature unstable steady state and osciUatory steady state for Figure 4.94.
CMo
YD YD
XI
Conv
PE,
g/s
(min)
YD
(max)
Conv (osc. )
0.247 1
0.0 1 886
1 .334
0.8 1 8
0. 1 8
2444.5
l.l8
1 . 330
0. 82 1 8
1 . 93
0.0 1 8 1
0. 1 8
2297 .05
1 . 8996
0.24
0.0 1 65.
1.312
0.827
1.19
0. 1 7
2032.89
1 .19
1 . 804
0.225
0.0 1 5 6
1 . 30
0.83
0. 1 7
1 8 8 8 .68
1.19
1 .739
0.2 1 5
PE, g/s (osc. )
33 1 9.6 3 09 2 .4
2 648 . 1
23 8 8 .96
S.S.E.H. ELNASHAIE and S.S. ELSHISHINI
602
e ee 1 10-130°C, e u (Te)max 130°C.
t
should not xc d d pending on the quali y of the poly ethylene and its struct re (i.e. linear or branched). The results
are for
=
I- Mathematical Programming
e
The differential terms dX1 /dt, dX2/dt and dYnldt wer set equal to zero in order to reduce the problem to its s te dy state form. Subrouti ne in t lib rary which solves general nonlinear programming problems using the successive quadratic programming algorithm and finite difference gradients, is used. This subroutine calls user supplied subrou tine to evaluate the functions at a given point. subroutine finds the minimum value of the objective function and its related parameters. However, in our case the objective function is t maximum PE production rate. refore the e u ati on PE production rate is multiplied by ( so that its minimum value would correspond to the maximum desired value. The operating parameters found under c onditio ns of (the lower bounds o the variables) and (the upper bounds of the variables) are as follows:
NCONF a
a
he IMSLNEW
The
FCN,
he calculates the f
-
The
XUB
,
q -1)
that
XLB
temperature TtfTret = 1 .16 = 0.13888 U01Umt = 10.0 = 0.02 g/cm3 t e a e Tw!Tref= 1 . 16
Dimensionless feed Catalyst injection rate Qc g/s Relative gas velocity Monomer feed concentration CMo Dimensionless wall emp r tur
The polyethylene production rate under these conditions is PE = 10.980 kg/s, which is much higher than the usual industrial production rate. This is e quiv alent to a polymer to c atalys t ratio for this case, of 10.98/0. 1 3888 = 79.1 kg PE/g-catalyst whereas the usual industrial ratio is 5-25 kg PE/g-catalyst. It is clear that the optimization procedure gives a relatively high value of U01Umt= 10.0, whic h is higher than the industrial practice that restri cts U01Umf to 6 becau se of solid entrainment. Therefore, in the following section we will look at the effect of U0/Umf un der these optimum c on di tions . ll-
Effect of U01Umf for the optimum case The effect of Uo!Umf for the optimum conditions is shown in Figure 4.96 where U01Umf is the bifurcation parameter. The diagram shows two HB po in ts without any SLPs wi th a periodic s oluti on existing between HB l =4. 1 1 8 and HB2 = 12.706. Tables 4.22 and 4.23 summa rize the effect of the relative gas vel oc ity on the different variables at the low temperature ste ad y state and the unstable and oscillatory ste ady states respectively.
STATIC AND DY N A MIC BIFURCATION BEHAVIOUR
603
1 .00 ...-----...,
.
· . . . . . . . . . . . . . . · -· .
0 80
0 60 -
0. 40
><
0. 20
0
� 0
0
[;1' 0 0 0
8
g_ °
' '
,
'
..
/
.
,,
, "'
...
.
---
•
•
-
-- - -
•
• •
•
•
-
. ���---------�
7 !
- - -- - -
. . ...
...
• •
•
•
••
0.00 +---r--...----.--r-.---' 0 00020 -r-------,
O ()(X)l 5 ><
0 000 10
0 00005
· . . .
�0
0 0
.
,'
.. . . .
/
,,
� . . ..
.."'
..
e • • • •
e e
,. ,
..
"" ...... "
• • •
• •
e I
� - --....
• • •
:.,•:..... .: : .
• • • •
0.00000 +----.--...--r--�---'
2 . 2 5 ,-------, 2 .00
a
>
1 .75 1 . 50
1.25
. .
..
. .
HB 1 Q'
0
g
0
0
8
r I
'..
'
.
. ....
............
.
•
'
••,
t I
•
•
..
".lHB 2
•
- - - - - - -- - • .• • . .
:r!,
. . . .. . . . . . . . . . . . . . . . .
I I
--1
------
I I
1 . 00 -!-....J.---,.--..---...---,r'---r--.--4 1 7. 5 12.5 5 .0 20.0 100 7.5 15.0 2 .5 HB : 1 2. 706 HB1 : 4 .1 1 8 Uo / U m f
FIGURE 4.96
Bifurcation diagram with U.JUmf as bifurcation parameter.
The entire region can be divided into six different subregions. 1.
2.
A region for U01Umr< 4.0398, the value corresponding to a periodic
limit point (PLP) . In this re g i on unique steady state s exist. The stable steady state corresponds to an unrealistically high temperature . A region between U0/Umf= 4.0398-4. 1 1 8, the latter value corres ponding to HB 1 in this region there exists a stable steady state surrounded by an unstable limit cycle and a stable periodic attractor. The temperature in this re gi on is again too high for practical purp o se s .
604
S.S .E.H. ELNASHAIE and S.S. ELSHISHINI
Table 4.22 Effect of relative gas velocity on the different variables at the low temperature steady state for optimum conditions.
Urlllmf
PE, g/s
YD
XI
X (conv. )
1 0.080
1 .3 3 3
0.788
0 . 220
1 0870.00
1 0. 5 80
1 .326
0.795
0 . 2 05
1 1 049.00
1 1 .080
1 .322
0.799
0 . 20 1
1 1 334.00
1 1 .5 80
1 .3 1 7
0.803
0 . 1 97
1 1 603.00
1 2 . 080
1 .3 1 2
0. 807
0. 1 9 3
1 1 857.80
1 2 . 700
1 . 307
0.8 1 2
0. 1 8 8
1 2 1 4 3 . 00
1 3 . 000
1 . 3 05
0.8 1 4
0. 1 8 6
1 229 7 . 00
1 4 .000
1 .298
0.820
0. 1 80
1 28 1 3 . 87
1 6 . 000
1 . 287
0. 829
0. 1 70
1 3 898 . 00
1 8 . 000
1 .279
0.836
0. 1 63
1 4962.50
3. A region between UofUmJ = 4. 1 1 8- 1 0.08, the latter value corres ponding to the point where the unstable steady state gives YD= 1 .333. In this region an unstable steady state and a periodic solution exist. However, both give Yn > 1 .3 3 3 which makes it unfeasible to work in this region. 4. A region between U01Umf = 1 0.08-1 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 kg-PE/g-cat is achieved which is a high value compared to the usual industrial range of 2-20 kg-PE/g-cat. 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 kg-PE/g-cat 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 kg-PE/g-cat 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 Non-Porous Catalyst Pellet
Consider a film of thickness 8 surrounding a non-porous 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 ·' d-r
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 - f-Ji) - (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 non-linear 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 (a-1A1 - 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 co-ordinate system,
d
i_ = j - v
'r = t
�
v
i s the ve loci ty of the moving co-ordinates (velocity of creep), <'1 is the
where i is the cell number with respect to the moving co-ordinate,
cell length and j denotes the cell number with respect to the stationary co-ordinate. The variables in one co-ordinate are related to the variables in the other co-ordinate system by, df,j dt
= -�(Ysr ysr-I . )+ _
<'1
d
Y,; d'r
- = -- ( Y - LI ) + r d d lj dt
dY
v
<'1
I
I
'f
(E. I )
(£.2)
Rhee et al. neglect the terms dYs;ld-r, dY; Id-r in equations E. I and E.2 in order to develop the analytical expression for v.
(1973)
The terms dYs;ld-r, dY;Id-r repre s e nt the natural transients of the system with respect to the moving c o-ordinates . In ot her words the transients can be divided into:
1 . Perfect creep (shock wave) . 2. Natural transients. 618
APPENDIX E
619
The part of the perfect creep will obviously be zero in a system of co-ordinates moving with the same velocity. However, the natural transients in general will not. This effect of the natural transients will tend to change the shape of the reaction zone. Therefore, neglecting the natural transients in E. l and E.2 is reasonably valid only when there is no appreciable change in reaction zone shape during the transient period (perfect creep). The results shown in Chapter 3 show that the distortion in the shape of the reaction zone is negligible when the mass capacity of the bed is lower than its heat capacity. However, when the mass capacity is higher than the heat capacity, the shape distortion of the reaction zone, during the transient period, is appreciable and therefore the natural transients terms in E. I and E.2 cannot be neglected. There is no single velocity in this case, i.e. the reaction zone is dispersed. Obviously, the natural transients cannot be neglected when starting from a non-steady state initial profile. Peifect creep
For the cases of low mass capacity, for which no appreciable distortion in the shape of the reaction zone takes place, we can neglect the natural transient terms and obtain, d �j
-
dt
= -� ( f:Jl. v
- fSl. - 1 )
_ ; = -� ( l-:I - f1 - 1 )
dY
v
dt
(E. 3)
(E.4)
Inserting equations (£. 3 ) and (£. 4) into the heat balance equation of the system gives,
( H- �v)cr ,:1
1-
1 -
YI ) - ( YI - YSl· ) = O
(E.5)
(E.6) where 1) = a [ exp ( - y I
�i )]X;*
S.S.E.H. ELNASHAIE and S . S . ELSHIS HINI
620
Su
mmi ng E.6 over i from i
=
l to i = r gi v e s ,
(£. 7 )
u pos the s a
the downstream cell r are sufficiently so that the profile s in their neighbourhood are flat, we can make the approximation,
S p e up tre m cell / and far from the reaction zone
ll-1 = lJ ;
�r+1 = � r
�/- 1 = �[
ti
Therefore equa on E.7 becomes,
(H- a2 ) V
�
f3
a� ( Y.s1 - Y.sr ) + -a (}-;1 - Yr ) - (X1 - Xr ) = 4V
J13A V
If we apply equation at cell I, Y r we sr · Sub titu in these relations in
obtain Y, = s t g [ Y]
(£.5)
*
*
�
T
r L (£. 8) _
r=l
r:1
we find Y1 = Y51 , and the same for
(£.8) gives,
(H- v � + a4 )+ a3f3A v
[X] = -fir
�
�
i i=l
'i
(£.9)
bracket [ ] denotes the jump of the quantity enclosed across ea t o zone. Notice that und er these conditions,
Where the the r c i n
[ Y] = [ Ys]
Similarly for
[X] = [x*]
the mass balance equations , [X]
(M- v at + a3 ) �
=
i i=l
'i
(£. 10)
Combining E.9 and E. l 0 and rearrangin g we get, (E. l l )
APPENDIX E
which can
be written as
where N is the number of cells travelled per unit time.
62 1
APPENDIX F
Computation of Ly apunov Exponents In this sec tio n the theoretical bases ( S himada and Nagashima, 1 974; Benettin et al. , 1 980; Wright, 1 984; Wolf et a/. , 1 985) on which the algorith m developed by Wolf et al. (Wolf et al. , 1 985) for c omputation of all n Lyapunov exponents (Lys), are presented. The method r equire s the i ntegration of the variational equations and the reorthonormalization of n vectors . The Gram-Schmidt reorthonormalization (GSR) procedure is used. The variational equations are appended to the o rig i n al system to obtain the combined system:
{ddrX} �
J(X) �{Dxf( X)}
(F. l )
{XU= O)} ={x0}
This system i s integrated from the initial condition s :
A(r=O)
(F. 2)
I
Let ( VJ , V2 , . . . , � ) be a se t of n perturbation vectors obtained by inte grating equations (F. l ) from the i l!itial conditions (F.2) for a period T time ( r) units . The selection of e ss en ti for the success of the procedure . Too small a value results in exc e ss orthonormalization, and if normi (l ) are all ne arl y it leads to inaccuracies. Too large a value of f results in an ill-conditioned sy stem of vectors and could lead to numerical overflow in i ntegrati on . For a three dimensional system the se vectors can be written as:
T is
al
I,
V]
where i, j and k are
= =
A1 1 i + A2 Ji + A3 1k
(F. 3)
V2 = A1 2i + A22 j + A32k
(F. 4)
ltj
A1 3i + A23j + A33k
unit vectors .
622
(F. 5)
APPENDIX F
623
If the integration continues with ( VJ , Vz , . . . , Vn ) as the initial condi tions for the next integration step, numerical difficulties arise, specially for chaotic attractors, therefore a renormalization procedure is used to , , replace ( VJ , \'2 , . . . , V, ) , by (v:,I ' � . . . . , Vn ) as �.allows : At the I th stage of the orthonormalization the norm (magnitude) of V1 is given by: (F. 6)
Normalizing the first vector to give ( VJ'):
\1,' - J:l - At t i + A2 d + A3 1 k -- . ' ' 1 - II V.I II - � A121 + A212 + A321 At t l + A2 1]. + A'3 t k
( F. 7)
The dot ( inner) product of V2 and VJ' is given by:
Multiplying
VJ'
by the scalar quantity
GSR 1 :
Construct a new vector:
V2 - (V2, "1) V!'= (A1 2 - GSR1 · A{1 )i + (A22 - GSR1 · A2 1 )j + ( A32 - GSR1 · A3 1 )k = A{2i + A22J + A3zk
(F. l O)
Calculate the vector norm:
Normalize the above vector to give ( V:{):
(F. l 2) The dot (inner) product of V3 with \.)' and V{ are given by:
( "3 , Vi? = GSR2 = A 3 A{ 1 + A23A2 1 + A33A} 1 1
( \t3, V{) = GSR3 = A1 3A{2 + Az3A2'2 + A33A32
(F. l 3) (F. 14)
S . S .E.H. ELNASHAIE and S.S. ELSHISHINI
624
Multiply
l'J.' and Vi by the scalar quantities GSR2 and GSR3 respectively: ( ":� . "!� V)' = GSR2 A{1i + GSR2A?.d + GSR2 A:i1 k
(F. 1 5 )
( ":�, Vi) Vi = GSR3A{2i + GSR3A2'zi + GSR3 A]'z k Construct
(F. 1 6)
a new vector:
":1 -
( "3, Vi) Vi - ( ":1 , "!� "!' = (A1 3 - GSR3A{2_ - GSR2A{ ) i 1
+ ( A2 3 - GSR3 A2.2 - GSR2 A2 1 )j
+ (A33 - GSR3 A3'z - GSR2A3 1 ) k (F. l 7)
Calculate the norm of the above vector: norm3 ( l ) = ll � - ( � , v;) v; - ( � . �,) � ' II = � (A;3 )2 + ( A;3 )2 ( A;3 )2 (F. 1 8)
Normalize the above vector to give V{: V'
":1 - ( V3, vi) Vi - ( "3 , "!� "!'
_
3 - Il l':! ( ":1. Vi) Vi - ( ":� , "'� "''ll 231. + A"3 3 k = A" 1 3l. + A"
(F. 1 9)
In general form GSR provides the foll o win g orthonormal set of vectors:
V' = � - (�. v;_1 )v;_� - . . . -(�. "!�"!' II� - (� . v,;_I )v;_l - . . . - ( vn , "'� "''ll n
(F. 20)
V.' replace V; (for i = 1 , 2, . . . , n) as the i ni tial conditions for the next stage of in tegration and new values of V; are generated after T time (r) units. The Aabove proc edure of integration and orthonormalization stages every T time ( r) units is repeated L times. The Lyapunov exponents at the Vh stage are given by: A
I
_
Ly. 1
1_
_
�
A £...
L. T
1=1
Ln ( norm i (/)) Ln 2 . 0
( F. 2 1 )
The calculati on continues until the Lyapunov exponents (Lys) con verge or the maximum iteration count is reached.
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Index
Chaotic 1 - 1 2, 57, 1 33 , 1 52- 1 69, 1 8 1 -1 83, 20 1 -222, 406--4 5 3 , 463-485 , 578-5 8 1 Chaotic attractors 1 3 , 57, 1 73 , 2 1 3-2 1 5 , 440, 464-485 Characteristic equations 1 35 , 1 5 3 , 234, 3 1 2 Characteristic roots 1 36, 234, 252 Chemisorption 22 1 , 243-254, 279, 337, 406--4 1 0, 465 Co oxidation 476--5 82 Coke 49 1 -534 Consecutive reactions 407, 408 Conservative systems 1 74- 1 79 Continuous stirred tank reactor (CSTR) I , 5, 9, 1 4-2 1 , 27-40, 60, 80, 1 52, 232, 3 1 2, 397-4 1 2, 503-508 Continuous systems 1 76-- 1 82, 206--2 1 7 Cracking 4, 5 , 49 1 -505 Creeping reaction zone 337, 354, 618 directional creep 345-35 1 initial creep 346--3 48 velocity of creep 347 Crises 3 37-344, 352-3 57 Cyclic fold 1 60
Activation energy 253-260, 3 1 5 Asymmetrical 1 1 7, 223-225, 245250, 32 1 Attractors 54-60, 80-95, 1 04- 1 1 2, 1 54, 2 1 4 Autonomous 56, 90, 1 1 9- 1 23 , 1 32- 1 35, 1 52-1 72, 207-2 1 5 , 32 1 , 407-463 Non-autonomous 1 5 2- 1 69, 208-2 1 8, 32 1 , 407, 423-438, 464
Banded Chaos 47 1 , 483 Bifurcation diagrams 64-80, 97- 1 92, 204-2 1 8 , 359-365 , 4 1 8-47 8 , 525-605 Bifurcation 1 - 1 6, 56-- 1 74, 202-207, 377-406, 454-5 87 Bubble phase 14, 3 89-398, 409, 4 1 4, 49 1 -499, 5 1 9-5 39, 590-592 Bubbles 3 8 9-398, 407-409, 508, 5 3 5-595 Bursting 432
Catalyst circulation 490-5 16, 5 2 1 -550 Catalyst 1 1 , 22 1 , 222, 237, 25 1 -257, 266--2 68, 290, 3 1 5 , 325-404, 480-5 1 7, 5 2 1 -550 Catalytic 4, 1 1 , 2 1 5 , 289-3 1 1 , 3 8 9-4 1 2, 49 1 , 6 1 8 Cell model 222, 308-3 14, 3 29-334, 613 ' Chaos 3 1 2, 1 6 1 , 200, 207, 40 6--4 3 8 , 47 1-475
Degeneracy 1 3 3- 1 50 Degenerate 1 43- 1 48 Dense phase 3 89-409, 49 1 -559, 590-592 Desorption 22 1 -225 , 25 1 -255, 272-3 1 5 Diffusion 226, 296, 5 1 9
-
641
642
INDEX
Discrete time sy s tem s
1 73-1 80,
Dissipative systems
Gas oil 49 1 -522
Gasoline 4, 49 1-567
206--209 1 74-- 1 76,
427-480 Distributed parameter model 28 1 ,
309 Domain of attraction 37, 55, 80,
86, 1 80, 1 8 1 1 -32, 80-89, 99, 1 25-1 3 1 , 1 82-239, 25 1 -256, 269-402, 555-563
Dynamics
He at g e ne ration function 44, 48 Heat removal function 44 Heat transfer 9, I I , 220-3 1 6 Histogram 429, 430-434, 472 Homoc li ni cal orbit 1 0 1 - 1 07, 1 45 ,
358-438, 452-464
Hopf Bifurcation (HB) 4, 89- 1 64,
370-440 , 453-605 Hydrodynamics 535, 590
Effectiveness factor 286-3 1 1 Eigenvalues 50-68, 82- 1 58, 234,
235, 249, 250, 288 Emulsion phase 389-398 Equilibrium 1 1 , 1 7 , 1 8, 22,
222-225, 255-259, 270-275, 283, 3 1 5 Non-Equilibrium 1 1 , 20, 26, 33, 59, 270, 6 1 0 Ethylene 4 , 5 , 588
Feed-back control 9, 1 4, 405-4 1 3 ,
465, 522, 554, 557-575, 197-199, 443-448 Fixed bed 2, 1 3 , 219-222, 308-3 14, 326-33 1 , 375-388, 6 1 3 Fixed point 68-95 , 1 05- 1 1 8 , 1 3 3 , 1 34, 1 73-194, 204, 44 1 Flip bifurcation 1 6 1 Floquete Multipliers 1 5 , 1 02, 1 22-1 24, 1 3 3, 1 64 , 44 1 -450, 464 Fluid c atal yt ic cracking (FCC) 1 , 4, 1 1 , 1 3 , 1 5 , 78, 202-205, 2 19-222, 406, 485-5 19 Fluidization 508, 535, 558-590 Fluidized bed 1 0, 1 1 , 1 3 5 , 1 6 1 , 2 1 9-222, 386-398, 404-4 1 8 , 439, 440, 463-5 1 7 , 587 Focus 52-54, 90- 1 0 3 , 1 33- 1 5 8, 1 79 Forcing 1 54-- 1 62, 2 1 0, 407, 424-447, 45 1 -463 Fully developed chaos 47 1 -473, 483
Infinite period
I 00, 370, 42 1-423 I ns tab ility 50, 323-352, 377 Intermittency 203-205, 428-434, 472 Intermittent 207, 429, 473 , 557 lntraparticle diffusion 353-357, 385 Intraparticle heat transfer 220, 257, 353, 379 Intraparticle mass transfer 220, 257, 353, 379 Intrinsic rate 260, 3 1 5 Iterate maps 190-197 , 204--207, 2 1 7 , 428-43 3 , 472
Fe i g e nbaum Universality
Jacobian matrix
64, 65, 86-89, 1 20,
1 63 , 1 64, 4 1 6
Kinetics 57, 255, 302, 520
Linearization 56, 1 34, 1 35 , 233, 320 Logistic m ap 1 73 , 1 86-197 , 202, 203 Lumped parameter model 1 5 , 223 ,
25 1 , 27 8, 309, 503 Lyapunov exponent 5 5 , 56, 2 1 4,
428-485, 622-624
Map
Return 1 72, 472 1 70- 1 77, 206-2 1 8 ,
Poincare 469-485
INDEX
Mass transfer 9, 1 1 , 220, 308-3 1 6 Modelling 1 , 7 , 1 1 , 2 1 9-223, 257-3 3 3 , 485 Mutiple ste ady states (see steady state) Mu l ti plicity 6, 9, 3 1 , 55, 57--6 1 , 84-86, 222-398, 464-5 3 6
Neimark 1 5 8 Node 50-55, 69-74, 83-96, 1 33-1 3 8, 1 57 , 1 5 8 , 44 1 , 475 Non-autonomous 1 52- 1 69, 208-2 1 8, 32 1 , 407, 423-43 8, 464 Non-Equilibrium (see Equilibrium)
Orthogonal Collocation 228-23 1 , 29 1 -305 Oscillation 4, 9, 27, 3 1 , 32 50-52, 82-1 6 1 , 556-5 8 1
Peri o d doub ling 1 22- 1 3 3 , 1 56- 1 64, 1 73-2 1 7 , 4 1 2-449, 452-483
Perio dic attractor 1 3 , 54-59, 90, 1 1 6, 1 3 1 - 1 3 3 , 148- 1 8 1 , 2 1 1 -279, 308-407, 468 Periodic limit po i nt (PLP) 85, 94- 1 06, 1 22- 1 47 , 4 1 9-423 , 467-483 Poincare map (see map) Poincare plane 1 3 , 1 1 3, 1 70- 1 80, 2 1 5, 2 1 6, 469-480 Point attractor 1 3 , 89, 95 , 480, 485 Polyethy lene 1 , 5 87--605 Polymerization 1 , 3, 43, 5 8 8 , 589
Quasi-periodic 1 23- 1 35 , 1 5 2- 1 62, 2 1 2-2 1 5 , 440-446, 45 1 -45 6
Radiation cell model 222, 308, 3 3 1-336 Reaction zone 222, 49 1 -494, 5 87, 620 Regeneration 489-5 3 3 , 569
643
Regenerator 486-5 39, 556 Region of stabi l ity 480-4 90 Return Map (see map)
Saddle 5 3- 1 5 8 , 44 1 , 475 Separatrix 55, 57, 8 1 -84, 1 00 Simple cell model 3 6 1 Stability 8, 3 6-4 8 , 222-375 , 40 1 , 402, 464--6 1 2 Static limit point (SLP) 363-420, 45 3--605 static bifurcation 359 Steady states 1 0, 1 2 , 26-44 , 50-1 34, 246-27 1 , 285-3 1 8, 393-4 1 5 Unique 39, 44 , 54, 77, 84, 103, 269-27 8, 440 ignited 3 5 1 , 352 Mutip1e 39, 44, 55, 23 8-245, 273-279, 402, 403 , 502 quenching 352 Stroboscopic maps 1 53-1 63, 1 80, 209-2 1 8, 424-43 1 , 434-44 1 Subcritica1 Hopf 99, 1 1 3 , 145, 146 Supercritica1 Hopf 99- 1 1 3 Symmetrical 204, 225, 235-237, 247 , 248
Tangent bifurcation 1 23, 1 45 , 203-205, 473, 474 Torus 1 23 , 1 3 3 , 1 45- 1 62, 2 1 2, 2 1 3, 440-450 Trajectory 50-57, 88- 1 00, 2 1 1 -2 1 3, 402, 427-449, 462
Unipol process 4, 1 5 , 1 6, 598, 5 87, 602 Unique steady state (see steady state) Universality 1 84
Werther model
3 92
Ziegler-Natta catalyst 587
Acknowledgements for Figures and Tables
The followi ng fi g ures and table s have been repro du ced from the p apers and
books g i v en below with permission from the c o rre spondin g publi shers given in the follo w i ng list:
1. With permission from Elsevier Science Ltd. (Pergamon), Oxford, U.K.
1 . 1 . Figs. 3 . 2-3 .24 from: Elnashaie, S . S .E.H. and C re s s well, D.L., Dynamic behaviour and s t ab i lity of nonporous catalyst particles. Chern. Engng. Sci ., vol. 28, pp.
1 3 87-1 399, 1 973.
1 .2 . Figs. 3 . 23-3 . 27 and Tables 3 .7-3 .9 from: Elnashaie, S . S .E.H. and Cre s s well, D.L., The influence of reactant ad s orptio n on the multipli c ity and stability of the s teady state of a catalyst particle. Chern. Engng. Sci., vol. 29, pp. 753-760, 1 974. 1 . 3 . Figs . 3 .46-3 . 60 from : Elnashaie, S . S .E.H. and Cresswell, D.L., Dynamic behaviour and s tabil i ty of adiabatic fixed bed reactors. Chern. Engng. Sci., vol. 29, pp. 1 889- 1 900, 1 974.
1 .4. Figs . 3 .45a from: Uppal, A., Ray, W.H. and Poore, A.B ., On the dynamic behaviour of continuous stirred tank reactors. Chern. Engng. Sci . , vol. 29, pp. 967-985 , 1 974.
1 . 5. Figs. 3 .28-3 . 3 8 and Tabl e 3 . 1 0 from: Elnashaie, S . S .E.H. and C re s s w ell , D.L., On the dynamic modelling of porous catalyst pellets . D is tribute d model . Chern Engn g . Sci . , vol. 30, pp. 355-3 5 8 , .
1 975.
1 .6. Table 3 . 1 1 from: Elnashaie, S . S .E.H., The effect of intrap arti cle resistance on the v el o ci ty of creep in adiabatic fixed bed reac tors. Chern. Engng. S ci , vol . 30, pp. 1 2971 299, 1 975. .
1 .7 . Fig s . 3 . 82-3 .90 from: Elnashaie, S .S.E.H., Multiplicity of the s te ady state in fluidized bed reactors III. Yield of the consecutive reac ti o n A � B � C. Chern. Engng . Sci. , vol. 32,
.
pp. 295-30 1 , 1 97 7 .
644
ACKNOWLEDGEMENTS
645
1 . 8 . Fi g . 2.45b from : Vaganov, D . A . , Somoilenko, N.G. and Abraham v G p ·odi . . , en c regunes of contmuous stmed tank re ac tors . Chern. Eng ng. S ci., vol . 33 33 1 1 1 1 40 ' pp. 1 978. .
•
.
·
-
•
1 .9. Figs. 4.50b, 4. 5 3-4.56 and Table 4 .8 from: Elnashaie, S .S.E.H. and El -Hennawi , I. M., Multipli · city of the steady state �n fluidized bed reactors. IV. Fluid Catalytic Cracking (FCC) . Chern Engng. Sct., v oL 34, pp. 1 1 1 3- 1 1 2 1 , 1 97 9 . ·
1 . 1 0. Figs . 2.70--2 .82 from:
Planeaux, J.B. and Je n sen, K.F. , Bifurcation phenomena in CSTR dynamics: A system with extraneous thermal capaci tan ce. Chern. Engng. Sci., vol. 4 1 , pp. 1 497- 1 523, 1 986. 1 . 1 1 . Fi g . 4 .50c from: Edwards, W.M. and Kim, H.N., Multiple steady states Chern. Engng. Sci. , vol. 43, pp. 1 825-1 830, 1 9 8 8. 1 . 1 2.
Fig s
4.52, 4.57-4.59 and
in FCC unit operations.
Tables 4.7, 4.9-4. 1 2 from:
Elshishini, S . S . and Elnashaie, S.S.E.H., Digital simulation of industrial fluid catalytic cracking units. Bifurcation and its implications. Chern. Engng. S ci. ,
voL 45, pp. 553-559, 1 990.
1 . 1 3. Figs. 4.5 1 , 4.60-4.62 and Tables 4. 1 3-4. 17 from: Elshishini, S . S . and Eln a sh ai e , S.S.E.H., Digital s imulation of industrial fluid catalytic cracking units. II. Effect of ch arge stock composition on bifurcation and gasoline yield. Chern. Engng. Sci . , vol. 45, pp. 2959-2964 1990.
1 . 14. Figs. 4.63-4.65 from: Elshishini, S . S . , Elnashaie, S . S. E . H. and Alzahrani,
,
S., Digital simulation of
industrial fluid catalyti c crac king units. ill . Effect of hydrodynamics. Chern. Engng. Sci . , vol. 47, pp. 3 1 5 2-3 1 56 , 1 992.
1 . 1 5 . Figs. 4.66-4.80 from: . . . . mdu�tnal flwd Elnashaie, S.S.E.H. and Elshishini, S.S., Digital simulation of . SCI., vol. 48, catal ytic cracking units. IV. Dynamic behaviour. Chern. Engng pp. 567-5 83 , 1 993. uerent between di" · " , s . s ., C ompari son. s . s . E . H . an d El s h"Is h101 El nas ha1e, · cracking atal mathematical mode l s for the simulation of i n du strial 1 ' (FCC) units. M ath l . & Compu t . Modi., vol. 1 8 , PP · 9 1-
1 . 1 6. Table 4.6 from:
·
flwt � 1:�
·
1 . 1 7 . Figs. 4.7-4. 1 8 from: . . Elnashaie, S . S .E.H. and Abashar, M . E. , forced fluidized bed catalytic reac tors with consecutive reactions. Chern. Engng . Sci. , vol. 49, pp. 2483-2498,
of periodically c chemical Chaotic �haVI�ermi � �� ·
ACKNOWLEDGEMENTS
646
1 . 1 8. Figs. 4. 1 -4.6 and Table 4.2 from: Elnashaie, S .S .E.H. and Abashar, M . E . , Chaotic behaviour of forced flu idized bed c atalytic reactors. Chaos, Solitons and Fractals, vol. 5 . pp. 797-83 1 , 1 995 . 1 . 1 9 . Figs.
4.37-4.48 and Tables 4.4, 4.5 from: ,
,
Elnashaie, S . S .E.H. and Abashar M.E. and Te ymour F.A., C haoti c behaviour
of fluidized bed cat al y ti c reactors with consecutive exothennic chemical
reactions. Chern. Engng. Sci. , vol . 50, pp. 49-67, 1 995 . 1 .20. Figs . 4. 8 1 -4 . 87 from: Elnashaie, S . S .E.H., Abasaeed, A.E. an d Elshishini, S . S ., Digital s imulat i on
of in dus trial fluid catalytic cracking units. V. Static and Dynamic bifurcation.
Chern. Engng. Sci. , vol. 50, pp. 1 635- 1 644, 1 995 .
1 . 2 1 . Figs . 3 . 6 1 -3 . 8 1 from: Wagialla, K.M. and Elnashaie, S . S . E . H . , B ifurcation and complex dynamics
in fixed bed catalytic reactors. Chern. Eng ng. Sci . , v o l . 50, pp. 28 1 3-2832,
1 995 .
2. With permission from Springer- Verlag, New York, USA. 2. 1 . Fig . 2. 1 3 1 from:
Parker, T.S. and Chua, L.O. , Practical numerical al go rithms for chaotic systems, Springer-Verlag, New York, 1 989.
2.2. Figs. 2.46--2 .49 from: Wiggins, S., Introduction to applied nonlinear dynamical systems and chaos, -
Spring er Verl ag , New York, 1 990.
3. With permission from Cambridge University Press, Cambridge, U.K. 3 . 1 . Figs 2.62-2.69 and 2 . 1 20-2. 1 22 from: .
Jackson, E.A. , Pers pec ti ve s of nonlinear dynamics , vol. I , 2, Cambridge University Press, Cambridge, 1 990.
4. With permission from Marcel Dekker Inc., New York, USA.
4. 1 . Figs. 4 . 8 8-4.90 from: Razon, L.F. and Schmitz, R.A., Intri nsical ly unstable behaviour during the oxidation of carbon mo no x i de on platinum. Catal. Rev. Sci. Engn g . , vol . 28, no. 1, pp. 89- 1 64, 1 986. 5. With permission from the Canadian Society for Chemical Engineers, Ottawa, Canada.
5 . 1 . Figs. 3 . 1 9-3 .22 from: Elnashaie, S . S . E . H . and Cresswell, D.L., On the dynamic modelling
of porous
catalyst particles. Can . J. Chern. Eng . , vol . 5 1 , pp. 20 1 -205 , 1 973 .
